[ { "id": "Calculus_-_single_variable_0000", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Finding limits using graphs", "level": "2", "keywords": [ "calculus", "derivative", "limits", "continuity", "piecewise functions" ], "problem_v1": "Use a graph to estimate the limit \\lim_{\\theta \\rightarrow 0} \\frac{\\sin(8\\theta)}{\\theta}. Note: $\\theta$ is measured in radians. All angles will be in radians in this class unless otherwise specified. $\\lim\\limits_{\\theta \\rightarrow 0} \\frac{\\sin(8\\theta)}{\\theta}=$ [ANS]", "answer_v1": [ "8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use a graph to estimate the limit \\lim_{\\theta \\rightarrow 0} \\frac{\\sin(2\\theta)}{\\theta}. Note: $\\theta$ is measured in radians. All angles will be in radians in this class unless otherwise specified. $\\lim\\limits_{\\theta \\rightarrow 0} \\frac{\\sin(2\\theta)}{\\theta}=$ [ANS]", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use a graph to estimate the limit \\lim_{\\theta \\rightarrow 0} \\frac{\\sin(4\\theta)}{\\theta}. Note: $\\theta$ is measured in radians. All angles will be in radians in this class unless otherwise specified. $\\lim\\limits_{\\theta \\rightarrow 0} \\frac{\\sin(4\\theta)}{\\theta}=$ [ANS]", "answer_v3": [ "4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0001", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "3", "keywords": [ "limits", "laws" ], "problem_v1": "Evaluate \\lim_{x \\rightarrow-1}(x+1)^{5} (2x^2) Answer: [ANS]\nUse the space below to enter the letters corresponding to the Limit Laws that you used to find this limit:\nLimit Laws\nA. Product Law B. Quotient Law C. Difference Law D. Constant Multiple Law E. Power Law F. Sum Law G. Root Law Answer: [ANS]", "answer_v1": [ "0", "ADEF" ], "answer_type_v1": [ "NV", "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Evaluate \\lim_{x \\rightarrow-2}(x-2)^{5} (5x^2) Answer: [ANS]\nUse the space below to enter the letters corresponding to the Limit Laws that you used to find this limit:\nLimit Laws\nA. Root Law B. Difference Law C. Sum Law D. Product Law E. Quotient Law F. Constant Multiple Law G. Power Law Answer: [ANS]", "answer_v2": [ "-20480", "BDFG" ], "answer_type_v2": [ "NV", "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Evaluate \\lim_{x \\rightarrow-1}(x-1)^{5} (-2x^2) Answer: [ANS]\nUse the space below to enter the letters corresponding to the Limit Laws that you used to find this limit:\nLimit Laws\nA. Power Law B. Root Law C. Quotient Law D. Sum Law E. Difference Law F. Constant Multiple Law G. Product Law Answer: [ANS]", "answer_v3": [ "64", "AEFG" ], "answer_type_v3": [ "NV", "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Calculus_-_single_variable_0002", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "2", "keywords": [ "limits", "factoring", "Calculus", "Limit" ], "problem_v1": "Evaluate the limit \\lim_{x \\rightarrow 8} \\frac {x^2+14x+48}{x+8} Answer: [ANS]", "answer_v1": [ "8 + 6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the limit \\lim_{x \\rightarrow 1} \\frac {x^2+11x+10}{x+1} Answer: [ANS]", "answer_v2": [ "1 + 10" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the limit \\lim_{x \\rightarrow 4} \\frac {x^2+11x+28}{x+4} Answer: [ANS]", "answer_v3": [ "4 + 7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0003", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "2", "keywords": [], "problem_v1": "Evaluate the limit \\lim_{x \\to 2} 4^{x^2-5x} Enter DNE if the limit does not exist. Limit=[ANS]", "answer_v1": [ "0.000244141" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the limit \\lim_{x \\to-4} 5^{x^2-3x} Enter DNE if the limit does not exist. Limit=[ANS]", "answer_v2": [ "3.72529E+19" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the limit \\lim_{x \\to-2} 4^{x^2-4x} Enter DNE if the limit does not exist. Limit=[ANS]", "answer_v3": [ "1.67772E+7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0004", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "3", "keywords": [], "problem_v1": "Using: $\\lim\\limits_{x \\to 1} f(x)=3$ and $\\lim\\limits_{x \\to 1} g(x)=7$, evaluate the limits,\n$\\lim\\limits_{x \\to 1} f(x) g(x)=$ [ANS]\n$\\lim\\limits_{x \\to 1} \\frac{f(x)}{g(x)}=$ [ANS]", "answer_v1": [ "21", "0.428571428571429" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Using: $\\lim\\limits_{x \\to-3} f(x)=-5$ and $\\lim\\limits_{x \\to-3} g(x)=10$, evaluate the limits,\n$\\lim\\limits_{x \\to-3} f(x) g(x)=$ [ANS]\n$\\lim\\limits_{x \\to-3} \\frac{f(x)}{g(x)}=$ [ANS]", "answer_v2": [ "-50", "-0.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Using: $\\lim\\limits_{x \\to-2} f(x)=-3$ and $\\lim\\limits_{x \\to-2} g(x)=7$, evaluate the limits,\n$\\lim\\limits_{x \\to-2} f(x) g(x)=$ [ANS]\n$\\lim\\limits_{x \\to-2} \\frac{f(x)}{g(x)}=$ [ANS]", "answer_v3": [ "-21", "-0.428571428571429" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0005", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "2", "keywords": [ "calculus", "limits", "basic limit laws", "linear functions" ], "problem_v1": "Evaluate the limit using the Limit Laws: $\\lim\\limits_{x \\to 3} (6x+3)=$ [ANS]", "answer_v1": [ "21" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the limit using the Limit Laws: $\\lim\\limits_{x \\to-5} (9x-7)=$ [ANS]", "answer_v2": [ "-52" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the limit using the Limit Laws: $\\lim\\limits_{x \\to-2} (6x-5)=$ [ANS]", "answer_v3": [ "-17" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0006", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "2", "keywords": [ "calculus", "limits", "basic limit laws", "rational functions" ], "problem_v1": "Evaluate the limit using the Limit Laws: $ \\lim_{t \\to 8} t^{-1}=$ [ANS]", "answer_v1": [ "0.125" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the limit using the Limit Laws: $ \\lim_{t \\to 2} t^{-1}=$ [ANS]", "answer_v2": [ "0.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the limit using the Limit Laws: $ \\lim_{t \\to 4} t^{-1}=$ [ANS]", "answer_v3": [ "0.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0007", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "2", "keywords": [ "Limit", "Root", "calculus", "limits", "derivatives" ], "problem_v1": "Given $\\lim_{x \\to 7} g(x)=6$, evaluate \\lim_{x \\to 7} \\sqrt{g(x)}. (If the limit does not exist, enter \"DNE\".) (If the limit does not exist, enter \"DNE\".) Limit=[ANS]", "answer_v1": [ "2.44949" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given $\\lim_{x \\to 3} g(x)=3$, evaluate \\lim_{x \\to 3} \\sqrt{g(x)}. (If the limit does not exist, enter \"DNE\".) (If the limit does not exist, enter \"DNE\".) Limit=[ANS]", "answer_v2": [ "1.73205" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given $\\lim_{x \\to 4} g(x)=4$, evaluate \\lim_{x \\to 4} \\sqrt{g(x)}. (If the limit does not exist, enter \"DNE\".) (If the limit does not exist, enter \"DNE\".) Limit=[ANS]", "answer_v3": [ "2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0008", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "2", "keywords": [ "Derivative", "Polynomial" ], "problem_v1": "Find $\\> \\lim_{x \\rightarrow 6} x\\!\\left(x-7\\right)\\!\\left(x+7\\right)$.\nEnter I for $\\infty$,-I for $-\\infty$, and DNE if the limit does not exist.\nAnswer: [ANS]", "answer_v1": [ "-78" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $\\> \\lim_{x \\rightarrow 9} x\\!\\left(x-1\\right)\\!\\left(x+1\\right)$.\nEnter I for $\\infty$,-I for $-\\infty$, and DNE if the limit does not exist.\nAnswer: [ANS]", "answer_v2": [ "720" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $\\> \\lim_{x \\rightarrow 6} x\\!\\left(x-3\\right)\\!\\left(x+3\\right)$.\nEnter I for $\\infty$,-I for $-\\infty$, and DNE if the limit does not exist.\nAnswer: [ANS]", "answer_v3": [ "162" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0009", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "2", "keywords": [], "problem_v1": "Part 1: Evaluate the limit Part 1: Evaluate the limit\n\\$(\"#Prob-1_SC-1_SECT-1\").canopen() Evaluate the following limit by simplifying the expression (first answer box) and then evaluating the limit (second answer box). ${\\lim_{x\\to 0} \\ {x\\!\\left(18+\\frac{6}{x}\\right)}=\\lim_{x\\to 0}}$ [ANS] ${=}$ [ANS]. Note: In your written solution, you should write the limit statement ${\\lim_{x \\to 0}}$ in every step except the last one, where the limit is finally evaluated. Part 2: Follow-up question Part 2: Follow-up question\n\\$(\"#Prob-1_SC-1_SECT-2\").canopen() Select the true statement [ANS] A. $ x\\!\\left(18+\\frac{6}{x}\\right)=18x+6$ for all $x \\not=0$. B. $ x\\!\\left(18+\\frac{6}{x}\\right)=18x+6$ for all $x$.\n\\$(\"#Prob-1_SC-1_SECT-1\").opensection(); \\$(\"#Prob-1_SC-1_SECT-2\").opensection();", "answer_v1": [ "18*x+6", "6", "A" ], "answer_type_v1": [ "EX", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "Part 1: Evaluate the limit Part 1: Evaluate the limit\n\\$(\"#Prob-1_SC-1_SECT-1\").canopen() Evaluate the following limit by simplifying the expression (first answer box) and then evaluating the limit (second answer box). ${\\lim_{x\\to 0} \\ {x\\!\\left(10+\\frac{9}{x}\\right)}=\\lim_{x\\to 0}}$ [ANS] ${=}$ [ANS]. Note: In your written solution, you should write the limit statement ${\\lim_{x \\to 0}}$ in every step except the last one, where the limit is finally evaluated. Part 2: Follow-up question Part 2: Follow-up question\n\\$(\"#Prob-1_SC-1_SECT-2\").canopen() Select the true statement [ANS] A. $ x\\!\\left(10+\\frac{9}{x}\\right)=10x+9$ for all $x$. B. $ x\\!\\left(10+\\frac{9}{x}\\right)=10x+9$ for all $x \\not=0$.\n\\$(\"#Prob-1_SC-1_SECT-1\").opensection(); \\$(\"#Prob-1_SC-1_SECT-2\").opensection();", "answer_v2": [ "10*x+9", "9", "B" ], "answer_type_v2": [ "EX", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "Part 1: Evaluate the limit Part 1: Evaluate the limit\n\\$(\"#Prob-1_SC-1_SECT-1\").canopen() Evaluate the following limit by simplifying the expression (first answer box) and then evaluating the limit (second answer box). ${\\lim_{x\\to 0} \\ {x\\!\\left(13+\\frac{6}{x}\\right)}=\\lim_{x\\to 0}}$ [ANS] ${=}$ [ANS]. Note: In your written solution, you should write the limit statement ${\\lim_{x \\to 0}}$ in every step except the last one, where the limit is finally evaluated. Part 2: Follow-up question Part 2: Follow-up question\n\\$(\"#Prob-1_SC-1_SECT-2\").canopen() Select the true statement [ANS] A. $ x\\!\\left(13+\\frac{6}{x}\\right)=13x+6$ for all $x$. B. $ x\\!\\left(13+\\frac{6}{x}\\right)=13x+6$ for all $x \\not=0$.\n\\$(\"#Prob-1_SC-1_SECT-1\").opensection(); \\$(\"#Prob-1_SC-1_SECT-2\").opensection();", "answer_v3": [ "13*x+6", "6", "B" ], "answer_type_v3": [ "EX", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Calculus_-_single_variable_0010", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "3", "keywords": [], "problem_v1": "Suppose $f$ and $g$ are functions satisfying\n${\\lim_{x \\to 6} f(x)=0}$,\n${\\lim_{x \\to 6} g(x)=0}$, and\n${\\lim_{x \\to 6} \\frac{f(x)}{g(x)}=8}$. What can be said about the relative sizes of $f(x)$ and $g(x)$ as $x$ approaches $6$? Select all that apply. [ANS] A. $f(x) \\approx 8 g(x)$ for $x$ near $6$. B. $g(x) \\approx 8 f(x)$ for $x$ near $6$. C. Values of $f(x)$ are about 8 times as large as values of $g(x)$ as $x$ approaches $6$. D. Values of $g(x)$ are about 8 times as large as values of $f(x)$ as $x$ approaches $6$. E. Nothing definitive can be said.", "answer_v1": [ "AC" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Suppose $f$ and $g$ are functions satisfying\n${\\lim_{x \\to 9} f(x)=0}$,\n${\\lim_{x \\to 9} g(x)=0}$, and\n${\\lim_{x \\to 9} \\frac{f(x)}{g(x)}=2}$. What can be said about the relative sizes of $f(x)$ and $g(x)$ as $x$ approaches $9$? Select all that apply. [ANS] A. Values of $g(x)$ are about 2 times as large as values of $f(x)$ as $x$ approaches $9$. B. $g(x) \\approx 2 f(x)$ for $x$ near $9$. C. $f(x) \\approx 2 g(x)$ for $x$ near $9$. D. Values of $f(x)$ are about 2 times as large as values of $g(x)$ as $x$ approaches $9$. E. Nothing definitive can be said.", "answer_v2": [ "CD" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Suppose $f$ and $g$ are functions satisfying\n${\\lim_{x \\to 6} f(x)=0}$,\n${\\lim_{x \\to 6} g(x)=0}$, and\n${\\lim_{x \\to 6} \\frac{f(x)}{g(x)}=4}$. What can be said about the relative sizes of $f(x)$ and $g(x)$ as $x$ approaches $6$? Select all that apply. [ANS] A. $f(x) \\approx 4 g(x)$ for $x$ near $6$. B. $g(x) \\approx 4 f(x)$ for $x$ near $6$. C. Values of $g(x)$ are about 4 times as large as values of $f(x)$ as $x$ approaches $6$. D. Values of $f(x)$ are about 4 times as large as values of $g(x)$ as $x$ approaches $6$. E. Nothing definitive can be said.", "answer_v3": [ "AD" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Calculus_-_single_variable_0011", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "3", "keywords": [ "secant", "difference quotient" ], "problem_v1": "Evaluate the limit below in two steps by using algebra to simplify the difference quotient and then evaluating the limit.\n$ \\lim_{h \\to 0^+} \\Bigg(\\frac{\\sqrt{h^{2}+14h+5}-\\sqrt{5}}{h} \\Bigg)=\\lim_{h \\to 0^+} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].", "answer_v1": [ "(h+14)/[sqrt(h^2+14*h+5)+sqrt(5)]", "14/[2*sqrt(5)]" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Evaluate the limit below in two steps by using algebra to simplify the difference quotient and then evaluating the limit.\n$ \\lim_{h \\to 0^+} \\Bigg(\\frac{\\sqrt{h^{2}+8h+7}-\\sqrt{7}}{h} \\Bigg)=\\lim_{h \\to 0^+} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].", "answer_v2": [ "(h+8)/[sqrt(h^2+8*h+7)+sqrt(7)]", "8/[2*sqrt(7)]" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Evaluate the limit below in two steps by using algebra to simplify the difference quotient and then evaluating the limit.\n$ \\lim_{h \\to 0^+} \\Bigg(\\frac{\\sqrt{h^{2}+10h+5}-\\sqrt{5}}{h} \\Bigg)=\\lim_{h \\to 0^+} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].", "answer_v3": [ "(h+10)/[sqrt(h^2+10*h+5)+sqrt(5)]", "10/[2*sqrt(5)]" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0012", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "2", "keywords": [ "limit", "trig limit", "trigonometric limit", "limit laws", "continuity" ], "problem_v1": "Evaluate the following limit.\n$ \\lim_{t \\to \\pi} \\sqrt{t+7}\\cos\\!\\left(t+\\pi \\right)=$ [ANS].", "answer_v1": [ "\\sqrt{\\pi+7}" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the following limit.\n$ \\lim_{t \\to-\\pi} \\sqrt{t+9}\\cos\\!\\left(t+\\pi \\right)=$ [ANS].", "answer_v2": [ "\\sqrt{-pi+9}" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the following limit.\n$ \\lim_{t \\to-\\pi} \\sqrt{t+7}\\cos\\!\\left(t+\\pi \\right)=$ [ANS].", "answer_v3": [ "\\sqrt{-pi+7}" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0013", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "2", "keywords": [ "limits", "derivatives", "Product", "Quotient", "Differentiation" ], "problem_v1": "Evaluate the limit \\lim_{u \\rightarrow 3} \\sqrt{u^4+3 u+6} Enter I for $\\infty$,-I for $-\\infty$, and DNE if the limit does not exist. Limit=[ANS]", "answer_v1": [ "9.79795897113271" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the limit \\lim_{u \\rightarrow 2} \\sqrt{u^2+2 u+8} Enter I for $\\infty$,-I for $-\\infty$, and DNE if the limit does not exist. Limit=[ANS]", "answer_v2": [ "4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the limit \\lim_{u \\rightarrow 2} \\sqrt{u^2+2 u+6} Enter I for $\\infty$,-I for $-\\infty$, and DNE if the limit does not exist. Limit=[ANS]", "answer_v3": [ "3.74165738677394" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0014", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "2", "keywords": [ "calculus", "limits", "continuous", "Limit" ], "problem_v1": "Calculate the following limits by direct substitution. ${\\lim_{t\\to-6}\\left(\\sqrt{t^2+3t-2}\\right)^3=}$ [ANS]\n${\\lim_{b\\to 7} \\frac{24}{b+1}-(b-4)^2=}$ [ANS]\n${\\lim_{a\\to-4} \\frac{a^2-3a+4}{a-12}=}$ [ANS]\n${\\lim_{s\\to 8} \\sqrt{\\frac{13-s}{s+12}}=}$ [ANS]\n${\\lim_{y\\to-1} (6-y)(y^2+1)^3=}$ [ANS]\n${\\lim_{t\\to 5} \\frac{(1-t)(t+5)}{3t-7}=}$ [ANS]", "answer_v1": [ "64", "-6", "-2", "0.5", "56", "-5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Calculate the following limits by direct substitution. ${\\lim_{x\\to 3} 2x^3-4x-10=}$ [ANS]\n${\\lim_{a\\to-10} \\frac{(a+7)^4}{a+1}=}$ [ANS]\n${\\lim_{t\\to 5} \\frac{(1-t)(t+5)}{3t-7}=}$ [ANS]\n${\\lim_{a\\to-4} \\frac{a^2-3a+4}{a-12}=}$ [ANS]\n${\\lim_{s\\to 8} \\sqrt{\\frac{13-s}{s+12}}=}$ [ANS]\n${\\lim_{x\\to 0} \\sqrt{3(x^2+12)}=}$ [ANS]", "answer_v2": [ "32", "-9", "-5", "-2", "0.5", "6" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Calculate the following limits by direct substitution. ${\\lim_{x\\to 0} \\sqrt{3(x^2+12)}=}$ [ANS]\n${\\lim_{y\\to-2} y^3(5-3y^2)=}$ [ANS]\n${\\lim_{t\\to 5} \\frac{(1-t)(t+5)}{3t-7}=}$ [ANS]\n${\\lim_{b\\to 7} \\frac{24}{b+1}-(b-4)^2=}$ [ANS]\n${\\lim_{y\\to-1} (6-y)(y^2+1)^3=}$ [ANS]\n${\\lim_{a\\to-4} \\frac{a^2-3a+4}{a-12}=}$ [ANS]", "answer_v3": [ "6", "56", "-5", "-6", "56", "-2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0015", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Rules of limits - basic", "level": "2", "keywords": [ "calculus", "limits", "sequences", "Limit" ], "problem_v1": "$\\begin{array}{ccccccc}\\hline a &-1 & 0 & 1 & 2 & 3 & 4 \\\\ \\hline \\lim_{x\\to a^-}f(x) & DNE & 1 & 0 & 3 & 3 & 1 \\\\ \\hline \\lim_{x\\to a^+}f(x) & 3 & 2 & 0 & 3 & 3 & DNE \\\\ \\hline f(a) & 3 & 1 & 0 & 3 & 3 & 1 \\\\ \\hline \\lim_{x\\to a^-}g(x) & DNE & 2 & 2 & 0 & 0 & 2 \\\\ \\hline \\lim_{x\\to a^+}g(x) & 1 & 2 & 2 & 0 & 3 & DNE \\\\ \\hline g(a) & 1 & 2 & 2 & 0 & 3 & 2 \\\\ \\hline \\end{array}$\nUsing the table above calcuate the limits below. Enter 'DNE' if the limit doesn't exist OR if limit can't be determined from the information given. [ANS] 1. $ \\lim_{x\\to 0^+} [f(g(x))]$ [ANS] 2. $f(3)+g(3)$ [ANS] 3. $ \\lim_{x\\to 3^-} [f(x)/g(x)]$ [ANS] 4. $f(g(3))$", "answer_v1": [ "3", "6", "DNE", "3" ], "answer_type_v1": [ "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "$\\begin{array}{ccccccc}\\hline a &-1 & 0 & 1 & 2 & 3 & 4 \\\\ \\hline \\lim_{x\\to a^-}f(x) & DNE & 2 & 2 & 1 & 0 & 2 \\\\ \\hline \\lim_{x\\to a^+}f(x) & 0 & 2 & 2 & 3 & 0 & DNE \\\\ \\hline f(a) & 0 & 2 & 2 & 1 & 0 & 2 \\\\ \\hline \\lim_{x\\to a^-}g(x) & DNE & 3 & 0 & 1 & 3 & 0 \\\\ \\hline \\lim_{x\\to a^+}g(x) & 3 & 3 & 0 & 3 & 3 & DNE \\\\ \\hline g(a) & 3 & 3 & 0 & 4 & 3 & 0 \\\\ \\hline \\end{array}$\nUsing the table above calcuate the limits below. Enter 'DNE' if the limit doesn't exist OR if limit can't be determined from the information given. [ANS] 1. $ \\lim_{x\\to 2^+} [f(g(x))]$ [ANS] 2. $ \\lim_{x\\to 2^-} [f(x)+g(x)]$ [ANS] 3. $ \\lim_{x\\to 2^-} [f(g(x))]$ [ANS] 4. $ \\lim_{x\\to 2^+} [f(g(x))]$", "answer_v2": [ "0", "2", "2", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "$\\begin{array}{ccccccc}\\hline a &-1 & 0 & 1 & 2 & 3 & 4 \\\\ \\hline \\lim_{x\\to a^-}f(x) & DNE & 3 & 1 & 2 & 0 & 2 \\\\ \\hline \\lim_{x\\to a^+}f(x) & 1 & 1 & 1 & 2 & 0 & DNE \\\\ \\hline f(a) & 1 & 1 & 1 & 2 & 0 & 2 \\\\ \\hline \\lim_{x\\to a^-}g(x) & DNE & 0 & 2 & 0 & 2 & 0 \\\\ \\hline \\lim_{x\\to a^+}g(x) & 0 & 0 & 1 & 0 & 2 & DNE \\\\ \\hline g(a) & 0 & 0 & 4 & 0 & 2 & 0 \\\\ \\hline \\end{array}$\nUsing the table above calcuate the limits below. Enter 'DNE' if the limit doesn't exist OR if limit can't be determined from the information given. [ANS] 1. $f(0)/g(0)$ [ANS] 2. $ \\lim_{x\\to 1^+} [f(x)g(x)]$ [ANS] 3. $ \\lim_{x\\to 0^-} [f(x)g(x)]$ [ANS] 4. $ \\lim_{x\\to 0^-} [f(x)+g(x)]$", "answer_v3": [ "DNE", "1", "0", "3" ], "answer_type_v3": [ "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0016", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Evaluating limits - factoring", "level": "3", "keywords": [ "limits", "factoring" ], "problem_v1": "Use factoring to calculate the following limit.\n\\lim_{s \\rightarrow a} \\frac {{s}^4-a^4} {{s}^5-a^5} Answer: [ANS]\nHint: Try doing this numerically for a couple of values of $s$ and $a$.", "answer_v1": [ " (4/5) * a ^ (-1) " ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use factoring to calculate the following limit.\n\\lim_{x \\rightarrow t} \\frac {{x}^2-t^2} {{x}^4-t^4} Answer: [ANS]\nHint: Try doing this numerically for a couple of values of $x$ and $t$.", "answer_v2": [ " (2/4) * t ^ (-2) " ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use factoring to calculate the following limit.\n\\lim_{y \\rightarrow s} \\frac {{y}^3-s^3} {{y}^4-s^4} Answer: [ANS]\nHint: Try doing this numerically for a couple of values of $y$ and $s$.", "answer_v3": [ " (3/4) * s ^ (-1) " ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0017", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Evaluating limits - factoring", "level": "2", "keywords": [ "calculus", "limits", "continuity", "rational functions", "identities" ], "problem_v1": "Evaluate the limit using the identity $a^3-b^3=(a-b)(a^2+ab+b^2)$: $\\lim\\limits_{x \\to 7} \\frac {x^3-343} {x-7}=$ [ANS]", "answer_v1": [ "147" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the limit using the identity $a^3-b^3=(a-b)(a^2+ab+b^2)$: $\\lim\\limits_{x \\to 1} \\frac {x^3-1} {x-1}=$ [ANS]", "answer_v2": [ "3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the limit using the identity $a^3-b^3=(a-b)(a^2+ab+b^2)$: $\\lim\\limits_{x \\to 3} \\frac {x^3-27} {x-3}=$ [ANS]", "answer_v3": [ "27" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0018", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Evaluating limits - factoring", "level": "3", "keywords": [ "Limit", "Difference Quotient" ], "problem_v1": "Let $f(x)=7x^{2}+5$. Evaluate \\lim_{h \\to 0} \\frac{f(1+h)-f(1)}{h}. (If the limit does not exist, enter \"DNE\".) (If the limit does not exist, enter \"DNE\".) Limit=[ANS]", "answer_v1": [ "14" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)=2x^{2}+8$. Evaluate \\lim_{h \\to 0} \\frac{f(-3+h)-f(-3)}{h}. (If the limit does not exist, enter \"DNE\".) (If the limit does not exist, enter \"DNE\".) Limit=[ANS]", "answer_v2": [ "-12" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(x)=4x^{2}+5$. Evaluate \\lim_{h \\to 0} \\frac{f(-2+h)-f(-2)}{h}. (If the limit does not exist, enter \"DNE\".) (If the limit does not exist, enter \"DNE\".) Limit=[ANS]", "answer_v3": [ "-16" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0019", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Evaluating limits - factoring", "level": "3", "keywords": [ "Calculus", "limits", "factoring", "Limit", "Factor" ], "problem_v1": "Let $f(x)=\\frac{7x+28}{x^2-4x-32}$. Calculate ${\\lim_{x\\to-4}f(x)}$ by first finding a continuous function which is equal to $f$ everywhere except $x=-4$. ${\\lim_{x\\to-4}f(x)}=$ [ANS]", "answer_v1": [ "-0.583333333333333" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)=\\frac{3x+3}{x^2-5x-6}$. Calculate ${\\lim_{x\\to-1}f(x)}$ by first finding a continuous function which is equal to $f$ everywhere except $x=-1$. ${\\lim_{x\\to-1}f(x)}=$ [ANS]", "answer_v2": [ "-0.428571428571429" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(x)=\\frac{4x+8}{x^2-4x-12}$. Calculate ${\\lim_{x\\to-2}f(x)}$ by first finding a continuous function which is equal to $f$ everywhere except $x=-2$. ${\\lim_{x\\to-2}f(x)}=$ [ANS]", "answer_v3": [ "-0.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0020", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Evaluating limits - factoring", "level": "3", "keywords": [ "Derivatives' 'Limits' 'Rates of Change" ], "problem_v1": "Let $f(x)=4-x^2$. Find each of the following:\n(A) ${\\frac{f(-2)-f(-6)}{-2-(-6)}=}$ [ANS]\n(B) ${\\frac{f(-6+h)-f(-6)}{h}=}$ [ANS]\n(C) ${\\lim_{h\\rightarrow 0}\\frac{f(-6+h)-f(-6)}{h}=}$ [ANS]", "answer_v1": [ "8", "((-6)^2 - (-6+h)^2)/h", "12" ], "answer_type_v1": [ "NV", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $f(x)=-7-x^2$. Find each of the following:\n(A) ${\\frac{f(-1)-f(-8)}{-1-(-8)}=}$ [ANS]\n(B) ${\\frac{f(-8+h)-f(-8)}{h}=}$ [ANS]\n(C) ${\\lim_{h\\rightarrow 0}\\frac{f(-8+h)-f(-8)}{h}=}$ [ANS]", "answer_v2": [ "9", "((-8)^2 - (-8+h)^2)/h", "16" ], "answer_type_v2": [ "NV", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $f(x)=-3-x^2$. Find each of the following:\n(A) ${\\frac{f(-2)-f(-7)}{-2-(-7)}=}$ [ANS]\n(B) ${\\frac{f(-7+h)-f(-7)}{h}=}$ [ANS]\n(C) ${\\lim_{h\\rightarrow 0}\\frac{f(-7+h)-f(-7)}{h}=}$ [ANS]", "answer_v3": [ "9", "((-7)^2 - (-7+h)^2)/h", "14" ], "answer_type_v3": [ "NV", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0021", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Evaluating limits - rational expressions", "level": "3", "keywords": [ "Calculus", "limits", "factoring", "Limit" ], "problem_v1": "Let $f(s)=\\frac{5}{s-6}-\\frac{60}{s^2-36}$ Calculate ${\\lim_{s \\to 6}f(s)}$ by first finding a continuous function which is equal to $f$ everywhere except $s=6$. ${\\lim_{s\\to 6}f(s)}=$ [ANS]", "answer_v1": [ "0.416666666666667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)=\\frac{2}{x-9}-\\frac{36}{x^2-81}$ Calculate ${\\lim_{x \\to 9}f(x)}$ by first finding a continuous function which is equal to $f$ everywhere except $x=9$. ${\\lim_{x\\to 9}f(x)}=$ [ANS]", "answer_v2": [ "0.111111111111111" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(y)=\\frac{3}{y-6}-\\frac{36}{y^2-36}$ Calculate ${\\lim_{y \\to 6}f(y)}$ by first finding a continuous function which is equal to $f$ everywhere except $y=6$. ${\\lim_{y\\to 6}f(y)}=$ [ANS]", "answer_v3": [ "0.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0022", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Evaluating limits - trigonometric", "level": "3", "keywords": [ "limit", "limits", "continuity" ], "problem_v1": "Find the limit. Notes: Enter \"DNE\" if limit Does Not Exist. $ \\lim_{h\\to 0} \\frac{1-\\cos\\!\\left(10h\\right)}{\\cos^{2}\\!\\left(6h\\right)-1}$=[ANS]", "answer_v1": [ "-25/18" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the limit. Notes: Enter \"DNE\" if limit Does Not Exist. $ \\lim_{h\\to 0} \\frac{1-\\cos\\!\\left(2h\\right)}{\\cos^{2}\\!\\left(8h\\right)-1}$=[ANS]", "answer_v2": [ "-1/32" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the limit. Notes: Enter \"DNE\" if limit Does Not Exist. $ \\lim_{h\\to 0} \\frac{1-\\cos\\!\\left(5h\\right)}{\\cos^{2}\\!\\left(6h\\right)-1}$=[ANS]", "answer_v3": [ "-25/72" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0023", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Evaluating limits - trigonometric", "level": "3", "keywords": [ "calculus", "differentiation", "Differentiation", "Trigonometric", "Transcendental", "Derivative" ], "problem_v1": "Evaluate \\lim_{\\theta \\to 0} \\frac{\\sin (5 \\cos \\theta)}{5 \\sec \\theta}. Limit=[ANS]", "answer_v1": [ "-0.191784854932628" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate \\lim_{\\theta \\to 0} \\frac{\\sin (2 \\cos \\theta)}{7 \\sec \\theta}. Limit=[ANS]", "answer_v2": [ "0.129899632403669" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate \\lim_{\\theta \\to 0} \\frac{\\sin (3 \\cos \\theta)}{5 \\sec \\theta}. Limit=[ANS]", "answer_v3": [ "0.0282240016119734" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0025", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "One-sided limits - concept of", "level": "2", "keywords": [], "problem_v1": "Evaluate the limits. f(x)=\\begin{cases} 8x^2+7x+4 & x < 0 \\\\ 4 \\sin x & x \\geq 0 \\end{cases} Enter DNE if the limit does not exist. $ \\lim_{x \\to 0^-} f(x)$=[ANS]\n$ \\lim_{x \\to 0^+} f(x)$=[ANS]\n$ \\lim_{x \\to 0} f(x)$=[ANS]", "answer_v1": [ "4", "0", "DNE" ], "answer_type_v1": [ "NV", "NV", "OE" ], "options_v1": [ [], [], [] ], "problem_v2": "Evaluate the limits. f(x)=\\begin{cases} 2x^2-3x+10 & x < 0 \\\\-4 \\sin x & x \\geq 0 \\end{cases} Enter DNE if the limit does not exist. $ \\lim_{x \\to 0^-} f(x)$=[ANS]\n$ \\lim_{x \\to 0^+} f(x)$=[ANS]\n$ \\lim_{x \\to 0} f(x)$=[ANS]", "answer_v2": [ "10", "0", "DNE" ], "answer_type_v2": [ "NV", "NV", "OE" ], "options_v2": [ [], [], [] ], "problem_v3": "Evaluate the limits. f(x)=\\begin{cases} 4x^2+4x+3 & x < 0 \\\\ 4 \\sin x & x \\geq 0 \\end{cases} Enter DNE if the limit does not exist. $ \\lim_{x \\to 0^-} f(x)$=[ANS]\n$ \\lim_{x \\to 0^+} f(x)$=[ANS]\n$ \\lim_{x \\to 0} f(x)$=[ANS]", "answer_v3": [ "3", "0", "DNE" ], "answer_type_v3": [ "NV", "NV", "OE" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0026", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "One-sided limits - concept of", "level": "3", "keywords": [ "calculus", "limit", "one-sided" ], "problem_v1": "Find the one-sided limit \\lim_{t\\to 9^+}\\frac{|81-t^2|}{9-t} Limit: [ANS]\nUse INF to denote $\\infty$ and-INF to denote $-\\infty$.", "answer_v1": [ "-18" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the one-sided limit \\lim_{t\\to 3^+}\\frac{|9-t^2|}{3-t} Limit: [ANS]\nUse INF to denote $\\infty$ and-INF to denote $-\\infty$.", "answer_v2": [ "-6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the one-sided limit \\lim_{t\\to 5^+}\\frac{|25-t^2|}{5-t} Limit: [ANS]\nUse INF to denote $\\infty$ and-INF to denote $-\\infty$.", "answer_v3": [ "-10" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0028", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "One-sided limits - concept of", "level": "2", "keywords": [ "limits", "derivatives" ], "problem_v1": "Let f(x)=\\begin{cases} 6, & x < 5, \\\\ 4x, & x=5,\\\\ 10+x, & x > 5. \\end{cases} Evaluate each of the following:\nNote: You use INF for $\\infty$ and-INF for $-\\infty$. (A) ${\\lim_{x \\rightarrow 5^{-}} f(x)}$=[ANS]\n(B) ${\\lim_{x \\rightarrow 5^{+}} f(x)}$=[ANS]\n(C) $f(5)=$ [ANS]", "answer_v1": [ "6", "15", "20" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let f(x)=\\begin{cases} 2, & x < 7, \\\\-3x, & x=7,\\\\ 10+x, & x > 7. \\end{cases} Evaluate each of the following:\nNote: You use INF for $\\infty$ and-INF for $-\\infty$. (A) ${\\lim_{x \\rightarrow 7^{-}} f(x)}$=[ANS]\n(B) ${\\lim_{x \\rightarrow 7^{+}} f(x)}$=[ANS]\n(C) $f(7)=$ [ANS]", "answer_v2": [ "2", "17", "-21" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let f(x)=\\begin{cases} 3, & x < 5, \\\\-3x, & x=5,\\\\ 10+x, & x > 5. \\end{cases} Evaluate each of the following:\nNote: You use INF for $\\infty$ and-INF for $-\\infty$. (A) ${\\lim_{x \\rightarrow 5^{-}} f(x)}$=[ANS]\n(B) ${\\lim_{x \\rightarrow 5^{+}} f(x)}$=[ANS]\n(C) $f(5)=$ [ANS]", "answer_v3": [ "3", "15", "-15" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0029", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "2", "keywords": [ "limit", "continuity" ], "problem_v1": "Use continuity to evaluate\n$ \\lim _{x\\to 1} e^{x^2-5x+3}$=[ANS]", "answer_v1": [ "e^{-1}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use continuity to evaluate\n$ \\lim _{x\\to 1} e^{x^2-2x+5}$=[ANS]", "answer_v2": [ "e^4" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use continuity to evaluate\n$ \\lim _{x\\to 1} e^{x^2-3x+4}$=[ANS]", "answer_v3": [ "1" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0031", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "4", "keywords": [], "problem_v1": "Give the interval(s) on which the function is continuous. g(t)=\\frac{1}{\\sqrt{64-t^2}} [ANS]", "answer_v1": [ "(-8,8)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Give the interval(s) on which the function is continuous. g(t)=\\frac{1}{\\sqrt{1-t^2}} [ANS]", "answer_v2": [ "(-1,1)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Give the interval(s) on which the function is continuous. g(t)=\\frac{1}{\\sqrt{16-t^2}} [ANS]", "answer_v3": [ "(-4,4)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0032", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "2", "keywords": [], "problem_v1": "Given the interval(s) on which the function is continuous. f(x)=e^{6x}-\\ln(x-8) [ANS]", "answer_v1": [ "(8,infinity)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Given the interval(s) on which the function is continuous. f(x)=e^{8x}-\\ln(x-1) [ANS]", "answer_v2": [ "(1,infinity)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Given the interval(s) on which the function is continuous. f(x)=e^{6x}-\\ln(x-4) [ANS]", "answer_v3": [ "(4,infinity)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0033", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "3", "keywords": [ "Continuous", "Piecewise", "Calculus", "continuity", "limits" ], "problem_v1": "Let f(x)=\\begin{cases}{b-2x}&\\text{if}\\ x < 5\\cr {-\\frac{150}{x-b}}&\\text{if}\\ x \\ge 5.\\end{cases} Find the two values of $b$ for which $f$ is a continuous function at $5$. The one with the greater absolute value is $b=$ [ANS]. Now draw a graph of $f$.", "answer_v1": [ "20" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let f(x)=\\begin{cases}{b-2x}&\\text{if}\\ x <-4\\cr {-\\frac{96}{x-b}}&\\text{if}\\ x \\ge-4.\\end{cases} Find the two values of $b$ for which $f$ is a continuous function at $-4$. The one with the greater absolute value is $b=$ [ANS]. Now draw a graph of $f$.", "answer_v2": [ "-16" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let f(x)=\\begin{cases}{b-2x}&\\text{if}\\ x <-1\\cr {-\\frac{6}{x-b}}&\\text{if}\\ x \\ge-1.\\end{cases} Find the two values of $b$ for which $f$ is a continuous function at $-1$. The one with the greater absolute value is $b=$ [ANS]. Now draw a graph of $f$.", "answer_v3": [ "-4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0034", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "4", "keywords": [ "Continuous", "Piecewise", "continuity", "limits", "Calculus" ], "problem_v1": "Find the value of the constant $m$ that makes the following function continuous on $(-\\infty,\\infty)$. f(x)=\\begin{cases}{mx-12}&\\text{if}\\ x <-7\\cr {x^{2}+8x-5}&\\text{if}\\ x \\ge-7\\end{cases} $m=$ [ANS]\nNow draw a graph of $f$.", "answer_v1": [ "0" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the value of the constant $m$ that makes the following function continuous on $(-\\infty,\\infty)$. f(x)=\\begin{cases}{mx-11}&\\text{if}\\ x <-3\\cr {x^{2}+2x-8}&\\text{if}\\ x \\ge-3\\end{cases} $m=$ [ANS]\nNow draw a graph of $f$.", "answer_v2": [ "-2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the value of the constant $m$ that makes the following function continuous on $(-\\infty,\\infty)$. f(x)=\\begin{cases}{mx-9}&\\text{if}\\ x <-4\\cr {x^{2}+4x-5}&\\text{if}\\ x \\ge-4\\end{cases} $m=$ [ANS]\nNow draw a graph of $f$.", "answer_v3": [ "-1" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0035", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "4", "keywords": [ "Continuous", "Piecewise", "calculus", "continuity" ], "problem_v1": "Find the value of the constant $c$ that makes the following function continuous on $(-\\infty,\\infty)$. f(x)=\\left\\lbrace \\begin{array}{ll} x^2-c &\\mbox{if}\\-\\infty < x < 6\\\\ cx+6 &\\mbox{if}\\ x\\ge 6 \\end{array}\\right. $c=$ [ANS]", "answer_v1": [ "4.28571" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the value of the constant $c$ that makes the following function continuous on $(-\\infty,\\infty)$. f(x)=\\left\\lbrace \\begin{array}{ll} x^2-c &\\mbox{if}\\-\\infty < x < 9\\\\ cx+2 &\\mbox{if}\\ x\\ge 9 \\end{array}\\right. $c=$ [ANS]", "answer_v2": [ "7.9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the value of the constant $c$ that makes the following function continuous on $(-\\infty,\\infty)$. f(x)=\\left\\lbrace \\begin{array}{ll} x^2-c &\\mbox{if}\\-\\infty < x < 6\\\\ cx+3 &\\mbox{if}\\ x\\ge 6 \\end{array}\\right. $c=$ [ANS]", "answer_v3": [ "4.71429" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0036", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "4", "keywords": [ "Continuous", "Piecewise", "Calculus", "continuity" ], "problem_v1": "Find the value of the constant $a$ that makes the following function continuous on $(-\\infty,\\infty)$. f(x)=\\begin{cases}{\\frac{5x^{3}-15x^{2}+9x-27}{x-3}}&\\text{if}\\ x < 3\\cr {-2x^{2}+x+a}&\\text{if}\\ x \\ge 3\\end{cases} $a=$ [ANS]", "answer_v1": [ "69" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the value of the constant $a$ that makes the following function continuous on $(-\\infty,\\infty)$. f(x)=\\begin{cases}{\\frac{2x^{3}+5x^{2}+9x+14}{x+2}}&\\text{if}\\ x <-2\\cr {-2x^{2}-4x+a}&\\text{if}\\ x \\ge-2\\end{cases} $a=$ [ANS]", "answer_v2": [ "13" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the value of the constant $a$ that makes the following function continuous on $(-\\infty,\\infty)$. f(x)=\\begin{cases}{\\frac{3x^{3}-3x^{2}+8x-8}{x-1}}&\\text{if}\\ x < 1\\cr {-2x^{2}+4x+a}&\\text{if}\\ x \\ge 1\\end{cases} $a=$ [ANS]", "answer_v3": [ "9" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0038", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "3", "keywords": [ "Calculus", "Discontinuity", "Removable", "limits", "factoring", "continuity" ], "problem_v1": "A function $f$ is said to have a removable discontinuity at $a$ if: 1. $f$ is either not defined or not continuous at $a$. 2. $f(a)$ could either be defined or redefined so that the new function is continuous at $a$. Let $f(x)=\\frac{2x^2+4x-48}{x-4}$. Show that $f$ has a removable discontinuity at $4$ and determine the value for $f(4)$ that would make $f$ continuous at $4$. Need to redefine $f(4)=$ [ANS].", "answer_v1": [ "20" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A function $f$ is said to have a removable discontinuity at $a$ if: 1. $f$ is either not defined or not continuous at $a$. 2. $f(a)$ could either be defined or redefined so that the new function is continuous at $a$. Let $f(x)=\\frac{2x^2+6x-8}{x-1}$. Show that $f$ has a removable discontinuity at $1$ and determine the value for $f(1)$ that would make $f$ continuous at $1$. Need to redefine $f(1)=$ [ANS].", "answer_v2": [ "10" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A function $f$ is said to have a removable discontinuity at $a$ if: 1. $f$ is either not defined or not continuous at $a$. 2. $f(a)$ could either be defined or redefined so that the new function is continuous at $a$. Let $f(x)=\\frac{2x^2+5x-18}{x-2}$. Show that $f$ has a removable discontinuity at $2$ and determine the value for $f(2)$ that would make $f$ continuous at $2$. Need to redefine $f(2)=$ [ANS].", "answer_v3": [ "13" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0039", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "4", "keywords": [ "Calculus" ], "problem_v1": "Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several examples, especially examples which might show that the statement is false! You must get all of the answers correct to receive credit. [ANS] 1. If a differentiable function has a maximum value then its domain must be a bounded, closed interval. [ANS] 2. Every differentiable function whose domain is a bounded, closed interval has a maximum value. [ANS] 3. If a differentiable function has a maximum value then it also has a minimum value. [ANS] 4. If the linear approximation of a differentiable function is increasing at a point $a$ then the function is also increasing near the point $a$.", "answer_v1": [ "F", "T", "F", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several examples, especially examples which might show that the statement is false! You must get all of the answers correct to receive credit. [ANS] 1. If $f(x)$ is a continuous function and the sequence $f(a_{1}), f(a_{2}), f(a_{3}),...$ converges to a finite limit, then the sequence $a_{1}, a_{2}, a_{3},...$ also converges to a limit. [ANS] 2. If the linear approximation of a differentiable function is constant at a point $a$ then the function could be increasing near the point $a$. [ANS] 3. Every continuous function whose domain is a bounded, closed interval has a maximum value. [ANS] 4. Every differentiable function is continuous.", "answer_v2": [ "F", "T", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several examples, especially examples which might show that the statement is false! You must get all of the answers correct to receive credit. [ANS] 1. If a continuous function $f(x)$ has a maximum value on an interval then the function $-f(x)$ has a minimum on that same interval. [ANS] 2. Every differentiable function whose domain is a bounded, closed interval and which has a maximum value also has a minimum value. [ANS] 3. Every continuous function whose domain is a bounded, closed interval and which has a maximum value also has a minimum value. [ANS] 4. Every differentiable function whose domain is a bounded, closed interval has a maximum value.", "answer_v3": [ "T", "T", "T", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0040", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "4", "keywords": [ "calculus", "derivative", "continuity", "functions" ], "problem_v1": "If possible, choose $k$ so that the following function is continuous on any interval: f(x)=\\begin{cases} \\frac{7x^{4}-28x^{3}}{x-4}\\quad & x \\ne 4 \\\\ k & x=4. \\end{cases} $k=$ [ANS]\n(If no k will make the function continuous, enter none)", "answer_v1": [ "7*4^3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If possible, choose $k$ so that the following function is continuous on any interval: f(x)=\\begin{cases} \\frac{3x^{5}-6x^{4}}{x-2}\\quad & x \\ne 2 \\\\ k & x=2. \\end{cases} $k=$ [ANS]\n(If no k will make the function continuous, enter none)", "answer_v2": [ "3*2^4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If possible, choose $k$ so that the following function is continuous on any interval: f(x)=\\begin{cases} \\frac{4x^{4}-12x^{3}}{x-3}\\quad & x \\ne 3 \\\\ k & x=3. \\end{cases} $k=$ [ANS]\n(If no k will make the function continuous, enter none)", "answer_v3": [ "4*3^3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0041", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "2", "keywords": [ "limit", "limits", "Continuity" ], "problem_v1": "Find values of $x$, if any, at which f(x)=\\frac{5}{x-9}-\\frac{3x}{x+7} is not continuous (enter as comma separated list, smallest to largest). $x=$ [ANS]", "answer_v1": [ "(-7, 9)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find values of $x$, if any, at which f(x)=-\\frac{1}{x-14}+\\frac{3x}{x+2} is not continuous (enter as comma separated list, smallest to largest). $x=$ [ANS]", "answer_v2": [ "(-2, 14)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find values of $x$, if any, at which f(x)=\\frac{2}{x-10}-\\frac{4x}{x+3} is not continuous (enter as comma separated list, smallest to largest). $x=$ [ANS]", "answer_v3": [ "(-3, 10)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0042", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "4", "keywords": [ "calculus", "limit", "continuity", "piecewise" ], "problem_v1": "Shown below are six statements about functions. Match each statement to one of the functions shown below which BEST matches that statement. [ANS] 1. $\\lim_{x\\to 8^+} f(x)$ and $\\lim_{x\\to 8^-} f(x)$ both exist and are finite, but they are not equal. [ANS] 2. The graph of $y=f(x)$ has vertical tangent line at $(8,f(8))$ [ANS] 3. $\\lim_{x\\to 8^-}f(x)=-\\infty$. [ANS] 4. $\\lim_{x\\to 8^+}f(x)$ exists but $\\lim_{x\\to 8^-}f(x)$ does not. [ANS] 5. $\\lim_{x\\to 8}f(x)=\\infty$. [ANS] 6. $\\lim_{x\\to 8}f(x)$ exists but $f$ is not continuous at 8. A. $f(x)=\\left\\lbrace\\begin{array}{ll}\\cos\\left(\\frac{1}{x-8}\\right) &\\mbox{if}x<8\\\\ 0 &\\mbox{if}x=8\\\\ 4x+64 &\\mbox{if}x>8\\\\ \\end{array}\\right.$ B. $f(x)=\\frac{1}{(x-8)^2}$ C. $f(x)=\\sqrt[3]{x-8}$ D. $f(x)=\\left\\lbrace\\begin{array}{ll}4x &\\mbox{if}x<8\\\\ 0 &\\mbox{if}x=8\\\\ 4x-64 &\\mbox{if}x>8 \\\\ \\end{array}\\right.$ E. $f(x)=\\left\\lbrace\\begin{array}{ll}4x &\\mbox{if}x<8\\\\ 0 &\\mbox{if}x=8\\\\ 64-4x &\\mbox{if}x>8\\\\ \\end{array}\\right.$ F. $f(x)=\\frac{1}{x-8}$", "answer_v1": [ "D", "C", "F", "A", "B", "E" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Shown below are six statements about functions. Match each statement to one of the functions shown below which BEST matches that statement. [ANS] 1. $\\lim_{x\\to 2^+} f(x)$ and $\\lim_{x\\to 2^-} f(x)$ both exist and are finite, but they are not equal. [ANS] 2. The graph of $y=f(x)$ has vertical tangent line at $(2,f(2))$ [ANS] 3. $\\lim_{x\\to 2^-}f(x)=-\\infty$. [ANS] 4. $\\lim_{x\\to 2^+}f(x)$ exists but $\\lim_{x\\to 2^-}f(x)$ does not. [ANS] 5. $\\lim_{x\\to 2}f(x)=\\infty$. [ANS] 6. $\\lim_{x\\to 2}f(x)$ exists but $f$ is not continuous at 2. A. $f(x)=\\left\\lbrace\\begin{array}{ll}5x &\\mbox{if}x<2\\\\ 0 &\\mbox{if}x=2\\\\ 5x-20 &\\mbox{if}x>2 \\\\ \\end{array}\\right.$ B. $f(x)=\\frac{1}{x-2}$ C. $f(x)=\\left\\lbrace\\begin{array}{ll}5x &\\mbox{if}x<2\\\\ 0 &\\mbox{if}x=2\\\\ 20-5x &\\mbox{if}x>2\\\\ \\end{array}\\right.$ D. $f(x)=\\sqrt[3]{x-2}$ E. $f(x)=\\left\\lbrace\\begin{array}{ll}\\cos\\left(\\frac{1}{x-2}\\right) &\\mbox{if}x<2\\\\ 0 &\\mbox{if}x=2\\\\ 5x+20 &\\mbox{if}x>2\\\\ \\end{array}\\right.$ F. $f(x)=\\frac{1}{(x-2)^2}$", "answer_v2": [ "A", "D", "B", "E", "F", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Shown below are six statements about functions. Match each statement to one of the functions shown below which BEST matches that statement. [ANS] 1. $\\lim_{x\\to 4^+} f(x)$ and $\\lim_{x\\to 4^-} f(x)$ both exist and are finite, but they are not equal. [ANS] 2. The graph of $y=f(x)$ has vertical tangent line at $(4,f(4))$ [ANS] 3. $\\lim_{x\\to 4^-}f(x)=-\\infty$. [ANS] 4. $\\lim_{x\\to 4^+}f(x)$ exists but $\\lim_{x\\to 4^-}f(x)$ does not. [ANS] 5. $\\lim_{x\\to 4}f(x)=\\infty$. [ANS] 6. $\\lim_{x\\to 4}f(x)$ exists but $f$ is not continuous at 4. A. $f(x)=\\sqrt[3]{x-4}$ B. $f(x)=\\left\\lbrace\\begin{array}{ll}\\cos\\left(\\frac{1}{x-4}\\right) &\\mbox{if}x<4\\\\ 0 &\\mbox{if}x=4\\\\ 4x+32 &\\mbox{if}x>4\\\\ \\end{array}\\right.$ C. $f(x)=\\left\\lbrace\\begin{array}{ll}4x &\\mbox{if}x<4\\\\ 0 &\\mbox{if}x=4\\\\ 4x-32 &\\mbox{if}x>4 \\\\ \\end{array}\\right.$ D. $f(x)=\\frac{1}{(x-4)^2}$ E. $f(x)=\\left\\lbrace\\begin{array}{ll}4x &\\mbox{if}x<4\\\\ 0 &\\mbox{if}x=4\\\\ 32-4x &\\mbox{if}x>4\\\\ \\end{array}\\right.$ F. $f(x)=\\frac{1}{x-4}$", "answer_v3": [ "C", "A", "F", "B", "D", "E" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Calculus_-_single_variable_0043", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "4", "keywords": [ "calculus", "function", "continuous", "continuity" ], "problem_v1": "Suppose $f$ and $g$ are continuous functions such that $g(6)=3$ and such that $f$ is defined at $x=6$. Assume also that \\lim_{x \\to 6} [3 f(x)+f(x)g(x)]=24\\;. Find $f(6)$. $f(6)$=[ANS]", "answer_v1": [ "4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose $f$ and $g$ are continuous functions such that $g(4)=5$ and such that $f$ is defined at $x=4$. Assume also that \\lim_{x \\to 4} [2 f(x)+f(x)g(x)]=14\\;. Find $f(4)$. $f(4)$=[ANS]", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose $f$ and $g$ are continuous functions such that $g(5)=4$ and such that $f$ is defined at $x=5$. Assume also that \\lim_{x \\to 5} [3 f(x)+f(x)g(x)]=21\\;. Find $f(5)$. $f(5)$=[ANS]", "answer_v3": [ "3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0044", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "3", "keywords": [ "limits", "derivatives" ], "problem_v1": "Use continuity to evaluate \\lim_{x \\rightarrow 2} \\arctan \\left(\\frac{5x^2-20}{5x^2-10x} \\right) Enter I for $\\infty$,-I for $-\\infty$, and DNE if the limit does not exist. Use at least 4 decimals. Limit=[ANS]", "answer_v1": [ "1.10714871779409" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use continuity to evaluate \\lim_{x \\rightarrow 2} \\arctan \\left(\\frac{2x^2-8}{7x^2-14x} \\right) Enter I for $\\infty$,-I for $-\\infty$, and DNE if the limit does not exist. Use at least 4 decimals. Limit=[ANS]", "answer_v2": [ "0.519146114246523" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use continuity to evaluate \\lim_{x \\rightarrow 2} \\arctan \\left(\\frac{3x^2-12}{5x^2-10x} \\right) Enter I for $\\infty$,-I for $-\\infty$, and DNE if the limit does not exist. Use at least 4 decimals. Limit=[ANS]", "answer_v3": [ "0.876058050598193" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0045", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - concept of", "level": "3", "keywords": [ "calculus", "continuity" ], "problem_v1": "Let f(x)=\\sqrt{x-7}. Use to indicate where $f$ is continuous.\nDomain of continuity: [ANS]", "answer_v1": [ "[7,infinity)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Let f(x)=\\sqrt{x-1}. Use to indicate where $f$ is continuous.\nDomain of continuity: [ANS]", "answer_v2": [ "[1,infinity)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Let f(x)=\\sqrt{x-3}. Use to indicate where $f$ is continuous.\nDomain of continuity: [ANS]", "answer_v3": [ "[3,infinity)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0046", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - intermediate value theorem", "level": "2", "keywords": [ "calculus", "derivative", "continuity", "functions" ], "problem_v1": "Consider the function $f(x)=7x-\\cos(x)+6$ on the interval $0\\le x\\le 1$. The Intermediate Value Theorem guarantees that for certain values of $k$ there is a number $c$ such that $f(c)=k$. In the case of the function above, what, exactly, does the intermediate value theorem say? To answer, fill in the following mathematical statements, giving an interval with non-zero length in each case. For every $k$ in the interval [ANS] $\\le k \\le$ [ANS], there is a $c$ in the interval [ANS] $\\le c \\le$ [ANS]\nsuch that $f(c)=k$.", "answer_v1": [ "6-1", "7-cos(1)+6", "0", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the function $f(x)=1x-\\cos(x)+8$ on the interval $0\\le x\\le 1$. The Intermediate Value Theorem guarantees that for certain values of $k$ there is a number $c$ such that $f(c)=k$. In the case of the function above, what, exactly, does the intermediate value theorem say? To answer, fill in the following mathematical statements, giving an interval with non-zero length in each case. For every $k$ in the interval [ANS] $\\le k \\le$ [ANS], there is a $c$ in the interval [ANS] $\\le c \\le$ [ANS]\nsuch that $f(c)=k$.", "answer_v2": [ "8-1", "1-cos(1)+8", "0", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the function $f(x)=3x-\\cos(x)+6$ on the interval $0\\le x\\le 1$. The Intermediate Value Theorem guarantees that for certain values of $k$ there is a number $c$ such that $f(c)=k$. In the case of the function above, what, exactly, does the intermediate value theorem say? To answer, fill in the following mathematical statements, giving an interval with non-zero length in each case. For every $k$ in the interval [ANS] $\\le k \\le$ [ANS], there is a $c$ in the interval [ANS] $\\le c \\le$ [ANS]\nsuch that $f(c)=k$.", "answer_v3": [ "6-1", "3-cos(1)+6", "0", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0047", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Continuity - intermediate value theorem", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Suppose that we use the bisection algorithm to approximate $r=\\sqrt{42}$ which the greatest zero of the function $f(x)=x^2-42.$ We begin by finding two numbers, say, $a_1=6$ and $b_1=7$ which bracket the zero. This is because $f(a_1)<0$ and $f(b_1)>0.$ Then we find $m_1=(a_1+b_1)/2=6.5$ and $h_1=(b_1-a_1)/2=0.5.$ We proceed with the bisection algorithm. Suppose that $a_n$ and $b_n$ bracket the zero. Then we compute $m_{n}=(a_n+b_n)/2$ and $h_n=|b_n-a_n|/2.$ If $f(m_{n})=0$ we stop because $r=m_{n}$ is the desired zero. If $f(m_n) > 0$ then $m_n$ becomes the new right endpoint, so we set $a_{n+1}:=a_n$ and $b_{n+1}:=m_n.$ If $f(m_n) < 0$ then $m_n$ becomes the new left endpoint, so we set $a_{n+1}:=m_n$ and $b_{n+1}:=b_n.$ Then $m_n$ is an approximation to $r$ with an error of $h_n$ Complete the following table:\n$\\begin{array}{ccccc}\\hline n & a_n & b_n & h_n & m_n \\\\ \\hline 1 & 6 & 7 & 0.5 & 6.5 \\\\ \\hline 2 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 5 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nThen [ANS] is an approximation so far to $r$ with an error of [ANS].", "answer_v1": [ "6", "6.5", "0.25", "6.25", "6.25", "6.5", "0.125", "6.375", "6.375", "6.5", "0.0625", "6.4375", "6.4375", "6.5", "0.03125", "6.46875", "6.46875", "0.03125" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Suppose that we use the bisection algorithm to approximate $r=\\sqrt{12}$ which the greatest zero of the function $f(x)=x^2-12.$ We begin by finding two numbers, say, $a_1=3$ and $b_1=4$ which bracket the zero. This is because $f(a_1)<0$ and $f(b_1)>0.$ Then we find $m_1=(a_1+b_1)/2=3.5$ and $h_1=(b_1-a_1)/2=0.5.$ We proceed with the bisection algorithm. Suppose that $a_n$ and $b_n$ bracket the zero. Then we compute $m_{n}=(a_n+b_n)/2$ and $h_n=|b_n-a_n|/2.$ If $f(m_{n})=0$ we stop because $r=m_{n}$ is the desired zero. If $f(m_n) > 0$ then $m_n$ becomes the new right endpoint, so we set $a_{n+1}:=a_n$ and $b_{n+1}:=m_n.$ If $f(m_n) < 0$ then $m_n$ becomes the new left endpoint, so we set $a_{n+1}:=m_n$ and $b_{n+1}:=b_n.$ Then $m_n$ is an approximation to $r$ with an error of $h_n$ Complete the following table:\n$\\begin{array}{ccccc}\\hline n & a_n & b_n & h_n & m_n \\\\ \\hline 1 & 3 & 4 & 0.5 & 3.5 \\\\ \\hline 2 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 5 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nThen [ANS] is an approximation so far to $r$ with an error of [ANS].", "answer_v2": [ "3", "3.5", "0.25", "3.25", "3.25", "3.5", "0.125", "3.375", "3.375", "3.5", "0.0625", "3.4375", "3.4375", "3.5", "0.03125", "3.46875", "3.46875", "0.03125" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Suppose that we use the bisection algorithm to approximate $r=\\sqrt{20}$ which the greatest zero of the function $f(x)=x^2-20.$ We begin by finding two numbers, say, $a_1=4$ and $b_1=5$ which bracket the zero. This is because $f(a_1)<0$ and $f(b_1)>0.$ Then we find $m_1=(a_1+b_1)/2=4.5$ and $h_1=(b_1-a_1)/2=0.5.$ We proceed with the bisection algorithm. Suppose that $a_n$ and $b_n$ bracket the zero. Then we compute $m_{n}=(a_n+b_n)/2$ and $h_n=|b_n-a_n|/2.$ If $f(m_{n})=0$ we stop because $r=m_{n}$ is the desired zero. If $f(m_n) > 0$ then $m_n$ becomes the new right endpoint, so we set $a_{n+1}:=a_n$ and $b_{n+1}:=m_n.$ If $f(m_n) < 0$ then $m_n$ becomes the new left endpoint, so we set $a_{n+1}:=m_n$ and $b_{n+1}:=b_n.$ Then $m_n$ is an approximation to $r$ with an error of $h_n$ Complete the following table:\n$\\begin{array}{ccccc}\\hline n & a_n & b_n & h_n & m_n \\\\ \\hline 1 & 4 & 5 & 0.5 & 4.5 \\\\ \\hline 2 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 3 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 4 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 5 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nThen [ANS] is an approximation so far to $r$ with an error of [ANS].", "answer_v3": [ "4", "4.5", "0.25", "4.25", "4.25", "4.5", "0.125", "4.375", "4.375", "4.5", "0.0625", "4.4375", "4.4375", "4.5", "0.03125", "4.46875", "4.46875", "0.03125" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0048", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Infinite limits and vertical asymptotes", "level": "2", "keywords": [ "Calculus", "infinite", "limit", "vertical", "asymptote", "Limit", "Vertical Asymptote" ], "problem_v1": "A function is said to have a vertical asymptote wherever the limit on the left or right (or both) is either positive or negative infinity. For example, the function $f(x)=\\frac{x^2-4}{(x-3)^2}$ has a vertical asymptote at $x=3$. For each of the following limits, enter either 'P' for positive infinity, 'N' for negative infinity, or 'D' when the limit simply does not exist. ${\\lim_{x\\to 3^-} \\frac{x^2-4}{(x-3)^2}=}$ [ANS]\n${\\lim_{x\\to 3^+} \\frac{x^2-4}{(x-3)^2}=}$ [ANS]\n${\\lim_{x\\to 3} \\frac{x^2-4}{(x-3)^2}=}$ [ANS]", "answer_v1": [ "P", "P", "P" ], "answer_type_v1": [ "OE", "OE", "OE" ], "options_v1": [ [], [], [] ], "problem_v2": "A function is said to have a vertical asymptote wherever the limit on the left or right (or both) is either positive or negative infinity. For example, the function $f(x)=\\frac{-2x+1}{(x-7)^2}$ has a vertical asymptote at $x=7$. For each of the following limits, enter either 'P' for positive infinity, 'N' for negative infinity, or 'D' when the limit simply does not exist. ${\\lim_{x\\to 7^-} \\frac{-2x+1}{(x-7)^2}=}$ [ANS]\n${\\lim_{x\\to 7^+} \\frac{-2x+1}{(x-7)^2}=}$ [ANS]\n${\\lim_{x\\to 7} \\frac{-2x+1}{(x-7)^2}=}$ [ANS]", "answer_v2": [ "N", "N", "N" ], "answer_type_v2": [ "OE", "OE", "OE" ], "options_v2": [ [], [], [] ], "problem_v3": "A function is said to have a vertical asymptote wherever the limit on the left or right (or both) is either positive or negative infinity. For example, the function $f(x)=\\frac{(x-1)^2}{x(x+4)}$ has a vertical asymptote at $x=-4$. For each of the following limits, enter either 'P' for positive infinity, 'N' for negative infinity, or 'D' when the limit simply does not exist. ${\\lim_{x\\to-4^-} \\frac{(x-1)^2}{x(x+4)}=}$ [ANS]\n${\\lim_{x\\to-4^+} \\frac{(x-1)^2}{x(x+4)}=}$ [ANS]\n${\\lim_{x\\to-4} \\frac{(x-1)^2}{x(x+4)}=}$ [ANS]", "answer_v3": [ "P", "N", "D" ], "answer_type_v3": [ "OE", "OE", "OE" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0050", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Infinite limits and vertical asymptotes", "level": "3", "keywords": [ "calculus", "limit", "asymptote" ], "problem_v1": "Consider the function f(x)=\\frac{x-\\frac{60}{x}-4}{x^2-5x-66} Find the vertical asymptotes of $f(x)$. Separate multiple answers by commas. $x=$ [ANS]", "answer_v1": [ "(0, 11)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Consider the function f(x)=\\frac{x-\\frac{24}{x}-5}{x^2-2x-15} Find the vertical asymptotes of $f(x)$. Separate multiple answers by commas. $x=$ [ANS]", "answer_v2": [ "(0, 5)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Consider the function f(x)=\\frac{x-\\frac{32}{x}-4}{x^2-3x-28} Find the vertical asymptotes of $f(x)$. Separate multiple answers by commas. $x=$ [ANS]", "answer_v3": [ "(0, 7)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0051", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Limits at infinity, horizontal and oblique asymptotes", "level": "4", "keywords": [ "limits", "infinite" ], "problem_v1": "Suppose the function $f(x)$ is an odd function and $\\lim_{x\\to\\infty} f(x)=33$. Use this information to evaluate $\\lim_{x\\to-\\infty} f(x).$\nAnswer: [ANS]", "answer_v1": [ "-33" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose the function $f(x)$ is an odd function and $\\lim_{x\\to\\infty} f(x)=12$. Use this information to evaluate $\\lim_{x\\to-\\infty} f(x).$\nAnswer: [ANS]", "answer_v2": [ "-12" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose the function $f(x)$ is an odd function and $\\lim_{x\\to\\infty} f(x)=19$. Use this information to evaluate $\\lim_{x\\to-\\infty} f(x).$\nAnswer: [ANS]", "answer_v3": [ "-19" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0052", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Limits at infinity, horizontal and oblique asymptotes", "level": "2", "keywords": [ "limits", "sequences" ], "problem_v1": "Determine the limit of the sequence $f(k)$ generated by the sequence $k=1,2,3,4,5...$ when f(x)=\\frac{(37.7x-29.2)(31.2x+36.2)}{15.3x^2-16.7} Answer: [ANS]", "answer_v1": [ "37.7*31.2/15.3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the limit of the sequence $f(k)$ generated by the sequence $k=1,2,3,4,5...$ when f(x)=\\frac{(4.4x-46.6)(7.7x+16.9)}{47.4x^2-16} Answer: [ANS]", "answer_v2": [ "4.4*7.7/47.4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the limit of the sequence $f(k)$ generated by the sequence $k=1,2,3,4,5...$ when f(x)=\\frac{(15.9x-30.4)(14.1x+27.6)}{10.5x^2-17.5} Answer: [ANS]", "answer_v3": [ "15.9*14.1/10.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0053", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Limits at infinity, horizontal and oblique asymptotes", "level": "2", "keywords": [ "exponential functions", "continuous growth", "natural base", "e" ], "problem_v1": "Compute the following limit. If the limit goes to $\\infty$ or $-\\infty$ enter INFINITY or-INFINITY, respectively. $ \\lim_{t \\to \\infty} \\left(14e^{-0.2 t}+13 \\right)=$ [ANS]", "answer_v1": [ "13" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the following limit. If the limit goes to $\\infty$ or $-\\infty$ enter INFINITY or-INFINITY, respectively. $ \\lim_{t \\to \\infty} \\left(2e^{-0.05 t}+5 \\right)=$ [ANS]", "answer_v2": [ "5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the following limit. If the limit goes to $\\infty$ or $-\\infty$ enter INFINITY or-INFINITY, respectively. $ \\lim_{t \\to \\infty} \\left(6e^{-0.2 t}+7 \\right)=$ [ANS]", "answer_v3": [ "7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0054", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Limits at infinity, horizontal and oblique asymptotes", "level": "2", "keywords": [ "slant", "oblique", "asymptote", "long division" ], "problem_v1": "The oblique or slant asymptote to\nf(x)=\\frac{x^3-9x^2+18x-9}{x^2-2x+2} is given by y=[ANS].", "answer_v1": [ "x-7" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The oblique or slant asymptote to\nf(x)=\\frac{x^3-3x^2+6x+6}{x^2-2x+2} is given by y=[ANS].", "answer_v2": [ "x-1" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The oblique or slant asymptote to\nf(x)=\\frac{x^3-5x^2+10x-1}{x^2-2x+2} is given by y=[ANS].", "answer_v3": [ "x-3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0055", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Limits at infinity, horizontal and oblique asymptotes", "level": "3", "keywords": [ "Calculus" ], "problem_v1": "The horizontal asymptotes of the curve y=\\frac{16x}{(x^4+1)^{\\frac{1}{4}}} are given by $y_1=$ [ANS] and $y_2=$ [ANS]\nwhere $y_1 > y_2$.", "answer_v1": [ "16", "-16" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The horizontal asymptotes of the curve y=\\frac{3x}{(x^4+1)^{\\frac{1}{4}}} are given by $y_1=$ [ANS] and $y_2=$ [ANS]\nwhere $y_1 > y_2$.", "answer_v2": [ "3", "-3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The horizontal asymptotes of the curve y=\\frac{7x}{(x^4+1)^{\\frac{1}{4}}} are given by $y_1=$ [ANS] and $y_2=$ [ANS]\nwhere $y_1 > y_2$.", "answer_v3": [ "7", "-7" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0056", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Limits at infinity, horizontal and oblique asymptotes", "level": "2", "keywords": [ "long division", "asymptote", "slant", "oblique" ], "problem_v1": "Using long division, you can calculate that f(x)=\\frac{2x^3-14x^2+7x-30}{2x^2+5}=Q(x)+\\frac{R(x)}{2x^2+5}. The quotient $Q(x)$ is [ANS]. The remainder $R(x)$ is [ANS]. From this you can conclude that $y=f(x)$ has an oblique (or slant) asymptote given by the line with equation y=[ANS] because $ \\lim_{x \\to \\pm \\infty} \\frac{R(x)}{2x^2+5}=$ [ANS].", "answer_v1": [ "x-7", "2*x+5", "Q(X)", "0" ], "answer_type_v1": [ "EX", "EX", "MCS", "NV" ], "options_v1": [ [], [], [ "Q(x)", "R(x)" ], [] ], "problem_v2": "Using long division, you can calculate that f(x)=\\frac{2x^3-4x^2+7x-2}{2x^2+5}=Q(x)+\\frac{R(x)}{2x^2+5}. The quotient $Q(x)$ is [ANS]. The remainder $R(x)$ is [ANS]. From this you can conclude that $y=f(x)$ has an oblique (or slant) asymptote given by the line with equation y=[ANS] because $ \\lim_{x \\to \\pm \\infty} \\frac{R(x)}{2x^2+5}=$ [ANS].", "answer_v2": [ "x-2", "2*x+8", "Q(X)", "0" ], "answer_type_v2": [ "EX", "EX", "MCS", "NV" ], "options_v2": [ [], [], [ "Q(x)", "R(x)" ], [] ], "problem_v3": "Using long division, you can calculate that f(x)=\\frac{2x^3-8x^2+7x-15}{2x^2+5}=Q(x)+\\frac{R(x)}{2x^2+5}. The quotient $Q(x)$ is [ANS]. The remainder $R(x)$ is [ANS]. From this you can conclude that $y=f(x)$ has an oblique (or slant) asymptote given by the line with equation y=[ANS] because $ \\lim_{x \\to \\pm \\infty} \\frac{R(x)}{2x^2+5}=$ [ANS].", "answer_v3": [ "x-4", "2*x+5", "Q(X)", "0" ], "answer_type_v3": [ "EX", "EX", "MCS", "NV" ], "options_v3": [ [], [], [ "Q(x)", "R(x)" ], [] ] }, { "id": "Calculus_-_single_variable_0057", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Limits at infinity, horizontal and oblique asymptotes", "level": "2", "keywords": [ "limits", "infinite" ], "problem_v1": "Find the horizontal limit(s) of the following function: f(x)=\\frac {9x^3-7x^2-8x}{9-5x-5x^3} [ANS] and [ANS]", "answer_v1": [ "-1.8", "-1.8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the horizontal limit(s) of the following function: f(x)=\\frac {2x^3-11x^2-3x}{5-11x-5x^3} [ANS] and [ANS]", "answer_v2": [ "-0.4", "-0.4" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the horizontal limit(s) of the following function: f(x)=\\frac {5x^3-8x^2-4x}{7-4x-5x^3} [ANS] and [ANS]", "answer_v3": [ "-1", "-1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0058", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Limits at infinity, horizontal and oblique asymptotes", "level": "3", "keywords": [ "calculus", "derivative", "limits", "continuity", "piecewise functions" ], "problem_v1": "Find a value of the constant $k$ such that the indicated limit exists.\n\\lim_{x\\to\\infty}\\,\\frac{7^x-5}{e^{kx}+5}. $k=$ [ANS]\n(If any value of k will work, enter any ; if none will work, enter none.) Are there any other values of $k$ that will work? [ANS]", "answer_v1": [ "ln(7)", "yes" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "yes", "no" ] ], "problem_v2": "Find a value of the constant $k$ such that the indicated limit exists.\n\\lim_{x\\to\\infty}\\,\\frac{2^x-8}{e^{kx}+2}. $k=$ [ANS]\n(If any value of k will work, enter any ; if none will work, enter none.) Are there any other values of $k$ that will work? [ANS]", "answer_v2": [ "ln(2)", "yes" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "yes", "no" ] ], "problem_v3": "Find a value of the constant $k$ such that the indicated limit exists.\n\\lim_{x\\to\\infty}\\,\\frac{4^x-5}{e^{kx}+3}. $k=$ [ANS]\n(If any value of k will work, enter any ; if none will work, enter none.) Are there any other values of $k$ that will work? [ANS]", "answer_v3": [ "ln(4)", "yes" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "yes", "no" ] ] }, { "id": "Calculus_-_single_variable_0059", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Limits at infinity, horizontal and oblique asymptotes", "level": "3", "keywords": [ "limit", "limits" ], "problem_v1": "Let $ g(t)=\\begin{cases} \\frac{8+6t}{7t^{2}+8} && t< 620000\\\\ \\frac{\\sqrt{16t^{2}-16}}{6-t}&&t>620000\\end{cases}$ Find Notes: Enter \"DNE\" if limit Does Not Exist.\n(a) $\\lim_{t\\to-\\infty} g(t)$=[ANS]\n(b) $\\lim_{t\\to \\infty} g(t)$=[ANS]", "answer_v1": [ "0", "-4" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $ g(t)=\\begin{cases} \\frac{1+10t}{2t^{2}+4} && t< 390000\\\\ \\frac{\\sqrt{100t^{2}-16}}{2-t}&&t>390000\\end{cases}$ Find Notes: Enter \"DNE\" if limit Does Not Exist.\n(a) $\\lim_{t\\to-\\infty} g(t)$=[ANS]\n(b) $\\lim_{t\\to \\infty} g(t)$=[ANS]", "answer_v2": [ "0", "-10" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $ g(t)=\\begin{cases} \\frac{4+7t}{3t^{2}+6} && t< 920000\\\\ \\frac{\\sqrt{9t^{2}-16}}{9-t}&&t>920000\\end{cases}$ Find Notes: Enter \"DNE\" if limit Does Not Exist.\n(a) $\\lim_{t\\to-\\infty} g(t)$=[ANS]\n(b) $\\lim_{t\\to \\infty} g(t)$=[ANS]", "answer_v3": [ "0", "-3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0060", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Limits at infinity, horizontal and oblique asymptotes", "level": "3", "keywords": [ "calculus", "limit", "asymptote" ], "problem_v1": "Evaluate \\lim_{x \\rightarrow \\infty} \\sqrt{x^2+5x+1}-x (Type INF for $\\infty$ and MINF for $-\\infty$) [ANS] Thus y=\\sqrt{x^2+5x+1}-x has a corresponding horizontal asymptote $y=$ [ANS]\n(Type in DNE if there is no horizontal asymptote.)\nEvaluate \\lim_{x \\rightarrow-\\infty} \\sqrt{x^2+5x+1}-x (Type INF for $\\infty$ and MINF for $-\\infty$) [ANS]\nThus y=\\sqrt{x^2+5x+1}-x has a corresponding horizontal asymptote $y=$ [ANS]\n(Type in DNE if there is no corresponding horizontal asymptote.)", "answer_v1": [ "2.5", "2.5", "INF", "DNE" ], "answer_type_v1": [ "NV", "NV", "NV", "OE" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Evaluate \\lim_{x \\rightarrow \\infty} \\sqrt{x^2-9x+1}-x (Type INF for $\\infty$ and MINF for $-\\infty$) [ANS] Thus y=\\sqrt{x^2-9x+1}-x has a corresponding horizontal asymptote $y=$ [ANS]\n(Type in DNE if there is no horizontal asymptote.)\nEvaluate \\lim_{x \\rightarrow-\\infty} \\sqrt{x^2-9x+1}-x (Type INF for $\\infty$ and MINF for $-\\infty$) [ANS]\nThus y=\\sqrt{x^2-9x+1}-x has a corresponding horizontal asymptote $y=$ [ANS]\n(Type in DNE if there is no corresponding horizontal asymptote.)", "answer_v2": [ "-4.5", "-4.5", "INF", "DNE" ], "answer_type_v2": [ "NV", "NV", "NV", "OE" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Evaluate \\lim_{x \\rightarrow \\infty} \\sqrt{x^2-4x+1}-x (Type INF for $\\infty$ and MINF for $-\\infty$) [ANS] Thus y=\\sqrt{x^2-4x+1}-x has a corresponding horizontal asymptote $y=$ [ANS]\n(Type in DNE if there is no horizontal asymptote.)\nEvaluate \\lim_{x \\rightarrow-\\infty} \\sqrt{x^2-4x+1}-x (Type INF for $\\infty$ and MINF for $-\\infty$) [ANS]\nThus y=\\sqrt{x^2-4x+1}-x has a corresponding horizontal asymptote $y=$ [ANS]\n(Type in DNE if there is no corresponding horizontal asymptote.)", "answer_v3": [ "-2", "-2", "INF", "DNE" ], "answer_type_v3": [ "NV", "NV", "NV", "OE" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0061", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Limits at infinity, horizontal and oblique asymptotes", "level": "3", "keywords": [ "calculus", "limit", "trigonometric" ], "problem_v1": "Find the limit of \\( \\frac{1-\\cos(x)}{x^{4}} \\) as \\(x \\) goes to 0. Enter INF for \\(\\infty \\), MINF for \\(-\\infty \\) or DNE for does not exist. You should also try using identities to transform the expressions algebraically so that you can identify the limits. [ANS]", "answer_v1": [ "INF" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the limit of \\( \\frac{1-\\cos(x)}{x} \\) as \\(x \\) goes to 0. Enter INF for \\(\\infty \\), MINF for \\(-\\infty \\) or DNE for does not exist. You should also try using identities to transform the expressions algebraically so that you can identify the limits. [ANS]", "answer_v2": [ "0" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the limit of \\( \\frac{|-8x|}{4x} \\) as \\(x \\) goes to 0. Enter INF for \\(\\infty \\), MINF for \\(-\\infty \\) or DNE for does not exist. You should also try using identities to transform the expressions algebraically so that you can identify the limits. [ANS]", "answer_v3": [ "DNE" ], "answer_type_v3": [ "OE" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0062", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Estimating limits numerically", "level": "2", "keywords": [], "problem_v1": "A function $f$ and value $a$ are given. Approximate the limit of the difference quotient, $\\lim\\limits_{h \\to 0} \\dfrac{f(a+h)-f(a)}{h},$ using $h=\\pm 0.1, \\pm 0.01$.\n$f(x)=x^2+6x+2, \\qquad a=3$\nWhen $h=0.1$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]\nWhen $h=-0.1$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]\nWhen $h=0.01$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]\nWhen $h=-0.01$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]", "answer_v1": [ "12.1", "11.9", "12.01", "11.99" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A function $f$ and value $a$ are given. Approximate the limit of the difference quotient, $\\lim\\limits_{h \\to 0} \\dfrac{f(a+h)-f(a)}{h},$ using $h=\\pm 0.1, \\pm 0.01$.\n$f(x)=x^2+9x-6, \\qquad a=-5$\nWhen $h=0.1$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]\nWhen $h=-0.1$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]\nWhen $h=0.01$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]\nWhen $h=-0.01$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]", "answer_v2": [ "-0.9", "-1.1", "-0.99", "-1.01" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A function $f$ and value $a$ are given. Approximate the limit of the difference quotient, $\\lim\\limits_{h \\to 0} \\dfrac{f(a+h)-f(a)}{h},$ using $h=\\pm 0.1, \\pm 0.01$.\n$f(x)=x^2+6x-4, \\qquad a=-2$\nWhen $h=0.1$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]\nWhen $h=-0.1$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]\nWhen $h=0.01$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]\nWhen $h=-0.01$, $\\dfrac{f(a+h)-f(a)}{h}$=[ANS]", "answer_v3": [ "2.1", "1.9", "2.01", "1.99" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0063", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Estimating limits numerically", "level": "1", "keywords": [], "problem_v1": "Approximate the given limit both numerically and graphically. $\\lim\\limits_{x \\to 2} x^2+7x+7=$ [ANS]\n(Enter DNE if the limit doesn't exist)", "answer_v1": [ "25" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Approximate the given limit both numerically and graphically. $\\lim\\limits_{x \\to-4} x^2+10x+3=$ [ANS]\n(Enter DNE if the limit doesn't exist)", "answer_v2": [ "-21" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Approximate the given limit both numerically and graphically. $\\lim\\limits_{x \\to-2} x^2+7x+4=$ [ANS]\n(Enter DNE if the limit doesn't exist)", "answer_v3": [ "-6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0064", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Estimating limits numerically", "level": "2", "keywords": [ "calculus", "logarithms", "rates of change" ], "problem_v1": "Estimate the instantaneous rate of change at the point $x=5$ for $f(x)=\\ln x$ ROC=[ANS]", "answer_v1": [ "1/5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Estimate the instantaneous rate of change at the point $x=2$ for $f(x)=\\ln x$ ROC=[ANS]", "answer_v2": [ "1/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Estimate the instantaneous rate of change at the point $x=3$ for $f(x)=\\ln x$ ROC=[ANS]", "answer_v3": [ "1/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0065", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Estimating limits numerically", "level": "3", "keywords": [ "calculus", "rates of change" ], "problem_v1": "Let $v=60 \\sqrt{T}$. Estimate the instantaneous rate of change of $v$ with respect to $T$ when $T=400$. ROC=[ANS]", "answer_v1": [ "1.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $v=20 \\sqrt{T}$. Estimate the instantaneous rate of change of $v$ with respect to $T$ when $T=500$. ROC=[ANS]", "answer_v2": [ "0.447214" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $v=30 \\sqrt{T}$. Estimate the instantaneous rate of change of $v$ with respect to $T$ when $T=400$. ROC=[ANS]", "answer_v3": [ "0.750001" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0066", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Estimating limits numerically", "level": "2", "keywords": [ "calculus", "limits", "approximation", "table question" ], "problem_v1": "Fill in the table and guess the value of the limit: $\\lim\\limits_{y \\to 7} f(y)$, where $f(y)=\\frac {y^2-6 y-7} {y^2-4 y-21}$\n$\\begin{array}{cccc}\\hline y & f(y) & y & f(y) \\\\ \\hline 7.002 & \\qquad \\qquad & 6.998 & \\qquad \\qquad \\\\ \\hline 7.001 & \\qquad \\qquad & 6.999 & \\qquad \\qquad \\\\ \\hline 7.0005 & \\qquad \\qquad & 6.9995 & \\qquad \\qquad \\\\ \\hline 7.0001 & \\qquad \\qquad & 6.9999 & \\qquad \\qquad \\\\ \\hline \\end{array}$\nThe limit as $y \\to 7$ is [ANS]", "answer_v1": [ "0.8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Fill in the table and guess the value of the limit: $\\lim\\limits_{y \\to-2} f(y)$, where $f(y)=\\frac {y^2+3 y+2} {y^2+5 y+6}$\n$\\begin{array}{cccc}\\hline y & f(y) & y & f(y) \\\\ \\hline-1.998 & \\qquad \\qquad &-2.002 & \\qquad \\qquad \\\\ \\hline-1.999 & \\qquad \\qquad &-2.001 & \\qquad \\qquad \\\\ \\hline-1.9995 & \\qquad \\qquad &-2.0005 & \\qquad \\qquad \\\\ \\hline-1.9999 & \\qquad \\qquad &-2.0001 & \\qquad \\qquad \\\\ \\hline \\end{array}$\nThe limit as $y \\to-2$ is [ANS]", "answer_v2": [ "-1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Fill in the table and guess the value of the limit: $\\lim\\limits_{y \\to 5} f(y)$, where $f(y)=\\frac {y^2-4 y-5} {y^2-2 y-15}$\n$\\begin{array}{cccc}\\hline y & f(y) & y & f(y) \\\\ \\hline 5.002 & \\qquad \\qquad & 4.998 & \\qquad \\qquad \\\\ \\hline 5.001 & \\qquad \\qquad & 4.999 & \\qquad \\qquad \\\\ \\hline 5.0005 & \\qquad \\qquad & 4.9995 & \\qquad \\qquad \\\\ \\hline 5.0001 & \\qquad \\qquad & 4.9999 & \\qquad \\qquad \\\\ \\hline \\end{array}$\nThe limit as $y \\to 5$ is [ANS]", "answer_v3": [ "0.75" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0067", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Estimating limits numerically", "level": "2", "keywords": [ "calculus", "limits", "inverse functions", "trigonometric functions" ], "problem_v1": "Estimate the limit numerically or state that the limit doesn't exist: $\\lim\\limits_{x \\to 1+} \\frac{\\sec^{-1}\\!\\left(x\\right)}{\\sqrt{x^{6}-1}}$ [ANS]\nEnter F if the limit doesn't exist.", "answer_v1": [ "0.57735" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Estimate the limit numerically or state that the limit doesn't exist: $\\lim\\limits_{x \\to 1+} \\frac{\\sec^{-1}\\!\\left(x\\right)}{\\sqrt{x^{2}-1}}$ [ANS]\nEnter F if the limit doesn't exist.", "answer_v2": [ "1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Estimate the limit numerically or state that the limit doesn't exist: $\\lim\\limits_{x \\to 1+} \\frac{\\sec^{-1}\\!\\left(x\\right)}{\\sqrt{x^{3}-1}}$ [ANS]\nEnter F if the limit doesn't exist.", "answer_v3": [ "0.816497" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0068", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Estimating limits numerically", "level": "2", "keywords": [ "calculator", "tangent line", "Tangent" ], "problem_v1": "Let $p(x)=5.7x^{1.60000}$. Use a calculator or a graphing program to find the slope of the tangent line to the point (x,p(x)) when $x=2$. Give the answer to 3 places. [ANS]", "answer_v1": [ "13.8233350865748" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $p(x)=4.6x^{1.20000}$. Use a calculator or a graphing program to find the slope of the tangent line to the point (x,p(x)) when $x=2.9$. Give the answer to 3 places. [ANS]", "answer_v2": [ "6.82996813201007" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $p(x)=5x^{1.30000}$. Use a calculator or a graphing program to find the slope of the tangent line to the point (x,p(x)) when $x=2$. Give the answer to 3 places. [ANS]", "answer_v3": [ "8.00243868674196" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0069", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Estimating limits numerically", "level": "3", "keywords": [ "calculus", "derivative", "limits", "continuity", "piecewise functions" ], "problem_v1": "Consider the function $f(x)=\\frac{5^{x}-1}{x}$.\n(a) Fill in the following table of values for $f(x)$:\n$\\begin{array}{ccccccccc}\\hline x=&-0.1 &-0.01 &-0.001 &-0.0001 & 0.0001 & 0.001 & 0.01 & 0.1 \\\\ \\hline f(x)=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) Based on your table of values, what would you expect the limit of $f(x)$ as $x$ approaches zero to be? $\\lim\\limits_{x\\to0}\\,\\frac{5^{x}-1}{x}=$ [ANS]\n(c) Graph the function to see if it is consistent with your answers to parts\n(a) and (b). By graphing, find an interval for $x$ near zero such that the difference between your conjectured limit and the value of the function is less than 0.01. In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom. What is the window? [ANS] $\\le x \\le$ [ANS], [ANS] $\\le y \\le$ [ANS].", "answer_v1": [ "1.4866", "1.59656", "1.60814", "1.60931", "1.60957", "1.61073", "1.62246", "1.74619", "ln(5)", "-0.00775339", "0.00768942", "ln(5)-0.01", "ln(5)+0.01" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the function $f(x)=\\frac{2^{x}-1}{x}$.\n(a) Fill in the following table of values for $f(x)$:\n$\\begin{array}{ccccccccc}\\hline x=&-0.1 &-0.01 &-0.001 &-0.0001 & 0.0001 & 0.001 & 0.01 & 0.1 \\\\ \\hline f(x)=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) Based on your table of values, what would you expect the limit of $f(x)$ as $x$ approaches zero to be? $\\lim\\limits_{x\\to0}\\,\\frac{2^{x}-1}{x}=$ [ANS]\n(c) Graph the function to see if it is consistent with your answers to parts\n(a) and (b). By graphing, find an interval for $x$ near zero such that the difference between your conjectured limit and the value of the function is less than 0.01. In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom. What is the window? [ANS] $\\le x \\le$ [ANS], [ANS] $\\le y \\le$ [ANS].", "answer_v2": [ "0.66967", "0.69075", "0.692907", "0.693123", "0.693171", "0.693387", "0.695555", "0.717735", "ln(2)", "-0.0420356", "0.0412345", "ln(2)-0.01", "ln(2)+0.01" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the function $f(x)=\\frac{3^{x}-1}{x}$.\n(a) Fill in the following table of values for $f(x)$:\n$\\begin{array}{ccccccccc}\\hline x=&-0.1 &-0.01 &-0.001 &-0.0001 & 0.0001 & 0.001 & 0.01 & 0.1 \\\\ \\hline f(x)=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(b) Based on your table of values, what would you expect the limit of $f(x)$ as $x$ approaches zero to be? $\\lim\\limits_{x\\to0}\\,\\frac{3^{x}-1}{x}=$ [ANS]\n(c) Graph the function to see if it is consistent with your answers to parts\n(a) and (b). By graphing, find an interval for $x$ near zero such that the difference between your conjectured limit and the value of the function is less than 0.01. In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom. What is the window? [ANS] $\\le x \\le$ [ANS], [ANS] $\\le y \\le$ [ANS].", "answer_v3": [ "1.04042", "1.0926", "1.09801", "1.09855", "1.09867", "1.09922", "1.10467", "1.16123", "ln(3)", "-0.0166725", "0.0164714", "ln(3)-0.01", "ln(3)+0.01" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0070", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Estimating limits numerically", "level": "2", "keywords": [ "calculus", "derivative", "instantaneous velocity", "rate of change" ], "problem_v1": "Estimate $f'(4)$ for $f(x)=4^x$. Be sure your answer is accurate to within 0.1 of the actual value. $f'(4) \\approx$ [ANS]\nBe sure that you can explain your reasoning. Be sure that you can explain your reasoning.", "answer_v1": [ "354.9" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Estimate $f'(2)$ for $f(x)=8^x$. Be sure your answer is accurate to within 0.1 of the actual value. $f'(2) \\approx$ [ANS]\nBe sure that you can explain your reasoning. Be sure that you can explain your reasoning.", "answer_v2": [ "133.1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Estimate $f'(3)$ for $f(x)=6^x$. Be sure your answer is accurate to within 0.1 of the actual value. $f'(3) \\approx$ [ANS]\nBe sure that you can explain your reasoning. Be sure that you can explain your reasoning.", "answer_v3": [ "387" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0072", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Estimating limits numerically", "level": "2", "keywords": [ "Derivatives' 'Rates of Change", "limits", "tangent line" ], "problem_v1": "The slope of the tangent line to the graph of $y=5x^{3}$ at the point $(3, 135)$ is \\lim _ {x \\to 3} \\frac {5x^{3}-135}{x-3}. By trying values of $x$ near $3,$ estimate the slope of the tangent line.\nThe slope of the tangent line is (roughly) [ANS].", "answer_v1": [ "135" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The slope of the tangent line to the graph of $y=2x^{3}$ at the point $(4, 128)$ is \\lim _ {x \\to 4} \\frac {2x^{3}-128}{x-4}. By trying values of $x$ near $4,$ estimate the slope of the tangent line.\nThe slope of the tangent line is (roughly) [ANS].", "answer_v2": [ "96" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The slope of the tangent line to the graph of $y=3x^{3}$ at the point $(3, 81)$ is \\lim _ {x \\to 3} \\frac {3x^{3}-81}{x-3}. By trying values of $x$ near $3,$ estimate the slope of the tangent line.\nThe slope of the tangent line is (roughly) [ANS].", "answer_v3": [ "81" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0073", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - instantaneous rate of change", "level": "2", "keywords": [ "calculus", "rates of change", "average rates of change" ], "problem_v1": "With an initial deposit of 300 dollars, the balance in a bank account after $t$ years is $f(t)=300 (1.05)^t$ dollars. $1.$ What are the units of the rate of change of $f(t)$? [ANS] $2.$ Find the average rate of change over $[0, 1]$ [ANS]", "answer_v1": [ "Dollars/year", "300*(1.05-1)" ], "answer_type_v1": [ "MCS", "NV" ], "options_v1": [ [ "Dollars", "Years", "Dollars/year", "Years/dollar" ], [] ], "problem_v2": "With an initial deposit of 500 dollars, the balance in a bank account after $t$ years is $f(t)=500 (1.05)^t$ dollars. $1.$ What are the units of the rate of change of $f(t)$? [ANS] $2.$ Find the average rate of change over $[0, 1]$ [ANS]", "answer_v2": [ "Dollars/year", "500*(1.05-1)" ], "answer_type_v2": [ "MCS", "NV" ], "options_v2": [ [ "Dollars", "Years", "Dollars/year", "Years/dollar" ], [] ], "problem_v3": "With an initial deposit of 400 dollars, the balance in a bank account after $t$ years is $f(t)=400 (1.05)^t$ dollars. $1.$ What are the units of the rate of change of $f(t)$? [ANS] $2.$ Find the average rate of change over $[0, 1]$ [ANS]", "answer_v3": [ "Dollars/year", "400*(1.05-1)" ], "answer_type_v3": [ "MCS", "NV" ], "options_v3": [ [ "Dollars", "Years", "Dollars/year", "Years/dollar" ], [] ] }, { "id": "Calculus_-_single_variable_0074", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - instantaneous rate of change", "level": "3", "keywords": [ "Calculus", "Secant", "Tangent", "average velocity", "instantaneous velocity" ], "problem_v1": "The displacement (in meters) of a particle moving in a straight line is given by $s=4 t^3$ where $t$ is measured in seconds.\nFind the average velocity of the particle over the time interval $[8, 11]$. [ANS]\nFind the (instantaneous) velocity of the particle when $t=8$. [ANS]", "answer_v1": [ "4*(11^3-8^3)/3", "3*4*8*8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The displacement (in meters) of a particle moving in a straight line is given by $s=2 t^3$ where $t$ is measured in seconds.\nFind the average velocity of the particle over the time interval $[10, 12]$. [ANS]\nFind the (instantaneous) velocity of the particle when $t=10$. [ANS]", "answer_v2": [ "2*(12^3-10^3)/2", "3*2*10*10" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The displacement (in meters) of a particle moving in a straight line is given by $s=2 t^3$ where $t$ is measured in seconds.\nFind the average velocity of the particle over the time interval $[8, 10]$. [ANS]\nFind the (instantaneous) velocity of the particle when $t=8$. [ANS]", "answer_v3": [ "2*(10^3-8^3)/2", "3*2*8*8" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0075", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - instantaneous rate of change", "level": "2", "keywords": [ "Calculus", "Average Rate of Change", "Secant", "rate of change", "average" ], "problem_v1": "The following chart shows \"living wage\" jobs in Rochester per 1000 working age adults over a 5 year period.\n\\begin{array}{| c | c | c | c | c | c |} \\hline \\mathrm{Year} & 1997 & 1998 & 1999 & 2000 & 2001 \\\\ \\hline \\mathrm{Jobs} & 665 & 715 & 760 & 790 & 810 \\\\ \\hline \\end{array} What is the average rate of change in the number of living wage jobs from 1997 to 1999? [ANS] Jobs/Year What is the average rate of change in the number of living wage jobs from 1999 to 2001? [ANS] Jobs/Year Based on these two answers, should the mayor from the last two years be reelected? (These numbers are made up. Please do not actually hold the mayor accountable.)", "answer_v1": [ "95/2", "50/2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The following chart shows \"living wage\" jobs in Rochester per 1000 working age adults over a 5 year period.\n\\begin{array}{| c | c | c | c | c | c |} \\hline \\mathrm{Year} & 1997 & 1998 & 1999 & 2000 & 2001 \\\\ \\hline \\mathrm{Jobs} & 625 & 685 & 730 & 765 & 795 \\\\ \\hline \\end{array} What is the average rate of change in the number of living wage jobs from 1997 to 1999? [ANS] Jobs/Year What is the average rate of change in the number of living wage jobs from 1999 to 2001? [ANS] Jobs/Year Based on these two answers, should the mayor from the last two years be reelected? (These numbers are made up. Please do not actually hold the mayor accountable.)", "answer_v2": [ "105/2", "65/2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The following chart shows \"living wage\" jobs in Rochester per 1000 working age adults over a 5 year period.\n\\begin{array}{| c | c | c | c | c | c |} \\hline \\mathrm{Year} & 1997 & 1998 & 1999 & 2000 & 2001 \\\\ \\hline \\mathrm{Jobs} & 640 & 685 & 725 & 755 & 780 \\\\ \\hline \\end{array} What is the average rate of change in the number of living wage jobs from 1997 to 1999? [ANS] Jobs/Year What is the average rate of change in the number of living wage jobs from 1999 to 2001? [ANS] Jobs/Year Based on these two answers, should the mayor from the last two years be reelected? (These numbers are made up. Please do not actually hold the mayor accountable.)", "answer_v3": [ "85/2", "55/2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0076", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - instantaneous rate of change", "level": "2", "keywords": [ "Derivative", "Tangent' 'Tangent Line", "Rate of Change", "Slope" ], "problem_v1": "For the function $\\small{y=x^{2}+8x+6}$:\n(a) Find a formula for the slope of the tangent line to the graph of $\\small{f}$ at a general point $\\small{x=a}$. Be sure that your formula is in terms of $\\small{a}$. (b) Using this formula, find the slope of the tangent line to $\\small{f}$ at $\\small{x=4}$.\n$\\begin{array}{cc}\\hline Slope at \\small{x=a:} & [ANS] \\\\ \\hline Slope at \\small{x=4:} & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "2*a+8", "16" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "For the function $\\small{y=x^{2}+x+10}$:\n(a) Find a formula for the slope of the tangent line to the graph of $\\small{f}$ at a general point $\\small{x=a}$. Be sure that your formula is in terms of $\\small{a}$. (b) Using this formula, find the slope of the tangent line to $\\small{f}$ at $\\small{x=1}$.\n$\\begin{array}{cc}\\hline Slope at \\small{x=a:} & [ANS] \\\\ \\hline Slope at \\small{x=1:} & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "2*a+1", "3" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "For the function $\\small{y=x^{2}+4x+7}$:\n(a) Find a formula for the slope of the tangent line to the graph of $\\small{f}$ at a general point $\\small{x=a}$. Be sure that your formula is in terms of $\\small{a}$. (b) Using this formula, find the slope of the tangent line to $\\small{f}$ at $\\small{x=2}$.\n$\\begin{array}{cc}\\hline Slope at \\small{x=a:} & [ANS] \\\\ \\hline Slope at \\small{x=2:} & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "2*a+4", "8" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0077", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - instantaneous rate of change", "level": "3", "keywords": [ "calculus", "limits", "derivatives" ], "problem_v1": "The cost (in dollars) of producing $x$ units of a certain commodity is C(x)=7000+14x+0.7x^2.\n(a) Find the average rate of change of $C$ with respect to $x$ when the production level is changed (i) from $x=100$ to $x=105$. Average rate of change=[ANS]\n(ii) from $x=100$ to $x=101$. Average rate of change=[ANS]\n(b) Find the instantaneous rate of change of $C$ with respect to $x$ when $x=100$. (This is called the marginal cost.) Instantaneous rate of change=[ANS]", "answer_v1": [ "157.5", "154.7", "154" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The cost (in dollars) of producing $x$ units of a certain commodity is C(x)=1500+19x+0.2x^2.\n(a) Find the average rate of change of $C$ with respect to $x$ when the production level is changed (i) from $x=100$ to $x=105$. Average rate of change=[ANS]\n(ii) from $x=100$ to $x=101$. Average rate of change=[ANS]\n(b) Find the instantaneous rate of change of $C$ with respect to $x$ when $x=100$. (This is called the marginal cost.) Instantaneous rate of change=[ANS]", "answer_v2": [ "60", "59.2", "59" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The cost (in dollars) of producing $x$ units of a certain commodity is C(x)=3500+14x+0.3x^2.\n(a) Find the average rate of change of $C$ with respect to $x$ when the production level is changed (i) from $x=100$ to $x=105$. Average rate of change=[ANS]\n(ii) from $x=100$ to $x=101$. Average rate of change=[ANS]\n(b) Find the instantaneous rate of change of $C$ with respect to $x$ when $x=100$. (This is called the marginal cost.) Instantaneous rate of change=[ANS]", "answer_v3": [ "75.5", "74.3", "74" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0078", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - instantaneous rate of change", "level": "4", "keywords": [ "calculus", "differentiation" ], "problem_v1": "(A) Find the average rate of change of the area of a circle with respect to its radius $r$ as $r$ changes from (i) 5 to 6 $\\rightarrow$ Average rate of change=[ANS]\n(ii) 5 to 5.5 $\\rightarrow$ Average rate of change=[ANS]\n(iii) 5 to 5.1 $\\rightarrow$ Average rate of change=[ANS]\n(B) Find the instantaneous rate of change when $r$=5. Instantaneous rate of change=[ANS]", "answer_v1": [ "34.5575", "32.9867", "31.7301", "31.4159" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "(A) Find the average rate of change of the area of a circle with respect to its radius $r$ as $r$ changes from (i) 2 to 3 $\\rightarrow$ Average rate of change=[ANS]\n(ii) 2 to 2.5 $\\rightarrow$ Average rate of change=[ANS]\n(iii) 2 to 2.1 $\\rightarrow$ Average rate of change=[ANS]\n(B) Find the instantaneous rate of change when $r$=2. Instantaneous rate of change=[ANS]", "answer_v2": [ "15.708", "14.1372", "12.8805", "12.5664" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "(A) Find the average rate of change of the area of a circle with respect to its radius $r$ as $r$ changes from (i) 3 to 4 $\\rightarrow$ Average rate of change=[ANS]\n(ii) 3 to 3.5 $\\rightarrow$ Average rate of change=[ANS]\n(iii) 3 to 3.1 $\\rightarrow$ Average rate of change=[ANS]\n(B) Find the instantaneous rate of change when $r$=3. Instantaneous rate of change=[ANS]", "answer_v3": [ "21.9911", "20.4204", "19.1637", "18.8496" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0079", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - instantaneous rate of change", "level": "2", "keywords": [ "Derivatives' 'Rates of Change" ], "problem_v1": "Determine the average rate of change of the following function between the given values of the variable: f(x)=x^{4}+2x; \\qquad x=-2,\\ x=3 Average rate of change=[ANS]", "answer_v1": [ "[3^4+2*3-(-2)^4-2*-2]/(3--2)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the average rate of change of the following function between the given values of the variable: f(x)=x^{4}-2x; \\qquad x=-1,\\ x=2 Average rate of change=[ANS]", "answer_v2": [ "[2^4+-2*2-(-1)^4--2*-1]/(2--1)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the average rate of change of the following function between the given values of the variable: f(x)=x^{4}-x; \\qquad x=-2,\\ x=1 Average rate of change=[ANS]", "answer_v3": [ "[1^4+-1*1-(-2)^4--1*-2]/(1--2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0080", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - instantaneous rate of change", "level": "5", "keywords": [ "Differentiation' 'Rates of Change" ], "problem_v1": "Suppose that an object moves along the $y$-axis so that its location is $y=x^2+7x$ at time $x$. (Here $y$ is in meters and $x$ is in seconds.) (A) Find the average velocity (the average rate of change of $y$ with respect to $x$) for $x$ changing from 2 to 8 seconds. Don't forget to include in all your answers. Average velocity=[ANS]\n(B) Find the average velocity for $x$ changing from $6$ to $6+h$ seconds. Average velocity=[ANS]\n(C) Find the instantaneous velocity at $x=6$ seconds. Instantaneous velocity=[ANS]", "answer_v1": [ "17", "19+h", "19" ], "answer_type_v1": [ "NV", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose that an object moves along the $y$-axis so that its location is $y=x^2+2x$ at time $x$. (Here $y$ is in meters and $x$ is in seconds.) (A) Find the average velocity (the average rate of change of $y$ with respect to $x$) for $x$ changing from 3 to 7 seconds. Don't forget to include in all your answers. Average velocity=[ANS]\n(B) Find the average velocity for $x$ changing from $5$ to $5+h$ seconds. Average velocity=[ANS]\n(C) Find the instantaneous velocity at $x=5$ seconds. Instantaneous velocity=[ANS]", "answer_v2": [ "12", "12+h", "12" ], "answer_type_v2": [ "NV", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose that an object moves along the $y$-axis so that its location is $y=x^2+4x$ at time $x$. (Here $y$ is in meters and $x$ is in seconds.) (A) Find the average velocity (the average rate of change of $y$ with respect to $x$) for $x$ changing from 2 to 7 seconds. Don't forget to include in all your answers. Average velocity=[ANS]\n(B) Find the average velocity for $x$ changing from $5$ to $5+h$ seconds. Average velocity=[ANS]\n(C) Find the instantaneous velocity at $x=5$ seconds. Instantaneous velocity=[ANS]", "answer_v3": [ "13", "14+h", "14" ], "answer_type_v3": [ "NV", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0081", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - instantaneous rate of change", "level": "4", "keywords": [ "Differentiation' 'Rates of Change" ], "problem_v1": "The position of a cat running from a dog down a dark alley is given by the values of the table.\n$\\begin{array}{ccccccc}\\hline t(seconds) & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline s(feet) & 0 & 17 & 38 & 69 & 95 & 107 \\\\ \\hline \\end{array}$\nBetween which two times is the cat moving the fastest?\nBetween $t=$ [ANS] and $t=$ [ANS]\nBetween which two times is the cat moving the slowest?\nBetween $t=$ [ANS] and $t=$ [ANS]", "answer_v1": [ "2", "2+1", "4", "4+1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The position of a cat running from a dog down a dark alley is given by the values of the table.\n$\\begin{array}{ccccccc}\\hline t(seconds) & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline s(feet) & 0 & 6 & 48 & 55 & 87 & 107 \\\\ \\hline \\end{array}$\nBetween which two times is the cat moving the fastest?\nBetween $t=$ [ANS] and $t=$ [ANS]\nBetween which two times is the cat moving the slowest?\nBetween $t=$ [ANS] and $t=$ [ANS]", "answer_v2": [ "1", "1+1", "0", "0+1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The position of a cat running from a dog down a dark alley is given by the values of the table.\n$\\begin{array}{ccccccc}\\hline t(seconds) & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline s(feet) & 0 & 10 & 39 & 59 & 91 & 105 \\\\ \\hline \\end{array}$\nBetween which two times is the cat moving the fastest?\nBetween $t=$ [ANS] and $t=$ [ANS]\nBetween which two times is the cat moving the slowest?\nBetween $t=$ [ANS] and $t=$ [ANS]", "answer_v3": [ "3", "3+1", "0", "0+1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0082", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - instantaneous rate of change", "level": "4", "keywords": [ "Differentiation' 'Rates of Change" ], "problem_v1": "The table below gives the population in a small coastal community for the period 1990-1999. Figures shown are for January 1 in each year.\n$\\begin{array}{ccccccccccc}\\hline Year & 1990 & 1991 & 1992 & 1993 & 1994 & 1995 & 1996 & 1997 & 1998 & 1999 \\\\ \\hline Population & 638 & 859 & 1331 & 1572 & 1599 & 1466 & 978 & 829 & 774 & 740 \\\\ \\hline \\end{array}$\n(a). What was the average rate of change of population between 1991 and 1994? [ANS]\n(b). What was the average rate of change of population between 1995 and 1997? [ANS]\n(c). For what period of time was the population increasing? From [ANS] to [ANS]\n(d). For what period of time was the population decreasing? From [ANS] to [ANS]\nNote: For (c) and (d), give the starting and ending years for the periods where the population was increasing/decreasing.", "answer_v1": [ "(1599-859)/3", "(829-1466)/2", "1990", "1994", "1994", "1999" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The table below gives the population in a small coastal community for the period 1990-1999. Figures shown are for January 1 in each year.\n$\\begin{array}{ccccccccccc}\\hline Year & 1990 & 1991 & 1992 & 1993 & 1994 & 1995 & 1996 & 1997 & 1998 & 1999 \\\\ \\hline Population & 604 & 877 & 1307 & 1560 & 1619 & 1466 & 959 & 816 & 782 & 731 \\\\ \\hline \\end{array}$\n(a). What was the average rate of change of population between 1991 and 1994? [ANS]\n(b). What was the average rate of change of population between 1995 and 1997? [ANS]\n(c). For what period of time was the population increasing? From [ANS] to [ANS]\n(d). For what period of time was the population decreasing? From [ANS] to [ANS]\nNote: For (c) and (d), give the starting and ending years for the periods where the population was increasing/decreasing.", "answer_v2": [ "(1619-877)/3", "(816-1466)/2", "1990", "1994", "1994", "1999" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The table below gives the population in a small coastal community for the period 1990-1999. Figures shown are for January 1 in each year.\n$\\begin{array}{ccccccccccc}\\hline Year & 1990 & 1991 & 1992 & 1993 & 1994 & 1995 & 1996 & 1997 & 1998 & 1999 \\\\ \\hline Population & 615 & 860 & 1314 & 1567 & 1596 & 1467 & 991 & 846 & 794 & 734 \\\\ \\hline \\end{array}$\n(a). What was the average rate of change of population between 1991 and 1994? [ANS]\n(b). What was the average rate of change of population between 1995 and 1997? [ANS]\n(c). For what period of time was the population increasing? From [ANS] to [ANS]\n(d). For what period of time was the population decreasing? From [ANS] to [ANS]\nNote: For (c) and (d), give the starting and ending years for the periods where the population was increasing/decreasing.", "answer_v3": [ "(1596-860)/3", "(846-1467)/2", "1990", "1994", "1994", "1999" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0083", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - instantaneous rate of change", "level": "5", "keywords": [ "average rate of change" ], "problem_v1": "It took 43 minutes for Usain the snail to reach the juicy rain drop that he discovered 38 centimeters from his home. Unfortunately, as soon as he reached the raindrop he was frightened by an aggressive caterpillar so he immediately turned back. He was tired, so after his second 43 minutes he is only half way home. Hint: Speed is always nonnegative but velocity can be negative if the movement is backward. What is his average velocity? [ANS] cm/min What is his average speed? [ANS] cm/min", "answer_v1": [ "0.22093", "0.662791" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "It took 22 minutes for Usain the snail to reach the juicy rain drop that he discovered 48 centimeters from his home. Unfortunately, as soon as he reached the raindrop he was frightened by an aggressive caterpillar so he immediately turned back. He was tired, so after his second 22 minutes he is only half way home. Hint: Speed is always nonnegative but velocity can be negative if the movement is backward. What is his average velocity? [ANS] cm/min What is his average speed? [ANS] cm/min", "answer_v2": [ "0.545455", "1.63636" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "It took 29 minutes for Usain the snail to reach the juicy rain drop that he discovered 38 centimeters from his home. Unfortunately, as soon as he reached the raindrop he was frightened by an aggressive caterpillar so he immediately turned back. He was tired, so after his second 29 minutes he is only half way home. Hint: Speed is always nonnegative but velocity can be negative if the movement is backward. What is his average velocity? [ANS] cm/min What is his average speed? [ANS] cm/min", "answer_v3": [ "0.327586", "0.982759" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0084", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - tangent lines and slopes", "level": "3", "keywords": [], "problem_v1": "Let $f(x)=\\frac{1}{x}$. a) Find the tangent line to the graph of the function at $x=7$. $y=$ [ANS]\nb) Find the normal line to the graph of the function at $x=7$. $y=$ [ANS]", "answer_v1": [ "(-1/49) x + 2/7", "49 x + -342.857142857143" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)=\\frac{1}{x}$. a) Find the tangent line to the graph of the function at $x=1$. $y=$ [ANS]\nb) Find the normal line to the graph of the function at $x=1$. $y=$ [ANS]", "answer_v2": [ "(-1/1) x + 2/1", "1 x + 0" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)=\\frac{1}{x}$. a) Find the tangent line to the graph of the function at $x=3$. $y=$ [ANS]\nb) Find the normal line to the graph of the function at $x=3$. $y=$ [ANS]", "answer_v3": [ "(-1/9) x + 2/3", "9 x + -26.6666666666667" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0085", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - tangent lines and slopes", "level": "3", "keywords": [ "derivative' 'limit' 'tangent" ], "problem_v1": "Let $ f(x)=5x^{3}$.\n(a) Use the limit process to find the slope of the line tangent to the graph of $f$ at $x=-1$. Slope at $x=-1$: [ANS]\n(b) Find an equation of the line tangent to the graph of $f$ at $x=-1$. Tangent line: $y$=[ANS]", "answer_v1": [ "15", "15*[x-(-1)]+(-5)" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let $ f(x)=2x^{3}$.\n(a) Use the limit process to find the slope of the line tangent to the graph of $f$ at $x=1$. Slope at $x=1$: [ANS]\n(b) Find an equation of the line tangent to the graph of $f$ at $x=1$. Tangent line: $y$=[ANS]", "answer_v2": [ "6", "6*(x-1)+2" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let $ f(x)=3x^{3}$.\n(a) Use the limit process to find the slope of the line tangent to the graph of $f$ at $x=-1$. Slope at $x=-1$: [ANS]\n(b) Find an equation of the line tangent to the graph of $f$ at $x=-1$. Tangent line: $y$=[ANS]", "answer_v3": [ "9", "9*[x-(-1)]+(-3)" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0086", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - tangent lines and slopes", "level": "3", "keywords": [ "Calculus", "Secant", "Tangent" ], "problem_v1": "The experimental data in the table below define y as a function of x. $\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline y & 3.5 & 2.2 & 1.8 & 2.1 & 2.4 & 3.4 \\\\ \\hline \\end{array}$\nA. Let P be the point (3, 2.1). Find the slopes of the secant lines PQ when Q is the point of the graph with x coordinate x1.\n$\\begin{array}{cccccc}\\hline x1 & 0 & 1 & 2 & 4 & 5 \\\\ \\hline slope & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nB. Draw the graph of the function for yourself and estimate the slope of the tangent line at P. [ANS]", "answer_v1": [ "-0.466666666666667", "-0.0499999999999998", "0.3", "0.3", "0.65", "0.3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The experimental data in the table below define y as a function of x. $\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline y & 2.1 & 0.6 & 0.5 & 0.7 & 1.5 & 2.5 \\\\ \\hline \\end{array}$\nA. Let P be the point (1, 0.6). Find the slopes of the secant lines PQ when Q is the point of the graph with x coordinate x1.\n$\\begin{array}{cccccc}\\hline x1 & 0 & 2 & 3 & 4 & 5 \\\\ \\hline slope & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nB. Draw the graph of the function for yourself and estimate the slope of the tangent line at P. [ANS]", "answer_v2": [ "-1.5", "-0.1", "0.05", "0.3", "0.475", "-0.8" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The experimental data in the table below define y as a function of x. $\\begin{array}{ccccccc}\\hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\ \\hline y & 2.6 & 1.3 & 1.1 & 1.4 & 1.6 & 2.6 \\\\ \\hline \\end{array}$\nA. Let P be the point (4, 1.6). Find the slopes of the secant lines PQ when Q is the point of the graph with x coordinate x1.\n$\\begin{array}{cccccc}\\hline x1 & 0 & 1 & 2 & 3 & 5 \\\\ \\hline slope & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nB. Draw the graph of the function for yourself and estimate the slope of the tangent line at P. [ANS]", "answer_v3": [ "-0.25", "0.1", "0.25", "0.2", "1", "0.6" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0087", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - tangent lines and slopes", "level": "3", "keywords": [ "Derivatives", "limits", "slope", "tangent", "derivative" ], "problem_v1": "Let $f(x)=4x^2-10x+10$ The slope of the tangent line to the graph of $f(x)$ at the point $(4,34)$ is [ANS]. The equation of the tangent line to the graph of $f(x)$ at $(4,34)$ is $y=mx+b$ for $m=$ [ANS]\nand $b=$ [ANS]. Hint: the slope is given by the derivative at $x=4$, ie. ${\\lim_{x\\to4}\\frac{f(4+h)-f(4)}{h}}$", "answer_v1": [ "22", "22", "-54" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $f(x)=2x^2-15x+3$ The slope of the tangent line to the graph of $f(x)$ at the point $(2,-19)$ is [ANS]. The equation of the tangent line to the graph of $f(x)$ at $(2,-19)$ is $y=mx+b$ for $m=$ [ANS]\nand $b=$ [ANS]. Hint: the slope is given by the derivative at $x=2$, ie. ${\\lim_{x\\to2}\\frac{f(2+h)-f(2)}{h}}$", "answer_v2": [ "-7", "-7", "-5" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $f(x)=2x^2-10x+5$ The slope of the tangent line to the graph of $f(x)$ at the point $(3,-7)$ is [ANS]. The equation of the tangent line to the graph of $f(x)$ at $(3,-7)$ is $y=mx+b$ for $m=$ [ANS]\nand $b=$ [ANS]. Hint: the slope is given by the derivative at $x=3$, ie. ${\\lim_{x\\to3}\\frac{f(3+h)-f(3)}{h}}$", "answer_v3": [ "2", "2", "-13" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0088", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - tangent lines and slopes", "level": "2", "keywords": [ "calculus", "derivative", "rate of change", "velocity", "average velocity", "instantaneous velocity" ], "problem_v1": "Estimate the following limit by substituting smaller and smaller values of $h$. $\\lim\\limits_{h \\rightarrow 0} \\frac{(9+h)^3-729}{h}=$ [ANS]\n(Your answer should be accurate within 0.001.) (Your answer should be accurate within 0.001.)", "answer_v1": [ "243" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Estimate the following limit by substituting smaller and smaller values of $h$. $\\lim\\limits_{h \\rightarrow 0} \\frac{(3+h)^3-27}{h}=$ [ANS]\n(Your answer should be accurate within 0.001.) (Your answer should be accurate within 0.001.)", "answer_v2": [ "27" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Estimate the following limit by substituting smaller and smaller values of $h$. $\\lim\\limits_{h \\rightarrow 0} \\frac{(5+h)^3-125}{h}=$ [ANS]\n(Your answer should be accurate within 0.001.) (Your answer should be accurate within 0.001.)", "answer_v3": [ "75" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0090", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - other", "level": "2", "keywords": [ "calculus", "derivative", "continuity", "functions" ], "problem_v1": "An electrical circuit switches instantaneously from a 7 volt battery to a 18 volt battery 6 seconds after being turned on. Sketch on a sheet of paper a graph the battery voltage against time. Then fill in the formulas below for the function represented by your graph. For $t <$ [ANS], $V(t)=$ [ANS]\nFor $t \\ge$ [ANS], $V(t)=$ [ANS]. At what point or points is your function discontinuous? $t=$ [ANS]\n(Give your answer as a comma-separated list, or enter the word none if there are no discontinuities.)", "answer_v1": [ "6", "7", "6", "18", "6" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "An electrical circuit switches instantaneously from a 2 volt battery to a 23 volt battery 3 seconds after being turned on. Sketch on a sheet of paper a graph the battery voltage against time. Then fill in the formulas below for the function represented by your graph. For $t <$ [ANS], $V(t)=$ [ANS]\nFor $t \\ge$ [ANS], $V(t)=$ [ANS]. At what point or points is your function discontinuous? $t=$ [ANS]\n(Give your answer as a comma-separated list, or enter the word none if there are no discontinuities.)", "answer_v2": [ "3", "2", "3", "23", "3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "An electrical circuit switches instantaneously from a 4 volt battery to a 19 volt battery 4 seconds after being turned on. Sketch on a sheet of paper a graph the battery voltage against time. Then fill in the formulas below for the function represented by your graph. For $t <$ [ANS], $V(t)=$ [ANS]\nFor $t \\ge$ [ANS], $V(t)=$ [ANS]. At what point or points is your function discontinuous? $t=$ [ANS]\n(Give your answer as a comma-separated list, or enter the word none if there are no discontinuities.)", "answer_v3": [ "4", "4", "4", "19", "4" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0091", "subject": "Calculus_-_single_variable", "topic": "Limits and continuity", "subtopic": "Applications - other", "level": "3", "keywords": [ "concavity", "increasing and decreasing functions", "maxima", "minima" ], "problem_v1": "The number, $N$, of people who have heard a rumor spread by mass media at time, $t$, is given by N(t)=a (1-e^{-kt}). There are 450000 people in the population who hear the rumor eventually. 17 percent of them heard it on the first day. Find $a$ and $k$, assuming $t$ is measured in days. $a=$ [ANS]\n$k=$ [ANS]", "answer_v1": [ "450000", "-1*ln(1 - 17/100)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The number, $N$, of people who have heard a rumor spread by mass media at time, $t$, is given by N(t)=a (1-e^{-kt}). There are 150000 people in the population who hear the rumor eventually. 24 percent of them heard it on the first day. Find $a$ and $k$, assuming $t$ is measured in days. $a=$ [ANS]\n$k=$ [ANS]", "answer_v2": [ "150000", "-1*ln(1 - 24/100)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The number, $N$, of people who have heard a rumor spread by mass media at time, $t$, is given by N(t)=a (1-e^{-kt}). There are 250000 people in the population who hear the rumor eventually. 17 percent of them heard it on the first day. Find $a$ and $k$, assuming $t$ is measured in days. $a=$ [ANS]\n$k=$ [ANS]", "answer_v3": [ "250000", "-1*ln(1 - 17/100)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0092", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "3", "keywords": [ "derivatives" ], "problem_v1": "If $f(x)=8 \\sqrt{x}$, use the definition of derivative to find $f'(x)$.\nAnswer: [ANS]", "answer_v1": [ "8/(2*sqrt(x))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If $f(x)=2 \\sqrt{x}$, use the definition of derivative to find $f'(x)$.\nAnswer: [ANS]", "answer_v2": [ "2/(2*sqrt(x))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If $f(x)=4 \\sqrt{x}$, use the definition of derivative to find $f'(x)$.\nAnswer: [ANS]", "answer_v3": [ "4/(2*sqrt(x))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0093", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [ "calculus", "derivatives", "definition of derivative", "rational functions" ], "problem_v1": "Let $f(x)=\\frac{1}{x}$. Compute the difference quotient for $f(x)$ at $x=4$ with $h=0.4$ difference quotient=[ANS]", "answer_v1": [ "-0.0568182" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)=\\frac{1}{x}$. Compute the difference quotient for $f(x)$ at $x=1$ with $h=0.5$ difference quotient=[ANS]", "answer_v2": [ "-0.666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(x)=\\frac{1}{x}$. Compute the difference quotient for $f(x)$ at $x=2$ with $h=0.4$ difference quotient=[ANS]", "answer_v3": [ "-0.208333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0094", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "1", "keywords": [ "calculus", "derivatives", "definition of derivative", "linear functions" ], "problem_v1": "Use the definition of the derivative to find the derivative of: $f(x)=12x+3$. $f'(x)=$ [ANS]", "answer_v1": [ "12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the definition of the derivative to find the derivative of: $f(x)=3x+13$. $f'(x)=$ [ANS]", "answer_v2": [ "3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the definition of the derivative to find the derivative of: $f(x)=6x+3$. $f'(x)=$ [ANS]", "answer_v3": [ "6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0095", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "3", "keywords": [], "problem_v1": "Given the following table: $\\begin{array}{cccccc}\\hline x & 0.0076 & 0.0077 & 0.0078 & 0.0079 & 0.008 \\\\ \\hline f(x) &-0.35969788 &-0.87470707 & 0.56478326 & 0.7947359 &-0.61604046 \\\\ \\hline \\end{array}$ Calculate the value of $f'(0.0078)$=[ANS] to two places of accuracy. To obtain more precise information about the value of $f$ near $0.0078$ enter a new increment value for $x$ here and then press the Submit Answer button. How will you tell when your increment is small enough to give you a good answer for the problem?", "answer_v1": [ "13564.0907196237" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given the following table: $\\begin{array}{cccccc}\\hline x & 0.0015 & 0.0016 & 0.0017 & 0.0018 & 0.0019 \\\\ \\hline f(x) & 0.60440922 & 0.17601627 &-0.68708388 & 0.48498335 &-0.99510152 \\\\ \\hline \\end{array}$ Calculate the value of $f'(0.0017)$=[ANS] to two places of accuracy. To obtain more precise information about the value of $f$ near $0.0017$ enter a new increment value for $x$ here and then press the Submit Answer button. How will you tell when your increment is small enough to give you a good answer for the problem?", "answer_v2": [ "251411.109245363" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given the following table: $\\begin{array}{cccccc}\\hline x & 0.0036 & 0.0037 & 0.0038 & 0.0039 & 0.004 \\\\ \\hline f(x) & 0.96812277 & 0.09316675 &-0.67124576 &-0.93216266 &-0.97052802 \\\\ \\hline \\end{array}$ Calculate the value of $f'(0.0038)$=[ANS] to two places of accuracy. To obtain more precise information about the value of $f$ near $0.0038$ enter a new increment value for $x$ here and then press the Submit Answer button. How will you tell when your increment is small enough to give you a good answer for the problem?", "answer_v3": [ "-51332.0546432667" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0096", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [ "Calculus", "Derivatives" ], "problem_v1": "Let $f(x)$ be the function $10x^2-8x+8$. Then the quotient $\\frac{f(8+h)-f(8)}{h}$ can be simplified to $ah+b$ for: $a=$ [ANS]\nand $b=$ [ANS]", "answer_v1": [ "10", "152" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)$ be the function $2x^2-12x+3$. Then the quotient $\\frac{f(4+h)-f(4)}{h}$ can be simplified to $ah+b$ for: $a=$ [ANS]\nand $b=$ [ANS]", "answer_v2": [ "2", "4" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)$ be the function $5x^2-8x+5$. Then the quotient $\\frac{f(6+h)-f(6)}{h}$ can be simplified to $ah+b$ for: $a=$ [ANS]\nand $b=$ [ANS]", "answer_v3": [ "5", "52" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0097", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [], "problem_v1": "Let $\\>f(t)=5t^{2}+2t$.\na) Find $\\>f(t+h)$: [ANS]\nb) Find $\\>f(t+h)-f(t)$: [ANS]\nc) Find $\\> \\frac {f(t+h)-f(t)} {h}$: [ANS]\nd) Find $\\> f'(t)$: [ANS]", "answer_v1": [ "5*(t+h)^2+2*(t+h)", "5*(t+h)^2+2*(t+h)-(5*t^2+2*t)", "[5*(t+h)^2+2*(t+h)-(5*t^2+2*t)]/h", "5*2*t+2" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $\\>f(t)=9t-9t^{2}$.\na) Find $\\>f(t+h)$: [ANS]\nb) Find $\\>f(t+h)-f(t)$: [ANS]\nc) Find $\\> \\frac {f(t+h)-f(t)} {h}$: [ANS]\nd) Find $\\> f'(t)$: [ANS]", "answer_v2": [ "9*(t+h)-9*(t+h)^2", "9*(t+h)-9*(t+h)^2-(9*t-9*t^2)", "[9*(t+h)-9*(t+h)^2-(9*t-9*t^2)]/h", "9-9*2*t" ], "answer_type_v2": [ "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $\\>f(t)=2t-4t^{2}$.\na) Find $\\>f(t+h)$: [ANS]\nb) Find $\\>f(t+h)-f(t)$: [ANS]\nc) Find $\\> \\frac {f(t+h)-f(t)} {h}$: [ANS]\nd) Find $\\> f'(t)$: [ANS]", "answer_v3": [ "2*(t+h)-4*(t+h)^2", "2*(t+h)-4*(t+h)^2-(2*t-4*t^2)", "[2*(t+h)-4*(t+h)^2-(2*t-4*t^2)]/h", "2-4*2*t" ], "answer_type_v3": [ "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0098", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [ "calculus" ], "problem_v1": "For f(x)=$4x^{2}+5x+5$ fill in the following table at x=3:\n$\\begin{array}{ccccc}\\hline x & f(x) & \\Delta x=x-3 & \\Delta f=f(x)-f(3) & \\frac{\\Delta f}{\\Delta x} \\\\ \\hline 2.9 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 2.99 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 2.999 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 2.9999 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 3.0001 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 3.001 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 3.01 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 3.1 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nUse the table to estimate $f^\\prime(3)$=[ANS]", "answer_v1": [ "53.14", "-0.1", "-2.86", "28.6", "55.7104", "-0.01", "-0.2896", "28.96", "55.971", "-0.001", "-0.028996", "28.996", "55.9971", "-0.0001", "-0.00289996", "28.9996", "56.0029", "0.0001", "0.00290004", "29.0004", "56.029", "0.001", "0.029004", "29.004", "56.2904", "0.01", "0.2904", "29.04", "58.94", "0.1", "2.94", "29.4", "29" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "For f(x)=$6x^{2}+2x+3$ fill in the following table at x=1:\n$\\begin{array}{ccccc}\\hline x & f(x) & \\Delta x=x-1 & \\Delta f=f(x)-f(1) & \\frac{\\Delta f}{\\Delta x} \\\\ \\hline 0.9 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 0.99 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 0.999 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 0.9999 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 1.0001 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 1.001 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 1.01 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 1.1 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nUse the table to estimate $f^\\prime(1)$=[ANS]", "answer_v2": [ "9.66", "-0.1", "-1.34", "13.4", "10.8606", "-0.01", "-0.1394", "13.94", "10.986", "-0.001", "-0.013994", "13.994", "10.9986", "-0.0001", "-0.00139994", "13.9994", "11.0014", "0.0001", "0.00140006", "14.0006", "11.014", "0.001", "0.014006", "14.006", "11.1406", "0.01", "0.1406", "14.06", "12.46", "0.1", "1.46", "14.6", "14" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "For f(x)=$5x^{2}+3x+4$ fill in the following table at x=1:\n$\\begin{array}{ccccc}\\hline x & f(x) & \\Delta x=x-1 & \\Delta f=f(x)-f(1) & \\frac{\\Delta f}{\\Delta x} \\\\ \\hline 0.9 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 0.99 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 0.999 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 0.9999 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 1.0001 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 1.001 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 1.01 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 1.1 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nUse the table to estimate $f^\\prime(1)$=[ANS]", "answer_v3": [ "10.75", "-0.1", "-1.25", "12.5", "11.8705", "-0.01", "-0.1295", "12.95", "11.987", "-0.001", "-0.012995", "12.995", "11.9987", "-0.0001", "-0.00129995", "12.9995", "12.0013", "0.0001", "0.00130005", "13.0005", "12.013", "0.001", "0.013005", "13.005", "12.1305", "0.01", "0.1305", "13.05", "13.35", "0.1", "1.35", "13.5", "13" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0099", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [ "convergence" ], "problem_v1": "Let $ f(x)=\\frac{5}{x-8}$. Then according to the definition of derivative\n$ f'(x)=\\lim_{t\\to x}$ [ANS]\n(Your answer above and the next few answers below will involve the variables $t$ and $x$.)\nThe expression inside the limit simplifies to a simple fraction with:\nNumerator $=$ [ANS]\nDenominator $=$ [ANS]\nWe can cancel the factor [ANS] appearing in the denominator against a similar factor appearing in the numerator leaving a simpler fraction with:\nNumerator $=$ [ANS]\nDenominator $=$ [ANS]\nTaking the limit of this fractional expression gives us $f'(x)=$ [ANS]", "answer_v1": [ "(5/(t- 8)-5/(x- 8))/(t-x)", "5*(x-t)", "(t-8)*(x-8)*(t-x)", "t-x", "-5", "(t-8)*(x-8)", "- 5/(x- 8)^2" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Let $ f(x)=\\frac{7}{x-2}$. Then according to the definition of derivative\n$ f'(x)=\\lim_{t\\to x}$ [ANS]\n(Your answer above and the next few answers below will involve the variables $t$ and $x$.)\nThe expression inside the limit simplifies to a simple fraction with:\nNumerator $=$ [ANS]\nDenominator $=$ [ANS]\nWe can cancel the factor [ANS] appearing in the denominator against a similar factor appearing in the numerator leaving a simpler fraction with:\nNumerator $=$ [ANS]\nDenominator $=$ [ANS]\nTaking the limit of this fractional expression gives us $f'(x)=$ [ANS]", "answer_v2": [ "(7/(t- 2)-7/(x- 2))/(t-x)", "7*(x-t)", "(t-2)*(x-2)*(t-x)", "t-x", "-7", "(t-2)*(x-2)", "- 7/(x- 2)^2" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Let $ f(x)=\\frac{5}{x-4}$. Then according to the definition of derivative\n$ f'(x)=\\lim_{t\\to x}$ [ANS]\n(Your answer above and the next few answers below will involve the variables $t$ and $x$.)\nThe expression inside the limit simplifies to a simple fraction with:\nNumerator $=$ [ANS]\nDenominator $=$ [ANS]\nWe can cancel the factor [ANS] appearing in the denominator against a similar factor appearing in the numerator leaving a simpler fraction with:\nNumerator $=$ [ANS]\nDenominator $=$ [ANS]\nTaking the limit of this fractional expression gives us $f'(x)=$ [ANS]", "answer_v3": [ "(5/(t- 4)-5/(x- 4))/(t-x)", "5*(x-t)", "(t-4)*(x-4)*(t-x)", "t-x", "-5", "(t-4)*(x-4)", "- 5/(x- 4)^2" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0100", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [ "calculus", "derivative", "instantaneous velocity", "difference quotient", "definition of derivative" ], "problem_v1": "Suppose that $f$ is given for $x$ in the interval $[0,12]$ by\n$\\begin{array}{cccccccc}\\hline x=& 0 & 2 & 4 & 6 & 8 & 10 & 12 \\\\ \\hline f(x)=& 13 & 15 & 18 & 19 & 18 & 15 & 12 \\\\ \\hline \\end{array}$\nA. Estimate $f'(2)$ using the values of $f$ in the table. $f'(2)\\approx$ [ANS]\nB. For what values of $x$ does $f'(x)$ appear to be positive? [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10).).) C. For what values of $x$ does $f'(x)$ appear to be negative? [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10).).)", "answer_v1": [ "1.25", "(0,6)", "(6,12)" ], "answer_type_v1": [ "NV", "INT", "INT" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose that $f$ is given for $x$ in the interval $[0,12]$ by\n$\\begin{array}{cccccccc}\\hline x=& 0 & 2 & 4 & 6 & 8 & 10 & 12 \\\\ \\hline f(x)=&-13 &-17 &-19 &-21 &-22 &-21 &-19 \\\\ \\hline \\end{array}$\nA. Estimate $f'(2)$ using the values of $f$ in the table. $f'(2)\\approx$ [ANS]\nB. For what values of $x$ does $f'(x)$ appear to be positive? [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10).).) C. For what values of $x$ does $f'(x)$ appear to be negative? [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10).).)", "answer_v2": [ "-1.5", "(8,12)", "(0,8)" ], "answer_type_v2": [ "NV", "INT", "INT" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose that $f$ is given for $x$ in the interval $[0,12]$ by\n$\\begin{array}{cccccccc}\\hline x=& 0 & 2 & 4 & 6 & 8 & 10 & 12 \\\\ \\hline f(x)=&-16 &-18 &-21 &-22 &-21 &-17 &-13 \\\\ \\hline \\end{array}$\nA. Estimate $f'(2)$ using the values of $f$ in the table. $f'(2)\\approx$ [ANS]\nB. For what values of $x$ does $f'(x)$ appear to be positive? [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10).).) C. For what values of $x$ does $f'(x)$ appear to be negative? [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10).).)", "answer_v3": [ "-1.25", "(6,12)", "(0,6)" ], "answer_type_v3": [ "NV", "INT", "INT" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0101", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [ "calculus", "derivative", "second derivative", "acceleration" ], "problem_v1": "The position of a particle moving along the $x$-axis is given by $s(t)=7t^{2}+6$. Use difference quotients to find the velocity $v(t)$ and acceleration $a(t)$, filling in the following expressions as you do so: $v(t)=\\lim\\limits_{h\\to0}$ [[ANS]/h]=[ANS]\n$a(t)=\\lim\\limits_{h\\to0}$ [[ANS]/h]=[ANS]", "answer_v1": [ "7*(t+h)^2+6-(7*t^2+6)", "7*2*t", "7*2*(t+h)-7*2*t", "14" ], "answer_type_v1": [ "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The position of a particle moving along the $x$-axis is given by $s(t)=2t^{2}+8$. Use difference quotients to find the velocity $v(t)$ and acceleration $a(t)$, filling in the following expressions as you do so: $v(t)=\\lim\\limits_{h\\to0}$ [[ANS]/h]=[ANS]\n$a(t)=\\lim\\limits_{h\\to0}$ [[ANS]/h]=[ANS]", "answer_v2": [ "2*(t+h)^2+8-(2*t^2+8)", "2*2*t", "2*2*(t+h)-2*2*t", "4" ], "answer_type_v2": [ "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The position of a particle moving along the $x$-axis is given by $s(t)=4t^{2}+6$. Use difference quotients to find the velocity $v(t)$ and acceleration $a(t)$, filling in the following expressions as you do so: $v(t)=\\lim\\limits_{h\\to0}$ [[ANS]/h]=[ANS]\n$a(t)=\\lim\\limits_{h\\to0}$ [[ANS]/h]=[ANS]", "answer_v3": [ "4*(t+h)^2+6-(4*t^2+6)", "4*2*t", "4*2*(t+h)-4*2*t", "8" ], "answer_type_v3": [ "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0102", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [ "derivative" ], "problem_v1": "Find the limit by interpreting the expression as an appropriate derivative. ${\\lim_{\\Delta x\\to 0}}\\frac{\\ln{(e^{4}+\\Delta x)}-4}{\\Delta x}=$ [ANS]", "answer_v1": [ "e^{-4}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the limit by interpreting the expression as an appropriate derivative. ${\\lim_{\\Delta x\\to 0}}\\frac{\\ln{(e^{-7}+\\Delta x)}+7}{\\Delta x}=$ [ANS]", "answer_v2": [ "e^7" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the limit by interpreting the expression as an appropriate derivative. ${\\lim_{\\Delta x\\to 0}}\\frac{\\ln{(e^{-3}+\\Delta x)}+3}{\\Delta x}=$ [ANS]", "answer_v3": [ "e^3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0103", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "", "keywords": [ "Derivative", "Tangent' 'Tangent Line", "Rate of Change" ], "problem_v1": "For the function $\\small {y=5x^{2}}$:\n(a) Find the average rate of change of $\\small {y}$ with respect to $\\small {x}$ over the interval $\\small {[3,5]}$. (b) Find the instantaneous rate of change of $\\small{y}$ with respect to $\\small{x}$ at the value $\\small{x=3}$.\n$\\begin{array}{cc}\\hline Average Rate of Change: & [ANS] \\\\ \\hline Instantaneous Rate of Change at \\small{x=3:} & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "80/2", "30" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "For the function $\\small {y=2x^{2}}$:\n(a) Find the average rate of change of $\\small {y}$ with respect to $\\small {x}$ over the interval $\\small {[5,6]}$. (b) Find the instantaneous rate of change of $\\small{y}$ with respect to $\\small{x}$ at the value $\\small{x=5}$.\n$\\begin{array}{cc}\\hline Average Rate of Change: & [ANS] \\\\ \\hline Instantaneous Rate of Change at \\small{x=5:} & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "22", "20" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "For the function $\\small {y=3x^{2}}$:\n(a) Find the average rate of change of $\\small {y}$ with respect to $\\small {x}$ over the interval $\\small {[4,5]}$. (b) Find the instantaneous rate of change of $\\small{y}$ with respect to $\\small{x}$ at the value $\\small{x=4}$.\n$\\begin{array}{cc}\\hline Average Rate of Change: & [ANS] \\\\ \\hline Instantaneous Rate of Change at \\small{x=4:} & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "(75-48)", "24" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0104", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [ "Derivative", "Tangent' 'Tangent Line", "Rate of Change", "Slope" ], "problem_v1": "For $\\small{f(x)=6x^{4}}$, find $\\large{\\frac{dy}{dx}}$ using the definition, \\small{\\frac{dy}{dx}=\\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}=\\lim_{\\Delta x \\to 0} \\frac{f(x+\\Delta x)-f(x)}{\\Delta x}.}\n(a) First, evaluate $\\small{\\Delta y=f(x+\\Delta x)-f(x)}$ and express it in the form $\\small{\\Delta y=L(x,\\Delta x) \\cdot \\Delta x}$. Use $\\small{dx}$ to represent $\\small{\\Delta x}$. $\\small{\\Delta y}$=[ANS] $\\small{\\Delta x}$ (b) Using the $\\small{L(x,\\Delta x)\\; \\Delta x} \\;$ above, find the simplified derivative $\\large{\\frac{dy}{dx}} \\small{=\\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}}$. $\\large{\\frac{dy}{dx}} \\small{=\\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}=}$ [ANS]", "answer_v1": [ "24*x^3+36*x^2*dx+24*x*dx^2+6*dx^3", "24*x^3" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "For $\\small{f(x)=9x^{2}}$, find $\\large{\\frac{dy}{dx}}$ using the definition, \\small{\\frac{dy}{dx}=\\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}=\\lim_{\\Delta x \\to 0} \\frac{f(x+\\Delta x)-f(x)}{\\Delta x}.}\n(a) First, evaluate $\\small{\\Delta y=f(x+\\Delta x)-f(x)}$ and express it in the form $\\small{\\Delta y=L(x,\\Delta x) \\cdot \\Delta x}$. Use $\\small{dx}$ to represent $\\small{\\Delta x}$. $\\small{\\Delta y}$=[ANS] $\\small{\\Delta x}$ (b) Using the $\\small{L(x,\\Delta x)\\; \\Delta x} \\;$ above, find the simplified derivative $\\large{\\frac{dy}{dx}} \\small{=\\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}}$. $\\large{\\frac{dy}{dx}} \\small{=\\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}=}$ [ANS]", "answer_v2": [ "18*x+9*dx", "18*x" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "For $\\small{f(x)=6x^{2}}$, find $\\large{\\frac{dy}{dx}}$ using the definition, \\small{\\frac{dy}{dx}=\\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}=\\lim_{\\Delta x \\to 0} \\frac{f(x+\\Delta x)-f(x)}{\\Delta x}.}\n(a) First, evaluate $\\small{\\Delta y=f(x+\\Delta x)-f(x)}$ and express it in the form $\\small{\\Delta y=L(x,\\Delta x) \\cdot \\Delta x}$. Use $\\small{dx}$ to represent $\\small{\\Delta x}$. $\\small{\\Delta y}$=[ANS] $\\small{\\Delta x}$ (b) Using the $\\small{L(x,\\Delta x)\\; \\Delta x} \\;$ above, find the simplified derivative $\\large{\\frac{dy}{dx}} \\small{=\\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}}$. $\\large{\\frac{dy}{dx}} \\small{=\\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x}=}$ [ANS]", "answer_v3": [ "12*x+6*dx", "12*x" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0105", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [ "Derivative", "Tangent' 'Tangent Line", "Rate of Change", "Slope" ], "problem_v1": "Find $\\large{\\left.\\frac{dy}{dx}\\right|_{x=1}}$, given that $\\small{y=\\large{\\frac{x+8}{x}}}$. $\\large{\\left.\\frac{dy}{dx}\\right|_{x=1}}$=[ANS]", "answer_v1": [ "-8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $\\large{\\left.\\frac{dy}{dx}\\right|_{x=5}}$, given that $\\small{y=\\large{\\frac{x+1}{x}}}$. $\\large{\\left.\\frac{dy}{dx}\\right|_{x=5}}$=[ANS]", "answer_v2": [ "-1/25" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $\\large{\\left.\\frac{dy}{dx}\\right|_{x=1}}$, given that $\\small{y=\\large{\\frac{x+4}{x}}}$. $\\large{\\left.\\frac{dy}{dx}\\right|_{x=1}}$=[ANS]", "answer_v3": [ "-4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0106", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [ "Derivative", "Tangent' 'Tangent Line", "Rate of Change", "Slope" ], "problem_v1": "For $\\small{f(x)=\\sqrt{x+24}}$, find $\\small{f\\;'(x)}$ using the definition $\\small{f\\;'(x)= \\lim_{h\\to 0}} \\large{\\frac{f(x+h)-f(x)}{h}}$. $\\small{f\\;'(x)=}$ [ANS]\nUsing this, find the tangent line to the graph of $\\small{y=\\sqrt{x+24}}$ at $\\small{x=1}$. Write the equation of the line in slope-intercept form. $\\small{y=}$ [ANS]", "answer_v1": [ "1/[2*sqrt(x+24)]", "0.1*x+4.9" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "For $\\small{f(x)=\\sqrt{x+1}}$, find $\\small{f\\;'(x)}$ using the definition $\\small{f\\;'(x)= \\lim_{h\\to 0}} \\large{\\frac{f(x+h)-f(x)}{h}}$. $\\small{f\\;'(x)=}$ [ANS]\nUsing this, find the tangent line to the graph of $\\small{y=\\sqrt{x+1}}$ at $\\small{x=3}$. Write the equation of the line in slope-intercept form. $\\small{y=}$ [ANS]", "answer_v2": [ "1/[2*sqrt(x+1)]", "0.25*x+1.25" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "For $\\small{f(x)=\\sqrt{x+8}}$, find $\\small{f\\;'(x)}$ using the definition $\\small{f\\;'(x)= \\lim_{h\\to 0}} \\large{\\frac{f(x+h)-f(x)}{h}}$. $\\small{f\\;'(x)=}$ [ANS]\nUsing this, find the tangent line to the graph of $\\small{y=\\sqrt{x+8}}$ at $\\small{x=1}$. Write the equation of the line in slope-intercept form. $\\small{y=}$ [ANS]", "answer_v3": [ "1/[2*sqrt(x+8)]", "0.166667*x+2.83333" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0108", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "4", "keywords": [ "calculus", "limits", "derivatives" ], "problem_v1": "The limit \\lim_{x\\rightarrow 8 \\pi}\\frac{\\cos(x)-1}{x-8 \\pi} represents the derivative of some function $f(x)$ at some number $a$. Find $f$ and $a$.\n$f(x)$=[ANS]\n$a$=[ANS]", "answer_v1": [ "cos(x)", "8 pi" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The limit \\lim_{x\\rightarrow 2 \\pi}\\frac{\\cos(x)-1}{x-2 \\pi} represents the derivative of some function $f(x)$ at some number $a$. Find $f$ and $a$.\n$f(x)$=[ANS]\n$a$=[ANS]", "answer_v2": [ "cos(x)", "2 pi" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The limit \\lim_{x\\rightarrow 4 \\pi}\\frac{\\cos(x)-1}{x-4 \\pi} represents the derivative of some function $f(x)$ at some number $a$. Find $f$ and $a$.\n$f(x)$=[ANS]\n$a$=[ANS]", "answer_v3": [ "cos(x)", "4 pi" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0109", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "3", "keywords": [ "calculus", "limits", "derivatives" ], "problem_v1": "Use the definition of the derivative (don't be tempted to take shortcuts!) to find the derivative of the function f(x)=8x-6. Then state the domain of the function and the domain of the derivative. \n$f'(x)$=[ANS]\nDomain of $f(x)$=[ANS]\nDomain of $f'(x)$=[ANS]", "answer_v1": [ "8", "(-infinity,infinity)", "(-infinity,infinity)" ], "answer_type_v1": [ "NV", "INT", "INT" ], "options_v1": [ [], [], [] ], "problem_v2": "Use the definition of the derivative (don't be tempted to take shortcuts!) to find the derivative of the function f(x)=2x-9. Then state the domain of the function and the domain of the derivative. \n$f'(x)$=[ANS]\nDomain of $f(x)$=[ANS]\nDomain of $f'(x)$=[ANS]", "answer_v2": [ "2", "(-infinity,infinity)", "(-infinity,infinity)" ], "answer_type_v2": [ "NV", "INT", "INT" ], "options_v2": [ [], [], [] ], "problem_v3": "Use the definition of the derivative (don't be tempted to take shortcuts!) to find the derivative of the function f(x)=4x-6. Then state the domain of the function and the domain of the derivative. \n$f'(x)$=[ANS]\nDomain of $f(x)$=[ANS]\nDomain of $f'(x)$=[ANS]", "answer_v3": [ "4", "(-infinity,infinity)", "(-infinity,infinity)" ], "answer_type_v3": [ "NV", "INT", "INT" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0110", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "4", "keywords": [ "calculus", "derivatives", "Polynomials" ], "problem_v1": "Let f(x)=3x^3+7x-2 Use the limit definition of the derivative to calculate the derivative of $f$: $f'(x)=$ [ANS]. Use the same formula from above to calculate the derivative of this new function (i.e. the second derivative of $f$): $f''(x)=$ [ANS].", "answer_v1": [ "3*3*x^2+7", "3*3*2*x" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let f(x)=-3x^3-4x+5 Use the limit definition of the derivative to calculate the derivative of $f$: $f'(x)=$ [ANS]. Use the same formula from above to calculate the derivative of this new function (i.e. the second derivative of $f$): $f''(x)=$ [ANS].", "answer_v2": [ "-(3*3*x^2+4)", "-(3*3*2*x)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let f(x)=-3x^3-6x-3 Use the limit definition of the derivative to calculate the derivative of $f$: $f'(x)=$ [ANS]. Use the same formula from above to calculate the derivative of this new function (i.e. the second derivative of $f$): $f''(x)=$ [ANS].", "answer_v3": [ "-(3*3*x^2+6)", "-(3*3*2*x)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0111", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "3", "keywords": [ "derivative", "definition" ], "problem_v1": "Suppose that f(x+h)-f(x)=4 h x^2+1 h x+2 h^2x+4 h^2-3 h^3. Find $f'(x)$.\n$f'(x)=$ [ANS]", "answer_v1": [ "4*x^2 + 1*x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Suppose that f(x+h)-f(x)=-7 h x^2+7 h x-6 h^2x-3 h^2+8 h^3. Find $f'(x)$.\n$f'(x)=$ [ANS]", "answer_v2": [ "-7*x^2 + 7*x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Suppose that f(x+h)-f(x)=-3 h x^2+2 h x-4 h^2x+1 h^2-5 h^3. Find $f'(x)$.\n$f'(x)=$ [ANS]", "answer_v3": [ "-3*x^2 + 2*x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0112", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "3", "keywords": [ "derivative", "definition" ], "problem_v1": "Let $f(x)={4} x+C$, where $C$ is any real number. Then the expression \\frac{f(x+h)-f(x)}{h} can be written in the form $Ah+Bx+D$, where $A$, $B$, and $D$ are constants. (Note: It's possible for one or more of these constants to be 0.) Find the constants. $A=$ [ANS]\n$B=$ [ANS]\n$D=$ [ANS]\nUse your answer from above to find $f'(x)$.\n$f'(x)=$ [ANS]", "answer_v1": [ "0", "0", "4", "4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $f(x)={-7} x+C$, where $C$ is any real number. Then the expression \\frac{f(x+h)-f(x)}{h} can be written in the form $Ah+Bx+D$, where $A$, $B$, and $D$ are constants. (Note: It's possible for one or more of these constants to be 0.) Find the constants. $A=$ [ANS]\n$B=$ [ANS]\n$D=$ [ANS]\nUse your answer from above to find $f'(x)$.\n$f'(x)=$ [ANS]", "answer_v2": [ "0", "0", "-7", "-7" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $f(x)={-3} x+C$, where $C$ is any real number. Then the expression \\frac{f(x+h)-f(x)}{h} can be written in the form $Ah+Bx+D$, where $A$, $B$, and $D$ are constants. (Note: It's possible for one or more of these constants to be 0.) Find the constants. $A=$ [ANS]\n$B=$ [ANS]\n$D=$ [ANS]\nUse your answer from above to find $f'(x)$.\n$f'(x)=$ [ANS]", "answer_v3": [ "0", "0", "-3", "-3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0113", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Definition of the derivative", "level": "2", "keywords": [ "calculator", "tangent line", "derivatives", "applications" ], "problem_v1": "The definition of the derivative, applied to $f(x)=9^x$, results in the formula $f'(x)=k \\cdot 9^x$ where k=\\lim_{h \\rightarrow 0} \\frac{9^h-1}{h}. The value of the constant $k$ cannot be determined using the usual tricks for evaluating limits. Using your calculator, approximate the constant $k$, rounded to three significant figures. $\\vphantom{A^A}k \\approx$ [ANS].", "answer_v1": [ "2.2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The definition of the derivative, applied to $f(x)=3^x$, results in the formula $f'(x)=k \\cdot 3^x$ where k=\\lim_{h \\rightarrow 0} \\frac{3^h-1}{h}. The value of the constant $k$ cannot be determined using the usual tricks for evaluating limits. Using your calculator, approximate the constant $k$, rounded to three significant figures. $\\vphantom{A^A}k \\approx$ [ANS].", "answer_v2": [ "1.1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The definition of the derivative, applied to $f(x)=5^x$, results in the formula $f'(x)=k \\cdot 5^x$ where k=\\lim_{h \\rightarrow 0} \\frac{5^h-1}{h}. The value of the constant $k$ cannot be determined using the usual tricks for evaluating limits. Using your calculator, approximate the constant $k$, rounded to three significant figures. $\\vphantom{A^A}k \\approx$ [ANS].", "answer_v3": [ "1.61" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0114", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Conceptual understanding of derivatives", "level": "4", "keywords": [ "tangent line", "derivatives", "Calculus", "Derivative", "Tangent", "derivative", "Product", "Quotient", "Differentiation", "limits" ], "problem_v1": "The tangent line to $y=f(x)$ at $(5, 2)$ passes through the point $(3, 5)$. Compute the following.\na.) $f(5)=$ [ANS]\nb.) $f'(5)=$ [ANS]", "answer_v1": [ "2", "-3/2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The tangent line to $y=f(x)$ at $(-9, 9)$ passes through the point $(-7,-3)$. Compute the following.\na.) $f(-9)=$ [ANS]\nb.) $f'(-9)=$ [ANS]", "answer_v2": [ "9", "-12/2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The tangent line to $y=f(x)$ at $(-4, 2)$ passes through the point $(-5, 1)$. Compute the following.\na.) $f(-4)=$ [ANS]\nb.) $f'(-4)=$ [ANS]", "answer_v3": [ "2", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0115", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Conceptual understanding of derivatives", "level": "3", "keywords": [ "calculus", "derivatives", "absolute value", "limits" ], "problem_v1": "Find the points $c$ (if any) such that $f'(c)$ does not exist.\nf(x)=\\left|x-2\\right| $c=$ [ANS]", "answer_v1": [ "2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the points $c$ (if any) such that $f'(c)$ does not exist.\nf(x)=\\left|x+4\\right| $c=$ [ANS]", "answer_v2": [ "-4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the points $c$ (if any) such that $f'(c)$ does not exist.\nf(x)=\\left|x+2\\right| $c=$ [ANS]", "answer_v3": [ "-2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0116", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Conceptual understanding of derivatives", "level": "2", "keywords": [ "calculus", "derivative", "instantaneous velocity", "difference quotient", "definition of derivative" ], "problem_v1": "Given the numerical values shown for the function $f(x)$, find approximate values for the derivative of $f(x)$ at each of the $x$-values given.\n$\\begin{array}{cccccccccc}\\hline x=& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline f(x)=& 13 & 15 & 17 & 18 & 20 & 20 & 18 & 16 & 15 \\\\ \\hline f'(x)=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nWhat seems to be the greatest rate of change of $f(x)$? rate of change=[ANS]", "answer_v1": [ "2", "2", "1.5", "1.5", "1", "-1", "-2", "-1.5", "-1", "2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Given the numerical values shown for the function $f(x)$, find approximate values for the derivative of $f(x)$ at each of the $x$-values given.\n$\\begin{array}{cccccccccc}\\hline x=& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline f(x)=&-13 &-14 &-15 &-16 &-18 &-19 &-20 &-20 &-18 \\\\ \\hline f'(x)=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nWhat seems to be the greatest rate of change of $f(x)$? rate of change=[ANS]", "answer_v2": [ "-1", "-1", "-1", "-1.5", "-1.5", "-1", "-0.5", "1", "2", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Given the numerical values shown for the function $f(x)$, find approximate values for the derivative of $f(x)$ at each of the $x$-values given.\n$\\begin{array}{cccccccccc}\\hline x=& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\\ \\hline f(x)=&-16 &-18 &-21 &-22 &-25 &-25 &-22 &-21 &-20 \\\\ \\hline f'(x)=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nWhat seems to be the greatest rate of change of $f(x)$? rate of change=[ANS]", "answer_v3": [ "-2", "-2.5", "-2", "-2", "-1.5", "1.5", "2", "1", "1", "2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0117", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Conceptual understanding of derivatives", "level": "2", "keywords": [ "calculus", "derivative" ], "problem_v1": "A laboratory study investigating the relationship between diet and weight in adult humans found that the weight of a subject, $W$, in pounds, was a function, $W=f(c)$, of the average number of Calories, $c$, consumed by the subject in a day.\n(a) In the statement $f(1800)=155$ what are the units of 1800? [ANS] what are the units of 155? [ANS] (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (b) In the statement $f'(2000)=0$, what are the units of 2000? [ANS] what are the units of 0? [ANS] (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (c) In the statement $f^{-1}(164)=2400$, what are the units of 164? [ANS] what are the units of 2400? [ANS] (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (d) What are the units of $f'(c)=dW/dc$? [ANS] (e) Suppose that Sam reads about $f'$ in this study and draws the following conclusion: If Sam increases her average calorie intake from 2700 to 2730 calories per day, then her weight will increase by approximately 0.3 pounds. Fill in the blanks below so that the equation supports her conclusion. $f'\\Big($ [ANS] $\\Big)=$ [ANS]", "answer_v1": [ "cal", "lb", "cal", "lb/cal", "lb", "cal", "lb/cal", "2700", "0.01" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "NV", "NV" ], "options_v1": [ [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [], [] ], "problem_v2": "A laboratory study investigating the relationship between diet and weight in adult humans found that the weight of a subject, $W$, in pounds, was a function, $W=f(c)$, of the average number of Calories, $c$, consumed by the subject in a day.\n(a) In the statement $f(1500)=165$ what are the units of 1500? [ANS] what are the units of 165? [ANS] (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (b) In the statement $f'(2000)=0$, what are the units of 2000? [ANS] what are the units of 0? [ANS] (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (c) In the statement $f^{-1}(172)=2100$, what are the units of 172? [ANS] what are the units of 2100? [ANS] (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (d) What are the units of $f'(c)=dW/dc$? [ANS] (e) Suppose that Sam reads about $f'$ in this study and draws the following conclusion: If Sam increases his average calorie intake from 3000 to 3020 calories per day, then his weight will increase by approximately 0.2 pounds. Fill in the blanks below so that the equation supports his conclusion. $f'\\Big($ [ANS] $\\Big)=$ [ANS]", "answer_v2": [ "cal", "lb", "cal", "lb/cal", "lb", "cal", "lb/cal", "3000", "0.01" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "NV", "NV" ], "options_v2": [ [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [], [] ], "problem_v3": "A laboratory study investigating the relationship between diet and weight in adult humans found that the weight of a subject, $W$, in pounds, was a function, $W=f(c)$, of the average number of Calories, $c$, consumed by the subject in a day.\n(a) In the statement $f(1600)=160$ what are the units of 1600? [ANS] what are the units of 160? [ANS] (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (b) In the statement $f'(2000)=0$, what are the units of 2000? [ANS] what are the units of 0? [ANS] (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (c) In the statement $f^{-1}(168)=2200$, what are the units of 168? [ANS] what are the units of 2200? [ANS] (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (Think about what this statement means in terms of the weight of the subject and the number of calories that the subject consumes.) (d) What are the units of $f'(c)=dW/dc$? [ANS] (e) Suppose that Sam reads about $f'$ in this study and draws the following conclusion: If Sam increases her average calorie intake from 2700 to 2740 calories per day, then her weight will increase by approximately 0.6 pounds. Fill in the blanks below so that the equation supports her conclusion. $f'\\Big($ [ANS] $\\Big)=$ [ANS]", "answer_v3": [ "cal", "lb", "cal", "lb/cal", "lb", "cal", "lb/cal", "2700", "0.015" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "NV", "NV" ], "options_v3": [ [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [ "lb", "cal", "day", "lb/cal", "cal/lb", "cal/day", "lb/day", "day/lb", "day/cal" ], [], [] ] }, { "id": "Calculus_-_single_variable_0118", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Conceptual understanding of derivatives", "level": "2", "keywords": [ "calculus", "derivative" ], "problem_v1": "Let $f(t)$ be the number of centimeters of rainfall that has fallen since midnight, where $t$ is the time in hours. Match the following statements to their interpretations, given below.\n(a) $f(10)=2.7$: [ANS] (b) $f^{-1}(6)=10$: [ANS] (c) $f'(10)=0.2$: [ANS] (d) $(f^{-1})'(6)=2$: [ANS] Interpretations: 1: at 0.2 hours after midnight, rain is falling at a rate of 10 cm/hr, so approximately 0.83 cm of rain falls in the next five minutes 2: at 10 hours after midnight, 0.2 centimeters of rain have fallen 3: at 10 AM, rain is falling at a rate of 2.7 cm/hr, so approximately 0.23 cm of rain falls in the next five minutes 4: at 10 hours after midnight, rain is falling at a rate of 6 cm/hr, so approximately 0.1000 cm of rain falls in the next minute 5: when 2 centimeters of rain have fallen, rain is falling at a rate of 6 cm/hr, so it takes approximately 10.0 min for the total accumulation to go from 2 to 3 cm 6: at 10 AM, the rain is falling at a rate of 0.2 cm/hr, so approximately 0.0033 cm of rain falls between 10 AM and 10:01 AM 7: 0.2 centimeters of rain have fallen 10 hours after midnight 8: at 2.7 hours past midnight, 10 centimeters of rain have fallen 9: at 6 hours after midnight, 10 centimeters of rain have fallen 10: at 6 hours after midnight, rain is falling at a rate of 2 cm/hr, so approximately 0.33 cm of rain falls in the next ten minutes 11: at 2 hours after midnight, rain is falling at a rate of 6 cm/hr, so approximately 1.00 cm of rain falls between 2 AM and 2:10 AM 12: at 10 AM, 2.7 centimeters of rain have fallen 13: 6 centimeters of rain have fallen 10 hours after midnight 14: when 2 centimeters have fallen, rain is accumulating at a rate of 1/6 cm/hr, so it takes approximately 36 min for the total accumulation to go from 2 to 2.1 cm 15: when 2 centimeters have fallen, rain is accumulating at a rate of 1/6 hr/cm, so it takes approximately 10.0 min for the total accumulation to go from 2 to 3 cm 16: when 6 centimeters of rain have fallen, it is accumulating at a rate of 1/2 cm/hr, so it takes approximately 12 min for the total accumulation to go from 6 to 6.1 cm", "answer_v1": [ "12", "13", "6", "16" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ] ], "problem_v2": "Let $f(t)$ be the number of centimeters of rainfall that has fallen since midnight, where $t$ is the time in hours. Match the following statements to their interpretations, given below.\n(a) $f(5)=3.1$: [ANS] (b) $f^{-1}(4)=6$: [ANS] (c) $f'(5)=0.5$: [ANS] (d) $(f^{-1})'(4)=2$: [ANS] Interpretations: 1: at 5 AM, the rain is falling at a rate of 0.5 cm/hr, so approximately 0.0083 cm of rain falls between 5 AM and 5:01 AM 2: at 5 AM, rain is falling at a rate of 3.1 cm/hr, so approximately 0.26 cm of rain falls in the next five minutes 3: at 0.5 hours after midnight, rain is falling at a rate of 5 cm/hr, so approximately 0.42 cm of rain falls in the next five minutes 4: at 5 AM, 3.1 centimeters of rain have fallen 5: when 2 centimeters of rain have fallen, rain is falling at a rate of 4 cm/hr, so it takes approximately 15.0 min for the total accumulation to go from 2 to 3 cm 6: at 6 hours after midnight, rain is falling at a rate of 4 cm/hr, so approximately 0.0667 cm of rain falls in the next minute 7: when 2 centimeters have fallen, rain is accumulating at a rate of 1/4 cm/hr, so it takes approximately 24 min for the total accumulation to go from 2 to 2.1 cm 8: when 4 centimeters of rain have fallen, it is accumulating at a rate of 1/2 cm/hr, so it takes approximately 12 min for the total accumulation to go from 4 to 4.1 cm 9: at 3.1 hours past midnight, 5 centimeters of rain have fallen 10: at 4 hours after midnight, 6 centimeters of rain have fallen 11: at 4 hours after midnight, rain is falling at a rate of 2 cm/hr, so approximately 0.33 cm of rain falls in the next ten minutes 12: 4 centimeters of rain have fallen 6 hours after midnight 13: 0.5 centimeters of rain have fallen 5 hours after midnight 14: at 2 hours after midnight, rain is falling at a rate of 4 cm/hr, so approximately 0.67 cm of rain falls between 2 AM and 2:10 AM 15: at 5 hours after midnight, 0.5 centimeters of rain have fallen 16: when 2 centimeters have fallen, rain is accumulating at a rate of 1/4 hr/cm, so it takes approximately 15.0 min for the total accumulation to go from 2 to 3 cm", "answer_v2": [ "4", "12", "1", "8" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ] ], "problem_v3": "Let $f(t)$ be the number of centimeters of rainfall that has fallen since midnight, where $t$ is the time in hours. Match the following statements to their interpretations, given below.\n(a) $f(7)=2.7$: [ANS] (b) $f^{-1}(5)=8$: [ANS] (c) $f'(7)=0.2$: [ANS] (d) $(f^{-1})'(5)=2$: [ANS] Interpretations: 1: at 5 hours after midnight, rain is falling at a rate of 2 cm/hr, so approximately 0.33 cm of rain falls in the next ten minutes 2: when 2 centimeters have fallen, rain is accumulating at a rate of 1/5 cm/hr, so it takes approximately 30 min for the total accumulation to go from 2 to 2.1 cm 3: at 2 hours after midnight, rain is falling at a rate of 5 cm/hr, so approximately 0.83 cm of rain falls between 2 AM and 2:10 AM 4: at 7 AM, the rain is falling at a rate of 0.2 cm/hr, so approximately 0.0033 cm of rain falls between 7 AM and 7:01 AM 5: at 2.7 hours past midnight, 7 centimeters of rain have fallen 6: when 5 centimeters of rain have fallen, it is accumulating at a rate of 1/2 cm/hr, so it takes approximately 12 min for the total accumulation to go from 5 to 5.1 cm 7: at 7 AM, 2.7 centimeters of rain have fallen 8: at 0.2 hours after midnight, rain is falling at a rate of 7 cm/hr, so approximately 0.58 cm of rain falls in the next five minutes 9: when 2 centimeters have fallen, rain is accumulating at a rate of 1/5 hr/cm, so it takes approximately 12.0 min for the total accumulation to go from 2 to 3 cm 10: 0.2 centimeters of rain have fallen 7 hours after midnight 11: at 8 hours after midnight, rain is falling at a rate of 5 cm/hr, so approximately 0.0833 cm of rain falls in the next minute 12: 5 centimeters of rain have fallen 8 hours after midnight 13: at 5 hours after midnight, 8 centimeters of rain have fallen 14: at 7 hours after midnight, 0.2 centimeters of rain have fallen 15: when 2 centimeters of rain have fallen, rain is falling at a rate of 5 cm/hr, so it takes approximately 12.0 min for the total accumulation to go from 2 to 3 cm 16: at 7 AM, rain is falling at a rate of 2.7 cm/hr, so approximately 0.23 cm of rain falls in the next five minutes", "answer_v3": [ "7", "12", "4", "6" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ], [ "1", "2", "3", "4", "5", "6", "7", "8", "9", "10", "11", "12", "13", "14", "15", "16" ] ] }, { "id": "Calculus_-_single_variable_0120", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Conceptual understanding of derivatives", "level": "2", "keywords": [ "derivative", "derivatives", "differentiable function" ], "problem_v1": "Find all points where f(x)=|20x-9| fails to be differentiable. Points $x=$ [ANS]\nNote: Enter points as a comma separated list, with smaller $x$ coordinates first. If no solution enter \"NA\".", "answer_v1": [ "9/20" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find all points where f(x)=|3x-14| fails to be differentiable. Points $x=$ [ANS]\nNote: Enter points as a comma separated list, with smaller $x$ coordinates first. If no solution enter \"NA\".", "answer_v2": [ "14/3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find all points where f(x)=|9x-10| fails to be differentiable. Points $x=$ [ANS]\nNote: Enter points as a comma separated list, with smaller $x$ coordinates first. If no solution enter \"NA\".", "answer_v3": [ "10/9" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0121", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Conceptual understanding of derivatives", "level": "3", "keywords": [ "derivative", "derivatives", "differentiable function", "continuity" ], "problem_v1": "Let f(x)=\\begin{cases} x^{2}-5x & x \\leq 9 \\\\ 4x & x>9 \\end{cases}\n(a) Is $f$ continuous at $x=9$ [ANS] (b) Is $f$ differentiable at $x=9$ [ANS] (c) If $f$ is differentiable at $x=9$ enter the value of $f'(9)$. If $f$ is NOT differentiable at $x=9$ enter the value of $f(9)$ instead. $f'(9)$ or $f(9)$=[ANS]", "answer_v1": [ "Yes", "No", "36" ], "answer_type_v1": [ "TF", "TF", "NV" ], "options_v1": [ [ "Yes", "No" ], [ "Yes", "No" ], [] ], "problem_v2": "Let f(x)=\\begin{cases} x^{2}-2x & x \\leq 7 \\\\ 5x & x>7 \\end{cases}\n(a) Is $f$ continuous at $x=7$ [ANS] (b) Is $f$ differentiable at $x=7$ [ANS] (c) If $f$ is differentiable at $x=7$ enter the value of $f'(7)$. If $f$ is NOT differentiable at $x=7$ enter the value of $f(7)$ instead. $f'(7)$ or $f(7)$=[ANS]", "answer_v2": [ "Yes", "No", "35" ], "answer_type_v2": [ "TF", "TF", "NV" ], "options_v2": [ [ "Yes", "No" ], [ "Yes", "No" ], [] ], "problem_v3": "Let f(x)=\\begin{cases} x^{2}-3x & x \\leq 7 \\\\ 4x & x>7 \\end{cases}\n(a) Is $f$ continuous at $x=7$ [ANS] (b) Is $f$ differentiable at $x=7$ [ANS] (c) If $f$ is differentiable at $x=7$ enter the value of $f'(7)$. If $f$ is NOT differentiable at $x=7$ enter the value of $f(7)$ instead. $f'(7)$ or $f(7)$=[ANS]", "answer_v3": [ "Yes", "No", "28" ], "answer_type_v3": [ "TF", "TF", "NV" ], "options_v3": [ [ "Yes", "No" ], [ "Yes", "No" ], [] ] }, { "id": "Calculus_-_single_variable_0122", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Conceptual understanding of derivatives", "level": "4", "keywords": [ "calculus", "differentiation" ], "problem_v1": "Find $a$ and $b$ so that the function f(x)=\\begin{cases} 7x^3-6x^2+6, & x <-2, \\\\ a x+b, & x \\geq-2 \\end{cases} is both continuous and differentiable.\n$a$=[ANS]\n$b$=[ANS]", "answer_v1": [ "12*7+4*6", "16*7+4*6+6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find $a$ and $b$ so that the function f(x)=\\begin{cases} 2x^3-8x^2+3, & x <-2, \\\\ a x+b, & x \\geq-2 \\end{cases} is both continuous and differentiable.\n$a$=[ANS]\n$b$=[ANS]", "answer_v2": [ "12*2+4*8", "16*2+4*8+3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find $a$ and $b$ so that the function f(x)=\\begin{cases} 4x^3-6x^2+3, & x <-2, \\\\ a x+b, & x \\geq-2 \\end{cases} is both continuous and differentiable.\n$a$=[ANS]\n$b$=[ANS]", "answer_v3": [ "12*4+4*6", "16*4+4*6+3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0123", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Conceptual understanding of derivatives", "level": "3", "keywords": [ "calculus", "differentiation" ], "problem_v1": "Let $f(x)=|x-7|+|x+6|.$\nFind the set of values of $x$ where $f$ is differentiable.\nAnswer [ANS]", "answer_v1": [ "(-infinity,-6) U (-6,7) U (7,infinity)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)=|x-1|+|x+9|.$\nFind the set of values of $x$ where $f$ is differentiable.\nAnswer [ANS]", "answer_v2": [ "(-infinity,-9) U (-9,1) U (1,infinity)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Let $f(x)=|x-3|+|x+6|.$\nFind the set of values of $x$ where $f$ is differentiable.\nAnswer [ANS]", "answer_v3": [ "(-infinity,-6) U (-6,3) U (3,infinity)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0124", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Conceptual understanding of derivatives", "level": "3", "keywords": [ "differential equation", "modeling" ], "problem_v1": "Recall that one model for population growth states that a population grows at a rate proportional to its size.\n(a) We begin with the differential equation ${\\frac{dP}{dt}=\\frac{1}{2} P}$. Find an equilibrium solution: $P=$ [ANS]\nIs this equilibrium solution stable or unstable? [ANS] A. stable B. unstable\nDescribe the long-term behavior of the solution to $\\frac{dP}{dt}=\\frac{1}{2}P$. [ANS] A. The value of $P$ increases without bound. B. The value of $P$ oscillates and does not approach a limit. C. The value of $P$ approaches a nonzero constant. D. The value of $P$ approaches zero. E. None of the above\n(b) Let's now consider a modified differential equation given by ${\\frac{dP}{dt}=\\frac{1}{2} P(3-P)}$. Find a stable equilibrium solution: $P=$ [ANS]\nFind an unstable equilibrium solution: $P=$ [ANS]\nIf $P(0)$ is positive, describe the long-term behavior of the solution to $\\frac{dP}{dt}=\\frac{1}{2}P(3-P)$. [ANS] A. The value of $P$ increases without bound. B. The value of $P$ approaches zero. C. The value of $P$ approaches a nonzero constant. D. The value of $P$ oscillates and does not approach a limit. E. None of the above", "answer_v1": [ "0", "B", "A", "3", "0", "C" ], "answer_type_v1": [ "NV", "MCS", "MCS", "NV", "NV", "MCS" ], "options_v1": [ [], [ "A", "B" ], [ "A", "B", "C", "D", "E" ], [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Recall that one model for population growth states that a population grows at a rate proportional to its size.\n(a) We begin with the differential equation ${\\frac{dP}{dt}=\\frac{1}{2} P}$. Find an equilibrium solution: $P=$ [ANS]\nIs this equilibrium solution stable or unstable? [ANS] A. stable B. unstable\nDescribe the long-term behavior of the solution to $\\frac{dP}{dt}=\\frac{1}{2}P$. [ANS] A. The value of $P$ approaches a nonzero constant. B. The value of $P$ oscillates and does not approach a limit. C. The value of $P$ approaches zero. D. The value of $P$ increases without bound. E. None of the above\n(b) Let's now consider a modified differential equation given by ${\\frac{dP}{dt}=\\frac{1}{2} P(3-P)}$. Find a stable equilibrium solution: $P=$ [ANS]\nFind an unstable equilibrium solution: $P=$ [ANS]\nIf $P(0)$ is positive, describe the long-term behavior of the solution to $\\frac{dP}{dt}=\\frac{1}{2}P(3-P)$. [ANS] A. The value of $P$ increases without bound. B. The value of $P$ approaches zero. C. The value of $P$ oscillates and does not approach a limit. D. The value of $P$ approaches a nonzero constant. E. None of the above", "answer_v2": [ "0", "B", "D", "3", "0", "D" ], "answer_type_v2": [ "NV", "MCS", "MCS", "NV", "NV", "MCS" ], "options_v2": [ [], [ "A", "B" ], [ "A", "B", "C", "D", "E" ], [], [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Recall that one model for population growth states that a population grows at a rate proportional to its size.\n(a) We begin with the differential equation ${\\frac{dP}{dt}=\\frac{1}{2} P}$. Find an equilibrium solution: $P=$ [ANS]\nIs this equilibrium solution stable or unstable? [ANS] A. stable B. unstable\nDescribe the long-term behavior of the solution to $\\frac{dP}{dt}=\\frac{1}{2}P$. [ANS] A. The value of $P$ approaches a nonzero constant. B. The value of $P$ increases without bound. C. The value of $P$ oscillates and does not approach a limit. D. The value of $P$ approaches zero. E. None of the above\n(b) Let's now consider a modified differential equation given by ${\\frac{dP}{dt}=\\frac{1}{2} P(3-P)}$. Find a stable equilibrium solution: $P=$ [ANS]\nFind an unstable equilibrium solution: $P=$ [ANS]\nIf $P(0)$ is positive, describe the long-term behavior of the solution to $\\frac{dP}{dt}=\\frac{1}{2}P(3-P)$. [ANS] A. The value of $P$ increases without bound. B. The value of $P$ approaches a nonzero constant. C. The value of $P$ oscillates and does not approach a limit. D. The value of $P$ approaches zero. E. None of the above", "answer_v3": [ "0", "B", "B", "3", "0", "B" ], "answer_type_v3": [ "NV", "MCS", "MCS", "NV", "NV", "MCS" ], "options_v3": [ [], [ "A", "B" ], [ "A", "B", "C", "D", "E" ], [], [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Calculus_-_single_variable_0125", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Conceptual understanding of derivatives", "level": "4", "keywords": [ "differential equation", "slope field" ], "problem_v1": "Consider the differential equation ${\\frac{dy}{dt}=t-2}$. Suppose that $y(t)$ is a solution to this differential equation corresponding to the initial value $y(0)=1$. On the graph of $y$ versus $t$, what is the slope of the tangent line to the curve at the point $(0,1)$? Slope=[ANS]\nSuppose (instead) that $y(t)$ is a solution to this differential equation corresponding to the initial value $y(0)=5$. On the graph of $y$ versus $t$, what is the slope of the tangent line to the curve at the point $(0,5)$? Slope=[ANS]\nSuppose that $y(t)$ is a solution to this differential equation that passes through the point $(684, 149)$. On the graph of $y$ versus $t$, what is the slope of the tangent line to the curve at the point $(684,149)$? Slope=[ANS]\nIf we only know the differential equation, (but not any points on the solution curve), which of the following could be a solution to the equation? Select all that apply. [ANS] A. $y=\\frac{1}{2}(t-2)^2$ B. $y=t^2/2-2t+1$ C. $y=t^2/2-2t+149$ D. $y=t^2/2-t$ E. None of the above\nGiven this differential equation (but not an initial value or the solution) what information is sufficient to determine the slope of the tangent line to the solution curve at a point on the curve? [ANS] A. The $t$-coordinate of the point. B. The $y$-coordinate of the point. C. The slope of the solution curve at $t=0$. D. None of the above", "answer_v1": [ "-2", "-2", "684-2", "ABC", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "MCM", "MCS" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Consider the differential equation ${\\frac{dy}{dt}=t-2}$. Suppose that $y(t)$ is a solution to this differential equation corresponding to the initial value $y(0)=1$. On the graph of $y$ versus $t$, what is the slope of the tangent line to the curve at the point $(0,1)$? Slope=[ANS]\nSuppose (instead) that $y(t)$ is a solution to this differential equation corresponding to the initial value $y(0)=5$. On the graph of $y$ versus $t$, what is the slope of the tangent line to the curve at the point $(0,5)$? Slope=[ANS]\nSuppose that $y(t)$ is a solution to this differential equation that passes through the point $(283, 209)$. On the graph of $y$ versus $t$, what is the slope of the tangent line to the curve at the point $(283,209)$? Slope=[ANS]\nIf we only know the differential equation, (but not any points on the solution curve), which of the following could be a solution to the equation? Select all that apply. [ANS] A. $y=t^2/2-t$ B. $y=t^2/2-2t+1$ C. $y=t^2/2-2t+209$ D. $y=\\frac{1}{2}(t-2)^2$ E. None of the above\nGiven this differential equation (but not an initial value or the solution) what information is sufficient to determine the slope of the tangent line to the solution curve at a point on the curve? [ANS] A. The slope of the solution curve at $t=0$. B. The $y$-coordinate of the point. C. The $t$-coordinate of the point. D. None of the above", "answer_v2": [ "-2", "-2", "283-2", "BCD", "C" ], "answer_type_v2": [ "NV", "NV", "NV", "MCM", "MCS" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Consider the differential equation ${\\frac{dy}{dt}=t-2}$. Suppose that $y(t)$ is a solution to this differential equation corresponding to the initial value $y(0)=1$. On the graph of $y$ versus $t$, what is the slope of the tangent line to the curve at the point $(0,1)$? Slope=[ANS]\nSuppose (instead) that $y(t)$ is a solution to this differential equation corresponding to the initial value $y(0)=5$. On the graph of $y$ versus $t$, what is the slope of the tangent line to the curve at the point $(0,5)$? Slope=[ANS]\nSuppose that $y(t)$ is a solution to this differential equation that passes through the point $(421, 153)$. On the graph of $y$ versus $t$, what is the slope of the tangent line to the curve at the point $(421,153)$? Slope=[ANS]\nIf we only know the differential equation, (but not any points on the solution curve), which of the following could be a solution to the equation? Select all that apply. [ANS] A. $y=t^2/2-2t+1$ B. $y=t^2/2-2t+153$ C. $y=t^2/2-t$ D. $y=\\frac{1}{2}(t-2)^2$ E. None of the above\nGiven this differential equation (but not an initial value or the solution) what information is sufficient to determine the slope of the tangent line to the solution curve at a point on the curve? [ANS] A. The $t$-coordinate of the point. B. The $y$-coordinate of the point. C. The slope of the solution curve at $t=0$. D. None of the above", "answer_v3": [ "-2", "-2", "421-2", "ABD", "A" ], "answer_type_v3": [ "NV", "NV", "NV", "MCM", "MCS" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_single_variable_0126", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "derivative", "tangent line", "calculus", "derivative", "tangent line" ], "problem_v1": "Let $f(x)=x+\\sqrt{x}$. The equation for the tangent line at the point $(49, 56)$ is $y=m(x-49)+56$, where $m$ is a constant. Find the value of $m$.\nAnswer: [ANS]", "answer_v1": [ "1 + 0.5/7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)=x+\\sqrt{x}$. The equation for the tangent line at the point $(1, 2)$ is $y=m(x-1)+2$, where $m$ is a constant. Find the value of $m$.\nAnswer: [ANS]", "answer_v2": [ "1 + 0.5/1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(x)=x+\\sqrt{x}$. The equation for the tangent line at the point $(9, 12)$ is $y=m(x-9)+12$, where $m$ is a constant. Find the value of $m$.\nAnswer: [ANS]", "answer_v3": [ "1 + 0.5/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0127", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "derivatives", "Derivative", "Power" ], "problem_v1": "Let $f(x)= \\frac {6x^2+6x+5} {\\sqrt{x}}$. Find the following: $f'(x)=$ [ANS]\n$f'(9)=$ [ANS]", "answer_v1": [ "3/2*6*x^{1/2}+6/2*x^{-1/2}-5/2*x^{-3/2}", "1507/54" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)= \\frac {2x^2+8x+2} {\\sqrt{x}}$. Find the following: $f'(x)=$ [ANS]\n$f'(9)=$ [ANS]", "answer_v2": [ "3/2*2*x^{1/2}+8/2*x^{-1/2}-2/2*x^{-3/2}", "278/27" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)= \\frac {3x^2+6x+3} {\\sqrt{x}}$. Find the following: $f'(x)=$ [ANS]\n$f'(9)=$ [ANS]", "answer_v3": [ "3/2*3*x^{1/2}+6/2*x^{-1/2}-3/2*x^{-3/2}", "130/9" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0128", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "derivatives", "tangent line", "min/max", "calculus", "derivatives" ], "problem_v1": "For what values of $x$ does the graph of f(x)=7x^{3}-28.35x^{2}+27.983x+37.8 have a horizontal tangent?\nAnswer $=$ [ANS]\nNote: Enter the $x$ values as a comma-separated list.", "answer_v1": [ "(2.05, 0.65)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "For what values of $x$ does the graph of f(x)=3.2x^{3}-0.72x^{2}-112.56x-12.96 have a horizontal tangent?\nAnswer $=$ [ANS]\nNote: Enter the $x$ values as a comma-separated list.", "answer_v2": [ "(-3.35, 3.5)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "For what values of $x$ does the graph of f(x)=4.2x^{3}+4.095x^{2}-16.065x+5.04 have a horizontal tangent?\nAnswer $=$ [ANS]\nNote: Enter the $x$ values as a comma-separated list.", "answer_v3": [ "(-1.5, 0.85)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0129", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "derivatives", "Derivative", "Polynomial" ], "problem_v1": "Let $f(t)=6 t^{-4}$. Determine $f'(t)$.\nAnswer: [ANS]\nFind $f'(2)$.\nAnswer: [ANS]", "answer_v1": [ "6*(-4)*t^{-5}", "6*(-4)*2^{-5}" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(t)=2 t^{-7}$. Determine $f'(t)$.\nAnswer: [ANS]\nFind $f'(2)$.\nAnswer: [ANS]", "answer_v2": [ "2*(-7)*t^{-8}", "2*(-7)*2^{-8}" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(t)=3 t^{-4}$. Determine $f'(t)$.\nAnswer: [ANS]\nFind $f'(2)$.\nAnswer: [ANS]", "answer_v3": [ "3*(-4)*t^{-5}", "3*(-4)*2^{-5}" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0130", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "4", "keywords": [ "tangent line", "normal line", "derivative", "Derivatives" ], "problem_v1": "At what point does the normal to $y=3+x+2x^{2}$ at $(1, 6)$ intersect the parabola a second time?\nAnswer: [ANS]\nNote: You should enter a cartesian coordinate.\nThe normal line is perpendicular to the tangent line. If two lines are perpendicular their slopes are negative reciprocals--i.e. if the slope of the first line is $m$ then the slope of the second line is $-1/m$", "answer_v1": [ "(-1.6,6.52)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "At what point does the normal to $y=5x-5-x^{2}$ at $(1,-1)$ intersect the parabola a second time?\nAnswer: [ANS]\nNote: You should enter a cartesian coordinate.\nThe normal line is perpendicular to the tangent line. If two lines are perpendicular their slopes are negative reciprocals--i.e. if the slope of the first line is $m$ then the slope of the second line is $-1/m$", "answer_v2": [ "(4.33333,-2.11111)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "At what point does the normal to $y=x-2+2x^{2}$ at $(1, 1)$ intersect the parabola a second time?\nAnswer: [ANS]\nNote: You should enter a cartesian coordinate.\nThe normal line is perpendicular to the tangent line. If two lines are perpendicular their slopes are negative reciprocals--i.e. if the slope of the first line is $m$ then the slope of the second line is $-1/m$", "answer_v3": [ "(-1.6,1.52)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0131", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "derivatives" ], "problem_v1": "If $f(x)=6x^{2}-8x-25$, find $f'(a)$.\nAnswer: [ANS]", "answer_v1": [ "6*2*a-8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If $f(x)=2x^{2}-12x-6$, find $f'(a)$.\nAnswer: [ANS]", "answer_v2": [ "2*2*a-12" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If $f(x)=3x^{2}-8x-12$, find $f'(a)$.\nAnswer: [ANS]", "answer_v3": [ "3*2*a-8" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0132", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [], "problem_v1": "Compute the derivative of the given function. f(x)=6x^{\\pi}+7.5x^{7.7}+\\pi^{7.7}. Note: Use pi for $\\pi$ in your answer. $f'(x)=$ [ANS].", "answer_v1": [ "6*pi x^{pi - 1} + 57.75 x^{6.7}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Compute the derivative of the given function. f(x)=-9x^{\\pi}+4.1x^{1.7}+\\pi^{1.7}. Note: Use pi for $\\pi$ in your answer. $f'(x)=$ [ANS].", "answer_v2": [ "-9*pi x^{pi - 1} + 6.97 x^{0.7}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Compute the derivative of the given function. f(x)=-6x^{\\pi}+5.9x^{3.9}+\\pi^{3.9}. Note: Use pi for $\\pi$ in your answer. $f'(x)=$ [ANS].", "answer_v3": [ "-6*pi x^{pi - 1} + 23.01 x^{2.9}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0133", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [], "problem_v1": "Let $f(x)=5x^2+20x-5$. Find the $x$-value(s) where the graph of the function has a horizontal tangent line. Separate multiple answers with commas. Enter DNE if there are no such $x$. $x=$ [ANS]", "answer_v1": [ "-2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)=2x^2-2x-15$. Find the $x$-value(s) where the graph of the function has a horizontal tangent line. Separate multiple answers with commas. Enter DNE if there are no such $x$. $x=$ [ANS]", "answer_v2": [ "0.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(x)=3x^2+6x-4$. Find the $x$-value(s) where the graph of the function has a horizontal tangent line. Separate multiple answers with commas. Enter DNE if there are no such $x$. $x=$ [ANS]", "answer_v3": [ "-1" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0134", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "calculus", "derivatives", "power rule", "algebraic functions" ], "problem_v1": "Use the Power Rule to compute the derivative: $\\frac{d}{dt}t^{2/3}|_{t=5}=$ [ANS]", "answer_v1": [ "0.389869" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the Power Rule to compute the derivative: $\\frac{d}{dt}t^{2/3}|_{t=2}=$ [ANS]", "answer_v2": [ "0.529134" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the Power Rule to compute the derivative: $\\frac{d}{dt}t^{2/3}|_{t=3}=$ [ANS]", "answer_v3": [ "0.462241" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0135", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "calculus", "derivatives", "power rule", "algebraic functions" ], "problem_v1": "Find the derivative of the function $f(x)=11x^{-3}+x^{2}+14$.\n$f'(x)=$ [ANS]", "answer_v1": [ "2*x-11*3*x^{-4}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of the function $f(x)=3x^{-3}+x^{2}+14$.\n$f'(x)=$ [ANS]", "answer_v2": [ "2*x-3*3*x^{-4}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of the function $f(x)=6x^{-3}+x^{2}+14$.\n$f'(x)=$ [ANS]", "answer_v3": [ "2*x-6*3*x^{-4}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0136", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "calculus", "derivatives", "tangent lines" ], "problem_v1": "Find all values of $x$ where the tangent lines to $y=x^{7}$ and $y=x^{8}$ are parallel. $x=$ [ANS]", "answer_v1": [ "(0, 7/8)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all values of $x$ where the tangent lines to $y=x^{2}$ and $y=x^{3}$ are parallel. $x=$ [ANS]", "answer_v2": [ "(0, 2/3)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all values of $x$ where the tangent lines to $y=x^{4}$ and $y=x^{5}$ are parallel. $x=$ [ANS]", "answer_v3": [ "(0, 4/5)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0137", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "calculus", "derivatives", "power rule", "polynomial functions" ], "problem_v1": "Determine coefficients $a$ and $b$ such that $p(x)=x^{2}+ax+b$ satisfies $p(1)=8$ and $p'(1)=7$.\n$a=$ [ANS]\n$b=$ [ANS]", "answer_v1": [ "5", "2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Determine coefficients $a$ and $b$ such that $p(x)=x^{2}+ax+b$ satisfies $p(1)=1$ and $p'(1)=-7$.\n$a=$ [ANS]\n$b=$ [ANS]", "answer_v2": [ "-9", "9" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Determine coefficients $a$ and $b$ such that $p(x)=x^{2}+ax+b$ satisfies $p(1)=-1$ and $p'(1)=-2$.\n$a=$ [ANS]\n$b=$ [ANS]", "answer_v3": [ "-4", "2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0138", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "derivative", "polynomial" ], "problem_v1": "Let $f(x)=x^{-5}$. Find $f'(x)$. $f'(x)$=[ANS]", "answer_v1": [ "-5*x^{-6}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)=x^{-15}$. Find $f'(x)$. $f'(x)$=[ANS]", "answer_v2": [ "-15*x^{-16}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $f(x)=x^{-11}$. Find $f'(x)$. $f'(x)$=[ANS]", "answer_v3": [ "-11*x^{-12}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0139", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "derivative", "polynomial" ], "problem_v1": "Find ${\\frac{d}{dx}\\left(\\frac{1}{x^{13}} \\right)}$. ${\\frac{d}{dx}\\left(\\frac{1}{x^{13}} \\right)}$=[ANS]", "answer_v1": [ "-(13*x^12/[(x^13)^2])" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find ${\\frac{d}{dx}\\left(\\frac{1}{x^{3}} \\right)}$. ${\\frac{d}{dx}\\left(\\frac{1}{x^{3}} \\right)}$=[ANS]", "answer_v2": [ "-(3*x^2/[(x^3)^2])" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find ${\\frac{d}{dx}\\left(\\frac{1}{x^{7}} \\right)}$. ${\\frac{d}{dx}\\left(\\frac{1}{x^{7}} \\right)}$=[ANS]", "answer_v3": [ "-(7*x^6/[(x^7)^2])" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0140", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "derivative' 'limit' 'tangent" ], "problem_v1": "Find the $x$-coordinates of all points where the graph of $f(x)=2x^{3}+12x^{2}-126x+17$ has a horizontal tangent line. $x$=[ANS]\nHint: What is the slope of a horizontal line? Note: If there is more that one $x$, give the answer as a list separated by commas (e.g.: 1,2). If there are no such $x$, enter none.", "answer_v1": [ "(-7, 3)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find the $x$-coordinates of all points where the graph of $f(x)=2x^{3}+3x^{2}-120x+7$ has a horizontal tangent line. $x$=[ANS]\nHint: What is the slope of a horizontal line? Note: If there is more that one $x$, give the answer as a list separated by commas (e.g.: 1,2). If there are no such $x$, enter none.", "answer_v2": [ "(-5, 4)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find the $x$-coordinates of all points where the graph of $f(x)=2x^{3}+6x^{2}-90x+9$ has a horizontal tangent line. $x$=[ANS]\nHint: What is the slope of a horizontal line? Note: If there is more that one $x$, give the answer as a list separated by commas (e.g.: 1,2). If there are no such $x$, enter none.", "answer_v3": [ "(-5, 3)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0141", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "4", "keywords": [ "derivative' 'limit' 'tangent", "derivatives", "tangent line" ], "problem_v1": "On a separate piece of paper, sketch the graph of the parabola $f(x)=x^{2}+8$. On the same set of axes, plot the point $(0,-4)$. Notice that there are two points on the parabola [of the form $(\\pm a, f(a))$] at which the tangent lines to the parabola pass through $(0,-4)$. Draw these two points and these two lines. The point with positive $x$-coordinate that is on the parabola and has its tangent line to the parabola passing through $(0,-4)$ is [ANS].", "answer_v1": [ "(3.4641,20)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "On a separate piece of paper, sketch the graph of the parabola $f(x)=x^{2}+3$. On the same set of axes, plot the point $(0,-6)$. Notice that there are two points on the parabola [of the form $(\\pm a, f(a))$] at which the tangent lines to the parabola pass through $(0,-6)$. Draw these two points and these two lines. The point with positive $x$-coordinate that is on the parabola and has its tangent line to the parabola passing through $(0,-6)$ is [ANS].", "answer_v2": [ "(3,12)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "On a separate piece of paper, sketch the graph of the parabola $f(x)=x^{2}+5$. On the same set of axes, plot the point $(0,-5)$. Notice that there are two points on the parabola [of the form $(\\pm a, f(a))$] at which the tangent lines to the parabola pass through $(0,-5)$. Draw these two points and these two lines. The point with positive $x$-coordinate that is on the parabola and has its tangent line to the parabola passing through $(0,-5)$ is [ANS].", "answer_v3": [ "(3.16228,15)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0142", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "Derivatives", "tangent line", "derivatives", "Product", "Quotient", "Differentiation" ], "problem_v1": "Let $h(x)=5-4x^{3}$, $h'(2)=$ [ANS]\nUse this to find the equation of the tangent line to the curve $y=5-4x^{3}$ at the point $(2,-27)$ and write your answer in the form: $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept. [ANS]", "answer_v1": [ "-48", "y = -48*x + 69" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let $h(x)=7-2x^{3}$, $h'(1)=$ [ANS]\nUse this to find the equation of the tangent line to the curve $y=7-2x^{3}$ at the point $(1, 5)$ and write your answer in the form: $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept. [ANS]", "answer_v2": [ "-6", "y = -6*x + 11" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let $h(x)=5-2x^{3}$, $h'(1)=$ [ANS]\nUse this to find the equation of the tangent line to the curve $y=5-2x^{3}$ at the point $(1, 3)$ and write your answer in the form: $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept. [ANS]", "answer_v3": [ "-6", "y = -6*x + 9" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0143", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "Derivative", "Polynomial" ], "problem_v1": "Find $\\>y'(1)\\>$ if $\\>y=5x^{2}+2x+3$.\nAnswer: [ANS]", "answer_v1": [ "12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $\\>y'(1)\\>$ if $\\>y=9x-9x^{2}-7$.\nAnswer: [ANS]", "answer_v2": [ "-9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $\\>y'(1)\\>$ if $\\>y=2x-4x^{2}-5$.\nAnswer: [ANS]", "answer_v3": [ "-6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0144", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "4", "keywords": [ "derivatives", "tangent line", "min/max", "calculus", "derivatives" ], "problem_v1": "For what values of x is the tangent line of the graph of f(x)=8x^{3}-36x^{2}+47x+48 parallel to the line $y=-\\left(x+0.8\\right)$? (If there are multiple values then separate them with commas.) $x=$ [ANS]", "answer_v1": [ "(2, 1)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "For what values of x is the tangent line of the graph of f(x)=2x^{3}-94x-6 parallel to the line $y=2x-0.8$? (If there are multiple values then separate them with commas.) $x=$ [ANS]", "answer_v2": [ "(-4, 4)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "For what values of x is the tangent line of the graph of f(x)=4x^{3}+6x^{2}-25x parallel to the line $y=-\\left(x+0.8\\right)$? (If there are multiple values then separate them with commas.) $x=$ [ANS]", "answer_v3": [ "(-2, 1)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0145", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "calculus", "derivative", "powers", "polynomials" ], "problem_v1": "Find the derivative of f(z)=-\\frac{1}{z^{12.5}}. $f'(z)=$ [ANS]", "answer_v1": [ "12.5*z^{-1*(12.5+1)}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of f(z)=-\\frac{1}{z^{3.2}}. $f'(z)=$ [ANS]", "answer_v2": [ "3.2*z^{-1*(3.2+1)}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of f(z)=-\\frac{1}{z^{6.4}}. $f'(z)=$ [ANS]", "answer_v3": [ "6.4*z^{-1*(6.4+1)}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0146", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "calculus", "derivative", "powers", "polynomials" ], "problem_v1": "Find the derivative of $y=ax^{12}+bx^{5}+c$. Assume that $a$, $b$ and $c$ are constants. ${dy\\over dx}=$ [ANS]", "answer_v1": [ "12*a*x^11+5*b*x^4" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of $y=ax^{3}+bx^{1}+c$. Assume that $a$, $b$ and $c$ are constants. ${dy\\over dx}=$ [ANS]", "answer_v2": [ "3*a*x^2+1*b*x^0" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of $y=ax^{6}+bx^{2}+c$. Assume that $a$, $b$ and $c$ are constants. ${dy\\over dx}=$ [ANS]", "answer_v3": [ "6*a*x^5+2*b*x^1" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0147", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "calculus", "derivative", "powers", "polynomials", "Product", "Quotient", "Differentiation", "derivative", "power rule" ], "problem_v1": "Find the derivative of $y=\\sqrt{x}(x^{4}+6)$. ${dy\\over dx}=$ [ANS]", "answer_v1": [ "(2*4+1)/2*x^{(2*4-1)/2}+6/2*x^{-0.5}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of $y=\\sqrt{x}(x+9)$. ${dy\\over dx}=$ [ANS]", "answer_v2": [ "(2*1+1)/2*x^{(2*1-1)/2}+9/2*x^{-0.5}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of $y=\\sqrt{x}(x^{2}+6)$. ${dy\\over dx}=$ [ANS]", "answer_v3": [ "(2*2+1)/2*x^{(2*2-1)/2}+6/2*x^{-0.5}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0148", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "calculus", "derivative", "powers", "polynomials" ], "problem_v1": "Find the derivative of $h(w)=-12 w^{-10}+10\\sqrt{w}$. $h'(w)=$ [ANS]", "answer_v1": [ "12*10*w^{-11}+10/2*w^{-0.5}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of $h(w)=-3 w^{-15}+4\\sqrt{w}$. $h'(w)=$ [ANS]", "answer_v2": [ "3*15*w^{-16}+4/2*w^{-0.5}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of $h(w)=-6 w^{-10}+5\\sqrt{w}$. $h'(w)=$ [ANS]", "answer_v3": [ "6*10*w^{-11}+5/2*w^{-0.5}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0149", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "calculus", "derivative", "powers", "polynomials" ], "problem_v1": "Find the derivative of $V={8 \\over 7} \\pi r^6 b$.\nAssume that $b$ is a constant. ${dV\\over dr}=$ [ANS]", "answer_v1": [ "8*6/7*pi*r^5*b" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of $V={3 \\over 2} \\pi r^9 b$.\nAssume that $b$ is a constant. ${dV\\over dr}=$ [ANS]", "answer_v2": [ "3*9/2*pi*r^8*b" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of $V={5 \\over 4} \\pi r^6 b$.\nAssume that $b$ is a constant. ${dV\\over dr}=$ [ANS]", "answer_v3": [ "5*6/4*pi*r^5*b" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0150", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "calculus", "derivative", "powers", "polynomials" ], "problem_v1": "Given a power function of the form $f(x)=ax^n$, with $f'(2)=30$ and $f'(6)=2430$, find $n$ and $a$.\n$n=$ [ANS]\n$a=$ [ANS]", "answer_v1": [ "5", "0.375" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Given a power function of the form $f(x)=ax^n$, with $f'(3)=6$ and $f'(6)=12$, find $n$ and $a$.\n$n=$ [ANS]\n$a=$ [ANS]", "answer_v2": [ "2", "1" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Given a power function of the form $f(x)=ax^n$, with $f'(2)=12$ and $f'(4)=48$, find $n$ and $a$.\n$n=$ [ANS]\n$a=$ [ANS]", "answer_v3": [ "3", "1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0151", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "calculus", "derivative", "powers", "polynomials" ], "problem_v1": "If $f(x)=x^3+6x^2-288x+16$, find analytically all values of $x$ for which $f'(x)=0$. (Enter your answer as a comma separated list of numbers, e.g.,-1,0,2) $x=$ [ANS]", "answer_v1": [ "(8, -12)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "If $f(x)=x^3+6x^2-36x+4$, find analytically all values of $x$ for which $f'(x)=0$. (Enter your answer as a comma separated list of numbers, e.g.,-1,0,2) $x=$ [ANS]", "answer_v2": [ "(2, -6)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "If $f(x)=x^3+6x^2-96x+7$, find analytically all values of $x$ for which $f'(x)=0$. (Enter your answer as a comma separated list of numbers, e.g.,-1,0,2) $x=$ [ANS]", "answer_v3": [ "(4, -8)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0152", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "calculus", "derivative", "powers", "polynomials" ], "problem_v1": "Find the derivative of $h(q)=\\frac{1}{\\sqrt[8]{q}}$. $h'(q)=$ [ANS]", "answer_v1": [ "-(1/8)*q^{-1.125}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of $h(q)=\\frac{1}{\\sqrt{q}}$. $h'(q)=$ [ANS]", "answer_v2": [ "-(1/2)*q^{-1.5}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of $h(q)=\\frac{1}{\\sqrt[4]{q}}$. $h'(q)=$ [ANS]", "answer_v3": [ "-(1/4)*q^{-1.25}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0153", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "calculus", "derivative", "powers", "polynomials" ], "problem_v1": "Find the derivative of $ h(x)=\\frac{a x+b}{p}$. Assume that $a$, $b$ and $p$ are constants. $h'(x)=$ [ANS]", "answer_v1": [ "a/p" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of $ h(z)=\\frac{a z+b}{c}$. Assume that $a$, $b$ and $c$ are constants. $h'(z)=$ [ANS]", "answer_v2": [ "a/c" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of $ h(x)=\\frac{a x+b}{k}$. Assume that $a$, $b$ and $k$ are constants. $h'(x)=$ [ANS]", "answer_v3": [ "a/k" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0154", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "calculus", "derivative", "instantaneous velocity", "rate of change" ], "problem_v1": "Find the derivative of $g(t)=7 t^2+6 t$ at $t=2$ algebraically. $g'(2)=$ [ANS]", "answer_v1": [ "34" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of $g(t)=t^2+9 t$ at $t=-7$ algebraically. $g'(-7)=$ [ANS]", "answer_v2": [ "-5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of $g(t)=3 t^2+6 t$ at $t=-4$ algebraically. $g'(-4)=$ [ANS]", "answer_v3": [ "-18" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0155", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "derivative", "derivatives", "slope", "tangent", "tangent line", "tangent lines" ], "problem_v1": "Let $f(x)=x^{9}+x+3$. Find $\\lim_{w\\to1} \\frac{f'(w)-f'(1)}{w-1}$=[ANS]", "answer_v1": [ "72" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)=x^{6}+5x-7$. Find $\\lim_{w\\to-1} \\frac{f'(w)-f'(-1)}{w+1}$=[ANS]", "answer_v2": [ "30" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(x)=x^{7}+x-5$. Find $\\lim_{w\\to-1} \\frac{f'(w)-f'(-1)}{w+1}$=[ANS]", "answer_v3": [ "-42" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0156", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "calculus", "derivative", "tangent line" ], "problem_v1": "The line $y=-5x-1075$ is tangent to the graph of y=2x^3-6x^2-215x+3 at the point [ANS].", "answer_v1": [ "(7,-1110)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "The line $y=-3x+1207$ is tangent to the graph of y=2x^3+21x^2-111x-8 at the point [ANS].", "answer_v2": [ "(-9,1234)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "The line $y=-6x+786$ is tangent to the graph of y=2x^3+12x^2-132x+2 at the point [ANS].", "answer_v3": [ "(-7,828)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0157", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "calculus", "limits", "derivatives" ], "problem_v1": "Find an equation of the line tangent to the graph of y=\\frac{5}{x^{2}} at the point $(4, 5/16)$.\nTangent line: $y=$ [ANS]", "answer_v1": [ "5/16 + -0.15625*(x-4)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an equation of the line tangent to the graph of y=\\frac{-9}{x^{2}} at the point $(5,-9/25)$.\nTangent line: $y=$ [ANS]", "answer_v2": [ "-9/25 + 0.144*(x-5)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an equation of the line tangent to the graph of y=\\frac{-5}{x^{2}} at the point $(4,-5/16)$.\nTangent line: $y=$ [ANS]", "answer_v3": [ "-5/16 + 0.15625*(x-4)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0158", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "secant", "difference quotient" ], "problem_v1": "Does the parabola $y=7x^{2}-11x+1$ have a tangent whose slope is $-1$? (Hint: sketch a graph of it.) If so, find the point of tangency and an equation for the tangent line. If not, enter NONE in both answer blanks, and be sure that you can explain your answer.\nPoint $(x,y)$=[ANS] (For example, enter (1,2) including the parentheses.) Tangent line: $y$=[ANS]", "answer_v1": [ "(0.714286,-3.28571)", "-3.28571-(x-0.714286)" ], "answer_type_v1": [ "OL", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Does the parabola $y=2x^{2}-7x-4$ have a tangent whose slope is $-1$? (Hint: sketch a graph of it.) If so, find the point of tangency and an equation for the tangent line. If not, enter NONE in both answer blanks, and be sure that you can explain your answer.\nPoint $(x,y)$=[ANS] (For example, enter (1,2) including the parentheses.) Tangent line: $y$=[ANS]", "answer_v2": [ "(1.5,-10)", "-10-(x-1.5)" ], "answer_type_v2": [ "OL", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Does the parabola $y=4x^{2}-9x-2$ have a tangent whose slope is $-1$? (Hint: sketch a graph of it.) If so, find the point of tangency and an equation for the tangent line. If not, enter NONE in both answer blanks, and be sure that you can explain your answer.\nPoint $(x,y)$=[ANS] (For example, enter (1,2) including the parentheses.) Tangent line: $y$=[ANS]", "answer_v3": [ "(1,-7)", "-7-(x-1)" ], "answer_type_v3": [ "OL", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0159", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "calculus" ], "problem_v1": "This a simple exercise in computing derivatives of polynomials. The derivative of p(x)=6x^2+6x+8 is $p'(x)=$ [ANS]. The derivative of q(x)=4x^5-6x^4+6x^3-4x^2+4x+7 is $q'(x)=$ [ANS].", "answer_v1": [ "2*6 x + 6", "5*4*x^4 - 4*6 * x^3 + 3*6* x^2-2*4 x + 4" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "This a simple exercise in computing derivatives of polynomials. The derivative of p(x)=3x^2+9x+2 is $p'(x)=$ [ANS]. The derivative of q(x)=6x^5-4x^4+3x^3-4x^2+9x+4 is $q'(x)=$ [ANS].", "answer_v2": [ "2*3 x + 9", "5*6*x^4 - 4*4 * x^3 + 3*3* x^2-2*4 x + 9" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "This a simple exercise in computing derivatives of polynomials. The derivative of p(x)=4x^2+6x+4 is $p'(x)=$ [ANS]. The derivative of q(x)=8x^5-9x^4+8x^3-4x^2+3x+6 is $q'(x)=$ [ANS].", "answer_v3": [ "2*4 x + 6", "5*8*x^4 - 4*9 * x^3 + 3*8* x^2-2*4 x + 3" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0160", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "calculus", "differentiation", "Product", "Quotient", "Differentiation" ], "problem_v1": "Find a cubic polynomial f(x)=a x^3+b x^2+c x+d that has horizontal tangents at the points (-2,4) and (4,-3).\n$f(x)$=[ANS]", "answer_v1": [ "0.0648148*x^3+-0.194444*x^2+-1.55556*x+2.18519" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a cubic polynomial f(x)=a x^3+b x^2+c x+d that has horizontal tangents at the points (-5,3) and (5,-5).\n$f(x)$=[ANS]", "answer_v2": [ "0.016*x^3+0*x^2+-1.2*x+-1" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a cubic polynomial f(x)=a x^3+b x^2+c x+d that has horizontal tangents at the points (-4,4) and (4,-4).\n$f(x)$=[ANS]", "answer_v3": [ "0.03125*x^3+0*x^2+-1.5*x+0" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0161", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "derivative" ], "problem_v1": "Let $f$ and $g$ be functions that satisfy $f'(2)=6$ and $g'(2)=5$. Find $h'(2)$ for each function $h$ given below:\n(A) $h(x)=8 f(x)$.\n$h'(2)$=[ANS]\n(B) $h(x)=-5 g(x)$. $h'(2)$=[ANS]\n(C) $h(x)=4 f(x)+5 g(x)$.\n$h'(2)$=[ANS]\n(D) $h(x)=8 g(x)-9 f(x)$.\n$h'(2)$=[ANS]\n(E) $h(x)=6 f(x)+7 g(x)-2$.\n$h'(2)$=[ANS]\n(F) $h(x)=-5 g(x)-5 f(x)-2x$.\n$h'(2)$=[ANS]", "answer_v1": [ "48", "-25", "49", "-14", "71", "-57" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let $f$ and $g$ be functions that satisfy $f'(2)=-10$ and $g'(2)=15$. Find $h'(2)$ for each function $h$ given below:\n(A) $h(x)=2 f(x)$.\n$h'(2)$=[ANS]\n(B) $h(x)=-9 g(x)$. $h'(2)$=[ANS]\n(C) $h(x)=12 f(x)+5 g(x)$.\n$h'(2)$=[ANS]\n(D) $h(x)=4 g(x)-5 f(x)$.\n$h'(2)$=[ANS]\n(E) $h(x)=8 f(x)+3 g(x)+8$.\n$h'(2)$=[ANS]\n(F) $h(x)=-5 g(x)-7 f(x)-8x$.\n$h'(2)$=[ANS]", "answer_v2": [ "-20", "-135", "-45", "110", "-35", "-13" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let $f$ and $g$ be functions that satisfy $f'(2)=-4$ and $g'(2)=5$. Find $h'(2)$ for each function $h$ given below:\n(A) $h(x)=4 f(x)$.\n$h'(2)$=[ANS]\n(B) $h(x)=-7 g(x)$. $h'(2)$=[ANS]\n(C) $h(x)=4 f(x)+7 g(x)$.\n$h'(2)$=[ANS]\n(D) $h(x)=10 g(x)-13 f(x)$.\n$h'(2)$=[ANS]\n(E) $h(x)=12 f(x)+5 g(x)-12$.\n$h'(2)$=[ANS]\n(F) $h(x)=-9 g(x)-5 f(x)+2x$.\n$h'(2)$=[ANS]", "answer_v3": [ "-16", "-35", "19", "102", "-23", "-23" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0162", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "", "keywords": [ "derivative", "polynomial" ], "problem_v1": "Find $y'$ for $y=2.6x^{13}$. $y'$=[ANS]", "answer_v1": [ "(13*2.6)*(x^12)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find $y'$ for $y=3.6x^{3}$. $y'$=[ANS]", "answer_v2": [ "(3*3.6)*(x^2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find $y'$ for $y=2.6x^{7}$. $y'$=[ANS]", "answer_v3": [ "(7*2.6)*(x^6)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0163", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "derivative", "polynomial" ], "problem_v1": "Suppose that $f(x)=8x^{-3}+1x^{-1}$. Evaluate each of the following:\n$f'(3)$=[ANS]\n$f'(-3)$=[ANS]", "answer_v1": [ "-0.407407407407407", "-0.407407407407407" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that $f(x)=2x^{-3}+9x^{-1}$. Evaluate each of the following:\n$f'(2)$=[ANS]\n$f'(-1)$=[ANS]", "answer_v2": [ "-2.625", "-15" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that $f(x)=4x^{-3}+3x^{-1}$. Evaluate each of the following:\n$f'(3)$=[ANS]\n$f'(-3)$=[ANS]", "answer_v3": [ "-0.481481481481481", "-0.481481481481481" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0164", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "Algebra", "Functions", "Polynomial", "polynomial' 'extreme value", "function", "local minimum or maximum" ], "problem_v1": "For the function $f(x)=x^3-48x$, its local maximum is the point: ([ANS], [ANS]); its local minimum is the point: ([ANS], [ANS]).", "answer_v1": [ "-4", "128", "4", "-128" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "For the function $f(x)=x^3-3x$, its local maximum is the point: ([ANS], [ANS]); its local minimum is the point: ([ANS], [ANS]).", "answer_v2": [ "-1", "2", "1", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "For the function $f(x)=x^3-12x$, its local maximum is the point: ([ANS], [ANS]); its local minimum is the point: ([ANS], [ANS]).", "answer_v3": [ "-2", "16", "2", "-16" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0165", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "2", "keywords": [ "Calculus", "Derivatives", "tangent line", "derivatives" ], "problem_v1": "If $f(x)=20x+17$, find $f'(3)$. [ANS]", "answer_v1": [ "20" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $f(x)=4x+26$, find $f'(-8)$. [ANS]", "answer_v2": [ "4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $f(x)=10x+17$, find $f'(-5)$. [ANS]", "answer_v3": [ "10" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0166", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "3", "keywords": [ "tangent line", "derivatives", "applications" ], "problem_v1": "Let $f$ be defined by $f(x)=3x^{2}+x+1$, and let $a$ be any constant.\n$f(a)=$ [ANS] and\n$f'(a)=$ [ANS].\nAn equation for the tangent line to $y=f(x)$ at $x=a$ is\n$y=$ [ANS].\n(This answer will depend on the variable $x$ and the constant $a$.)", "answer_v1": [ "3*a^2+a+1", "3*2*a+1", "(3*2*a+1)*(x-a)+3*a^2+a+1" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $f$ be defined by $f(x)=5x-5x^{2}-4$, and let $a$ be any constant.\n$f(a)=$ [ANS] and\n$f'(a)=$ [ANS].\nAn equation for the tangent line to $y=f(x)$ at $x=a$ is\n$y=$ [ANS].\n(This answer will depend on the variable $x$ and the constant $a$.)", "answer_v2": [ "5*a-5*a^2-4", "5-5*2*a", "(5-5*2*a)*(x-a)+5*a-5*a^2-4" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $f$ be defined by $f(x)=x-2x^{2}-2$, and let $a$ be any constant.\n$f(a)=$ [ANS] and\n$f'(a)=$ [ANS].\nAn equation for the tangent line to $y=f(x)$ at $x=a$ is\n$y=$ [ANS].\n(This answer will depend on the variable $x$ and the constant $a$.)", "answer_v3": [ "a-2*a^2-2", "1-2*2*a", "(1-2*2*a)*(x-a)+a-2*a^2-2" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0167", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of polynomials and power functions", "level": "4", "keywords": [ "tangent", "tangent line" ], "problem_v1": "Let $f(x)=x^2$. Find the first coordinate of the intersection point of the two tangent lines of $f$ at $5$ and at $2$. [ANS]", "answer_v1": [ "3.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)=x^2$. Find the first coordinate of the intersection point of the two tangent lines of $f$ at $-9$ and at $9$. [ANS]", "answer_v2": [ "0" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(x)=x^2$. Find the first coordinate of the intersection point of the two tangent lines of $f$ at $-4$ and at $2$. [ANS]", "answer_v3": [ "-1" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0168", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of trigonometric functions", "level": "4", "keywords": [ "trigonometry", "tangent line", "derivatives" ], "problem_v1": "Find the equation of the tangent line to the curve $y=5x \\cos x$ at the point $(\\pi,-5 \\pi)$. The equation of this tangent line can be written in the form $y=mx+b$. Compute $m$ and $b$.\n$m=$ [ANS]\n$b=$ [ANS]", "answer_v1": [ "-5", "0" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the equation of the tangent line to the curve $y=2x \\cos x$ at the point $(\\pi,-2 \\pi)$. The equation of this tangent line can be written in the form $y=mx+b$. Compute $m$ and $b$.\n$m=$ [ANS]\n$b=$ [ANS]", "answer_v2": [ "-2", "0" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the equation of the tangent line to the curve $y=3x \\cos x$ at the point $(\\pi,-3 \\pi)$. The equation of this tangent line can be written in the form $y=mx+b$. Compute $m$ and $b$.\n$m=$ [ANS]\n$b=$ [ANS]", "answer_v3": [ "-3", "0" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0169", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of trigonometric functions", "level": "3", "keywords": [ "calculus", "derivatives", "trigonometric functions", "quotient rule" ], "problem_v1": "The ratio \\frac{\\frac{d}{dx} (9 \\cot(x))}{\\csc^2(x)} is a constant number. Its value is [ANS].", "answer_v1": [ "-9" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The ratio \\frac{\\frac{d}{dx} (3 \\cot(x))}{\\csc^2(x)} is a constant number. Its value is [ANS].", "answer_v2": [ "-3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The ratio \\frac{\\frac{d}{dx} (5 \\cot(x))}{\\csc^2(x)} is a constant number. Its value is [ANS].", "answer_v3": [ "-5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0170", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of trigonometric functions", "level": "2", "keywords": [ "derivative' 'trig function" ], "problem_v1": "Let $f(x)= 10\\cos\\!\\left(x\\right)+8\\tan\\!\\left(x\\right)$. Find the following:\n$\\begin{array}{cccc}\\hline 1. & f'(x) &=& [ANS] \\\\ \\hline 2. & f'(\\frac {\\pi} {4}) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "8*[sec(x)]^2-10*sin(x)", "8.92893" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)= 2\\cos\\!\\left(x\\right)+3\\tan\\!\\left(x\\right)$. Find the following:\n$\\begin{array}{cccc}\\hline 1. & f'(x) &=& [ANS] \\\\ \\hline 2. & f'(\\frac {7 \\pi} {4}) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "3*[sec(x)]^2-2*sin(x)", "7.41421" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)= 5\\cos\\!\\left(x\\right)+5\\tan\\!\\left(x\\right)$. Find the following:\n$\\begin{array}{cccc}\\hline 1. & f'(x) &=& [ANS] \\\\ \\hline 2. & f'(\\frac {\\pi} {3}) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "5*[sec(x)]^2-5*sin(x)", "15.6699" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0171", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of trigonometric functions", "level": "2", "keywords": [ "derivatives", "trigonometry" ], "problem_v1": "If $f(x)=\\cos x-6 \\tan x$, then $f'(x)=$ [ANS]", "answer_v1": [ "-[sin(x)]-6*[sec(x)]^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If $f(x)=\\cos x-2 \\tan x$, then $f'(x)=$ [ANS]", "answer_v2": [ "-[sin(x)]-2*[sec(x)]^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If $f(x)=\\cos x-3 \\tan x$, then $f'(x)=$ [ANS]", "answer_v3": [ "-[sin(x)]-3*[sec(x)]^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0172", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of trigonometric functions", "level": "4", "keywords": [ "trigonometry", "tangent line", "derivatives" ], "problem_v1": "In this problem, please evaluate the trig functions without a calculator and do not use a decimal point in your answer. An equation of the tangent line to the curve $y=\\sin (x)$ at $x=7\\pi/6 \\,$ is $y=$ [ANS] $+$ [ANS] $\\cdot (x-7\\pi/6)$.\nAn equation of the tangent line to the curve $y=\\cos (x)$ at $x=4\\pi/3 \\,$ is $y=$ [ANS] $+$ [ANS] $\\cdot (x-4\\pi/3)$.", "answer_v1": [ "sin(14*pi/12)", "cos(14*pi/12)", "cos(16*pi/12)", "-sin(16*pi/12)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "In this problem, please evaluate the trig functions without a calculator and do not use a decimal point in your answer. An equation of the tangent line to the curve $y=\\sin (x)$ at $x=7\\pi/4 \\,$ is $y=$ [ANS] $+$ [ANS] $\\cdot (x-7\\pi/4)$.\nAn equation of the tangent line to the curve $y=\\cos (x)$ at $x=\\pi/4 \\,$ is $y=$ [ANS] $+$ [ANS] $\\cdot (x-\\pi/4)$.", "answer_v2": [ "sin(21*pi/12)", "cos(21*pi/12)", "cos(3*pi/12)", "-sin(3*pi/12)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "In this problem, please evaluate the trig functions without a calculator and do not use a decimal point in your answer. An equation of the tangent line to the curve $y=\\sin (x)$ at $x=\\pi/2 \\,$ is $y=$ [ANS] $+$ [ANS] $\\cdot (x-\\pi/2)$.\nAn equation of the tangent line to the curve $y=\\cos (x)$ at $x=\\pi \\,$ is $y=$ [ANS] $+$ [ANS] $\\cdot (x-\\pi)$.", "answer_v3": [ "sin(6*pi/12)", "cos(6*pi/12)", "cos(12*pi/12)", "-sin(12*pi/12)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0173", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of exponential functions", "level": "4", "keywords": [ "derivatives", "tangent line" ], "problem_v1": "Find the equation of the tangent line to the curve $y=12x e^x$ at the point $(0, 0)$. The equation of this tangent line can be written in the form $y=mx+b$. Find $m$ and $b$.\n$m=$ [ANS]\n$b=$ [ANS]", "answer_v1": [ "12", "0" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the equation of the tangent line to the curve $y=3x e^x$ at the point $(0, 0)$. The equation of this tangent line can be written in the form $y=mx+b$. Find $m$ and $b$.\n$m=$ [ANS]\n$b=$ [ANS]", "answer_v2": [ "3", "0" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the equation of the tangent line to the curve $y=6x e^x$ at the point $(0, 0)$. The equation of this tangent line can be written in the form $y=mx+b$. Find $m$ and $b$.\n$m=$ [ANS]\n$b=$ [ANS]", "answer_v3": [ "6", "0" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0174", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of exponential functions", "level": "2", "keywords": [], "problem_v1": "Compute the derivatives of the given functions. a) $f(r)=16^{r}.$ $f'(r)=$ [ANS]. b) $g(s)=13^{6}.$ $g'(s)=$ [ANS]. b) $h(t)=\\frac{15^t}{14^t}.$ $h'(t)=$ [ANS].", "answer_v1": [ "ln(16) * 16^r", "0", "ln(15/14) * (15/14)^t" ], "answer_type_v1": [ "EX", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Compute the derivatives of the given functions. a) $f(r)=3^{r}.$ $f'(r)=$ [ANS]. b) $g(s)=19^{3}.$ $g'(s)=$ [ANS]. b) $h(t)=\\frac{8^t}{9^t}.$ $h'(t)=$ [ANS].", "answer_v2": [ "ln(3) * 3^r", "0", "ln(8/9) * (8/9)^t" ], "answer_type_v2": [ "EX", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Compute the derivatives of the given functions. a) $f(r)=7^{r}.$ $f'(r)=$ [ANS]. b) $g(s)=13^{4}.$ $g'(s)=$ [ANS]. b) $h(t)=\\frac{12^t}{11^t}.$ $h'(t)=$ [ANS].", "answer_v3": [ "ln(7) * 7^r", "0", "ln(12/11) * (12/11)^t" ], "answer_type_v3": [ "EX", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0175", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of exponential functions", "level": "2", "keywords": [ "calculus", "derivative", "exponentials" ], "problem_v1": "Find the derivative of $f(t)=(\\ln8)^t$.\n$f'(t)=$ [ANS]", "answer_v1": [ "ln(ln(8))*[ln(8)]^t" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of $f(t)=(\\ln2)^t$.\n$f'(t)=$ [ANS]", "answer_v2": [ "ln(ln(2))*[ln(2)]^t" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of $f(t)=(\\ln4)^t$.\n$f'(t)=$ [ANS]", "answer_v3": [ "ln(ln(4))*[ln(4)]^t" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0176", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of exponential functions", "level": "2", "keywords": [ "calculus", "derivative", "exponentials" ], "problem_v1": "Find the derivative of $z=(\\ln 12) e^x$.\n${dz\\over dx}=$ [ANS]", "answer_v1": [ "ln(12)*e^x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of $z=(\\ln 3) e^x$.\n${dz\\over dx}=$ [ANS]", "answer_v2": [ "ln(3)*e^x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of $z=(\\ln 6) e^x$.\n${dz\\over dx}=$ [ANS]", "answer_v3": [ "ln(6)*e^x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0177", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of exponential functions", "level": "3", "keywords": [ "calculus", "derivative", "exponentials" ], "problem_v1": "Since January 1, 1960, the population of Slim Chance has been described by the formula $P=43000 (0.95)^t,$ where $P$ is the population of the city $t$ years after the start of 1960. At what rate was the population changing on January 1, 1983?\nrate=[ANS] people/yr", "answer_v1": [ "43000*ln(0.95)*0.95^23" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Since January 1, 1960, the population of Slim Chance has been described by the formula $P=22000 (0.98)^t,$ where $P$ is the population of the city $t$ years after the start of 1960. At what rate was the population changing on January 1, 1973?\nrate=[ANS] people/yr", "answer_v2": [ "22000*ln(0.98)*0.98^13" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Since January 1, 1960, the population of Slim Chance has been described by the formula $P=29000 (0.95)^t,$ where $P$ is the population of the city $t$ years after the start of 1960. At what rate was the population changing on January 1, 1975?\nrate=[ANS] people/yr", "answer_v3": [ "29000*ln(0.95)*0.95^15" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0178", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of exponential functions", "level": "2", "keywords": [ "calculus", "derivative", "exponentials" ], "problem_v1": "Find the derivative of $f(x)=e^n+n^x$. Assume that $n$ is a positive constant.\n$f'(x)=$ [ANS]", "answer_v1": [ "ln(n)*n^x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of $f(x)=e^a+a^x$. Assume that $a$ is a positive constant.\n$f'(x)=$ [ANS]", "answer_v2": [ "ln(a)*a^x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of $f(x)=e^c+c^x$. Assume that $c$ is a positive constant.\n$f'(x)=$ [ANS]", "answer_v3": [ "ln(c)*c^x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0179", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of exponential functions", "level": "2", "keywords": [ "calculus", "derivative", "exponentials" ], "problem_v1": "Certain pieces of antique furniture increased very rapidly in price in the 1970s and 1980s. For example, the value of a particular rocking chair is well approximated by V=125 (1.55)^t, where $V$ is in dollars and $t$ is the number of years since 1975. Find the rate, in dollars per year, at which the price is increasing.\nrate=[ANS] dollars/yr", "answer_v1": [ "125*ln(1.55)*1.55^t" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Certain pieces of antique furniture increased very rapidly in price in the 1970s and 1980s. For example, the value of a particular rocking chair is well approximated by V=55 (1.8)^t, where $V$ is in dollars and $t$ is the number of years since 1975. Find the rate, in dollars per year, at which the price is increasing.\nrate=[ANS] dollars/yr", "answer_v2": [ "55*ln(1.8)*1.8^t" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Certain pieces of antique furniture increased very rapidly in price in the 1970s and 1980s. For example, the value of a particular rocking chair is well approximated by V=80 (1.55)^t, where $V$ is in dollars and $t$ is the number of years since 1975. Find the rate, in dollars per year, at which the price is increasing.\nrate=[ANS] dollars/yr", "answer_v3": [ "80*ln(1.55)*1.55^t" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0180", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of exponential functions", "level": "4", "keywords": [ "calculus", "function", "rotate", "slope" ], "problem_v1": "The graph of f(x)=7x+e^{6x-50x^2} is rotated counterclockwise about the origin through an acute angle $\\theta$. What is the largest value of $\\theta$ for which the rotated graph is still the graph of a function? What about if the graph is rotated clockwise? To answer this question we need to find the maximal slope of $y=f(x)$, which is [ANS], and the minimal slope which is [ANS]. Thus the maximal acute angle through which the graph can be rotated counterclockwise is $\\theta=$ [ANS] degrees. Thus the maximal acute angle through which the graph can be rotated clockwise is $\\theta=$ [ANS] degrees. (Your answer should be negative to indicate the clockwise direction.) Note that a line $y=mx+b$ makes angle $\\alpha$ with the horizontal, where $\\tan(\\alpha)=m$. Hints: Recall that a graph of a function is characterized by the property that every vertical line intersects the graph in at most one point. In view of this: 1. If ALL lines $y=mx+b$ of a fixed slope $m$ intersect a graph of $y=f(x)$ in at most one point, what can you say about rotating the graph of $y=f(x)$? 2. If some line $y=mx+b$ intersects the graph of $y=f(x)$ in two or more points, what can you say about rotating the graph of $y=f(x)$? 3. If some line $y=mx+b$ intersects the graph of $y=f(x)$ in two or more points, what does the Mean Value Theorem tell us about $f'(x)$?", "answer_v1": [ "51.8168907033807", "-37.8168907033807", "1.10559833344655", "178.485268633067" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The graph of f(x)=-4x+e^{2x-98x^2} is rotated counterclockwise about the origin through an acute angle $\\theta$. What is the largest value of $\\theta$ for which the rotated graph is still the graph of a function? What about if the graph is rotated clockwise? To answer this question we need to find the maximal slope of $y=f(x)$, which is [ANS], and the minimal slope which is [ANS]. Thus the maximal acute angle through which the graph can be rotated counterclockwise is $\\theta=$ [ANS] degrees. Thus the maximal acute angle through which the graph can be rotated clockwise is $\\theta=$ [ANS] degrees. (Your answer should be negative to indicate the clockwise direction.) Note that a line $y=mx+b$ makes angle $\\alpha$ with the horizontal, where $\\tan(\\alpha)=m$. Hints: Recall that a graph of a function is characterized by the property that every vertical line intersects the graph in at most one point. In view of this: 1. If ALL lines $y=mx+b$ of a fixed slope $m$ intersect a graph of $y=f(x)$ in at most one point, what can you say about rotating the graph of $y=f(x)$? 2. If some line $y=mx+b$ intersects the graph of $y=f(x)$ in two or more points, what can you say about rotating the graph of $y=f(x)$? 3. If some line $y=mx+b$ intersects the graph of $y=f(x)$ in two or more points, what does the Mean Value Theorem tell us about $f'(x)$?", "answer_v2": [ "58.7436469847329", "-66.7436469847329", "0.975258583708509", "179.141618784678" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The graph of f(x)=-5x+e^{3x-50x^2} is rotated counterclockwise about the origin through an acute angle $\\theta$. What is the largest value of $\\theta$ for which the rotated graph is still the graph of a function? What about if the graph is rotated clockwise? To answer this question we need to find the maximal slope of $y=f(x)$, which is [ANS], and the minimal slope which is [ANS]. Thus the maximal acute angle through which the graph can be rotated counterclockwise is $\\theta=$ [ANS] degrees. Thus the maximal acute angle through which the graph can be rotated clockwise is $\\theta=$ [ANS] degrees. (Your answer should be negative to indicate the clockwise direction.) Note that a line $y=mx+b$ makes angle $\\alpha$ with the horizontal, where $\\tan(\\alpha)=m$. Hints: Recall that a graph of a function is characterized by the property that every vertical line intersects the graph in at most one point. In view of this: 1. If ALL lines $y=mx+b$ of a fixed slope $m$ intersect a graph of $y=f(x)$ in at most one point, what can you say about rotating the graph of $y=f(x)$? 2. If some line $y=mx+b$ intersects the graph of $y=f(x)$ in two or more points, what can you say about rotating the graph of $y=f(x)$? 3. If some line $y=mx+b$ intersects the graph of $y=f(x)$ in two or more points, what does the Mean Value Theorem tell us about $f'(x)$?", "answer_v3": [ "39.8168907033807", "-49.8168907033807", "1.43867932420551", "178.850026871127" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0181", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of exponential functions", "level": "2", "keywords": [ "calculus", "differentiation" ], "problem_v1": "Find an equation of the tangent line to the curve $y=8^{x}$ at the point $(2, 64)$.\nTangent line: $y=$ [ANS]", "answer_v1": [ "64+133.084258667509*(x-2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an equation of the tangent line to the curve $y=2^{x}$ at the point $(2, 4)$.\nTangent line: $y=$ [ANS]", "answer_v2": [ "4+2.77258872223978*(x-2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an equation of the tangent line to the curve $y=4^{x}$ at the point $(2, 16)$.\nTangent line: $y=$ [ANS]", "answer_v3": [ "16+22.1807097779182*(x-2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0182", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of logarithmic functions", "level": "4", "keywords": [], "problem_v1": "A property of logarithms is that $ \\log_a x=\\frac{\\log_b x}{\\log_b a}$ for all bases $a, b > 0, \\neq 1$. When $b=e$, this becomes $ \\log_a x=\\frac{\\ln x}{\\ln a}$. a) Using this identity, find the derivative of $y=\\log_a x.$ [ANS]\nb) Find the derivative of $y=\\log_{8} x.$ [ANS]", "answer_v1": [ "1/[x ln(a)]", "1/[x ln(8)]" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "A property of logarithms is that $ \\log_a x=\\frac{\\log_b x}{\\log_b a}$ for all bases $a, b > 0, \\neq 1$. When $b=e$, this becomes $ \\log_a x=\\frac{\\ln x}{\\ln a}$. a) Using this identity, find the derivative of $y=\\log_a x.$ [ANS]\nb) Find the derivative of $y=\\log_{2} x.$ [ANS]", "answer_v2": [ "1/[x ln(a)]", "1/[x ln(2)]" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "A property of logarithms is that $ \\log_a x=\\frac{\\log_b x}{\\log_b a}$ for all bases $a, b > 0, \\neq 1$. When $b=e$, this becomes $ \\log_a x=\\frac{\\ln x}{\\ln a}$. a) Using this identity, find the derivative of $y=\\log_a x.$ [ANS]\nb) Find the derivative of $y=\\log_{4} x.$ [ANS]", "answer_v3": [ "1/[x ln(a)]", "1/[x ln(4)]" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0183", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of logarithmic functions", "level": "2", "keywords": [ "derivatives", "logarithmic functions", "Calculus" ], "problem_v1": "If $f(x)=7 \\log_{7}(x)$, find $f'(4)$.\n$f'(4)=$ [ANS]", "answer_v1": [ "0.899322" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $f(x)=2 \\log_{10}(x)$, find $f'(1)$.\n$f'(1)=$ [ANS]", "answer_v2": [ "0.868589" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $f(x)=4 \\log_{7}(x)$, find $f'(2)$.\n$f'(2)=$ [ANS]", "answer_v3": [ "1.0278" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0184", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of logarithmic functions", "level": "2", "keywords": [ "derivative" ], "problem_v1": "Find $\\frac{dy}{dx}$ when y=\\frac{\\log\\!\\left(x\\right)}{6+\\log\\!\\left(x\\right)} $\\frac{dy}{dx}=$ [ANS]", "answer_v1": [ "6/[ln(10)*x*[6+log(x)]^2]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find $\\frac{dy}{dx}$ when y=\\frac{\\log\\!\\left(x\\right)}{\\log\\!\\left(x\\right)-8} $\\frac{dy}{dx}=$ [ANS]", "answer_v2": [ "-8/[ln(10)*x*[-8+log(x)]^2]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find $\\frac{dy}{dx}$ when y=\\frac{\\log\\!\\left(x\\right)}{\\log\\!\\left(x\\right)-3} $\\frac{dy}{dx}=$ [ANS]", "answer_v3": [ "-3/[ln(10)*x*[-3+log(x)]^2]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0185", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of logarithmic functions", "level": "2", "keywords": [ "derivative" ], "problem_v1": "Find the equation of the tangent line to the graph of $y=f(x)$ at $x=1$. f(x)=\\ln\\!\\left(\\left|x+6\\right|\\right) $y=$ [ANS]", "answer_v1": [ "x/7-1/7+ln(7)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the equation of the tangent line to the graph of $y=f(x)$ at $x=5$. f(x)=\\ln\\!\\left(\\left|x-8\\right|\\right) $y=$ [ANS]", "answer_v2": [ "-(x/3)+5/3+ln(3)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the equation of the tangent line to the graph of $y=f(x)$ at $x=1$. f(x)=\\ln\\!\\left(\\left|x-3\\right|\\right) $y=$ [ANS]", "answer_v3": [ "-(x/2)+1/2+ln(2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0186", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of logarithmic functions", "level": "4", "keywords": [ "derivative" ], "problem_v1": "Find a formula for the area $A(w)$ of the triangle bounded by the tangent line to the graph of $y=\\ln\\!\\left(x^{38}\\right)$ at $P(w,\\ln\\!\\left(w^{38}\\right))$, the horizontal line through $P$, and the $y$-axis. $A(w)=$ [ANS]", "answer_v1": [ "19*w" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a formula for the area $A(w)$ of the triangle bounded by the tangent line to the graph of $y=\\ln\\!\\left(x^{6}\\right)$ at $P(w,\\ln\\!\\left(w^{6}\\right))$, the horizontal line through $P$, and the $y$-axis. $A(w)=$ [ANS]", "answer_v2": [ "3*w" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a formula for the area $A(w)$ of the triangle bounded by the tangent line to the graph of $y=\\ln\\!\\left(x^{16}\\right)$ at $P(w,\\ln\\!\\left(w^{16}\\right))$, the horizontal line through $P$, and the $y$-axis. $A(w)=$ [ANS]", "answer_v3": [ "8*w" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0187", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of logarithmic functions", "level": "2", "keywords": [ "logarithm", "logarithmic differentiation" ], "problem_v1": "(a) Using laws of logarithms, write the expression below using sums and/or differences of logarithmic expressions which do not contain the logarithms of products, quotients, or powers.\n$ \\ln \\sqrt{(x-8)^{26} (6-x)^{30}}=$ [ANS]\nHint: $\\sqrt{u^2}=|u|$.\n(b) Use your answer from part (a) to evaluate the derivative.\n$ \\frac{d}{dx} \\left( \\ln \\sqrt{(x-8)^{26} (6-x)^{30}} \\right)=$ [ANS]", "answer_v1": [ "13*ln(|x-8|)+15*ln(|6-x|)", "13/(x-8)-15/(6-x)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "(a) Using laws of logarithms, write the expression below using sums and/or differences of logarithmic expressions which do not contain the logarithms of products, quotients, or powers.\n$ \\ln \\sqrt{(x-2)^{22} (9-x)^{26}}=$ [ANS]\nHint: $\\sqrt{u^2}=|u|$.\n(b) Use your answer from part (a) to evaluate the derivative.\n$ \\frac{d}{dx} \\left( \\ln \\sqrt{(x-2)^{22} (9-x)^{26}} \\right)=$ [ANS]", "answer_v2": [ "11*ln(|x-2|)+13*ln(|9-x|)", "11/(x-2)-13/(9-x)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "(a) Using laws of logarithms, write the expression below using sums and/or differences of logarithmic expressions which do not contain the logarithms of products, quotients, or powers.\n$ \\ln \\sqrt{(x-4)^{26} (6-x)^{30}}=$ [ANS]\nHint: $\\sqrt{u^2}=|u|$.\n(b) Use your answer from part (a) to evaluate the derivative.\n$ \\frac{d}{dx} \\left( \\ln \\sqrt{(x-4)^{26} (6-x)^{30}} \\right)=$ [ANS]", "answer_v3": [ "13*ln(|x-4|)+15*ln(|6-x|)", "13/(x-4)-15/(6-x)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0188", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of logarithmic functions", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find $y'$ if y=x^5 \\ln x [ANS]", "answer_v1": [ "5*x^4*ln(x)+x^4" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find $y'$ if y=x^2 \\ln x [ANS]", "answer_v2": [ "2*x*ln(x)+x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find $y'$ if y=x^3 \\ln x [ANS]", "answer_v3": [ "3*x^2*ln(x)+x^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0189", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of logarithmic functions", "level": "2", "keywords": [ "derivatives", "logarithmic functions" ], "problem_v1": "Suppose that f(x)=\\ln(15x+12). Find $f'(x)$, and use interval notation to give the domain of $f$.\n$f'(x)$=[ANS]\nDomain=[ANS]", "answer_v1": [ "15/(15*x+12)", "(-0.8,infinity)" ], "answer_type_v1": [ "EX", "INT" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that f(x)=\\ln(5x+18). Find $f'(x)$, and use interval notation to give the domain of $f$.\n$f'(x)$=[ANS]\nDomain=[ANS]", "answer_v2": [ "5/(5*x+18)", "(-3.6,infinity)" ], "answer_type_v2": [ "EX", "INT" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that f(x)=\\ln(9x+12). Find $f'(x)$, and use interval notation to give the domain of $f$.\n$f'(x)$=[ANS]\nDomain=[ANS]", "answer_v3": [ "9/(9*x+12)", "(-1.33333,infinity)" ], "answer_type_v3": [ "EX", "INT" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0190", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of inverse functions", "level": "2", "keywords": [ "calculus", "derivatives", "inverse functions" ], "problem_v1": "Calculate $g(b)$ and $g'(b)$ where $g(x)$ is the inverse of $f(x)=8x+7$. $g'(x)$=[ANS]", "answer_v1": [ "0.125" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate $g(b)$ and $g'(b)$ where $g(x)$ is the inverse of $f(x)=2x+13$. $g'(x)$=[ANS]", "answer_v2": [ "0.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate $g(b)$ and $g'(b)$ where $g(x)$ is the inverse of $f(x)=4x+7$. $g'(x)$=[ANS]", "answer_v3": [ "0.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0191", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of inverse functions", "level": "2", "keywords": [ "Calculus", "Derivatives", "inverse functions", "derivatives", "Polynomials", "Derivative", "Inverse" ], "problem_v1": "For each of the given functions $f(x)$, find the derivative $\\left(f^{-1}\\right)'(c)$ at the given point $c$, first finding $a=f^{-1}(c)$.\na) $f(x)=6x+8x^{17}$ ; $c=-14$ $a$=[ANS]\n$\\left(f^{-1}\\right)'(c)$=[ANS]\nb) $f(x)=x^2-13x+53$ on the interval $[6.5,\\infty)$ ; $c=13$ $a$=[ANS]\n$\\left(f^{-1}\\right)'(c)$=[ANS]", "answer_v1": [ "-1", "0.00704225352112676", "8", "0.333333333333333" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "For each of the given functions $f(x)$, find the derivative $\\left(f^{-1}\\right)'(c)$ at the given point $c$, first finding $a=f^{-1}(c)$.\na) $f(x)=3x+10x^{9}$ ; $c=-13$ $a$=[ANS]\n$\\left(f^{-1}\\right)'(c)$=[ANS]\nb) $f(x)=x^2-13x+49$ on the interval $[6.5,\\infty)$ ; $c=13$ $a$=[ANS]\n$\\left(f^{-1}\\right)'(c)$=[ANS]", "answer_v2": [ "-1", "0.010752688172043", "9", "0.2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "For each of the given functions $f(x)$, find the derivative $\\left(f^{-1}\\right)'(c)$ at the given point $c$, first finding $a=f^{-1}(c)$.\na) $f(x)=4x+9x^{11}$ ; $c=-13$ $a$=[ANS]\n$\\left(f^{-1}\\right)'(c)$=[ANS]\nb) $f(x)=x^2-12x+48$ on the interval $[6,\\infty)$ ; $c=13$ $a$=[ANS]\n$\\left(f^{-1}\\right)'(c)$=[ANS]", "answer_v3": [ "-1", "0.00970873786407767", "7", "0.5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0192", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of inverse functions", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find a formula for $f^{-1}(x)$ and $\\left(f^{-1}\\right)'(x)$ if $f(x)=\\sqrt{\\frac{1}{x-5}}$.\n$f^{-1}(x)$=[ANS]\n$\\left(f^{-1}\\right)'(x)=$ [ANS]", "answer_v1": [ "1/x^2 + 5", "-2/x^3" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find a formula for $f^{-1}(x)$ and $\\left(f^{-1}\\right)'(x)$ if $f(x)=\\sqrt{\\frac{1}{x-2}}$.\n$f^{-1}(x)$=[ANS]\n$\\left(f^{-1}\\right)'(x)=$ [ANS]", "answer_v2": [ "1/x^2 + 2", "-2/x^3" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find a formula for $f^{-1}(x)$ and $\\left(f^{-1}\\right)'(x)$ if $f(x)=\\sqrt{\\frac{1}{x-3}}$.\n$f^{-1}(x)$=[ANS]\n$\\left(f^{-1}\\right)'(x)=$ [ANS]", "answer_v3": [ "1/x^2 + 3", "-2/x^3" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0193", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of inverse trigonometric functions", "level": "2", "keywords": [], "problem_v1": "Compute the derivatives of the given function. $\\dfrac{d}{dx}\\left[\\tan^{-1}(\\sqrt[4]{x}) \\right]=$ [ANS]", "answer_v1": [ "1/(1 + (x^{0.25})^2) * 1/[4 x^{0.75}]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Compute the derivatives of the given function. $\\dfrac{d}{dx}\\left[\\cot^{-1}(\\sqrt{x}) \\right]=$ [ANS]", "answer_v2": [ "-1/(1 + (sqrt(x))^2) * 1/[2 x^{0.5}]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Compute the derivatives of the given function. $\\dfrac{d}{dx}\\left[\\tan^{-1}(\\sqrt{x}) \\right]=$ [ANS]", "answer_v3": [ "1/(1 + (sqrt(x))^2) * 1/[2 x^{0.5}]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0194", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of inverse trigonometric functions", "level": "3", "keywords": [], "problem_v1": "Compute the derivatives of the given functions. a) $\\dfrac{d}{dx}\\left[x\\arccos (x) \\right]=$ [ANS]\nb) $\\dfrac{d}{dx}\\left[\\cot (x) \\arcsin (x) \\right]=$ [ANS]", "answer_v1": [ "x *(-1/sqrt(1-x^2)) + arccos(x)", "(cot(x))(1/sqrt(1-x^2)) + (-csc(x)^2)(arcsin(x))" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Compute the derivatives of the given functions. a) $\\dfrac{d}{dx}\\left[x\\arcsin (x) \\right]=$ [ANS]\nb) $\\dfrac{d}{dx}\\left[\\csc (x) \\arccos (x) \\right]=$ [ANS]", "answer_v2": [ "x *(1/sqrt(1-x^2)) + arcsin(x)", "(csc(x))(-1/sqrt(1-x^2)) + (-csc(x)*cot(x))(arccos(x))" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Compute the derivatives of the given functions. a) $\\dfrac{d}{dx}\\left[x\\arcsin (x) \\right]=$ [ANS]\nb) $\\dfrac{d}{dx}\\left[\\cot (x) \\arccos (x) \\right]=$ [ANS]", "answer_v3": [ "x *(1/sqrt(1-x^2)) + arcsin(x)", "(cot(x))(-1/sqrt(1-x^2)) + (-csc(x)^2)(arccos(x))" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0195", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of inverse trigonometric functions", "level": "3", "keywords": [ "Derivatives", "Calculus", "inverse trig functions" ], "problem_v1": "Let y=\\tan^{-1}\\left(\\sqrt{7x^2-1}\\right), then $ \\frac{dy}{dx}$=[ANS]", "answer_v1": [ "1/[x*sqrt(7*x^2-1)]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let y=\\tan^{-1}\\left(\\sqrt{3x^2-1}\\right), then $ \\frac{dy}{dx}$=[ANS]", "answer_v2": [ "1/[x*sqrt(3*x^2-1)]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let y=\\tan^{-1}\\left(\\sqrt{4x^2-1}\\right), then $ \\frac{dy}{dx}$=[ANS]", "answer_v3": [ "1/[x*sqrt(4*x^2-1)]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0196", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of inverse trigonometric functions", "level": "2", "keywords": [ "derivatives", "inverse trig functions", "Calculus" ], "problem_v1": "Let f(x)=7\\sin^{-1}\\!\\left(x^{3}\\right) $f'(x)=$ [ANS]", "answer_v1": [ "7*1/[sqrt(1-(x^3)^2)]*3*x^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let f(x)=2\\sin^{-1}\\!\\left(x^{4}\\right) $f'(x)=$ [ANS]", "answer_v2": [ "2*1/[sqrt(1-(x^4)^2)]*4*x^3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let f(x)=4\\sin^{-1}\\!\\left(x^{3}\\right) $f'(x)=$ [ANS]", "answer_v3": [ "4*1/[sqrt(1-(x^3)^2)]*3*x^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0197", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of inverse trigonometric functions", "level": "2", "keywords": [ "calculus" ], "problem_v1": "a) Let $ | x | < \\frac{\\pi}{2}$ and $y=\\left(\\cos^{-1}x\\right)^{8}$. Then the derivative\n$D_{x}y$=[ANS]. b) Let $ |x|<\\frac{\\pi}{2}$ and $z=\\ln\\left(\\sec x+\\tan x \\right)$. Then the derivative\n$D_{x}z$=[ANS].", "answer_v1": [ "-8*[acos(x)]^7/[sqrt(1-x^2)]", "sec(x)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "a) Let $ | x | < \\frac{\\pi}{2}$ and $y=\\left(\\cos^{-1}x\\right)^{2}$. Then the derivative\n$D_{x}y$=[ANS]. b) Let $ |x|<\\frac{\\pi}{2}$ and $z=\\ln\\left(\\sec x+\\tan x \\right)$. Then the derivative\n$D_{x}z$=[ANS].", "answer_v2": [ "-2*[acos(x)]/[sqrt(1-x^2)]", "sec(x)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "a) Let $ | x | < \\frac{\\pi}{2}$ and $y=\\left(\\cos^{-1}x\\right)^{4}$. Then the derivative\n$D_{x}y$=[ANS]. b) Let $ |x|<\\frac{\\pi}{2}$ and $z=\\ln\\left(\\sec x+\\tan x \\right)$. Then the derivative\n$D_{x}z$=[ANS].", "answer_v3": [ "-4*[acos(x)]^3/[sqrt(1-x^2)]", "sec(x)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0198", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Hyperbolic functions", "level": "2", "keywords": [ "calculus", "hyperbolic functions", "inverse functions" ], "problem_v1": "Compute $\\cosh (x)$ and $\\tanh\\!\\left(x\\right)$, assuming $\\sinh (x)=0.75$. $\\cosh (x)=$ [ANS]\n$\\tanh\\!\\left(x\\right)=$ [ANS]", "answer_v1": [ "1.25", "0.6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Compute $\\cosh (x)$ and $\\coth\\!\\left(x\\right)$, assuming $\\sinh (x)=0.09$. $\\cosh (x)=$ [ANS]\n$\\coth\\!\\left(x\\right)=$ [ANS]", "answer_v2": [ "1.00404", "11.156" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Compute $\\cosh (x)$ and $\\tanh\\!\\left(x\\right)$, assuming $\\sinh (x)=0.32$. $\\cosh (x)=$ [ANS]\n$\\tanh\\!\\left(x\\right)=$ [ANS]", "answer_v3": [ "1.04995", "0.304776" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0199", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Hyperbolic functions", "level": "2", "keywords": [ "trigonometry", "hyperbolic" ], "problem_v1": "(a) $\\sinh (7)=$ [ANS]\n(b) $\\sinh (-7)=$ [ANS]\n(c) $\\cosh (7)=$ [ANS]\n(d) $\\cosh (-7)=$ [ANS]", "answer_v1": [ "sinh(7)", "sinh(-7)", "cosh(7)", "cosh(-7)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "(a) $\\sinh (2)=$ [ANS]\n(b) $\\sinh (-2)=$ [ANS]\n(c) $\\cosh (2)=$ [ANS]\n(d) $\\cosh (-2)=$ [ANS]", "answer_v2": [ "sinh(2)", "sinh(-2)", "cosh(2)", "cosh(-2)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "(a) $\\sinh (4)=$ [ANS]\n(b) $\\sinh (-4)=$ [ANS]\n(c) $\\cosh (4)=$ [ANS]\n(d) $\\cosh (-4)=$ [ANS]", "answer_v3": [ "sinh(4)", "sinh(-4)", "cosh(4)", "cosh(-4)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0200", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Hyperbolic functions", "level": "2", "keywords": [ "trigonometry", "hyperbolic" ], "problem_v1": "The hyperbolic sine function is denoted $\\sinh(x)$ and the hyperbolic cosine function is denoted as $\\cosh(x)$. These two functions are both defined using either the difference or sum of exponential functions and then dividing by 2:\n$ \\sinh(x)=\\frac {e^x-e^{-x}} {2}$\n$ \\cosh(x)=\\frac {e^x+e^{-x}} {2}$\n$\\sinh (1.5)=$ [ANS]\n$\\cosh (1.5)=$ [ANS]", "answer_v1": [ "2.12928", "2.35241" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The hyperbolic sine function is denoted $\\sinh(x)$ and the hyperbolic cosine function is denoted as $\\cosh(x)$. These two functions are both defined using either the difference or sum of exponential functions and then dividing by 2:\n$ \\sinh(x)=\\frac {e^x-e^{-x}} {2}$\n$ \\cosh(x)=\\frac {e^x+e^{-x}} {2}$\n$\\sinh (0.2)=$ [ANS]\n$\\cosh (0.2)=$ [ANS]", "answer_v2": [ "0.201336", "1.02007" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The hyperbolic sine function is denoted $\\sinh(x)$ and the hyperbolic cosine function is denoted as $\\cosh(x)$. These two functions are both defined using either the difference or sum of exponential functions and then dividing by 2:\n$ \\sinh(x)=\\frac {e^x-e^{-x}} {2}$\n$ \\cosh(x)=\\frac {e^x+e^{-x}} {2}$\n$\\sinh (0.6)=$ [ANS]\n$\\cosh (0.6)=$ [ANS]", "answer_v3": [ "0.636654", "1.18547" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0201", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Hyperbolic functions", "level": "2", "keywords": [ "Hyperbolic Functions" ], "problem_v1": "Find the numerical value of the following expression:\n$\\tanh 4=$ [ANS]", "answer_v1": [ "0.999329" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the numerical value of the following expression:\n$\\tanh 0=$ [ANS]", "answer_v2": [ "0" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the numerical value of the following expression:\n$\\tanh 1=$ [ANS]", "answer_v3": [ "0.761594" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0202", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Hyperbolic functions", "level": "2", "keywords": [ "Hyperbolic Functions" ], "problem_v1": "Find the numerical value of the following expression:\n$\\cosh (\\ln 5)=$ [ANS]", "answer_v1": [ "5/2+1/(2*5)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the numerical value of the following expression:\n$\\cosh (\\ln 2)=$ [ANS]", "answer_v2": [ "2/2+1/(2*2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the numerical value of the following expression:\n$\\cosh (\\ln 3)=$ [ANS]", "answer_v3": [ "3/2+1/(2*3)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0203", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Hyperbolic functions", "level": "2", "keywords": [ "Hyperbolic Functions" ], "problem_v1": "Evaluate the following as a rational number $n/m$:\n$9 \\sinh {(\\ln 10)}=$ [ANS]", "answer_v1": [ "9*[10/2-1/(2*10)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the following as a rational number $n/m$:\n$13 \\sinh {(\\ln 2)}=$ [ANS]", "answer_v2": [ "13*[2/2-1/(2*2)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the following as a rational number $n/m$:\n$9 \\sinh {(\\ln 4)}=$ [ANS]", "answer_v3": [ "9*[4/2-1/(2*4)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0204", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of hyperbolic functions", "level": "3", "keywords": [ "Derivative", "Hyperbolic" ], "problem_v1": "If f(x)=e^{\\cosh (8x)} then $f'(x)=$ [ANS].", "answer_v1": [ "2.71828182845905^{cosh(8*x)} * sinh(8*x) * 8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If f(x)=e^{\\cosh (2x)} then $f'(x)=$ [ANS].", "answer_v2": [ "2.71828182845905^{cosh(2*x)} * sinh(2*x) * 2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If f(x)=e^{\\cosh (4x)} then $f'(x)=$ [ANS].", "answer_v3": [ "2.71828182845905^{cosh(4*x)} * sinh(4*x) * 4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0205", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of hyperbolic functions", "level": "2", "keywords": [ "calculus", "derivative" ], "problem_v1": "Find the derivative of the function $f(t)=\\cosh (e^{5 t})$ $f'(t)=$ [ANS]", "answer_v1": [ "5*e^{5*t}*sinh(e^{5*t})" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of the function $f(t)=\\cosh (e^{2 t})$ $f'(t)=$ [ANS]", "answer_v2": [ "2*e^{2*t}*sinh(e^{2*t})" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of the function $f(t)=\\cosh (e^{3 t})$ $f'(t)=$ [ANS]", "answer_v3": [ "3*e^{3*t}*sinh(e^{3*t})" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0206", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of hyperbolic functions", "level": "4", "keywords": [ "calculus", "derivative" ], "problem_v1": "Using a calculator or computer, sketch the graph of $y=5 e^x+4 e^{-x}$ for $-3 \\le x \\le 3$, $0\\le y\\le 20$. Observe that it looks like the graph of $y=\\cosh x$. Approximately where is its minimum? Minimum at $x=$ [ANS]\nShow algebraically that $y=5 e^x+4 e^{-x}$ can be written in the form $y=A\\cosh (x-c)$. Calculate the values of $A$ and $c$. $A=$ [ANS]\n$c=$ [ANS]\n(Think what this tells you about the graph you obtained!) (Think what this tells you about the graph you obtained!)", "answer_v1": [ "0.5*ln(4/5)", "2*sqrt(5*4)", "0.5*ln(4/5)" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Using a calculator or computer, sketch the graph of $y=2 e^x+5 e^{-x}$ for $-3 \\le x \\le 3$, $0\\le y\\le 20$. Observe that it looks like the graph of $y=\\cosh x$. Approximately where is its minimum? Minimum at $x=$ [ANS]\nShow algebraically that $y=2 e^x+5 e^{-x}$ can be written in the form $y=A\\cosh (x-c)$. Calculate the values of $A$ and $c$. $A=$ [ANS]\n$c=$ [ANS]\n(Think what this tells you about the graph you obtained!) (Think what this tells you about the graph you obtained!)", "answer_v2": [ "0.5*ln(5/2)", "2*sqrt(2*5)", "0.5*ln(5/2)" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Using a calculator or computer, sketch the graph of $y=3 e^x+4 e^{-x}$ for $-3 \\le x \\le 3$, $0\\le y\\le 20$. Observe that it looks like the graph of $y=\\cosh x$. Approximately where is its minimum? Minimum at $x=$ [ANS]\nShow algebraically that $y=3 e^x+4 e^{-x}$ can be written in the form $y=A\\cosh (x-c)$. Calculate the values of $A$ and $c$. $A=$ [ANS]\n$c=$ [ANS]\n(Think what this tells you about the graph you obtained!) (Think what this tells you about the graph you obtained!)", "answer_v3": [ "0.5*ln(4/3)", "2*sqrt(3*4)", "0.5*ln(4/3)" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0207", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Derivatives of hyperbolic functions", "level": "2", "keywords": [], "problem_v1": "Find each derivative. $\\frac{d}{dx}\\sinh(6x)=$ [ANS]\n$\\frac{d}{dx}\\cosh(5x)=$ [ANS]\n$\\frac{d}{dx}\\tanh(5x)=$ [ANS]", "answer_v1": [ "6*cosh(6 x)", "5*sinh(5 x)", "5/cosh(5 x)^2" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Find each derivative. $\\frac{d}{dx}\\sinh(2x)=$ [ANS]\n$\\frac{d}{dx}\\cosh(7x)=$ [ANS]\n$\\frac{d}{dx}\\tanh(2x)=$ [ANS]", "answer_v2": [ "2*cosh(2 x)", "7*sinh(7 x)", "2/cosh(2 x)^2" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Find each derivative. $\\frac{d}{dx}\\sinh(3x)=$ [ANS]\n$\\frac{d}{dx}\\cosh(5x)=$ [ANS]\n$\\frac{d}{dx}\\tanh(3x)=$ [ANS]", "answer_v3": [ "3*cosh(3 x)", "5*sinh(5 x)", "3/cosh(3 x)^2" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0208", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "2", "keywords": [ "derivatives", "Higher Derivative" ], "problem_v1": "Let $g(s)=(5 s-5)^ 8$. Find the following derivatives.\n$g'(s)=$ [ANS]\n$g'(4)=$ [ANS]\n$g''(s)=$ [ANS]\n$g''(4)=$ [ANS]", "answer_v1": [ "8 * (5*s -5)^7 * 5", "8 * (5*4 -5)^7 * 5", "8*7*(5*s -5)^6 * 5^2", "8*7*(5*4 -5)^6 * 5*5" ], "answer_type_v1": [ "EX", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $g(s)=(2 s-8)^ 6$. Find the following derivatives.\n$g'(s)=$ [ANS]\n$g'(2)=$ [ANS]\n$g''(s)=$ [ANS]\n$g''(2)=$ [ANS]", "answer_v2": [ "6 * (2*s -8)^5 * 2", "6 * (2*2 -8)^5 * 2", "6*5*(2*s -8)^4 * 2^2", "6*5*(2*2 -8)^4 * 2*2" ], "answer_type_v2": [ "EX", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $g(s)=(3 s-5)^ 7$. Find the following derivatives.\n$g'(s)=$ [ANS]\n$g'(3)=$ [ANS]\n$g''(s)=$ [ANS]\n$g''(3)=$ [ANS]", "answer_v3": [ "7 * (3*s -5)^6 * 3", "7 * (3*3 -5)^6 * 3", "7*6*(3*s -5)^5 * 3^2", "7*6*(3*3 -5)^5 * 3*3" ], "answer_type_v3": [ "EX", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0209", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "4", "keywords": [ "higher derivatives", "implicit function" ], "problem_v1": "Let $x^3+y^3=126$. Find $y''(x)$ at the point $(5,1)$.\n$y''(5)=$ [ANS]", "answer_v1": [ "-2*5*126" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $x^3+y^3=9$. Find $y''(x)$ at the point $(2,1)$.\n$y''(2)=$ [ANS]", "answer_v2": [ "-2*2*9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $x^3+y^3=28$. Find $y''(x)$ at the point $(3,1)$.\n$y''(3)=$ [ANS]", "answer_v3": [ "-2*3*28" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0210", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "2", "keywords": [ "calculus", "derivatives", "polynomial functions", "power rule" ], "problem_v1": "Calculate the second and third derivatives. y=7x^{4}-6x^{2}+6x $y''=$ [ANS]\n$y'''=$ [ANS]", "answer_v1": [ "84*x^2-12", "168*x" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Calculate the second and third derivatives. y=x^{4}-9x^{2}+2x $y''=$ [ANS]\n$y'''=$ [ANS]", "answer_v2": [ "12*x^2-18", "24*x" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Calculate the second and third derivatives. y=3x^{4}-6x^{2}+3x $y''=$ [ANS]\n$y'''=$ [ANS]", "answer_v3": [ "36*x^2-12", "72*x" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0211", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "2", "keywords": [ "calculus", "derivatives", "polynomial functions", "power rule" ], "problem_v1": "Calculate $y^{(k)}(0)$ for $0 \\le k \\le 5$, where $y=8x^4+a x^3+b x^2+c x+d$ (with a,b,c,d the constants) $y^{(0)}(0)=$ [ANS]\n$y^{(1)}(0)=$ [ANS]\n$y^{(2)}(0)=$ [ANS]\n$y^{(3)}(0)=$ [ANS]\n$y^{(4)}(0)=$ [ANS]\n$y^{(5)}(0)=$ [ANS]", "answer_v1": [ "d", "c", "2*b", "6*a", "192", "0" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Calculate $y^{(k)}(0)$ for $0 \\le k \\le 5$, where $y=2x^4+a x^3+b x^2+c x+d$ (with a,b,c,d the constants) $y^{(0)}(0)=$ [ANS]\n$y^{(1)}(0)=$ [ANS]\n$y^{(2)}(0)=$ [ANS]\n$y^{(3)}(0)=$ [ANS]\n$y^{(4)}(0)=$ [ANS]\n$y^{(5)}(0)=$ [ANS]", "answer_v2": [ "d", "c", "2*b", "6*a", "48", "0" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Calculate $y^{(k)}(0)$ for $0 \\le k \\le 5$, where $y=4x^4+a x^3+b x^2+c x+d$ (with a,b,c,d the constants) $y^{(0)}(0)=$ [ANS]\n$y^{(1)}(0)=$ [ANS]\n$y^{(2)}(0)=$ [ANS]\n$y^{(3)}(0)=$ [ANS]\n$y^{(4)}(0)=$ [ANS]\n$y^{(5)}(0)=$ [ANS]", "answer_v3": [ "d", "c", "2*b", "6*a", "96", "0" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0212", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "calculus", "derivatives", "trigonometric functions" ], "problem_v1": "Calculate the first five derivatives of $f(x)=-\\left(\\cos\\!\\left(x\\right)\\right)$. Then determine $f^{(12)}$ and $f^{(33)}$. $f'(x)=$ [ANS]\n$f''(x)=$ [ANS]\n$f^{(3)} (x)=$ [ANS]\n$f^{(4)} (x)=$ [ANS]\n$f^{(5)} (x)=$ [ANS]\n$f^{(12)} (x)=$ [ANS]\n$f^{(33)} (x)=$ [ANS]", "answer_v1": [ "sin(x)", "cos(x)", "-[sin(x)]", "-[cos(x)]", "sin(x)", "-[cos(x)]", "sin(x)" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Calculate the first five derivatives of $f(x)=\\sin\\!\\left(x\\right)$. Then determine $f^{(16)}$ and $f^{(21)}$. $f'(x)=$ [ANS]\n$f''(x)=$ [ANS]\n$f^{(3)} (x)=$ [ANS]\n$f^{(4)} (x)=$ [ANS]\n$f^{(5)} (x)=$ [ANS]\n$f^{(16)} (x)=$ [ANS]\n$f^{(21)} (x)=$ [ANS]", "answer_v2": [ "cos(x)", "-[sin(x)]", "-[cos(x)]", "sin(x)", "cos(x)", "sin(x)", "cos(x)" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Calculate the first five derivatives of $f(x)=\\cos\\!\\left(x\\right)$. Then determine $f^{(12)}$ and $f^{(25)}$. $f'(x)=$ [ANS]\n$f''(x)=$ [ANS]\n$f^{(3)} (x)=$ [ANS]\n$f^{(4)} (x)=$ [ANS]\n$f^{(5)} (x)=$ [ANS]\n$f^{(12)} (x)=$ [ANS]\n$f^{(25)} (x)=$ [ANS]", "answer_v3": [ "-[sin(x)]", "-[cos(x)]", "sin(x)", "cos(x)", "-[sin(x)]", "cos(x)", "-[sin(x)]" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0213", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "2", "keywords": [], "problem_v1": "Find the $59$ th derivative of $\\sin(x)$ by finding the first few derivatives and observing the pattern that occurs. $(\\sin(x))^{(59)}=$ [ANS]", "answer_v1": [ "- cos x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the $76$ th derivative of $\\sin(x)$ by finding the first few derivatives and observing the pattern that occurs. $(\\sin(x))^{(76)}=$ [ANS]", "answer_v2": [ "sin x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the $57$ th derivative of $\\sin(x)$ by finding the first few derivatives and observing the pattern that occurs. $(\\sin(x))^{(57)}=$ [ANS]", "answer_v3": [ "cos x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0214", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "Higher Derivative", "higher derivatives", "" ], "problem_v1": "Let f(x)=\\frac{8x^{4}}{1-x} $f^{(4)}(x)=$ [ANS]\nNote: There is a way of doing this problem without using the quotient rule 4 times.", "answer_v1": [ "8 * 24 * (1-x)^{-5}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let f(x)=\\frac{2x^{4}}{1-x} $f^{(4)}(x)=$ [ANS]\nNote: There is a way of doing this problem without using the quotient rule 4 times.", "answer_v2": [ "2 * 24 * (1-x)^{-5}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let f(x)=\\frac{4x^{4}}{1-x} $f^{(4)}(x)=$ [ANS]\nNote: There is a way of doing this problem without using the quotient rule 4 times.", "answer_v3": [ "4 * 24 * (1-x)^{-5}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0215", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "2", "keywords": [ "Higher Derivative", "derivatives" ], "problem_v1": "If $g(t)=3 t^4+6 t^2-4$ find\n$g(0)=$ [ANS]\n$g'(0)=$ [ANS]\n$g''(0)=$ [ANS]\n$g'''(0)=$ [ANS]\n$g^{(4)}(0)=$ [ANS]\n$g^{(5)}(0)=$ [ANS]", "answer_v1": [ "-4", "0", "12", "0", "72", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "If $g(t)=1 t^4-3 t^2-9$ find\n$g(0)=$ [ANS]\n$g'(0)=$ [ANS]\n$g''(0)=$ [ANS]\n$g'''(0)=$ [ANS]\n$g^{(4)}(0)=$ [ANS]\n$g^{(5)}(0)=$ [ANS]", "answer_v2": [ "-9", "0", "-6", "0", "24", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "If $g(t)=1 t^4+4 t^2-3$ find\n$g(0)=$ [ANS]\n$g'(0)=$ [ANS]\n$g''(0)=$ [ANS]\n$g'''(0)=$ [ANS]\n$g^{(4)}(0)=$ [ANS]\n$g^{(5)}(0)=$ [ANS]", "answer_v3": [ "-3", "0", "8", "0", "24", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0216", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "calculus", "derivative", "exponentials" ], "problem_v1": "Find the quadratic polynomial $g(x)=a x^2+b x+c$ which best fits the function $f(x)=7^x$ at $x=0$, in the sense that $g(0)=f(0)$, and $g'(0)=f'(0)$, and $g''(0)=f''(0)$. $g(x)=$ [ANS]\n(Using a computer or calculator, sketch graphs of $f$ and $g$ on the same axes. What do you notice?)", "answer_v1": [ "1/2*[ln(7)*x]^2+ln(7)*x+1" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the quadratic polynomial $g(x)=a x^2+b x+c$ which best fits the function $f(x)=e^x$ at $x=0$, in the sense that $g(0)=f(0)$, and $g'(0)=f'(0)$, and $g''(0)=f''(0)$. $g(x)=$ [ANS]\n(Using a computer or calculator, sketch graphs of $f$ and $g$ on the same axes. What do you notice?)", "answer_v2": [ "1/2*[ln(2.71828)*x]^2+ln(2.71828)*x+1" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the quadratic polynomial $g(x)=a x^2+b x+c$ which best fits the function $f(x)=3^x$ at $x=0$, in the sense that $g(0)=f(0)$, and $g'(0)=f'(0)$, and $g''(0)=f''(0)$. $g(x)=$ [ANS]\n(Using a computer or calculator, sketch graphs of $f$ and $g$ on the same axes. What do you notice?)", "answer_v3": [ "1/2*[ln(3)*x]^2+ln(3)*x+1" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0217", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "derivative" ], "problem_v1": "Find $\\frac{d^2y}{dx^2}$ for y=16x^{2}\\cos\\!\\left(x\\right)+15\\sin\\!\\left(x\\right) $\\frac{d^2y}{dx^2}=$ [ANS]", "answer_v1": [ "32*cos(x)-64*x*sin(x)-16*x^2*cos(x)-15*sin(x)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find $\\frac{d^2y}{dx^2}$ for y=2x^{2}\\cos\\!\\left(x\\right)+24\\sin\\!\\left(x\\right) $\\frac{d^2y}{dx^2}=$ [ANS]", "answer_v2": [ "4*cos(x)-8*x*sin(x)-2*x^2*cos(x)-24*sin(x)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find $\\frac{d^2y}{dx^2}$ for y=7x^{2}\\cos\\!\\left(x\\right)+16\\sin\\!\\left(x\\right) $\\frac{d^2y}{dx^2}=$ [ANS]", "answer_v3": [ "14*cos(x)-28*x*sin(x)-7*x^2*cos(x)-16*sin(x)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0218", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "derivative", "derivatives", "slope", "tangent", "tangent line", "tangent lines" ], "problem_v1": "Compute $f'(x)$, $f''(x)$, $f'''(x)$, and then state a formula for $f^{(n)}(x)$, when f(x)=\\frac{1}{x^{4}} $f'(x)=$ [ANS]\n$f''(x)=$ [ANS]\n$f'''(x)=$ [ANS]\n$f^{(n)}(x)=$ [ANS]\n[Hint: The expression $(-1)^n$ has value $1$ if $n$ is even and $-1$ if $n$ is odd. This expression can be used in your answer for the last part.]", "answer_v1": [ "-[4/(x^5)]", "20/(x^6)", "-[120/(x^7)]", "(-1)^n*[(n+3)!]/[6*x^{n+4}]" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Compute $f'(x)$, $f''(x)$, $f'''(x)$, and then state a formula for $f^{(n)}(x)$, when f(x)=\\frac{4}{x^{2}} $f'(x)=$ [ANS]\n$f''(x)=$ [ANS]\n$f'''(x)=$ [ANS]\n$f^{(n)}(x)=$ [ANS]\n[Hint: The expression $(-1)^n$ has value $1$ if $n$ is even and $-1$ if $n$ is odd. This expression can be used in your answer for the last part.]", "answer_v2": [ "-[8/(x^3)]", "24/(x^4)", "-[96/(x^5)]", "4*(-1)^n*[(n+1)!]/[x^{n+2}]" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Compute $f'(x)$, $f''(x)$, $f'''(x)$, and then state a formula for $f^{(n)}(x)$, when f(x)=\\frac{1}{x^{2}} $f'(x)=$ [ANS]\n$f''(x)=$ [ANS]\n$f'''(x)=$ [ANS]\n$f^{(n)}(x)=$ [ANS]\n[Hint: The expression $(-1)^n$ has value $1$ if $n$ is even and $-1$ if $n$ is odd. This expression can be used in your answer for the last part.]", "answer_v3": [ "-[2/(x^3)]", "6/(x^4)", "-[24/(x^5)]", "(-1)^n*[(n+1)!]/[x^{n+2}]" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0219", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "calculus", "derivative", "logarithm", "trigonometric", "chain rule", "derivatives", "logarithmic functions" ], "problem_v1": "Let f(x)=5 \\ln(\\cos x) Then $f'(x)=$ [ANS]\nand $f''(x)=$ [ANS]", "answer_v1": [ "- 5*sin(x)/cos(x)", "- 5/(cos(x))^2" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let f(x)=2 \\ln(\\sin x) Then $f'(x)=$ [ANS]\nand $f''(x)=$ [ANS]", "answer_v2": [ "2*cos(x)/sin(x)", "- 2/(sin(x))^2" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let f(x)=3 \\ln(\\sin x) Then $f'(x)=$ [ANS]\nand $f''(x)=$ [ANS]", "answer_v3": [ "3*cos(x)/sin(x)", "- 3/(sin(x))^2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0220", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Compute the first three derivatives of $f(t)=\\sqrt{10t^{2}+9}$.\n$f'(t)=$ [ANS]\n$f''(t)=$ [ANS]\n$f'''(t)=$ [ANS]", "answer_v1": [ "10*t/[sqrt(10*t^2+9)]", "90/[(10*t^2+9)^{1.5}]", "-2700*t/[(10*t^2+9)^{2.5}]" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Compute the first three derivatives of $f(t)=\\sqrt{14-17t^{2}}$.\n$f'(t)=$ [ANS]\n$f''(t)=$ [ANS]\n$f'''(t)=$ [ANS]", "answer_v2": [ "-17*t/[sqrt(14-17*t^2)]", "-238/[(14-17*t^2)^{1.5}]", "-12138*t/[(14-17*t^2)^{2.5}]" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Compute the first three derivatives of $f(t)=\\sqrt{10-8t^{2}}$.\n$f'(t)=$ [ANS]\n$f''(t)=$ [ANS]\n$f'''(t)=$ [ANS]", "answer_v3": [ "-8*t/[sqrt(10-8*t^2)]", "-80/[(10-8*t^2)^{1.5}]", "-1920*t/[(10-8*t^2)^{2.5}]" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0221", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "calculus", "differentiation" ], "problem_v1": "Let $ g(u)=\\frac{5}{\\sqrt{6-3 u}}$.\n$g'(u)$=[ANS]\n$g''(u)$=[ANS]", "answer_v1": [ "(5)*(1/2)*(3)*((6-3*u)^{-1.5})", "(5)*(3/4)*((3)^2)*((6-3*u)^{-2.5})" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let $ g(u)=\\frac{2}{\\sqrt{8-2 u}}$.\n$g'(u)$=[ANS]\n$g''(u)$=[ANS]", "answer_v2": [ "(2)*(1/2)*(2)*((8-2*u)^{-1.5})", "(2)*(3/4)*((2)^2)*((8-2*u)^{-2.5})" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let $ g(u)=\\frac{3}{\\sqrt{6-2 u}}$.\n$g'(u)$=[ANS]\n$g''(u)$=[ANS]", "answer_v3": [ "(3)*(1/2)*(2)*((6-2*u)^{-1.5})", "(3)*(3/4)*((2)^2)*((6-2*u)^{-2.5})" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0222", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "2", "keywords": [ "calculus", "differentiation" ], "problem_v1": "Let $f(r)=5 \\sqrt{r}+6 \\sqrt[3]{r}$.\n$f'(r)=$ [ANS]\n$f'(4)=$ [ANS],\n$f''(r)=$ [ANS]\n$f''(4)=$ [ANS]", "answer_v1": [ "2.5*r^{-0.5}+2*r^{-0.666667}", "2.0437", "(-1.25)*r^{-1.5}+(-1.33333)*r^{-1.66667}", "-0.288533" ], "answer_type_v1": [ "EX", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $f(r)=2 \\sqrt{r}+8 \\sqrt[3]{r}$.\n$f'(r)=$ [ANS]\n$f'(2)=$ [ANS],\n$f''(r)=$ [ANS]\n$f''(2)=$ [ANS]", "answer_v2": [ "1*r^{-0.5}+2.66667*r^{-0.666667}", "2.387", "(-0.5)*r^{-1.5}+(-1.77778)*r^{-1.66667}", "-0.736742" ], "answer_type_v2": [ "EX", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $f(r)=3 \\sqrt{r}+6 \\sqrt[3]{r}$.\n$f'(r)=$ [ANS]\n$f'(3)=$ [ANS],\n$f''(r)=$ [ANS]\n$f''(3)=$ [ANS]", "answer_v3": [ "1.5*r^{-0.5}+2*r^{-0.666667}", "1.82753", "(-0.75)*r^{-1.5}+(-1.33333)*r^{-1.66667}", "-0.358004" ], "answer_type_v3": [ "EX", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0223", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "Trigonometry", "Derivative" ], "problem_v1": "Find the 83rd derivative of $\\sin(x)$ at $x=4$ by finding the first few derivatives and observing the pattern that occurs. [ANS]", "answer_v1": [ "0.653643620863612" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the 83rd derivative of $\\sin(x)$ at $x=1$ by finding the first few derivatives and observing the pattern that occurs. [ANS]", "answer_v2": [ "-0.54030230586814" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the 83rd derivative of $\\sin(x)$ at $x=2$ by finding the first few derivatives and observing the pattern that occurs. [ANS]", "answer_v3": [ "0.416146836547142" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0224", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "Higher Derivative", "Trigonometry", "trigonometry" ], "problem_v1": "Let $f(x)=x \\sin(x)$. Find $f''(3.8)$. (Remember--radian mode!) [ANS]", "answer_v1": [ "0.743124561753502" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)=x \\sin(x)$. Find $f''(-2.3)$. (Remember--radian mode!) [ANS]", "answer_v2": [ "-3.0476740305661" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(x)=x \\sin(x)$. Find $f''(-0.2)$. (Remember--radian mode!) [ANS]", "answer_v3": [ "1.92039928952347" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0225", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "derivatives" ], "problem_v1": "$ \\frac{d}{dx} [(3x^2+x+1) e^x]=$ [ANS].\n$ \\frac{d^2}{dx^2} [(3x^2+x+1) e^x]=$ [ANS].", "answer_v1": [ "(3*x^2+(2*3+1)*x + 1+1)*e^{x}", "(3*x^2 + (4*3+1)*x + 2*3 + 2*1+1)*e^{x}" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "$ \\frac{d}{dx} [(-5x^2+5x-4) e^x]=$ [ANS].\n$ \\frac{d^2}{dx^2} [(-5x^2+5x-4) e^x]=$ [ANS].", "answer_v2": [ "(-5*x^2+(2*-5+5)*x + 5+-4)*e^{x}", "(-5*x^2 + (4*-5+5)*x + 2*-5 + 2*5+-4)*e^{x}" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "$ \\frac{d}{dx} [(-2x^2+x-2) e^x]=$ [ANS].\n$ \\frac{d^2}{dx^2} [(-2x^2+x-2) e^x]=$ [ANS].", "answer_v3": [ "(-2*x^2+(2*-2+1)*x + 1+-2)*e^{x}", "(-2*x^2 + (4*-2+1)*x + 2*-2 + 2*1+-2)*e^{x}" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0226", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "Derivatives" ], "problem_v1": "A mass attached to a vertical spring has position function given by $s(t)=5 \\sin (4 t+3.9)+4$ where $t$ is measured in seconds and $s$ in inches. This is an example of simple harmonic motion. Find the velocity at time $t$. $v(t)=$ [ANS] inches per second. Find the acceleration at time $t$. $a(t)=$ [ANS] inches per second.", "answer_v1": [ "5 *4 * cos(4*t+3.9)", "-5 *4*4 * sin(4*t + 3.9)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "A mass attached to a vertical spring has position function given by $s(t)=2 \\sin (5 t+1)+2$ where $t$ is measured in seconds and $s$ in inches. This is an example of simple harmonic motion. Find the velocity at time $t$. $v(t)=$ [ANS] inches per second. Find the acceleration at time $t$. $a(t)=$ [ANS] inches per second.", "answer_v2": [ "2 *5 * cos(5*t+1)", "-2 *5*5 * sin(5*t + 1)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "A mass attached to a vertical spring has position function given by $s(t)=3 \\sin (4 t+1.8)+3$ where $t$ is measured in seconds and $s$ in inches. This is an example of simple harmonic motion. Find the velocity at time $t$. $v(t)=$ [ANS] inches per second. Find the acceleration at time $t$. $a(t)=$ [ANS] inches per second.", "answer_v3": [ "3 *4 * cos(4*t+1.8)", "-3 *4*4 * sin(4*t + 1.8)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0227", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "3", "keywords": [ "derivatives", "chain rule", "Concavity and Points of Inflection" ], "problem_v1": "Let $f(x)=e^{-4x^2}$.\n$f''(x)=$ [ANS].\nThe solutions to $f''(x)=0$ are $x=\\pm$ [ANS].", "answer_v1": [ "(4*4^2*x^2-2*4)*exp(-4*x^2)", "1/sqrt(2*4)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)=e^{-x^2}$.\n$f''(x)=$ [ANS].\nThe solutions to $f''(x)=0$ are $x=\\pm$ [ANS].", "answer_v2": [ "(4*1^2*x^2-2*1)*exp(-1*x^2)", "1/sqrt(2*1)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)=e^{-2x^2}$.\n$f''(x)=$ [ANS].\nThe solutions to $f''(x)=0$ are $x=\\pm$ [ANS].", "answer_v3": [ "(4*2^2*x^2-2*2)*exp(-2*x^2)", "1/sqrt(2*2)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0228", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Higher-order derivatives", "level": "", "keywords": [ "calculus", "derivative", "higher order derivatives", "trigonometric functions" ], "problem_v1": "Find $ \\frac{d^{90}}{dx^{90}}{\\textrm{cos}} ({0.5} x).$\nAnswer: [ANS]", "answer_v1": [ "-0.5^90*cos(0.5*x)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find $ \\frac{d^{63}}{dx^{63}}{\\textrm{sin}} ({3} x).$\nAnswer: [ANS]", "answer_v2": [ "-3^63*cos(3*x)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find $ \\frac{d^{72}}{dx^{72}}{\\textrm{sin}} ({0.5} x).$\nAnswer: [ANS]", "answer_v3": [ "0.5^72*sin(0.5*x)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0229", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Logarithmic differentiation", "level": "2", "keywords": [ "Derivative", "Polynomial" ], "problem_v1": "Let $\\>y=\\left(x^{3}-8x\\right)^{\\ln\\!\\left(x\\right)}$. Find $\\>dy/dx\\>$ using the method of logarithmic differentiation.\nAnswer: [ANS]", "answer_v1": [ "[1/x*ln(x^3-8*x)+1/(x^3-8*x)*(3*x^2-8)*ln(x)]*exp(ln(x)*ln(x^3-8*x))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $\\>y=\\left(x^{3}-2x\\right)^{\\ln\\!\\left(x\\right)}$. Find $\\>dy/dx\\>$ using the method of logarithmic differentiation.\nAnswer: [ANS]", "answer_v2": [ "[1/x*ln(x^3-2*x)+1/(x^3-2*x)*(3*x^2-2)*ln(x)]*exp(ln(x)*ln(x^3-2*x))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $\\>y=\\left(x^{3}-4x\\right)^{\\ln\\!\\left(x\\right)}$. Find $\\>dy/dx\\>$ using the method of logarithmic differentiation.\nAnswer: [ANS]", "answer_v3": [ "[1/x*ln(x^3-4*x)+1/(x^3-4*x)*(3*x^2-4)*ln(x)]*exp(ln(x)*ln(x^3-4*x))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0230", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Logarithmic differentiation", "level": "4", "keywords": [ "Calculus", "Derivatives" ], "problem_v1": "Find the indicated derivatives.\n(a) $\\frac{d}{dx}\\left(e^{x^6}+\\log_{4}(\\pi)\\right)$=[ANS]\n(b) $\\frac{d}{dx}\\left(\\left(\\sqrt[6]{x} \\right) ^{\\ln(x)}\\right)$=[ANS]", "answer_v1": [ "6*x^5*e^{x^6}", "2*ln(x)*x^{[ln(x)]/6}/(6*x)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find the indicated derivatives.\n(a) $\\frac{d}{dx}\\left(e^{x^3}+\\log_{5}(\\pi)\\right)$=[ANS]\n(b) $\\frac{d}{dx}\\left(\\left(\\sqrt[3]{x} \\right) ^{\\ln(x)}\\right)$=[ANS]", "answer_v2": [ "3*x^2*e^{x^3}", "2*ln(x)*x^{[ln(x)]/3}/(3*x)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find the indicated derivatives.\n(a) $\\frac{d}{dx}\\left(e^{x^4}+\\log_{4}(\\pi)\\right)$=[ANS]\n(b) $\\frac{d}{dx}\\left(\\left(\\sqrt[4]{x} \\right) ^{\\ln(x)}\\right)$=[ANS]", "answer_v3": [ "4*x^3*e^{x^4}", "2*ln(x)*x^{[ln(x)]/4}/(4*x)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0231", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Logarithmic differentiation", "level": "3", "keywords": [], "problem_v1": "$\\frac{d}{dx}\\left(\\left(\\sqrt[6]{x} \\right) ^{\\ln(x)}\\right)$=[ANS]", "answer_v1": [ "[6*1/x/36*ln(x)+[ln(x)]/6*1/x]*exp([ln(x)]/6*ln(x))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "$\\frac{d}{dx}\\left(\\left(\\sqrt[3]{x} \\right) ^{\\ln(x)}\\right)$=[ANS]", "answer_v2": [ "[3*1/x/9*ln(x)+[ln(x)]/3*1/x]*exp([ln(x)]/3*ln(x))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "$\\frac{d}{dx}\\left(\\left(\\sqrt[4]{x} \\right) ^{\\ln(x)}\\right)$=[ANS]", "answer_v3": [ "[4*1/x/16*ln(x)+[ln(x)]/4*1/x]*exp([ln(x)]/4*ln(x))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0232", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "derivatives", "implicit", "Implicit", "Derivative", "Trigonometry", "Implicit Derivative", "calculus", "derivative", "implicit differentiation" ], "problem_v1": "Find $y'$ by implicit differentiation. Match the equations defining $y$ implicitly with the letters labeling the expressions for $y'$. [ANS] 1. $6x\\sin y+5 \\sin 2y=5 \\cos y$ [ANS] 2. $6x\\sin y+5 \\cos 2y=5 \\cos y$ [ANS] 3. $6x\\cos y+5 \\sin 2y=5 \\sin y$ [ANS] 4. $6x\\cos y+5 \\cos 2y=5 \\sin y$\nA. $ y'=\\frac {6 \\cos y} {6x \\sin y-10 \\cos 2y+5 \\cos y}$ B. $ y'=\\frac {6 \\sin y} {10 \\sin 2y-6x \\cos y-5 \\sin y}$ C. $ y'=\\frac {6 \\sin y} {-6x \\cos y-10 \\cos 2y-5 \\sin y}$ D. $ y'=\\frac {6 \\cos y} {6x \\sin y+10 \\sin 2y+5 \\cos y}$", "answer_v1": [ "C", "B", "A", "D" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Find $y'$ by implicit differentiation. Match the equations defining $y$ implicitly with the letters labeling the expressions for $y'$. [ANS] 1. $2x\\cos y+7 \\sin 2y=2 \\sin y$ [ANS] 2. $2x\\cos y+7 \\cos 2y=2 \\sin y$ [ANS] 3. $2x\\sin y+7 \\cos 2y=2 \\cos y$ [ANS] 4. $2x\\sin y+7 \\sin 2y=2 \\cos y$\nA. $ y'=\\frac {2 \\cos y} {2x \\sin y+14 \\sin 2y+2 \\cos y}$ B. $ y'=\\frac {2 \\sin y} {14 \\sin 2y-2x \\cos y-2 \\sin y}$ C. $ y'=\\frac {2 \\cos y} {2x \\sin y-14 \\cos 2y+2 \\cos y}$ D. $ y'=\\frac {2 \\sin y} {-2x \\cos y-14 \\cos 2y-2 \\sin y}$", "answer_v2": [ "C", "A", "B", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Find $y'$ by implicit differentiation. Match the equations defining $y$ implicitly with the letters labeling the expressions for $y'$. [ANS] 1. $3x\\sin y+5 \\sin 2y=3 \\cos y$ [ANS] 2. $3x\\sin y+5 \\cos 2y=3 \\cos y$ [ANS] 3. $3x\\cos y+5 \\sin 2y=3 \\sin y$ [ANS] 4. $3x\\cos y+5 \\cos 2y=3 \\sin y$\nA. $ y'=\\frac {3 \\cos y} {3x \\sin y+10 \\sin 2y+3 \\cos y}$ B. $ y'=\\frac {3 \\cos y} {3x \\sin y-10 \\cos 2y+3 \\cos y}$ C. $ y'=\\frac {3 \\sin y} {-3x \\cos y-10 \\cos 2y-3 \\sin y}$ D. $ y'=\\frac {3 \\sin y} {10 \\sin 2y-3x \\cos y-3 \\sin y}$", "answer_v3": [ "C", "D", "B", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_single_variable_0233", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "derivatives", "tangent line", "implicit differentiation", "calculus", "Implicit", "Derivative", "Tangent", "Implicit Derivative" ], "problem_v1": "Find the slope of the tangent line to the curve defined by $5x^{5}+3xy-4y^{4}=-65$ at the point $(1,-2)$.\nThe slope of the curve at the point $(1,-2)$ is [ANS].", "answer_v1": [ "-19/131" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the slope of the tangent line to the curve defined by $8x^{3}-xy+2y^{4}=740$ at the point $(3,-4)$.\nThe slope of the curve at the point $(3,-4)$ is [ANS].", "answer_v2": [ "44/103" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the slope of the tangent line to the curve defined by $4x^{3}+4xy-5y^{2}=880$ at the point $(6,4)$.\nThe slope of the curve at the point $(6, 4)$ is [ANS].", "answer_v3": [ "28" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0234", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [], "problem_v1": "Find $ \\frac{dy}{dx}$ using implicit differentiation. \\dfrac{8x^2+6 y}{6x+7 y^2}=-10 Hint: First clear the denominator to get $8x^2+6 y=-10(6x+7 y^2)$. $\\dfrac{dy}{dx}=$ [ANS]", "answer_v1": [ "(-60 - 16 x)/(6 - -140 y)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find $ \\frac{dy}{dx}$ using implicit differentiation. \\dfrac{2x^2+9 y}{3x+4 y^2}=-29 Hint: First clear the denominator to get $2x^2+9 y=-29(3x+4 y^2)$. $\\dfrac{dy}{dx}=$ [ANS]", "answer_v2": [ "(-87 - 4 x)/(9 - -232 y)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find $ \\frac{dy}{dx}$ using implicit differentiation. \\dfrac{3x^2+4 y}{8x+9 y^2}=-27 Hint: First clear the denominator to get $3x^2+4 y=-27(8x+9 y^2)$. $\\dfrac{dy}{dx}=$ [ANS]", "answer_v3": [ "(-216 - 6 x)/(4 - -486 y)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0235", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "calculus", "derivatives", "implicit differentiation" ], "problem_v1": "Calculate the derivative of $y$ with respect to $x$. $xe^{y}=5xy+4y^{4}$ $\\frac{dy}{dx}=$ [ANS]", "answer_v1": [ "(e^y-5*y)/(5*x+16*y^3-x*e^y)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Calculate the derivative of $y$ with respect to $x$. $xe^{y}=2xy+6y^{2}$ $\\frac{dy}{dx}=$ [ANS]", "answer_v2": [ "(e^y-2*y)/(2*x+12*y-x*e^y)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Calculate the derivative of $y$ with respect to $x$. $xe^{y}=3xy+4y^{3}$ $\\frac{dy}{dx}=$ [ANS]", "answer_v3": [ "(e^y-3*y)/(3*x+12*y^2-x*e^y)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0236", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "derivative' 'implicit" ], "problem_v1": "For the equation given below, evaluate $\\frac{dy}{dx}$ at the point $(2,1)$. 3x^{2}+2xy+2y^{3}=18 $\\frac{dy}{dx}$ at $(2,1)$=[ANS]", "answer_v1": [ "-1.4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "For the equation given below, evaluate $\\frac{dy}{dx}$ at the point $(-4,4)$. x^{2}-xy+4y^{3}=288 $\\frac{dy}{dx}$ at $(-4,4)$=[ANS]", "answer_v2": [ "0.0612245" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "For the equation given below, evaluate $\\frac{dy}{dx}$ at the point $(-2,1)$. 2x^{2}-3xy+2y^{3}=16 $\\frac{dy}{dx}$ at $(-2,1)$=[ANS]", "answer_v3": [ "0.916667" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0237", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "Implicit", "Derivative", "calculus", "derivatives", "implicit", "Implicit Derivative" ], "problem_v1": "If $\\sqrt{x}+\\sqrt{y}=11$ and $y(49)=16$, find $y'(49)$ by implicit differentiation.\n$y'(49)=$ [ANS]", "answer_v1": [ "-4/7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $\\sqrt{x}+\\sqrt{y}=6$ and $y(25)=1$, find $y'(25)$ by implicit differentiation.\n$y'(25)=$ [ANS]", "answer_v2": [ "-1/5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $\\sqrt{x}+\\sqrt{y}=8$ and $y(25)=9$, find $y'(25)$ by implicit differentiation.\n$y'(25)=$ [ANS]", "answer_v3": [ "-3/5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0239", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "calculus", "derivative", "implicit functions", "implicit differentiation" ], "problem_v1": "Find the slope of the tangent to the curve $x y^{10}=1$ at $(1,-1)$.\n$ {dy\\over dx} \\bigg \\vert _ {(1,-1)}=$ [ANS].\n(Enter undef if the slope is not defined at this point.)", "answer_v1": [ "1/10" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the slope of the tangent to the curve $x y^{2}=1$ at $(1,-1)$.\n$ {dy\\over dx} \\bigg \\vert _ {(1,-1)}=$ [ANS].\n(Enter undef if the slope is not defined at this point.)", "answer_v2": [ "1/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the slope of the tangent to the curve $x y^{4}=1$ at $(1,-1)$.\n$ {dy\\over dx} \\bigg \\vert _ {(1,-1)}=$ [ANS].\n(Enter undef if the slope is not defined at this point.)", "answer_v3": [ "1/4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0240", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "calculus", "derivative", "implicit functions", "implicit differentiation" ], "problem_v1": "Find $dy/dx$ in terms of $x$ and $y$ if $(x-a)^8+y^8=a^8$. Assume that $a$ is a constant. ${dy\\over dx}=$ [ANS]", "answer_v1": [ "-1*8*(x-a)^7/[8*y^7]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find $dy/dx$ in terms of $x$ and $y$ if $(x-a)^2+y^2=a^2$. Assume that $a$ is a constant. ${dy\\over dx}=$ [ANS]", "answer_v2": [ "-1*2*(x-a)/[2*y]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find $dy/dx$ in terms of $x$ and $y$ if $(x-a)^4+y^4=a^4$. Assume that $a$ is a constant. ${dy\\over dx}=$ [ANS]", "answer_v3": [ "-1*4*(x-a)^3/[4*y^3]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0241", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "4", "keywords": [ "calculus", "derivative", "implicit functions", "implicit differentiation" ], "problem_v1": "If $x^{5}+y^3-2x y^2=10$, find $dy/dx$ in terms of $x$ and $y$. ${dy\\over dx}=$ [ANS]\nUsing your answer for $dy/dx$, fill in the following table of approximate $y$-values of points on the curve near $x=1, y=3$.\n$\\begin{array}{cccc}\\hline 0.96 & 0.98 & 1.02 & 1.04 \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nFinally, find the $y$-value for $x=0.96$ by substituting $x=0.96$ in the original equation and solving for $y$ using a computer or calculator. $y(0.96) \\approx$ [ANS]\nHow large (in magnitude) is the difference between your estimate for $y(0.96)$ using $dy/dx$ and your solution with a computer or calculator? [ANS]", "answer_v1": [ "[2*y^2-5*x^4]/(3*y^2-2*2*x*y)", "3+0.866667*(0.96-1)", "3+0.866667*(0.98-1)", "3+0.866667*(1.02-1)", "3+0.866667*(1.04-1)", "2.96485", "|3+0.866667*(0.96-1)-2.96485|" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "If $x^{2}+y^3-3x y^2=-1$, find $dy/dx$ in terms of $x$ and $y$. ${dy\\over dx}=$ [ANS]\nUsing your answer for $dy/dx$, fill in the following table of approximate $y$-values of points on the curve near $x=1, y=1$.\n$\\begin{array}{cccc}\\hline 0.96 & 0.98 & 1.02 & 1.04 \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nFinally, find the $y$-value for $x=0.96$ by substituting $x=0.96$ in the original equation and solving for $y$ using a computer or calculator. $y(0.96) \\approx$ [ANS]\nHow large (in magnitude) is the difference between your estimate for $y(0.96)$ using $dy/dx$ and your solution with a computer or calculator? [ANS]", "answer_v2": [ "[3*y^2-2*x^1]/(3*y^2-2*3*x*y)", "1+-0.333333*(0.96-1)", "1+-0.333333*(0.98-1)", "1+-0.333333*(1.02-1)", "1+-0.333333*(1.04-1)", "1.01508", "|1+-0.333333*(0.96-1)-1.01508|" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "If $x^{3}+y^3-4x y^2=-7$, find $dy/dx$ in terms of $x$ and $y$. ${dy\\over dx}=$ [ANS]\nUsing your answer for $dy/dx$, fill in the following table of approximate $y$-values of points on the curve near $x=1, y=2$.\n$\\begin{array}{cccc}\\hline 0.96 & 0.98 & 1.02 & 1.04 \\\\ \\hline [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nFinally, find the $y$-value for $x=0.96$ by substituting $x=0.96$ in the original equation and solving for $y$ using a computer or calculator. $y(0.96) \\approx$ [ANS]\nHow large (in magnitude) is the difference between your estimate for $y(0.96)$ using $dy/dx$ and your solution with a computer or calculator? [ANS]", "answer_v3": [ "[4*y^2-3*x^2]/(3*y^2-2*4*x*y)", "2+-3.25*(0.96-1)", "2+-3.25*(0.98-1)", "2+-3.25*(1.02-1)", "2+-3.25*(1.04-1)", "2.17829", "|2+-3.25*(0.96-1)-2.17829|" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0242", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "4", "keywords": [ "derivative" ], "problem_v1": "Find equations for two lines through the origin that are tangent to the ellipse 5x^{2}-160x+y^{2}+79=0 Enter the equation of the line with the smaller slope first. $y=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "-[sqrt(6005)*x]/[1*sqrt(79)]", "sqrt(6005)*x/[1*sqrt(79)]" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find equations for two lines through the origin that are tangent to the ellipse 2x^{2}-96x+y^{2}+71=0 Enter the equation of the line with the smaller slope first. $y=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "-[sqrt(2162)*x]/[1*sqrt(71)]", "sqrt(2162)*x/[1*sqrt(71)]" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find equations for two lines through the origin that are tangent to the ellipse 3x^{2}-96x+y^{2}+47=0 Enter the equation of the line with the smaller slope first. $y=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "-[sqrt(2163)*x]/[1*sqrt(47)]", "sqrt(2163)*x/[1*sqrt(47)]" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0243", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "4", "keywords": [ "calculus", "derivative", "tangent line", "implicit differentiation" ], "problem_v1": "Let $y$ be defined implicitly by the equation $\\ln(7 y)=4xy$. Use implicit differentiation to find the first derivative of $y$ with respect to $x$.\n$ \\frac{dy}{dx}=$ [ANS]\nUse implicit differentiation to find the second derivative of $y$ with respect to $x$.\n$ \\frac{d^2y}{dx^2}=$ [ANS]\nNote: Your answer should only involve the variables $x$ and $y$.\nFind the point on the curve where $ \\frac{d^2y}{dx^2}=0$.\n$ \\frac{d^2y}{dx^2}=0$ at the point $(x,y)=($ [ANS] $)$.", "answer_v1": [ "4*y^2/(1-4*x*y)", "16*y^3*(8*x*y-3)/[(4*x*y-1)^3]", "(21/8*e^{-1.5},1/7*e^{1.5})" ], "answer_type_v1": [ "EX", "EX", "OL" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $y$ be defined implicitly by the equation $\\ln(2 y)=5xy$. Use implicit differentiation to find the first derivative of $y$ with respect to $x$.\n$ \\frac{dy}{dx}=$ [ANS]\nUse implicit differentiation to find the second derivative of $y$ with respect to $x$.\n$ \\frac{d^2y}{dx^2}=$ [ANS]\nNote: Your answer should only involve the variables $x$ and $y$.\nFind the point on the curve where $ \\frac{d^2y}{dx^2}=0$.\n$ \\frac{d^2y}{dx^2}=0$ at the point $(x,y)=($ [ANS] $)$.", "answer_v2": [ "5*y^2/(1-5*x*y)", "25*y^3*(10*x*y-3)/[(5*x*y-1)^3]", "(3/5*e^{-1.5},1/2*e^{1.5})" ], "answer_type_v2": [ "EX", "EX", "OL" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $y$ be defined implicitly by the equation $\\ln(4 y)=4xy$. Use implicit differentiation to find the first derivative of $y$ with respect to $x$.\n$ \\frac{dy}{dx}=$ [ANS]\nUse implicit differentiation to find the second derivative of $y$ with respect to $x$.\n$ \\frac{d^2y}{dx^2}=$ [ANS]\nNote: Your answer should only involve the variables $x$ and $y$.\nFind the point on the curve where $ \\frac{d^2y}{dx^2}=0$.\n$ \\frac{d^2y}{dx^2}=0$ at the point $(x,y)=($ [ANS] $)$.", "answer_v3": [ "4*y^2/(1-4*x*y)", "16*y^3*(8*x*y-3)/[(4*x*y-1)^3]", "(3/2*e^{-1.5},1/4*e^{1.5})" ], "answer_type_v3": [ "EX", "EX", "OL" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0244", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "calculus" ], "problem_v1": "One standard equation of an elliptic curve is $y^2=x(x-a)(x-b)$, where $0 < a < b$.\nFind the slope of the elliptic curve $y^2=x \\bigl(x-\\frac{1}{6} \\bigr) \\bigl(x-\\frac{1}{4} \\bigr)$ at the point $\\bigl(1, \\frac{1}{4}\\sqrt{10} \\, \\bigr)$.\nThe slope of the elliptic curve at the given point is [ANS].", "answer_v1": [ "53/120*sqrt(10)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "One standard equation of an elliptic curve is $y^2=x(x-a)(x-b)$, where $0 < a < b$.\nFind the slope of the elliptic curve $y^2=x \\bigl(x-\\frac{1}{3} \\bigr) \\bigl(x-\\frac{1}{2} \\bigr)$ at the point $\\bigl(1,-\\frac{1}{3}\\sqrt{3} \\, \\bigr)$.\nThe slope of the elliptic curve at the given point is [ANS].", "answer_v2": [ "-(3/4)*sqrt(3)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "One standard equation of an elliptic curve is $y^2=x(x-a)(x-b)$, where $0 < a < b$.\nFind the slope of the elliptic curve $y^2=x \\bigl(x-\\frac{1}{4} \\bigr) \\bigl(x-\\frac{1}{3} \\bigr)$ at the point $\\bigl(1,-\\frac{1}{2}\\sqrt{2} \\, \\bigr)$.\nThe slope of the elliptic curve at the given point is [ANS].", "answer_v3": [ "-(23/24)*sqrt(2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0245", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "calculus", "differentiation" ], "problem_v1": "Find an equation of the tangent line to the curve \\frac{x^2}{64}-\\frac{y^2}{36}=1 (a hyperbola) at the point $(-11, \\frac{3}{4}\\sqrt{57})$.\nAn equation of this tangent line to the hyperbola at the given point is $y=mx+b$, where $m=$ [ANS] and $b=$ [ANS].", "answer_v1": [ "-(11/76)*sqrt(57)", "-(16/19)*sqrt(57)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find an equation of the tangent line to the curve \\frac{x^2}{4}+\\frac{y^2}{81}=1 (an ellipse) at the point $(1,-\\frac{9}{2}\\sqrt{3})$.\nAn equation of this tangent line to the ellipse at the given point is $y=mx+b$, where $m=$ [ANS] and $b=$ [ANS].", "answer_v2": [ "3/2*sqrt(3)", "-6*sqrt(3)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find an equation of the tangent line to the curve \\frac{x^2}{16}+\\frac{y^2}{36}=1 (an ellipse) at the point $(-2, 3\\sqrt{3})$.\nAn equation of this tangent line to the ellipse at the given point is $y=mx+b$, where $m=$ [ANS] and $b=$ [ANS].", "answer_v3": [ "1/2*sqrt(3)", "4*sqrt(3)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0246", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "calculus", "differentiation" ], "problem_v1": "Find $f'(-1)$ if $7x \\left(f(x) \\right)^{4}+6x^{4} f(x)=-13$ and $f(-1)=-1$.\n$f'(-1)=$ [ANS]", "answer_v1": [ "-31/34" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $f'(1)$ if $2x \\left(f(x) \\right)^{4}+3x^{2} f(x)=38$ and $f(1)=2$.\n$f'(1)=$ [ANS]", "answer_v2": [ "-44/67" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $f'(-2)$ if $3x \\left(f(x) \\right)^{3}+6x^{4} f(x)=-90$ and $f(-2)=-1$.\n$f'(-2)=$ [ANS]", "answer_v3": [ "-63/26" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0247", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "calculus", "derivative", "implicit" ], "problem_v1": "For the equation given below, evaluate $y'$ at the point $(2,2)$. 5 e^{x y}-4x=y+263.\n$y'$ at $(2,2)$=[ANS]", "answer_v1": [ "-0.994495225988083" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "For the equation given below, evaluate $y'$ at the point $(1,2)$. 2 e^{x y}-3x=y+9.78.\n$y'$ at $(1,2)$=[ANS]", "answer_v2": [ "-1.92742111650425" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "For the equation given below, evaluate $y'$ at the point $(1,2)$. 3 e^{x y}-4x=y+16.2.\n$y'$ at $(1,2)$=[ANS]", "answer_v3": [ "-1.90551405025191" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0248", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "calculus", "derivative", "implicit" ], "problem_v1": "For the equation given below, evaluate $y'$ at the point $(-2, 3)$. 4x^{3}-3y=\\ln\\!\\left(y\\right)-41-\\ln\\!\\left(3\\right)\n$y'(-2)=$ [ANS]", "answer_v1": [ "72/5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "For the equation given below, evaluate $y'$ at the point $(1, 2)$. 5x^{3}-3y=\\ln\\!\\left(y\\right)-1-\\ln\\!\\left(2\\right)\n$y'(1)=$ [ANS]", "answer_v2": [ "30/7" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "For the equation given below, evaluate $y'$ at the point $(-2, 4)$. 3x^{3}-3y=\\ln\\!\\left(y\\right)-36-\\ln\\!\\left(4\\right)\n$y'(-2)=$ [ANS]", "answer_v3": [ "144/13" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0249", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "derivatives", "chain rule" ], "problem_v1": "Use implicit differentiation to find the derivative of the family of curves \\sin\\!\\left(xy\\right)+x^{4}+y^{3}=c. $ {\\frac{dy}{dx}=}$ [ANS].\nNote: your answer will be a function of $x$ and $y$. If you take differential equations, you will learn how to get the family of curves starting with the formula for $\\frac{dy}{dx}$.", "answer_v1": [ "-(cos(x y)y+4 x^3)/(cos(x y) x + 3 y^2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use implicit differentiation to find the derivative of the family of curves \\sin\\!\\left(xy\\right)+x+y^{5}=c. $ {\\frac{dy}{dx}=}$ [ANS].\nNote: your answer will be a function of $x$ and $y$. If you take differential equations, you will learn how to get the family of curves starting with the formula for $\\frac{dy}{dx}$.", "answer_v2": [ "-(cos(x y)y+1 x^0)/(cos(x y) x + 5 y^4)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use implicit differentiation to find the derivative of the family of curves \\sin\\!\\left(xy\\right)+x^{2}+y^{4}=c. $ {\\frac{dy}{dx}=}$ [ANS].\nNote: your answer will be a function of $x$ and $y$. If you take differential equations, you will learn how to get the family of curves starting with the formula for $\\frac{dy}{dx}$.", "answer_v3": [ "-(cos(x y)y+2 x^1)/(cos(x y) x + 4 y^3)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0250", "subject": "Calculus_-_single_variable", "topic": "Differentiation", "subtopic": "Implicit differentiation", "level": "3", "keywords": [ "Differentiation", "Product", "Quotient" ], "problem_v1": "Assume that the function $y$ is defined implicitly as a function of $x$ by the given equation. Use the technique of implicit differentiation to find a differential equation for which $y$ is a solution.\n(x^2+y^2)^2=8xy^2 $y^\\prime=$ [ANS]", "answer_v1": [ " (8 y^2-4x(x^2+y^2))/(4y(x^2+y^2)-2(8)xy) " ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Assume that the function $y$ is defined implicitly as a function of $x$ by the given equation. Use the technique of implicit differentiation to find a differential equation for which $y$ is a solution.\n(x^2+y^2)^2=1xy^2 $y^\\prime=$ [ANS]", "answer_v2": [ " (1 y^2-4x(x^2+y^2))/(4y(x^2+y^2)-2(1)xy) " ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Assume that the function $y$ is defined implicitly as a function of $x$ by the given equation. Use the technique of implicit differentiation to find a differential equation for which $y$ is a solution.\n(x^2+y^2)^2=4xy^2 $y^\\prime=$ [ANS]", "answer_v3": [ " (4 y^2-4x(x^2+y^2))/(4y(x^2+y^2)-2(4)xy) " ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0251", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Mean value theorem", "level": "3", "keywords": [ "Mean Value Theorem" ], "problem_v1": "Suppose $f(x)$ is continuous on $[4,7]$ and $-3 \\le f'(x)\\le 4$ for all $x$ in $(4,7)$. Use the Mean Value Theorem to estimate $f(7)-f(4)$.\nAnswer: [ANS] $\\le f(7)-f(4) \\le$ [ANS]", "answer_v1": [ "-9", "12" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose $f(x)$ is continuous on $[2,8]$ and $-5 \\le f'(x)\\le 3$ for all $x$ in $(2,8)$. Use the Mean Value Theorem to estimate $f(8)-f(2)$.\nAnswer: [ANS] $\\le f(8)-f(2) \\le$ [ANS]", "answer_v2": [ "-30", "18" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose $f(x)$ is continuous on $[2,7]$ and $-4 \\le f'(x)\\le 4$ for all $x$ in $(2,7)$. Use the Mean Value Theorem to estimate $f(7)-f(2)$.\nAnswer: [ANS] $\\le f(7)-f(2) \\le$ [ANS]", "answer_v3": [ "-20", "20" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0252", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Mean value theorem", "level": "3", "keywords": [ "derivatives", "Rolle's theorem" ], "problem_v1": "Consider the function $f(x)=x^3-3x^2+2x-15$ on the interval $[0, 2]$. Verify that this function satisfies the three hypotheses of Rolle's Theorem on the inverval:\n$f(x)$ is [ANS] on $[0,2]$ ; $f(x)$ is [ANS] on $(0,2)$ ; $f(0)=f(2)=$ [ANS].\nThen by Rolle's theorem, there exists a $c$ such that $f'(c)=0$. Find all values $c$ that satisfy the conclusion of Rolle's theorem and give then in a comma-separated list.\nValues of $c$: [ANS]", "answer_v1": [ "CONTINUOUS", "DIFFERENTIABLE", "-15", "(0.42265, 1.57735)" ], "answer_type_v1": [ "OE", "OE", "NV", "UOL" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the function $f(x)=x^3-3x^2+2x-2$ on the interval $[0, 2]$. Verify that this function satisfies the three hypotheses of Rolle's Theorem on the inverval:\n$f(x)$ is [ANS] on $[0,2]$ ; $f(x)$ is [ANS] on $(0,2)$ ; $f(0)=f(2)=$ [ANS].\nThen by Rolle's theorem, there exists a $c$ such that $f'(c)=0$. Find all values $c$ that satisfy the conclusion of Rolle's theorem and give then in a comma-separated list.\nValues of $c$: [ANS]", "answer_v2": [ "CONTINUOUS", "DIFFERENTIABLE", "-2", "(0.42265, 1.57735)" ], "answer_type_v2": [ "OE", "OE", "NV", "UOL" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the function $f(x)=x^3-3x^2+2x-6$ on the interval $[0, 2]$. Verify that this function satisfies the three hypotheses of Rolle's Theorem on the inverval:\n$f(x)$ is [ANS] on $[0,2]$ ; $f(x)$ is [ANS] on $(0,2)$ ; $f(0)=f(2)=$ [ANS].\nThen by Rolle's theorem, there exists a $c$ such that $f'(c)=0$. Find all values $c$ that satisfy the conclusion of Rolle's theorem and give then in a comma-separated list.\nValues of $c$: [ANS]", "answer_v3": [ "CONTINUOUS", "DIFFERENTIABLE", "-6", "(0.42265, 1.57735)" ], "answer_type_v3": [ "OE", "OE", "NV", "UOL" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0253", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Mean value theorem", "level": "3", "keywords": [ "Mean Value Theorem" ], "problem_v1": "Let $f(x)=8 \\sin(x)$.\na.) $|f'(x)| \\le$ [ANS]\nb.) By the Mean Value Theorem, $|f(a)-f(b)|\\le$ [ANS] $|a-b|$ for all $a$ and $b$.", "answer_v1": [ "8", "8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)=2 \\sin(x)$.\na.) $|f'(x)| \\le$ [ANS]\nb.) By the Mean Value Theorem, $|f(a)-f(b)|\\le$ [ANS] $|a-b|$ for all $a$ and $b$.", "answer_v2": [ "2", "2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)=4 \\sin(x)$.\na.) $|f'(x)| \\le$ [ANS]\nb.) By the Mean Value Theorem, $|f(a)-f(b)|\\le$ [ANS] $|a-b|$ for all $a$ and $b$.", "answer_v3": [ "4", "4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0254", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Mean value theorem", "level": "2", "keywords": [], "problem_v1": "A function $f(x)$ and interval $[a, b]$ are given. Check if the Mean Value Theorem can be applied to $f$ on $[a, b]$. If so, find all values $c$ in $[a, b]$ guaranteed by the Mean Value Theorem Note, if the Mean Value Theorem does not apply, enter DNE for the $c$ value.\nf(x)=\\sqrt{36-x^2} \\qquad \\textrm{on} \\; [0, 6] $c=$ [ANS] (Separate multiple answers by commas.)", "answer_v1": [ "6/sqrt(2)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A function $f(x)$ and interval $[a, b]$ are given. Check if the Mean Value Theorem can be applied to $f$ on $[a, b]$. If so, find all values $c$ in $[a, b]$ guaranteed by the Mean Value Theorem Note, if the Mean Value Theorem does not apply, enter DNE for the $c$ value.\nf(x)=\\sqrt{1-x^2} \\qquad \\textrm{on} \\; [0, 1] $c=$ [ANS] (Separate multiple answers by commas.)", "answer_v2": [ "1/sqrt(2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A function $f(x)$ and interval $[a, b]$ are given. Check if the Mean Value Theorem can be applied to $f$ on $[a, b]$. If so, find all values $c$ in $[a, b]$ guaranteed by the Mean Value Theorem Note, if the Mean Value Theorem does not apply, enter DNE for the $c$ value.\nf(x)=\\sqrt{9-x^2} \\qquad \\textrm{on} \\; [0, 3] $c=$ [ANS] (Separate multiple answers by commas.)", "answer_v3": [ "3/sqrt(2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0255", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Mean value theorem", "level": "3", "keywords": [], "problem_v1": "A function $f(x)$ and interval $[a, b]$ are given. Check if Rolle's Theorem can be applied to $f$ on $[a, b]$.\na) $f(x)=16$ on $[-19, 19]$. $f$ [ANS] continuous on $[-19, 19]$. $f$ [ANS] differentiable on $(-19, 19)$. $f(-19)=$ [ANS] and $f(19)=$ [ANS]. The two values are [ANS]. Rolle's Theorem [ANS] apply in this situation.\nb) $f(x)=16x$ on $[-19, 19]$. $f$ [ANS] continuous on $[-19, 19]$. $f$ [ANS] differentiable on $(-19, 19)$. $f(-19)=$ [ANS] and $f(19)=$ [ANS]. The two values are [ANS]. Rolle's Theorem [ANS] apply in this situation.\nb) $f(x)=16 |x|$ on $[-19, 19]$. $f$ [ANS] continuous on $[-19, 19]$. $f$ [ANS] differentiable on $(-19, 19)$. $f(-19)=$ [ANS] and $f(19)=$ [ANS]. The two values are [ANS]. Rolle's Theorem [ANS] apply in this situation.", "answer_v1": [ "IS", "IS", "16", "16", "equal", "does", "is", "is", "-304", "304", "not equal", "does not", "is", "is not", "304", "304", "equal", "does not" ], "answer_type_v1": [ "MCS", "MCS", "NV", "NV", "MCS", "MCS", "MCS", "MCS", "NV", "NV", "MCS", "MCS", "MCS", "MCS", "NV", "NV", "MCS", "MCS" ], "options_v1": [ [ "is", "is not" ], [ "is", "is not" ], [], [], [ "equal", "not equal" ], [ "does", "does not" ], [ "is", "is not" ], [ "is", "is not" ], [], [], [ "equal", "not equal" ], [ "does", "does not" ], [ "is", "is not" ], [ "is", "is not" ], [], [], [ "equal", "not equal" ], [ "does", "does not" ] ], "problem_v2": "A function $f(x)$ and interval $[a, b]$ are given. Check if Rolle's Theorem can be applied to $f$ on $[a, b]$.\na) $f(x)=3$ on $[-5, 5]$. $f$ [ANS] continuous on $[-5, 5]$. $f$ [ANS] differentiable on $(-5, 5)$. $f(-5)=$ [ANS] and $f(5)=$ [ANS]. The two values are [ANS]. Rolle's Theorem [ANS] apply in this situation.\nb) $f(x)=3x$ on $[-5, 5]$. $f$ [ANS] continuous on $[-5, 5]$. $f$ [ANS] differentiable on $(-5, 5)$. $f(-5)=$ [ANS] and $f(5)=$ [ANS]. The two values are [ANS]. Rolle's Theorem [ANS] apply in this situation.\nb) $f(x)=3 |x|$ on $[-5, 5]$. $f$ [ANS] continuous on $[-5, 5]$. $f$ [ANS] differentiable on $(-5, 5)$. $f(-5)=$ [ANS] and $f(5)=$ [ANS]. The two values are [ANS]. Rolle's Theorem [ANS] apply in this situation.", "answer_v2": [ "IS", "IS", "3", "3", "equal", "does", "is", "is", "-15", "15", "not equal", "does not", "is", "is not", "15", "15", "equal", "does not" ], "answer_type_v2": [ "MCS", "MCS", "NV", "NV", "MCS", "MCS", "MCS", "MCS", "NV", "NV", "MCS", "MCS", "MCS", "MCS", "NV", "NV", "MCS", "MCS" ], "options_v2": [ [ "is", "is not" ], [ "is", "is not" ], [], [], [ "equal", "not equal" ], [ "does", "does not" ], [ "is", "is not" ], [ "is", "is not" ], [], [], [ "equal", "not equal" ], [ "does", "does not" ], [ "is", "is not" ], [ "is", "is not" ], [], [], [ "equal", "not equal" ], [ "does", "does not" ] ], "problem_v3": "A function $f(x)$ and interval $[a, b]$ are given. Check if Rolle's Theorem can be applied to $f$ on $[a, b]$.\na) $f(x)=7$ on $[-9, 9]$. $f$ [ANS] continuous on $[-9, 9]$. $f$ [ANS] differentiable on $(-9, 9)$. $f(-9)=$ [ANS] and $f(9)=$ [ANS]. The two values are [ANS]. Rolle's Theorem [ANS] apply in this situation.\nb) $f(x)=7x$ on $[-9, 9]$. $f$ [ANS] continuous on $[-9, 9]$. $f$ [ANS] differentiable on $(-9, 9)$. $f(-9)=$ [ANS] and $f(9)=$ [ANS]. The two values are [ANS]. Rolle's Theorem [ANS] apply in this situation.\nb) $f(x)=7 |x|$ on $[-9, 9]$. $f$ [ANS] continuous on $[-9, 9]$. $f$ [ANS] differentiable on $(-9, 9)$. $f(-9)=$ [ANS] and $f(9)=$ [ANS]. The two values are [ANS]. Rolle's Theorem [ANS] apply in this situation.", "answer_v3": [ "IS", "IS", "7", "7", "equal", "does", "is", "is", "-63", "63", "not equal", "does not", "is", "is not", "63", "63", "equal", "does not" ], "answer_type_v3": [ "MCS", "MCS", "NV", "NV", "MCS", "MCS", "MCS", "MCS", "NV", "NV", "MCS", "MCS", "MCS", "MCS", "NV", "NV", "MCS", "MCS" ], "options_v3": [ [ "is", "is not" ], [ "is", "is not" ], [], [], [ "equal", "not equal" ], [ "does", "does not" ], [ "is", "is not" ], [ "is", "is not" ], [], [], [ "equal", "not equal" ], [ "does", "does not" ], [ "is", "is not" ], [ "is", "is not" ], [], [], [ "equal", "not equal" ], [ "does", "does not" ] ] }, { "id": "Calculus_-_single_variable_0256", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Mean value theorem", "level": "3", "keywords": [ "calculus", "derivatives", "mean value theorem" ], "problem_v1": "Find a point $c$ satisfying the conclusion of the Mean Value Theorem for the function $f(x)=x^{-{7}}$ on the interval $[1, 7]$.\n$c=$ [ANS]", "answer_v1": [ "1.59553" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find a point $c$ satisfying the conclusion of the Mean Value Theorem for the function $f(x)=x^{-{1}}$ on the interval $[1, 10]$.\n$c=$ [ANS]", "answer_v2": [ "3.16228" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find a point $c$ satisfying the conclusion of the Mean Value Theorem for the function $f(x)=x^{-{3}}$ on the interval $[1, 7]$.\n$c=$ [ANS]", "answer_v3": [ "2.06127" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0257", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Mean value theorem", "level": "3", "keywords": [ "Mean Value Theorem", "real roots" ], "problem_v1": "In this problem you will use Rolle's theorem to determine whether it is possible for the function f(x)=6x^{7}+7x-7 to have two or more real roots (or, equivalently, whether the graph of $y=f(x)$ crosses the $x$-axis two or more times).\nSuppose that $f(x)$ has at least two real roots. Choose two of these roots and call the smaller one $a$ and the larger one $b$. By applying Rolle's theorem to $f(x)$ on the interval $[a,b]$, there exists at least one number $c$ in the interval $(a,b)$ so that $f'(c)=$ [ANS].\nThe values of the derivative $f'(x)=$ [ANS] are always [ANS], and therefore it is [ANS] for $f(x)$ to have two or more real roots.", "answer_v1": [ "0", "42*x^6+7", "POSITIVE", "impossible" ], "answer_type_v1": [ "NV", "EX", "MCS", "MCS" ], "options_v1": [ [], [], [ "changing", "negative", "zero", "positive", "undefined" ], [ "plausible", "unlikely", "possible", "impossible" ] ], "problem_v2": "In this problem you will use Rolle's theorem to determine whether it is possible for the function f(x)=-\\left(9x^{5}+4x+19\\right) to have two or more real roots (or, equivalently, whether the graph of $y=f(x)$ crosses the $x$-axis two or more times).\nSuppose that $f(x)$ has at least two real roots. Choose two of these roots and call the smaller one $a$ and the larger one $b$. By applying Rolle's theorem to $f(x)$ on the interval $[a,b]$, there exists at least one number $c$ in the interval $(a,b)$ so that $f'(c)=$ [ANS].\nThe values of the derivative $f'(x)=$ [ANS] are always [ANS], and therefore it is [ANS] for $f(x)$ to have two or more real roots.", "answer_v2": [ "0", "-(45*x^4+4)", "NEGATIVE", "impossible" ], "answer_type_v2": [ "NV", "EX", "MCS", "MCS" ], "options_v2": [ [], [], [ "changing", "negative", "zero", "positive", "undefined" ], [ "plausible", "unlikely", "possible", "impossible" ] ], "problem_v3": "In this problem you will use Rolle's theorem to determine whether it is possible for the function f(x)=-\\left(6x^{5}+6x+5\\right) to have two or more real roots (or, equivalently, whether the graph of $y=f(x)$ crosses the $x$-axis two or more times).\nSuppose that $f(x)$ has at least two real roots. Choose two of these roots and call the smaller one $a$ and the larger one $b$. By applying Rolle's theorem to $f(x)$ on the interval $[a,b]$, there exists at least one number $c$ in the interval $(a,b)$ so that $f'(c)=$ [ANS].\nThe values of the derivative $f'(x)=$ [ANS] are always [ANS], and therefore it is [ANS] for $f(x)$ to have two or more real roots.", "answer_v3": [ "0", "-(30*x^4+6)", "NEGATIVE", "impossible" ], "answer_type_v3": [ "NV", "EX", "MCS", "MCS" ], "options_v3": [ [], [], [ "changing", "negative", "zero", "positive", "undefined" ], [ "plausible", "unlikely", "possible", "impossible" ] ] }, { "id": "Calculus_-_single_variable_0258", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Mean value theorem", "level": "3", "keywords": [ "Mean Value", "Slope", "Calculus", "Derivatives" ], "problem_v1": "Consider the function $f(x)=2x^3-6x^2-90x+7$ on the interval $[-4, 8]$. Find the average or mean slope of the function on this interval.\nAverage slope: [ANS]\nBy the Mean Value Theorem, we know there exists at least one value $c$ in the open interval $(-4, 8)$ such that $f'(c)$ is equal to this mean slope. List all values $c$ that work. If there are none, enter none.\nValues of $c$: [ANS]", "answer_v1": [ "-18", "(-2.60555, 4.60555)" ], "answer_type_v1": [ "NV", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "Consider the function $f(x)=2x^3-15x^2-36x+2$ on the interval $[-5, 10]$. Find the average or mean slope of the function on this interval.\nAverage slope: [ANS]\nBy the Mean Value Theorem, we know there exists at least one value $c$ in the open interval $(-5, 10)$ such that $f'(c)$ is equal to this mean slope. List all values $c$ that work. If there are none, enter none.\nValues of $c$: [ANS]", "answer_v2": [ "39", "(-1.83013, 6.83013)" ], "answer_type_v2": [ "NV", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "Consider the function $f(x)=2x^3-12x^2-30x+3$ on the interval $[-5, 7]$. Find the average or mean slope of the function on this interval.\nAverage slope: [ANS]\nBy the Mean Value Theorem, we know there exists at least one value $c$ in the open interval $(-5, 7)$ such that $f'(c)$ is equal to this mean slope. List all values $c$ that work. If there are none, enter none.\nValues of $c$: [ANS]", "answer_v3": [ "24", "(-1.60555, 5.60555)" ], "answer_type_v3": [ "NV", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0259", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Mean value theorem", "level": "4", "keywords": [ "calculus", "derivative", "mean value theorem", "theory" ], "problem_v1": "Suppose that $f(t)$ is continuous and twice-differentiable for $t\\ge 0$. Further suppose $f''(t) \\ge 9$ for all $t\\ge 0$ and $f(0)=f'(0)=0$. Using the Racetrack Principle, what linear function $g(t)$ can we prove is less than $f'(t)$ (for $t\\ge 0$)? $g(t)=$ [ANS]\nThen, also using the Racetrack Principle, what quadratic function $h(t)$ can we prove is less than $f(t)$ (for $t\\ge 0$)? $h(t)=$ [ANS]\nFor both parts of this problem, be sure you can clearly state how the theorem is applied to prove the indicated inequalities. For both parts of this problem, be sure you can clearly state how the theorem is applied to prove the indicated inequalities.", "answer_v1": [ "9*t", "9/2*t^2" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that $f(t)$ is continuous and twice-differentiable for $t\\ge 0$. Further suppose $f''(t) \\ge 3$ for all $t\\ge 0$ and $f(0)=f'(0)=0$. Using the Racetrack Principle, what linear function $g(t)$ can we prove is less than $f'(t)$ (for $t\\ge 0$)? $g(t)=$ [ANS]\nThen, also using the Racetrack Principle, what quadratic function $h(t)$ can we prove is less than $f(t)$ (for $t\\ge 0$)? $h(t)=$ [ANS]\nFor both parts of this problem, be sure you can clearly state how the theorem is applied to prove the indicated inequalities. For both parts of this problem, be sure you can clearly state how the theorem is applied to prove the indicated inequalities.", "answer_v2": [ "3*t", "3/2*t^2" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that $f(t)$ is continuous and twice-differentiable for $t\\ge 0$. Further suppose $f''(t) \\ge 5$ for all $t\\ge 0$ and $f(0)=f'(0)=0$. Using the Racetrack Principle, what linear function $g(t)$ can we prove is less than $f'(t)$ (for $t\\ge 0$)? $g(t)=$ [ANS]\nThen, also using the Racetrack Principle, what quadratic function $h(t)$ can we prove is less than $f(t)$ (for $t\\ge 0$)? $h(t)=$ [ANS]\nFor both parts of this problem, be sure you can clearly state how the theorem is applied to prove the indicated inequalities. For both parts of this problem, be sure you can clearly state how the theorem is applied to prove the indicated inequalities.", "answer_v3": [ "5*t", "5/2*t^2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0260", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - general", "level": "4", "keywords": [ "derivatives", "rate of change", "volume" ], "problem_v1": "A spherical balloon is being inflated. Find the rate of increase of the surface area ($S=4\\pi r^2$) with respect to the radius $r$ when $r=14$: [ANS]\nwhen $r=11$: [ANS]\nNote: You may input pi for $\\pi$.", "answer_v1": [ "8*pi*14", "8*pi*11" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A spherical balloon is being inflated. Find the rate of increase of the surface area ($S=4\\pi r^2$) with respect to the radius $r$ when $r=4$: [ANS]\nwhen $r=16$: [ANS]\nNote: You may input pi for $\\pi$.", "answer_v2": [ "8*pi*4", "8*pi*16" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A spherical balloon is being inflated. Find the rate of increase of the surface area ($S=4\\pi r^2$) with respect to the radius $r$ when $r=7$: [ANS]\nwhen $r=12$: [ANS]\nNote: You may input pi for $\\pi$.", "answer_v3": [ "8*pi*7", "8*pi*12" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0261", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - general", "level": "4", "keywords": [ "derivatives", "rate of change", "area" ], "problem_v1": "The area of a rectangle with one of its sides $s$ is $A(s)=11 s^2$. What is the rate of change of the area of the rectangle with respect to the side length when $s=14$?\nAnswer: [ANS]", "answer_v1": [ "2*11*14" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The area of a rectangle with one of its sides $s$ is $A(s)=16 s^2$. What is the rate of change of the area of the rectangle with respect to the side length when $s=4$?\nAnswer: [ANS]", "answer_v2": [ "2*16*4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The area of a rectangle with one of its sides $s$ is $A(s)=12 s^2$. What is the rate of change of the area of the rectangle with respect to the side length when $s=7$?\nAnswer: [ANS]", "answer_v3": [ "2*12*7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0262", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - general", "level": "3", "keywords": [ "Differentiation' 'Rates of Change" ], "problem_v1": "Find the slope of the graph of f(x)=6 \\sqrt{x} at the point $(36, 36)$.\nIf the limit does not exist enter 'DNE'. Slope of graph=[ANS]", "answer_v1": [ "0.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the slope of the graph of f(x)=2 \\sqrt{x} at the point $(64, 16)$.\nIf the limit does not exist enter 'DNE'. Slope of graph=[ANS]", "answer_v2": [ "0.125" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the slope of the graph of f(x)=3 \\sqrt{x} at the point $(36, 18)$.\nIf the limit does not exist enter 'DNE'. Slope of graph=[ANS]", "answer_v3": [ "0.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0263", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "4", "keywords": [ "derivatives", "applications", "marginal cost" ], "problem_v1": "Suppose that the cost, in dollars, for a company to produce $x$ pairs of a new line of jeans is C(x)=6300+6x+0.01x^2+0.0002x^3.\n(a) Find the marginal cost function. Answer: [ANS]\n(b) Find the marginal cost at $x=100$. Answer: [ANS]\n(c) Find the cost at $x=100$. Answer: [ANS]", "answer_v1": [ "6+0.02*x+0.0006*x^2", "6+0.02*100+0.0006*100^2", "6300+6*100+0.01*100^2+0.0002*100^3" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose that the cost, in dollars, for a company to produce $x$ pairs of a new line of jeans is C(x)=1500+9x+0.01x^2+0.0002x^3.\n(a) Find the marginal cost function. Answer: [ANS]\n(b) Find the marginal cost at $x=100$. Answer: [ANS]\n(c) Find the cost at $x=100$. Answer: [ANS]", "answer_v2": [ "9+0.02*x+0.0006*x^2", "9+0.02*100+0.0006*100^2", "1500+9*100+0.01*100^2+0.0002*100^3" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose that the cost, in dollars, for a company to produce $x$ pairs of a new line of jeans is C(x)=3200+6x+0.01x^2+0.0002x^3.\n(a) Find the marginal cost function. Answer: [ANS]\n(b) Find the marginal cost at $x=100$. Answer: [ANS]\n(c) Find the cost at $x=100$. Answer: [ANS]", "answer_v3": [ "6+0.02*x+0.0006*x^2", "6+0.02*100+0.0006*100^2", "3200+6*100+0.01*100^2+0.0002*100^3" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0264", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "4", "keywords": [ "calculus", "derivatives", "rates of change" ], "problem_v1": "The demand for a commodity generally decreases as the price is raised. Suppose that the demand for oil (per capita per year) is $D(p)=1050/p$ barrels, where $p$ is the price per barrel in dollars. Find the demand when $p=45$. Estimate the decrease in demand if $p$ rises to $46$ and the increase in demand if $p$ is decreased to $44$. The demand $D(45)$=[ANS]. The decrease in demand=[ANS] barrels. The increase in demand=[ANS] barrels.", "answer_v1": [ "23.3333", "0.518519", "0.518519" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The demand for a commodity generally decreases as the price is raised. Suppose that the demand for oil (per capita per year) is $D(p)=750/p$ barrels, where $p$ is the price per barrel in dollars. Find the demand when $p=55$. Estimate the decrease in demand if $p$ rises to $56$ and the increase in demand if $p$ is decreased to $54$. The demand $D(55)$=[ANS]. The decrease in demand=[ANS] barrels. The increase in demand=[ANS] barrels.", "answer_v2": [ "13.6364", "0.247934", "0.247934" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The demand for a commodity generally decreases as the price is raised. Suppose that the demand for oil (per capita per year) is $D(p)=850/p$ barrels, where $p$ is the price per barrel in dollars. Find the demand when $p=50$. Estimate the decrease in demand if $p$ rises to $51$ and the increase in demand if $p$ is decreased to $49$. The demand $D(50)$=[ANS]. The decrease in demand=[ANS] barrels. The increase in demand=[ANS] barrels.", "answer_v3": [ "17", "0.34", "0.34" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0265", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "4", "keywords": [ "derivatives", "economics", "marginal cost", "calculus", "differentiation" ], "problem_v1": "The cost of producing $x$ units of stuffed alligator toys is $c(x)=0.004x^{2}+8x+7000$. Find the marginal cost at the production level of $1000$ units. [ANS]", "answer_v1": [ "0.004*2*1000+8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The cost of producing $x$ units of stuffed alligator toys is $c(x)=0.001x^{2}+10x+4000$. Find the marginal cost at the production level of $1000$ units. [ANS]", "answer_v2": [ "0.001*2*1000+10" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The cost of producing $x$ units of stuffed alligator toys is $c(x)=0.002x^{2}+9x+5000$. Find the marginal cost at the production level of $1000$ units. [ANS]", "answer_v3": [ "0.002*2*1000+9" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0266", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "2", "keywords": [ "calculus", "derivative", "second derivative", "acceleration" ], "problem_v1": "Let $P(t)$ represent the price of a share of stock of a corporation at time $t$. What does each of the following statements tell us about the signs of the first and second derivatives of $P(t)$?\n(a) The price of the stock is falling slower and slower. The first derivative of $P(t)$ is [ANS] The second derivative of $P(t)$ is [ANS] (b) The price of the stock is just past where it bottomed out. The first derivative of $P(t)$ is [ANS] The second derivative of $P(t)$ is [ANS]", "answer_v1": [ "negative", "positive", "positive", "positive" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "positive", "zero", "negative" ], [ "positive", "zero", "negative" ], [ "positive", "zero", "negative" ], [ "positive", "zero", "negative" ] ], "problem_v2": "Let $P(t)$ represent the price of a share of stock of a corporation at time $t$. What does each of the following statements tell us about the signs of the first and second derivatives of $P(t)$?\n(a) The price of the stock is rising faster and faster. The first derivative of $P(t)$ is [ANS] The second derivative of $P(t)$ is [ANS] (b) The price of the stock is just past where it maxed out. The first derivative of $P(t)$ is [ANS] The second derivative of $P(t)$ is [ANS]", "answer_v2": [ "positive", "positive", "negative", "negative" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "positive", "zero", "negative" ], [ "positive", "zero", "negative" ], [ "positive", "zero", "negative" ], [ "positive", "zero", "negative" ] ], "problem_v3": "Let $P(t)$ represent the price of a share of stock of a corporation at time $t$. What does each of the following statements tell us about the signs of the first and second derivatives of $P(t)$?\n(a) The price of the stock is falling faster and faster. The first derivative of $P(t)$ is [ANS] The second derivative of $P(t)$ is [ANS] (b) The price of the stock is just past where it bottomed out. The first derivative of $P(t)$ is [ANS] The second derivative of $P(t)$ is [ANS]", "answer_v3": [ "negative", "negative", "positive", "positive" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "positive", "zero", "negative" ], [ "positive", "zero", "negative" ], [ "positive", "zero", "negative" ], [ "positive", "zero", "negative" ] ] }, { "id": "Calculus_-_single_variable_0267", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "4", "keywords": [], "problem_v1": "A company that makes thing-a-ma-bobs has a start up cost of \\$40108. It costs the company \\$2.75 to make each thing-a-ma-bob. The company charges \\$4.86 for each thing-a-ma-bob. Let $x$ denote the number of thing-a-ma-bobs produced.\nWrite the cost function for this company. $C(x)$=[ANS]\nWrite the revenue function for this company. $R(x)$=[ANS]\nWhat is the minumum number of thing-a-ma-bobs that the company must produce and sell to make a profit? [ANS]", "answer_v1": [ "2.75 x + 40108", "4.86 x", "19009" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A company that makes thing-a-ma-bobs has a start up cost of \\$13320. It costs the company \\$3.80 to make each thing-a-ma-bob. The company charges \\$4.00 for each thing-a-ma-bob. Let $x$ denote the number of thing-a-ma-bobs produced.\nWrite the cost function for this company. $C(x)$=[ANS]\nWrite the revenue function for this company. $R(x)$=[ANS]\nWhat is the minumum number of thing-a-ma-bobs that the company must produce and sell to make a profit? [ANS]", "answer_v2": [ "3.8 x + 13320", "4 x", "66601" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A company that makes thing-a-ma-bobs has a start up cost of \\$22538. It costs the company \\$2.82 to make each thing-a-ma-bob. The company charges \\$3.83 for each thing-a-ma-bob. Let $x$ denote the number of thing-a-ma-bobs produced.\nWrite the cost function for this company. $C(x)$=[ANS]\nWrite the revenue function for this company. $R(x)$=[ANS]\nWhat is the minumum number of thing-a-ma-bobs that the company must produce and sell to make a profit? [ANS]", "answer_v3": [ "2.82 x + 22538", "3.83 x", "22315" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0268", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "5", "keywords": [ "differentiation", "economics", "marginal", "cost" ], "problem_v1": "The total cost (in dollars) of producing $x$ motorcycle frames is C(x)=11000+500x. (A) Find the average cost per motorcycle if 500 frames are produced. Ave. cost per motorcycle=[ANS]\n(B) Find the marginal average cost at a production level of 500 frames. Marginal average cost=[ANS]\n(C) Use the results from parts (A) and (B) to estimate the average cost per frame if 501 frames are produced. Average cost=[ANS]", "answer_v1": [ "522", "-0.044", "521.956" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The total cost (in dollars) of producing $x$ motorcycle frames is C(x)=15000+700x. (A) Find the average cost per motorcycle if 500 frames are produced. Ave. cost per motorcycle=[ANS]\n(B) Find the marginal average cost at a production level of 500 frames. Marginal average cost=[ANS]\n(C) Use the results from parts (A) and (B) to estimate the average cost per frame if 501 frames are produced. Average cost=[ANS]", "answer_v2": [ "730", "-0.06", "729.94" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The total cost (in dollars) of producing $x$ motorcycle frames is C(x)=14000+600x. (A) Find the average cost per motorcycle if 500 frames are produced. Ave. cost per motorcycle=[ANS]\n(B) Find the marginal average cost at a production level of 500 frames. Marginal average cost=[ANS]\n(C) Use the results from parts (A) and (B) to estimate the average cost per frame if 501 frames are produced. Average cost=[ANS]", "answer_v3": [ "628", "-0.056", "627.944" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0269", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "5", "keywords": [ "differentiation", "economics", "marginal", "cost" ], "problem_v1": "Recall that to find the cost to produce the $n$ th unit in a production process, we calculate the difference\nC(n)-C(n-1) where $C(n)$ is the cost of producing $n$ units.\nThe total cost (in dollars) for a fast food franchise of producing $x$ thousand hamburgers might be\nC(x)=22000+800x-70x^2. Find the exact cost of producing the 2,001st burger. Be careful though. The units for $x$ are thousand of hamburgers. The 2,001st burger is represented by 2.001!\nExact cost of 2,001st burger=\\$ [ANS]\nUse marginal cost to approximate the cost of producing the 2001st burger.\nApprox. cost of 2001st burger=\\$ [ANS]", "answer_v1": [ "0.519929999998567", "0.52" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Recall that to find the cost to produce the $n$ th unit in a production process, we calculate the difference\nC(n)-C(n-1) where $C(n)$ is the cost of producing $n$ units.\nThe total cost (in dollars) for a fast food franchise of producing $x$ thousand hamburgers might be\nC(x)=11000+1000x-20x^2. Find the exact cost of producing the 2,001st burger. Be careful though. The units for $x$ are thousand of hamburgers. The 2,001st burger is represented by 2.001!\nExact cost of 2,001st burger=\\$ [ANS]\nUse marginal cost to approximate the cost of producing the 2001st burger.\nApprox. cost of 2001st burger=\\$ [ANS]", "answer_v2": [ "0.919980000000578", "0.92" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Recall that to find the cost to produce the $n$ th unit in a production process, we calculate the difference\nC(n)-C(n-1) where $C(n)$ is the cost of producing $n$ units.\nThe total cost (in dollars) for a fast food franchise of producing $x$ thousand hamburgers might be\nC(x)=15000+900x-30x^2. Find the exact cost of producing the 2,001st burger. Be careful though. The units for $x$ are thousand of hamburgers. The 2,001st burger is represented by 2.001!\nExact cost of 2,001st burger=\\$ [ANS]\nUse marginal cost to approximate the cost of producing the 2001st burger.\nApprox. cost of 2001st burger=\\$ [ANS]", "answer_v3": [ "0.779970000003232", "0.78" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0270", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "5", "keywords": [ "differentiation", "economics", "marginal", "cost" ], "problem_v1": "The total cost (in dollars) of producing $x$ coffee machines is C(x)=2300+50x-0.7x^2. (A) Find the exact cost of producing the 21st machine. Exact cost of 21st machine=[ANS]\n(B) Use marginal cost to approximate the cost of producing the 21st machine. Approx. cost of 21st machine=[ANS]", "answer_v1": [ "21.3000000000002", "22" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The total cost (in dollars) of producing $x$ coffee machines is C(x)=1500+70x-0.2x^2. (A) Find the exact cost of producing the 21st machine. Exact cost of 21st machine=[ANS]\n(B) Use marginal cost to approximate the cost of producing the 21st machine. Approx. cost of 21st machine=[ANS]", "answer_v2": [ "61.8000000000002", "62" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The total cost (in dollars) of producing $x$ coffee machines is C(x)=1800+60x-0.3x^2. (A) Find the exact cost of producing the 21st machine. Exact cost of 21st machine=[ANS]\n(B) Use marginal cost to approximate the cost of producing the 21st machine. Approx. cost of 21st machine=[ANS]", "answer_v3": [ "47.6999999999998", "48" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0271", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "5", "keywords": [ "differentiation", "economics", "marginal", "profit" ], "problem_v1": "The total profit (in dollars) from the sale of $x$ charcoal grills is P(x)=70x-0.6x^2-265 (A) Find the average profit per grill if 40 grills are produced. Ave. profit=[ANS]\nFind the marginal average profit at a production level of 40 grills. (B) Marginal average profit=[ANS]\nUse the results from parts (A) and (B) to estimate the average profit per grill if 41 grills are produced. (C) Estimated average profit=[ANS]", "answer_v1": [ "70 - 40*0.6 - 265/40", "-0.434375", "38.940625" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The total profit (in dollars) from the sale of $x$ charcoal grills is P(x)=10x-0.1x^2-220 (A) Find the average profit per grill if 40 grills are produced. Ave. profit=[ANS]\nFind the marginal average profit at a production level of 40 grills. (B) Marginal average profit=[ANS]\nUse the results from parts (A) and (B) to estimate the average profit per grill if 41 grills are produced. (C) Estimated average profit=[ANS]", "answer_v2": [ "10 - 40*0.1 - 220/40", "0.0375", "0.5375" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The total profit (in dollars) from the sale of $x$ charcoal grills is P(x)=30x-0.6x^2-225 (A) Find the average profit per grill if 40 grills are produced. Ave. profit=[ANS]\nFind the marginal average profit at a production level of 40 grills. (B) Marginal average profit=[ANS]\nUse the results from parts (A) and (B) to estimate the average profit per grill if 41 grills are produced. (C) Estimated average profit=[ANS]", "answer_v3": [ "30 - 40*0.6 - 225/40", "-0.459375", "-0.084375" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0272", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "4", "keywords": [ "differentiation", "economics", "marginal" ], "problem_v1": "The total profit (in dollars) from the sale of $x$ espresso machines is P(x)=80x-0.6x^2-260. Evaluate the marginal profit at the following values: (A) $P'(200)$=[ANS]\n(B) $P'(350)$=[ANS]", "answer_v1": [ "-160", "-340" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The total profit (in dollars) from the sale of $x$ espresso machines is P(x)=30x-0.9x^2-210. Evaluate the marginal profit at the following values: (A) $P'(200)$=[ANS]\n(B) $P'(350)$=[ANS]", "answer_v2": [ "-330", "-600" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The total profit (in dollars) from the sale of $x$ espresso machines is P(x)=50x-0.6x^2-230. Evaluate the marginal profit at the following values: (A) $P'(200)$=[ANS]\n(B) $P'(350)$=[ANS]", "answer_v3": [ "-190", "-370" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0273", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "4", "keywords": [ "differentiation", "economics", "marginal" ], "problem_v1": "The total revenue and total cost functions for the production and sale of $x$ TV's are given as R(x)=160x-0.7x^2 and C(x)=3580+20x. (A) Find the value of $x$ where the graph of $R(x)$ has a horizontal tangent line. $x$ values is [ANS]\n(B) Find the profit function in terms of $x$. $P(x)$=[ANS]\n(C) Find the value of $x$ where the graph of $P(x)$ has a horizontal tangent line. $x$ values=[ANS]\n(D) List all the $x$ values of the break-even point(s). If there are no break-even points, enter 'NONE'. List of $x$ values=[ANS]", "answer_v1": [ "114.285714285714", "160*x - 0.7*(x^2) - 3580 - 20*x", "100", "(30.1021152987139, 169.897884701286)" ], "answer_type_v1": [ "NV", "EX", "NV", "UOL" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The total revenue and total cost functions for the production and sale of $x$ TV's are given as R(x)=110x-0.1x^2 and C(x)=3940+15x. (A) Find the value of $x$ where the graph of $R(x)$ has a horizontal tangent line. $x$ values is [ANS]\n(B) Find the profit function in terms of $x$. $P(x)$=[ANS]\n(C) Find the value of $x$ where the graph of $P(x)$ has a horizontal tangent line. $x$ values=[ANS]\n(D) List all the $x$ values of the break-even point(s). If there are no break-even points, enter 'NONE'. List of $x$ values=[ANS]", "answer_v2": [ "550", "110*x - 0.1*(x^2) - 3940 - 15*x", "475", "(43.4620526535355, 906.537947346465)" ], "answer_type_v2": [ "NV", "EX", "NV", "UOL" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The total revenue and total cost functions for the production and sale of $x$ TV's are given as R(x)=130x-0.3x^2 and C(x)=3610+18x. (A) Find the value of $x$ where the graph of $R(x)$ has a horizontal tangent line. $x$ values is [ANS]\n(B) Find the profit function in terms of $x$. $P(x)$=[ANS]\n(C) Find the value of $x$ where the graph of $P(x)$ has a horizontal tangent line. $x$ values=[ANS]\n(D) List all the $x$ values of the break-even point(s). If there are no break-even points, enter 'NONE'. List of $x$ values=[ANS]", "answer_v3": [ "216.666666666667", "130*x - 0.3*(x^2) - 3610 - 18*x", "186.666666666667", "(35.6331898772108, 337.700143456122)" ], "answer_type_v3": [ "NV", "EX", "NV", "UOL" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0274", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "4", "keywords": [ "differentiation", "economics", "marginal" ], "problem_v1": "The total revenue (in dollars) from the sale of $x$ toasters is R(x)=130-0.6x^2. Evaluate the marginal revenue at the following values: (A) $R'(400)$=[ANS]\n(B) $R'(650)$=[ANS]", "answer_v1": [ "-480", "-780" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The total revenue (in dollars) from the sale of $x$ toasters is R(x)=50-1x^2. Evaluate the marginal revenue at the following values: (A) $R'(400)$=[ANS]\n(B) $R'(650)$=[ANS]", "answer_v2": [ "-800", "-1300" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The total revenue (in dollars) from the sale of $x$ toasters is R(x)=80-0.7x^2. Evaluate the marginal revenue at the following values: (A) $R'(400)$=[ANS]\n(B) $R'(650)$=[ANS]", "answer_v3": [ "-560", "-910" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0275", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "5", "keywords": [ "differentiation", "economics", "marginal" ], "problem_v1": "The price-demand and cost functions for the production of microwaves are given as p=265-\\frac{x}{80} and C(x)=56000+90x, where $x$ is the number of microwaves that can be sold at a price of $p$ dollars per unit and $C(x)$ is the total cost (in dollars) of producing $x$ units.\n(A) Find the marginal cost as a function of $x$. $C'(x)$=[ANS]\n(B) Find the revenue function in terms of $x$. $R(x)$=[ANS]\n(C) Find the marginal revenue function in terms of $x$. $R'(x)$=[ANS]\n(D) Evaluate the marginal revenue function at $x=1500$. $R'(1500)$=[ANS]\n(E) Find the profit function in terms of $x$. $P(x)$=[ANS]\n(F) Evaluate the marginal profit function at $x=1500$. $P'(1500)$=[ANS]", "answer_v1": [ "90", "265*x - (x^2)/80", "265 - (2*(x))/80", "227.5", "265*x-x^2/80-(56000+90*x)", "137.5" ], "answer_type_v1": [ "NV", "EX", "EX", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The price-demand and cost functions for the production of microwaves are given as p=215-\\frac{x}{30} and C(x)=86000+60x, where $x$ is the number of microwaves that can be sold at a price of $p$ dollars per unit and $C(x)$ is the total cost (in dollars) of producing $x$ units.\n(A) Find the marginal cost as a function of $x$. $C'(x)$=[ANS]\n(B) Find the revenue function in terms of $x$. $R(x)$=[ANS]\n(C) Find the marginal revenue function in terms of $x$. $R'(x)$=[ANS]\n(D) Evaluate the marginal revenue function at $x=1500$. $R'(1500)$=[ANS]\n(E) Find the profit function in terms of $x$. $P(x)$=[ANS]\n(F) Evaluate the marginal profit function at $x=1500$. $P'(1500)$=[ANS]", "answer_v2": [ "60", "215*x - (x^2)/30", "215 - (2*(x))/30", "115", "215*x-x^2/30-(86000+60*x)", "55" ], "answer_type_v2": [ "NV", "EX", "EX", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The price-demand and cost functions for the production of microwaves are given as p=225-\\frac{x}{50} and C(x)=58000+70x, where $x$ is the number of microwaves that can be sold at a price of $p$ dollars per unit and $C(x)$ is the total cost (in dollars) of producing $x$ units.\n(A) Find the marginal cost as a function of $x$. $C'(x)$=[ANS]\n(B) Find the revenue function in terms of $x$. $R(x)$=[ANS]\n(C) Find the marginal revenue function in terms of $x$. $R'(x)$=[ANS]\n(D) Evaluate the marginal revenue function at $x=1500$. $R'(1500)$=[ANS]\n(E) Find the profit function in terms of $x$. $P(x)$=[ANS]\n(F) Evaluate the marginal profit function at $x=1500$. $P'(1500)$=[ANS]", "answer_v3": [ "70", "225*x - (x^2)/50", "225 - (2*(x))/50", "165", "225*x-x^2/50-(58000+70*x)", "95" ], "answer_type_v3": [ "NV", "EX", "EX", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0276", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "4", "keywords": [ "maxima", "minima", "inflection point" ], "problem_v1": "A company estimates that it will sell $N(x)$ units of a product after spending $x$ thousand dollars on advertising, as given by N(x)=-5x^3+260x^2-3600x+19000, \\qquad 10 \\leq x \\leq 40. (A) Use interval notation to indicate when the rate of change of sales $N'(x)$ is increasing.\n$N'(x)$ increasing: [ANS]\n(B) Use interval notation to indicate when the rate of change of sales $N'(x)$ is decreasing. $N'(x)$ decreasing: [ANS]\n(C) Find the average of the $x$ values of all inflection points of $N(x)$. Note: If there are no inflection points, enter-1000. Average of inflection points=[ANS]\n(D) Find the maximum rate of change of sales.\nMaximum rate of change of sales=[ANS]", "answer_v1": [ "(10,17.3333333333333)", "(17.3333333333333,40)", "17.3333333333333", "906.666666666666" ], "answer_type_v1": [ "INT", "INT", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A company estimates that it will sell $N(x)$ units of a product after spending $x$ thousand dollars on advertising, as given by N(x)=-3x^3+295x^2-3100x+17000, \\qquad 10 \\leq x \\leq 40. (A) Use interval notation to indicate when the rate of change of sales $N'(x)$ is increasing.\n$N'(x)$ increasing: [ANS]\n(B) Use interval notation to indicate when the rate of change of sales $N'(x)$ is decreasing. $N'(x)$ decreasing: [ANS]\n(C) Find the average of the $x$ values of all inflection points of $N(x)$. Note: If there are no inflection points, enter-1000. Average of inflection points=[ANS]\n(D) Find the maximum rate of change of sales.\nMaximum rate of change of sales=[ANS]", "answer_v2": [ "(10,32.7777777777778)", "(32.7777777777778,40)", "32.7777777777778", "6569.44444444445" ], "answer_type_v2": [ "INT", "INT", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A company estimates that it will sell $N(x)$ units of a product after spending $x$ thousand dollars on advertising, as given by N(x)=-3x^3+260x^2-3300x+18000, \\qquad 10 \\leq x \\leq 40. (A) Use interval notation to indicate when the rate of change of sales $N'(x)$ is increasing.\n$N'(x)$ increasing: [ANS]\n(B) Use interval notation to indicate when the rate of change of sales $N'(x)$ is decreasing. $N'(x)$ decreasing: [ANS]\n(C) Find the average of the $x$ values of all inflection points of $N(x)$. Note: If there are no inflection points, enter-1000. Average of inflection points=[ANS]\n(D) Find the maximum rate of change of sales.\nMaximum rate of change of sales=[ANS]", "answer_v3": [ "(10,28.8888888888889)", "(28.8888888888889,40)", "28.8888888888889", "4211.11111111111" ], "answer_type_v3": [ "INT", "INT", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0277", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "3", "keywords": [ "algebra", "inequality", "fraction" ], "problem_v1": "Suppose that the marginal cost function of a handbag manufacturer is C'(x)=0.046875x^{2}-x+325 dollars per unit at production level $x$ (where $x$ is measured in units of 100 handbags). Find the total cost of producing $8$ additional units if $6$ units are currently being produced. Total cost of producing the additional units: [ANS]\nNote: Your answer should be a dollar amount and include a dollar sign and be correct to two decimal places.", "answer_v1": [ "2559.50" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that the marginal cost function of a handbag manufacturer is C'(x)=0.375x^{2}-x+475 dollars per unit at production level $x$ (where $x$ is measured in units of 100 handbags). Find the total cost of producing $6$ additional units if $2$ units are currently being produced. Total cost of producing the additional units: [ANS]\nNote: Your answer should be a dollar amount and include a dollar sign and be correct to two decimal places.", "answer_v2": [ "2883.00" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that the marginal cost function of a handbag manufacturer is C'(x)=0.1875x^{2}-x+350 dollars per unit at production level $x$ (where $x$ is measured in units of 100 handbags). Find the total cost of producing $8$ additional units if $4$ units are currently being produced. Total cost of producing the additional units: [ANS]\nNote: Your answer should be a dollar amount and include a dollar sign and be correct to two decimal places.", "answer_v3": [ "2840.00" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0278", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "3", "keywords": [], "problem_v1": "The price (in dollars) $p$ and the quantity demanded $q$ are related by the equation: $p^2+2q^2=1100$. If $R=pq$ is revenue, $\\frac{dR}{dt}$ can be expressed by the following equation: $\\frac{dR}{dt}=A\\frac{dp}{dt}$, where $A$ is a function of just $q$. $A=$ [ANS]\nFind $\\frac{dR}{dt}$ when $q=20$ and $\\frac{dp}{dt}=4$. $\\frac{dR}{dt}=$ [ANS]", "answer_v1": [ "(4*q^2-1100)/(2*q)", "50" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The price (in dollars) $p$ and the quantity demanded $q$ are related by the equation: $p^2+2q^2=1100$. If $R=pq$ is revenue, $\\frac{dR}{dt}$ can be expressed by the following equation: $\\frac{dR}{dt}=A\\frac{dp}{dt}$, where $A$ is a function of just $q$. $A=$ [ANS]\nFind $\\frac{dR}{dt}$ when $q=5$ and $\\frac{dp}{dt}=5$. $\\frac{dR}{dt}=$ [ANS]", "answer_v2": [ "(4*q^2-1100)/(2*q)", "-500" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The price (in dollars) $p$ and the quantity demanded $q$ are related by the equation: $p^2+2q^2=1100$. If $R=pq$ is revenue, $\\frac{dR}{dt}$ can be expressed by the following equation: $\\frac{dR}{dt}=A\\frac{dp}{dt}$, where $A$ is a function of just $q$. $A=$ [ANS]\nFind $\\frac{dR}{dt}$ when $q=10$ and $\\frac{dp}{dt}=4$. $\\frac{dR}{dt}=$ [ANS]", "answer_v3": [ "(4*q^2-1100)/(2*q)", "-140" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0279", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - business and economics", "level": "5", "keywords": [], "problem_v1": "Nick has lost faith in the banks, and has decided to diversify his portfolio by keeping some money under his mattress. He decides to put \\$4500 under his mattress and \\$4000 in a GIC with a 10\\% annual interest rate (compounded continuously). If there is a 6\\% annual inflation rate, when will the real value of Nick's investments be at a minimum? NOTE: An inflation rate of 6\\% means that the real value of money is decreasing at this rate (compounded continuously). You should also consider what inflation does to the interest rate. If $I(t)$ is the total real value of the investments after $t$ years: $I(t)=$ [ANS]\nIf $t^*$ is the number of years until the value of the assets is a minimum: $t^*=$ [ANS]\nNOTE: Use at least one decimal in your answer.\n(you will lose 50\\% of your points if you do)", "answer_v1": [ "4500*e^{-0.06*t}+4000*e^{0.04*t}", "[ln(270)-ln(160)]/0.1" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Nick has lost faith in the banks, and has decided to diversify his portfolio by keeping some money under his mattress. He decides to put \\$2000 under his mattress and \\$2500 in a GIC with a 11\\% annual interest rate (compounded continuously). If there is a 8\\% annual inflation rate, when will the real value of Nick's investments be at a minimum? NOTE: An inflation rate of 8\\% means that the real value of money is decreasing at this rate (compounded continuously). You should also consider what inflation does to the interest rate. If $I(t)$ is the total real value of the investments after $t$ years: $I(t)=$ [ANS]\nIf $t^*$ is the number of years until the value of the assets is a minimum: $t^*=$ [ANS]\nNOTE: Use at least one decimal in your answer.\n(you will lose 50\\% of your points if you do)", "answer_v2": [ "2000*e^{-0.08*t}+2500*e^{0.03*t}", "[ln(160)-ln(75)]/0.11" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Nick has lost faith in the banks, and has decided to diversify his portfolio by keeping some money under his mattress. He decides to put \\$3000 under his mattress and \\$2500 in a GIC with a 10\\% annual interest rate (compounded continuously). If there is a 6\\% annual inflation rate, when will the real value of Nick's investments be at a minimum? NOTE: An inflation rate of 6\\% means that the real value of money is decreasing at this rate (compounded continuously). You should also consider what inflation does to the interest rate. If $I(t)$ is the total real value of the investments after $t$ years: $I(t)=$ [ANS]\nIf $t^*$ is the number of years until the value of the assets is a minimum: $t^*=$ [ANS]\nNOTE: Use at least one decimal in your answer.\n(you will lose 50\\% of your points if you do)", "answer_v3": [ "3000*e^{-0.06*t}+2500*e^{0.04*t}", "[ln(180)-ln(100)]/0.1" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0280", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "4", "keywords": [ "derivatives", "applications" ], "problem_v1": "The mass of the part of a rod that lies between its left end and a point $x$ meters to the right is $5x^{4}$ kg. Compute linear density of the rod at $7$ meters.\nAnswer: [ANS] kg/meter\nCompute the density at $5$ meters. Answer: [ANS] kg/meter", "answer_v1": [ "5*4*7^3", "5*4*5^3" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The mass of the part of a rod that lies between its left end and a point $x$ meters to the right is $2x^{3}$ kg. Compute linear density of the rod at $1$ meters.\nAnswer: [ANS] kg/meter\nCompute the density at $8$ meters. Answer: [ANS] kg/meter", "answer_v2": [ "2*3*1^2", "2*3*8^2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The mass of the part of a rod that lies between its left end and a point $x$ meters to the right is $3x^{4}$ kg. Compute linear density of the rod at $3$ meters.\nAnswer: [ANS] kg/meter\nCompute the density at $5$ meters. Answer: [ANS] kg/meter", "answer_v3": [ "3*4*3^3", "3*4*5^3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0281", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "4", "keywords": [ "derivatives", "velocity", "distance" ], "problem_v1": "If a ball is thrown vertically upward from the roof of $48$ foot building with a velocity of $96$ ft/sec, its height after $t$ seconds is $s(t)=48+96 t-16 t^2$.\na.) What is the maximum height the ball reaches? Answer: [ANS]\nb.) What is the velocity of the ball when it hits the ground (height $0$)? Answer: [ANS]", "answer_v1": [ "48 + 96*6/2 -4*6^2", "-16*(6*6+4*3)^{0.5}" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If a ball is thrown vertically upward from the roof of $64$ foot building with a velocity of $16$ ft/sec, its height after $t$ seconds is $s(t)=64+16 t-16 t^2$.\na.) What is the maximum height the ball reaches? Answer: [ANS]\nb.) What is the velocity of the ball when it hits the ground (height $0$)? Answer: [ANS]", "answer_v2": [ "64 + 16*1/2 -4*1^2", "-16*(1*1+4*4)^{0.5}" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If a ball is thrown vertically upward from the roof of $48$ foot building with a velocity of $48$ ft/sec, its height after $t$ seconds is $s(t)=48+48 t-16 t^2$.\na.) What is the maximum height the ball reaches? Answer: [ANS]\nb.) What is the velocity of the ball when it hits the ground (height $0$)? Answer: [ANS]", "answer_v3": [ "48 + 48*3/2 -4*3^2", "-16*(3*3+4*3)^{0.5}" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0282", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "5", "keywords": [ "calculus", "derivatives", "rates of change", "velocity", "initial velocity", "maximum/minimum" ], "problem_v1": "A ball is tossed up vertically from ground level and returns to earth 7 s later. What was the initial velocity of the ball and how high did it go? $v_0$=[ANS] m/s $h$=[ANS] m", "answer_v1": [ "34.3", "60.025" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A ball is tossed up vertically from ground level and returns to earth 1 s later. What was the initial velocity of the ball and how high did it go? $v_0$=[ANS] m/s $h$=[ANS] m", "answer_v2": [ "4.9", "1.225" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A ball is tossed up vertically from ground level and returns to earth 3 s later. What was the initial velocity of the ball and how high did it go? $v_0$=[ANS] m/s $h$=[ANS] m", "answer_v3": [ "14.7", "11.025" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0283", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "4", "keywords": [ "calculus", "derivatives", "rates of change", "velocity" ], "problem_v1": "What is the velocity of an object dropped from a height of 330 m when it hits the ground? [ANS]", "answer_v1": [ "-80.4239" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What is the velocity of an object dropped from a height of 120 m when it hits the ground? [ANS]", "answer_v2": [ "-48.4975" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What is the velocity of an object dropped from a height of 190 m when it hits the ground? [ANS]", "answer_v3": [ "-61.0246" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0284", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "4", "keywords": [ "calculus", "derivatives", "rates of change" ], "problem_v1": "Let $F(s)=1.1s+0.03s^{2}$ represent the stopping distance (in feet) of a car travelling at $s$ miles per hour. Calculate $F(55)$ and estimate the increase in stopping distance if speed is increased from $55$ to $56$. Compare your estimate with the actual increase. $F(55)=$ [ANS] feet Estimated increase in stopping distance: [ANS] feet Actual increase in stopping distance: [ANS] feet", "answer_v1": [ "151.25", "4.4", "4.43" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $F(s)=1.1s+0.03s^{2}$ represent the stopping distance (in feet) of a car travelling at $s$ miles per hour. Calculate $F(25)$ and estimate the increase in stopping distance if speed is increased from $25$ to $26$. Compare your estimate with the actual increase. $F(25)=$ [ANS] feet Estimated increase in stopping distance: [ANS] feet Actual increase in stopping distance: [ANS] feet", "answer_v2": [ "46.25", "2.6", "2.63" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $F(s)=1.1s+0.03s^{2}$ represent the stopping distance (in feet) of a car travelling at $s$ miles per hour. Calculate $F(35)$ and estimate the increase in stopping distance if speed is increased from $35$ to $36$. Compare your estimate with the actual increase. $F(35)=$ [ANS] feet Estimated increase in stopping distance: [ANS] feet Actual increase in stopping distance: [ANS] feet", "answer_v3": [ "75.25", "3.2", "3.23" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0285", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "2", "keywords": [ "calculus", "velocity", "average velocity", "rates of change", "average rates of change" ], "problem_v1": "A stone is tossed in the air from ground level with an initial velocity of $30$ $m/s$. Its height at time $t$ is $h(t)=30 t-4.9 t^2$ $m$. Compute the stone's average velocity over the time interval $[1, 3]$. Average velocity $=$ [ANS]", "answer_v1": [ "10.4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A stone is tossed in the air from ground level with an initial velocity of $15$ $m/s$. Its height at time $t$ is $h(t)=15 t-4.9 t^2$ $m$. Compute the stone's average velocity over the time interval $[1.5, 2.5]$. Average velocity $=$ [ANS]", "answer_v2": [ "-4.6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A stone is tossed in the air from ground level with an initial velocity of $20$ $m/s$. Its height at time $t$ is $h(t)=20 t-4.9 t^2$ $m$. Compute the stone's average velocity over the time interval $[1, 2.5]$. Average velocity $=$ [ANS]", "answer_v3": [ "2.85" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0286", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "2", "keywords": [ "derivative' 'limit' 'velocity" ], "problem_v1": "An automobile starts from rest and travels down a straight section of road. The distance $s$ (in feet) of the car from the starting position after $t$ seconds is given by $s(t)=7t^{3}.$\n(a) Find the instantaneous velocity (in feet per second) at $t=4$ seconds. Instantaneous velocity at $t=4$ is [ANS] ft/s. (b) Find the instantaneous velocity in (feet per second) at $t=8$ seconds. Instantaneous velocity at $t=8$ is [ANS] ft/s.", "answer_v1": [ "336", "1344" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "An automobile starts from rest and travels down a straight section of road. The distance $s$ (in feet) of the car from the starting position after $t$ seconds is given by $s(t)=2t^{3}.$\n(a) Find the instantaneous velocity (in feet per second) at $t=5$ seconds. Instantaneous velocity at $t=5$ is [ANS] ft/s. (b) Find the instantaneous velocity in (feet per second) at $t=6$ seconds. Instantaneous velocity at $t=6$ is [ANS] ft/s.", "answer_v2": [ "150", "216" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "An automobile starts from rest and travels down a straight section of road. The distance $s$ (in feet) of the car from the starting position after $t$ seconds is given by $s(t)=4t^{3}.$\n(a) Find the instantaneous velocity (in feet per second) at $t=4$ seconds. Instantaneous velocity at $t=4$ is [ANS] ft/s. (b) Find the instantaneous velocity in (feet per second) at $t=7$ seconds. Instantaneous velocity at $t=7$ is [ANS] ft/s.", "answer_v3": [ "192", "588" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0287", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "4", "keywords": [ "derivative' 'limit' 'velocity" ], "problem_v1": "If an arrow is shot straight upward on the moon with a velocity of 73 m/s, its height (in meters) after t seconds is given by $s(t)=73t-0.84t^{2}$.\n(a) What is the velocity of the arrow in meters per second after 8 seconds? [ANS]\n(b) How many seconds will it take for the arrow to return and hit the moon? [ANS]\n(c) With what velocity in meters per second will the arrow hit the moon? [ANS]", "answer_v1": [ "59.56", "86.9048", "-73" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "If an arrow is shot straight upward on the moon with a velocity of 52 m/s, its height (in meters) after t seconds is given by $s(t)=52t-0.88t^{2}$.\n(a) What is the velocity of the arrow in meters per second after 5 seconds? [ANS]\n(b) How many seconds will it take for the arrow to return and hit the moon? [ANS]\n(c) With what velocity in meters per second will the arrow hit the moon? [ANS]", "answer_v2": [ "43.2", "59.0909", "-52" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "If an arrow is shot straight upward on the moon with a velocity of 59 m/s, its height (in meters) after t seconds is given by $s(t)=59t-0.86t^{2}$.\n(a) What is the velocity of the arrow in meters per second after 6 seconds? [ANS]\n(b) How many seconds will it take for the arrow to return and hit the moon? [ANS]\n(c) With what velocity in meters per second will the arrow hit the moon? [ANS]", "answer_v3": [ "48.68", "68.6047", "-59" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0288", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "2", "keywords": [ "Calculus", "Velocity", "Derivative", "derivative" ], "problem_v1": "A rock is thrown off of a 100 foot cliff with an upward velocity of 45 m/s. As a result its height after t seconds is given by the formula: $h(t)=100+45 t-5t^2$ What is its height after 5 seconds? [ANS]\nWhat is its velocity after 5 seconds? [ANS]\n(Positive velocity means it is on the way up, negative velocity means it is on the way down.)", "answer_v1": [ "200", "-5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A rock is thrown off of a 100 foot cliff with an upward velocity of 30 m/s. As a result its height after t seconds is given by the formula: $h(t)=100+30 t-5t^2$ What is its height after 7 seconds? [ANS]\nWhat is its velocity after 7 seconds? [ANS]\n(Positive velocity means it is on the way up, negative velocity means it is on the way down.)", "answer_v2": [ "65", "-40" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A rock is thrown off of a 100 foot cliff with an upward velocity of 35 m/s. As a result its height after t seconds is given by the formula: $h(t)=100+35 t-5t^2$ What is its height after 5 seconds? [ANS]\nWhat is its velocity after 5 seconds? [ANS]\n(Positive velocity means it is on the way up, negative velocity means it is on the way down.)", "answer_v3": [ "150", "-15" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0289", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "2", "keywords": [ "Derivatives", "applications" ], "problem_v1": "A particle moves along a straight line with equation of motion $s=t^{5}-5 t^{4}$ Find the value of $t$ (other than 0) at which the acceleration is equal to zero. [ANS]", "answer_v1": [ "3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A particle moves along a straight line with equation of motion $s=t^{6}-1 t^{5}$ Find the value of $t$ (other than 0) at which the acceleration is equal to zero. [ANS]", "answer_v2": [ "0.666666666666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A particle moves along a straight line with equation of motion $s=t^{5}-2 t^{4}$ Find the value of $t$ (other than 0) at which the acceleration is equal to zero. [ANS]", "answer_v3": [ "1.2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0292", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "2", "keywords": [ "Functions", "Applied Problems", "Velocity", "Acceleration", "Speed", "Position" ], "problem_v1": "Let $\\small{s=\\large{\\frac{80}{t^{2}+18}}}$ be the position function of a particle moving along a coordinate line, where $\\small{s}$ is in feet and $\\small{t}$ is in seconds.\n(a) Find the maximum speed of the particle for $\\small{t \\ge 0}$. If appropriate, leave your answer in radical form. Speed (ft/sec): [ANS]\n(b) Find the direction of the particle when it has its maximum speed. [ANS]", "answer_v1": [ "0.680414", "Decreasing" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "Increasing", "Decreasing" ] ], "problem_v2": "Let $\\small{s=\\large{\\frac{10}{t^{2}+30}}}$ be the position function of a particle moving along a coordinate line, where $\\small{s}$ is in feet and $\\small{t}$ is in seconds.\n(a) Find the maximum speed of the particle for $\\small{t \\ge 0}$. If appropriate, leave your answer in radical form. Speed (ft/sec): [ANS]\n(b) Find the direction of the particle when it has its maximum speed. [ANS]", "answer_v2": [ "0.0395285", "Decreasing" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "Increasing", "Decreasing" ] ], "problem_v3": "Let $\\small{s=\\large{\\frac{40}{t^{2}+21}}}$ be the position function of a particle moving along a coordinate line, where $\\small{s}$ is in feet and $\\small{t}$ is in seconds.\n(a) Find the maximum speed of the particle for $\\small{t \\ge 0}$. If appropriate, leave your answer in radical form. Speed (ft/sec): [ANS]\n(b) Find the direction of the particle when it has its maximum speed. [ANS]", "answer_v3": [ "0.269975", "Decreasing" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "Increasing", "Decreasing" ] ] }, { "id": "Calculus_-_single_variable_0293", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "3", "keywords": [ "rate of change", "displacement", "velocity", "acceleration" ], "problem_v1": "The equations for free fall at the surfaces of Uranus and Jupiter ($s$ in meters, $t$ in seconds) are $s=5.335 t^2$ on Uranus and $s=12.975 t^2$ on Jupiter.\n(a) How long does it take a rock falling from rest to reach a velocity of $27.8 \\ \\mathrm{m/s}$ (about $100 \\ \\mathrm{km/hr}$) on each planet? On Uranus: [ANS] seconds On Jupiter: [ANS] seconds\n(b) Using derivatives, what is the acceleration due to gravity on each planet? On Uranus: [ANS] $\\mathrm{m/s^2}$ On Jupiter: [ANS] $\\mathrm{m/s^2}$", "answer_v1": [ "2.60544", "1.07129", "10.67", "25.95" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The equations for free fall at the surfaces of Mercury and Jupiter ($s$ in meters, $t$ in seconds) are $s=1.795 t^2$ on Mercury and $s=12.975 t^2$ on Jupiter.\n(a) How long does it take a rock falling from rest to reach a velocity of $27.8 \\ \\mathrm{m/s}$ (about $100 \\ \\mathrm{km/hr}$) on each planet? On Mercury: [ANS] seconds On Jupiter: [ANS] seconds\n(b) Using derivatives, what is the acceleration due to gravity on each planet? On Mercury: [ANS] $\\mathrm{m/s^2}$ On Jupiter: [ANS] $\\mathrm{m/s^2}$", "answer_v2": [ "7.74373", "1.07129", "3.59", "25.95" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The equations for free fall at the surfaces of Venus and Jupiter ($s$ in meters, $t$ in seconds) are $s=4.435 t^2$ on Venus and $s=12.975 t^2$ on Jupiter.\n(a) How long does it take a rock falling from rest to reach a velocity of $27.8 \\ \\mathrm{m/s}$ (about $100 \\ \\mathrm{km/hr}$) on each planet? On Venus: [ANS] seconds On Jupiter: [ANS] seconds\n(b) Using derivatives, what is the acceleration due to gravity on each planet? On Venus: [ANS] $\\mathrm{m/s^2}$ On Jupiter: [ANS] $\\mathrm{m/s^2}$", "answer_v3": [ "3.13416", "1.07129", "8.87", "25.95" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0294", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "5", "keywords": [ "calculus" ], "problem_v1": "Let's now combine our rock tossing with a horizontal motion. This kind of problem can be handled by considering the vertical and horizontal motions separately. The horizontal velocity is constant. (Again, we ignore air resistance.) Recall from your study of trigonometry that if you release a rock at a speed $v$ in a direction that makes an angle $\\alpha$ with the horizontal, then the initial vertical velocity $v_v$ and the horizontal velocity $v_h$ are given by v_v=v\\sin\\alpha \\quad\\hbox{and}\\quad v_h=v\\cos\\alpha. You shoot a rifle at an angle of 38 degrees. The bullet leaves your rifle at a height of 6 feet and a speed of 876 feet per second. It hits the ground after [ANS] seconds at a distance of [ANS] feet. Note: Again, ignore air resistance. When shooting a rifle this is of course a gross simplification. Even so, that's some rifle!", "answer_v1": [ "33.7185872364759", "23275.853780762" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let's now combine our rock tossing with a horizontal motion. This kind of problem can be handled by considering the vertical and horizontal motions separately. The horizontal velocity is constant. (Again, we ignore air resistance.) Recall from your study of trigonometry that if you release a rock at a speed $v$ in a direction that makes an angle $\\alpha$ with the horizontal, then the initial vertical velocity $v_v$ and the horizontal velocity $v_h$ are given by v_v=v\\sin\\alpha \\quad\\hbox{and}\\quad v_h=v\\cos\\alpha. You shoot a rifle at an angle of 48 degrees. The bullet leaves your rifle at a height of 6 feet and a speed of 808 feet per second. It hits the ground after [ANS] seconds at a distance of [ANS] feet. Note: Again, ignore air resistance. When shooting a rifle this is of course a gross simplification. Even so, that's some rifle!", "answer_v2": [ "37.5388033497446", "20295.6366958996" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let's now combine our rock tossing with a horizontal motion. This kind of problem can be handled by considering the vertical and horizontal motions separately. The horizontal velocity is constant. (Again, we ignore air resistance.) Recall from your study of trigonometry that if you release a rock at a speed $v$ in a direction that makes an angle $\\alpha$ with the horizontal, then the initial vertical velocity $v_v$ and the horizontal velocity $v_h$ are given by v_v=v\\sin\\alpha \\quad\\hbox{and}\\quad v_h=v\\cos\\alpha. You shoot a rifle at an angle of 38 degrees. The bullet leaves your rifle at a height of 6 feet and a speed of 831 feet per second. It hits the ground after [ANS] seconds at a distance of [ANS] feet. Note: Again, ignore air resistance. When shooting a rifle this is of course a gross simplification. Even so, that's some rifle!", "answer_v3": [ "31.9876411524202", "20946.6889300252" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0295", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "3", "keywords": [ "calculus", "limits", "derivatives" ], "problem_v1": "If a cylindrical tank holds 390000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume $V$ of water remaining in the tank after $t$ minutes as V(t)=390000\\left(1-\\frac{t}{60}\\right)^2, \\qquad 0 \\leq t \\leq 60\n(a) Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of $V$ with respect to $t$) as a function of $t$. Rate of Flow=[ANS]\n(b) Find both the flow rate and the amount of water remaining in the tank for the times given below. (i) $t=0$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(ii) $t=10$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(iii) $t=20$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(iv) $t=30$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(v) $t=40$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(c) At what time is the flow rate the greatest? $t$=[ANS]\nAt what time is the flow rate the least? $t$=[ANS]", "answer_v1": [ "390000/30*(t/60-1)", "-13000", "390000", "-10833.3333333333", "270833.333333333", "-8666.66666666667", "173333.333333333", "-6500", "97500", "-4333.33333333333", "43333.3333333333", "0", "60" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "If a cylindrical tank holds 80000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume $V$ of water remaining in the tank after $t$ minutes as V(t)=80000\\left(1-\\frac{t}{60}\\right)^2, \\qquad 0 \\leq t \\leq 60\n(a) Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of $V$ with respect to $t$) as a function of $t$. Rate of Flow=[ANS]\n(b) Find both the flow rate and the amount of water remaining in the tank for the times given below. (i) $t=0$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(ii) $t=10$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(iii) $t=20$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(iv) $t=30$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(v) $t=40$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(c) At what time is the flow rate the greatest? $t$=[ANS]\nAt what time is the flow rate the least? $t$=[ANS]", "answer_v2": [ "80000/30*(t/60-1)", "-2666.66666666667", "80000", "-2222.22222222222", "55555.5555555556", "-1777.77777777778", "35555.5555555556", "-1333.33333333333", "20000", "-888.888888888889", "8888.88888888889", "0", "60" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "If a cylindrical tank holds 190000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume $V$ of water remaining in the tank after $t$ minutes as V(t)=190000\\left(1-\\frac{t}{60}\\right)^2, \\qquad 0 \\leq t \\leq 60\n(a) Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of $V$ with respect to $t$) as a function of $t$. Rate of Flow=[ANS]\n(b) Find both the flow rate and the amount of water remaining in the tank for the times given below. (i) $t=0$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(ii) $t=10$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(iii) $t=20$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(iv) $t=30$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(v) $t=40$: Flow rate=[ANS] Amount of water remaining=[ANS]\n(c) At what time is the flow rate the greatest? $t$=[ANS]\nAt what time is the flow rate the least? $t$=[ANS]", "answer_v3": [ "190000/30*(t/60-1)", "-6333.33333333333", "190000", "-5277.77777777778", "131944.444444444", "-4222.22222222222", "84444.4444444445", "-3166.66666666667", "47500", "-2111.11111111111", "21111.1111111111", "0", "60" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0296", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - engineering and physics", "level": "4", "keywords": [ "calculus", "differentiation" ], "problem_v1": "The position function of a particle is given by $s(t)=2t^{3}-15t^{2}-73t$, $t\\geq 0$.\nFind all values of $t$ for which the particle is moving at a velocity of 11 meters/second. (If there are no such values, enter 0. If there are more than one value, list them separated by commas.) $t$=[ANS]", "answer_v1": [ "7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The position function of a particle is given by $s(t)=2t^{3}-3t^{2}-68t$, $t\\geq 0$.\nFind all values of $t$ for which the particle is moving at a velocity of 4 meters/second. (If there are no such values, enter 0. If there are more than one value, list them separated by commas.) $t$=[ANS]", "answer_v2": [ "4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The position function of a particle is given by $s(t)=2t^{3}-9t^{2}-54t$, $t\\geq 0$.\nFind all values of $t$ for which the particle is moving at a velocity of 6 meters/second. (If there are no such values, enter 0. If there are more than one value, list them separated by commas.) $t$=[ANS]", "answer_v3": [ "5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0297", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - natural and social sciences", "level": "4", "keywords": [ "derivatives", "rate of change", "biology" ], "problem_v1": "The population of a slowly growing bacterial colony after $t$ hours is given by $p(t)=5t^{2}+29t+150$. Find the growth rate after $4$ hours.\nAnswer: [ANS]", "answer_v1": [ "5*2*4+29" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The population of a slowly growing bacterial colony after $t$ hours is given by $p(t)=3t^{2}+34t+100$. Find the growth rate after $3$ hours.\nAnswer: [ANS]", "answer_v2": [ "3*2*3+34" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The population of a slowly growing bacterial colony after $t$ hours is given by $p(t)=3t^{2}+29t+100$. Find the growth rate after $3$ hours.\nAnswer: [ANS]", "answer_v3": [ "3*2*3+29" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0299", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - natural and social sciences", "level": "4", "keywords": [], "problem_v1": "The population of mice in Alfred is given by $P(t)=2621 e^{8 t}$, where t is in years since 1986. The rate of change of the population is given by the formula [ANS] mice/yr. In year 1990 the population changes by approximately [ANS] mice. In 1990 538799946654077088 mice died, which means that [ANS] mice were born that year.", "answer_v1": [ "2621*8*e^{8*t}", "1.65569534911045E+18", "2.19449529576453E+18" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The population of mice in Alfred is given by $P(t)=2149 e^{2 t}$, where t is in years since 1986. The rate of change of the population is given by the formula [ANS] mice/yr. In year 1988 the population changes by approximately [ANS] mice. In 1988 114209 mice died, which means that [ANS] mice were born that year.", "answer_v2": [ "2149*2*e^{2*t}", "234662.848842454", "348871.848842454" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The population of mice in Alfred is given by $P(t)=2278 e^{4 t}$, where t is in years since 1986. The rate of change of the population is given by the formula [ANS] mice/yr. In year 1989 the population changes by approximately [ANS] mice. In 1989 447196153 mice died, which means that [ANS] mice were born that year.", "answer_v3": [ "2278*4*e^{4*t}", "1483021659.40996", "1930217812.40996" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0300", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - natural and social sciences", "level": "5", "keywords": [ "calculus", "differentiation" ], "problem_v1": "If $R$ denotes the reaction of the body to some stimulus of strength $x$, the \\ sensitivity $S$ is defined to be the rate of change of the reaction with respect to $x$. A particular example is that when the brightness $x$ of a light source is increased, the eye reacts by decreasing the area $R$ of the pupil. The experimental formula R=\\frac{43+21x^{5}}{1+0.5x^{5}} can be used to model the dependence of $R$ on $x$ when $R$ is measured in square millimeters and $x$ is measured in appropriate units of brightness. Find the sensitivity corresponding to $x=2$. Sensitivity=[ANS]", "answer_v1": [ "-0.13840830449827" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $R$ denotes the reaction of the body to some stimulus of strength $x$, the \\ sensitivity $S$ is defined to be the rate of change of the reaction with respect to $x$. A particular example is that when the brightness $x$ of a light source is increased, the eye reacts by decreasing the area $R$ of the pupil. The experimental formula R=\\frac{35+24x^{3}}{1+0.4x^{3}} can be used to model the dependence of $R$ on $x$ when $R$ is measured in square millimeters and $x$ is measured in appropriate units of brightness. Find the sensitivity corresponding to $x=2$. Sensitivity=[ANS]", "answer_v2": [ "6.80272108843537" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $R$ denotes the reaction of the body to some stimulus of strength $x$, the \\ sensitivity $S$ is defined to be the rate of change of the reaction with respect to $x$. A particular example is that when the brightness $x$ of a light source is increased, the eye reacts by decreasing the area $R$ of the pupil. The experimental formula R=\\frac{38+21x^{4}}{1+0.4x^{4}} can be used to model the dependence of $R$ on $x$ when $R$ is measured in square millimeters and $x$ is measured in appropriate units of brightness. Find the sensitivity corresponding to $x=2$. Sensitivity=[ANS]", "answer_v3": [ "3.38933528122718" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0301", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - natural and social sciences", "level": "5", "keywords": [ "calculus", "differentiation" ], "problem_v1": "Sodium chlorate crystals are easy to grow in the shape of a cube by allowing a solution of water and sodium chlorate to evaporate slowly. If $V$ is the volume of such a cube with side length $x$, calculate $dV/dx$ when $x$=10 mm. $dV/dx$=[ANS]", "answer_v1": [ "300" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sodium chlorate crystals are easy to grow in the shape of a cube by allowing a solution of water and sodium chlorate to evaporate slowly. If $V$ is the volume of such a cube with side length $x$, calculate $dV/dx$ when $x$=4 mm. $dV/dx$=[ANS]", "answer_v2": [ "48" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sodium chlorate crystals are easy to grow in the shape of a cube by allowing a solution of water and sodium chlorate to evaporate slowly. If $V$ is the volume of such a cube with side length $x$, calculate $dV/dx$ when $x$=6 mm. $dV/dx$=[ANS]", "answer_v3": [ "108" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0302", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - natural and social sciences", "level": "5", "keywords": [ "derivatives", "Business and Economics" ], "problem_v1": "A commercial cherry grower estimates from past records that if 51 trees are planted per acre, each tree will yield 33 pounds of cherries for a growing season. Each additional tree per acre (up to 20) results in a decrease in yield per tree of 1 pound. How many trees per acre should be planted to maximize yield per acre, and what is the maximum yield? Number of Trees=[ANS]\nMaximum Yield=[ANS]", "answer_v1": [ "42", "1764" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A commercial cherry grower estimates from past records that if 55 trees are planted per acre, each tree will yield 27 pounds of cherries for a growing season. Each additional tree per acre (up to 20) results in a decrease in yield per tree of 1 pound. How many trees per acre should be planted to maximize yield per acre, and what is the maximum yield? Number of Trees=[ANS]\nMaximum Yield=[ANS]", "answer_v2": [ "41", "1681" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A commercial cherry grower estimates from past records that if 51 trees are planted per acre, each tree will yield 29 pounds of cherries for a growing season. Each additional tree per acre (up to 20) results in a decrease in yield per tree of 1 pound. How many trees per acre should be planted to maximize yield per acre, and what is the maximum yield? Number of Trees=[ANS]\nMaximum Yield=[ANS]", "answer_v3": [ "40", "1600" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0303", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - natural and social sciences", "level": "4", "keywords": [ "instantaneous", "rate of change", "application", "derivative" ], "problem_v1": "A person $x$ inches tall has a pulse rate approximately given by the function y=590x^{-1/2}. The instantaneous rate of change of the pulse rate for a person that is: (A) 38 inches tall=[ANS]\n(B) 69 inches tall=[ANS]", "answer_v1": [ "-1.25935050614671", "-0.514693140004376" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A person $x$ inches tall has a pulse rate approximately given by the function y=560x^{-1/2}. The instantaneous rate of change of the pulse rate for a person that is: (A) 30 inches tall=[ANS]\n(B) 74 inches tall=[ANS]", "answer_v2": [ "-1.70402573446052", "-0.439855930382019" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A person $x$ inches tall has a pulse rate approximately given by the function y=570x^{-1/2}. The instantaneous rate of change of the pulse rate for a person that is: (A) 33 inches tall=[ANS]\n(B) 69 inches tall=[ANS]", "answer_v3": [ "-1.50339793779921", "-0.497245914919482" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0304", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - natural and social sciences", "level": "4", "keywords": [ "instantaneous", "rate of change", "application", "derivative" ], "problem_v1": "A coal-burning electrical generating plant emits sulfur dioxide into the surrounding air. The concentration $C(x)$, in parts per million, is approximately given by the function C(x)=\\frac{0.8}{x^2}, where $x$ is the distance away from the plant in miles. The instantaneous rate of change of the sulfur dioxide concentration: (A) 3 miles from the plant=[ANS]\n(B) 10 miles from the plant=[ANS]", "answer_v1": [ "-0.0592592592592593", "-0.0016" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A coal-burning electrical generating plant emits sulfur dioxide into the surrounding air. The concentration $C(x)$, in parts per million, is approximately given by the function C(x)=\\frac{0.3}{x^2}, where $x$ is the distance away from the plant in miles. The instantaneous rate of change of the sulfur dioxide concentration: (A) 5 miles from the plant=[ANS]\n(B) 7 miles from the plant=[ANS]", "answer_v2": [ "-0.0048", "-0.00174927113702624" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A coal-burning electrical generating plant emits sulfur dioxide into the surrounding air. The concentration $C(x)$, in parts per million, is approximately given by the function C(x)=\\frac{0.5}{x^2}, where $x$ is the distance away from the plant in miles. The instantaneous rate of change of the sulfur dioxide concentration: (A) 4 miles from the plant=[ANS]\n(B) 8 miles from the plant=[ANS]", "answer_v3": [ "-0.015625", "-0.001953125" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0305", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Rates of change - natural and social sciences", "level": "4", "keywords": [ "instantaneous", "rate of change", "application", "derivative" ], "problem_v1": "If a person learns $y$ items in $x$ hours, as given by y=23 \\sqrt[3]{x^2}, find the rate of learning for a person at the end of: (A) 3 hours: [ANS]\n(B) 7 hours: [ANS]", "answer_v1": [ "10.6315395400431", "8.01562203147889" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If a person learns $y$ items in $x$ hours, as given by y=15 \\sqrt[3]{x^2}, find the rate of learning for a person at the end of: (A) 4 hours: [ANS]\n(B) 6 hours: [ANS]", "answer_v2": [ "6.29960524947437", "5.50321208149104" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If a person learns $y$ items in $x$ hours, as given by y=18 \\sqrt[3]{x^2}, find the rate of learning for a person at the end of: (A) 3 hours: [ANS]\n(B) 6 hours: [ANS]", "answer_v3": [ "8.32033529220762", "6.60385449778925" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0306", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "3", "keywords": [ "critical numbers" ], "problem_v1": "The critical numbers of the function f(t)=8 t^{2/3}+t^{5/3} are $t_1=$ [ANS] and $t_2=$ [ANS] with $t_10$ Critical Point=[ANS]\nIs $f$ a maximum or minumum at the critical point? [ANS] The interval on the left of the critical point is [ANS]. On this interval, $f$ is [ANS] while $f'$ is [ANS].\nThe interval on the right of the critical point is [ANS]. On this interval, $f$ is [ANS] while $f'$ is [ANS].", "answer_v1": [ "3", "LOCAL MIN", "(0,3)", "Decreasing", "Negative", "(3,infinity)", "Increasing", "Positive" ], "answer_type_v1": [ "NV", "MCS", "INT", "MCS", "MCS", "INT", "MCS", "MCS" ], "options_v1": [ [], [ "Local Max", "Local Min", "Neither" ], [], [ "Increasing", "Decreasing" ], [ "Positive", "Negative" ], [], [ "Increasing", "Decreasing" ], [ "Positive", "Negative" ] ], "problem_v2": "Find the critical point and the interval on which the given function is increasing or decreasing, and apply the First Derivative Test to the critical point. Let $f(x)=20\\ln\\!\\left(2x\\right)-4x, x>0$ Critical Point=[ANS]\nIs $f$ a maximum or minumum at the critical point? [ANS] The interval on the left of the critical point is [ANS]. On this interval, $f$ is [ANS] while $f'$ is [ANS].\nThe interval on the right of the critical point is [ANS]. On this interval, $f$ is [ANS] while $f'$ is [ANS].", "answer_v2": [ "5", "LOCAL MAX", "(0,5)", "Increasing", "Positive", "(5,infinity)", "Decreasing", "Negative" ], "answer_type_v2": [ "NV", "MCS", "INT", "MCS", "MCS", "INT", "MCS", "MCS" ], "options_v2": [ [], [ "Local Max", "Local Min", "Neither" ], [], [ "Increasing", "Decreasing" ], [ "Positive", "Negative" ], [], [ "Increasing", "Decreasing" ], [ "Positive", "Negative" ] ], "problem_v3": "Find the critical point and the interval on which the given function is increasing or decreasing, and apply the First Derivative Test to the critical point. Let $f(x)=4\\ln\\!\\left(3x\\right)-x, x>0$ Critical Point=[ANS]\nIs $f$ a maximum or minumum at the critical point? [ANS] The interval on the left of the critical point is [ANS]. On this interval, $f$ is [ANS] while $f'$ is [ANS].\nThe interval on the right of the critical point is [ANS]. On this interval, $f$ is [ANS] while $f'$ is [ANS].", "answer_v3": [ "4", "LOCAL MAX", "(0,4)", "Increasing", "Positive", "(4,infinity)", "Decreasing", "Negative" ], "answer_type_v3": [ "NV", "MCS", "INT", "MCS", "MCS", "INT", "MCS", "MCS" ], "options_v3": [ [], [ "Local Max", "Local Min", "Neither" ], [], [ "Increasing", "Decreasing" ], [ "Positive", "Negative" ], [], [ "Increasing", "Decreasing" ], [ "Positive", "Negative" ] ] }, { "id": "Calculus_-_single_variable_0315", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "3", "keywords": [ "calculus", "increasing", "decreasing" ], "problem_v1": "Let $ f(x)=8-\\frac{6}{x}+\\frac{6}{x^{2}}$. Find the open intervals on which $f$ is increasing (decreasing). Then determine the $x$-coordinates of all relative maxima (minima).\n$\\begin{array}{ccc}\\hline 1. & fis increasing on the intervals & [ANS] \\\\ \\hline 2. & fis decreasing on the intervals & [ANS] \\\\ \\hline 3. & The relative maxima of foccur at x=& [ANS] \\\\ \\hline 4. & The relative minima of foccur at x=& [ANS] \\\\ \\hline \\end{array}$\nNotes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word \"none\". In the last two, your answer should be a comma separated list of $x$ values or the word \"none\".", "answer_v1": [ "(-infinity,0) \\cup (2,infinity)", "(0,2)", "NONE", "2" ], "answer_type_v1": [ "INT", "INT", "OE", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $ f(x)=2-\\frac{8}{x}+\\frac{2}{x^{2}}$. Find the open intervals on which $f$ is increasing (decreasing). Then determine the $x$-coordinates of all relative maxima (minima).\n$\\begin{array}{ccc}\\hline 1. & fis increasing on the intervals & [ANS] \\\\ \\hline 2. & fis decreasing on the intervals & [ANS] \\\\ \\hline 3. & The relative maxima of foccur at x=& [ANS] \\\\ \\hline 4. & The relative minima of foccur at x=& [ANS] \\\\ \\hline \\end{array}$\nNotes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word \"none\". In the last two, your answer should be a comma separated list of $x$ values or the word \"none\".", "answer_v2": [ "(-infinity,0) \\cup (0.5,infinity)", "(0,0.5)", "NONE", "0.5" ], "answer_type_v2": [ "INT", "INT", "OE", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $ f(x)=4-\\frac{6}{x}+\\frac{4}{x^{2}}$. Find the open intervals on which $f$ is increasing (decreasing). Then determine the $x$-coordinates of all relative maxima (minima).\n$\\begin{array}{ccc}\\hline 1. & fis increasing on the intervals & [ANS] \\\\ \\hline 2. & fis decreasing on the intervals & [ANS] \\\\ \\hline 3. & The relative maxima of foccur at x=& [ANS] \\\\ \\hline 4. & The relative minima of foccur at x=& [ANS] \\\\ \\hline \\end{array}$\nNotes: In the first two, your answer should either be a single interval, such as (0,1), a comma separated list of intervals, such as (-inf, 2), (3,4), or the word \"none\". In the last two, your answer should be a comma separated list of $x$ values or the word \"none\".", "answer_v3": [ "(-infinity,0) \\cup (1.33333,infinity)", "(0,1.33333)", "NONE", "1.33333" ], "answer_type_v3": [ "INT", "INT", "OE", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0316", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "3", "keywords": [ "Graph", "Critical", "derivatives", "critical points", "Calculus" ], "problem_v1": "The function $f(x)=(6x+5)e^{-6x}$ has one critical number. Find it. [ANS]", "answer_v1": [ "-0.666666666666667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The function $f(x)=(3x-8)e^{5x}$ has one critical number. Find it. [ANS]", "answer_v2": [ "2.46666666666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The function $f(x)=(3x-4)e^{-6x}$ has one critical number. Find it. [ANS]", "answer_v3": [ "1.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0317", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "3", "keywords": [ "Increasing", "Decreasing", "Local", "Extrema" ], "problem_v1": "The function f(x)=4x^{3}-18x^{2}-120x+4 is decreasing on the interval [ANS]. Enter your answer using the interval notation for open intervals. Enter your answer using the interval notation for open intervals. It is increasing on the interval(s) [ANS]. The function has a local maximum at [ANS].", "answer_v1": [ "(-2,5)", "(-infinity,-2) \\cup (5,infinity)", "-2" ], "answer_type_v1": [ "INT", "INT", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The function f(x)=2x^{3}+3x^{2}-336x-3 is decreasing on the interval [ANS]. Enter your answer using the interval notation for open intervals. Enter your answer using the interval notation for open intervals. It is increasing on the interval(s) [ANS]. The function has a local maximum at [ANS].", "answer_v2": [ "(-8,7)", "(-infinity,-8) \\cup (7,infinity)", "-8" ], "answer_type_v2": [ "INT", "INT", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The function f(x)=2x^{3}+3x^{2}-180x+1 is decreasing on the interval [ANS]. Enter your answer using the interval notation for open intervals. Enter your answer using the interval notation for open intervals. It is increasing on the interval(s) [ANS]. The function has a local maximum at [ANS].", "answer_v3": [ "(-6,5)", "(-infinity,-6)\\cup (5,infinity)", "-6" ], "answer_type_v3": [ "INT", "INT", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0318", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "2", "keywords": [ "critical numbers" ], "problem_v1": "Find all critical numbers of the polynomial f(x)=4x^{3}-18x^{2}-120x+4. If there are no critical numbers, enter None. List of critical numbers: [ANS]", "answer_v1": [ "(-2, 5)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find all critical numbers of the polynomial f(x)=2x^{3}-384x-3. If there are no critical numbers, enter None. List of critical numbers: [ANS]", "answer_v2": [ "(-8, 8)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find all critical numbers of the polynomial f(x)=2x^{3}+3x^{2}-180x+1. If there are no critical numbers, enter None. List of critical numbers: [ANS]", "answer_v3": [ "(-6, 5)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0319", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "3", "keywords": [ "Graph", "Critical", "Increasing", "Decreasing", "Calculus", "Derivatives", "first derivative", "critical points", "minimum", "maximum", "minimum", "maximum" ], "problem_v1": "Consider the function $f(x)=x^{2}e^{16x}$. For this function there are three important intervals: $(-\\infty, A]$, $[A,B]$, and $[B,\\infty)$ where $A$ and $B$ are the critical numbers. Find $A$ [ANS]\nand $B$ [ANS]\nFor each of the following intervals, tell whether $f(x)$ is increasing (type in INC) or decreasing (type in DEC). $(-\\infty, A]$: [ANS]\n$[A,B]$: [ANS]\n$[B,\\infty)$ [ANS]", "answer_v1": [ "-0.125", "0", "INC", "DEC", "INC" ], "answer_type_v1": [ "NV", "NV", "MCS", "MCS", "MCS" ], "options_v1": [ [], [], [ "INC", "DEC" ], [ "INC", "DEC" ], [ "INC", "DEC" ] ], "problem_v2": "Consider the function $f(x)=x^{2}e^{3x}$. For this function there are three important intervals: $(-\\infty, A]$, $[A,B]$, and $[B,\\infty)$ where $A$ and $B$ are the critical numbers. Find $A$ [ANS]\nand $B$ [ANS]\nFor each of the following intervals, tell whether $f(x)$ is increasing (type in INC) or decreasing (type in DEC). $(-\\infty, A]$: [ANS]\n$[A,B]$: [ANS]\n$[B,\\infty)$ [ANS]", "answer_v2": [ "-0.666667", "0", "INC", "DEC", "INC" ], "answer_type_v2": [ "NV", "NV", "MCS", "MCS", "MCS" ], "options_v2": [ [], [], [ "INC", "DEC" ], [ "INC", "DEC" ], [ "INC", "DEC" ] ], "problem_v3": "Consider the function $f(x)=x^{2}e^{7x}$. For this function there are three important intervals: $(-\\infty, A]$, $[A,B]$, and $[B,\\infty)$ where $A$ and $B$ are the critical numbers. Find $A$ [ANS]\nand $B$ [ANS]\nFor each of the following intervals, tell whether $f(x)$ is increasing (type in INC) or decreasing (type in DEC). $(-\\infty, A]$: [ANS]\n$[A,B]$: [ANS]\n$[B,\\infty)$ [ANS]", "answer_v3": [ "-0.285714", "0", "INC", "DEC", "INC" ], "answer_type_v3": [ "NV", "NV", "MCS", "MCS", "MCS" ], "options_v3": [ [], [], [ "INC", "DEC" ], [ "INC", "DEC" ], [ "INC", "DEC" ] ] }, { "id": "Calculus_-_single_variable_0320", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "5", "keywords": [ "concavity", "increasing and decreasing functions", "maxima", "minima" ], "problem_v1": "Find constants $a$ and $b$ in the function $f(x)=a x e^{b x}$ such that $f(\\frac{1}{8})=1$ and the function has a local maximum at $x=\\frac{1}{8}$. $a=$ [ANS]\n$b=$ [ANS]", "answer_v1": [ "8*e", "-1*8" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find constants $a$ and $b$ in the function $f(x)=a x e^{b x}$ such that $f(\\frac{1}{2})=1$ and the function has a local maximum at $x=\\frac{1}{2}$. $a=$ [ANS]\n$b=$ [ANS]", "answer_v2": [ "2*e", "-1*2" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find constants $a$ and $b$ in the function $f(x)=a x e^{b x}$ such that $f(\\frac{1}{4})=1$ and the function has a local maximum at $x=\\frac{1}{4}$. $a=$ [ANS]\n$b=$ [ANS]", "answer_v3": [ "4*e", "-1*4" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0322", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "3", "keywords": [ "concavity", "increasing and decreasing functions", "maxima", "minima" ], "problem_v1": "Find and classify the critical points of $f(x)=8x^{8}(6-x)^7$ as local maxima and minima. Critical points: $x=$ [ANS]\nClassifications: [ANS]\n(Enter your critical points and classifications as comma-separated lists, and enter the types in the same order as your critical points. Note that you must enter something in both blanks for either to be evaluated. For the types, enter min, max, or neither.", "answer_v1": [ "(0, 3.2, 6)", "(min, max, neither)" ], "answer_type_v1": [ "UOL", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "Find and classify the critical points of $f(x)=2x^{5}(9-x)^4$ as local maxima and minima. Critical points: $x=$ [ANS]\nClassifications: [ANS]\n(Enter your critical points and classifications as comma-separated lists, and enter the types in the same order as your critical points. Note that you must enter something in both blanks for either to be evaluated. For the types, enter min, max, or neither.", "answer_v2": [ "(0, 5, 9)", "(neither, max, min)" ], "answer_type_v2": [ "UOL", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "Find and classify the critical points of $f(x)=4x^{4}(6-x)^5$ as local maxima and minima. Critical points: $x=$ [ANS]\nClassifications: [ANS]\n(Enter your critical points and classifications as comma-separated lists, and enter the types in the same order as your critical points. Note that you must enter something in both blanks for either to be evaluated. For the types, enter min, max, or neither.", "answer_v3": [ "(0, 2.66667, 6)", "(min, max, neither)" ], "answer_type_v3": [ "UOL", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0324", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "4", "keywords": [ "Increasing", "Decreasing", "Concavity" ], "problem_v1": "Let $y=-\\frac{1}{x^{2}+7}$. Find the values of $x$ for which $y$ is increasing most rapidly or decreasing most rapidly. $x=$ [ANS] is where $y$ is increasing most rapidly. $x=$ [ANS] is where $y$ is decreasing most rapidly.", "answer_v1": [ "sqrt(7/3)", "-[sqrt(7/3)]" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $y=-\\frac{1}{x^{2}+1}$. Find the values of $x$ for which $y$ is increasing most rapidly or decreasing most rapidly. $x=$ [ANS] is where $y$ is increasing most rapidly. $x=$ [ANS] is where $y$ is decreasing most rapidly.", "answer_v2": [ "sqrt(1/3)", "-[sqrt(1/3)]" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $y=-\\frac{1}{x^{2}+2}$. Find the values of $x$ for which $y$ is increasing most rapidly or decreasing most rapidly. $x=$ [ANS] is where $y$ is increasing most rapidly. $x=$ [ANS] is where $y$ is decreasing most rapidly.", "answer_v3": [ "sqrt(2/3)", "-[sqrt(2/3)]" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0325", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "3", "keywords": [ "Critical point", "Stationary point" ], "problem_v1": "Locate the critical points and identify which critical points are stationary points for the function $f(x)=x^2(x-9)^\\frac{2}{3}$. critical point(s) [ANS]\nstationary point(s) [ANS]\nNotes: Your answers should be a comma separated list of $x$ values or the word \"none\".", "answer_v1": [ "(0, 6.75, 9)", "(0, 6.75)" ], "answer_type_v1": [ "UOL", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "Locate the critical points and identify which critical points are stationary points for the function $f(x)=x^2(x-3)^\\frac{2}{3}$. critical point(s) [ANS]\nstationary point(s) [ANS]\nNotes: Your answers should be a comma separated list of $x$ values or the word \"none\".", "answer_v2": [ "(0, 2.25, 3)", "(0, 2.25)" ], "answer_type_v2": [ "UOL", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "Locate the critical points and identify which critical points are stationary points for the function $f(x)=x^2(x-5)^\\frac{2}{3}$. critical point(s) [ANS]\nstationary point(s) [ANS]\nNotes: Your answers should be a comma separated list of $x$ values or the word \"none\".", "answer_v3": [ "(0, 3.75, 5)", "(0, 3.75)" ], "answer_type_v3": [ "UOL", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0326", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "3", "keywords": [ "calculus", "critical number", "minimum", "maximum" ], "problem_v1": "Consider the function $ f(x)=\\frac{\\ln(x)}{x^{6}}$. $f(x)$ has a critical number $A=$ [ANS]\n$f''(A)=$ [ANS]\nThus we conclude that $f(x)$ has a local [ANS] at $A$ (type in MAX or MIN).", "answer_v1": [ "1.18136041286565", "-1.58158282869436", "MAX" ], "answer_type_v1": [ "NV", "NV", "MCS" ], "options_v1": [ [], [], [ "MAX", "MIN" ] ], "problem_v2": "Consider the function $ f(x)=\\frac{\\ln(x)}{x^{2}}$. $f(x)$ has a critical number $A=$ [ANS]\n$f''(A)=$ [ANS]\nThus we conclude that $f(x)$ has a local [ANS] at $A$ (type in MAX or MIN).", "answer_v2": [ "1.64872127070013", "-0.270670566473225", "MAX" ], "answer_type_v2": [ "NV", "NV", "MCS" ], "options_v2": [ [], [], [ "MAX", "MIN" ] ], "problem_v3": "Consider the function $ f(x)=\\frac{\\ln(x)}{x^{3}}$. $f(x)$ has a critical number $A=$ [ANS]\n$f''(A)=$ [ANS]\nThus we conclude that $f(x)$ has a local [ANS] at $A$ (type in MAX or MIN).", "answer_v3": [ "1.39561242508609", "-0.566626808512686", "MAX" ], "answer_type_v3": [ "NV", "NV", "MCS" ], "options_v3": [ [], [], [ "MAX", "MIN" ] ] }, { "id": "Calculus_-_single_variable_0327", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "3", "keywords": [ "derivatives", "minimum", "maximum" ], "problem_v1": "Find all critical values for the function f(x)=x e^{5x}. List of critical numbers: [ANS]", "answer_v1": [ "-0.2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find all critical values for the function f(x)=x e^{2x}. List of critical numbers: [ANS]", "answer_v2": [ "-0.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find all critical values for the function f(x)=x e^{3x}. List of critical numbers: [ANS]", "answer_v3": [ "-0.333333333333333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0328", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "3", "keywords": [ "derivatives", "shape of graph" ], "problem_v1": "Suppose that f(x)=x^3-8x^2+1.\n(A) Find all critical values of $f$. If there are no critical values, enter-1000. If there are more than one, enter them separated by commas. Critical value(s)=[ANS]\n(B) Use interval notation to indicate where $f(x)$ is decreasing. Decreasing: [ANS]\n(C) Find the $x$-coordinates of all local maxima of $f$. If there are no local maxima, enter-1000. If there are more than one, enter them separated by commas. Local maxima at $x$=[ANS]\n(D) Find the $x$-coordinates of all local minima of $f$. If there are no local minima, enter-1000. If there are more than one, enter them separated by commas. Local minima at $x$=[ANS]", "answer_v1": [ "(0, 5.33333333333333)", "(0,5.33333333333333)", "0", "5.33333333333333" ], "answer_type_v1": [ "UOL", "UOL", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose that f(x)=x^3-2x^2+3.\n(A) Find all critical values of $f$. If there are no critical values, enter-1000. If there are more than one, enter them separated by commas. Critical value(s)=[ANS]\n(B) Use interval notation to indicate where $f(x)$ is decreasing. Decreasing: [ANS]\n(C) Find the $x$-coordinates of all local maxima of $f$. If there are no local maxima, enter-1000. If there are more than one, enter them separated by commas. Local maxima at $x$=[ANS]\n(D) Find the $x$-coordinates of all local minima of $f$. If there are no local minima, enter-1000. If there are more than one, enter them separated by commas. Local minima at $x$=[ANS]", "answer_v2": [ "(0, 1.33333333333333)", "(0,1.33333333333333)", "0", "1.33333333333333" ], "answer_type_v2": [ "UOL", "UOL", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose that f(x)=x^3-4x^2+1.\n(A) Find all critical values of $f$. If there are no critical values, enter-1000. If there are more than one, enter them separated by commas. Critical value(s)=[ANS]\n(B) Use interval notation to indicate where $f(x)$ is decreasing. Decreasing: [ANS]\n(C) Find the $x$-coordinates of all local maxima of $f$. If there are no local maxima, enter-1000. If there are more than one, enter them separated by commas. Local maxima at $x$=[ANS]\n(D) Find the $x$-coordinates of all local minima of $f$. If there are no local minima, enter-1000. If there are more than one, enter them separated by commas. Local minima at $x$=[ANS]", "answer_v3": [ "(0, 2.66666666666667)", "(0,2.66666666666667)", "0", "2.66666666666667" ], "answer_type_v3": [ "UOL", "UOL", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0329", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "4", "keywords": [ "calculus", "first derivative", "critical points", "minimum", "maximum", "average cost", "increasing", "decreasing" ], "problem_v1": "The cost of manufacturing $x$ toasters in one day is given by C(x)=0.09x^2+21x+330, \\qquad 0 < x < 150. (A) Find the average cost function $\\bar{C}(x)$. [ANS]\n(B) List all the critical values of $\\bar{C}(x)$. Note: If there are no critical values, enter 'NONE'. [ANS]\n(C) Use interval notation to indicate where $\\bar{C}(x)$ is increasing. Note: Use 'INF' for $\\infty$, '-INF' for $-\\infty$, and use 'U' for the union symbol. Increasing: [ANS]\n(D) Use interval notation to indicate where $\\bar{C}(x)$ is decreasing. Decreasing: [ANS]\n(E) List the $x$ values of all local maxima of $\\bar{C}(x)$. If there are no local maxima, enter 'NONE'. $x$ values of local maximums=[ANS]\n(F) List the $x$ values of all local minima of $\\bar{C}(x)$. If there are no local minima, enter 'NONE'. $x$ values of local minimums=[ANS]", "answer_v1": [ "0.09 x + 21 + 330/x", "60.5530070819498", "(60.5530070819498,150)", "(0,60.5530070819498)", "NONE", "60.5530070819498" ], "answer_type_v1": [ "EX", "NV", "UOL", "UOL", "OE", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The cost of manufacturing $x$ toasters in one day is given by C(x)=0.05x^2+25x+300, \\qquad 0 < x < 150. (A) Find the average cost function $\\bar{C}(x)$. [ANS]\n(B) List all the critical values of $\\bar{C}(x)$. Note: If there are no critical values, enter 'NONE'. [ANS]\n(C) Use interval notation to indicate where $\\bar{C}(x)$ is increasing. Note: Use 'INF' for $\\infty$, '-INF' for $-\\infty$, and use 'U' for the union symbol. Increasing: [ANS]\n(D) Use interval notation to indicate where $\\bar{C}(x)$ is decreasing. Decreasing: [ANS]\n(E) List the $x$ values of all local maxima of $\\bar{C}(x)$. If there are no local maxima, enter 'NONE'. $x$ values of local maximums=[ANS]\n(F) List the $x$ values of all local minima of $\\bar{C}(x)$. If there are no local minima, enter 'NONE'. $x$ values of local minimums=[ANS]", "answer_v2": [ "0.05 x + 25 + 300/x", "77.4596669241483", "(77.4596669241483,150)", "(0,77.4596669241483)", "NONE", "77.4596669241483" ], "answer_type_v2": [ "EX", "NV", "UOL", "UOL", "OE", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The cost of manufacturing $x$ toasters in one day is given by C(x)=0.06x^2+21x+310, \\qquad 0 < x < 150. (A) Find the average cost function $\\bar{C}(x)$. [ANS]\n(B) List all the critical values of $\\bar{C}(x)$. Note: If there are no critical values, enter 'NONE'. [ANS]\n(C) Use interval notation to indicate where $\\bar{C}(x)$ is increasing. Note: Use 'INF' for $\\infty$, '-INF' for $-\\infty$, and use 'U' for the union symbol. Increasing: [ANS]\n(D) Use interval notation to indicate where $\\bar{C}(x)$ is decreasing. Decreasing: [ANS]\n(E) List the $x$ values of all local maxima of $\\bar{C}(x)$. If there are no local maxima, enter 'NONE'. $x$ values of local maximums=[ANS]\n(F) List the $x$ values of all local minima of $\\bar{C}(x)$. If there are no local minima, enter 'NONE'. $x$ values of local minimums=[ANS]", "answer_v3": [ "0.06 x + 21 + 310/x", "71.8795288428261", "(71.8795288428261,150)", "(0,71.8795288428261)", "NONE", "71.8795288428261" ], "answer_type_v3": [ "EX", "NV", "UOL", "UOL", "OE", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0330", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "4", "keywords": [ "differentiation", "second derivative", "maxima", "minima" ], "problem_v1": "Let f(x)=8x+\\frac{6}{x}. Use either the first derivative test or the second derivative test to find the following:\n(A) The average of the $x$ values of all local maxima of $f$. Note: If there are no local maxima, enter-1000. Average of $x$ values=[ANS]\n(B) The average of the $x$ values of all local minima of $f$. Note: If there are no local minima, enter-1000. Average of $x$ values=[ANS]", "answer_v1": [ "-0.866025403784439", "0.866025403784439" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let f(x)=2x+\\frac{8}{x}. Use either the first derivative test or the second derivative test to find the following:\n(A) The average of the $x$ values of all local maxima of $f$. Note: If there are no local maxima, enter-1000. Average of $x$ values=[ANS]\n(B) The average of the $x$ values of all local minima of $f$. Note: If there are no local minima, enter-1000. Average of $x$ values=[ANS]", "answer_v2": [ "-2", "2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let f(x)=4x+\\frac{6}{x}. Use either the first derivative test or the second derivative test to find the following:\n(A) The average of the $x$ values of all local maxima of $f$. Note: If there are no local maxima, enter-1000. Average of $x$ values=[ANS]\n(B) The average of the $x$ values of all local minima of $f$. Note: If there are no local minima, enter-1000. Average of $x$ values=[ANS]", "answer_v3": [ "-1.22474487139159", "1.22474487139159" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0331", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Increasing/decreasing functions and local extrema", "level": "3", "keywords": [ "derivatives" ], "problem_v1": "The function $f$ has a continuous second derivative, and it satisfies $f(-4)=2$, $f'(-4)=0$ and $f''(-4)=0$. We can conclude that [ANS] A. $f$ has a local minimum at-4 B. $f$ has a local maximum at-4 C. $f$ has neither a local maximum nor a local minimum at-4 D. We cannot determine if A, B, or C hold without more information.", "answer_v1": [ "D" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "The function $f$ has a continuous second derivative, and it satisfies $f(8)=-7$, $f'(8)=0$ and $f''(8)=2$. We can conclude that [ANS] A. $f$ has a local minimum at 8 B. $f$ has a local maximum at 8 C. $f$ has neither a local maximum nor a local minimum at 8 D. We cannot determine if A, B, or C hold without more information.", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "The function $f$ has a continuous second derivative, and it satisfies $f(-6)=-4$, $f'(-6)=0$ and $f''(-6)=1$. We can conclude that [ANS] A. $f$ has neither a local maximum nor a local minimum at-6 B. $f$ has a local minimum at-6 C. $f$ has a local maximum at-6 D. We cannot determine if A, B, or C hold without more information.", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_single_variable_0332", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Concavity and points of inflection", "level": "4", "keywords": [ "calculus", "derivatives", "inflection points", "concavity", "second derivative test" ], "problem_v1": "Let f(x)=x\\!\\left(x-8\\sqrt{x}\\right) Determine the intervals on which the given function is concave up or down and find the point of inflection. (Help with.) The x-coordinate of the point of inflection is [ANS]\nIn the next two questions give the largest interval in the domain of $f$ that fits the description. The interval on the left of the inflection point is [ANS]. On this interval $f$ is [ANS]. The interval on the right of the inflection point is [ANS]. On this interval $f$ is [ANS].", "answer_v1": [ "9*8^2/64", "(0,9)", "Concave Down", "(9,infinity)", "Concave Up" ], "answer_type_v1": [ "NV", "INT", "MCS", "INT", "MCS" ], "options_v1": [ [], [], [ "Concave Up", "Concave Down" ], [], [ "Concave Up", "Concave Down" ] ], "problem_v2": "Let f(x)=x\\!\\left(x-1\\sqrt{x}\\right) Determine the intervals on which the given function is concave up or down and find the point of inflection. (Help with.) The x-coordinate of the point of inflection is [ANS]\nIn the next two questions give the largest interval in the domain of $f$ that fits the description. The interval on the left of the inflection point is [ANS]. On this interval $f$ is [ANS]. The interval on the right of the inflection point is [ANS]. On this interval $f$ is [ANS].", "answer_v2": [ "9*1^2/64", "(0,0.140625)", "Concave Down", "(0.140625,infinity)", "Concave Up" ], "answer_type_v2": [ "NV", "INT", "MCS", "INT", "MCS" ], "options_v2": [ [], [], [ "Concave Up", "Concave Down" ], [], [ "Concave Up", "Concave Down" ] ], "problem_v3": "Let f(x)=x\\!\\left(x-4\\sqrt{x}\\right) Determine the intervals on which the given function is concave up or down and find the point of inflection. (Help with.) The x-coordinate of the point of inflection is [ANS]\nIn the next two questions give the largest interval in the domain of $f$ that fits the description. The interval on the left of the inflection point is [ANS]. On this interval $f$ is [ANS]. The interval on the right of the inflection point is [ANS]. On this interval $f$ is [ANS].", "answer_v3": [ "9*4^2/64", "(0,2.25)", "Concave Down", "(2.25,infinity)", "Concave Up" ], "answer_type_v3": [ "NV", "INT", "MCS", "INT", "MCS" ], "options_v3": [ [], [], [ "Concave Up", "Concave Down" ], [], [ "Concave Up", "Concave Down" ] ] }, { "id": "Calculus_-_single_variable_0333", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Concavity and points of inflection", "level": "4", "keywords": [ "calculus", "derivative", "chain rule" ], "problem_v1": "For what values of $x$ is the graph of $y=x e^{-4x}$ concave down? values=[ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10).).)", "answer_v1": [ "(-infinity,2/4)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "For what values of $x$ is the graph of $y=e^{-x^2}$ concave down? values=[ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10).).)", "answer_v2": [ "(-1/[sqrt(2)],1/[sqrt(2)])" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "For what values of $x$ is the graph of $y=x e^{-2x}$ concave down? values=[ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10).).)", "answer_v3": [ "(-infinity,2/2)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0334", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Concavity and points of inflection", "level": "3", "keywords": [ "concavity", "increasing and decreasing functions", "maxima", "minima" ], "problem_v1": "Find the inflection points of $f(x)=6x^4+58x^3-30x^2+11$. (Give your answers as a comma separated list, e.g., 3,-2.) inflection points=[ANS]", "answer_v1": [ "(-5, 0.166666666666667)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "Find the inflection points of $f(x)=4x^4+23x^3-9x^2+6$. (Give your answers as a comma separated list, e.g., 3,-2.) inflection points=[ANS]", "answer_v2": [ "(-3, 0.125)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "Find the inflection points of $f(x)=3x^4+17x^3-9x^2+9$. (Give your answers as a comma separated list, e.g., 3,-2.) inflection points=[ANS]", "answer_v3": [ "(-3, 0.166666666666667)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0335", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Concavity and points of inflection", "level": "3", "keywords": [ "derivatives", "shape of graph" ], "problem_v1": "Suppose that f(x)=\\frac{6 e^x}{6 e^x+5}.\n(A) Find all critical values of $f$. If there are no critical values, enter None. If there are more than one, enter them separated by commas.\nCritical value(s)=[ANS]\n(B) Use to indicate where $f(x)$ is concave up.\nConcave up: [ANS]\n(C) Use to indicate where $f(x)$ is concave down.\nConcave down: [ANS]\n(D) Find all inflection points of $f$. If there are no inflection points, enter None. If there are more than one, enter them separated by commas. Inflection point(s) at $x$=[ANS]", "answer_v1": [ "None", "(-infinity,-0.182321556793955)", "(-0.182321556793955,infinity)", "-0.182321556793955" ], "answer_type_v1": [ "OE", "INT", "INT", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose that f(x)=\\frac{3 e^x}{3 e^x+7}.\n(A) Find all critical values of $f$. If there are no critical values, enter None. If there are more than one, enter them separated by commas.\nCritical value(s)=[ANS]\n(B) Use to indicate where $f(x)$ is concave up.\nConcave up: [ANS]\n(C) Use to indicate where $f(x)$ is concave down.\nConcave down: [ANS]\n(D) Find all inflection points of $f$. If there are no inflection points, enter None. If there are more than one, enter them separated by commas. Inflection point(s) at $x$=[ANS]", "answer_v2": [ "None", "(-infinity,0.847297860387204)", "(0.847297860387204,infinity)", "0.847297860387204" ], "answer_type_v2": [ "OE", "INT", "INT", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose that f(x)=\\frac{4 e^x}{4 e^x+6}.\n(A) Find all critical values of $f$. If there are no critical values, enter None. If there are more than one, enter them separated by commas.\nCritical value(s)=[ANS]\n(B) Use to indicate where $f(x)$ is concave up.\nConcave up: [ANS]\n(C) Use to indicate where $f(x)$ is concave down.\nConcave down: [ANS]\n(D) Find all inflection points of $f$. If there are no inflection points, enter None. If there are more than one, enter them separated by commas. Inflection point(s) at $x$=[ANS]", "answer_v3": [ "None", "(-infinity,0.405465108108164)", "(0.405465108108164,infinity)", "0.405465108108164" ], "answer_type_v3": [ "OE", "INT", "INT", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0336", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Concavity and points of inflection", "level": "3", "keywords": [ "calculus", "derivative", "inflection point" ], "problem_v1": "Consider the function f(x)=\\frac{x}{7x^2+6}. List the $x$ values of the inflection points of $f$. If there are no inflection points, enter 'NONE'. [ANS]", "answer_v1": [ "(0, -1.60356745147455, 1.60356745147455)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Consider the function f(x)=\\frac{x}{2x^2+8}. List the $x$ values of the inflection points of $f$. If there are no inflection points, enter 'NONE'. [ANS]", "answer_v2": [ "(0, -3.46410161513775, 3.46410161513775)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Consider the function f(x)=\\frac{x}{4x^2+6}. List the $x$ values of the inflection points of $f$. If there are no inflection points, enter 'NONE'. [ANS]", "answer_v3": [ "(0, -2.12132034355964, 2.12132034355964)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0337", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Global extrema", "level": "4", "keywords": [ "derivatives", "critical points", "minimum", "maximum" ], "problem_v1": "Consider the function $f(x)=xe^{-8x}, \\quad 0\\le x\\le 2.$\nThis function has an absolute minimum value equal to: [ANS]\nwhich is attained at $x=$ [ANS]\nand an absolute maximum value equal to: [ANS]\nwhich is attained at $x=$ [ANS]", "answer_v1": [ "0", "0", "(1/8)*e^{-1}", "1/8" ], "answer_type_v1": [ "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the function $f(x)=xe^{-2x}, \\quad 0\\le x\\le 2.$\nThis function has an absolute minimum value equal to: [ANS]\nwhich is attained at $x=$ [ANS]\nand an absolute maximum value equal to: [ANS]\nwhich is attained at $x=$ [ANS]", "answer_v2": [ "0", "0", "(1/2)*e^{-1}", "1/2" ], "answer_type_v2": [ "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the function $f(x)=xe^{-4x}, \\quad 0\\le x\\le 2.$\nThis function has an absolute minimum value equal to: [ANS]\nwhich is attained at $x=$ [ANS]\nand an absolute maximum value equal to: [ANS]\nwhich is attained at $x=$ [ANS]", "answer_v3": [ "0", "0", "(1/4)*e^{-1}", "1/4" ], "answer_type_v3": [ "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0338", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Global extrema", "level": "4", "keywords": [], "problem_v1": "Find the extreme values of the function $f$ on the interval $(0, \\infty)$, and the $x$-value(s) at which they occur. If an extreme value does not exist, enter DNE for both the value and location. f(x)=\\frac{3+8 \\ln x}{x} Absolute minimum value: [ANS], located at $x=$ [ANS]. Absolute maximum value: [ANS], located at $x=$ [ANS].", "answer_v1": [ "DNE", "DNE", "4.28209142815192", "1.86824595743222" ], "answer_type_v1": [ "OE", "OE", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the extreme values of the function $f$ on the interval $(0, \\infty)$, and the $x$-value(s) at which they occur. If an extreme value does not exist, enter DNE for both the value and location. f(x)=\\frac{4+5 \\ln x}{x} Absolute minimum value: [ANS], located at $x=$ [ANS]. Absolute maximum value: [ANS], located at $x=$ [ANS].", "answer_v2": [ "DNE", "DNE", "4.09365376538991", "1.22140275816017" ], "answer_type_v2": [ "OE", "OE", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the extreme values of the function $f$ on the interval $(0, \\infty)$, and the $x$-value(s) at which they occur. If an extreme value does not exist, enter DNE for both the value and location. f(x)=\\frac{4+9 \\ln x}{x} Absolute minimum value: [ANS], located at $x=$ [ANS]. Absolute maximum value: [ANS], located at $x=$ [ANS].", "answer_v3": [ "DNE", "DNE", "5.1637807866369", "1.74290899863346" ], "answer_type_v3": [ "OE", "OE", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0339", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Global extrema", "level": "4", "keywords": [ "calculus", "derivatives", "critical points", "maximum/minimum" ], "problem_v1": "Find the minimum and maximum values of $y=\\sqrt{12} \\theta-\\sqrt{6} \\sec \\theta$ on the interval $[0, \\frac {\\pi}{3}]$ $f_{\\text{min}}=$ [ANS]\n$f_{\\text{max}}=$ [ANS]", "answer_v1": [ "-2.44949", "-0.743403" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the minimum and maximum values of $y=\\sqrt{6} \\theta-\\sqrt{3} \\sec \\theta$ on the interval $[0, \\frac {\\pi}{3}]$ $f_{\\text{min}}=$ [ANS]\n$f_{\\text{max}}=$ [ANS]", "answer_v2": [ "-1.73205", "-0.525665" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the minimum and maximum values of $y=\\sqrt{8} \\theta-\\sqrt{4} \\sec \\theta$ on the interval $[0, \\frac {\\pi}{3}]$ $f_{\\text{min}}=$ [ANS]\n$f_{\\text{max}}=$ [ANS]", "answer_v3": [ "-2", "-0.606986" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0340", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Global extrema", "level": "3", "keywords": [ "calculus", "derivatives", "critical points", "maximum/minimum" ], "problem_v1": "Find the maximum and minimum values of the function $f(x)=x-\\frac{125x}{x+5}$ on the interval [0,22]. The minimum value=[ANS]\nThe maximum value=[ANS]", "answer_v1": [ "-80", "0" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the maximum and minimum values of the function $f(x)=x-\\frac{8x}{x+2}$ on the interval [0,5]. The minimum value=[ANS]\nThe maximum value=[ANS]", "answer_v2": [ "-2", "0" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the maximum and minimum values of the function $f(x)=x-\\frac{27x}{x+3}$ on the interval [0,8]. The minimum value=[ANS]\nThe maximum value=[ANS]", "answer_v3": [ "-12", "0" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0341", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Global extrema", "level": "4", "keywords": [ "derivative' 'extrema' 'optimization", "calculus", "differentiation", "maximum", "minimum", "derivatives" ], "problem_v1": "Find the absolute maximum and absolute minimum values of the function f(x)=\\left(x-3\\right)\\!\\left(x-6\\right)^{3}+8 on each of the indicated intervals.\n(a) Interval=$[1, 4]$.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum &=& [ANS] \\\\ \\hline 2. & Absolute minimum &=& [ANS] \\\\ \\hline \\end{array}$\n(b) Interval=$[1,8]$.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum &=& [ANS] \\\\ \\hline 2. & Absolute minimum &=& [ANS] \\\\ \\hline \\end{array}$\n(c) Interval=$[4,9]$.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum &=& [ANS] \\\\ \\hline 2. & Absolute minimum &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "258", "-0.542969", "258", "-0.542969", "170", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find the absolute maximum and absolute minimum values of the function f(x)=\\left(x-1\\right)\\!\\left(x-7\\right)^{3}+3 on each of the indicated intervals.\n(a) Interval=$[1, 4]$.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum &=& [ANS] \\\\ \\hline 2. & Absolute minimum &=& [ANS] \\\\ \\hline \\end{array}$\n(b) Interval=$[1,8]$.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum &=& [ANS] \\\\ \\hline 2. & Absolute minimum &=& [ANS] \\\\ \\hline \\end{array}$\n(c) Interval=$[4,9]$.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum &=& [ANS] \\\\ \\hline 2. & Absolute minimum &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "3", "-133.688", "10", "-133.688", "67", "-78" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find the absolute maximum and absolute minimum values of the function f(x)=\\left(x-1\\right)\\!\\left(x-6\\right)^{3}+5 on each of the indicated intervals.\n(a) Interval=$[1, 4]$.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum &=& [ANS] \\\\ \\hline 2. & Absolute minimum &=& [ANS] \\\\ \\hline \\end{array}$\n(b) Interval=$[1,8]$.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum &=& [ANS] \\\\ \\hline 2. & Absolute minimum &=& [ANS] \\\\ \\hline \\end{array}$\n(c) Interval=$[4,9]$.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum &=& [ANS] \\\\ \\hline 2. & Absolute minimum &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "5", "-60.918", "61", "-60.918", "221", "-19" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0342", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Global extrema", "level": "3", "keywords": [ "Optimization", "Absolute", "Extrema" ], "problem_v1": "Let $g(s)=\\frac{1}{s-2}$ on the interval $\\left[0.5,1.3\\right]$. Find the absolute maximum and absolute minimum of $g(s)$ on this interval. Enter DNE if the absolute maximum or minimum does not exist. The absolute max occurs at $s=$ [ANS]. The absolute min occurs at $s=$ [ANS].", "answer_v1": [ "0.5", "1.3" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $g(s)=\\frac{1}{s-2}$ on the interval $\\left[-0.9,1.7\\right)$. Find the absolute maximum and absolute minimum of $g(s)$ on this interval. Enter DNE if the absolute maximum or minimum does not exist. The absolute max occurs at $s=$ [ANS]. The absolute min occurs at $s=$ [ANS].", "answer_v2": [ "-0.9", "DNE" ], "answer_type_v2": [ "NV", "OE" ], "options_v2": [ [], [] ], "problem_v3": "Let $g(s)=\\frac{1}{s-2}$ on the interval $\\left[-0.4,1\\right)$. Find the absolute maximum and absolute minimum of $g(s)$ on this interval. Enter DNE if the absolute maximum or minimum does not exist. The absolute max occurs at $s=$ [ANS]. The absolute min occurs at $s=$ [ANS].", "answer_v3": [ "-0.4", "DNE" ], "answer_type_v3": [ "NV", "OE" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0343", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Global extrema", "level": "2", "keywords": [ "calculus", "derivative", "local maxima and minima", "maxima", "minima" ], "problem_v1": "Find the exact global maximum and minimum values of the function $g(x)=8x-x^2-9$ if its domain is all real numbers. global maximum at $x=$ [ANS]\nglobal minimum at $x=$ [ANS]\n(Enter (Enter none if there is no global maximum or global minimum for this function.) if there is no global maximum or global minimum for this function.)", "answer_v1": [ "8/2", "none" ], "answer_type_v1": [ "NV", "OE" ], "options_v1": [ [], [] ], "problem_v2": "Find the exact global maximum and minimum values of the function $g(x)=2x-x^2-14$ if its domain is all real numbers. global maximum at $x=$ [ANS]\nglobal minimum at $x=$ [ANS]\n(Enter (Enter none if there is no global maximum or global minimum for this function.) if there is no global maximum or global minimum for this function.)", "answer_v2": [ "2/2", "none" ], "answer_type_v2": [ "NV", "OE" ], "options_v2": [ [], [] ], "problem_v3": "Find the exact global maximum and minimum values of the function $g(x)=4x-x^2-10$ if its domain is all real numbers. global maximum at $x=$ [ANS]\nglobal minimum at $x=$ [ANS]\n(Enter (Enter none if there is no global maximum or global minimum for this function.) if there is no global maximum or global minimum for this function.)", "answer_v3": [ "4/2", "none" ], "answer_type_v3": [ "NV", "OE" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0344", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Global extrema", "level": "4", "keywords": [ "Optimization' 'Maximum' 'Minimum" ], "problem_v1": "Find the $x$-coordinate of the absolute maximum and absolute minimum for the function f(x)=10-{8} x-\\frac{16}{x^2}, \\qquad x > 0. Enter None for any absolute extrema that does not exist.\n$x$-coordinate of absolute maximum=[ANS]\n$x$-coordinate of absolute minimum=[ANS]", "answer_v1": [ "1.5874010519682", "None" ], "answer_type_v1": [ "NV", "OE" ], "options_v1": [ [], [] ], "problem_v2": "Find the $x$-coordinate of the absolute maximum and absolute minimum for the function f(x)=2-{12} x-\\frac{5}{x^2}, \\qquad x > 0. Enter None for any absolute extrema that does not exist.\n$x$-coordinate of absolute maximum=[ANS]\n$x$-coordinate of absolute minimum=[ANS]", "answer_v2": [ "0.941036028881029", "None" ], "answer_type_v2": [ "NV", "OE" ], "options_v2": [ [], [] ], "problem_v3": "Find the $x$-coordinate of the absolute maximum and absolute minimum for the function f(x)=5-{8} x-\\frac{8}{x^2}, \\qquad x > 0. Enter None for any absolute extrema that does not exist.\n$x$-coordinate of absolute maximum=[ANS]\n$x$-coordinate of absolute minimum=[ANS]", "answer_v3": [ "1.25992104989487", "None" ], "answer_type_v3": [ "NV", "OE" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0345", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Global extrema", "level": "4", "keywords": [ "Optimization' 'Maximum' 'Minimum" ], "problem_v1": "Find the $x$-coordinate of the absolute maximum for the function f(x)=11x-4x \\ln(x), \\qquad x > 0.\n$x$-coordinate of absolute maximum=[ANS]", "answer_v1": [ "5.75460267600573" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the $x$-coordinate of the absolute maximum for the function f(x)=6x-5x \\ln(x), \\qquad x > 0.\n$x$-coordinate of absolute maximum=[ANS]", "answer_v2": [ "1.22140275816017" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the $x$-coordinate of the absolute maximum for the function f(x)=8x-4x \\ln(x), \\qquad x > 0.\n$x$-coordinate of absolute maximum=[ANS]", "answer_v3": [ "2.71828182845905" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0346", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Summary of curve sketching", "level": "3", "keywords": [ "maximum", "minimum", "increase", "decrease", "convex", "trigonometry", "Graphing", "Increasing", "Decreasing", "Concavity", "Calculus", "Derivatives" ], "problem_v1": "Answer the following questions for the function f(x)=\\sin^2 \\left(\\frac{x}{5} \\right) defined on the interval $[-15.11, 3.33]$.\nRemember that you can enter pi for $\\pi$ as part of your answer.\na.) $f(x)$ is concave down on the region(s) [ANS]\nb.) A global minimum for this function occurs at [ANS]\nc.) A local maximum for this function which is not a global maximum occurs at [ANS]\nd.) The function is increasing on the region(s) [ANS]\nNote: In some cases, you may need to give a comma-separated list of intervals, and intervals should be given in.", "answer_v1": [ "[-5*pi*3/4,-5*pi/4]", "0", "3.33", "[-15.11,-5*pi/2] \\cup [0,3.33]" ], "answer_type_v1": [ "INT", "NV", "NV", "INT" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Answer the following questions for the function f(x)=\\sin^2 \\left(\\frac{x}{2} \\right) defined on the interval $[-5.28, 1.47]$.\nRemember that you can enter pi for $\\pi$ as part of your answer.\na.) $f(x)$ is concave down on the region(s) [ANS]\nb.) A global minimum for this function occurs at [ANS]\nc.) A local maximum for this function which is not a global maximum occurs at [ANS]\nd.) The function is increasing on the region(s) [ANS]\nNote: In some cases, you may need to give a comma-separated list of intervals, and intervals should be given in.", "answer_v2": [ "[-2*pi*3/4,-2*pi/4]", "0", "1.47", "[-5.28,-2*pi/2] \\cup [0,1.47]" ], "answer_type_v2": [ "INT", "NV", "NV", "INT" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Answer the following questions for the function f(x)=\\sin^2 \\left(\\frac{x}{3} \\right) defined on the interval $[-8.82, 2.06]$.\nRemember that you can enter pi for $\\pi$ as part of your answer.\na.) $f(x)$ is concave down on the region(s) [ANS]\nb.) A global minimum for this function occurs at [ANS]\nc.) A local maximum for this function which is not a global maximum occurs at [ANS]\nd.) The function is increasing on the region(s) [ANS]\nNote: In some cases, you may need to give a comma-separated list of intervals, and intervals should be given in.", "answer_v3": [ "[-3*pi*3/4,-3*pi/4]", "0", "2.06", "[-8.82,-3*pi/2] \\cup [0,2.06]" ], "answer_type_v3": [ "INT", "NV", "NV", "INT" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0347", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Summary of curve sketching", "level": "3", "keywords": [ "exponentials", "increase", "decrease", "concavity" ], "problem_v1": "Consider the function $f(x)= \\frac{e^x}{7+e^x}$. Compute the derivative of this function.\n$f'(x)$=[ANS]\nFor the following questions, write inf for $\\infty$,-inf-inf for $-\\infty$, and NA (ie. not applicable) if no such answer exists.\na.) $f(x$ is increasing on the region [ANS].\nb.) $f(x)$ is decreasing on the region [ANS].\nc.) $f(x)$ has a local minimum at [ANS].\nd.) $f(x)$ has a local maximum at [ANS].\ne.) $f''(x)$=[ANS]\nf.) $f(x)$ is concave up on the region [ANS]\ng.) $f(x)$ concave down on the region [ANS].\nh.) $f(x)$ has a point of inflection at [ANS].\nNote: Many of your answers must be given in.", "answer_v1": [ "7*e^x/(7+e^x)^2", "(-infinity,infinity)", "NA", "NA", "NA", "7*e^x*(7- e^x)/(7+e^x)^3", "(-infinity,1.94591014905531)", "(1.94591014905531,infinity)", "1.94591014905531" ], "answer_type_v1": [ "EX", "INT", "EX", "EX", "EX", "EX", "INT", "INT", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the function $f(x)= \\frac{e^x}{3+e^x}$. Compute the derivative of this function.\n$f'(x)$=[ANS]\nFor the following questions, write inf for $\\infty$,-inf-inf for $-\\infty$, and NA (ie. not applicable) if no such answer exists.\na.) $f(x$ is increasing on the region [ANS].\nb.) $f(x)$ is decreasing on the region [ANS].\nc.) $f(x)$ has a local minimum at [ANS].\nd.) $f(x)$ has a local maximum at [ANS].\ne.) $f''(x)$=[ANS]\nf.) $f(x)$ is concave up on the region [ANS]\ng.) $f(x)$ concave down on the region [ANS].\nh.) $f(x)$ has a point of inflection at [ANS].\nNote: Many of your answers must be given in.", "answer_v2": [ "3*e^x/(3+e^x)^2", "(-infinity,infinity)", "NA", "NA", "NA", "3*e^x*(3- e^x)/(3+e^x)^3", "(-infinity,1.09861228866811)", "(1.09861228866811,infinity)", "1.09861228866811" ], "answer_type_v2": [ "EX", "INT", "EX", "EX", "EX", "EX", "INT", "INT", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the function $f(x)= \\frac{e^x}{4+e^x}$. Compute the derivative of this function.\n$f'(x)$=[ANS]\nFor the following questions, write inf for $\\infty$,-inf-inf for $-\\infty$, and NA (ie. not applicable) if no such answer exists.\na.) $f(x$ is increasing on the region [ANS].\nb.) $f(x)$ is decreasing on the region [ANS].\nc.) $f(x)$ has a local minimum at [ANS].\nd.) $f(x)$ has a local maximum at [ANS].\ne.) $f''(x)$=[ANS]\nf.) $f(x)$ is concave up on the region [ANS]\ng.) $f(x)$ concave down on the region [ANS].\nh.) $f(x)$ has a point of inflection at [ANS].\nNote: Many of your answers must be given in.", "answer_v3": [ "4*e^x/(4+e^x)^2", "(-infinity,infinity)", "NA", "NA", "NA", "4*e^x*(4- e^x)/(4+e^x)^3", "(-infinity,1.38629436111989)", "(1.38629436111989,infinity)", "1.38629436111989" ], "answer_type_v3": [ "EX", "INT", "EX", "EX", "EX", "EX", "INT", "INT", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0348", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Summary of curve sketching", "level": "4", "keywords": [ "concavity", "increasing and decreasing functions", "maxima", "minima" ], "problem_v1": "Find a formula for the fourth degree polynomial $p(x)$ whose graph is symmetric about the $y$-axis, and which has a $y$-intercept of 8, and global maxima at $(3,170)$ and $(-3, 170)$. $p(x)=$ [ANS]", "answer_v1": [ "36*x^2-2*x^4+8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a formula for the fourth degree polynomial $p(x)$ whose graph is symmetric about the $y$-axis, and which has a $y$-intercept of 0, and global maxima at $(5,625)$ and $(-5, 625)$. $p(x)=$ [ANS]", "answer_v2": [ "50*x^2-x^4" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a formula for the fourth degree polynomial $p(x)$ whose graph is symmetric about the $y$-axis, and which has a $y$-intercept of 3, and global maxima at $(4,259)$ and $(-4, 259)$. $p(x)=$ [ANS]", "answer_v3": [ "32*x^2-x^4+3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0349", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Summary of curve sketching", "level": "3", "keywords": [ "concavity", "increasing and decreasing functions", "maxima", "minima" ], "problem_v1": "Find a formula for a curve of the form $y=e^{-(x-a)^2/b}$ for $b>0$ with a local maximum at $x=1$ and points of inflection at $x=-3$ and $x=5$. $y=$ [ANS]", "answer_v1": [ "e^{-1*(x-1)^2/32}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a formula for a curve of the form $y=e^{-(x-a)^2/b}$ for $b>0$ with a local maximum at $x=7$ and points of inflection at $x=6$ and $x=8$. $y=$ [ANS]", "answer_v2": [ "e^{-1*(x-7)^2/2}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a formula for a curve of the form $y=e^{-(x-a)^2/b}$ for $b>0$ with a local maximum at $x=3$ and points of inflection at $x=1$ and $x=5$. $y=$ [ANS]", "answer_v3": [ "e^{-1*(x-3)^2/8}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0350", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Summary of curve sketching", "level": "4", "keywords": [ "concavity", "increasing and decreasing functions", "maxima", "minima" ], "problem_v1": "Find the formula for a function of the form $y=A \\sin (Bx)+C$ with a maximum at $(2.5, 8)$, a minimum at $(7.5,-6)$, and no critical points between these two points. $y=$ [ANS]", "answer_v1": [ "7*sin(pi*x/5)+1" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the formula for a function of the form $y=A \\sin (Bx)+C$ with a maximum at $(4,-3)$, a minimum at $(12,-5)$, and no critical points between these two points. $y=$ [ANS]", "answer_v2": [ "1*sin(pi*x/8)+-4" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the formula for a function of the form $y=A \\sin (Bx)+C$ with a maximum at $(2.5, 1)$, a minimum at $(7.5,-5)$, and no critical points between these two points. $y=$ [ANS]", "answer_v3": [ "3*sin(pi*x/5)+-2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0352", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Summary of curve sketching", "level": "4", "keywords": [ "calculus", "relative extrema", "minimum", "maximum", "inflection point" ], "problem_v1": "Suppose that it is given to you that f'(x)=(x+6)(10-x)(x-15) Then the first relative extremum (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe function $f(x)$ has a relative [ANS] at this point.\nThe second relative extremum (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe function $f(x)$ has a relative [ANS] at this point.\nThe third relative extremum (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe function $f(x)$ has a relative [ANS] at this point.\nThe first inflection point (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe second inflection point (from the left) for $f(x)$ occurs at $x=$ [ANS]", "answer_v1": [ "-6", "MAX", "10", "min", "15", "max", "0", "12.6666666666667" ], "answer_type_v1": [ "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "NV" ], "options_v1": [ [], [ "max", "min" ], [], [ "max", "min" ], [], [ "max", "min" ], [], [] ], "problem_v2": "Suppose that it is given to you that f'(x)=(x+3)(12-x)(13-x) Then the first relative extremum (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe function $f(x)$ has a relative [ANS] at this point.\nThe second relative extremum (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe function $f(x)$ has a relative [ANS] at this point.\nThe third relative extremum (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe function $f(x)$ has a relative [ANS] at this point.\nThe first inflection point (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe second inflection point (from the left) for $f(x)$ occurs at $x=$ [ANS]", "answer_v2": [ "-3", "MIN", "12", "max", "13", "min", "2.15860843457999", "12.5080582320867" ], "answer_type_v2": [ "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "NV" ], "options_v2": [ [], [ "max", "min" ], [], [ "max", "min" ], [], [ "max", "min" ], [], [] ], "problem_v3": "Suppose that it is given to you that f'(x)=(x+4)(9-x)(x-12) Then the first relative extremum (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe function $f(x)$ has a relative [ANS] at this point.\nThe second relative extremum (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe function $f(x)$ has a relative [ANS] at this point.\nThe third relative extremum (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe function $f(x)$ has a relative [ANS] at this point.\nThe first inflection point (from the left) for $f(x)$ occurs at $x=$ [ANS]\nThe second inflection point (from the left) for $f(x)$ occurs at $x=$ [ANS]", "answer_v3": [ "-4", "MAX", "9", "min", "12", "max", "0.756360045781255", "10.5769732875521" ], "answer_type_v3": [ "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "NV" ], "options_v3": [ [], [ "max", "min" ], [], [ "max", "min" ], [], [ "max", "min" ], [], [] ] }, { "id": "Calculus_-_single_variable_0353", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Summary of curve sketching", "level": "5", "keywords": [ "differentiation", "second derivative", "maxima", "minima" ], "problem_v1": "The marketing research department for a computer company used a large city to test market their new product. They found that the relationship between the price $p$ and the demand $x$ was given approximately by p=1326-0.15x^2. Find $R(x)$, the revenue function. $R(x)$=[ANS]\nNow use the revenue function to do the following:\n(A) Find the average of the $x$ values of all local maxima of $R(x)$. Note: If there are no local maxima, enter-1000. Average of $x$ values=[ANS]\n(B) Find the average of the $x$ values of all local minima of $R(x)$. Note: If there are no local minima, enter-1000. Average of $x$ values=[ANS]\n(C) Use interval notation to indicate where $R(x)$ is concave up.\nNote: Enter 'I' for $\\infty$, '-I' for $-\\infty$, and 'U' for the union symbol. If you have extra boxes, fill each in with an 'x'. Concave up: [ANS]\n(D) Use interval notation to indicate where $R(x)$ is concave down. Concave down: [ANS]", "answer_v1": [ "1326*x - 0.15*x^3", "54.2832079621927", "-1000", "x", "(0,infinity)" ], "answer_type_v1": [ "EX", "NV", "NV", "EX", "INT" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "The marketing research department for a computer company used a large city to test market their new product. They found that the relationship between the price $p$ and the demand $x$ was given approximately by p=1124-0.2x^2. Find $R(x)$, the revenue function. $R(x)$=[ANS]\nNow use the revenue function to do the following:\n(A) Find the average of the $x$ values of all local maxima of $R(x)$. Note: If there are no local maxima, enter-1000. Average of $x$ values=[ANS]\n(B) Find the average of the $x$ values of all local minima of $R(x)$. Note: If there are no local minima, enter-1000. Average of $x$ values=[ANS]\n(C) Use interval notation to indicate where $R(x)$ is concave up.\nNote: Enter 'I' for $\\infty$, '-I' for $-\\infty$, and 'U' for the union symbol. If you have extra boxes, fill each in with an 'x'. Concave up: [ANS]\n(D) Use interval notation to indicate where $R(x)$ is concave down. Concave down: [ANS]", "answer_v2": [ "1124*x - 0.2*x^3", "43.2820209016785", "-1000", "x", "(0,infinity)" ], "answer_type_v2": [ "EX", "NV", "NV", "EX", "INT" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "The marketing research department for a computer company used a large city to test market their new product. They found that the relationship between the price $p$ and the demand $x$ was given approximately by p=1194-0.15x^2. Find $R(x)$, the revenue function. $R(x)$=[ANS]\nNow use the revenue function to do the following:\n(A) Find the average of the $x$ values of all local maxima of $R(x)$. Note: If there are no local maxima, enter-1000. Average of $x$ values=[ANS]\n(B) Find the average of the $x$ values of all local minima of $R(x)$. Note: If there are no local minima, enter-1000. Average of $x$ values=[ANS]\n(C) Use interval notation to indicate where $R(x)$ is concave up.\nNote: Enter 'I' for $\\infty$, '-I' for $-\\infty$, and 'U' for the union symbol. If you have extra boxes, fill each in with an 'x'. Concave up: [ANS]\n(D) Use interval notation to indicate where $R(x)$ is concave down. Concave down: [ANS]", "answer_v3": [ "1194*x - 0.15*x^3", "51.5105167255516", "-1000", "x", "(0,infinity)" ], "answer_type_v3": [ "EX", "NV", "NV", "EX", "INT" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0354", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Summary of curve sketching", "level": "3", "keywords": [ "calculus", "derivatives", "curve sketching", "function analysis" ], "problem_v1": "Fill in the function analysis table.\n$\\begin{array}{ccccccccc}\\hline x & & x< &-4 & $)$ and its distance from the origin is [ANS]", "answer_v1": [ "-(2*18*x+21*y)/(21*x+2*y*18)", "sqrt(x^2 + y^2)", "-0.561951486949016", "-1.09544511501033", "0.561951486949016", "1.09544511501033", "2", "-1.09544511501033", "-1.09544511501033", "0.794719414239026" ], "answer_type_v1": [ "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Find the distance from the origin to the closest point(s) on the graph of 10x^2+18xy+10 y^2=8 We begin by computing $\\frac{dy}{dx}$ implicitly for points $(x,y)$ on the graph, obtaining: $\\frac{dy}{dx}=$ [ANS]\nWe need to minimize the following function of $x$ and $y$: $f(x,y)=$ [ANS]\nThere are four critical points, in increasing order: $x_1=$ [ANS]\n$x_2=$ [ANS]\n$x_3=$ [ANS]\n$x_4=$ [ANS]\nThus there is/are [ANS] point(s) closest to the origin, one of which is $(x,y)=($ [ANS] $,$ $)$ and its distance from the origin is [ANS]", "answer_v2": [ "-(2*10*x+18*y)/(18*x+2*y*10)", "sqrt(x^2 + y^2)", "-0.458831467741124", "-2", "0.458831467741124", "2", "2", "-2", "-2", "0.64888568452305" ], "answer_type_v2": [ "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Find the distance from the origin to the closest point(s) on the graph of 13x^2+16xy+13 y^2=10 We begin by computing $\\frac{dy}{dx}$ implicitly for points $(x,y)$ on the graph, obtaining: $\\frac{dy}{dx}=$ [ANS]\nWe need to minimize the following function of $x$ and $y$: $f(x,y)=$ [ANS]\nThere are four critical points, in increasing order: $x_1=$ [ANS]\n$x_2=$ [ANS]\n$x_3=$ [ANS]\n$x_4=$ [ANS]\nThus there is/are [ANS] point(s) closest to the origin, one of which is $(x,y)=($ [ANS] $,$ $)$ and its distance from the origin is [ANS]", "answer_v3": [ "-(2*13*x+16*y)/(16*x+2*y*13)", "sqrt(x^2 + y^2)", "-0.487950036474267", "-1", "0.487950036474267", "1", "2", "-1", "-1", "0.690065559342354" ], "answer_type_v3": [ "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0395", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "5", "keywords": [ "calculus", "optimization" ], "problem_v1": "The diagonals of a convex quadrilateral are mutually perpendicular. The sum of the lengths of the diagonals is 20. We want to find the maximum possible area of such a quadrilateral. Let us denote by $x$ and $y$ the lengths of the two diagonals. Then the area of the quadrilateral is the following function of $x$ and $y$: [ANS]\nIf we solve for $y$ in terms of $x$, we can reexpress this area as the following function of $x$ alone: [ANS]\nThus we find that the maximum area is [ANS]", "answer_v1": [ "x*y/2", "x*(20 - x)/2", "50" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The diagonals of a convex quadrilateral are mutually perpendicular. The sum of the lengths of the diagonals is 6. We want to find the maximum possible area of such a quadrilateral. Let us denote by $x$ and $y$ the lengths of the two diagonals. Then the area of the quadrilateral is the following function of $x$ and $y$: [ANS]\nIf we solve for $y$ in terms of $x$, we can reexpress this area as the following function of $x$ alone: [ANS]\nThus we find that the maximum area is [ANS]", "answer_v2": [ "x*y/2", "x*(6 - x)/2", "4.5" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The diagonals of a convex quadrilateral are mutually perpendicular. The sum of the lengths of the diagonals is 12. We want to find the maximum possible area of such a quadrilateral. Let us denote by $x$ and $y$ the lengths of the two diagonals. Then the area of the quadrilateral is the following function of $x$ and $y$: [ANS]\nIf we solve for $y$ in terms of $x$, we can reexpress this area as the following function of $x$ alone: [ANS]\nThus we find that the maximum area is [ANS]", "answer_v3": [ "x*y/2", "x*(12 - x)/2", "18" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0396", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "5", "keywords": [ "calculus", "optimization" ], "problem_v1": "The gasoline consumption in gallons per hour of a certain vehicle is known to be the following function of velocity: f(v)=\\frac{v^3-86 v^2+5100 v}{150000} What is the optimal velocity which minimizes the fuel consumption of the vehicle in gallons PER MILE? To solve this problem, we need to minimize the following function of $v$: $g(v)=$ [ANS]\nHint for the above: Assume the vehicle is moving at constant velocity $v$. How long will it take to travel 1 mile? How much gas will it use during that time? We find that this function has one critical number at $v=$ [ANS]. To verify that $g(v)$ has a minimum at this critical number we compute the second derivative $g''(x)$ and find that its value at the critical number is [ANS], a positive number.", "answer_v1": [ "(v^2 - 86*v + 5100)/150000", "43", "1.33333333333333E-05" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The gasoline consumption in gallons per hour of a certain vehicle is known to be the following function of velocity: f(v)=\\frac{v^3-70 v^2+5300 v}{150000} What is the optimal velocity which minimizes the fuel consumption of the vehicle in gallons PER MILE? To solve this problem, we need to minimize the following function of $v$: $g(v)=$ [ANS]\nHint for the above: Assume the vehicle is moving at constant velocity $v$. How long will it take to travel 1 mile? How much gas will it use during that time? We find that this function has one critical number at $v=$ [ANS]. To verify that $g(v)$ has a minimum at this critical number we compute the second derivative $g''(x)$ and find that its value at the critical number is [ANS], a positive number.", "answer_v2": [ "(v^2 - 70*v + 5300)/150000", "35", "1.33333333333333E-05" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The gasoline consumption in gallons per hour of a certain vehicle is known to be the following function of velocity: f(v)=\\frac{v^3-76 v^2+5100 v}{150000} What is the optimal velocity which minimizes the fuel consumption of the vehicle in gallons PER MILE? To solve this problem, we need to minimize the following function of $v$: $g(v)=$ [ANS]\nHint for the above: Assume the vehicle is moving at constant velocity $v$. How long will it take to travel 1 mile? How much gas will it use during that time? We find that this function has one critical number at $v=$ [ANS]. To verify that $g(v)$ has a minimum at this critical number we compute the second derivative $g''(x)$ and find that its value at the critical number is [ANS], a positive number.", "answer_v3": [ "(v^2 - 76*v + 5100)/150000", "38", "1.33333333333333E-05" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0397", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "5", "keywords": [ "calculus", "optimization" ], "problem_v1": "A wire of length 15 is cut into two pieces which are then bent into the shape of a circle of radius $r$ and a square of side $s$. Then the total area enclosed by the circle and square is the following function of $s$ and $r$ [ANS]\nIf we solve for $s$ in terms of $r$, we can reexpress this area as the following function of $r$ alone: [ANS]\nThus we find that to obtain maximal area we should let $r=$ [ANS]\nTo obtain minimal area we should let $r=$ [ANS]", "answer_v1": [ "s^2 + 3.14159265358979*r^2", "(15-2*3.14159265358979*r)^2/16+3.14159265358979*r^2", "2.38732414637843", "1.05018591283417" ], "answer_type_v1": [ "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A wire of length 4 is cut into two pieces which are then bent into the shape of a circle of radius $r$ and a square of side $s$. Then the total area enclosed by the circle and square is the following function of $s$ and $r$ [ANS]\nIf we solve for $s$ in terms of $r$, we can reexpress this area as the following function of $r$ alone: [ANS]\nThus we find that to obtain maximal area we should let $r=$ [ANS]\nTo obtain minimal area we should let $r=$ [ANS]", "answer_v2": [ "s^2 + 3.14159265358979*r^2", "(4-2*3.14159265358979*r)^2/16+3.14159265358979*r^2", "0.636619772367581", "0.280049576755779" ], "answer_type_v2": [ "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A wire of length 8 is cut into two pieces which are then bent into the shape of a circle of radius $r$ and a square of side $s$. Then the total area enclosed by the circle and square is the following function of $s$ and $r$ [ANS]\nIf we solve for $s$ in terms of $r$, we can reexpress this area as the following function of $r$ alone: [ANS]\nThus we find that to obtain maximal area we should let $r=$ [ANS]\nTo obtain minimal area we should let $r=$ [ANS]", "answer_v3": [ "s^2 + 3.14159265358979*r^2", "(8-2*3.14159265358979*r)^2/16+3.14159265358979*r^2", "1.27323954473516", "0.560099153511557" ], "answer_type_v3": [ "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0398", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "5", "keywords": [ "calculus", "optimization" ], "problem_v1": "A printed poster is to have a total area of 733 square inches with top and bottom margins of 5 inches and side margins of 4 inches. What should be the dimensions of the poster so that the printed area be as large as possible? To solve this problem let $x$ denote the width of the poster in inches and let $y$ denote the length in inches. We need to maximize the following function of $x$ and $y$: [ANS]\nWe can reexpress this as the following function of $x$ alone: $f(x)=$ [ANS]\nWe find that $f(x)$ has a critical number at $x=$ [ANS]\nTo verify that $f(x)$ has a maximum at this critical number we compute the second derivative $f''(x)$ and find that its value at the critical number is [ANS], a negative number. Thus the optimal dimensions of the poster are [ANS] inches in width and [ANS] inches in height giving us a maximumal printed area of [ANS] square inches.", "answer_v1": [ "(x - 2*4)*(y - 2*5)", "(x - 2*4)*(733/x - 2*5)", "24.2156973882645", "-0.825910552123619", "24.2156973882645", "30.2696217353306", "328.68605223471" ], "answer_type_v1": [ "EX", "EX", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "A printed poster is to have a total area of 602 square inches with top and bottom margins of 2 inches and side margins of 6 inches. What should be the dimensions of the poster so that the printed area be as large as possible? To solve this problem let $x$ denote the width of the poster in inches and let $y$ denote the length in inches. We need to maximize the following function of $x$ and $y$: [ANS]\nWe can reexpress this as the following function of $x$ alone: $f(x)=$ [ANS]\nWe find that $f(x)$ has a critical number at $x=$ [ANS]\nTo verify that $f(x)$ has a maximum at this critical number we compute the second derivative $f''(x)$ and find that its value at the critical number is [ANS], a negative number. Thus the optimal dimensions of the poster are [ANS] inches in width and [ANS] inches in height giving us a maximumal printed area of [ANS] square inches.", "answer_v2": [ "(x - 2*6)*(y - 2*2)", "(x - 2*6)*(602/x - 2*2)", "42.4970587217516", "-0.188248322134005", "42.4970587217516", "14.1656862405839", "310.023530225988" ], "answer_type_v2": [ "EX", "EX", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "A printed poster is to have a total area of 638 square inches with top and bottom margins of 3 inches and side margins of 5 inches. What should be the dimensions of the poster so that the printed area be as large as possible? To solve this problem let $x$ denote the width of the poster in inches and let $y$ denote the length in inches. We need to maximize the following function of $x$ and $y$: [ANS]\nWe can reexpress this as the following function of $x$ alone: $f(x)=$ [ANS]\nWe find that $f(x)$ has a critical number at $x=$ [ANS]\nTo verify that $f(x)$ has a maximum at this critical number we compute the second derivative $f''(x)$ and find that its value at the critical number is [ANS], a negative number. Thus the optimal dimensions of the poster are [ANS] inches in width and [ANS] inches in height giving us a maximumal printed area of [ANS] square inches.", "answer_v3": [ "(x - 2*5)*(y - 2*3)", "(x - 2*5)*(638/x - 2*3)", "32.6087922703883", "-0.367998909634476", "32.6087922703883", "19.565275362233", "306.694492755341" ], "answer_type_v3": [ "EX", "EX", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0399", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "5", "keywords": [ "Derivative", "Optimization" ], "problem_v1": "A truck has a minimum speed of 13 mph in high gear. When traveling $x$ mph, the truck burns diesel fuel at the rate of 0.0091145\\left(\\frac{961}{x}+x\\right)\\frac{\\mbox{gal}}{\\mbox{mile}} Assuming that the truck can not be driven over 54 mph and that diesel fuel costs \\$1.44 a gallon, find the following.\n(a) The steady speed that will minimize the cost of the fuel for a 620 mile trip. $x=$ [ANS]\n(b) The steady speed that will minimize the total cost of a 620 mile trip if the driver is paid \\$17 an hour. $x=$ [ANS]\n(c) The steady speed that will minimize the total cost of a 550 mile trip if the driver is paid \\$31 an hour. $x=$ [ANS]", "answer_v1": [ "31", "47.5", "54" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A truck has a minimum speed of 10 mph in high gear. When traveling $x$ mph, the truck burns diesel fuel at the rate of 0.0030173\\left(\\frac{729}{x}+x\\right)\\frac{\\mbox{gal}}{\\mbox{mile}} Assuming that the truck can not be driven over 65 mph and that diesel fuel costs \\$1.44 a gallon, find the following.\n(a) The steady speed that will minimize the cost of the fuel for a 370 mile trip. $x=$ [ANS]\n(b) The steady speed that will minimize the total cost of a 370 mile trip if the driver is paid \\$13 an hour. $x=$ [ANS]\n(c) The steady speed that will minimize the total cost of a 680 mile trip if the driver is paid \\$17 an hour. $x=$ [ANS]", "answer_v2": [ "27", "61", "65" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A truck has a minimum speed of 12 mph in high gear. When traveling $x$ mph, the truck burns diesel fuel at the rate of 0.0121036\\left(\\frac{1089}{x}+x\\right)\\frac{\\mbox{gal}}{\\mbox{mile}} Assuming that the truck can not be driven over 53 mph and that diesel fuel costs \\$1.44 a gallon, find the following.\n(a) The steady speed that will minimize the cost of the fuel for a 460 mile trip. $x=$ [ANS]\n(b) The steady speed that will minimize the total cost of a 460 mile trip if the driver is paid \\$14 an hour. $x=$ [ANS]\n(c) The steady speed that will minimize the total cost of a 560 mile trip if the driver is paid \\$36 an hour. $x=$ [ANS]", "answer_v3": [ "33", "43.5", "53" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0400", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "5", "keywords": [ "calculus" ], "problem_v1": "I have enough pure silver to coat $4$ square meters of surface area. I plan to coat a sphere and a cube, and to use all of the silver in doing so. Allowing for the possibility of all of the silver going onto one of the solids, what dimensions should they be if the total volume of the silvered solids is to be a maximum? The radius of the sphere is [ANS] meters, and the length of the sides of the cube is [ANS] meters.\nAgain, allowing for the possibility of all of the silver going onto one of the solids, what dimensions should they be if the total volume of the silvered solids is to be a minimum? The radius of the sphere is [ANS] meters, and the length of the sides of the cube is [ANS] meters.", "answer_v1": [ "0.564189583547756", "0", "0.330741786049482", "0.661483572098963" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "I have enough pure silver to coat $1$ square meters of surface area. I plan to coat a sphere and a cube, and to use all of the silver in doing so. Allowing for the possibility of all of the silver going onto one of the solids, what dimensions should they be if the total volume of the silvered solids is to be a maximum? The radius of the sphere is [ANS] meters, and the length of the sides of the cube is [ANS] meters.\nAgain, allowing for the possibility of all of the silver going onto one of the solids, what dimensions should they be if the total volume of the silvered solids is to be a minimum? The radius of the sphere is [ANS] meters, and the length of the sides of the cube is [ANS] meters.", "answer_v2": [ "0.282094791773878", "0", "0.165370893024741", "0.330741786049482" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "I have enough pure silver to coat $2$ square meters of surface area. I plan to coat a sphere and a cube, and to use all of the silver in doing so. Allowing for the possibility of all of the silver going onto one of the solids, what dimensions should they be if the total volume of the silvered solids is to be a maximum? The radius of the sphere is [ANS] meters, and the length of the sides of the cube is [ANS] meters.\nAgain, allowing for the possibility of all of the silver going onto one of the solids, what dimensions should they be if the total volume of the silvered solids is to be a minimum? The radius of the sphere is [ANS] meters, and the length of the sides of the cube is [ANS] meters.", "answer_v3": [ "0.398942280401433", "0", "0.233869759737339", "0.467739519474677" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0401", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "4", "keywords": [ "Optimization Problems", "Optimization' 'Maximum' 'Minimum" ], "problem_v1": "Find two positive numbers $A$ and $B$ (with $A \\leq B$) whose sum is 88 and whose product is maximized. $A$=[ANS]\n$B$=[ANS]", "answer_v1": [ "44", "44" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find two positive numbers $A$ and $B$ (with $A \\leq B$) whose sum is 44 and whose product is maximized. $A$=[ANS]\n$B$=[ANS]", "answer_v2": [ "22", "22" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find two positive numbers $A$ and $B$ (with $A \\leq B$) whose sum is 60 and whose product is maximized. $A$=[ANS]\n$B$=[ANS]", "answer_v3": [ "30", "30" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0402", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "5", "keywords": [], "problem_v1": "Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle with sides of length 16 if one side of the rectangle lies on the base of the triangle. Width=[ANS] Height=[ANS]", "answer_v1": [ "8", "6.92820323027551" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle with sides of length 3 if one side of the rectangle lies on the base of the triangle. Width=[ANS] Height=[ANS]", "answer_v2": [ "1.5", "1.29903810567666" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle with sides of length 7 if one side of the rectangle lies on the base of the triangle. Width=[ANS] Height=[ANS]", "answer_v3": [ "3.5", "3.03108891324554" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0403", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "5", "keywords": [ "Optimization' 'Maximum' 'Minimum" ], "problem_v1": "Find the length $L$ and width $W$ (with $W \\leq L$) of the rectangle with perimeter 88 that has maximum area, and then find the maximum area. $L$=[ANS]\n$W$=[ANS]\nMaximum area=[ANS]", "answer_v1": [ "22", "22", "484" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the length $L$ and width $W$ (with $W \\leq L$) of the rectangle with perimeter 44 that has maximum area, and then find the maximum area. $L$=[ANS]\n$W$=[ANS]\nMaximum area=[ANS]", "answer_v2": [ "11", "11", "121" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the length $L$ and width $W$ (with $W \\leq L$) of the rectangle with perimeter 60 that has maximum area, and then find the maximum area. $L$=[ANS]\n$W$=[ANS]\nMaximum area=[ANS]", "answer_v3": [ "15", "15", "225" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0404", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "5", "keywords": [ "Optimization' 'Maximum' 'Minimum" ], "problem_v1": "How would you divide a 24 inch line into two parts of length $A$ and $B$ so that $A+B=24$ and the product $AB$ is maximized? (Assume that $A \\leq B$. $A$=[ANS]\n$B$=[ANS]", "answer_v1": [ "12", "12" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "How would you divide a 11 inch line into two parts of length $A$ and $B$ so that $A+B=11$ and the product $AB$ is maximized? (Assume that $A \\leq B$. $A$=[ANS]\n$B$=[ANS]", "answer_v2": [ "5.5", "5.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "How would you divide a 15 inch line into two parts of length $A$ and $B$ so that $A+B=15$ and the product $AB$ is maximized? (Assume that $A \\leq B$. $A$=[ANS]\n$B$=[ANS]", "answer_v3": [ "7.5", "7.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0405", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "5", "keywords": [], "problem_v1": "In this question you will derive a simple labour demand curve. Suppose that the number of calculators a firm can produce per hour (TP) given a certain number of workers (L), is given by: TP=532\\ln(L+22)+20 L The cost of using each worker is just their hourly wage (w). So the total labour cost is $C=w L$. If the price of each calculator is \\$16, find the profit maximizing wage as a function of number of workers used (L). $w=$ [ANS]", "answer_v1": [ "[532/(L+22)+20]*16" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "In this question you will derive a simple labour demand curve. Suppose that the number of calculators a firm can produce per hour (TP) given a certain number of workers (L), is given by: TP=86\\ln(L+7)+3 L The cost of using each worker is just their hourly wage (w). So the total labour cost is $C=w L$. If the price of each calculator is \\$11, find the profit maximizing wage as a function of number of workers used (L). $w=$ [ANS]", "answer_v2": [ "[86/(L+7)+3]*11" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "In this question you will derive a simple labour demand curve. Suppose that the number of calculators a firm can produce per hour (TP) given a certain number of workers (L), is given by: TP=232\\ln(L+21)+9 L The cost of using each worker is just their hourly wage (w). So the total labour cost is $C=w L$. If the price of each calculator is \\$13, find the profit maximizing wage as a function of number of workers used (L). $w=$ [ANS]", "answer_v3": [ "[232/(L+21)+9]*13" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0406", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - general", "level": "4", "keywords": [], "problem_v1": "Let $f(x)$ be the perimeter of a rectangle with an area $49 units^2$ and one side with length $x$. $f(x)$=[ANS]\nWhat is the minimum perimeter of all rectangles with this area? Perimeter $=$ [ANS]\n(you will lose 50\\% of your points if you do)", "answer_v1": [ "(2*x^2+98)/x", "28" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x)$ be the perimeter of a rectangle with an area $9 units^2$ and one side with length $x$. $f(x)$=[ANS]\nWhat is the minimum perimeter of all rectangles with this area? Perimeter $=$ [ANS]\n(you will lose 50\\% of your points if you do)", "answer_v2": [ "(2*x^2+18)/x", "12" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x)$ be the perimeter of a rectangle with an area $16 units^2$ and one side with length $x$. $f(x)$=[ANS]\nWhat is the minimum perimeter of all rectangles with this area? Perimeter $=$ [ANS]\n(you will lose 50\\% of your points if you do)", "answer_v3": [ "(2*x^2+32)/x", "16" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0407", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [ "calculus", "derivative", "local maxima and minima", "maxima", "minima", "marginal cost", "marginal profit" ], "problem_v1": "The revenue from selling $q$ items is $R(q)=550 q-q^2$, and the total cost is $C(q)=150+11 q$. Write a function that gives the total profit earned, and find the quantity which maximizes the profit. Profit $\\pi(q)=$ [ANS]\nQuantity maximizing profit $q=$ [ANS]", "answer_v1": [ "550*q-q^2-150-11*q", "269.5" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The revenue from selling $q$ items is $R(q)=150 q-q^2$, and the total cost is $C(q)=200+6 q$. Write a function that gives the total profit earned, and find the quantity which maximizes the profit. Profit $\\pi(q)=$ [ANS]\nQuantity maximizing profit $q=$ [ANS]", "answer_v2": [ "150*q-q^2-200-6*q", "72" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The revenue from selling $q$ items is $R(q)=275 q-q^2$, and the total cost is $C(q)=150+8 q$. Write a function that gives the total profit earned, and find the quantity which maximizes the profit. Profit $\\pi(q)=$ [ANS]\nQuantity maximizing profit $q=$ [ANS]", "answer_v3": [ "275*q-q^2-150-8*q", "133.5" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0408", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [ "calculus", "derivative", "local maxima and minima", "maxima", "minima", "marginal cost", "marginal profit" ], "problem_v1": "The total cost $C(q)$ of producing $q$ goods is given by: C(q)=0.01q^{3}-0.6q^{2}+14q What is the fixed cost? fixed cost=[ANS] dollars What is the maximum profit if each item is sold for 10 dollars? (Assume you sell everything you produce. Also note that you can only produce a whole number of goods.) (Assume you sell everything you produce. Also note that you can only produce a whole number of goods.) maximum profit=[ANS] dollars Suppose we fix production at 36 goods produced, and that they all sell when the price is 10 dollars each. Also suppose that for each 1 dollar increase in price, 2 fewer goods are sold (so if the price is 11 dollars, 34 of the 36 goods being produced are sold.) To maximize profit in this case, we should [ANS] the price by [ANS] dollars. (Enter zero if the price should remain at 10 dollars.)", "answer_v1": [ "0", "167.04", "increase", "4" ], "answer_type_v1": [ "NV", "NV", "MCS", "NV" ], "options_v1": [ [], [], [ "decrease", "increase", "neither increase nor decrease" ], [] ], "problem_v2": "The total cost $C(q)$ of producing $q$ goods is given by: C(q)=0.01q^{3}-0.6q^{2}+12q What is the fixed cost? fixed cost=[ANS] dollars What is the maximum profit if each item is sold for 6 dollars? (Assume you sell everything you produce. Also note that you can only produce a whole number of goods.) (Assume you sell everything you produce. Also note that you can only produce a whole number of goods.) maximum profit=[ANS] dollars Suppose we fix production at 34 goods produced, and that they all sell when the price is 6 dollars each. Also suppose that for each 1 dollar increase in price, 1 fewer goods are sold (so if the price is 7 dollars, 33 of the 34 goods being produced are sold.) To maximize profit in this case, we should [ANS] the price by [ANS] dollars. (Enter zero if the price should remain at 6 dollars.)", "answer_v2": [ "0", "96.56", "increase", "14" ], "answer_type_v2": [ "NV", "NV", "MCS", "NV" ], "options_v2": [ [], [], [ "decrease", "increase", "neither increase nor decrease" ], [] ], "problem_v3": "The total cost $C(q)$ of producing $q$ goods is given by: C(q)=0.01q^{3}-0.6q^{2}+14q What is the fixed cost? fixed cost=[ANS] dollars What is the maximum profit if each item is sold for 6 dollars? (Assume you sell everything you produce. Also note that you can only produce a whole number of goods.) (Assume you sell everything you produce. Also note that you can only produce a whole number of goods.) maximum profit=[ANS] dollars Suppose we fix production at 32 goods produced, and that they all sell when the price is 6 dollars each. Also suppose that for each 1 dollar increase in price, 2 fewer goods are sold (so if the price is 7 dollars, 30 of the 32 goods being produced are sold.) To maximize profit in this case, we should [ANS] the price by [ANS] dollars. (Enter zero if the price should remain at 6 dollars.)", "answer_v3": [ "0", "30.72", "increase", "5" ], "answer_type_v3": [ "NV", "NV", "MCS", "NV" ], "options_v3": [ [], [], [ "decrease", "increase", "neither increase nor decrease" ], [] ] }, { "id": "Calculus_-_single_variable_0409", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [ "maximization' 'minimization' 'derivatives' 'optimization", "derivatives", "Business and Economics", "economics", "Optimization", "Economics", "Calculus" ], "problem_v1": "For the given cost function $C(x)=250 \\sqrt{x}+\\frac{x^2}{125000}$ find a) The cost at the production level 1650 [ANS]\nb) The average cost at the production level 1650 [ANS]\nc) The marginal cost at the production level 1650 [ANS]\nd) The production level that will minimize the average cost. [ANS]\ne) The minimal average cost. [ANS]", "answer_v1": [ "10176.828005795", "6.16777454896664", "3.10368727448332", "62500", "1.5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "For the given cost function $C(x)=16 \\sqrt{x}+\\frac{x^2}{421875}$ find a) The cost at the production level 1150 [ANS]\nb) The average cost at the production level 1150 [ANS]\nc) The marginal cost at the production level 1150 [ANS]\nd) The production level that will minimize the average cost. [ANS]\ne) The minimal average cost. [ANS]", "answer_v2": [ "545.721213464836", "0.474540185621597", "0.241358981699687", "22500", "0.16" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "For the given cost function $C(x)=54 \\sqrt{x}+\\frac{x^2}{166375}$ find a) The cost at the production level 1250 [ANS]\nb) The average cost at the production level 1250 [ANS]\nc) The marginal cost at the production level 1250 [ANS]\nd) The production level that will minimize the average cost. [ANS]\ne) The minimal average cost. [ANS]", "answer_v3": [ "1918.57974421495", "1.53486379537196", "0.778701619699503", "27225", "0.490909090909091" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0410", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [ "maximization' 'minimization' 'derivatives' 'optimization", "Optimization", "Economics", "Calculus", "Derivatives" ], "problem_v1": "The manager of a large apartment complex knows from experience that 110 units will be occupied if the rent is 416 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 8 dollar increase in rent. Similarly, one additional unit will be occupied for each 8 dollar decrease in rent. What rent should the manager charge to maximize revenue? [ANS]", "answer_v1": [ "648" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The manager of a large apartment complex knows from experience that 80 units will be occupied if the rent is 500 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 1 dollar increase in rent. Similarly, one additional unit will be occupied for each 1 dollar decrease in rent. What rent should the manager charge to maximize revenue? [ANS]", "answer_v2": [ "290" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The manager of a large apartment complex knows from experience that 90 units will be occupied if the rent is 424 dollars per month. A market survey suggests that, on the average, one additional unit will remain vacant for each 4 dollar increase in rent. Similarly, one additional unit will be occupied for each 4 dollar decrease in rent. What rent should the manager charge to maximize revenue? [ANS]", "answer_v3": [ "392" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0411", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [ "derivatives", "Business and Economics", "Optimization", "Economics", "economics", "Calculus" ], "problem_v1": "A baseball team plays in a stadium that holds 66000 spectators. With the ticket price at \\$11 the average attendance has been 28000. When the price dropped to \\$8, the average attendance rose to 33000.\na) Find the demand function $p(x)$, where $x$ is the number of the spectators. (Assume that $p(x)$ is linear.)\n$p(x)=$ [ANS]\nb) How should ticket prices be set to maximize revenue?\nThe revenue is maximized by charging \\$ [ANS] per ticket.", "answer_v1": [ "139/5-3/5000*x", "139/10" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A baseball team plays in a stadium that holds 52000 spectators. With the ticket price at \\$8 the average attendance has been 19000. When the price dropped to \\$6, the average attendance rose to 26000.\na) Find the demand function $p(x)$, where $x$ is the number of the spectators. (Assume that $p(x)$ is linear.)\n$p(x)=$ [ANS]\nb) How should ticket prices be set to maximize revenue?\nThe revenue is maximized by charging \\$ [ANS] per ticket.", "answer_v2": [ "94/7-1/3500*x", "47/7" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A baseball team plays in a stadium that holds 56000 spectators. With the ticket price at \\$9 the average attendance has been 22000. When the price dropped to \\$7, the average attendance rose to 28000.\na) Find the demand function $p(x)$, where $x$ is the number of the spectators. (Assume that $p(x)$ is linear.)\n$p(x)=$ [ANS]\nb) How should ticket prices be set to maximize revenue?\nThe revenue is maximized by charging \\$ [ANS] per ticket.", "answer_v3": [ "49/3-1/3000*x", "49/6" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0412", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [ "derivatives", "Business and Economics", "Optimization' 'Maximum' 'Minimum" ], "problem_v1": "A company manufacturers and sells $x$ electric drills per month. The monthly cost and price-demand equations are \\begin{align*} C(x) &=72000+60x,\\\\ p &=210-\\frac{x}{30}, \\qquad 0 \\leq x\\leq 5000. \\end{align*} (A) Find the production level that results in the maximum revenue. Production Level=[ANS]\n(B) Find the price that the company should charge for each drill in order to maximize profit.\nPrice=[ANS]\n(C) Suppose that a 5 dollar per drill tax is imposed. Determine the number of drills that should be produced and sold in order to maximize profit under these new circumstances.\nNumber of drills=[ANS]", "answer_v1": [ "3150", "135", "2175" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A company manufacturers and sells $x$ electric drills per month. The monthly cost and price-demand equations are \\begin{align*} C(x) &=61000+80x,\\\\ p &=180-\\frac{x}{30}, \\qquad 0 \\leq x\\leq 5000. \\end{align*} (A) Find the production level that results in the maximum revenue. Production Level=[ANS]\n(B) Find the price that the company should charge for each drill in order to maximize profit.\nPrice=[ANS]\n(C) Suppose that a 5 dollar per drill tax is imposed. Determine the number of drills that should be produced and sold in order to maximize profit under these new circumstances.\nNumber of drills=[ANS]", "answer_v2": [ "2700", "130", "1425" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A company manufacturers and sells $x$ electric drills per month. The monthly cost and price-demand equations are \\begin{align*} C(x) &=65000+70x,\\\\ p &=190-\\frac{x}{30}, \\qquad 0 \\leq x\\leq 5000. \\end{align*} (A) Find the production level that results in the maximum revenue. Production Level=[ANS]\n(B) Find the price that the company should charge for each drill in order to maximize profit.\nPrice=[ANS]\n(C) Suppose that a 5 dollar per drill tax is imposed. Determine the number of drills that should be produced and sold in order to maximize profit under these new circumstances.\nNumber of drills=[ANS]", "answer_v3": [ "2850", "130", "1725" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0413", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [ "derivatives", "Business and Economics", "economics", "Optimization", "Calculus" ], "problem_v1": "For the cost function $C(x)=7050+570x+1.2x^2$ and the demand function $p(x)=1710$, find the production level that will maximaze profit. Production Level=[ANS]", "answer_v1": [ "475" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "For the cost function $C(x)=1650+850x+0.3x^2$ and the demand function $p(x)=2550$, find the production level that will maximaze profit. Production Level=[ANS]", "answer_v2": [ "2833.33333333333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "For the cost function $C(x)=3500+590x+0.6x^2$ and the demand function $p(x)=1770$, find the production level that will maximaze profit. Production Level=[ANS]", "answer_v3": [ "983.333333333333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0414", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [ "Differentiation' 'Rates of Change" ], "problem_v1": "The profit function for a computer company is given by $P(x)=-x^2+35x-18$ where $x$ is the number of units produced (in thousands) and the profit is in thousand of dollars. a) Determine how many (thousands of) units must be produced to yield maximum profit. Determine the maximum profit. (thousands of) units=[ANS]\nmaximum profit=[ANS] thousand dollars b) Determine how many units should be produced for a profit of at least 40 thousand. more than [ANS] (thousands of) units less than [ANS] (thousands of) units", "answer_v1": [ "17.5", "288.25", "1.74404874341127", "33.2559512565887" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The profit function for a computer company is given by $P(x)=-x^2+21x-11$ where $x$ is the number of units produced (in thousands) and the profit is in thousand of dollars. a) Determine how many (thousands of) units must be produced to yield maximum profit. Determine the maximum profit. (thousands of) units=[ANS]\nmaximum profit=[ANS] thousand dollars b) Determine how many units should be produced for a profit of at least 40 thousand. more than [ANS] (thousands of) units less than [ANS] (thousands of) units", "answer_v2": [ "10.5", "99.25", "2.80259784082967", "18.1974021591703" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The profit function for a computer company is given by $P(x)=-x^2+26x-18$ where $x$ is the number of units produced (in thousands) and the profit is in thousand of dollars. a) Determine how many (thousands of) units must be produced to yield maximum profit. Determine the maximum profit. (thousands of) units=[ANS]\nmaximum profit=[ANS] thousand dollars b) Determine how many units should be produced for a profit of at least 40 thousand. more than [ANS] (thousands of) units less than [ANS] (thousands of) units", "answer_v3": [ "13", "151", "2.46434624714726", "23.5356537528527" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0415", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [ "Optimization' 'Maximum' 'Minimum" ], "problem_v1": "A cosmetics company is planning the introduction and promotion of a new lipstick line. The marketing research department has found that the demand in a particular city is given approximately by p=\\frac{11}{e^x}, \\quad 0 \\leq x \\leq 2, where $x$ thousand lipsticks were sold per week at a price of $p$ dollars each. At what price will the weekly revenue be maximized?\nPrice=\\$ [ANS]\nNote: the answer must an actual value for money, like 7.19.", "answer_v1": [ "4.05" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A cosmetics company is planning the introduction and promotion of a new lipstick line. The marketing research department has found that the demand in a particular city is given approximately by p=\\frac{6}{e^x}, \\quad 0 \\leq x \\leq 2, where $x$ thousand lipsticks were sold per week at a price of $p$ dollars each. At what price will the weekly revenue be maximized?\nPrice=\\$ [ANS]\nNote: the answer must an actual value for money, like 7.19.", "answer_v2": [ "2.21" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A cosmetics company is planning the introduction and promotion of a new lipstick line. The marketing research department has found that the demand in a particular city is given approximately by p=\\frac{8}{e^x}, \\quad 0 \\leq x \\leq 2, where $x$ thousand lipsticks were sold per week at a price of $p$ dollars each. At what price will the weekly revenue be maximized?\nPrice=\\$ [ANS]\nNote: the answer must an actual value for money, like 7.19.", "answer_v3": [ "2.94" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0416", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [ "Optimization' 'Maximum' 'Minimum", "Optimization", "Economics", "derivatives", "economics", "Calculus", "Business and Economics" ], "problem_v1": "A manufacture has been selling 1750 television sets a week at \\$480 each. A market survey indicates that for each \\$23 rebate offered to a buyer, the number of sets sold will increase by 230 per week.\na) Find the function representing the demand $p(x)$, where $x$ is the number of the television sets sold per week and $p(x)$ is the corresponding price. $p(x)=$ [ANS]\nb) How large rebate should the company offer to a buyer, in order to maximize its revenue? [ANS] dollars\nc) If the weekly cost function is $140000+160x$, how should it set the size of the rebate to maximize its profit? [ANS] dollars", "answer_v1": [ "(1750-x)/10 +480", "152.5", "72.5" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A manufacture has been selling 1050 television sets a week at \\$540 each. A market survey indicates that for each \\$13 rebate offered to a buyer, the number of sets sold will increase by 130 per week.\na) Find the function representing the demand $p(x)$, where $x$ is the number of the television sets sold per week and $p(x)$ is the corresponding price. $p(x)=$ [ANS]\nb) How large rebate should the company offer to a buyer, in order to maximize its revenue? [ANS] dollars\nc) If the weekly cost function is $94500+180x$, how should it set the size of the rebate to maximize its profit? [ANS] dollars", "answer_v2": [ "(1050-x)/10 +540", "217.5", "127.5" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A manufacture has been selling 1300 television sets a week at \\$480 each. A market survey indicates that for each \\$15 rebate offered to a buyer, the number of sets sold will increase by 150 per week.\na) Find the function representing the demand $p(x)$, where $x$ is the number of the television sets sold per week and $p(x)$ is the corresponding price. $p(x)=$ [ANS]\nb) How large rebate should the company offer to a buyer, in order to maximize its revenue? [ANS] dollars\nc) If the weekly cost function is $104000+160x$, how should it set the size of the rebate to maximize its profit? [ANS] dollars", "answer_v3": [ "(1300-x)/10 +480", "175", "95" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0417", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [ "Optimization", "End point solution" ], "problem_v1": "(Once you have their money, never give it back.) An apartment complex on Ferenginar with 250 units currently has 180 occupants. The current rent for a unit is 1260 slips of Gold-Pressed Latinum. The owner of the complex knows from experience that he loses one occupant every time he raises the rent by 3.5 slips of Latinum. Since \"profit is its own reward\", the owner wants to maximize his profit so he asks for our help, even though he knows that \"free advice is seldom cheap\".\nWhat should be our recommendation for the optimal rent? [ANS] slips of Gold-Pressed Latinum", "answer_v1": [ "1015" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "(Once you have their money, never give it back.) An apartment complex on Ferenginar with 250 units currently has 234 occupants. The current rent for a unit is 702 slips of Gold-Pressed Latinum. The owner of the complex knows from experience that he loses one occupant every time he raises the rent by 1.5 slips of Latinum. Since \"profit is its own reward\", the owner wants to maximize his profit so he asks for our help, even though he knows that \"free advice is seldom cheap\".\nWhat should be our recommendation for the optimal rent? [ANS] slips of Gold-Pressed Latinum", "answer_v2": [ "678" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "(Once you have their money, never give it back.) An apartment complex on Ferenginar with 250 units currently has 197 occupants. The current rent for a unit is 788 slips of Gold-Pressed Latinum. The owner of the complex knows from experience that he loses one occupant every time he raises the rent by 2 slips of Latinum. Since \"profit is its own reward\", the owner wants to maximize his profit so he asks for our help, even though he knows that \"free advice is seldom cheap\".\nWhat should be our recommendation for the optimal rent? [ANS] slips of Gold-Pressed Latinum", "answer_v3": [ "682" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0418", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [], "problem_v1": "Suppose that you have two consumption choices: good X, and good Y. An indifference curve is the set of consumption choices with a CONSTANT utility. For example if $X=10$ and $Y=6$ gives the same utility as consuming $X=11$ and $Y=5$, than these are both points on the same indifference curve. An indifference map is the set of all indifference curves for EVERY given utility. Consider the following utility map: $U=\\ln(X-6)+\\ln(Y-7)$ A budget curve gives the set of possible consumption choices with a given income. If you have an income of \\$1060 and the price of good X is given by $p_x$, and the price of good Y given by $p_y$. The equation for the budget line is given by: $1060=p_x X+p_y Y$. A utility maximizing combination of goods X and Y occurs when the budget line is tangent to an indifference curve. Find X as a function of its price and the price of good Y: (If Y represents all other goods, than this function is just a demand curve for X). $X=$ [ANS]\n(Use px for $p_x$ and py for $p_y$) Let $X_0$ be the value for X when $p_x=13$ and $p_y=14$. $X_0=$ [ANS]\n(you will lose 25\\% of your points if you do)", "answer_v1": [ "(1060-7*py)/(2*px)", "40" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that you have two consumption choices: good X, and good Y. An indifference curve is the set of consumption choices with a CONSTANT utility. For example if $X=10$ and $Y=6$ gives the same utility as consuming $X=11$ and $Y=5$, than these are both points on the same indifference curve. An indifference map is the set of all indifference curves for EVERY given utility. Consider the following utility map: $U=\\ln(X-2)+\\ln(Y-4)$ A budget curve gives the set of possible consumption choices with a given income. If you have an income of \\$796 and the price of good X is given by $p_x$, and the price of good Y given by $p_y$. The equation for the budget line is given by: $796=p_x X+p_y Y$. A utility maximizing combination of goods X and Y occurs when the budget line is tangent to an indifference curve. Find X as a function of its price and the price of good Y: (If Y represents all other goods, than this function is just a demand curve for X). $X=$ [ANS]\n(Use px for $p_x$ and py for $p_y$) Let $X_0$ be the value for X when $p_x=5$ and $p_y=19$. $X_0=$ [ANS]\n(you will lose 25\\% of your points if you do)", "answer_v2": [ "(796-4*py)/(2*px)", "73" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that you have two consumption choices: good X, and good Y. An indifference curve is the set of consumption choices with a CONSTANT utility. For example if $X=10$ and $Y=6$ gives the same utility as consuming $X=11$ and $Y=5$, than these are both points on the same indifference curve. An indifference map is the set of all indifference curves for EVERY given utility. Consider the following utility map: $U=\\ln(X-4)+\\ln(Y-5)$ A budget curve gives the set of possible consumption choices with a given income. If you have an income of \\$598 and the price of good X is given by $p_x$, and the price of good Y given by $p_y$. The equation for the budget line is given by: $598=p_x X+p_y Y$. A utility maximizing combination of goods X and Y occurs when the budget line is tangent to an indifference curve. Find X as a function of its price and the price of good Y: (If Y represents all other goods, than this function is just a demand curve for X). $X=$ [ANS]\n(Use px for $p_x$ and py for $p_y$) Let $X_0$ be the value for X when $p_x=8$ and $p_y=14$. $X_0=$ [ANS]\n(you will lose 25\\% of your points if you do)", "answer_v3": [ "(598-5*py)/(2*px)", "35" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0419", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [], "problem_v1": "Suppose that a firm uses both labour (X) and capital (Y) to produce a certain good. An isoquant is a curve made up of the set of points $(X,Y)$ that gives the same level of output. For example, if 10 units of labour and 6 units of capital produce the same output as 11 units of labour and 5 units of capital, than (10,6) and (11,5) are on the same isoquant. The production function is the set of all isoquants given any level of output. The Cobb-Douglas production function is a simple example: $q=X^{\\alpha}Y^{\\alpha-1}$, where $0 \\le \\alpha \\le 1$. The cost function gives the cost of using a given combination of labour and capital. If $p_x$ is the cost of using labour and $p_y$ is the cost of using capital then: $C=p_x X+p_y Y$. A set of points with a constant cost is known as an isocost. To minimize cost given a certain level of output, an isocost will be tangent to an isoquant. This is not the same situation as with utility maximization where the total income was fixed. Here, you are minimizing cost for a given level of output. Let $\\alpha=.5$, $p_x=25$ and $p_y=100$. Suppose the firm wants to produce $Q$ units. Find the cost minimizing amount of labour and capital to use as a function of $Q$. $X=$ [ANS]\n$Y=$ [ANS]\nNow, express the cost of production as a function of output (Q). Note that this will be the long run cost function. $C=$ [ANS]\n(you will lose 25\\% of your points if you do)", "answer_v1": [ "Q*2", "Q*0.5", "Q*100" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose that a firm uses both labour (X) and capital (Y) to produce a certain good. An isoquant is a curve made up of the set of points $(X,Y)$ that gives the same level of output. For example, if 10 units of labour and 6 units of capital produce the same output as 11 units of labour and 5 units of capital, than (10,6) and (11,5) are on the same isoquant. The production function is the set of all isoquants given any level of output. The Cobb-Douglas production function is a simple example: $q=X^{\\alpha}Y^{\\alpha-1}$, where $0 \\le \\alpha \\le 1$. The cost function gives the cost of using a given combination of labour and capital. If $p_x$ is the cost of using labour and $p_y$ is the cost of using capital then: $C=p_x X+p_y Y$. A set of points with a constant cost is known as an isocost. To minimize cost given a certain level of output, an isocost will be tangent to an isoquant. This is not the same situation as with utility maximization where the total income was fixed. Here, you are minimizing cost for a given level of output. Let $\\alpha=.5$, $p_x=11$ and $p_y=44$. Suppose the firm wants to produce $Q$ units. Find the cost minimizing amount of labour and capital to use as a function of $Q$. $X=$ [ANS]\n$Y=$ [ANS]\nNow, express the cost of production as a function of output (Q). Note that this will be the long run cost function. $C=$ [ANS]\n(you will lose 25\\% of your points if you do)", "answer_v2": [ "Q*2", "Q*0.5", "Q*44" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose that a firm uses both labour (X) and capital (Y) to produce a certain good. An isoquant is a curve made up of the set of points $(X,Y)$ that gives the same level of output. For example, if 10 units of labour and 6 units of capital produce the same output as 11 units of labour and 5 units of capital, than (10,6) and (11,5) are on the same isoquant. The production function is the set of all isoquants given any level of output. The Cobb-Douglas production function is a simple example: $q=X^{\\alpha}Y^{\\alpha-1}$, where $0 \\le \\alpha \\le 1$. The cost function gives the cost of using a given combination of labour and capital. If $p_x$ is the cost of using labour and $p_y$ is the cost of using capital then: $C=p_x X+p_y Y$. A set of points with a constant cost is known as an isocost. To minimize cost given a certain level of output, an isocost will be tangent to an isoquant. This is not the same situation as with utility maximization where the total income was fixed. Here, you are minimizing cost for a given level of output. Let $\\alpha=.5$, $p_x=16$ and $p_y=64$. Suppose the firm wants to produce $Q$ units. Find the cost minimizing amount of labour and capital to use as a function of $Q$. $X=$ [ANS]\n$Y=$ [ANS]\nNow, express the cost of production as a function of output (Q). Note that this will be the long run cost function. $C=$ [ANS]\n(you will lose 25\\% of your points if you do)", "answer_v3": [ "Q*2", "Q*0.5", "Q*64" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0420", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "3", "keywords": [], "problem_v1": "An electronics store expects to sell 900 laptops at a steady rate next year. The manager of the store plans to order these laptops from the manufacturer by placing several orders of the same size spaced equally throughout the year. The ordering cost for each delivery is \\$26 for the setup costs and \\$4 per laptop. The carrying costs, based on the average number of laptops in inventory, amount to \\$13 per year for one laptop. If $C(x)$ is the inventory cost (which is the sum of the ordering costs and the carrying costs) and $x$ is the number of orders. $C(x)=$ [ANS]\nHow many orders should the store manager place? [ANS]\n(you will lose 50\\% of your points if you do)", "answer_v1": [ "26*x+5850/x+3600", "15" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "An electronics store expects to sell 1600 laptops at a steady rate next year. The manager of the store plans to order these laptops from the manufacturer by placing several orders of the same size spaced equally throughout the year. The ordering cost for each delivery is \\$10 for the setup costs and \\$1 per laptop. The carrying costs, based on the average number of laptops in inventory, amount to \\$5 per year for one laptop. If $C(x)$ is the inventory cost (which is the sum of the ordering costs and the carrying costs) and $x$ is the number of orders. $C(x)=$ [ANS]\nHow many orders should the store manager place? [ANS]\n(you will lose 50\\% of your points if you do)", "answer_v2": [ "10*x+4000/x+1600", "20" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "An electronics store expects to sell 1024 laptops at a steady rate next year. The manager of the store plans to order these laptops from the manufacturer by placing several orders of the same size spaced equally throughout the year. The ordering cost for each delivery is \\$14 for the setup costs and \\$2 per laptop. The carrying costs, based on the average number of laptops in inventory, amount to \\$7 per year for one laptop. If $C(x)$ is the inventory cost (which is the sum of the ordering costs and the carrying costs) and $x$ is the number of orders. $C(x)=$ [ANS]\nHow many orders should the store manager place? [ANS]\n(you will lose 50\\% of your points if you do)", "answer_v3": [ "14*x+3584/x+2048", "16" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0421", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "4", "keywords": [], "problem_v1": "Consider the demand for tickets to see a specific hockey team play. The price of the ticket can be related to the quantity demanded (q) by the function: $p=255-0.01 q$. When the arena is not close to full capacity the total cost can be expressed by the function: $Cost=79 q+5,000,000$. Find marginal revenue (MR) as a function of quantity demanded. $MR=$ [ANS]\nLet $p^*$ and $q^*$ be the price and quantity demanded where profit is maximized. $p^*=$ [ANS] $q^*=$ [ANS]\nThe hockey players union has negotiated a deal requiring the team owner to pay an extra \\$1,000,000 a year in salaries to the players. What should the new ticket price ($p_1$) be to ensure that profit is maximized. $p_1=$ [ANS]", "answer_v1": [ "255-0.02*q", "167", "8800", "167" ], "answer_type_v1": [ "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the demand for tickets to see a specific hockey team play. The price of the ticket can be related to the quantity demanded (q) by the function: $p=205-0.01 q$. When the arena is not close to full capacity the total cost can be expressed by the function: $Cost=97 q+5,000,000$. Find marginal revenue (MR) as a function of quantity demanded. $MR=$ [ANS]\nLet $p^*$ and $q^*$ be the price and quantity demanded where profit is maximized. $p^*=$ [ANS] $q^*=$ [ANS]\nThe hockey players union has negotiated a deal requiring the team owner to pay an extra \\$1,000,000 a year in salaries to the players. What should the new ticket price ($p_1$) be to ensure that profit is maximized. $p_1=$ [ANS]", "answer_v2": [ "205-0.02*q", "151", "5400", "151" ], "answer_type_v2": [ "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the demand for tickets to see a specific hockey team play. The price of the ticket can be related to the quantity demanded (q) by the function: $p=210-0.01 q$. When the arena is not close to full capacity the total cost can be expressed by the function: $Cost=80 q+5,000,000$. Find marginal revenue (MR) as a function of quantity demanded. $MR=$ [ANS]\nLet $p^*$ and $q^*$ be the price and quantity demanded where profit is maximized. $p^*=$ [ANS] $q^*=$ [ANS]\nThe hockey players union has negotiated a deal requiring the team owner to pay an extra \\$1,000,000 a year in salaries to the players. What should the new ticket price ($p_1$) be to ensure that profit is maximized. $p_1=$ [ANS]", "answer_v3": [ "210-0.02*q", "145", "6500", "145" ], "answer_type_v3": [ "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0422", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "4", "keywords": [], "problem_v1": "Suppose that a when a cell phone store charges $18$ \\$ for a car charger, it sells $39$ units. When it drops the price to $17$ \\$ it sells $42$ units. Assume that demand is a linear function of price. If each phone charger costs $1$ \\$ to make, what price should the store charge to maximize its profit? If $x$ is the number of times the price is reduced by one dollar. Find a function for total profit with respect to $x$. A negative value for $x$ will mean the price is increased. $f(x)=$ [ANS]\nPrice to maximize profit $=$ [ANS]", "answer_v1": [ "(39+3*x)*(18-x)-1*(39+3*x)", "16" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that a when a cell phone store charges $15$ \\$ for a car charger, it sells $120$ units. When it drops the price to $14$ \\$ it sells $125$ units. Assume that demand is a linear function of price. If each phone charger costs $1$ \\$ to make, what price should the store charge to maximize its profit? If $x$ is the number of times the price is reduced by one dollar. Find a function for total profit with respect to $x$. A negative value for $x$ will mean the price is increased. $f(x)=$ [ANS]\nPrice to maximize profit $=$ [ANS]", "answer_v2": [ "(120+5*x)*(15-x)-1*(120+5*x)", "20" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that a when a cell phone store charges $16$ \\$ for a car charger, it sells $76$ units. When it drops the price to $15$ \\$ it sells $80$ units. Assume that demand is a linear function of price. If each phone charger costs $1$ \\$ to make, what price should the store charge to maximize its profit? If $x$ is the number of times the price is reduced by one dollar. Find a function for total profit with respect to $x$. A negative value for $x$ will mean the price is increased. $f(x)=$ [ANS]\nPrice to maximize profit $=$ [ANS]", "answer_v3": [ "(76+4*x)*(16-x)-1*(76+4*x)", "18" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0423", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - business and economics", "level": "5", "keywords": [], "problem_v1": "Say you buy an house as an investment for 450000\\$ (assume that you did not need a mortgage). You estimate that the house will increase in value continuously by 56250\\$ per year. At any time in the future you can sell the house and invest the money in a fund with a yearly interest rate of 7.5\\% compounded bi-monthly. If you want to maximize your return, after how many years should you sell the house? Report your answer to 1 decimal place. $years=$ [ANS]", "answer_v1": [ "4.92" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Say you buy an house as an investment for 250000\\$ (assume that you did not need a mortgage). You estimate that the house will increase in value continuously by 31250\\$ per year. At any time in the future you can sell the house and invest the money in a fund with a yearly interest rate of 9\\% compounded quarterly. If you want to maximize your return, after how many years should you sell the house? Report your answer to 1 decimal place. $years=$ [ANS]", "answer_v2": [ "2.74" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Say you buy an house as an investment for 300000\\$ (assume that you did not need a mortgage). You estimate that the house will increase in value continuously by 37500\\$ per year. At any time in the future you can sell the house and invest the money in a fund with a yearly interest rate of 7.5\\% compounded monthly. If you want to maximize your return, after how many years should you sell the house? Report your answer to 1 decimal place. $years=$ [ANS]", "answer_v3": [ "4.88" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0424", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - engineering and physics", "level": "5", "keywords": [ "derivative' 'extrema' 'optimization" ], "problem_v1": "A Union student decided to depart from Earth after graduation to find work on Mars. Careful calculations made regarding the space shuttle to be built used the following mathematical model for the velocity (in ft/sec) of the shuttle from liftoff at $t=0$ seconds until the solid rocket boosters are jettisoned at $t=51.2$ seconds: v(t)=0.001753t^{3}-0.08582t^{2}+26.28t+11.12 Using this model, consider the acceleration of the shuttle. Find the absolute maximum and minimum values of acceleration between liftoff and the jettisoning of the boosters.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum of acceleration &=& [ANS] \\\\ \\hline 2. & Absolute minimum of acceleration &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "31.2782", "24.8795" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A Union student decided to depart from Earth after graduation to find work on Mars. Careful calculations made regarding the space shuttle to be built used the following mathematical model for the velocity (in ft/sec) of the shuttle from liftoff at $t=0$ seconds until the solid rocket boosters are jettisoned at $t=103$ seconds: v(t)=0.00108383t^{3}-0.089315t^{2}+15.44t+5.68 Using this model, consider the acceleration of the shuttle. Find the absolute maximum and minimum values of acceleration between liftoff and the jettisoning of the boosters.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum of acceleration &=& [ANS] \\\\ \\hline 2. & Absolute minimum of acceleration &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "31.5363", "12.9866" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A Union student decided to depart from Earth after graduation to find work on Mars. Careful calculations made regarding the space shuttle to be built used the following mathematical model for the velocity (in ft/sec) of the shuttle from liftoff at $t=0$ seconds until the solid rocket boosters are jettisoned at $t=57.9$ seconds: v(t)=0.00131417t^{3}-0.086055t^{2}+18.41t+8.68 Using this model, consider the acceleration of the shuttle. Find the absolute maximum and minimum values of acceleration between liftoff and the jettisoning of the boosters.\n$\\begin{array}{cccc}\\hline 1. & Absolute maximum of acceleration &=& [ANS] \\\\ \\hline 2. & Absolute minimum of acceleration &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "21.6617", "16.5316" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0425", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - engineering and physics", "level": "5", "keywords": [ "calculus", "derivative", "local maxima and minima", "maxima", "minima", "modeling" ], "problem_v1": "An electric current, $I$, in amps, is given by I=\\cos(w t)+\\sqrt{7} \\sin(w t), where $w\\ne 0$ is a constant. What are the maximum and minimum values of I? Minimum value of $I$: [ANS] amp Maximum value of $I$: [ANS] amp", "answer_v1": [ "-1*sqrt(1+7)", "sqrt(1+7)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "An electric current, $I$, in amps, is given by I=\\cos(w t)+\\sqrt{2} \\sin(w t), where $w\\ne 0$ is a constant. What are the maximum and minimum values of I? Minimum value of $I$: [ANS] amp Maximum value of $I$: [ANS] amp", "answer_v2": [ "-1*sqrt(1+2)", "sqrt(1+2)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "An electric current, $I$, in amps, is given by I=\\cos(w t)+\\sqrt{6} \\sin(w t), where $w\\ne 0$ is a constant. What are the maximum and minimum values of I? Minimum value of $I$: [ANS] amp Maximum value of $I$: [ANS] amp", "answer_v3": [ "-1*sqrt(1+6)", "sqrt(1+6)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0426", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - engineering and physics", "level": "5", "keywords": [ "calculus", "derivative", "local maxima and minima", "maxima", "minima", "modeling" ], "problem_v1": "A woman pulls a sled which, together with its load, has a mass of $m$ kg. Let $g$ be the acceleration due to gravity. If her arm makes an angle of $\\theta$ with her body (assumed vertical) and the coefficient of friction (a positive constant) is $\\mu$, the least force, $F$, she must exert to move the sled is given by F={mg\\mu\\over\\sin\\theta+\\mu\\cos\\theta}. If $\\mu=0.55$, find the maximum and minimum values of $F$ for $0\\leq\\theta\\leq\\pi/2$. maximum force=[ANS] Newtons minimum force=[ANS] Newtons", "answer_v1": [ "m*g", "0.55*m*g/[sqrt(1+0.55*0.55)]" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "A woman pulls a sled which, together with its load, has a mass of $m$ kg. Let $g$ be the acceleration due to gravity. If her arm makes an angle of $\\theta$ with her body (assumed vertical) and the coefficient of friction (a positive constant) is $\\mu$, the least force, $F$, she must exert to move the sled is given by F={mg\\mu\\over\\sin\\theta+\\mu\\cos\\theta}. If $\\mu=0.1$, find the maximum and minimum values of $F$ for $0\\leq\\theta\\leq\\pi/2$. maximum force=[ANS] Newtons minimum force=[ANS] Newtons", "answer_v2": [ "m*g", "0.1*m*g/[sqrt(1+0.1*0.1)]" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "A woman pulls a sled which, together with its load, has a mass of $m$ kg. Let $g$ be the acceleration due to gravity. If her arm makes an angle of $\\theta$ with her body (assumed vertical) and the coefficient of friction (a positive constant) is $\\mu$, the least force, $F$, she must exert to move the sled is given by F={mg\\mu\\over\\sin\\theta+\\mu\\cos\\theta}. If $\\mu=0.25$, find the maximum and minimum values of $F$ for $0\\leq\\theta\\leq\\pi/2$. maximum force=[ANS] Newtons minimum force=[ANS] Newtons", "answer_v3": [ "m*g", "0.25*m*g/[sqrt(1+0.25*0.25)]" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0427", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - engineering and physics", "level": "2", "keywords": [ "calculus", "maximum" ], "problem_v1": "A ball is thrown up on the surface of a moon. Its height above the lunar surface (in feet) after $t$ seconds is given by the formula h=559 t-\\frac{13}{5}t^2. Find the time that the ball reaches its maximum height. Answer=[ANS]\nFind the maximal height attained by the ball Answer=[ANS]", "answer_v1": [ "107.5", "30046.25" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A ball is thrown up on the surface of a moon. Its height above the lunar surface (in feet) after $t$ seconds is given by the formula h=198 t-\\frac{9}{7}t^2. Find the time that the ball reaches its maximum height. Answer=[ANS]\nFind the maximal height attained by the ball Answer=[ANS]", "answer_v2": [ "77", "7623" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A ball is thrown up on the surface of a moon. Its height above the lunar surface (in feet) after $t$ seconds is given by the formula h=290 t-\\frac{10}{6}t^2. Find the time that the ball reaches its maximum height. Answer=[ANS]\nFind the maximal height attained by the ball Answer=[ANS]", "answer_v3": [ "87", "12615" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0428", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Optimization - natural and social sciences", "level": "5", "keywords": [ "calculus", "derivative", "local maxima and minima", "maxima", "minima", "modeling" ], "problem_v1": "A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks $d$ miles apart, the concentration of the combined deposits on the line joining them, at a distance $x$ from one stack, is given by S=\\frac{c}{x^2}+\\frac{k}{(d-x)^2} where $c$ and $k$ are positive constants which depend on the quantity of smoke each stack is emitting. If $k=8 c$, find the point on the line joining the stacks where the concentration of the deposit is a minimum. $x_{\\hbox{min}}$=[ANS] mi", "answer_v1": [ "d/(1 + 8^{\\frac{1}{3}})" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks $d$ miles apart, the concentration of the combined deposits on the line joining them, at a distance $x$ from one stack, is given by S=\\frac{c}{x^2}+\\frac{k}{(d-x)^2} where $c$ and $k$ are positive constants which depend on the quantity of smoke each stack is emitting. If $k=2 c$, find the point on the line joining the stacks where the concentration of the deposit is a minimum. $x_{\\hbox{min}}$=[ANS] mi", "answer_v2": [ "d/(1 + 2^{\\frac{1}{3}})" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A smokestack deposits soot on the ground with a concentration inversely proportional to the square of the distance from the stack. With two smokestacks $d$ miles apart, the concentration of the combined deposits on the line joining them, at a distance $x$ from one stack, is given by S=\\frac{c}{x^2}+\\frac{k}{(d-x)^2} where $c$ and $k$ are positive constants which depend on the quantity of smoke each stack is emitting. If $k=4 c$, find the point on the line joining the stacks where the concentration of the deposit is a minimum. $x_{\\hbox{min}}$=[ANS] mi", "answer_v3": [ "d/(1 + 4^{\\frac{1}{3}})" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0429", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "derivatives", "linear approximation", "trigonometric functions", "Calculus", "Differentiation", "Product", "Quotient" ], "problem_v1": "The linearization at $a=0$ to $\\sin (8x)$ is $A+B x$. Compute $A$ and $B$.\n$A=$ [ANS]\n$B=$ [ANS]", "answer_v1": [ "0", "8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The linearization at $a=0$ to $\\sin (2x)$ is $A+B x$. Compute $A$ and $B$.\n$A=$ [ANS]\n$B=$ [ANS]", "answer_v2": [ "0", "2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The linearization at $a=0$ to $\\sin (4x)$ is $A+B x$. Compute $A$ and $B$.\n$A=$ [ANS]\n$B=$ [ANS]", "answer_v3": [ "0", "4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0430", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "derivatives", "linear approximation", "calculus", "derivative", "linear approximation", "tangent line" ], "problem_v1": "Use linear approximation to approximate $\\sqrt {64.3}$ as follows.\nLet $f(x)=\\sqrt x$. The equation of the tangent line to $f(x)$ at $x=64$ can be written in the form $y=mx+b$. Compute $m$ and $b$.\n$m=$ [ANS]\n$b=$ [ANS]\nUsing this find the approximation for $\\sqrt {64.3}$.\nAnswer: [ANS]", "answer_v1": [ "0.5*64^{-0.5}", "8 -0.5*64^{-0.5}*64", "0.5*64^{-0.5}*64.3 + 8 - (0.5*64^{-0.5}*64)" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Use linear approximation to approximate $\\sqrt {16.4}$ as follows.\nLet $f(x)=\\sqrt x$. The equation of the tangent line to $f(x)$ at $x=16$ can be written in the form $y=mx+b$. Compute $m$ and $b$.\n$m=$ [ANS]\n$b=$ [ANS]\nUsing this find the approximation for $\\sqrt {16.4}$.\nAnswer: [ANS]", "answer_v2": [ "0.5*16^{-0.5}", "4 -0.5*16^{-0.5}*16", "0.5*16^{-0.5}*16.4 + 4 - (0.5*16^{-0.5}*16)" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Use linear approximation to approximate $\\sqrt {25.3}$ as follows.\nLet $f(x)=\\sqrt x$. The equation of the tangent line to $f(x)$ at $x=25$ can be written in the form $y=mx+b$. Compute $m$ and $b$.\n$m=$ [ANS]\n$b=$ [ANS]\nUsing this find the approximation for $\\sqrt {25.3}$.\nAnswer: [ANS]", "answer_v3": [ "0.5*25^{-0.5}", "5 -0.5*25^{-0.5}*25", "0.5*25^{-0.5}*25.3 + 5 - (0.5*25^{-0.5}*25)" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0431", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "4", "keywords": [ "derivatives", "linear approximation", "Differentiation", "Product", "Quotient" ], "problem_v1": "Use the differential (i.e., linear approximation) to approximate the number $\\ln(1.04)$.\nAnswer: [ANS]", "answer_v1": [ "4/100" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the differential (i.e., linear approximation) to approximate the number $\\ln(1.01)$.\nAnswer: [ANS]", "answer_v2": [ "1/100" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the differential (i.e., linear approximation) to approximate the number $\\ln(1.02)$.\nAnswer: [ANS]", "answer_v3": [ "2/100" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0432", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [ "derivatives", "antiderivatives", "tangent line" ], "problem_v1": "Given that the graph of $f(x)$ passes through the point $(8, 7)$ and that the slope of its tangent line at $(x,f(x))$ is $5x+6$, what is $f(3)$?\nAnswer: [ANS]", "answer_v1": [ "5 * (3)^2/2 + 6 * 3 + 7 - (5*8^2/2) - (6*8)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given that the graph of $f(x)$ passes through the point $(2, 10)$ and that the slope of its tangent line at $(x,f(x))$ is $2x+4$, what is $f(5)$?\nAnswer: [ANS]", "answer_v2": [ "2 * (5)^2/2 + 4 * 5 + 10 - (2*2^2/2) - (4*2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given that the graph of $f(x)$ passes through the point $(4, 7)$ and that the slope of its tangent line at $(x,f(x))$ is $3x+5$, what is $f(2)$?\nAnswer: [ANS]", "answer_v3": [ "3 * (2)^2/2 + 5 * 2 + 7 - (3*4^2/2) - (5*4)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0433", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [], "problem_v1": "Given that $P(110)=55$ and $P'(110)=13$, approximate $P(107)$. $P(107) \\approx$ [ANS]", "answer_v1": [ "16" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given that $P(83)=76$ and $P'(83)=-3$, approximate $P(74)$. $P(74) \\approx$ [ANS]", "answer_v2": [ "103" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given that $P(92)=56$ and $P'(92)=6$, approximate $P(90)$. $P(90) \\approx$ [ANS]", "answer_v3": [ "44" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0434", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "5", "keywords": [], "problem_v1": "Given that $H(15)=29$ and $H(18)=32$, approximate $H'(15)$. $H'(15) \\approx$ [ANS]", "answer_v1": [ "1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given that $H(1)=14$ and $H(6)=20$, approximate $H'(1)$. $H'(1) \\approx$ [ANS]", "answer_v2": [ "1.2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given that $H(6)=18$ and $H(10)=27$, approximate $H'(6)$. $H'(6) \\approx$ [ANS]", "answer_v3": [ "2.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0435", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "5", "keywords": [], "problem_v1": "Given that $f(5.1)=2.4$ and $f(5.4)=4.5$, approximate $f'(5.1)$. $f'(5.1) \\approx$ [ANS]", "answer_v1": [ "7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Given that $f(-8.4)=-7$ and $f(-7.9)=-3.3$, approximate $f'(-8.4)$. $f'(-8.4) \\approx$ [ANS]", "answer_v2": [ "7.4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Given that $f(-3.7)=-4.4$ and $f(-3.3)=1$, approximate $f'(-3.7)$. $f'(-3.7) \\approx$ [ANS]", "answer_v3": [ "13.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0436", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [], "problem_v1": "Compute the differential $dy$ for $y=16^x \\ln x$. $dy=$ [ANS]", "answer_v1": [ "((2.77259*16^x*ln(x)+16^x*1/x))dx" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Compute the differential $dy$ for $y=3^x \\ln x$. $dy=$ [ANS]", "answer_v2": [ "((1.09861*3^x*ln(x)+3^x*1/x))dx" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Compute the differential $dy$ for $y=7^x \\ln x$. $dy=$ [ANS]", "answer_v3": [ "((1.94591*7^x*ln(x)+7^x*1/x))dx" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0437", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [], "problem_v1": "Compute the differential $dy$ for $y=7 \\cos(\\sin x)$. $dy=$ [ANS]", "answer_v1": [ "((-7*sin(sin(x))*cos(x)))dx" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Compute the differential $dy$ for $y=2 \\cos(\\sin x)$. $dy=$ [ANS]", "answer_v2": [ "((-2*sin(sin(x))*cos(x)))dx" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Compute the differential $dy$ for $y=4 \\cos(\\sin x)$. $dy=$ [ANS]", "answer_v3": [ "((-4*sin(sin(x))*cos(x)))dx" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0438", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [], "problem_v1": "Use differentials to approximate the given values by hand. Provide 4 decimal digits of accuracy. (Note: Only use a calculator for routine arithmetic (addition, subtraction, multiplication, division) as part of the approximation by differentials. If you use your calculator to obtain a result directly (i.e., taking powers or roots), then your answer may be different than the requested approximation.)\n$\\sqrt{64.06} \\approx$ [ANS]\n$\\sqrt{63.96} \\approx$ [ANS]\n$\\sqrt[3]{64.03} \\approx$ [ANS]\n$\\sqrt[3]{63.93} \\approx$ [ANS]", "answer_v1": [ "8.00375", "7.9975", "4.000625", "3.99854166666667" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Use differentials to approximate the given values by hand. Provide 4 decimal digits of accuracy. (Note: Only use a calculator for routine arithmetic (addition, subtraction, multiplication, division) as part of the approximation by differentials. If you use your calculator to obtain a result directly (i.e., taking powers or roots), then your answer may be different than the requested approximation.)\n$\\sqrt{4.09} \\approx$ [ANS]\n$\\sqrt{3.92} \\approx$ [ANS]\n$\\sqrt[3]{27.09} \\approx$ [ANS]\n$\\sqrt[3]{26.93} \\approx$ [ANS]", "answer_v2": [ "2.0225", "1.98", "3.00333333333333", "2.99740740740741" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Use differentials to approximate the given values by hand. Provide 4 decimal digits of accuracy. (Note: Only use a calculator for routine arithmetic (addition, subtraction, multiplication, division) as part of the approximation by differentials. If you use your calculator to obtain a result directly (i.e., taking powers or roots), then your answer may be different than the requested approximation.)\n$\\sqrt{16.06} \\approx$ [ANS]\n$\\sqrt{15.93} \\approx$ [ANS]\n$\\sqrt[3]{64.02} \\approx$ [ANS]\n$\\sqrt[3]{63.94} \\approx$ [ANS]", "answer_v3": [ "4.0075", "3.99125", "4.00041666666667", "3.99875" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0439", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "4", "keywords": [], "problem_v1": "A set of plastic spheres are to be made with a diameter of 16 cm. If the manufacturing process is accurate to 2 mm, what is the propagated error in volume of the spheres? Error=$\\pm$ [ANS] cm 3 3", "answer_v1": [ "80.4248" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A set of plastic spheres are to be made with a diameter of 2 cm. If the manufacturing process is accurate to 3 mm, what is the propagated error in volume of the spheres? Error=$\\pm$ [ANS] cm 3 3", "answer_v2": [ "1.88496" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A set of plastic spheres are to be made with a diameter of 7 cm. If the manufacturing process is accurate to 2 mm, what is the propagated error in volume of the spheres? Error=$\\pm$ [ANS] cm 3 3", "answer_v3": [ "15.3938" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0440", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "5", "keywords": [], "problem_v1": "The distance, in feet, a stone drops in $t$ seconds is given by $d(t)=16t^2$. The depth of a hole is to be approximated by dropping a rock and listening for it to hit the bottom. Assume that time measurement is accurate to 3/10 th of a second. Use a linear approximation to estimate an upper bound for the propagated error if the measured time is: a) 3 seconds. [ANS] feet b) 8 seconds. [ANS] feet", "answer_v1": [ "28.8", "76.8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The distance, in feet, a stone drops in $t$ seconds is given by $d(t)=16t^2$. The depth of a hole is to be approximated by dropping a rock and listening for it to hit the bottom. Assume that time measurement is accurate to 1/10 th of a second. Use a linear approximation to estimate an upper bound for the propagated error if the measured time is: a) 4 seconds. [ANS] feet b) 5 seconds. [ANS] feet", "answer_v2": [ "12.8", "16" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The distance, in feet, a stone drops in $t$ seconds is given by $d(t)=16t^2$. The depth of a hole is to be approximated by dropping a rock and listening for it to hit the bottom. Assume that time measurement is accurate to 1/10 th of a second. Use a linear approximation to estimate an upper bound for the propagated error if the measured time is: a) 3 seconds. [ANS] feet b) 6 seconds. [ANS] feet", "answer_v3": [ "9.6", "19.2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0441", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "differentials", "percentage", "relative" ], "problem_v1": "According to Poiseuille's Law, the volume of blood per unit time that flows past a given position in a blood vessel is proportional to the fourth power of the radius of the vessel. In other words, $F=kr^4$, where $F$ is the flux, $r$ is the radius, and $k$ is a constant. If $r$ changes by 7 percent, use differentials to approximate the percentage change in $F$.\nAnswer: [ANS] \\%", "answer_v1": [ "28" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "According to Poiseuille's Law, the volume of blood per unit time that flows past a given position in a blood vessel is proportional to the fourth power of the radius of the vessel. In other words, $F=kr^4$, where $F$ is the flux, $r$ is the radius, and $k$ is a constant. If $r$ changes by 2 percent, use differentials to approximate the percentage change in $F$.\nAnswer: [ANS] \\%", "answer_v2": [ "8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "According to Poiseuille's Law, the volume of blood per unit time that flows past a given position in a blood vessel is proportional to the fourth power of the radius of the vessel. In other words, $F=kr^4$, where $F$ is the flux, $r$ is the radius, and $k$ is a constant. If $r$ changes by 4 percent, use differentials to approximate the percentage change in $F$.\nAnswer: [ANS] \\%", "answer_v3": [ "16" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0442", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "calculus", "derivatives", "linearization" ], "problem_v1": "Find the linearization $L(x)$ of $y=e^{9x}\\ln(x)$ at $a=1$. $L(x)$=[ANS]", "answer_v1": [ "8103.08*(x-1)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the linearization $L(x)$ of $y=e^{3x}\\ln(x)$ at $a=1$. $L(x)$=[ANS]", "answer_v2": [ "20.0855*(x-1)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the linearization $L(x)$ of $y=e^{5x}\\ln(x)$ at $a=1$. $L(x)$=[ANS]", "answer_v3": [ "148.413*(x-1)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0443", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "calculus", "derivatives", "linearization", "Differentiation", "Product", "Quotient" ], "problem_v1": "$y=(11+x)^{-1/2}, \\qquad a={14}$ Find the Linearization at $x=a$. $L(x)=$ [ANS]", "answer_v1": [ "0.2+(-0.004)*(x-14)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "$y=(1+x)^{-1/2}, \\qquad a={3}$ Find the Linearization at $x=a$. $L(x)=$ [ANS]", "answer_v2": [ "0.5+(-0.0625)*(x-3)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "$y=(4+x)^{-1/2}, \\qquad a={5}$ Find the Linearization at $x=a$. $L(x)=$ [ANS]", "answer_v3": [ "0.333333+(-0.0185185)*(x-5)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0444", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "calculus", "derivatives", "linear approximation" ], "problem_v1": "Estimate ${\\Delta}f$ using the Linear Approximation and use a calculator to compute both the error and the percentage error. $f(x)=\\sqrt{12+x}.{\\qquad}a={13}.{\\qquad}{\\Delta}x={0.1}$ $\\Delta{f} \\approx$ [ANS]\nWith these calculations, we have determined that the square root of [ANS] is approximately [ANS]\nThe error in Linear Approximation is: [ANS]\nThe error in percentage terms is: [ANS] \\%", "answer_v1": [ "0.01", "25.1", "5.01", "9.9800498605182E-06", "0.099900199502393" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Estimate ${\\Delta}f$ using the Linear Approximation and use a calculator to compute both the error and the percentage error. $f(x)=\\sqrt{3+x}.{\\qquad}a={1}.{\\qquad}{\\Delta}x={-1}$ $\\Delta{f} \\approx$ [ANS]\nWith these calculations, we have determined that the square root of [ANS] is approximately [ANS]\nThe error in Linear Approximation is: [ANS]\nThe error in percentage terms is: [ANS] \\%", "answer_v2": [ "-0.25", "3", "1.75", "0.0179491924311228", "6.6987298107781" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Estimate ${\\Delta}f$ using the Linear Approximation and use a calculator to compute both the error and the percentage error. $f(x)=\\sqrt{5+x}.{\\qquad}a={4}.{\\qquad}{\\Delta}x={-0.5}$ $\\Delta{f} \\approx$ [ANS]\nWith these calculations, we have determined that the square root of [ANS] is approximately [ANS]\nThe error in Linear Approximation is: [ANS]\nThe error in percentage terms is: [ANS] \\%", "answer_v3": [ "-0.0833333333333333", "8.5", "2.91666666666667", "0.0011907192440163", "1.40873420962234" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0445", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "Calculus", "Derivatives", "linear approximation", "trigonometric functions" ], "problem_v1": "Find the linear approximation at $x=0$ to $\\frac{1}{\\sqrt{8-x}}$. Write your answer in the form $y=Ax+B$. [ANS]", "answer_v1": [ "y-0.0220971*x = 0.353553" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Find the linear approximation at $x=0$ to $\\frac{1}{\\sqrt{2-x}}$. Write your answer in the form $y=Ax+B$. [ANS]", "answer_v2": [ "y-0.176777*x = 0.707107" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Find the linear approximation at $x=0$ to $\\frac{1}{\\sqrt{4-x}}$. Write your answer in the form $y=Ax+B$. [ANS]", "answer_v3": [ "y-0.0625*x = 0.5" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0446", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "5", "keywords": [], "problem_v1": "Let $f(t)$ be the weight (in grams) of a solid sitting in a beaker of water. Suppose that the solid dissolves in such a way that the rate of change (in grams/minute) of the weight of the solid at any time $t$ can be determined from the weight using the formula: f'(t)=-2\\, f(t)\\, (4+f(t)) If there is 4 grams of solid at time $t=2$ estimate the amount of solid 1 second later. [ANS]", "answer_v1": [ "4+-2*4*(4+4)*1/60" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(t)$ be the weight (in grams) of a solid sitting in a beaker of water. Suppose that the solid dissolves in such a way that the rate of change (in grams/minute) of the weight of the solid at any time $t$ can be determined from the weight using the formula: f'(t)=-5\\, f(t)\\, (6+f(t)) If there is 1 grams of solid at time $t=2$ estimate the amount of solid 1 second later. [ANS]", "answer_v2": [ "1+-5*1*(6+1)*1/60" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(t)$ be the weight (in grams) of a solid sitting in a beaker of water. Suppose that the solid dissolves in such a way that the rate of change (in grams/minute) of the weight of the solid at any time $t$ can be determined from the weight using the formula: f'(t)=-4\\, f(t)\\, (4+f(t)) If there is 2 grams of solid at time $t=2$ estimate the amount of solid 1 second later. [ANS]", "answer_v3": [ "2+-4*2*(4+2)*1/60" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0447", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "5", "keywords": [ "derivatives", "differentials", "Calculus" ], "problem_v1": "Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint $0.080000$ cm thick to a hemispherical dome with a diameter of $60.000$ meters. [ANS]", "answer_v1": [ "2*pi*3000^2*0.08" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint $0.010000$ cm thick to a hemispherical dome with a diameter of $80.000$ meters. [ANS]", "answer_v2": [ "2*pi*4000^2*0.01" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use linear approximation to estimate the amount of paint in cubic centimeters needed to apply a coat of paint $0.040000$ cm thick to a hemispherical dome with a diameter of $60.000$ meters. [ANS]", "answer_v3": [ "2*pi*3000^2*0.04" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0448", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "4", "keywords": [ "derivatives", "differentials", "linear approximation", "Calculus" ], "problem_v1": "Suppose that you can calculate the derivative of a function using the formula $f'(x)=5f(x)+4x$. If the output value of the function at $x=3$ is $5$ estimate the value of the function at $3.008$. [ANS]", "answer_v1": [ "5.296" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that you can calculate the derivative of a function using the formula $f'(x)=2f(x)+6x$. If the output value of the function at $x=1$ is $3$ estimate the value of the function at $1.014$. [ANS]", "answer_v2": [ "3.168" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that you can calculate the derivative of a function using the formula $f'(x)=3f(x)+4x$. If the output value of the function at $x=2$ is $4$ estimate the value of the function at $2.007$. [ANS]", "answer_v3": [ "4.14" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0449", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "Calculus", "Derivatives", "differentials" ], "problem_v1": "Let $y=5 \\sqrt x$. Find the change in $y$, $\\Delta y$ when $x=3$ and $\\Delta x=0.3$ [ANS]\nFind the differential $dy$ when $x=3$ and $dx=0.3$ [ANS]", "answer_v1": [ "0.422697024448089", "0.433012701892219" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $y=2 \\sqrt x$. Find the change in $y$, $\\Delta y$ when $x=5$ and $\\Delta x=0.1$ [ANS]\nFind the differential $dy$ when $x=5$ and $dx=0.1$ [ANS]", "answer_v2": [ "0.044499961254906", "0.0447213595499958" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $y=3 \\sqrt x$. Find the change in $y$, $\\Delta y$ when $x=4$ and $\\Delta x=0.2$ [ANS]\nFind the differential $dy$ when $x=4$ and $dx=0.2$ [ANS]", "answer_v3": [ "0.14817045957576", "0.15" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0450", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "4", "keywords": [ "derivatives", "differentials", "Calculus", "derivatives", "Differentiation", "Product", "Quotient" ], "problem_v1": "The circumference of a sphere was measured to be $85.000$ cm with a possible error of $0.50000$ cm. Use linear approximation to estimate the maximum error in the calculated surface area. [ANS]\nEstimate the relative error in the calculated surface area. [ANS]", "answer_v1": [ "2*85*0.5/pi", "2*85*0.5/pi/(4*pi*[85/(2*pi)]^2)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The circumference of a sphere was measured to be $71.000$ cm with a possible error of $0.50000$ cm. Use linear approximation to estimate the maximum error in the calculated surface area. [ANS]\nEstimate the relative error in the calculated surface area. [ANS]", "answer_v2": [ "2*71*0.5/pi", "2*71*0.5/pi/(4*pi*[71/(2*pi)]^2)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The circumference of a sphere was measured to be $76.000$ cm with a possible error of $0.50000$ cm. Use linear approximation to estimate the maximum error in the calculated surface area. [ANS]\nEstimate the relative error in the calculated surface area. [ANS]", "answer_v3": [ "2*76*0.5/pi", "2*76*0.5/pi/(4*pi*[76/(2*pi)]^2)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0451", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [], "problem_v1": "Find the linear approximation of $f(x)=\\ln x$ at $x=1$ and use it to estimate $\\ln(1.37)$. $L(x)=$ [ANS]\n$\\ln 1.37 \\approx$ [ANS]", "answer_v1": [ "x-1", "0.37" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the linear approximation of $f(x)=\\ln x$ at $x=1$ and use it to estimate $\\ln(1.05)$. $L(x)=$ [ANS]\n$\\ln 1.05 \\approx$ [ANS]", "answer_v2": [ "x-1", "0.05" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the linear approximation of $f(x)=\\ln x$ at $x=1$ and use it to estimate $\\ln(1.16)$. $L(x)=$ [ANS]\n$\\ln 1.16 \\approx$ [ANS]", "answer_v3": [ "x-1", "0.16" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0452", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "derivatives", "linear approximation", "Calculus", "derivative", "linear approximation", "tangent line" ], "problem_v1": "Use linear approximation, i.e. the tangent line, to approximate $\\frac 1{0.503}$ as follows: Let $f(x)=\\frac 1x$ and find the equation of the tangent line to $f(x)$ at a \"nice\" point near $0.503$. Then use this to approximate $\\frac 1{0.503}$. [ANS]", "answer_v1": [ "1.988" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use linear approximation, i.e. the tangent line, to approximate $\\frac 1{0.104}$ as follows: Let $f(x)=\\frac 1x$ and find the equation of the tangent line to $f(x)$ at a \"nice\" point near $0.104$. Then use this to approximate $\\frac 1{0.104}$. [ANS]", "answer_v2": [ "9.6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use linear approximation, i.e. the tangent line, to approximate $\\frac 1{0.203}$ as follows: Let $f(x)=\\frac 1x$ and find the equation of the tangent line to $f(x)$ at a \"nice\" point near $0.203$. Then use this to approximate $\\frac 1{0.203}$. [ANS]", "answer_v3": [ "4.925" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0453", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "4", "keywords": [ "calculus", "derivative", "local maxima and minima", "maxima", "minima", "marginal cost", "marginal profit" ], "problem_v1": "When production is 4000, marginal revenue is 6 dollars per unit and marginal cost is 6.25 dollars per unit. Do you expect maximum profit to occur at a production level above or below 4000? [ANS] If production is increased by 80 units, what would you estimate the change in profit would be? $d\\pi \\approx$ [ANS] dollars", "answer_v1": [ "below", "80*(6-6.25)" ], "answer_type_v1": [ "MCS", "NV" ], "options_v1": [ [ "above", "below" ], [] ], "problem_v2": "When production is 1300, marginal revenue is 7.75 dollars per unit and marginal cost is 3.75 dollars per unit. Do you expect maximum profit to occur at a production level above or below 1300? [ANS] If production is increased by 40 units, what would you estimate the change in profit would be? $d\\pi \\approx$ [ANS] dollars", "answer_v2": [ "above", "40*(7.75-3.75)" ], "answer_type_v2": [ "MCS", "NV" ], "options_v2": [ [ "above", "below" ], [] ], "problem_v3": "When production is 2200, marginal revenue is 6 dollars per unit and marginal cost is 4.25 dollars per unit. Do you expect maximum profit to occur at a production level above or below 2200? [ANS] If production is increased by 60 units, what would you estimate the change in profit would be? $d\\pi \\approx$ [ANS] dollars", "answer_v3": [ "above", "60*(6-4.25)" ], "answer_type_v3": [ "MCS", "NV" ], "options_v3": [ [ "above", "below" ], [] ] }, { "id": "Calculus_-_single_variable_0454", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "calculus", "derivative", "linear approximation", "tangent line", "approximation" ], "problem_v1": "Find the tangent line approximation for $\\sqrt{7+x}$ near $x=3$. $y=$ [ANS]", "answer_v1": [ "sqrt(7+3)+(x-3)/[2*sqrt(7+3)]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the tangent line approximation for $\\sqrt{1+x}$ near $x=5$. $y=$ [ANS]", "answer_v2": [ "sqrt(1+5)+(x-5)/[2*sqrt(1+5)]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the tangent line approximation for $\\sqrt{3+x}$ near $x=3$. $y=$ [ANS]", "answer_v3": [ "sqrt(3+3)+(x-3)/[2*sqrt(3+3)]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0455", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "4", "keywords": [ "calculus", "derivative", "linear approximation", "tangent line", "approximation" ], "problem_v1": "The acceleration due to gravity, $g$, is given by g=\\frac{G M}{r^2}, where $M$ is the mass of the Earth, $r$ is the distance from the center of the Earth, and $G$ is the uniform gravitational constant.\n(a) Suppose that we increase from our distance from the center of the Earth by a distance $\\Delta r=x$. Use a linear approximation to find an approximation to the resulting change in $g$, as a fraction of the original acceleration: $\\Delta g \\approx$ $g \\times$ [ANS]\n(Your answer will be a function of $x$ and $r$.) (b) Is this change positive or negative? $\\Delta g$ is [ANS] (Think about what this tells you about the acceleration due to gravity.) (Think about what this tells you about the acceleration due to gravity.) (c) What is the percentage change in $g$ when moving from sea level to the top of Mount Elbert (a mountain over 14,000 feet tall in Colorado; in km, its height is 4.29 km; assume the radius of the Earth is 6400 km)? percent change=[ANS]", "answer_v1": [ "-2*x/r", "negative", "2*4.29/64" ], "answer_type_v1": [ "EX", "MCS", "NV" ], "options_v1": [ [], [ "positive", "negative" ], [] ], "problem_v2": "The acceleration due to gravity, $g$, is given by g=\\frac{G M}{r^2}, where $M$ is the mass of the Earth, $r$ is the distance from the center of the Earth, and $G$ is the uniform gravitational constant.\n(a) Suppose that we increase from our distance from the center of the Earth by a distance $\\Delta r=x$. Use a linear approximation to find an approximation to the resulting change in $g$, as a fraction of the original acceleration: $\\Delta g \\approx$ $g \\times$ [ANS]\n(Your answer will be a function of $x$ and $r$.) (b) Is this change positive or negative? $\\Delta g$ is [ANS] (Think about what this tells you about the acceleration due to gravity.) (Think about what this tells you about the acceleration due to gravity.) (c) What is the percentage change in $g$ when moving from sea level to the top of Mount Elbert (a mountain over 14,000 feet tall in Colorado; in km, its height is 4.4 km; assume the radius of the Earth is 6400 km)? percent change=[ANS]", "answer_v2": [ "-2*x/r", "negative", "2*4.4/64" ], "answer_type_v2": [ "EX", "MCS", "NV" ], "options_v2": [ [], [ "positive", "negative" ], [] ], "problem_v3": "The acceleration due to gravity, $g$, is given by g=\\frac{G M}{r^2}, where $M$ is the mass of the Earth, $r$ is the distance from the center of the Earth, and $G$ is the uniform gravitational constant.\n(a) Suppose that we increase from our distance from the center of the Earth by a distance $\\Delta r=x$. Use a linear approximation to find an approximation to the resulting change in $g$, as a fraction of the original acceleration: $\\Delta g \\approx$ $g \\times$ [ANS]\n(Your answer will be a function of $x$ and $r$.) (b) Is this change positive or negative? $\\Delta g$ is [ANS] (Think about what this tells you about the acceleration due to gravity.) (Think about what this tells you about the acceleration due to gravity.) (c) What is the percentage change in $g$ when moving from sea level to the top of Mount Elbert (a mountain over 14,000 feet tall in Colorado; in km, its height is 4.35 km; assume the radius of the Earth is 6400 km)? percent change=[ANS]", "answer_v3": [ "-2*x/r", "negative", "2*4.35/64" ], "answer_type_v3": [ "EX", "MCS", "NV" ], "options_v3": [ [], [ "positive", "negative" ], [] ] }, { "id": "Calculus_-_single_variable_0456", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "4", "keywords": [ "calculus", "derivative", "linear approximation", "tangent line", "approximation" ], "problem_v1": "Find a formula for the error $E(x)$ in the tangent line approximation to the function $f(x)=e^{x-3}$ near $x=a=3$. $E(x)=$ [ANS]\nThen fill in the following table of values for $E(x)/(x-a)$ near $x=a$\n$\\begin{array}{cccc}\\hline x=& 3.1 & 3.01 & 3.001 \\\\ \\hline \\frac{E(x)}{(x-a)}=& [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nUsing your table, find a value of $k$ such that $E(x)/(x-a) \\approx k(x-a)$. $k=$ [ANS]\n(Check that, approximately, $k=f''(a)/2$ and that $E(x) \\approx (f''(a)/2)(x-a)^2$.", "answer_v1": [ "e^{x-3}-1-(x-3)", "0.05", "0.005", "0.0005", "0.5" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Find a formula for the error $E(x)$ in the tangent line approximation to the function $f(x)=\\ln(x-5)$ near $x=a=6$. $E(x)=$ [ANS]\nThen fill in the following table of values for $E(x)/(x-a)$ near $x=a$\n$\\begin{array}{cccc}\\hline x=& 6.1 & 6.01 & 6.001 \\\\ \\hline \\frac{E(x)}{(x-a)}=& [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nUsing your table, find a value of $k$ such that $E(x)/(x-a) \\approx k(x-a)$. $k=$ [ANS]\n(Check that, approximately, $k=f''(a)/2$ and that $E(x) \\approx (f''(a)/2)(x-a)^2$.", "answer_v2": [ "ln(x-5)-(x-6)", "-0.047", "-0.005", "-0.0005", "-0.5" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Find a formula for the error $E(x)$ in the tangent line approximation to the function $f(x)=\\ln(x-3)$ near $x=a=4$. $E(x)=$ [ANS]\nThen fill in the following table of values for $E(x)/(x-a)$ near $x=a$\n$\\begin{array}{cccc}\\hline x=& 4.1 & 4.01 & 4.001 \\\\ \\hline \\frac{E(x)}{(x-a)}=& [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\nUsing your table, find a value of $k$ such that $E(x)/(x-a) \\approx k(x-a)$. $k=$ [ANS]\n(Check that, approximately, $k=f''(a)/2$ and that $E(x) \\approx (f''(a)/2)(x-a)^2$.", "answer_v3": [ "ln(x-3)-(x-4)", "-0.047", "-0.005", "-0.0005", "-0.5" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0457", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "calculus", "derivative", "linear approximation", "tangent line", "approximation", "Differentiation", "Product", "Quotient" ], "problem_v1": "Near $x=0$, the tangent line approximation gives $6 e^{-7x} \\approx$ [ANS].", "answer_v1": [ "6-6*7*x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Near $x=0$, the tangent line approximation gives $9 e^{-x} \\approx$ [ANS].", "answer_v2": [ "9-9*1*x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Near $x=0$, the tangent line approximation gives $6 e^{-3x} \\approx$ [ANS].", "answer_v3": [ "6-6*3*x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0458", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [ "calculus", "derivative", "instantaneous velocity", "rate of change" ], "problem_v1": "Suppose that $f(x)$ is a function with $f(140)=61$ and $f'(140)=6$. Estimate $f(142.5)$. $f(142.5)=$ [ANS]", "answer_v1": [ "76" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that $f(x)$ is a function with $f(90)=82$ and $f'(90)=2$. Estimate $f(88.5)$. $f(88.5)=$ [ANS]", "answer_v2": [ "79" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that $f(x)$ is a function with $f(105)=61$ and $f'(105)=3$. Estimate $f(105.5)$. $f(105.5)=$ [ANS]", "answer_v3": [ "62.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0459", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [ "derivative", "derivatives", "linear approximation", "differentials" ], "problem_v1": "Use the formula f(x)\\approx f(x_0)+f'(x_0)(x-x_0) to obtain the local linear approximation $(\\small{y})$ of $\\small{\\frac{4}{x^{4}}}$ at $\\small{x_0=1}$. $y \\approx$ [ANS]", "answer_v1": [ "20-16*x" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the formula f(x)\\approx f(x_0)+f'(x_0)(x-x_0) to obtain the local linear approximation $(\\small{y})$ of $\\small{-\\frac{7}{x^{6}}}$ at $\\small{x_0=-4}$. $y \\approx$ [ANS]", "answer_v2": [ "-49/4096+-21/8192*x" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the formula f(x)\\approx f(x_0)+f'(x_0)(x-x_0) to obtain the local linear approximation $(\\small{y})$ of $\\small{-\\frac{3}{x^{4}}}$ at $\\small{x_0=-2}$. $y \\approx$ [ANS]", "answer_v3": [ "-15/16+-3/8*x" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0460", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [ "derivative", "derivatives", "linear approximation", "differentials" ], "problem_v1": "Use an appropriate local linear approximation to estimate the value of the given quantity. $\\tan{(0.1)} \\approx$ [ANS]", "answer_v1": [ "0.1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use an appropriate local linear approximation to estimate the value of the given quantity. $\\tan{(-0.17)} \\approx$ [ANS]", "answer_v2": [ "-0.17" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use an appropriate local linear approximation to estimate the value of the given quantity. $\\tan{(-0.08)} \\approx$ [ANS]", "answer_v3": [ "-0.08" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0461", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "derivative", "derivatives", "linear approximation", "differentials" ], "problem_v1": "Use an appropriate local linear approximation to estimate the value of $\\cos(46^{\\circ})$. $\\cos(46^{\\circ}) \\approx$ [ANS]", "answer_v1": [ "0.694765" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use an appropriate local linear approximation to estimate the value of $\\sin(31^{\\circ})$. $\\sin(31^{\\circ}) \\approx$ [ANS]", "answer_v2": [ "0.515115" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use an appropriate local linear approximation to estimate the value of $\\sin(46^{\\circ})$. $\\sin(46^{\\circ}) \\approx$ [ANS]", "answer_v3": [ "0.719448" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0462", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [ "derivative", "derivatives", "linear approximation", "differentials" ], "problem_v1": "Use an appropriate local linear approximation to estimate the value of the given quantity. $\\sqrt{37} \\approx$ [ANS]", "answer_v1": [ "73/12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use an appropriate local linear approximation to estimate the value of the given quantity. $\\sqrt{99} \\approx$ [ANS]", "answer_v2": [ "199/20" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use an appropriate local linear approximation to estimate the value of the given quantity. $\\sqrt{48} \\approx$ [ANS]", "answer_v3": [ "97/14" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0463", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [ "derivative", "derivatives", "linear approximation", "differentials" ], "problem_v1": "The approximation $(1+x)^k\\approx 1+kx$ is commonly used by engineers for quick calculations.\n(a) Derive this result and use it to make a rough estimate of $(1.0001)^{38}$. $(1.0001)^{38}\\approx$ [ANS] Enter value to 6 decimal places. (b) The error between the computed and estimated values of $(1+x)^k$ is given by; E(x)=|(1+x)^k-(1+kx)| Using your answer to part (a) compute $E(1.0001)$ to 6 decimal places. $E(1.0001)$=[ANS]", "answer_v1": [ "1.0038", "7E-6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The approximation $(1+x)^k\\approx 1+kx$ is commonly used by engineers for quick calculations.\n(a) Derive this result and use it to make a rough estimate of $(1.1)^{48}$. $(1.1)^{48}\\approx$ [ANS] Enter value to 6 decimal places. (b) The error between the computed and estimated values of $(1+x)^k$ is given by; E(x)=|(1+x)^k-(1+kx)| Using your answer to part (a) compute $E(1.1)$ to 6 decimal places. $E(1.1)$=[ANS]", "answer_v2": [ "5.8", "91.2172" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The approximation $(1+x)^k\\approx 1+kx$ is commonly used by engineers for quick calculations.\n(a) Derive this result and use it to make a rough estimate of $(1.01)^{38}$. $(1.01)^{38}\\approx$ [ANS] Enter value to 6 decimal places. (b) The error between the computed and estimated values of $(1+x)^k$ is given by; E(x)=|(1+x)^k-(1+kx)| Using your answer to part (a) compute $E(1.01)$ to 6 decimal places. $E(1.01)$=[ANS]", "answer_v3": [ "1.38", "0.079527" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0464", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "linear approximation", "differentials" ], "problem_v1": "Use an appropriate local quadratic approximation to approximate $\\sqrt{4.05}$, and compare the result to that produced directly by your calculating utility. Enter the local quadratic approximation of $\\sqrt{4.05}$. $\\sqrt{4.05}\\approx$ [ANS]", "answer_v1": [ "2+1/4*0.05+-1/64*0.05^2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use an appropriate local quadratic approximation to approximate $\\sqrt{3.91}$, and compare the result to that produced directly by your calculating utility. Enter the local quadratic approximation of $\\sqrt{3.91}$. $\\sqrt{3.91}\\approx$ [ANS]", "answer_v2": [ "2+1/4*-0.09+-1/64*(-0.09)^2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use an appropriate local quadratic approximation to approximate $\\sqrt{3.96}$, and compare the result to that produced directly by your calculating utility. Enter the local quadratic approximation of $\\sqrt{3.96}$. $\\sqrt{3.96}\\approx$ [ANS]", "answer_v3": [ "2+1/4*-0.04+-1/64*(-0.04)^2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0465", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [ "differentials" ], "problem_v1": "If $2 y^{9/2}+xy-7x=16$, find the differential $dy$. Your answer will be in terms of x, y, and dx. Enter $dx$ as dx.\n$dy$=[ANS]", "answer_v1": [ "(7-y)/[9*y^{3.5}+x]*dx" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If $2 y^{3/2}+xy-9x=11$, find the differential $dy$. Your answer will be in terms of x, y, and dx. Enter $dx$ as dx.\n$dy$=[ANS]", "answer_v2": [ "(9-y)/[3*y^{0.5}+x]*dx" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If $2 y^{5/2}+xy-7x=13$, find the differential $dy$. Your answer will be in terms of x, y, and dx. Enter $dx$ as dx.\n$dy$=[ANS]", "answer_v3": [ "(7-y)/[5*y^{1.5}+x]*dx" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0466", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "differentials" ], "problem_v1": "The volume of a sphere of radius $r$ is $ V=\\frac{4}{3} \\pi r^3$.\n(a) Write a differential formula that estimates the change in volume of a sphere when the radius changes from $r_0$ to $r_0+dr$. Enter $r_0$ as r0 and $dr$ as dr. $dV$=[ANS]\n(b) Write a differential formula that estimates the change in volume of a sphere when the radius changes from $13$ to $13+dr$. $dV$=[ANS]\n(c) Use a differential to estimate the change in volume of a melting spherical snowball when the radius changes from $13 \\ \\mathrm{cm}$ to $12.6 \\ \\mathrm{cm}$. $dV$=[ANS] $\\mathrm{cm^3}$", "answer_v1": [ "4*pi*r0^2*dr", "4*pi*13^2*dr", "4*pi*13^2*-0.4" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The volume of a sphere of radius $r$ is $ V=\\frac{4}{3} \\pi r^3$.\n(a) Write a differential formula that estimates the change in volume of a sphere when the radius changes from $r_0$ to $r_0+dr$. Enter $r_0$ as r0 and $dr$ as dr. $dV$=[ANS]\n(b) Write a differential formula that estimates the change in volume of a sphere when the radius changes from $5$ to $5+dr$. $dV$=[ANS]\n(c) Use a differential to estimate the change in volume of a melting spherical snowball when the radius changes from $5 \\ \\mathrm{cm}$ to $4.4 \\ \\mathrm{cm}$. $dV$=[ANS] $\\mathrm{cm^3}$", "answer_v2": [ "4*pi*r0^2*dr", "4*pi*5^2*dr", "4*pi*5^2*-0.6" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The volume of a sphere of radius $r$ is $ V=\\frac{4}{3} \\pi r^3$.\n(a) Write a differential formula that estimates the change in volume of a sphere when the radius changes from $r_0$ to $r_0+dr$. Enter $r_0$ as r0 and $dr$ as dr. $dV$=[ANS]\n(b) Write a differential formula that estimates the change in volume of a sphere when the radius changes from $8$ to $8+dr$. $dV$=[ANS]\n(c) Use a differential to estimate the change in volume of a melting spherical snowball when the radius changes from $8 \\ \\mathrm{cm}$ to $7.5 \\ \\mathrm{cm}$. $dV$=[ANS] $\\mathrm{cm^3}$", "answer_v3": [ "4*pi*r0^2*dr", "4*pi*8^2*dr", "4*pi*8^2*-0.5" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0467", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "derivatives", "linear approximation" ], "problem_v1": "The edge of a cube was found to be $70$ cm with a possible error of $0.3$ cm. Use differentials to estimate the maximum possible error in the calculated volume of the cube.\nError $\\approx$ [ANS] $\\textrm{cm}^3$", "answer_v1": [ "3* 70^2 * 0.3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The edge of a cube was found to be $30$ cm with a possible error of $0.4$ cm. Use differentials to estimate the maximum possible error in the calculated volume of the cube.\nError $\\approx$ [ANS] $\\textrm{cm}^3$", "answer_v2": [ "3* 30^2 * 0.4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The edge of a cube was found to be $40$ cm with a possible error of $0.3$ cm. Use differentials to estimate the maximum possible error in the calculated volume of the cube.\nError $\\approx$ [ANS] $\\textrm{cm}^3$", "answer_v3": [ "3* 40^2 * 0.3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0468", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "3", "keywords": [ "derivatives", "linear approximation" ], "problem_v1": "Use differentials (or equivalently, a linear approximation) to approximate $\\sin(57^{\\circ})$ as follows: Let $f(x)=\\sin(x)$ and find the equation of the tangent line to $f(x)$ at a \"nice\" point near $57^{\\circ}$. Then use this to approximate $\\sin(57^{\\circ})$. Approximation=[ANS]", "answer_v1": [ "0.83984546500456" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use differentials (or equivalently, a linear approximation) to approximate $\\sin(25^{\\circ})$ as follows: Let $f(x)=\\sin(x)$ and find the equation of the tangent line to $f(x)$ at a \"nice\" point near $25^{\\circ}$. Then use this to approximate $\\sin(25^{\\circ})$. Approximation=[ANS]", "answer_v2": [ "0.424425026490267" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use differentials (or equivalently, a linear approximation) to approximate $\\sin(26^{\\circ})$ as follows: Let $f(x)=\\sin(x)$ and find the equation of the tangent line to $f(x)$ at a \"nice\" point near $26^{\\circ}$. Then use this to approximate $\\sin(26^{\\circ})$. Approximation=[ANS]", "answer_v3": [ "0.43954002119222" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0469", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Linear approximation and differentials", "level": "2", "keywords": [ "linearization" ], "problem_v1": "Find the local linearization of $f(x)=1/x$ at $5$. $l_{5}(x)=$ [ANS]", "answer_v1": [ "0.2-0.04*(x-5)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the local linearization of $f(x)=1/x$ at $-8$. $l_{-8}(x)=$ [ANS]", "answer_v2": [ "-[0.125+0.015625*(x+8)]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the local linearization of $f(x)=1/x$ at $-4$. $l_{-4}(x)=$ [ANS]", "answer_v3": [ "-[0.25+0.0625*(x+4)]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0470", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "derivatives", "related rates", "calculus", "Differentiation", "Product", "Quotient", "Related Rates", "Water Tank" ], "problem_v1": "Water is leaking out of an inverted conical tank at a rate of $12800.0$ $\\textrm{cm}^3/\\textrm{min}$ at the same time that water is being pumped into the tank at a constant rate. The tank has height $11.0 \\ \\textrm{m}$ and the diameter at the top is $5.5 \\ \\textrm{m}$. If the water level is rising at a rate of $19.0 \\ \\textrm{cm}/\\textrm{min}$ when the height of the water is $4.0 \\ \\textrm{m}$, find the rate at which water is being pumped into the tank in cubic centimeters per minute.\nAnswer: [ANS] $\\textrm{cm}^3/\\textrm{min}$", "answer_v1": [ "pi*5.5*5.5*400*400*19/(4*11*11)+12800" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Water is leaking out of an inverted conical tank at a rate of $6700.0$ $\\textrm{cm}^3/\\textrm{min}$ at the same time that water is being pumped into the tank at a constant rate. The tank has height $15.0 \\ \\textrm{m}$ and the diameter at the top is $3.5 \\ \\textrm{m}$. If the water level is rising at a rate of $30.0 \\ \\textrm{cm}/\\textrm{min}$ when the height of the water is $2.5 \\ \\textrm{m}$, find the rate at which water is being pumped into the tank in cubic centimeters per minute.\nAnswer: [ANS] $\\textrm{cm}^3/\\textrm{min}$", "answer_v2": [ "pi*3.5*3.5*250*250*30/(4*15*15)+6700" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Water is leaking out of an inverted conical tank at a rate of $8800.0$ $\\textrm{cm}^3/\\textrm{min}$ at the same time that water is being pumped into the tank at a constant rate. The tank has height $12.0 \\ \\textrm{m}$ and the diameter at the top is $4.0 \\ \\textrm{m}$. If the water level is rising at a rate of $18.0 \\ \\textrm{cm}/\\textrm{min}$ when the height of the water is $3.0 \\ \\textrm{m}$, find the rate at which water is being pumped into the tank in cubic centimeters per minute.\nAnswer: [ANS] $\\textrm{cm}^3/\\textrm{min}$", "answer_v3": [ "pi*4*4*300*300*18/(4*12*12)+8800" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0471", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "derivatives", "related rates", "Calculus" ], "problem_v1": "When air expands adiabatically (without gaining or losing heat), its pressure $P$ and volume $V$ are related by the equation $PV^{1.4}=C$ where $C$ is a constant. Suppose that at a certain instant the volume is $600$ cubic centimeters and the pressure is $89$ kPa and is decreasing at a rate of $12$ kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant?\nAnswer: [ANS]\nNote: Pa stands for Pascal. One Pa is equivalent to one $\\textrm{Newton}/\\textrm{m}^2$. kPa is a kiloPascal or 1000 Pascals.", "answer_v1": [ "-(600*(-12)/(1.4*89))" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "When air expands adiabatically (without gaining or losing heat), its pressure $P$ and volume $V$ are related by the equation $PV^{1.4}=C$ where $C$ is a constant. Suppose that at a certain instant the volume is $330$ cubic centimeters and the pressure is $99$ kPa and is decreasing at a rate of $8$ kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant?\nAnswer: [ANS]\nNote: Pa stands for Pascal. One Pa is equivalent to one $\\textrm{Newton}/\\textrm{m}^2$. kPa is a kiloPascal or 1000 Pascals.", "answer_v2": [ "-(330*(-8)/(1.4*99))" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "When air expands adiabatically (without gaining or losing heat), its pressure $P$ and volume $V$ are related by the equation $PV^{1.4}=C$ where $C$ is a constant. Suppose that at a certain instant the volume is $420$ cubic centimeters and the pressure is $89$ kPa and is decreasing at a rate of $9$ kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant?\nAnswer: [ANS]\nNote: Pa stands for Pascal. One Pa is equivalent to one $\\textrm{Newton}/\\textrm{m}^2$. kPa is a kiloPascal or 1000 Pascals.", "answer_v3": [ "-(420*(-9)/(1.4*89))" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0472", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [], "problem_v1": "Water flows onto a flat surface at a rate of 16 cm 3 3/s forming a circular puddle 10 mm deep. How fast is the radius growing when the radius is:\n$\\begin{array}{ccc}\\hline 1 cm? Answer &=& [ANS] \\\\ \\hline 10 cm? Answer &=& [ANS] \\\\ \\hline 100 cm? Answer &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "2.54648", "0.254648", "0.0254648" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Water flows onto a flat surface at a rate of 4 cm 3 3/s forming a circular puddle 10 mm deep. How fast is the radius growing when the radius is:\n$\\begin{array}{ccc}\\hline 1 cm? Answer &=& [ANS] \\\\ \\hline 10 cm? Answer &=& [ANS] \\\\ \\hline 100 cm? Answer &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "0.63662", "0.063662", "0.0063662" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Water flows onto a flat surface at a rate of 8 cm 3 3/s forming a circular puddle 10 mm deep. How fast is the radius growing when the radius is:\n$\\begin{array}{ccc}\\hline 1 cm? Answer &=& [ANS] \\\\ \\hline 10 cm? Answer &=& [ANS] \\\\ \\hline 100 cm? Answer &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "1.27324", "0.127324", "0.0127324" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0473", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [], "problem_v1": "A 20 ft. ladder is leaning against a house while the base is pulled away at a constant rate of 2 ft./s. At what rate is the top of the ladder sliding down the side of the house when the base is:\n$\\begin{array}{ccc}\\hline 1 foot from the house? Answer &=& [ANS] \\\\ \\hline 5 feet from the house? Answer &=& [ANS] \\\\ \\hline 19 feet from the house? Answer &=& [ANS] \\\\ \\hline \\end{array}$ (Answers should be positive.)", "answer_v1": [ "0.100125", "0.516398", "6.08487" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A 25 ft. ladder is leaning against a house while the base is pulled away at a constant rate of 1 ft./s. At what rate is the top of the ladder sliding down the side of the house when the base is:\n$\\begin{array}{ccc}\\hline 1 foot from the house? Answer &=& [ANS] \\\\ \\hline 5 feet from the house? Answer &=& [ANS] \\\\ \\hline 24 feet from the house? Answer &=& [ANS] \\\\ \\hline \\end{array}$ (Answers should be positive.)", "answer_v2": [ "0.040032", "0.204124", "3.42857" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A 21 ft. ladder is leaning against a house while the base is pulled away at a constant rate of 1 ft./s. At what rate is the top of the ladder sliding down the side of the house when the base is:\n$\\begin{array}{ccc}\\hline 1 foot from the house? Answer &=& [ANS] \\\\ \\hline 5 feet from the house? Answer &=& [ANS] \\\\ \\hline 20 feet from the house? Answer &=& [ANS] \\\\ \\hline \\end{array}$ (Answers should be positive.)", "answer_v3": [ "0.0476731", "0.245145", "3.12348" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0474", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "calculus", "derivatives", "related rates" ], "problem_v1": "The radius of a circular oil slick expands at a rate of 7 m/min.\n(a) How fast is the area of the oil slick increasing when the radius is 26 m? $\\frac{dA}{dt}=$ [ANS] $m^2/min$ (b) If the radius is 0 at time $t=0$, how fast is the area increasing after 4 mins? $\\frac{dA}{dt}=$ [ANS] $m^2/min$", "answer_v1": [ "1143.53972590668", "1231.5043202072" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The radius of a circular oil slick expands at a rate of 2 m/min.\n(a) How fast is the area of the oil slick increasing when the radius is 21 m? $\\frac{dA}{dt}=$ [ANS] $m^2/min$ (b) If the radius is 0 at time $t=0$, how fast is the area increasing after 5 mins? $\\frac{dA}{dt}=$ [ANS] $m^2/min$", "answer_v2": [ "263.893782901543", "125.663706143592" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The radius of a circular oil slick expands at a rate of 4 m/min.\n(a) How fast is the area of the oil slick increasing when the radius is 23 m? $\\frac{dA}{dt}=$ [ANS] $m^2/min$ (b) If the radius is 0 at time $t=0$, how fast is the area increasing after 4 mins? $\\frac{dA}{dt}=$ [ANS] $m^2/min$", "answer_v3": [ "578.053048260522", "402.123859659494" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0475", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "calculus", "derivatives", "related rates" ], "problem_v1": "A hot air balloon rising vertically is tracked by an observer located 4 miles from the lift-off point. At a certain moment, the angle between the observer's line-of-sight and the horizontal is $\\frac{\\pi}{6}$, and it is changing at a rate of 0.1 rad/min. How fast is the balloon rising at this moment? [ANS] miles/min", "answer_v1": [ "0.533333" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A hot air balloon rising vertically is tracked by an observer located 5 miles from the lift-off point. At a certain moment, the angle between the observer's line-of-sight and the horizontal is $\\frac{\\pi}{3}$, and it is changing at a rate of 0.1 rad/min. How fast is the balloon rising at this moment? [ANS] miles/min", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A hot air balloon rising vertically is tracked by an observer located 4 miles from the lift-off point. At a certain moment, the angle between the observer's line-of-sight and the horizontal is $\\frac{\\pi}{4}$, and it is changing at a rate of 0.1 rad/min. How fast is the balloon rising at this moment? [ANS] miles/min", "answer_v3": [ "0.8" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0476", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "calculus", "derivatives", "related rates" ], "problem_v1": "A road perpendicular to a highway leads to a farmhouse located $6$ mile away. An automobile traveling on the highway passes through this intersection at a speed of $70 \\textrm{mph}.$ How fast is the distance between the automobile and the farmhouse increasing when the automobile is $7$ miles past the intersection of the highway and the road? The distance between the automobile and the farmhouse is increasing at a rate of [ANS] miles per hour.", "answer_v1": [ "53.148" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A road perpendicular to a highway leads to a farmhouse located $10$ mile away. An automobile traveling on the highway passes through this intersection at a speed of $40 \\textrm{mph}.$ How fast is the distance between the automobile and the farmhouse increasing when the automobile is $2$ miles past the intersection of the highway and the road? The distance between the automobile and the farmhouse is increasing at a rate of [ANS] miles per hour.", "answer_v2": [ "7.84465" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A road perpendicular to a highway leads to a farmhouse located $7$ mile away. An automobile traveling on the highway passes through this intersection at a speed of $50 \\textrm{mph}.$ How fast is the distance between the automobile and the farmhouse increasing when the automobile is $3$ miles past the intersection of the highway and the road? The distance between the automobile and the farmhouse is increasing at a rate of [ANS] miles per hour.", "answer_v3": [ "19.696" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0477", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "derivative' 'related rates", "derivatives", "related rates", "Calculus" ], "problem_v1": "At noon, ship A is 40 nautical miles due west of ship B. Ship A is sailing west at 21 knots and ship B is sailing north at 21 knots. How fast (in knots) is the distance between the ships changing at 6 PM? The distance is changing at [ANS] knots. (Note: 1 knot is a speed of 1 nautical mile per hour.)", "answer_v1": [ "29.4237" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "At noon, ship A is 10 nautical miles due west of ship B. Ship A is sailing west at 25 knots and ship B is sailing north at 16 knots. How fast (in knots) is the distance between the ships changing at 4 PM? The distance is changing at [ANS] knots. (Note: 1 knot is a speed of 1 nautical mile per hour.)", "answer_v2": [ "29.655" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "At noon, ship A is 20 nautical miles due west of ship B. Ship A is sailing west at 21 knots and ship B is sailing north at 18 knots. How fast (in knots) is the distance between the ships changing at 5 PM? The distance is changing at [ANS] knots. (Note: 1 knot is a speed of 1 nautical mile per hour.)", "answer_v3": [ "27.5597" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0478", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "derivative' 'related rates" ], "problem_v1": "A spherical snowball is melting so that its diameter is decreasing at rate of 0.4 cm/min.\nAt what is the rate is the volume of the snowball changing when the diameter is 14 cm? The volume is changing at a rate of [ANS] ${\\rm cm}^3{\\rm/min}$.", "answer_v1": [ "-123.15" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A spherical snowball is melting so that its diameter is decreasing at rate of 0.1 cm/min.\nAt what is the rate is the volume of the snowball changing when the diameter is 18 cm? The volume is changing at a rate of [ANS] ${\\rm cm}^3{\\rm/min}$.", "answer_v2": [ "-50.8938" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A spherical snowball is melting so that its diameter is decreasing at rate of 0.2 cm/min.\nAt what is the rate is the volume of the snowball changing when the diameter is 14 cm? The volume is changing at a rate of [ANS] ${\\rm cm}^3{\\rm/min}$.", "answer_v3": [ "-61.5752" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0479", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "derivative' 'related rates", "derivatives", "related rates", "Calculus", "Differentiation", "Product", "Quotient" ], "problem_v1": "The altitude (i.e., height) of a triangle is increasing at a rate of 3 cm/minute while the area of the triangle is increasing at a rate of 4 square cm/minute. At what rate is the base of the triangle changing when the altitude is 11 centimeters and the area is 92 square centimeters? The base is changing at [ANS] cm/min.", "answer_v1": [ "-3.83471" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The altitude (i.e., height) of a triangle is increasing at a rate of 1.5 cm/minute while the area of the triangle is increasing at a rate of 2.5 square cm/minute. At what rate is the base of the triangle changing when the altitude is 7 centimeters and the area is 99 square centimeters? The base is changing at [ANS] cm/min.", "answer_v2": [ "-5.34694" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The altitude (i.e., height) of a triangle is increasing at a rate of 2 cm/minute while the area of the triangle is increasing at a rate of 3.5 square cm/minute. At what rate is the base of the triangle changing when the altitude is 8.5 centimeters and the area is 92 square centimeters? The base is changing at [ANS] cm/min.", "answer_v3": [ "-4.2699" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0480", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "derivative' 'related rates" ], "problem_v1": "A spherical balloon is inflated so that its volume is increasing at the rate of 3.5 ${\\rm ft}^3{\\rm/min}$. How rapidly is the diameter of the balloon increasing when the diameter is 1.5 feet? The diameter is increasing at [ANS] ft/min.", "answer_v1": [ "0.990297" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A spherical balloon is inflated so that its volume is increasing at the rate of 2.1 ${\\rm ft}^3{\\rm/min}$. How rapidly is the diameter of the balloon increasing when the diameter is 1.8 feet? The diameter is increasing at [ANS] ft/min.", "answer_v2": [ "0.412624" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A spherical balloon is inflated so that its volume is increasing at the rate of 2.6 ${\\rm ft}^3{\\rm/min}$. How rapidly is the diameter of the balloon increasing when the diameter is 1.5 feet? The diameter is increasing at [ANS] ft/min.", "answer_v3": [ "0.73565" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0481", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "derivative' 'related rates" ], "problem_v1": "A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 2.9 ft/s.\n(a) How rapidly is the area enclosed by the ripple increasing when the radius is 4 feet? The area is increasing at [ANS] ${\\rm ft}^2{\\rm/s}$. (b) How rapidly is the area enclosed by the ripple increasing at the end of 9 seconds? The area is increasing at [ANS] ${\\rm ft}^2{\\rm/s}$.", "answer_v1": [ "72.8849", "475.574" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 3.4 ft/s.\n(a) How rapidly is the area enclosed by the ripple increasing when the radius is 2 feet? The area is increasing at [ANS] ${\\rm ft}^2{\\rm/s}$. (b) How rapidly is the area enclosed by the ripple increasing at the end of 6.2 seconds? The area is increasing at [ANS] ${\\rm ft}^2{\\rm/s}$.", "answer_v2": [ "42.7257", "450.328" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 2.9 ft/s.\n(a) How rapidly is the area enclosed by the ripple increasing when the radius is 3 feet? The area is increasing at [ANS] ${\\rm ft}^2{\\rm/s}$. (b) How rapidly is the area enclosed by the ripple increasing at the end of 7.2 seconds? The area is increasing at [ANS] ${\\rm ft}^2{\\rm/s}$.", "answer_v3": [ "54.6637", "380.459" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0482", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "derivative' 'related rates", "derivatives", "related rates", "Differentiation", "Product", "Quotient" ], "problem_v1": "Gravel is being dumped from a conveyor belt at a rate of 40 ${\\rm ft}^3{\\rm/min}$. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same.\nHow fast is the height of the pile increasing when the pile is 19 ft high? The height is increasing at [ANS] ft/min.", "answer_v1": [ "0.141079" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Gravel is being dumped from a conveyor belt at a rate of 10 ${\\rm ft}^3{\\rm/min}$. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same.\nHow fast is the height of the pile increasing when the pile is 24 ft high? The height is increasing at [ANS] ft/min.", "answer_v2": [ "0.0221049" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Gravel is being dumped from a conveyor belt at a rate of 20 ${\\rm ft}^3{\\rm/min}$. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same.\nHow fast is the height of the pile increasing when the pile is 19 ft high? The height is increasing at [ANS] ft/min.", "answer_v3": [ "0.0705396" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0483", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "derivative' 'related rates" ], "problem_v1": "A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the front of the boat, which is 9 feet below the level of the pulley.\nIf the rope is pulled through the pulley at a rate of 16 ft/min, at what rate will the boat be approaching the dock when 110 ft of rope is out? The boat will be approaching the dock at [ANS] ft/min. Hint: Sketch a diagram of this situation.", "answer_v1": [ "16.0538" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the front of the boat, which is 5 feet below the level of the pulley.\nIf the rope is pulled through the pulley at a rate of 20 ft/min, at what rate will the boat be approaching the dock when 90 ft of rope is out? The boat will be approaching the dock at [ANS] ft/min. Hint: Sketch a diagram of this situation.", "answer_v2": [ "20.0309" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the front of the boat, which is 6 feet below the level of the pulley.\nIf the rope is pulled through the pulley at a rate of 16 ft/min, at what rate will the boat be approaching the dock when 100 ft of rope is out? The boat will be approaching the dock at [ANS] ft/min. Hint: Sketch a diagram of this situation.", "answer_v3": [ "16.0289" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0484", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "derivative' 'related rates", "Calculus", "Derivatives", "Related Rates", "Similar", "Differentiation", "Product", "Quotient", "related rates" ], "problem_v1": "A street light is at the top of a 18 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 6 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 45 feet from the base of the pole? The tip of the shadow is moving at [ANS] ft/sec.", "answer_v1": [ "9" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A street light is at the top of a 10 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 30 feet from the base of the pole? The tip of the shadow is moving at [ANS] ft/sec.", "answer_v2": [ "20" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A street light is at the top of a 13 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 35 feet from the base of the pole? The tip of the shadow is moving at [ANS] ft/sec.", "answer_v3": [ "13" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0485", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "derivative' 'related rates", "Derivatives", "related rates", "Calculus", "Differentiation", "Product", "Quotient", "related rates" ], "problem_v1": "A plane flying with a constant speed of 24 km/min passes over a ground radar station at an altitude of 11 km and climbs at an angle of 40 degrees.\nAt what rate is the distance from the plane to the radar station increasing 4 minutes later? The distance is increasing at [ANS] km/min.", "answer_v1": [ "23.9202" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A plane flying with a constant speed of 4 km/min passes over a ground radar station at an altitude of 16 km and climbs at an angle of 25 degrees.\nAt what rate is the distance from the plane to the radar station increasing 3 minutes later? The distance is increasing at [ANS] km/min.", "answer_v2": [ "3.16489" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A plane flying with a constant speed of 9 km/min passes over a ground radar station at an altitude of 11 km and climbs at an angle of 25 degrees.\nAt what rate is the distance from the plane to the radar station increasing 4 minutes later? The distance is increasing at [ANS] km/min.", "answer_v3": [ "8.74095" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0486", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "3", "keywords": [ "derivative' 'related rates' 'function" ], "problem_v1": "Suppose that a point is moving along the path $xy=4$ so that $\\frac {dy}{dt}=3$. Find $\\frac{dx}{dt}$ when $x=4$. $\\frac{dx}{dt}$=[ANS]", "answer_v1": [ "-12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that a point is moving along the path $xy=1$ so that $\\frac {dy}{dt}=5$. Find $\\frac{dx}{dt}$ when $x=1$. $\\frac{dx}{dt}$=[ANS]", "answer_v2": [ "-5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that a point is moving along the path $xy=2$ so that $\\frac {dy}{dt}=4$. Find $\\frac{dx}{dt}$ when $x=2$. $\\frac{dx}{dt}$=[ANS]", "answer_v3": [ "-8" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0487", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "derivative' 'related rates" ], "problem_v1": "A spherical balloon is to be deflated so that its radius decreases at a constant rate of 13 cm/min. At what rate must air be removed when the radius is 7 cm? Air must be removed at [ANS] ${\\rm cm}^3{\\rm/min}$.", "answer_v1": [ "8004.78" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A spherical balloon is to be deflated so that its radius decreases at a constant rate of 7 cm/min. At what rate must air be removed when the radius is 9 cm? Air must be removed at [ANS] ${\\rm cm}^3{\\rm/min}$.", "answer_v2": [ "7125.13" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A spherical balloon is to be deflated so that its radius decreases at a constant rate of 9 cm/min. At what rate must air be removed when the radius is 8 cm? Air must be removed at [ANS] ${\\rm cm}^3{\\rm/min}$.", "answer_v3": [ "7238.23" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0488", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [], "problem_v1": "A particle is moving along the curve $y=5 \\sqrt{4x+9}$. As the particle passes through the point $(4, 25)$, its $x$-coordinate increases at a rate of $4$ units per second. Find the rate of change of the distance from the particle to the origin at this instant. [ANS]", "answer_v1": [ "8.53148705976033" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A particle is moving along the curve $y=2 \\sqrt{5x+4}$. As the particle passes through the point $(1, 6)$, its $x$-coordinate increases at a rate of $3$ units per second. Find the rate of change of the distance from the particle to the origin at this instant. [ANS]", "answer_v2": [ "5.42516658107679" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A particle is moving along the curve $y=3 \\sqrt{4x+1}$. As the particle passes through the point $(2, 9)$, its $x$-coordinate increases at a rate of $4$ units per second. Find the rate of change of the distance from the particle to the origin at this instant. [ANS]", "answer_v3": [ "8.67721831274625" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0489", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "Derivatives", "applications", "Derivative", "Related Rates", "Focal Length" ], "problem_v1": "If $f$ is the focal length of a convex lens and an object is placed at a distance $p$ from the lens, then its image will be at a distance $q$ from the lens, where $f$, $p$, and $q$ are related by the lens equation \\frac{1}{f}=\\frac{1}{p}+\\frac{1}{q} What is the rate of change of $p$ with respect to $q$ if $q=5$ and $f=7$? (Make sure you have the correct sign for the rate.) [ANS]", "answer_v1": [ "-12.25" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $f$ is the focal length of a convex lens and an object is placed at a distance $p$ from the lens, then its image will be at a distance $q$ from the lens, where $f$, $p$, and $q$ are related by the lens equation \\frac{1}{f}=\\frac{1}{p}+\\frac{1}{q} What is the rate of change of $p$ with respect to $q$ if $q=8$ and $f=1$? (Make sure you have the correct sign for the rate.) [ANS]", "answer_v2": [ "-0.0204081632653061" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $f$ is the focal length of a convex lens and an object is placed at a distance $p$ from the lens, then its image will be at a distance $q$ from the lens, where $f$, $p$, and $q$ are related by the lens equation \\frac{1}{f}=\\frac{1}{p}+\\frac{1}{q} What is the rate of change of $p$ with respect to $q$ if $q=5$ and $f=3$? (Make sure you have the correct sign for the rate.) [ANS]", "answer_v3": [ "-2.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0490", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "Derivative", "Polynomial" ], "problem_v1": "Let $\\>\\theta\\>$ (in radians) be an acute angle in a right triangle and let $\\>x\\>$ and $\\>y\\>$, respectively, be the lengths of the sides adjacent to and opposite $\\>\\theta$. Suppose also that $\\>x\\>$ and $\\>y\\>$ vary with time. At a certain instant $\\>x=7\\>$ units and is increasing at $\\>6\\>$ unit/s, while $\\>y=6\\>$ and is decreasing at $\\>\\frac {1} {7}\\>$ units/s. How fast is $\\>\\theta\\>$ changing at that instant?\nAnswer: [ANS]", "answer_v1": [ "-0.435294" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $\\>\\theta\\>$ (in radians) be an acute angle in a right triangle and let $\\>x\\>$ and $\\>y\\>$, respectively, be the lengths of the sides adjacent to and opposite $\\>\\theta$. Suppose also that $\\>x\\>$ and $\\>y\\>$ vary with time. At a certain instant $\\>x=1\\>$ units and is increasing at $\\>9\\>$ unit/s, while $\\>y=2\\>$ and is decreasing at $\\>\\frac {1} {4}\\>$ units/s. How fast is $\\>\\theta\\>$ changing at that instant?\nAnswer: [ANS]", "answer_v2": [ "-3.65" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $\\>\\theta\\>$ (in radians) be an acute angle in a right triangle and let $\\>x\\>$ and $\\>y\\>$, respectively, be the lengths of the sides adjacent to and opposite $\\>\\theta$. Suppose also that $\\>x\\>$ and $\\>y\\>$ vary with time. At a certain instant $\\>x=3\\>$ units and is increasing at $\\>6\\>$ unit/s, while $\\>y=3\\>$ and is decreasing at $\\>\\frac {1} {6}\\>$ units/s. How fast is $\\>\\theta\\>$ changing at that instant?\nAnswer: [ANS]", "answer_v3": [ "-1.02778" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0491", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "Derivative", "Polynomial" ], "problem_v1": "A beacon that makes one revolution every $\\>11\\>$ s is located on a ship anchored $\\>8\\>$ kilometers from a straight shoreline. How fast is the beam moving when it makes an angle of $\\>45\\>$ degrees with the shore?\nAnswer: [ANS]", "answer_v1": [ "32*pi/11" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A beacon that makes one revolution every $\\>15\\>$ s is located on a ship anchored $\\>2\\>$ kilometers from a straight shoreline. How fast is the beam moving when it makes an angle of $\\>45\\>$ degrees with the shore?\nAnswer: [ANS]", "answer_v2": [ "8*pi/15" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A beacon that makes one revolution every $\\>11\\>$ s is located on a ship anchored $\\>4\\>$ kilometers from a straight shoreline. How fast is the beam moving when it makes an angle of $\\>45\\>$ degrees with the shore?\nAnswer: [ANS]", "answer_v3": [ "16*pi/11" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0492", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "Derivative", "Polynomial" ], "problem_v1": "A aircraft is climbing at a 30 degree angle to the horizontal. How fast is the aircraft gaining altitude if its speed is $\\>500\\>$ mi/h?\nAnswer: [ANS]", "answer_v1": [ "250" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A aircraft is climbing at a 30 degree angle to the horizontal. How fast is the aircraft gaining altitude if its speed is $\\>200\\>$ mi/h?\nAnswer: [ANS]", "answer_v2": [ "100" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A aircraft is climbing at a 30 degree angle to the horizontal. How fast is the aircraft gaining altitude if its speed is $\\>300\\>$ mi/h?\nAnswer: [ANS]", "answer_v3": [ "150" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0493", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "Derivative", "Polynomial" ], "problem_v1": "The minute hand of a certain clock is $\\>7\\>$ in long. Starting from the moment when the hand is pointing straight up, how fast is the area of the sector that is swept out by the hand increasing during the next revolution of the hand. Express your answer in square inches per minute.\nAnswer: [ANS]", "answer_v1": [ "49*pi/60" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The minute hand of a certain clock is $\\>1\\>$ in long. Starting from the moment when the hand is pointing straight up, how fast is the area of the sector that is swept out by the hand increasing during the next revolution of the hand. Express your answer in square inches per minute.\nAnswer: [ANS]", "answer_v2": [ "pi/60" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The minute hand of a certain clock is $\\>3\\>$ in long. Starting from the moment when the hand is pointing straight up, how fast is the area of the sector that is swept out by the hand increasing during the next revolution of the hand. Express your answer in square inches per minute.\nAnswer: [ANS]", "answer_v3": [ "9*pi/60" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0494", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "Derivative", "Polynomial" ], "problem_v1": "A point $\\>P\\>$ is moving along the curve whose equation is $\\>y=6x\\>$. How fast is the distance between $\\>P\\>$ and the point $\\> (8, 0) \\>$ changing at the instant when $\\>P\\>$ is at $\\> (8, 48) \\>$ if $\\>x\\>$ is decreasing at the rate of $\\>6\\>$ units/s at that instant?\nAnswer: [ANS]", "answer_v1": [ "-36" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A point $\\>P\\>$ is moving along the curve whose equation is $\\>y=9x\\>$. How fast is the distance between $\\>P\\>$ and the point $\\> (2, 0) \\>$ changing at the instant when $\\>P\\>$ is at $\\> (2, 18) \\>$ if $\\>x\\>$ is decreasing at the rate of $\\>3\\>$ units/s at that instant?\nAnswer: [ANS]", "answer_v2": [ "-27" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A point $\\>P\\>$ is moving along the curve whose equation is $\\>y=6x\\>$. How fast is the distance between $\\>P\\>$ and the point $\\> (4, 0) \\>$ changing at the instant when $\\>P\\>$ is at $\\> (4, 24) \\>$ if $\\>x\\>$ is decreasing at the rate of $\\>4\\>$ units/s at that instant?\nAnswer: [ANS]", "answer_v3": [ "-24" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0495", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "2", "keywords": [ "Derivative", "Polynomial" ], "problem_v1": "The side of a square is measured to be $\\>8\\>$ ft, with a possible error of $\\>\\pm 0.2\\>$ ft. Use differentials to estimate the error in the calculated area.\nError: $\\pm$ [ANS] square ft.", "answer_v1": [ "3.2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The side of a square is measured to be $\\>2\\>$ ft, with a possible error of $\\>\\pm 0.2\\>$ ft. Use differentials to estimate the error in the calculated area.\nError: $\\pm$ [ANS] square ft.", "answer_v2": [ "0.8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The side of a square is measured to be $\\>4\\>$ ft, with a possible error of $\\>\\pm 0.2\\>$ ft. Use differentials to estimate the error in the calculated area.\nError: $\\pm$ [ANS] square ft.", "answer_v3": [ "1.6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0496", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "2", "keywords": [ "Derivative", "Polynomial" ], "problem_v1": "The electrical resistance $\\>R\\>$ of a certain wire is given by $\\>R=\\frac {k}{r^2}\\>$, where $\\>k\\>$ is a constant and $\\>r\\>$ is the radius of the wire. Assuming that the radius $\\>r\\>$ has a possible error of $\\> \\pm 8 \\%$, use differentials to estimate the percentage error in $\\>R$.\nError: $\\pm$ [ANS] \\%", "answer_v1": [ "16" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The electrical resistance $\\>R\\>$ of a certain wire is given by $\\>R=\\frac {k}{r^2}\\>$, where $\\>k\\>$ is a constant and $\\>r\\>$ is the radius of the wire. Assuming that the radius $\\>r\\>$ has a possible error of $\\> \\pm 2 \\%$, use differentials to estimate the percentage error in $\\>R$.\nError: $\\pm$ [ANS] \\%", "answer_v2": [ "4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The electrical resistance $\\>R\\>$ of a certain wire is given by $\\>R=\\frac {k}{r^2}\\>$, where $\\>k\\>$ is a constant and $\\>r\\>$ is the radius of the wire. Assuming that the radius $\\>r\\>$ has a possible error of $\\> \\pm 4 \\%$, use differentials to estimate the percentage error in $\\>R$.\nError: $\\pm$ [ANS] \\%", "answer_v3": [ "8" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0497", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "calculus", "derivative", "related rates", "modeling" ], "problem_v1": "A rectangle has one side of $10$ cm. How fast is the area of the rectangle changing at the instant when the other side is $17$ cm and increasing at $4$ cm per minute? Answer: [ANS] cm^2/min", "answer_v1": [ "40" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A rectangle has one side of $6$ cm. How fast is the area of the rectangle changing at the instant when the other side is $20$ cm and increasing at $2$ cm per minute? Answer: [ANS] cm^2/min", "answer_v2": [ "12" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A rectangle has one side of $7$ cm. How fast is the area of the rectangle changing at the instant when the other side is $17$ cm and increasing at $3$ cm per minute? Answer: [ANS] cm^2/min", "answer_v3": [ "21" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0498", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "calculus", "derivative", "related rates", "modeling" ], "problem_v1": "A potter forms a piece of clay into a cylinder. As he rolls it, the length, $L$, of the cylinder increases and the radius, $r$, decreases. If the length of the cylinder is increasing by $0.7$ cm per second, find the rate at which the radius is changing when the radius is $3$ cm and the length is $9$ cm. rate=[ANS] cm/s", "answer_v1": [ "-0.116666666666667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A potter forms a piece of clay into a cylinder. As he rolls it, the length, $L$, of the cylinder increases and the radius, $r$, decreases. If the length of the cylinder is increasing by $0.1$ cm per second, find the rate at which the radius is changing when the radius is $4$ cm and the length is $6$ cm. rate=[ANS] cm/s", "answer_v2": [ "-0.0333333333333333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A potter forms a piece of clay into a cylinder. As he rolls it, the length, $L$, of the cylinder increases and the radius, $r$, decreases. If the length of the cylinder is increasing by $0.3$ cm per second, find the rate at which the radius is changing when the radius is $3$ cm and the length is $7$ cm. rate=[ANS] cm/s", "answer_v3": [ "-0.0642857142857143" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0499", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "calculus", "derivative", "related rates", "modeling" ], "problem_v1": "For positive constants $k$ and $g$, the velocity, $v$, of a particle of mass $m$ at time $t$ is given by v=\\frac{mg}{k}\\left(1-e^{-kt/m}\\right). At what rate is the velocity is changing at time 0? At $t=7$? What do your answers tell you about the motion? At what rate is the velocity changing at time 0? rate=[ANS]\nAt what rate is it changing at $t=7$? rate=[ANS]", "answer_v1": [ "g", "g*e^{-1*7*k/m}" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "For positive constants $k$ and $g$, the velocity, $v$, of a particle of mass $m$ at time $t$ is given by v=\\frac{mg}{k}\\left(1-e^{-kt/m}\\right). At what rate is the velocity is changing at time 0? At $t=1$? What do your answers tell you about the motion? At what rate is the velocity changing at time 0? rate=[ANS]\nAt what rate is it changing at $t=1$? rate=[ANS]", "answer_v2": [ "g", "g*e^{-1*1*k/m}" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "For positive constants $k$ and $g$, the velocity, $v$, of a particle of mass $m$ at time $t$ is given by v=\\frac{mg}{k}\\left(1-e^{-kt/m}\\right). At what rate is the velocity is changing at time 0? At $t=3$? What do your answers tell you about the motion? At what rate is the velocity changing at time 0? rate=[ANS]\nAt what rate is it changing at $t=3$? rate=[ANS]", "answer_v3": [ "g", "g*e^{-1*3*k/m}" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0500", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "calculus", "derivative", "related rates", "modeling" ], "problem_v1": "The power, $P$, dissipated when a 8-volt battery is put across a resistance of $R$ ohms is given by P=\\frac{64}R. What is the rate of change of power with respect to resistance? rate of change=[ANS] V/ohm ${}^2$", "answer_v1": [ "-1*64/R^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The power, $P$, dissipated when a 2-volt battery is put across a resistance of $R$ ohms is given by P=\\frac{4}R. What is the rate of change of power with respect to resistance? rate of change=[ANS] V/ohm ${}^2$", "answer_v2": [ "-1*4/R^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The power, $P$, dissipated when a 4-volt battery is put across a resistance of $R$ ohms is given by P=\\frac{16}R. What is the rate of change of power with respect to resistance? rate of change=[ANS] V/ohm ${}^2$", "answer_v3": [ "-1*16/R^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0501", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "calculus", "derivative", "related rates", "modeling" ], "problem_v1": "A voltage $V$ across a resistance $R$ generates a current $I=V/R$. If a constant voltage of 19 volts is put across a resistance that is increasing at a rate of 0.5 ohms per second when the resistance is 6 ohms, at what rate is the current changing? rate=[ANS] amp/s", "answer_v1": [ "-0.263888888888889" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A voltage $V$ across a resistance $R$ generates a current $I=V/R$. If a constant voltage of 3 volts is put across a resistance that is increasing at a rate of 0.8 ohms per second when the resistance is 3 ohms, at what rate is the current changing? rate=[ANS] amp/s", "answer_v2": [ "-0.266666666666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A voltage $V$ across a resistance $R$ generates a current $I=V/R$. If a constant voltage of 9 volts is put across a resistance that is increasing at a rate of 0.5 ohms per second when the resistance is 4 ohms, at what rate is the current changing? rate=[ANS] amp/s", "answer_v3": [ "-0.28125" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0502", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "calculus" ], "problem_v1": "A child is flying a kite. If the kite is 135 feet above the child's hand level and the wind is blowing it on a horizontal course at 7 feet per second, the child is paying out cord at [ANS] feet per second when 245 feet of cord are out. Assume that the cord remains straight from hand to kite. (If you have ever flown a kite you know that this is an unrealistic assumption.)", "answer_v1": [ "5.84144237150311" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A child is flying a kite. If the kite is 85 feet above the child's hand level and the wind is blowing it on a horizontal course at 4 feet per second, the child is paying out cord at [ANS] feet per second when 235 feet of cord are out. Assume that the cord remains straight from hand to kite. (If you have ever flown a kite you know that this is an unrealistic assumption.)", "answer_v2": [ "3.72917485960964" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A child is flying a kite. If the kite is 100 feet above the child's hand level and the wind is blowing it on a horizontal course at 4 feet per second, the child is paying out cord at [ANS] feet per second when 210 feet of cord are out. Assume that the cord remains straight from hand to kite. (If you have ever flown a kite you know that this is an unrealistic assumption.)", "answer_v3": [ "3.51736863097512" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0503", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "derivatives", "related rates", "Differentiation", "Product", "Quotient" ], "problem_v1": "A baseball diamond is a square with sides of length 90 ft. A batter hits the ball and runs toward first base with a speed of 25 ft/s.\nAt what rate is his distance from second base changing when he is halfway to first base? Answer=[ANS] ft/s\nAt what rate is his distance from third base changing at the same moment? Answer=[ANS] ft/s", "answer_v1": [ "-11.1803398874989", "11.1803398874989" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A baseball diamond is a square with sides of length 90 ft. A batter hits the ball and runs toward first base with a speed of 20 ft/s.\nAt what rate is his distance from second base changing when he is halfway to first base? Answer=[ANS] ft/s\nAt what rate is his distance from third base changing at the same moment? Answer=[ANS] ft/s", "answer_v2": [ "-8.94427190999916", "8.94427190999916" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A baseball diamond is a square with sides of length 90 ft. A batter hits the ball and runs toward first base with a speed of 22 ft/s.\nAt what rate is his distance from second base changing when he is halfway to first base? Answer=[ANS] ft/s\nAt what rate is his distance from third base changing at the same moment? Answer=[ANS] ft/s", "answer_v3": [ "-9.83869910099907", "9.83869910099907" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0504", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "5", "keywords": [ "calculus", "differentiation" ], "problem_v1": "A ladder 18 ft long rests against a vertical wall. Let $\\theta$ be the angle between the top of the ladder and the wall and let $x$ be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does $x$ change with respect to $\\theta$ when $\\theta=\\pi/3$? $x'(\\pi/3)=$ [ANS]", "answer_v1": [ "9" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A ladder 10 ft long rests against a vertical wall. Let $\\theta$ be the angle between the top of the ladder and the wall and let $x$ be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does $x$ change with respect to $\\theta$ when $\\theta=\\pi/3$? $x'(\\pi/3)=$ [ANS]", "answer_v2": [ "5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A ladder 12 ft long rests against a vertical wall. Let $\\theta$ be the angle between the top of the ladder and the wall and let $x$ be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does $x$ change with respect to $\\theta$ when $\\theta=\\pi/3$? $x'(\\pi/3)=$ [ANS]", "answer_v3": [ "6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0505", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "calculus", "derivative", "implicit", "related rates" ], "problem_v1": "A retail store estimates that weekly sales $s$ and weekly advertising costs $x$ (both in dollars) are related by s=70000-390000 e^{-0.0007x}. The current weekly advertising costs are 2000 dollars and these costs are increasing at the rate of 300 dollars per week. Find the current rate of change of sales.\nRate of change of sales=[ANS]", "answer_v1": [ "20196.2913468176" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A retail store estimates that weekly sales $s$ and weekly advertising costs $x$ (both in dollars) are related by s=50000-440000 e^{-0.0004x}. The current weekly advertising costs are 2000 dollars and these costs are increasing at the rate of 300 dollars per week. Find the current rate of change of sales.\nRate of change of sales=[ANS]", "answer_v2": [ "23724.5693053893" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A retail store estimates that weekly sales $s$ and weekly advertising costs $x$ (both in dollars) are related by s=50000-390000 e^{-0.0005x}. The current weekly advertising costs are 2000 dollars and these costs are increasing at the rate of 300 dollars per week. Find the current rate of change of sales.\nRate of change of sales=[ANS]", "answer_v3": [ "21520.9473085294" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0506", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Related rates", "level": "4", "keywords": [ "calculus", "derivative", "implicit", "related rates" ], "problem_v1": "A price $p$ (in dollars) and demand $x$ for a product are related by 2x^2+1x p+50 p^2=26000. If the price is increasing at a rate of 2 dollars per month when the price is 20 dollars, find the rate of change of the demand.\nRate of change of demand=[ANS]", "answer_v1": [ "-18.6363636363636" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A price $p$ (in dollars) and demand $x$ for a product are related by 2x^2-12x p+50 p^2=8200. If the price is increasing at a rate of 2 dollars per month when the price is 10 dollars, find the rate of change of the demand.\nRate of change of demand=[ANS]", "answer_v2": [ "-0.4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A price $p$ (in dollars) and demand $x$ for a product are related by 2x^2-4x p+50 p^2=8000. If the price is increasing at a rate of 2 dollars per month when the price is 10 dollars, find the rate of change of the demand.\nRate of change of demand=[ANS]", "answer_v3": [ "-10" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0507", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Indeterminate forms and L'Hopital's rule", "level": "3", "keywords": [ "derivatives", "L'Hopital's rule", "calculus", "derivative", "l'hospital's rule", "Indeterminant Forms", "LHopitals rule" ], "problem_v1": "Evaluate the following limit using L'Hospital's rule where appropriate.\n\\lim_{x \\rightarrow 0} \\frac{\\sin(12x)}{\\tan (10x)} Answer: [ANS]", "answer_v1": [ "12/10" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the following limit using L'Hospital's rule where appropriate.\n\\lim_{x \\rightarrow 0} \\frac{\\sin(3x)}{\\tan (15x)} Answer: [ANS]", "answer_v2": [ "3/15" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the following limit using L'Hospital's rule where appropriate.\n\\lim_{x \\rightarrow 0} \\frac{\\sin(6x)}{\\tan (10x)} Answer: [ANS]", "answer_v3": [ "6/10" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0508", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Indeterminate forms and L'Hopital's rule", "level": "3", "keywords": [], "problem_v1": "Evaluate the limit, using L'H\u00f4pital's Rule. Enter INF for $\\infty$,-INF for $-\\infty$, or DNE if the limit does not exist, but is neither $\\infty$ nor $-\\infty$. $ \\lim_{x\\to 1} \\dfrac{x^2+11x-12}{\\ln x}=$ [ANS]", "answer_v1": [ "13" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the limit, using L'H\u00f4pital's Rule. Enter INF for $\\infty$,-INF for $-\\infty$, or DNE if the limit does not exist, but is neither $\\infty$ nor $-\\infty$. $ \\lim_{x\\to 1} \\dfrac{x^2+2x-3}{\\ln x}=$ [ANS]", "answer_v2": [ "4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the limit, using L'H\u00f4pital's Rule. Enter INF for $\\infty$,-INF for $-\\infty$, or DNE if the limit does not exist, but is neither $\\infty$ nor $-\\infty$. $ \\lim_{x\\to 1} \\dfrac{x^2+5x-6}{\\ln x}=$ [ANS]", "answer_v3": [ "7" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0509", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Indeterminate forms and L'Hopital's rule", "level": "3", "keywords": [ "derivatives", "L'Hopital's rule" ], "problem_v1": "The function \\sqrt{x^2+9x+13}-x has one horizontal asymptote at $y=$ [ANS]", "answer_v1": [ "4.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The function \\sqrt{x^2+3x+19}-x has one horizontal asymptote at $y=$ [ANS]", "answer_v2": [ "1.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The function \\sqrt{x^2+5x+13}-x has one horizontal asymptote at $y=$ [ANS]", "answer_v3": [ "2.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0511", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Indeterminate forms and L'Hopital's rule", "level": "3", "keywords": [ "calculus", "derivative", "limits", "lhopitals rule" ], "problem_v1": "If $x$, and $y$ are both positive, evaluate $ \\lim_{p\\rightarrow 0}\\frac{\\ln(0.7\\, x^p+0.3\\, y^p)}{p}=$ [ANS]\nand $ \\lim_{p\\rightarrow 0}(0.7\\, x^p+0.3\\, y^p)^{{1}/{p}}=$ [ANS]", "answer_v1": [ "0.7*ln(x)+0.3*ln(y)", "x^0.7*y^0.3" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "If $x$, and $y$ are both positive, evaluate $ \\lim_{p\\rightarrow 0}\\frac{\\ln(0.1\\, x^p+0.9\\, y^p)}{p}=$ [ANS]\nand $ \\lim_{p\\rightarrow 0}(0.1\\, x^p+0.9\\, y^p)^{{1}/{p}}=$ [ANS]", "answer_v2": [ "0.1*ln(x)+0.9*ln(y)", "x^0.1*y^0.9" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "If $x$, and $y$ are both positive, evaluate $ \\lim_{p\\rightarrow 0}\\frac{\\ln(0.3\\, x^p+0.7\\, y^p)}{p}=$ [ANS]\nand $ \\lim_{p\\rightarrow 0}(0.3\\, x^p+0.7\\, y^p)^{{1}/{p}}=$ [ANS]", "answer_v3": [ "0.3*ln(x)+0.7*ln(y)", "x^0.3*y^0.7" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0512", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Indeterminate forms and L'Hopital's rule", "level": "3", "keywords": [ "calculus", "derivative", "limits", "lhopitals rule" ], "problem_v1": "Evaluate the limit below, given that f(t)=\\left(\\frac{4^t+6^t}{4}\\right)^{1/t}\\quad\\hbox{for} t\\ne 0. $\\lim\\limits_{t\\to+\\infty} f(t)=$ [ANS]", "answer_v1": [ "6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the limit below, given that f(t)=\\left(\\frac{2^t+5^t}{2}\\right)^{1/t}\\quad\\hbox{for} t\\ne 0. $\\lim\\limits_{t\\to+\\infty} f(t)=$ [ANS]", "answer_v2": [ "5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the limit below, given that f(t)=\\left(\\frac{2^t+4^t}{3}\\right)^{1/t}\\quad\\hbox{for} t\\ne 0. $\\lim\\limits_{t\\to+\\infty} f(t)=$ [ANS]", "answer_v3": [ "4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0513", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Indeterminate forms and L'Hopital's rule", "level": "3", "keywords": [ "calculus", "derivative", "limits", "lhopitals rule" ], "problem_v1": "Find the limit: $ \\lim_{x \\rightarrow 5}\\frac{\\ln(x/5)}{x^2-25}=$ [ANS]\n(Enter undefined if the limit does not exist.)", "answer_v1": [ "1/50" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the limit: $ \\lim_{x \\rightarrow 2}\\frac{\\ln(x/2)}{x^2-4}=$ [ANS]\n(Enter undefined if the limit does not exist.)", "answer_v2": [ "1/8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the limit: $ \\lim_{x \\rightarrow 3}\\frac{\\ln(x/3)}{x^2-9}=$ [ANS]\n(Enter undefined if the limit does not exist.)", "answer_v3": [ "1/18" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0514", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Indeterminate forms and L'Hopital's rule", "level": "4", "keywords": [ "calculus", "limit", "trigonometric", "indeterminant form", "lhopitals rule", "logarithm" ], "problem_v1": "Consider the limit \\lim_{x\\to 1}\\frac{\\sin(9 \\ln(x))}{\\ln(x)} To simplify this limit, we should substitute $y=$ [ANS]\nAs $x\\to 1$, $y\\to$ [ANS]\nThus we find that the value of limit is [ANS]", "answer_v1": [ "9*ln(x)", "0", "9" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the limit \\lim_{x\\to 1}\\frac{\\sin(3 \\ln(x))}{\\ln(x)} To simplify this limit, we should substitute $y=$ [ANS]\nAs $x\\to 1$, $y\\to$ [ANS]\nThus we find that the value of limit is [ANS]", "answer_v2": [ "3*ln(x)", "0", "3" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the limit \\lim_{x\\to 1}\\frac{\\sin(5 \\ln(x))}{\\ln(x)} To simplify this limit, we should substitute $y=$ [ANS]\nAs $x\\to 1$, $y\\to$ [ANS]\nThus we find that the value of limit is [ANS]", "answer_v3": [ "5*ln(x)", "0", "5" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0515", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Newton's method", "level": "2", "keywords": [ "derivatives", "Newton's method", "Newton's Method", "Root", "Approximate", "Trigonometry", "Calculus" ], "problem_v1": "Use Newton's method to approximate a root of the equation $\\cos(x^2+5)=x^3$ as follows.\nLet $x_1=2$ be the initial approximation.\nThe second approximation $x_2$ is [ANS].", "answer_v1": [ "1.34709695014787" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Newton's method to approximate a root of the equation $\\cos(x^2+2)=x^3$ as follows.\nLet $x_1=2$ be the initial approximation.\nThe second approximation $x_2$ is [ANS].", "answer_v2": [ "1.35309584128988" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Newton's method to approximate a root of the equation $\\cos(x^2+3)=x^3$ as follows.\nLet $x_1=2$ be the initial approximation.\nThe second approximation $x_2$ is [ANS].", "answer_v3": [ "1.50464012171966" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0516", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Newton's method", "level": "3", "keywords": [ "calculus", "derivatives", "newtons method" ], "problem_v1": "Use Newton's Method to approximate $7^{\\frac{1}{3}}$ and compare with the value obtained from a calculator. $7^{\\frac{1}{3}} \\approx$ [ANS]", "answer_v1": [ "1.91293" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Newton's Method to approximate $4^{\\frac{1}{3}}$ and compare with the value obtained from a calculator. $4^{\\frac{1}{3}} \\approx$ [ANS]", "answer_v2": [ "1.5874" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Newton's Method to approximate $5^{\\frac{1}{3}}$ and compare with the value obtained from a calculator. $5^{\\frac{1}{3}} \\approx$ [ANS]", "answer_v3": [ "1.70998" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0517", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Newton's method", "level": "2", "keywords": [ "calculus", "derivatives", "newtons method" ], "problem_v1": "Use Newton's Method to approximate $\\sqrt{14}$ to 6 significant figures starting with $x_0=3.5$. Compare with the value obtained from a calculator. $\\sqrt{14} \\approx$ [ANS]", "answer_v1": [ "3.74166" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Newton's Method to approximate $\\sqrt{10}$ to 6 significant figures starting with $x_0=3.5$. Compare with the value obtained from a calculator. $\\sqrt{10} \\approx$ [ANS]", "answer_v2": [ "3.16228" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Newton's Method to approximate $\\sqrt{11}$ to 6 significant figures starting with $x_0=3.5$. Compare with the value obtained from a calculator. $\\sqrt{11} \\approx$ [ANS]", "answer_v3": [ "3.31662" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0518", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Newton's method", "level": "3", "keywords": [ "derivative' 'newton", "derivatives", "Newton's method", "Calculus", "Newton's Method", "Root", "Approximate", "Trigonometry", "Newtons method" ], "problem_v1": "Find the positive value of $x$ that satisfies $x=4.100 \\sin(x)$. Give the answer to four places of accuracy. $x\\approx$ [ANS]\nRemember to calculate the trig functions in radian mode.", "answer_v1": [ "2.48916867983044" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the positive value of $x$ that satisfies $x=1.400 \\sin(x)$. Give the answer to four places of accuracy. $x\\approx$ [ANS]\nRemember to calculate the trig functions in radian mode.", "answer_v2": [ "1.37258984608488" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the positive value of $x$ that satisfies $x=2.300 \\sin(x)$. Give the answer to four places of accuracy. $x\\approx$ [ANS]\nRemember to calculate the trig functions in radian mode.", "answer_v3": [ "2.04559012035529" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0519", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Newton's method", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Suppose that we use Newton's Method to approximate $r=\\sqrt[3]{336}$, which is the zero of the function $f(x)=x^3-336$. We begin with a good guess, say $x_1=8$. Then Newton's Method proceeds by the recursion $x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}$.\nCompute the first few terms of the sequence $\\lbrace x_n \\rbrace$ obtained from Newton's method.\n$x_1=8$\n$x_2=$ [ANS]\n$x_3=$ [ANS]\n$x_4=$ [ANS]\n$x_5=$ [ANS]\n$x_6=$ [ANS]\n$x_7=$ [ANS]", "answer_v1": [ "7.08333333333333", "6.95447135717032", "6.95205413043669", "6.952053289773", "6.9520532897729", "6.9520532897729" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Suppose that we use Newton's Method to approximate $r=\\sqrt[3]{60}$, which is the zero of the function $f(x)=x^3-60$. We begin with a good guess, say $x_1=5$. Then Newton's Method proceeds by the recursion $x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}$.\nCompute the first few terms of the sequence $\\lbrace x_n \\rbrace$ obtained from Newton's method.\n$x_1=5$\n$x_2=$ [ANS]\n$x_3=$ [ANS]\n$x_4=$ [ANS]\n$x_5=$ [ANS]\n$x_6=$ [ANS]\n$x_7=$ [ANS]", "answer_v2": [ "4.13333333333333", "3.92621112267314", "3.91490038282432", "3.91486764144269", "3.91486764116886", "3.91486764116886" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Suppose that we use Newton's Method to approximate $r=\\sqrt[3]{120}$, which is the zero of the function $f(x)=x^3-120$. We begin with a good guess, say $x_1=6$. Then Newton's Method proceeds by the recursion $x_{n+1}=x_n-\\frac{f(x_n)}{f'(x_n)}$.\nCompute the first few terms of the sequence $\\lbrace x_n \\rbrace$ obtained from Newton's method.\n$x_1=6$\n$x_2=$ [ANS]\n$x_3=$ [ANS]\n$x_4=$ [ANS]\n$x_5=$ [ANS]\n$x_6=$ [ANS]\n$x_7=$ [ANS]", "answer_v3": [ "5.11111111111111", "4.9385983336834", "4.93243186434708", "4.93242414867301", "4.93242414866094", "4.93242414866094" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0520", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Newton's method", "level": "3", "keywords": [ "derivatives", "Newtons method" ], "problem_v1": "Use Newton's method to find the positive value of $x$ which satisfies $x=3.8 \\cos(x)$. Compute enough approximations so that your answer is within.05 of the exact answer. $x=$ [ANS]\n(Remember to calculate the trig functions in radian mode!)", "answer_v1": [ "1.23874289906441" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Newton's method to find the positive value of $x$ which satisfies $x=0.5 \\cos(x)$. Compute enough approximations so that your answer is within.05 of the exact answer. $x=$ [ANS]\n(Remember to calculate the trig functions in radian mode!)", "answer_v2": [ "0.450183611294874" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Newton's method to find the positive value of $x$ which satisfies $x=1.6 \\cos(x)$. Compute enough approximations so that your answer is within.05 of the exact answer. $x=$ [ANS]\n(Remember to calculate the trig functions in radian mode!)", "answer_v3": [ "0.941596635990477" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0521", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Newton's method", "level": "3", "keywords": [ "derivatives", "Newton's method" ], "problem_v1": "Use Newton's method to approximate a solution of the equation $e^{2x}=x+2$, starting with the initial guess indicated. $x_1=2$. $x_2=$ [ANS]. $x_3=$ [ANS]. The solution to the equation found by Newton's method is $x=$ [ANS].", "answer_v1": [ "1.5323486107922", "1.10479587072546", "0.447542160637616" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Use Newton's method to approximate a solution of the equation $e^{-3x}=5x-7$, starting with the initial guess indicated. $x_1=-2$. $x_2=$ [ANS]. $x_3=$ [ANS]. The solution to the equation found by Newton's method is $x=$ [ANS].", "answer_v2": [ "-1.65404961312304", "-1.28935005679198", "1.40297248973821" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Use Newton's method to approximate a solution of the equation $e^{-x}=x-4$, starting with the initial guess indicated. $x_1=1$. $x_2=$ [ANS]. $x_3=$ [ANS]. The solution to the equation found by Newton's method is $x=$ [ANS].", "answer_v3": [ "3.46211715726001", "4.01405277350198", "4.01798910282853" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0522", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Newton's method", "level": "4", "keywords": [ "inverse functions", "Newton's method" ], "problem_v1": "Find the approximate value of $f^{-1}(2)$ for the function $f(x)=0.7x+0.6x^3+0.6x^5$. $f^{-1}(2)=$ [ANS]. Hint: Do not try to find $f^{-1}(x)$ for arbitrary $x$ ; it's impossible. Use the intersect feature on your calculator or some other technology. Your answer needs to be correct to within one tenth of a percent.", "answer_v1": [ "1.01772930223755" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the approximate value of $f^{-1}(-1)$ for the function $f(x)=0.1x+0.9x^3+0.2x^5$. $f^{-1}(-1)=$ [ANS]. Hint: Do not try to find $f^{-1}(x)$ for arbitrary $x$ ; it's impossible. Use the intersect feature on your calculator or some other technology. Your answer needs to be correct to within one tenth of a percent.", "answer_v2": [ "-0.943563527078741" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the approximate value of $f^{-1}(-2)$ for the function $f(x)=0.3x+0.6x^3+0.3x^5$. $f^{-1}(-2)=$ [ANS]. Hint: Do not try to find $f^{-1}(x)$ for arbitrary $x$ ; it's impossible. Use the intersect feature on your calculator or some other technology. Your answer needs to be correct to within one tenth of a percent.", "answer_v3": [ "-1.17539846584523" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0523", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Newton's method", "level": "4", "keywords": [], "problem_v1": "A loan of \\$240000 is repaid over 20 years by monthly payments of \\$1800. Use two iterations of the Newton-Raphson method starting with the initial guess of an annual interest rate of 12\\% (compounded monthly), to determine the annual interest rate. NOTE: If P is borrowed at a rate of r per period, and is repaid over N periods by payments of Y, then: $Pr+Y[(1+r)^{-N}-1]=0$. If $f(r)=240000 r+1800 [(1+r)^{-400}-1]$ $f'(r)=$ [ANS]\nAnnual interest rate $=$ [ANS]\nReport your answer to 3 decimal places.\n(you will lose 25\\% of your points if you do)", "answer_v1": [ "240000-1800*400*(1+r)^{-401}", "8.459" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": " A loan of \\$45000 is repaid over 20 years by monthly payments of \\$337.5. Use two iterations of the Newton-Raphson method starting with the initial guess of an annual interest rate of 12\\% (compounded monthly), to determine the annual interest rate. NOTE: If P is borrowed at a rate of r per period, and is repaid over N periods by payments of Y, then: $Pr+Y[(1+r)^{-N}-1]=0$. If $f(r)=45000 r+337.5 [(1+r)^{-400}-1]$ $f'(r)=$ [ANS]\nAnnual interest rate $=$ [ANS]\nReport your answer to 3 decimal places.\n(you will lose 25\\% of your points if you do)", "answer_v2": [ "45000-337.5*400*(1+r)^{-401}", "8.459" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": " A loan of \\$105000 is repaid over 20 years by monthly payments of \\$787.5. Use two iterations of the Newton-Raphson method starting with the initial guess of an annual interest rate of 12\\% (compounded monthly), to determine the annual interest rate. NOTE: If P is borrowed at a rate of r per period, and is repaid over N periods by payments of Y, then: $Pr+Y[(1+r)^{-N}-1]=0$. If $f(r)=105000 r+787.5 [(1+r)^{-400}-1]$ $f'(r)=$ [ANS]\nAnnual interest rate $=$ [ANS]\nReport your answer to 3 decimal places.\n(you will lose 25\\% of your points if you do)", "answer_v3": [ "105000-787.5*400*(1+r)^{-401}", "8.459" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0524", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Elasticity of demand", "level": "3", "keywords": [ "calculus", "elasticity", "economics" ], "problem_v1": "The demand function for a certain item is $x=200(80-p^2)$\n(a) Evaluate the elasticity at 7. $E(7)=$ [ANS]\n(b) For what value of p, the demand is unitary. $p=$ [ANS]", "answer_v1": [ "2*7*7/(80-7*7)", "5.16397779494322" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The demand function for a certain item is $x=200(80-p^2)$\n(a) Evaluate the elasticity at 4. $E(4)=$ [ANS]\n(b) For what value of p, the demand is unitary. $p=$ [ANS]", "answer_v2": [ "2*4*4/(80-4*4)", "5.16397779494322" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The demand function for a certain item is $x=200(80-p^2)$\n(a) Evaluate the elasticity at 5. $E(5)=$ [ANS]\n(b) For what value of p, the demand is unitary. $p=$ [ANS]", "answer_v3": [ "2*5*5/(80-5*5)", "5.16397779494322" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0525", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Elasticity of demand", "level": "5", "keywords": [ "calculus", "elasticity", "economics" ], "problem_v1": "Currently, 10,000 people take city buses each day and pay 2 dollars for a ticket. The number of people taking city buses at price p dollars per ticket is given $x=5000\\sqrt{6-p}$\n(a) Evaluate the elasticity at 5. $E(5)=$ [ANS]\n(b) Currently, the price per ticket is 2 dollars. Should the price be raised in order to increase revenue? Enter 1 for Yes or 2 for No. $answer=$ [ANS]\n(c) For what value of p, the demand is unitary. $p=$ [ANS]\n(d) What is the maximum revenue? $Revenue=$ [ANS]", "answer_v1": [ "2.5", "1", "4", "20000*2^0.5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Currently, 10,000 people take city buses each day and pay 2 dollars for a ticket. The number of people taking city buses at price p dollars per ticket is given $x=5000\\sqrt{6-p}$\n(a) Evaluate the elasticity at 2. $E(2)=$ [ANS]\n(b) Currently, the price per ticket is 2 dollars. Should the price be raised in order to increase revenue? Enter 1 for Yes or 2 for No. $answer=$ [ANS]\n(c) For what value of p, the demand is unitary. $p=$ [ANS]\n(d) What is the maximum revenue? $Revenue=$ [ANS]", "answer_v2": [ "0.25", "1", "4", "20000*2^0.5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Currently, 10,000 people take city buses each day and pay 2 dollars for a ticket. The number of people taking city buses at price p dollars per ticket is given $x=5000\\sqrt{6-p}$\n(a) Evaluate the elasticity at 3. $E(3)=$ [ANS]\n(b) Currently, the price per ticket is 2 dollars. Should the price be raised in order to increase revenue? Enter 1 for Yes or 2 for No. $answer=$ [ANS]\n(c) For what value of p, the demand is unitary. $p=$ [ANS]\n(d) What is the maximum revenue? $Revenue=$ [ANS]", "answer_v3": [ "0.5", "1", "4", "20000*2^0.5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0526", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Elasticity of demand", "level": "3", "keywords": [], "problem_v1": "Let $Q(p)$ be the demand function for a certain product, where $p$ is price. If $R$ is a function of $p$ for the total revenue, $\\frac{dR}{dp}=MR$ is the marginal revenue. Remember: the price elasticity of demand can be defined as: $E=-\\frac{dQ}{dp}\\frac{p}{Q}$. $MR=$ [ANS]\nYour answer should be in terms of $Q$ and $E$.\nGiven that elasticity is a function of price; based on your answer to the previous question, how should price be set to maximize revenue? [ANS] A. $E > 1$ B. $0 < E < 1$ C. $E=1$ D. Not enough information\n(you will lose 25\\% of your points if you do)", "answer_v1": [ "Q*(1-E)", "C" ], "answer_type_v1": [ "EX", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D" ] ], "problem_v2": "Let $Q(p)$ be the demand function for a certain product, where $p$ is price. If $R$ is a function of $p$ for the total revenue, $\\frac{dR}{dp}=MR$ is the marginal revenue. Remember: the price elasticity of demand can be defined as: $E=-\\frac{dQ}{dp}\\frac{p}{Q}$. $MR=$ [ANS]\nYour answer should be in terms of $Q$ and $E$.\nGiven that elasticity is a function of price; based on your answer to the previous question, how should price be set to maximize revenue? [ANS] A. $E=1$ B. $0 < E < 1$ C. $E > 1$ D. Not enough information\n(you will lose 25\\% of your points if you do)", "answer_v2": [ "Q*(1-E)", "A" ], "answer_type_v2": [ "EX", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D" ] ], "problem_v3": "Let $Q(p)$ be the demand function for a certain product, where $p$ is price. If $R$ is a function of $p$ for the total revenue, $\\frac{dR}{dp}=MR$ is the marginal revenue. Remember: the price elasticity of demand can be defined as: $E=-\\frac{dQ}{dp}\\frac{p}{Q}$. $MR=$ [ANS]\nYour answer should be in terms of $Q$ and $E$.\nGiven that elasticity is a function of price; based on your answer to the previous question, how should price be set to maximize revenue? [ANS] A. $E > 1$ B. $E=1$ C. $0 < E < 1$ D. Not enough information\n(you will lose 25\\% of your points if you do)", "answer_v3": [ "Q*(1-E)", "B" ], "answer_type_v3": [ "EX", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_single_variable_0527", "subject": "Calculus_-_single_variable", "topic": "Applications of differentiation", "subtopic": "Elasticity of demand", "level": "3", "keywords": [], "problem_v1": "Shark Inc. has determined that demand for its newest netbook model is given by $\\ln{q}-3 \\ln{p}+0.004 p=7$, where q is the number of netbooks Shark can sell at a price of p dollars per unit. Shark has determined that this model is valid for prices $p \\ge 100$. You may \ufb01nd it useful in this problem to know that elasticity of demand is de\ufb01ned to be $E(p)=\\frac{dq}{dp}\\frac{p}{q}$ Find $E(p)=$ [ANS]\nYour answer should only be in terms of $p$. What price will maximize revenue. If the price is less than 100, write 'NA'. $p=$ [ANS]", "answer_v1": [ "-(0.004*p-3)", "(1+3)/0.004" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Shark Inc. has determined that demand for its newest netbook model is given by $\\ln{q}-5 \\ln{p}+0.001 p=7$, where q is the number of netbooks Shark can sell at a price of p dollars per unit. Shark has determined that this model is valid for prices $p \\ge 100$. You may \ufb01nd it useful in this problem to know that elasticity of demand is de\ufb01ned to be $E(p)=\\frac{dq}{dp}\\frac{p}{q}$ Find $E(p)=$ [ANS]\nYour answer should only be in terms of $p$. What price will maximize revenue. If the price is less than 100, write 'NA'. $p=$ [ANS]", "answer_v2": [ "-(0.001*p-5)", "(1+5)/0.001" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Shark Inc. has determined that demand for its newest netbook model is given by $\\ln{q}-4 \\ln{p}+0.002 p=7$, where q is the number of netbooks Shark can sell at a price of p dollars per unit. Shark has determined that this model is valid for prices $p \\ge 100$. You may \ufb01nd it useful in this problem to know that elasticity of demand is de\ufb01ned to be $E(p)=\\frac{dq}{dp}\\frac{p}{q}$ Find $E(p)=$ [ANS]\nYour answer should only be in terms of $p$. What price will maximize revenue. If the price is less than 100, write 'NA'. $p=$ [ANS]", "answer_v3": [ "-(0.002*p-4)", "(1+4)/0.002" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0528", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "2", "keywords": [ "integrals", "theory" ], "problem_v1": "If $ \\int_{4}^{8} f(x) dx=17$, then $ \\int_{8}^{4} f(t) dt=$ [ANS]", "answer_v1": [ "-17" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $ \\int_{1}^{10} f(x) dx=-49$, then $ \\int_{10}^{1} f(t) dt=$ [ANS]", "answer_v2": [ "49" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $ \\int_{2}^{9} f(x) dx=-31$, then $ \\int_{9}^{2} f(t) dt=$ [ANS]", "answer_v3": [ "31" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0529", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "2", "keywords": [ "integrals", "theory", "definite", "area" ], "problem_v1": "You are given the four points in the plane $A=(4,5)$, $B=(8,-3)$, $C=(11,5)$, and $D=(15,-3)$. The graph of the function $f(x)$ consists of the three line segments $AB$, $BC$ and $CD$. Find the integral $ \\int_{4}^{15} f(x)\\,dx$ by interpreting the integral in terms of sums and/or differences of areas of elementary figures.\n$ \\int_{4}^{15} f(x)\\,dx=$ [ANS]", "answer_v1": [ "11" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You are given the four points in the plane $A=(-7,2)$, $B=(-4,-8)$, $C=(-1,2)$, and $D=(2,-5)$. The graph of the function $f(x)$ consists of the three line segments $AB$, $BC$ and $CD$. Find the integral $ \\int_{-7}^{2} f(x)\\,dx$ by interpreting the integral in terms of sums and/or differences of areas of elementary figures.\n$ \\int_{-7}^{2} f(x)\\,dx=$ [ANS]", "answer_v2": [ "-22.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You are given the four points in the plane $A=(-3,3)$, $B=(1,-2)$, $C=(4,7)$, and $D=(9,-7)$. The graph of the function $f(x)$ consists of the three line segments $AB$, $BC$ and $CD$. Find the integral $ \\int_{-3}^{9} f(x)\\,dx$ by interpreting the integral in terms of sums and/or differences of areas of elementary figures.\n$ \\int_{-3}^{9} f(x)\\,dx=$ [ANS]", "answer_v3": [ "9.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0530", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "3", "keywords": [ "integrals", "theory" ], "problem_v1": "Let $ \\int_{5}^{9.5} f(x)\\, dx=7, \\ \\int_{5}^{6.5} f(x)\\, dx=7, \\ \\int_{8}^{9.5} f(x)\\, dx=8$.\nFind $ \\int_{6.5}^{8} f(x)\\, dx=$ [ANS]\nand $ \\int_{8}^{6.5} 7 f(x)-7 \\, dx=$ [ANS]", "answer_v1": [ "7-(7)-(8)", "-(7*(7-(7)-(8)))+7*1.5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $ \\int_{-9}^{0} f(x)\\, dx=10, \\ \\int_{-9}^{-6} f(x)\\, dx=3, \\ \\int_{-3}^{0} f(x)\\, dx=4$.\nFind $ \\int_{-6}^{-3} f(x)\\, dx=$ [ANS]\nand $ \\int_{-3}^{-6} 10 f(x)-3 \\, dx=$ [ANS]", "answer_v2": [ "10-(3)-(4)", "-(10*(10-(3)-(4)))+3*3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $ \\int_{-4}^{0.5} f(x)\\, dx=7, \\ \\int_{-4}^{-2.5} f(x)\\, dx=4, \\ \\int_{-1}^{0.5} f(x)\\, dx=6$.\nFind $ \\int_{-2.5}^{-1} f(x)\\, dx=$ [ANS]\nand $ \\int_{-1}^{-2.5} 7 f(x)-4 \\, dx=$ [ANS]", "answer_v3": [ "7-(4)-(6)", "-(7*(7-(4)-(6)))+4*1.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0531", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "2", "keywords": [ "integrals", "theory" ], "problem_v1": "If $13 \\le f(x) \\le 19$, then [ANS] $ \\le \\int_{4}^{8} f(x) dx \\le$ [ANS]", "answer_v1": [ "13*(8-(4))", "19*(8-(4))" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If $10 \\le f(x) \\le 17$, then [ANS] $ \\le \\int_{1}^{10} f(x) dx \\le$ [ANS]", "answer_v2": [ "10*(10-(1))", "17*(10-(1))" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If $11 \\le f(x) \\le 18$, then [ANS] $ \\le \\int_{2}^{9} f(x) dx \\le$ [ANS]", "answer_v3": [ "11*(9-(2))", "18*(9-(2))" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0532", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "2", "keywords": [ "integrals", "theory" ], "problem_v1": "Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry. \\int_{3}^{6} (2+2x) dx Answer: [ANS]", "answer_v1": [ "2*6+6^2-(2*3+3^2)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry. \\int_{1}^{7} (-7+2x) dx Answer: [ANS]", "answer_v2": [ "-7*7+7^2-(-7*1+1^2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the integral below by interpreting it in terms of areas. In other words, draw a picture of the region the integral represents, and find the area using high school geometry. \\int_{1}^{6} (-4+2x) dx Answer: [ANS]", "answer_v3": [ "-4*6+6^2-(-4*1+1^2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0533", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "3", "keywords": [], "problem_v1": "Let \\int_{0}^2 f(x) \\, dx=12, \\quad \\int_{0}^3 f(x) \\, dx=10, \\quad \\int_{0}^2 g(x) \\, dx=-5, \\quad \\int_{2}^3 g(x) \\, dx=9, \\quad Use these values to evaluate the given definite integrals. a) $ \\int_{0}^2 \\left(f(x)+g(x)\\right) \\, dx=$ [ANS]\nb) $ \\int_{0}^3 \\left(f(x)-g(x)\\right) \\, dx=$ [ANS]\nc) $ \\int_{2}^3 \\left(3f(x)+2g(x)\\right) \\, dx=$ [ANS]\nd) Find the value $a$ such that $ \\int_{0}^3 \\left(a f(x)+g(x)\\right) \\, dx=0$. $a=$ [ANS]", "answer_v1": [ "7", "6", "12", "-0.4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let \\int_{0}^2 f(x) \\, dx=2, \\quad \\int_{0}^3 f(x) \\, dx=-3, \\quad \\int_{0}^2 g(x) \\, dx=-15, \\quad \\int_{2}^3 g(x) \\, dx=-3, \\quad Use these values to evaluate the given definite integrals. a) $ \\int_{0}^2 \\left(f(x)+g(x)\\right) \\, dx=$ [ANS]\nb) $ \\int_{0}^3 \\left(f(x)-g(x)\\right) \\, dx=$ [ANS]\nc) $ \\int_{2}^3 \\left(3f(x)+2g(x)\\right) \\, dx=$ [ANS]\nd) Find the value $a$ such that $ \\int_{0}^3 \\left(a f(x)+g(x)\\right) \\, dx=0$. $a=$ [ANS]", "answer_v2": [ "-13", "15", "-21", "-6" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let \\int_{0}^2 f(x) \\, dx=5, \\quad \\int_{0}^3 f(x) \\, dx=5, \\quad \\int_{0}^2 g(x) \\, dx=-4, \\quad \\int_{2}^3 g(x) \\, dx=13, \\quad Use these values to evaluate the given definite integrals. a) $ \\int_{0}^2 \\left(f(x)+g(x)\\right) \\, dx=$ [ANS]\nb) $ \\int_{0}^3 \\left(f(x)-g(x)\\right) \\, dx=$ [ANS]\nc) $ \\int_{2}^3 \\left(3f(x)+2g(x)\\right) \\, dx=$ [ANS]\nd) Find the value $a$ such that $ \\int_{0}^3 \\left(a f(x)+g(x)\\right) \\, dx=0$. $a=$ [ANS]", "answer_v3": [ "1", "-4", "26", "-1.8" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0534", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "5", "keywords": [], "problem_v1": "An object is thrown straight up with a velocity, in ft/s, given by $v(t)=-32t+75$, where $t$ is in seconds, from a height of 40 feet. a) What is the object's initial velocity? [ANS] [ANS] b) What is the object's maximum velocity? [ANS] [ANS] c) What is the object's maximum displacement? [ANS] [ANS] d) When does the maximum displacement occur? [ANS] [ANS] e) When is the object's displacement 0? [ANS] [ANS] f) What is the object's maximum height? [ANS] [ANS]", "answer_v1": [ "75", "FT/S", "75", "ft/s", "87.890625", "ft", "2.34375", "s", "4.6875", "s", "127.890625", "ft" ], "answer_type_v1": [ "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v1": [ [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ] ], "problem_v2": "An object is thrown straight up with a velocity, in ft/s, given by $v(t)=-32t+61$, where $t$ is in seconds, from a height of 12 feet. a) What is the object's initial velocity? [ANS] [ANS] b) What is the object's maximum velocity? [ANS] [ANS] c) What is the object's maximum displacement? [ANS] [ANS] d) When does the maximum displacement occur? [ANS] [ANS] e) When is the object's displacement 0? [ANS] [ANS] f) What is the object's maximum height? [ANS] [ANS]", "answer_v2": [ "61", "FT/S", "61", "ft/s", "58.140625", "ft", "1.90625", "s", "3.8125", "s", "70.140625", "ft" ], "answer_type_v2": [ "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v2": [ [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ] ], "problem_v3": "An object is thrown straight up with a velocity, in ft/s, given by $v(t)=-32t+57$, where $t$ is in seconds, from a height of 22 feet. a) What is the object's initial velocity? [ANS] [ANS] b) What is the object's maximum velocity? [ANS] [ANS] c) What is the object's maximum displacement? [ANS] [ANS] d) When does the maximum displacement occur? [ANS] [ANS] e) When is the object's displacement 0? [ANS] [ANS] f) What is the object's maximum height? [ANS] [ANS]", "answer_v3": [ "57", "FT/S", "57", "ft/s", "50.765625", "ft", "1.78125", "s", "3.5625", "s", "72.765625", "ft" ], "answer_type_v3": [ "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v3": [ [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ], [], [ "ft", "s", "ft/s" ] ] }, { "id": "Calculus_-_single_variable_0535", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "5", "keywords": [ "density function' 'integral" ], "problem_v1": "The density function for the number of times the riders scream on a roller coaster is given by f(x)=\\begin{cases} \\frac{1}{8 \\pi}(1-\\cos(6x)) & \\text{if} \\;\\; 0\\leq x \\leq 8 \\pi\\\\ 0 & \\text{otherwise.} \\end{cases} Find the mean number of screams over the course of the ride. [ANS]", "answer_v1": [ "12.5663706143592" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The density function for the number of times the riders scream on a roller coaster is given by f(x)=\\begin{cases} \\frac{1}{\\pi}(1-\\cos(10x)) & \\text{if} \\;\\; 0\\leq x \\leq \\pi\\\\ 0 & \\text{otherwise.} \\end{cases} Find the mean number of screams over the course of the ride. [ANS]", "answer_v2": [ "1.5707963267949" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The density function for the number of times the riders scream on a roller coaster is given by f(x)=\\begin{cases} \\frac{1}{4 \\pi}(1-\\cos(8x)) & \\text{if} \\;\\; 0\\leq x \\leq 4 \\pi\\\\ 0 & \\text{otherwise.} \\end{cases} Find the mean number of screams over the course of the ride. [ANS]", "answer_v3": [ "6.28318530717959" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0536", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "2", "keywords": [ "density function' 'integral" ], "problem_v1": "Let $p(t)$ be the density function given by p(t)=\\begin{cases} \\frac{6}{117649}t^{5} & \\text{if} \\;\\; 0\\leq t \\leq 7\\\\ 0 & \\text{otherwise.} \\end{cases} Find the median value of $p(t)$. [ANS]", "answer_v1": [ "6.23629102698237" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $p(t)$ be the density function given by p(t)=\\begin{cases} \\frac{9}{1}t^{8} & \\text{if} \\;\\; 0\\leq t \\leq 1\\\\ 0 & \\text{otherwise.} \\end{cases} Find the median value of $p(t)$. [ANS]", "answer_v2": [ "0.92587471228729" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $p(t)$ be the density function given by p(t)=\\begin{cases} \\frac{6}{729}t^{5} & \\text{if} \\;\\; 0\\leq t \\leq 3\\\\ 0 & \\text{otherwise.} \\end{cases} Find the median value of $p(t)$. [ANS]", "answer_v3": [ "2.67269615442102" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0537", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "3", "keywords": [ "integral' 'summation' 'area' 'riemann", "integrals", "inverse functions", "area", "Definite", "Integral" ], "problem_v1": "Consider the function f(x)=x^3-12x^2+78x+7 By drawing a suitable picture, find a relation between the definite integrals $ \\int_1 ^2 f(x)\\,dx$ and $ \\int_{74}^{123}f^{-1}(x)\\,dx$. Use this relation to find the second of these two integrals $ \\int_{74}^{123}f^{-1}(x)\\,dx$=[ANS]", "answer_v1": [ "72.25" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider the function f(x)=x^3-3x^2+48x+3 By drawing a suitable picture, find a relation between the definite integrals $ \\int_1 ^2 f(x)\\,dx$ and $ \\int_{49}^{95}f^{-1}(x)\\,dx$. Use this relation to find the second of these two integrals $ \\int_{49}^{95}f^{-1}(x)\\,dx$=[ANS]", "answer_v2": [ "69.25" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider the function f(x)=x^3-6x^2+42x+4 By drawing a suitable picture, find a relation between the definite integrals $ \\int_1 ^2 f(x)\\,dx$ and $ \\int_{41}^{72}f^{-1}(x)\\,dx$. Use this relation to find the second of these two integrals $ \\int_{41}^{72}f^{-1}(x)\\,dx$=[ANS]", "answer_v3": [ "46.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0538", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "3", "keywords": [ "integral' 'summation' 'area", "integrals", "inverse functions", "area", "Definite", "Integral", "Inverse Function" ], "problem_v1": "Suppose $f(x)$ is continuous and decreasing on the closed interval $6\\le x\\le 12$, that $f(6)=11$, $f(12)=5$ and that $ \\int_{6}^{12}f(x)\\,dx=40.86079$. Then $ \\int_{5}^{11}f^{-1}(x)\\,dx$=[ANS]", "answer_v1": [ "46.86079" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose $f(x)$ is continuous and decreasing on the closed interval $3\\le x\\le 11$, that $f(3)=6$, $f(11)=2$ and that $ \\int_{3}^{11}f(x)\\,dx=46.296574$. Then $ \\int_{2}^{6}f^{-1}(x)\\,dx$=[ANS]", "answer_v2": [ "42.296574" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose $f(x)$ is continuous and decreasing on the closed interval $4\\le x\\le 11$, that $f(4)=8$, $f(11)=3$ and that $ \\int_{4}^{11}f(x)\\,dx=28.216148$. Then $ \\int_{3}^{8}f^{-1}(x)\\,dx$=[ANS]", "answer_v3": [ "27.216148" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0539", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "3", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "properties of integrals", "area" ], "problem_v1": "Let $\\int_a^b f(x) dx=25$ and $\\int_a^b (f(x))^2 dx=20$. Find the following integrals: $\\int_a^b cf(z) dz=$ [ANS]\n$\\int_a^b (f(x))^2 dx-(\\int_a^b f(x) dx)^2=$ [ANS]\n$\\int_{a+5}^{b+5} f(x-5) dx=$ [ANS].", "answer_v1": [ "25*c", "20 - 25*25", "25" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $\\int_a^b f(x) dx=4$ and $\\int_a^b (f(x))^2 dx=30$. Find the following integrals: $\\int_a^b cf(z) dz=$ [ANS]\n$\\int_a^b (f(x))^2 dx-(\\int_a^b f(x) dx)^2=$ [ANS]\n$\\int_{a+2}^{b+2} f(x-2) dx=$ [ANS].", "answer_v2": [ "4*c", "30 - 4*4", "4" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $\\int_a^b f(x) dx=11$ and $\\int_a^b (f(x))^2 dx=20$. Find the following integrals: $\\int_a^b cf(z) dz=$ [ANS]\n$\\int_a^b (f(x))^2 dx-(\\int_a^b f(x) dx)^2=$ [ANS]\n$\\int_{a+3}^{b+3} f(x-3) dx=$ [ANS].", "answer_v3": [ "11*c", "20 - 11*11", "11" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0540", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "3", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "properties of integrals", "area" ], "problem_v1": "If $f(x)$ is odd and $\\int_{-3}^{7} f(x) dx=11$, then $\\int_{3}^{7} f(x) dx=$ [ANS]", "answer_v1": [ "11" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $f(x)$ is odd and $\\int_{-1}^{6} f(x) dx=4$, then $\\int_{1}^{6} f(x) dx=$ [ANS]", "answer_v2": [ "4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $f(x)$ is odd and $\\int_{-1}^{5} f(x) dx=6$, then $\\int_{1}^{5} f(x) dx=$ [ANS]", "answer_v3": [ "6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0541", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "2", "keywords": [ "calculus", "definite integral" ], "problem_v1": "Evaluate the integral by interpreting it in terms of areas:\n${\\int_{0}^{4} (8-4x)\\, dx}=$ [ANS]", "answer_v1": [ "0" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the integral by interpreting it in terms of areas:\n${\\int_{-3}^{1} (2-5x)\\, dx}=$ [ANS]", "answer_v2": [ "28" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the integral by interpreting it in terms of areas:\n${\\int_{-2}^{2} (4-4x)\\, dx}=$ [ANS]", "answer_v3": [ "16" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0542", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "2", "keywords": [ "Definite integrals" ], "problem_v1": "Suppose f(x)=\\left\\lbrace \\begin{array}{rll} 4 && \\mbox{if} 2 \\leq x < 9, \\\\-2 && \\mbox{if} 9 \\leq x \\leq 16. \\end{array} \\right.\n(a) Evaluate the definite integral by interpreting it in terms of signed area. $ \\int_{2}^{16} f(x) \\, dx=$ [ANS]\nSuggestion: Draw a picture of the region whose signed area is represented by the integral. Then find the signed area using formulas from high school geometry.\n(b) Find the average value of $f(x)$ over the interval $\\lbrack 2, 16 \\rbrack$. Average value=[ANS]", "answer_v1": [ "14", "1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose f(x)=\\left\\lbrace \\begin{array}{rll} 1 && \\mbox{if} 5 \\leq x < 7, \\\\-2 && \\mbox{if} 7 \\leq x \\leq 9. \\end{array} \\right.\n(a) Evaluate the definite integral by interpreting it in terms of signed area. $ \\int_{5}^{9} f(x) \\, dx=$ [ANS]\nSuggestion: Draw a picture of the region whose signed area is represented by the integral. Then find the signed area using formulas from high school geometry.\n(b) Find the average value of $f(x)$ over the interval $\\lbrack 5, 9 \\rbrack$. Average value=[ANS]", "answer_v2": [ "-2", "-0.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose f(x)=\\left\\lbrace \\begin{array}{rll} 2 && \\mbox{if} 2 \\leq x < 6, \\\\-3 && \\mbox{if} 6 \\leq x \\leq 10. \\end{array} \\right.\n(a) Evaluate the definite integral by interpreting it in terms of signed area. $ \\int_{2}^{10} f(x) \\, dx=$ [ANS]\nSuggestion: Draw a picture of the region whose signed area is represented by the integral. Then find the signed area using formulas from high school geometry.\n(b) Find the average value of $f(x)$ over the interval $\\lbrack 2, 10 \\rbrack$. Average value=[ANS]", "answer_v3": [ "-4", "-0.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0543", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Conceptual understanding of integration", "level": "3", "keywords": [ "calculus", "integrals" ], "problem_v1": "Let f(x)=\\begin{cases} 0 & \\text{if} x <-3 \\\\ 3 & \\text{if}-3 \\leq x < 0 \\\\-4 & \\text{if} 0 \\leq x < 5 \\\\ 0 & \\text{if} x \\geq 5 \\end{cases} and g(x)=\\int_{-3}^{x}\\, f(t) dt Determine the value of each of the following:\n(a) $g(-5)=$ [ANS]\n(b) $g(-2)=$ [ANS]\n(c) $g(1)=$ [ANS]\n(d) $g(6)=$ [ANS]\n(e) The absolute maximum of $g(x)$ occurs when $x=$ [ANS] and is the value [ANS]\nIt may be helpful to make a graph of $f(x)$ when answering these questions.", "answer_v1": [ "0", "3", "5", "-11", "0", "9" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let f(x)=\\begin{cases} 0 & \\text{if} x <-5 \\\\ 5 & \\text{if}-5 \\leq x <-1 \\\\-4 & \\text{if}-1 \\leq x < 4 \\\\ 0 & \\text{if} x \\geq 4 \\end{cases} and g(x)=\\int_{-5}^{x}\\, f(t) dt Determine the value of each of the following:\n(a) $g(-6)=$ [ANS]\n(b) $g(-4)=$ [ANS]\n(c) $g(0)=$ [ANS]\n(d) $g(5)=$ [ANS]\n(e) The absolute maximum of $g(x)$ occurs when $x=$ [ANS] and is the value [ANS]\nIt may be helpful to make a graph of $f(x)$ when answering these questions.", "answer_v2": [ "0", "5", "16", "0", "-1", "20" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let f(x)=\\begin{cases} 0 & \\text{if} x <-5 \\\\ 2 & \\text{if}-5 \\leq x <-1 \\\\-4 & \\text{if}-1 \\leq x < 4 \\\\ 0 & \\text{if} x \\geq 4 \\end{cases} and g(x)=\\int_{-5}^{x}\\, f(t) dt Determine the value of each of the following:\n(a) $g(-7)=$ [ANS]\n(b) $g(-4)=$ [ANS]\n(c) $g(0)=$ [ANS]\n(d) $g(5)=$ [ANS]\n(e) The absolute maximum of $g(x)$ occurs when $x=$ [ANS] and is the value [ANS]\nIt may be helpful to make a graph of $f(x)$ when answering these questions.", "answer_v3": [ "0", "2", "4", "-12", "-1", "8" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0544", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "3", "keywords": [ "integrals", "theory", "Riemann sums", "integral' 'summation' 'area' 'riemann", "Definite", "Integral", "Approximate", "Riemann Sum", "Approximate", "Definite", "Riemann" ], "problem_v1": "The following sum \\frac{1}{1+\\frac{6}{n}} \\cdot \\frac{6}{n}+\\frac{1}{1+\\frac{12}{n}} \\cdot \\frac{6}{n}+\\frac{1}{1+\\frac{18}{n}} \\cdot \\frac{6}{n}+\\ldots+\\frac{1}{1+\\frac{6 n}{n}} \\cdot \\frac{6}{n} is a right Riemann sum for a certain definite integral \\int_1^b f(x)\\, dx using a partition of the interval $[1,b]$ into $n$ subintervals of equal length.\nThen the upper limit of integration must be: $b$=[ANS]\nand the integrand must be the function $f(x)$=[ANS]", "answer_v1": [ "1+6", "1/x" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "The following sum \\frac{1}{1+\\frac{2}{n}} \\cdot \\frac{2}{n}+\\frac{1}{1+\\frac{4}{n}} \\cdot \\frac{2}{n}+\\frac{1}{1+\\frac{6}{n}} \\cdot \\frac{2}{n}+\\ldots+\\frac{1}{1+\\frac{2 n}{n}} \\cdot \\frac{2}{n} is a right Riemann sum for a certain definite integral \\int_1^b f(x)\\, dx using a partition of the interval $[1,b]$ into $n$ subintervals of equal length.\nThen the upper limit of integration must be: $b$=[ANS]\nand the integrand must be the function $f(x)$=[ANS]", "answer_v2": [ "1+2", "1/x" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "The following sum \\frac{1}{1+\\frac{3}{n}} \\cdot \\frac{3}{n}+\\frac{1}{1+\\frac{6}{n}} \\cdot \\frac{3}{n}+\\frac{1}{1+\\frac{9}{n}} \\cdot \\frac{3}{n}+\\ldots+\\frac{1}{1+\\frac{3 n}{n}} \\cdot \\frac{3}{n} is a right Riemann sum for a certain definite integral \\int_1^b f(x)\\, dx using a partition of the interval $[1,b]$ into $n$ subintervals of equal length.\nThen the upper limit of integration must be: $b$=[ANS]\nand the integrand must be the function $f(x)$=[ANS]", "answer_v3": [ "1+3", "1/x" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0545", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "3", "keywords": [], "problem_v1": "\\int_{-2}^{3} 6x^2 \\, dx a) Find a formula to approximate the above integral using $n$ subintervals and using Right Hand Rule. [ANS] (enter a formula involving $n$ alone). b) Evalute the formula using the indicated $n$ values. $n=10$: [ANS]\n$n=100$: [ANS]\n$n=1000$: [ANS]\nc) Find the limit of the formula, as $n \\to \\infty$, to find the exact value of the integral. [ANS]", "answer_v1": [ "30/n*[4*n-10/n*n*(n+1)+(5/n)^2*n*(n+1)*(2*n+1)/6]", "78.75", "70.7625", "70.0751", "70" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "\\int_{-5}^{5} 3x^2 \\, dx a) Find a formula to approximate the above integral using $n$ subintervals and using Right Hand Rule. [ANS] (enter a formula involving $n$ alone). b) Evalute the formula using the indicated $n$ values. $n=10$: [ANS]\n$n=100$: [ANS]\n$n=1000$: [ANS]\nc) Find the limit of the formula, as $n \\to \\infty$, to find the exact value of the integral. [ANS]", "answer_v2": [ "30/n*[25*n-50/n*n*(n+1)+(10/n)^2*n*(n+1)*(2*n+1)/6]", "255", "250.05", "250", "250" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "\\int_{-4}^{4} 3x^2 \\, dx a) Find a formula to approximate the above integral using $n$ subintervals and using Right Hand Rule. [ANS] (enter a formula involving $n$ alone). b) Evalute the formula using the indicated $n$ values. $n=10$: [ANS]\n$n=100$: [ANS]\n$n=1000$: [ANS]\nc) Find the limit of the formula, as $n \\to \\infty$, to find the exact value of the integral. [ANS]", "answer_v3": [ "24/n*[16*n-32/n*n*(n+1)+(8/n)^2*n*(n+1)*(2*n+1)/6]", "130.56", "128.026", "128", "128" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0546", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "5", "keywords": [ "calculus", "integrals", "integration", "net change", "total change" ], "problem_v1": "The rate (in liters per minute) at which water drains from a tank is recorded at half-minute intervals. Use the average of the left-and right-endpoint approximations to estimate the total amount of water drained during the first 3 min.\n$\\begin{array}{cccccccc}\\hline tmin & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\\\ \\hline l/min & 56 & 54 & 52 & 50 & 48 & 46 & 44 \\\\ \\hline \\end{array}$\nAnswer: [ANS] liters.", "answer_v1": [ "150" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The rate (in liters per minute) at which water drains from a tank is recorded at half-minute intervals. Use the average of the left-and right-endpoint approximations to estimate the total amount of water drained during the first 3 min.\n$\\begin{array}{cccccccc}\\hline tmin & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\\\ \\hline l/min & 40 & 38 & 36 & 34 & 32 & 30 & 28 \\\\ \\hline \\end{array}$\nAnswer: [ANS] liters.", "answer_v2": [ "102" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The rate (in liters per minute) at which water drains from a tank is recorded at half-minute intervals. Use the average of the left-and right-endpoint approximations to estimate the total amount of water drained during the first 3 min.\n$\\begin{array}{cccccccc}\\hline tmin & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\\\ \\hline l/min & 44 & 42 & 40 & 38 & 36 & 34 & 32 \\\\ \\hline \\end{array}$\nAnswer: [ANS] liters.", "answer_v3": [ "114" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0547", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "2", "keywords": [ "calculus", "integrals", "integration", "riemann sums", "partial sums" ], "problem_v1": "Evaluate the sum: $\\sum\\limits_{k=100}^{250} k^3=$ [ANS]", "answer_v1": [ "959888125" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the sum: $\\sum\\limits_{k=50}^{300} k^3=$ [ANS]", "answer_v2": [ "2037021875" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the sum: $\\sum\\limits_{k=50}^{250} k^3=$ [ANS]", "answer_v3": [ "982890000" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0548", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "2", "keywords": [ "calculus", "integrals", "integration", "riemann sums", "partial sums" ], "problem_v1": "Evaluate the sum $\\sum_{k=1}^{45}(13 \\cdot k+6)$. $\\sum\\limits_{k=1}^{45} (13 \\cdot k+6)=$ [ANS]", "answer_v1": [ "13725" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the sum $\\sum_{k=1}^{31}(19 \\cdot k+3)$. $\\sum\\limits_{k=1}^{31} (19 \\cdot k+3)=$ [ANS]", "answer_v2": [ "9517" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the sum $\\sum_{k=1}^{36}(13 \\cdot k+3)$. $\\sum\\limits_{k=1}^{36} (13 \\cdot k+3)=$ [ANS]", "answer_v3": [ "8766" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0549", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "2", "keywords": [ "calculus", "integrals", "integration", "riemann sums", "partial sums" ], "problem_v1": "Evaluate the limit $\\lim\\limits_{n\\to\\infty}\\sum\\limits_{j=1}^{n}\\frac{7 j^3}{n^{4}}$. $\\lim\\limits_{n\\to\\infty}\\sum\\limits_{j=1}^{n}\\frac{7 j^3}{n^{4}}$=[ANS]", "answer_v1": [ "7/4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the limit $\\lim\\limits_{n\\to\\infty}\\sum\\limits_{j=1}^{n}\\frac{9 j}{n^{2}}$. $\\lim\\limits_{n\\to\\infty}\\sum\\limits_{j=1}^{n}\\frac{9 j}{n^{2}}$=[ANS]", "answer_v2": [ "9/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the limit $\\lim\\limits_{n\\to\\infty}\\sum\\limits_{j=1}^{n}\\frac{7 j}{n^{2}}$. $\\lim\\limits_{n\\to\\infty}\\sum\\limits_{j=1}^{n}\\frac{7 j}{n^{2}}$=[ANS]", "answer_v3": [ "7/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0550", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "2", "keywords": [ "calculus", "integration", "riemann sums", "definite integral" ], "problem_v1": "In this problem you will calculate $\\int_{0}^{\\,4} 4x^3 \\;dx$ by using the formal definition of the definite integral: \\int_{a}^{b} f(x) \\;dx=\\lim_{n \\to \\infty} \\left[\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x \\right].\n(a) The interval $[0,4]$ is divided into $n$ equal subintervals of length $\\Delta x$. What is $\\Delta x$ (in terms of $n$)? $\\Delta x$=\n(b) The right-hand endpoint of the $k$ th subinterval is denoted $x_{k}^{*}$. What is $x_{k}^{*}$ (in terms of $k$ and $n$)? $x_{k}^{*}$=\n(c) Using these choices for $x_{k}^{*}$ and $\\Delta x$, the definition tells us that \\int_{0}^{\\,4} 4x^3 \\;dx=\\lim_{n \\to \\infty} \\left[\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x \\right]. What is $f(x^{*}_{k}) \\Delta x$ (in terms of $k$ and $n$)? $f(x^{*}_{k}) \\Delta x$=\n(d) Express $\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x$ in closed form. (Your answer will be in terms of $n$.) $\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x$=[ANS]\n(e) Finally, complete the problem by taking the limit as $n \\rightarrow \\infty$ of the expression that you found in the previous part. $\\int_{0}^{\\,4} 4x^3 \\;dx=\\lim_{n \\to \\infty} \\left[\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x \\right]$=[ANS]", "answer_v1": [ "4/n", "0 + k*4/n", "(4(0 + k*4/n)^3)*(4/n)", "4(4/n)^3*(n(n+1)/2)^2*(4/n)", "256" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "In this problem you will calculate $\\int_{0}^{\\,2} 6x^3 \\;dx$ by using the formal definition of the definite integral: \\int_{a}^{b} f(x) \\;dx=\\lim_{n \\to \\infty} \\left[\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x \\right].\n(a) The interval $[0,2]$ is divided into $n$ equal subintervals of length $\\Delta x$. What is $\\Delta x$ (in terms of $n$)? $\\Delta x$=\n(b) The right-hand endpoint of the $k$ th subinterval is denoted $x_{k}^{*}$. What is $x_{k}^{*}$ (in terms of $k$ and $n$)? $x_{k}^{*}$=\n(c) Using these choices for $x_{k}^{*}$ and $\\Delta x$, the definition tells us that \\int_{0}^{\\,2} 6x^3 \\;dx=\\lim_{n \\to \\infty} \\left[\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x \\right]. What is $f(x^{*}_{k}) \\Delta x$ (in terms of $k$ and $n$)? $f(x^{*}_{k}) \\Delta x$=\n(d) Express $\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x$ in closed form. (Your answer will be in terms of $n$.) $\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x$=[ANS]\n(e) Finally, complete the problem by taking the limit as $n \\rightarrow \\infty$ of the expression that you found in the previous part. $\\int_{0}^{\\,2} 6x^3 \\;dx=\\lim_{n \\to \\infty} \\left[\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x \\right]$=[ANS]", "answer_v2": [ "2/n", "0 + k*2/n", "(6(0 + k*2/n)^3)*(2/n)", "6(2/n)^3*(n(n+1)/2)^2*(2/n)", "24" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "In this problem you will calculate $\\int_{0}^{\\,2} 5x^3 \\;dx$ by using the formal definition of the definite integral: \\int_{a}^{b} f(x) \\;dx=\\lim_{n \\to \\infty} \\left[\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x \\right].\n(a) The interval $[0,2]$ is divided into $n$ equal subintervals of length $\\Delta x$. What is $\\Delta x$ (in terms of $n$)? $\\Delta x$=\n(b) The right-hand endpoint of the $k$ th subinterval is denoted $x_{k}^{*}$. What is $x_{k}^{*}$ (in terms of $k$ and $n$)? $x_{k}^{*}$=\n(c) Using these choices for $x_{k}^{*}$ and $\\Delta x$, the definition tells us that \\int_{0}^{\\,2} 5x^3 \\;dx=\\lim_{n \\to \\infty} \\left[\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x \\right]. What is $f(x^{*}_{k}) \\Delta x$ (in terms of $k$ and $n$)? $f(x^{*}_{k}) \\Delta x$=\n(d) Express $\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x$ in closed form. (Your answer will be in terms of $n$.) $\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x$=[ANS]\n(e) Finally, complete the problem by taking the limit as $n \\rightarrow \\infty$ of the expression that you found in the previous part. $\\int_{0}^{\\,2} 5x^3 \\;dx=\\lim_{n \\to \\infty} \\left[\\sum\\limits_{k=1}^{n} f(x^{*}_{k}) \\Delta x \\right]$=[ANS]", "answer_v3": [ "2/n", "0 + k*2/n", "(5(0 + k*2/n)^3)*(2/n)", "5(2/n)^3*(n(n+1)/2)^2*(2/n)", "20" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0551", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "2", "keywords": [ "integral' 'summation' 'area' 'riemann" ], "problem_v1": "Consider the integral \\int_{4}^{10} (3x^2+4x+2)\\,dx\n(a) Find the Riemann sum for this integral using right endpoints and $n=3$. $R_3=$ [ANS]\n(b) Find the Riemann sum for this same integral, using left endpoints and $n=3$. $L_3=$ [ANS]", "answer_v1": [ "1404", "852" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider the integral \\int_{0}^{6} (2x^2+3x+6)\\,dx\n(a) Find the Riemann sum for this integral using right endpoints and $n=3$. $R_3=$ [ANS]\n(b) Find the Riemann sum for this same integral, using left endpoints and $n=3$. $L_3=$ [ANS]", "answer_v2": [ "332", "152" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider the integral \\int_{1}^{7} (2x^2+3x+2)\\,dx\n(a) Find the Riemann sum for this integral using right endpoints and $n=3$. $R_3=$ [ANS]\n(b) Find the Riemann sum for this same integral, using left endpoints and $n=3$. $L_3=$ [ANS]", "answer_v3": [ "434", "206" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0552", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "3", "keywords": [ "Definite", "Integral", "Approximate", "Riemann Sum" ], "problem_v1": "Estimate the area under the graph of $f(x)=x^2+4x$ from $x=4$ to $x=12$ using $4$ approximating rectangles and left endpoints. Approximation=[ANS]", "answer_v1": [ "2*(32+60+96+140)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Estimate the area under the graph of $f(x)=x^2+3x$ from $x=1$ to $x=7$ using $3$ approximating rectangles and left endpoints. Approximation=[ANS]", "answer_v2": [ "2*(4+18+40)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Estimate the area under the graph of $f(x)=x^2+3x$ from $x=2$ to $x=8$ using $3$ approximating rectangles and left endpoints. Approximation=[ANS]", "answer_v3": [ "2*(10+28+54)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0553", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "3", "keywords": [ "Sum", "Sigma Notation" ], "problem_v1": "$\\sum\\limits_{k=1}^{74} (4+k+2k^{2}+4k^{3})$=[ANS].", "answer_v1": [ "3.10812E+7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$\\sum\\limits_{k=1}^{145} (7k-7-6k^{2}-3k^{3})$=[ANS].", "answer_v2": [ "-3.42214E+8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$\\sum\\limits_{k=1}^{64} (2k-3-4k^{2}+k^{3})$=[ANS].", "answer_v3": [ "3.97261E+6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0554", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "2", "keywords": [ "integrals", "theory", "Riemann sums", "Calculus", "Riemann Integral" ], "problem_v1": "Approximate the definite integral \\int_{6}^{10} |8-t|\\, dt using midpoint Riemann sums with the following partitions:\n(a) $P=\\lbrace 6, 8, 10 \\rbrace$. Then midpoint Riemann sum=[ANS]\n(b) Using 4 subintervals of equal length. Then midpoint Riemann sum=[ANS]", "answer_v1": [ "4", "4" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Approximate the definite integral \\int_{3}^{7} |6-t|\\, dt using midpoint Riemann sums with the following partitions:\n(a) $P=\\lbrace 3, 6, 7 \\rbrace$. Then midpoint Riemann sum=[ANS]\n(b) Using 4 subintervals of equal length. Then midpoint Riemann sum=[ANS]", "answer_v2": [ "5", "5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Approximate the definite integral \\int_{4}^{7} |6-t|\\, dt using midpoint Riemann sums with the following partitions:\n(a) $P=\\lbrace 4, 6, 7 \\rbrace$. Then midpoint Riemann sum=[ANS]\n(b) Using 3 subintervals of equal length. Then midpoint Riemann sum=[ANS]", "answer_v3": [ "2.5", "2.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0555", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "2", "keywords": [ "calculus", "integral", "definite integral", "area" ], "problem_v1": "On a sketch of $y=\\ln\\!\\left(x\\right)$, represent the left Riemann sum with $n=2$ approximating $\\int_{3}^{4}\\,\\ln\\!\\left(x\\right)\\,dx$. Write out the terms of the sum, but do not evaluate it: Sum=[ANS]+[ANS]\nOn another sketch, represent the right Riemann sum with $n=2$ approximating $\\int_{3}^{4}\\,\\ln\\!\\left(x\\right)\\,dx$. Write out the terms of the sum, but do not evaluate it: Sum=[ANS]+[ANS]\nWhich sum is an overestimate? [ANS] A. the right Riemann sum B. the left Riemann sum C. neither sum\nWhich sum is an underestimate? [ANS] A. the left Riemann sum B. the right Riemann sum C. neither sum", "answer_v1": [ "0.549306", "0.626381", "0.626381", "0.693147", "A", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "MCS", "MCS" ], "options_v1": [ [], [], [], [], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "On a sketch of $y=e^{x}$, represent the left Riemann sum with $n=2$ approximating $\\int_{4}^{5}\\,e^{x}\\,dx$. Write out the terms of the sum, but do not evaluate it: Sum=[ANS]+[ANS]\nOn another sketch, represent the right Riemann sum with $n=2$ approximating $\\int_{4}^{5}\\,e^{x}\\,dx$. Write out the terms of the sum, but do not evaluate it: Sum=[ANS]+[ANS]\nWhich sum is an overestimate? [ANS] A. the left Riemann sum B. the right Riemann sum C. neither sum\nWhich sum is an underestimate? [ANS] A. the left Riemann sum B. the right Riemann sum C. neither sum", "answer_v2": [ "e^4*0.5", "e^4.5*0.5", "e^4.5*0.5", "e^5*0.5", "B", "A" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "MCS", "MCS" ], "options_v2": [ [], [], [], [], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "On a sketch of $y=e^{x}$, represent the left Riemann sum with $n=2$ approximating $\\int_{3}^{4}\\,e^{x}\\,dx$. Write out the terms of the sum, but do not evaluate it: Sum=[ANS]+[ANS]\nOn another sketch, represent the right Riemann sum with $n=2$ approximating $\\int_{3}^{4}\\,e^{x}\\,dx$. Write out the terms of the sum, but do not evaluate it: Sum=[ANS]+[ANS]\nWhich sum is an overestimate? [ANS] A. the left Riemann sum B. the right Riemann sum C. neither sum\nWhich sum is an underestimate? [ANS] A. the right Riemann sum B. the left Riemann sum C. neither sum", "answer_v3": [ "e^3*0.5", "e^3.5*0.5", "e^3.5*0.5", "e^4*0.5", "B", "B" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "MCS", "MCS" ], "options_v3": [ [], [], [], [], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Calculus_-_single_variable_0556", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "2", "keywords": [ "calculus", "integral", "definite integral", "area" ], "problem_v1": "Consider the following table:\n$\\begin{array}{ccccccc}\\hline x & 0 & 3 & 6 & 9 & 12 & 15 \\\\ \\hline f(x) & 50 & 49 & 47 & 42 & 31 & 8 \\\\ \\hline \\end{array}$\nUse this to estimate the integral: $\\int_0^{15} f(x) dx \\approx$ [ANS]", "answer_v1": [ "594" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider the following table:\n$\\begin{array}{ccccccc}\\hline x & 0 & 2 & 4 & 6 & 8 & 10 \\\\ \\hline f(x) & 54 & 53 & 50 & 43 & 29 & 1 \\\\ \\hline \\end{array}$\nUse this to estimate the integral: $\\int_0^{10} f(x) dx \\approx$ [ANS]", "answer_v2": [ "405" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider the following table:\n$\\begin{array}{ccccccc}\\hline x & 0 & 2 & 4 & 6 & 8 & 10 \\\\ \\hline f(x) & 46 & 45 & 43 & 38 & 27 & 4 \\\\ \\hline \\end{array}$\nUse this to estimate the integral: $\\int_0^{10} f(x) dx \\approx$ [ANS]", "answer_v3": [ "356" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0557", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "3", "keywords": [ "calculus", "integral", "definite integral", "area" ], "problem_v1": "Using the graph of $7+\\cos(5x)$, for $0\\le x\\le \\frac{3\\pi}{5}$, list the following quantities in increasing order: A. the value of the integral $\\int_0^{\\frac{3\\pi}{5}} (7+\\cos(5x)) dx$, B. the left sum with $n=3$ subdivisions, and C. the right sum with $n=3$ subdivisions. (Enter the letter of the value in each box.) (Enter the letter of the value in each box.) [ANS] $<$ [ANS] $<$ [ANS]\nWhat is the value of the integral $\\int_0^{\\frac{3\\pi}{5}} (7+\\cos(5x)) dx$? value=[ANS]", "answer_v1": [ "C", "A", "B", "3*pi*7/5" ], "answer_type_v1": [ "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Using the graph of $2+\\cos(8x)$, for $0\\le x\\le \\frac{3\\pi}{8}$, list the following quantities in increasing order: A. the value of the integral $\\int_0^{\\frac{3\\pi}{8}} (2+\\cos(8x)) dx$, B. the left sum with $n=3$ subdivisions, and C. the right sum with $n=3$ subdivisions. (Enter the letter of the value in each box.) (Enter the letter of the value in each box.) [ANS] $<$ [ANS] $<$ [ANS]\nWhat is the value of the integral $\\int_0^{\\frac{3\\pi}{8}} (2+\\cos(8x)) dx$? value=[ANS]", "answer_v2": [ "C", "A", "B", "3*pi*2/8" ], "answer_type_v2": [ "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Using the graph of $4+\\cos(5x)$, for $0\\le x\\le \\frac{3\\pi}{5}$, list the following quantities in increasing order: A. the value of the integral $\\int_0^{\\frac{3\\pi}{5}} (4+\\cos(5x)) dx$, B. the left sum with $n=3$ subdivisions, and C. the right sum with $n=3$ subdivisions. (Enter the letter of the value in each box.) (Enter the letter of the value in each box.) [ANS] $<$ [ANS] $<$ [ANS]\nWhat is the value of the integral $\\int_0^{\\frac{3\\pi}{5}} (4+\\cos(5x)) dx$? value=[ANS]", "answer_v3": [ "C", "A", "B", "3*pi*4/5" ], "answer_type_v3": [ "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0558", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "3", "keywords": [ "derivative" ], "problem_v1": "Approximate the integral using\n(a) The midpoint approximation $M_{10}$, (b) The trapezoidal approximation $T_{10}$, and (c) Simpson's rule approximation $S_{20}$. Find the exact value of the integral and approximate the absolute error for each case. Express your final approximation answers to six decimal places. $ \\int_{0}^{5} \\sqrt{x+4} \\;dx=$ [ANS]\n(a) $M_{10}=$ [ANS], $E_{M}=$ [ANS], (b) $T_{10}=$ [ANS], $E_{T}=$ [ANS], (c) $S_{20}=$ [ANS], $E_{S}=$ [ANS],", "answer_v1": [ "2/3*(27-8)", "12.6675", "0.000867", "12.6649", "0.001735", "12.6667", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Approximate the integral using\n(a) The midpoint approximation $M_{10}$, (b) The trapezoidal approximation $T_{10}$, and (c) Simpson's rule approximation $S_{20}$. Find the exact value of the integral and approximate the absolute error for each case. Express your final approximation answers to six decimal places. $ \\int_{0}^{1} \\sqrt{3x+1} \\;dx=$ [ANS]\n(a) $M_{10}=$ [ANS], $E_{M}=$ [ANS], (b) $T_{10}=$ [ANS], $E_{T}=$ [ANS], (c) $S_{20}=$ [ANS], $E_{S}=$ [ANS],", "answer_v2": [ "2/9*(8-1)", "1.55587", "0.000311", "1.55493", "0.000624", "1.55556", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Approximate the integral using\n(a) The midpoint approximation $M_{10}$, (b) The trapezoidal approximation $T_{10}$, and (c) Simpson's rule approximation $S_{20}$. Find the exact value of the integral and approximate the absolute error for each case. Express your final approximation answers to six decimal places. $ \\int_{0}^{2} \\sqrt{4x+1} \\;dx=$ [ANS]\n(a) $M_{10}=$ [ANS], $E_{M}=$ [ANS], (b) $T_{10}=$ [ANS], $E_{T}=$ [ANS], (c) $S_{20}=$ [ANS], $E_{S}=$ [ANS],", "answer_v3": [ "1/6*(27-1)", "4.33551", "0.002181", "4.32894", "0.004397", "4.33332", "1.2E-05" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0559", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "3", "keywords": [ "calculus", "sigma notation", "Riemann sums" ], "problem_v1": "In this problem you will calculate the area between $f(x)=x^2$ and the $x$-axis over the interval $\\lbrack 3, 11 \\rbrack$ using a limit of right-endpoint Riemann sums:\n\\mathrm{Area}=\\lim_{n \\to \\infty} \\left(\\sum_{k=1}^{n} f(x_k) \\Delta x \\right). Express the following quantities in terms of $n$, the number of rectangles in the Riemann sum, and $k$, the index for the rectangles in the Riemann sum.\nWe start by subdividing $\\lbrack 3, 11 \\rbrack$ into $n$ equal width subintervals $\\lbrack x_0, x_1 \\rbrack, \\lbrack x_1, x_2 \\rbrack, \\ldots, \\lbrack x_{n-1}, x_{n} \\rbrack$ each of width $\\Delta x$. Express the width of each subinterval $\\Delta x$ in terms of the number of subintervals $n$. $\\Delta x=$ [ANS]\nFind the right endpoints $x_1, x_2, x_3$ of the first, second, and third subintervals $\\lbrack x_0, x_1 \\rbrack, \\lbrack x_1, x_2 \\rbrack, \\lbrack x_2, x_3 \\rbrack$ and express your answers in terms of $n$. $x_1, x_2, x_3=$ [ANS] (Enter a comma separated list.)\nFind a general expression for the right endpoint $x_k$ of the $k$ th subinterval $\\lbrack x_{k-1}, x_{k} \\rbrack$, where $1 \\leq k \\leq n$. Express your answer in terms of $k$ and $n$. $x_k=$ [ANS]\nFind $f(x_k)$ in terms of $k$ and $n$. $f(x_k)=$ [ANS]\nFind $f(x_k) \\Delta x$ in terms of $k$ and $n$. $f(x_k) \\Delta x=$ [ANS]\nFind the value of the right-endpoint Riemann sum in terms of $n$. $ \\sum_{k=1}^{n} f(x_k) \\Delta x=$ [ANS]\nFind the limit of the right-endpoint Riemann sum. $ \\lim_{n\\to\\infty} \\left(\\sum_{k=1}^{n} f(x_k) \\Delta x \\right)=$ [ANS]", "answer_v1": [ "(11-3)/n", "(3+8/n, 3+16/n, 3+24/n)", "3+k*8/n", "(3+k*8/n)^2", "(3+k*8/n)^2*8/n", "3^2*n*8/n+2*3*n*(n+1)/2*(8/n)^2+n*(n+1)*(2*n+1)/6*(8/n)^3", "3^2*8+3*8^2+8^3/3" ], "answer_type_v1": [ "EX", "OL", "EX", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "In this problem you will calculate the area between $f(x)=x^2$ and the $x$-axis over the interval $\\lbrack 1, 10 \\rbrack$ using a limit of right-endpoint Riemann sums:\n\\mathrm{Area}=\\lim_{n \\to \\infty} \\left(\\sum_{k=1}^{n} f(x_k) \\Delta x \\right). Express the following quantities in terms of $n$, the number of rectangles in the Riemann sum, and $k$, the index for the rectangles in the Riemann sum.\nWe start by subdividing $\\lbrack 1, 10 \\rbrack$ into $n$ equal width subintervals $\\lbrack x_0, x_1 \\rbrack, \\lbrack x_1, x_2 \\rbrack, \\ldots, \\lbrack x_{n-1}, x_{n} \\rbrack$ each of width $\\Delta x$. Express the width of each subinterval $\\Delta x$ in terms of the number of subintervals $n$. $\\Delta x=$ [ANS]\nFind the right endpoints $x_1, x_2, x_3$ of the first, second, and third subintervals $\\lbrack x_0, x_1 \\rbrack, \\lbrack x_1, x_2 \\rbrack, \\lbrack x_2, x_3 \\rbrack$ and express your answers in terms of $n$. $x_1, x_2, x_3=$ [ANS] (Enter a comma separated list.)\nFind a general expression for the right endpoint $x_k$ of the $k$ th subinterval $\\lbrack x_{k-1}, x_{k} \\rbrack$, where $1 \\leq k \\leq n$. Express your answer in terms of $k$ and $n$. $x_k=$ [ANS]\nFind $f(x_k)$ in terms of $k$ and $n$. $f(x_k)=$ [ANS]\nFind $f(x_k) \\Delta x$ in terms of $k$ and $n$. $f(x_k) \\Delta x=$ [ANS]\nFind the value of the right-endpoint Riemann sum in terms of $n$. $ \\sum_{k=1}^{n} f(x_k) \\Delta x=$ [ANS]\nFind the limit of the right-endpoint Riemann sum. $ \\lim_{n\\to\\infty} \\left(\\sum_{k=1}^{n} f(x_k) \\Delta x \\right)=$ [ANS]", "answer_v2": [ "(10-1)/n", "(1+9/n, 1+18/n, 1+27/n)", "1+k*9/n", "(1+k*9/n)^2", "(1+k*9/n)^2*9/n", "1^2*n*9/n+2*1*n*(n+1)/2*(9/n)^2+n*(n+1)*(2*n+1)/6*(9/n)^3", "1^2*9+1*9^2+9^3/3" ], "answer_type_v2": [ "EX", "OL", "EX", "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "In this problem you will calculate the area between $f(x)=x^2$ and the $x$-axis over the interval $\\lbrack 1, 9 \\rbrack$ using a limit of right-endpoint Riemann sums:\n\\mathrm{Area}=\\lim_{n \\to \\infty} \\left(\\sum_{k=1}^{n} f(x_k) \\Delta x \\right). Express the following quantities in terms of $n$, the number of rectangles in the Riemann sum, and $k$, the index for the rectangles in the Riemann sum.\nWe start by subdividing $\\lbrack 1, 9 \\rbrack$ into $n$ equal width subintervals $\\lbrack x_0, x_1 \\rbrack, \\lbrack x_1, x_2 \\rbrack, \\ldots, \\lbrack x_{n-1}, x_{n} \\rbrack$ each of width $\\Delta x$. Express the width of each subinterval $\\Delta x$ in terms of the number of subintervals $n$. $\\Delta x=$ [ANS]\nFind the right endpoints $x_1, x_2, x_3$ of the first, second, and third subintervals $\\lbrack x_0, x_1 \\rbrack, \\lbrack x_1, x_2 \\rbrack, \\lbrack x_2, x_3 \\rbrack$ and express your answers in terms of $n$. $x_1, x_2, x_3=$ [ANS] (Enter a comma separated list.)\nFind a general expression for the right endpoint $x_k$ of the $k$ th subinterval $\\lbrack x_{k-1}, x_{k} \\rbrack$, where $1 \\leq k \\leq n$. Express your answer in terms of $k$ and $n$. $x_k=$ [ANS]\nFind $f(x_k)$ in terms of $k$ and $n$. $f(x_k)=$ [ANS]\nFind $f(x_k) \\Delta x$ in terms of $k$ and $n$. $f(x_k) \\Delta x=$ [ANS]\nFind the value of the right-endpoint Riemann sum in terms of $n$. $ \\sum_{k=1}^{n} f(x_k) \\Delta x=$ [ANS]\nFind the limit of the right-endpoint Riemann sum. $ \\lim_{n\\to\\infty} \\left(\\sum_{k=1}^{n} f(x_k) \\Delta x \\right)=$ [ANS]", "answer_v3": [ "(9-1)/n", "(1+8/n, 1+16/n, 1+24/n)", "1+k*8/n", "(1+k*8/n)^2", "(1+k*8/n)^2*8/n", "1^2*n*8/n+2*1*n*(n+1)/2*(8/n)^2+n*(n+1)*(2*n+1)/6*(8/n)^3", "1^2*8+1*8^2+8^3/3" ], "answer_type_v3": [ "EX", "UOL", "EX", "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0560", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "4", "keywords": [ "calculus", "sigma notation" ], "problem_v1": "In parts\n(a)-(d), find the values of the sums in terms of $n$. In part (e), evaluate the limit of the sum from part (d).\n$ \\sum_{k=1}^{n} k^2=$ [ANS]\n$ \\sum_{k=1}^{n} \\frac{k^2}{n^3}=$ [ANS]\n$ \\sum_{k=1}^{n} \\frac{6}{n}=$ [ANS]\n$ \\sum_{k=1}^{n} \\left(\\frac{8 k^2}{n^3}-\\frac{6}{n} \\right)=$ [ANS]\n$ \\lim_{n\\to\\infty} \\left(\\sum_{k=1}^{n} \\left(\\frac{8 k^2}{n^3}-\\frac{6}{n} \\right) \\right)=$ [ANS]", "answer_v1": [ "n*(n+1)*(2*n+1)/6", "n*(n+1)*(2*n+1)/(6*n^3)", "6/n*n", "8*n*(n+1)*(2*n+1)/(6*n^3)-6/n*n", "8/3-6" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "In parts\n(a)-(d), find the values of the sums in terms of $n$. In part (e), evaluate the limit of the sum from part (d).\n$ \\sum_{k=1}^{n} k^2=$ [ANS]\n$ \\sum_{k=1}^{n} \\frac{k^2}{n^3}=$ [ANS]\n$ \\sum_{k=1}^{n} \\frac{9}{n}=$ [ANS]\n$ \\sum_{k=1}^{n} \\left(\\frac{2 k^2}{n^3}-\\frac{9}{n} \\right)=$ [ANS]\n$ \\lim_{n\\to\\infty} \\left(\\sum_{k=1}^{n} \\left(\\frac{2 k^2}{n^3}-\\frac{9}{n} \\right) \\right)=$ [ANS]", "answer_v2": [ "n*(n+1)*(2*n+1)/6", "n*(n+1)*(2*n+1)/(6*n^3)", "9/n*n", "2*n*(n+1)*(2*n+1)/(6*n^3)-9/n*n", "2/3-9" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "In parts\n(a)-(d), find the values of the sums in terms of $n$. In part (e), evaluate the limit of the sum from part (d).\n$ \\sum_{k=1}^{n} k^2=$ [ANS]\n$ \\sum_{k=1}^{n} \\frac{k^2}{n^3}=$ [ANS]\n$ \\sum_{k=1}^{n} \\frac{6}{n}=$ [ANS]\n$ \\sum_{k=1}^{n} \\left(\\frac{4 k^2}{n^3}-\\frac{6}{n} \\right)=$ [ANS]\n$ \\lim_{n\\to\\infty} \\left(\\sum_{k=1}^{n} \\left(\\frac{4 k^2}{n^3}-\\frac{6}{n} \\right) \\right)=$ [ANS]", "answer_v3": [ "n*(n+1)*(2*n+1)/6", "n*(n+1)*(2*n+1)/(6*n^3)", "6/n*n", "4*n*(n+1)*(2*n+1)/(6*n^3)-6/n*n", "4/3-6" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0561", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "2", "keywords": [ "calculus", "areas", "distances" ], "problem_v1": "The value of the limit \\lim_{n\\rightarrow\\infty}\\sum_{i=1}^{n} \\frac{6}{n} \\sqrt{5+\\frac{6 i}{n}} is equal to the area below the graph of a function $f(x)$ on an interval $[A,B]$. Find $f$, $A$, and $B$. (Do not evaluate the limit.)\n$f(x)$=[ANS]\n$A$=[ANS] (use $A=0$)\n$B$=[ANS]", "answer_v1": [ "sqrt(5 + x)", "0", "6" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The value of the limit \\lim_{n\\rightarrow\\infty}\\sum_{i=1}^{n} \\frac{3}{n} \\sqrt{8+\\frac{3 i}{n}} is equal to the area below the graph of a function $f(x)$ on an interval $[A,B]$. Find $f$, $A$, and $B$. (Do not evaluate the limit.)\n$f(x)$=[ANS]\n$A$=[ANS] (use $A=0$)\n$B$=[ANS]", "answer_v2": [ "sqrt(8 + x)", "0", "3" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The value of the limit \\lim_{n\\rightarrow\\infty}\\sum_{i=1}^{n} \\frac{4}{n} \\sqrt{5+\\frac{4 i}{n}} is equal to the area below the graph of a function $f(x)$ on an interval $[A,B]$. Find $f$, $A$, and $B$. (Do not evaluate the limit.)\n$f(x)$=[ANS]\n$A$=[ANS] (use $A=0$)\n$B$=[ANS]", "answer_v3": [ "sqrt(5 + x)", "0", "4" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0562", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Riemann sums", "level": "4", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "area", "properties of integrals" ], "problem_v1": "The following table gives the approximate amount of emissions, $E$, of nitrogen oxides in millions of metric tons per year in the US. Let $t$ be the number of years since 1940 and $E=f(t)$.\n$\\begin{array}{ccccccc}\\hline t & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 \\\\ \\hline E & 7.2 & 9.5 & 13.1 & 18.7 & 20.7 & 19.4 \\\\ \\hline \\end{array}$\n(a) Using left endpoints, estimate the integral: $ \\int_0^{50} f(t) \\, dt \\approx$ [ANS]\n(b) Using right endpoints, estimate the integral: $ \\int_0^{50} f(t) \\, dt \\approx$ [ANS]\nBe sure that you know what the units of your answer are, and what its meaning is! Be sure that you know what the units of your answer are, and what its meaning is! (Original data from the Statistical Abstract of the US, 1992) (Original data from the Statistical Abstract of the US, 1992)", "answer_v1": [ "10*(7.2+9.5+13.1+18.7+20.7)", "10*(9.5+13.1+18.7+20.7+19.4)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The following table gives the approximate amount of emissions, $E$, of nitrogen oxides in millions of metric tons per year in the US. Let $t$ be the number of years since 1940 and $E=f(t)$.\n$\\begin{array}{ccccccc}\\hline t & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 \\\\ \\hline E & 6.4 & 9.9 & 12.6 & 18.3 & 21.4 & 19.4 \\\\ \\hline \\end{array}$\n(a) Using left endpoints, estimate the integral: $ \\int_0^{50} f(t) \\, dt \\approx$ [ANS]\n(b) Using right endpoints, estimate the integral: $ \\int_0^{50} f(t) \\, dt \\approx$ [ANS]\nBe sure that you know what the units of your answer are, and what its meaning is! Be sure that you know what the units of your answer are, and what its meaning is! (Original data from the Statistical Abstract of the US, 1992) (Original data from the Statistical Abstract of the US, 1992)", "answer_v2": [ "10*(6.4+9.9+12.6+18.3+21.4)", "10*(9.9+12.6+18.3+21.4+19.4)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The following table gives the approximate amount of emissions, $E$, of nitrogen oxides in millions of metric tons per year in the US. Let $t$ be the number of years since 1940 and $E=f(t)$.\n$\\begin{array}{ccccccc}\\hline t & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 \\\\ \\hline E & 6.7 & 9.5 & 12.8 & 18.6 & 20.6 & 19.4 \\\\ \\hline \\end{array}$\n(a) Using left endpoints, estimate the integral: $ \\int_0^{50} f(t) \\, dt \\approx$ [ANS]\n(b) Using right endpoints, estimate the integral: $ \\int_0^{50} f(t) \\, dt \\approx$ [ANS]\nBe sure that you know what the units of your answer are, and what its meaning is! Be sure that you know what the units of your answer are, and what its meaning is! (Original data from the Statistical Abstract of the US, 1992) (Original data from the Statistical Abstract of the US, 1992)", "answer_v3": [ "10*(6.7+9.5+12.8+18.6+20.6)", "10*(9.5+12.8+18.6+20.6+19.4)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0563", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "integrals", "theory", "Fundamental Theorem of Calculus", "Definite", "Integral", "Fundamental Theorem" ], "problem_v1": "Find the derivative of the following function F(x)=\\int_{x^5}^{x^6} (2t-1)^3\\, dt using the Fundamental Theorem of Calculus. $F'(x)$=[ANS]", "answer_v1": [ "6*x^5*(2*x^6-1)^3 - 5*x^4*(2*x^5-1)^3" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the derivative of the following function F(x)=\\int_{x^2}^{x^7} (2t-1)^3\\, dt using the Fundamental Theorem of Calculus. $F'(x)$=[ANS]", "answer_v2": [ "7*x^6*(2*x^7-1)^3 - 2*x^1*(2*x^2-1)^3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the derivative of the following function F(x)=\\int_{x^3}^{x^6} (2t-1)^3\\, dt using the Fundamental Theorem of Calculus. $F'(x)$=[ANS]", "answer_v3": [ "6*x^5*(2*x^6-1)^3 - 3*x^2*(2*x^3-1)^3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0564", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "integrals", "fundamental theorem of calculus", "calculus", "integrals" ], "problem_v1": "Use the Fundamental Theorem of Calculus to find the derivative of g(x)=\\int_{4x}^{3x} \\frac{u+4}{u-4} du Answer: [ANS]", "answer_v1": [ "3*(3*x+4)/(3*x-4)-4*(4*x+4)/(4*x-4)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the Fundamental Theorem of Calculus to find the derivative of g(x)=\\int_{1x}^{5x} \\frac{u+1}{u-2} du Answer: [ANS]", "answer_v2": [ "5*(5*x+1)/(5*x-2)-1*(1*x+1)/(1*x-2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the Fundamental Theorem of Calculus to find the derivative of g(x)=\\int_{2x}^{4x} \\frac{u+2}{u-3} du Answer: [ANS]", "answer_v3": [ "4*(4*x+2)/(4*x-3)-2*(2*x+2)/(2*x-3)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0565", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "calculus", "antiderivatives", "fundamental theorem of calculus", "integrals", "integration" ], "problem_v1": "The antiderivative $F(x)$ of $f(x)=\\sqrt{x^4+1}$ satisfying the initial condition $F(5)=0$ is given by: $F(x)=\\int_{a}^{b} \\sqrt{t^4+1} \\,dt$ Find $a$, $b$: $a=$ [ANS]\n$b=$ [ANS]", "answer_v1": [ "5", "x" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "The antiderivative $F(x)$ of $f(x)=\\sqrt{x^4+1}$ satisfying the initial condition $F(-9)=0$ is given by: $F(x)=\\int_{a}^{b} \\sqrt{t^4+1} \\,dt$ Find $a$, $b$: $a=$ [ANS]\n$b=$ [ANS]", "answer_v2": [ "-9", "x" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "The antiderivative $F(x)$ of $f(x)=\\sqrt{x^4+1}$ satisfying the initial condition $F(-4)=0$ is given by: $F(x)=\\int_{a}^{b} \\sqrt{t^4+1} \\,dt$ Find $a$, $b$: $a=$ [ANS]\n$b=$ [ANS]", "answer_v3": [ "-4", "x" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0566", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "2", "keywords": [ "calculus", "integrals", "integration", "antiderivatives", "fundamental theorem of calculus" ], "problem_v1": "$G(x)=\\int_{1}^{x} \\tan t \\,dt$ Find $G(1)=$ [ANS]\nFind $G'(\\frac{\\pi}{6})=$ [ANS]", "answer_v1": [ "0", "tan(pi/6)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "$G(x)=\\int_{1}^{x} \\tan t \\,dt$ Find $G(1)=$ [ANS]\nFind $G'(\\frac{\\pi}{3})=$ [ANS]", "answer_v2": [ "0", "tan(pi/3)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "$G(x)=\\int_{1}^{x} \\tan t \\,dt$ Find $G(1)=$ [ANS]\nFind $G'(\\frac{\\pi}{4})=$ [ANS]", "answer_v3": [ "0", "tan(pi/4)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0567", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "integrals", "definite", "Fundamental Theorem of Calculus", "Calculus", "Riemann Integral" ], "problem_v1": "Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $ \\int_{-18}^1 s\\, \\left|81-s^2\\right|\\, ds$=[ANS]\n$ \\int_{0}^{81 \\pi^2} \\frac{\\sin(\\sqrt{x})}{\\sqrt{x}}\\,dx$=[ANS]\n$ \\int_{9}^{18}\\frac{t-9}{t^2-18 t+82}\\,dt$=[ANS]", "answer_v1": [ "-16362.25", "4", "2.20335962363213" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $ \\int_{-6}^1 s\\, \\left|9-s^2\\right|\\, ds$=[ANS]\n$ \\int_{0}^{9 \\pi^2} \\frac{\\sin(\\sqrt{x})}{\\sqrt{x}}\\,dx$=[ANS]\n$ \\int_{3}^{6}\\frac{t-3}{t^2-6 t+10}\\,dt$=[ANS]", "answer_v2": [ "-198.25", "4", "1.15129254649702" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $ \\int_{-10}^1 s\\, \\left|25-s^2\\right|\\, ds$=[ANS]\n$ \\int_{0}^{25 \\pi^2} \\frac{\\sin(\\sqrt{x})}{\\sqrt{x}}\\,dx$=[ANS]\n$ \\int_{5}^{10}\\frac{t-5}{t^2-10 t+26}\\,dt$=[ANS]", "answer_v3": [ "-1550.25", "4", "1.62904826901074" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0568", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "2", "keywords": [ "integrals", "theory", "Fundamental Theorem of Calculus", "Calculus", "Riemann Integral" ], "problem_v1": "Evaluate the definite integral \\int_4^6 \\left(\\frac{d}{dt}\\sqrt{4+4 t^4}\\right)\\, dt using the Fundamental Theorem of Calculus. You will need accuracy to at least 4 decimal places for your numerical answer to be accepted. You can also leave your answer as an algebraic expression involving square roots. $ \\int_4^6 \\left(\\frac{d}{dt}\\sqrt{4+4 t^4}\\right)\\, dt$=[ANS]", "answer_v1": [ "39.9653333377136" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the definite integral \\int_1^4 \\left(\\frac{d}{dt}\\sqrt{2+2 t^4}\\right)\\, dt using the Fundamental Theorem of Calculus. You will need accuracy to at least 4 decimal places for your numerical answer to be accepted. You can also leave your answer as an algebraic expression involving square roots. $ \\int_1^4 \\left(\\frac{d}{dt}\\sqrt{2+2 t^4}\\right)\\, dt$=[ANS]", "answer_v2": [ "20.6715680975093" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the definite integral \\int_2^4 \\left(\\frac{d}{dt}\\sqrt{3+3 t^4}\\right)\\, dt using the Fundamental Theorem of Calculus. You will need accuracy to at least 4 decimal places for your numerical answer to be accepted. You can also leave your answer as an algebraic expression involving square roots. $ \\int_2^4 \\left(\\frac{d}{dt}\\sqrt{3+3 t^4}\\right)\\, dt$=[ANS]", "answer_v3": [ "20.6254583252868" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0569", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "integrals", "fundamental theorem of calculus", "Calculus", "Riemann Integral" ], "problem_v1": "Find a function $f$ and a number $a$ such that 2+\\int_{a}^{x} \\frac {f(t)} {t^{5}} dt=4x^{-1} $f(x)=$ [ANS]\n$a=$ [ANS]", "answer_v1": [ "-4*x^3", "2" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find a function $f$ and a number $a$ such that 1+\\int_{a}^{x} \\frac {f(t)} {t^{2}} dt=6x^{-3} $f(x)=$ [ANS]\n$a=$ [ANS]", "answer_v2": [ "-18*x^{-2}", "1.81712059283214" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find a function $f$ and a number $a$ such that 1+\\int_{a}^{x} \\frac {f(t)} {t^{3}} dt=5x^{-1} $f(x)=$ [ANS]\n$a=$ [ANS]", "answer_v3": [ "-5*x^1", "5" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0570", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "4", "keywords": [], "problem_v1": "Suppose $ f(x)=\\int_0^x \\frac{t^2-36}{3+\\cos^2(t)} \\, dt.$ For what value(s) of $x$ does $f(x)$ have a local minimum? $x=$ [ANS] (Enter a number, a list of numbers separated by commas, or NONE.)", "answer_v1": [ "6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose $ f(x)=\\int_0^x \\frac{1-t^2}{5+\\cos^2(t)} \\, dt.$ For what value(s) of $x$ does $f(x)$ have a local minimum? $x=$ [ANS] (Enter a number, a list of numbers separated by commas, or NONE.)", "answer_v2": [ "-1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose $ f(x)=\\int_0^x \\frac{9-t^2}{4+\\cos^2(t)} \\, dt.$ For what value(s) of $x$ does $f(x)$ have a local maximum? $x=$ [ANS] (Enter a number, a list of numbers separated by commas, or NONE.)", "answer_v3": [ "3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0571", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "properties of integrals", "area" ], "problem_v1": "Let $f(t)=F'(t)$. Write the integral $\\int_a^b f(t) dt$ and evaluate it using the Fundamental Theorem of Calculus: if $F(t)=t^{6}$, $a=3$, and $b=6$, then the integral is\n$\\int_a^b f(t) dt=\\int_{v_1}^{v_2}$ [ANS], where $v_1=$ [ANS]\n$v_2=$ [ANS]\nThen, after evaluation, $\\int_a^b f(t) dt=$ [ANS]", "answer_v1": [ "6*t^5 dt", "3", "6", "6^6 - 3^6" ], "answer_type_v1": [ "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $f(t)=F'(t)$. Write the integral $\\int_a^b f(t) dt$ and evaluate it using the Fundamental Theorem of Calculus: if $F(t)=t^{3}$, $a=4$, and $b=5$, then the integral is\n$\\int_a^b f(t) dt=\\int_{v_1}^{v_2}$ [ANS], where $v_1=$ [ANS]\n$v_2=$ [ANS]\nThen, after evaluation, $\\int_a^b f(t) dt=$ [ANS]", "answer_v2": [ "3*t^2 dt", "4", "5", "5^3 - 4^3" ], "answer_type_v2": [ "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $f(t)=F'(t)$. Write the integral $\\int_a^b f(t) dt$ and evaluate it using the Fundamental Theorem of Calculus: if $F(t)=t^{4}$, $a=3$, and $b=5$, then the integral is\n$\\int_a^b f(t) dt=\\int_{v_1}^{v_2}$ [ANS], where $v_1=$ [ANS]\n$v_2=$ [ANS]\nThen, after evaluation, $\\int_a^b f(t) dt=$ [ANS]", "answer_v3": [ "4*t^3 dt", "3", "5", "5^4 - 3^4" ], "answer_type_v3": [ "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0572", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "4", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "properties of integrals", "area" ], "problem_v1": "Suppose $f'(x)=\\sqrt{x}\\sin(x^2)$ and $f(0)=3$. A. Use a graph of $f'(x)$ to decide which is larger: [ANS] A. $f(0)$ B. $f(1.2)$\nUse a graph of $f'(x)$ to decide which is larger: [ANS] A. $f(2.5)$ B. $f(2.1)$\nB. Estimate $f(b)$ for: $b=0$: $f(b) \\approx$ [ANS]\n$b=1$: $f(b) \\approx$ [ANS]\n$b=2$: $f(b) \\approx$ [ANS]\n$b=3$: $f(b) \\approx$ [ANS]", "answer_v1": [ "B", "B", "3", "3 + 0.264204", "3 + 0.808583", "3 + 0.824944" ], "answer_type_v1": [ "MCS", "MCS", "NV", "NV", "NV", "NV" ], "options_v1": [ [ "A", "B" ], [ "A", "B" ], [], [], [], [] ], "problem_v2": "Suppose $f'(x)=\\sin(x^2)$ and $f(0)=5$. A. Use a graph of $f'(x)$ to decide which is larger: [ANS] A. $f(0.6)$ B. $f(0)$\nUse a graph of $f'(x)$ to decide which is larger: [ANS] A. $f(1.9)$ B. $f(2.5)$\nB. Estimate $f(b)$ for: $b=0$: $f(b) \\approx$ [ANS]\n$b=1$: $f(b) \\approx$ [ANS]\n$b=2$: $f(b) \\approx$ [ANS]\n$b=3$: $f(b) \\approx$ [ANS]", "answer_v2": [ "A", "A", "5", "5 + 0.310268", "5 + 0.804776", "5 + 0.773563" ], "answer_type_v2": [ "MCS", "MCS", "NV", "NV", "NV", "NV" ], "options_v2": [ [ "A", "B" ], [ "A", "B" ], [], [], [], [] ], "problem_v3": "Suppose $f'(x)=\\sin(x^2)$ and $f(0)=4$. A. Use a graph of $f'(x)$ to decide which is larger: [ANS] A. $f(0)$ B. $f(0.8)$\nUse a graph of $f'(x)$ to decide which is larger: [ANS] A. $f(2.5)$ B. $f(2)$\nB. Estimate $f(b)$ for: $b=0$: $f(b) \\approx$ [ANS]\n$b=1$: $f(b) \\approx$ [ANS]\n$b=2$: $f(b) \\approx$ [ANS]\n$b=3$: $f(b) \\approx$ [ANS]", "answer_v3": [ "B", "B", "4", "4 + 0.310268", "4 + 0.804776", "4 + 0.773563" ], "answer_type_v3": [ "MCS", "MCS", "NV", "NV", "NV", "NV" ], "options_v3": [ [ "A", "B" ], [ "A", "B" ], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0573", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "antiderivatives" ], "problem_v1": "Calculate the derivative: $ \\frac{d\\}{dt}\\int_1^{\\tan t} \\cos(x^2)\\,dx=$ [ANS]", "answer_v1": [ "[cos([tan(t)]^2)]/([cos(t)]^2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Calculate the derivative: $ \\frac{d\\}{dt}\\int_1^{\\sin t} \\cos(x^2)\\,dx=$ [ANS]", "answer_v2": [ "cos([sin(t)]^2)*cos(t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Calculate the derivative: $ \\frac{d\\}{dt}\\int_1^{\\cos t} \\sin(x^2)\\,dx=$ [ANS]", "answer_v3": [ "-1*sin([cos(t)]^2)*sin(t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0574", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "antiderivatives" ], "problem_v1": "Write an expression for the function, $f(x)$, with the properties $f'(x)={(\\tan(x))}/{x}$ and $f(4)=6$. $f(x)=\\int_{t\\,=\\,a}^{t\\,=\\,b}$ [ANS], where $a=$ [ANS]\nand $b=$ [ANS]", "answer_v1": [ "[tan(t)]/t*dt+6", "4", "x" ], "answer_type_v1": [ "EX", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Write an expression for the function, $f(x)$, with the properties $f'(x)={(\\sin(x))}/{x}$ and $f(1)=9$. $f(x)=\\int_{t\\,=\\,a}^{t\\,=\\,b}$ [ANS], where $a=$ [ANS]\nand $b=$ [ANS]", "answer_v2": [ "[sin(t)]/t*dt+9", "1", "x" ], "answer_type_v2": [ "EX", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Write an expression for the function, $f(x)$, with the properties $f'(x)={(\\cos(x))}/{x}$ and $f(2)=6$. $f(x)=\\int_{t\\,=\\,a}^{t\\,=\\,b}$ [ANS], where $a=$ [ANS]\nand $b=$ [ANS]", "answer_v3": [ "[cos(t)]/t*dt+6", "2", "x" ], "answer_type_v3": [ "EX", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0575", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "antiderivatives" ], "problem_v1": "Let $ F(x)=\\int_{8}^x \\frac{6}{\\ln(4 t)} \\, dt$, for $x\\geq {8}$. A. $F'(x)=$ [ANS]\nB. On what interval or intervals is $F$ increasing? $x \\in$ [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10), or enter none for no intervals.) for no intervals.) C. On what interval or intervals is the graph of $F$ concave up? $x\\in$ [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10), or enter none for no intervals.) for no intervals.)\nUse your answers above and your knowledge of derivatives and antiderivatives to sketch a graph of $F(x)$.", "answer_v1": [ "6/[ln(4*x)]", "[8,infinity)", "none" ], "answer_type_v1": [ "EX", "INT", "OE" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $ F(x)=\\int_{2}^x \\frac{9}{\\ln(t)} \\, dt$, for $x\\geq {2}$. A. $F'(x)=$ [ANS]\nB. On what interval or intervals is $F$ increasing? $x \\in$ [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10), or enter none for no intervals.) for no intervals.) C. On what interval or intervals is the graph of $F$ concave up? $x\\in$ [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10), or enter none for no intervals.) for no intervals.)\nUse your answers above and your knowledge of derivatives and antiderivatives to sketch a graph of $F(x)$.", "answer_v2": [ "9/[ln(1*x)]", "[2,infinity)", "none" ], "answer_type_v2": [ "EX", "INT", "OE" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $ F(x)=\\int_{4}^x \\frac{6}{\\ln(2 t)} \\, dt$, for $x\\geq {4}$. A. $F'(x)=$ [ANS]\nB. On what interval or intervals is $F$ increasing? $x \\in$ [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10), or enter none for no intervals.) for no intervals.) C. On what interval or intervals is the graph of $F$ concave up? $x\\in$ [ANS]\n(Give your answer as an interval or a list of intervals, e.g., (Give your answer as an interval or a list of intervals, e.g., (-infinity,8] or (1,5),(7,10), or enter none for no intervals.) for no intervals.)\nUse your answers above and your knowledge of derivatives and antiderivatives to sketch a graph of $F(x)$.", "answer_v3": [ "6/[ln(2*x)]", "[4,infinity)", "none" ], "answer_type_v3": [ "EX", "INT", "OE" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0576", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "2", "keywords": [ "derivative" ], "problem_v1": "Sketch and find the area of the region below the interval $[-3,-2]$ and above the curve $y=\\frac{x^{3}}{4}$. Area=[ANS]", "answer_v1": [ "65/16" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sketch and find the area of the region below the interval $[-6,-5]$ and above the curve $y=\\frac{x^{3}}{25}$. Area=[ANS]", "answer_v2": [ "671/100" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sketch and find the area of the region below the interval $[-5,-4]$ and above the curve $y=\\frac{x^{3}}{16}$. Area=[ANS]", "answer_v3": [ "369/64" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0577", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "antiderivatives" ], "problem_v1": "On what interval(s) is the curve y=\\int_0^x \\frac{t^2}{t^2+3 t+6}\\,dt concave upward? Answer (in interval notation): [ANS]", "answer_v1": [ "(-infinity,-4) U (0,infinity)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "On what interval(s) is the curve y=\\int_0^x \\frac{t^2}{t^2-4 t+7}\\,dt concave upward? Answer (in interval notation): [ANS]", "answer_v2": [ "(0,7/2)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "On what interval(s) is the curve y=\\int_0^x \\frac{t^2}{t^2-2 t+5}\\,dt concave downward? Answer (in interval notation): [ANS]", "answer_v3": [ "(-infinity,0) U (5,infinity)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0578", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "calculus", "integrals", "fundamental theorem of calculus" ], "problem_v1": "Use the Fundamental Theorem of Calculus to find the derivative of y=\\int_{-7}^{\\sqrt{x}} {\\frac{\\cos t}{t^{11}}} dt $\\frac{dy}{dx}$=[ANS]\n[NOTE: Enter a function as your answer. Make sure that your syntax is correct, i.e. remember to put all the necessary (,), etc.]", "answer_v1": [ "cos(sqrt(x))/(2*x^6" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the Fundamental Theorem of Calculus to find the derivative of y=\\int_{-1}^{\\sqrt{x}} {\\frac{\\cos t}{t^{5}}} dt $\\frac{dy}{dx}$=[ANS]\n[NOTE: Enter a function as your answer. Make sure that your syntax is correct, i.e. remember to put all the necessary (,), etc.]", "answer_v2": [ "cos(sqrt(x))/(2*x^3)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the Fundamental Theorem of Calculus to find the derivative of y=\\int_{-3}^{\\sqrt{x}} {\\frac{\\cos t}{t^{7}}} dt $\\frac{dy}{dx}$=[ANS]\n[NOTE: Enter a function as your answer. Make sure that your syntax is correct, i.e. remember to put all the necessary (,), etc.]", "answer_v3": [ "cos(sqrt(x))/(2*x^4)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0579", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "2", "keywords": [ "calculus", "integrals", "definite integrals", "fundamental theorem of calculus" ], "problem_v1": "Use Part 2 of the Fundamental Theorem of Calculus to decide if the definite integral exists and either evaluate the integral or enter DNE if it does not exist.\n\\int_{3}^{8} {5^t} dt [ANS]", "answer_v1": [ "1/ln(5)*(5^8-5^3)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Part 2 of the Fundamental Theorem of Calculus to decide if the definite integral exists and either evaluate the integral or enter DNE if it does not exist.\n\\int_{1}^{8} {2^t} dt [ANS]", "answer_v2": [ "1/ln(2)*(2^8-2^1)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Part 2 of the Fundamental Theorem of Calculus to decide if the definite integral exists and either evaluate the integral or enter DNE if it does not exist.\n\\int_{1}^{6} {3^t} dt [ANS]", "answer_v3": [ "1/ln(3)*(3^6-3^1)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0581", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "2", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "properties of integrals", "area" ], "problem_v1": "Part 1 of 2:\nThis is a multi-part problem. If $F(t)=4 t^2$, find $F^{\\prime}(t)$.\n$F^{\\prime}(t)=$ [ANS]", "answer_v1": [ "4*2*t" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Part 1 of 2:\nThis is a multi-part problem. If $F(t)=2 t^2$, find $F^{\\prime}(t)$.\n$F^{\\prime}(t)=$ [ANS]", "answer_v2": [ "2*2*t" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Part 1 of 2:\nThis is a multi-part problem. If $F(t)=3 t^2$, find $F^{\\prime}(t)$.\n$F^{\\prime}(t)=$ [ANS]", "answer_v3": [ "3*2*t" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0582", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "2", "keywords": [], "problem_v1": "Evaluate the definite integral. Your answer will be a function of $x$.\n$ \\int_{3}^x (3 t^2+2 t+4) dt=$ [ANS].", "answer_v1": [ "3*x^3/3 + 2*x^2/2 + 4*x - 48" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Evaluate the definite integral. Your answer will be a function of $x$.\n$ \\int_{-5}^x (9 t^2-6 t-3) dt=$ [ANS].", "answer_v2": [ "9*x^3/3 + -6*x^2/2 + -3*x - -435" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Evaluate the definite integral. Your answer will be a function of $x$.\n$ \\int_{-2}^x (3 t^2-4 t+1) dt=$ [ANS].", "answer_v3": [ "3*x^3/3 + -4*x^2/2 + 1*x - -18" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0583", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "algebra", "inequality", "fraction" ], "problem_v1": "The United States has been consuming iron ore at the rate of $R(t)$ million metric tons per year at time $t$, where $t$ is measured in years since 1980 (that is, $t=0$ corresponds to the year 1980), and R(t)=17.9e^{0.018t}. Find a formula $T(t)$ for the total U.S. consumption of iron ore, in millions of metric tons, from 1980 until time $t$. $T(t)=$ [ANS]", "answer_v1": [ "994.444*[e^{0.018*t}-1]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "The United States has been consuming iron ore at the rate of $R(t)$ million metric tons per year at time $t$, where $t$ is measured in years since 1980 (that is, $t=0$ corresponds to the year 1980), and R(t)=19.66e^{0.01t}. Find a formula $T(t)$ for the total U.S. consumption of iron ore, in millions of metric tons, from 1980 until time $t$. $T(t)=$ [ANS]", "answer_v2": [ "1966*[e^{0.01*t}-1]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "The United States has been consuming iron ore at the rate of $R(t)$ million metric tons per year at time $t$, where $t$ is measured in years since 1980 (that is, $t=0$ corresponds to the year 1980), and R(t)=18.02e^{0.013t}. Find a formula $T(t)$ for the total U.S. consumption of iron ore, in millions of metric tons, from 1980 until time $t$. $T(t)=$ [ANS]", "answer_v3": [ "1386.15*[e^{0.013*t}-1]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0584", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Fundamental theorem of calculus", "level": "3", "keywords": [ "calculus", "integration", "definite integrals", "substitution" ], "problem_v1": "If $f$ is continuous and $ \\int_0^{81}\\!\\! f(t)\\,dt=-15,$ find the following integrals. 1. $ \\int_0^{9}\\! f(9 t)\\,dt=$ [ANS]\n2. $ \\int_0^{9}\\! tf(t^2)\\,dt=$ [ANS]", "answer_v1": [ "-15/9", "-15/2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If $f$ is continuous and $ \\int_0^{9}\\!\\! f(t)\\,dt=-5,$ find the following integrals. 1. $ \\int_0^{3}\\! f(3 t)\\,dt=$ [ANS]\n2. $ \\int_0^{3}\\! tf(t^2)\\,dt=$ [ANS]", "answer_v2": [ "-5/3", "-5/2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If $f$ is continuous and $ \\int_0^{25}\\!\\! f(t)\\,dt=-10,$ find the following integrals. 1. $ \\int_0^{5}\\! f(5 t)\\,dt=$ [ANS]\n2. $ \\int_0^{5}\\! tf(t^2)\\,dt=$ [ANS]", "answer_v3": [ "-10/5", "-10/2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0585", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Determine if the improper integral converges and, if so, evaluate it. $\\int^0_{-\\infty}e^{5x}dx$=[ANS]\nWrite F if the integral doesn't converge", "answer_v1": [ "0.2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine if the improper integral converges and, if so, evaluate it. $\\int^0_{-\\infty}e^{2x}dx$=[ANS]\nWrite F if the integral doesn't converge", "answer_v2": [ "0.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine if the improper integral converges and, if so, evaluate it. $\\int^0_{-\\infty}e^{3x}dx$=[ANS]\nWrite F if the integral doesn't converge", "answer_v3": [ "0.333333333333333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0586", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "2", "keywords": [ "calculus", "integration", "fraction", "improper", "integral", "improper integral", "\\inftyinity" ], "problem_v1": "Determine if the improper integral converges and, if so, evaluate it. $\\int^{\\infty}_{10} \\frac{dx}{\\sqrt{x}-3}$. [ANS] A. Diverges B. 1 C. 10 D. 0", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Determine if the improper integral converges and, if so, evaluate it. $\\int^{\\infty}_{2} \\frac{dx}{\\sqrt{x}-1}$. [ANS] A. 2 B. 1 C. 0 D. Diverges", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Determine if the improper integral converges and, if so, evaluate it. $\\int^{\\infty}_{2} \\frac{dx}{\\sqrt{x}-1}$. [ANS] A. 2 B. Diverges C. 1 D. 0", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_single_variable_0587", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "2", "keywords": [ "Integral", "Improper Integral", "integrals", "improper", "calculus", "integral" ], "problem_v1": "Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as \"infinity\". If it diverges to negative infinity, state your answer as \"-infinity\". If it diverges without being infinity or negative infinity, state your answer as \"divergent\". $\\int_{5}^{\\,69} \\frac{6}{\\sqrt[3]{x-5}} \\,dx$=[ANS].", "answer_v1": [ "288/2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as \"infinity\". If it diverges to negative infinity, state your answer as \"-infinity\". If it diverges without being infinity or negative infinity, state your answer as \"divergent\". $\\int_{7}^{\\,15} \\frac{3}{\\sqrt[3]{x-7}} \\,dx$=[ANS].", "answer_v2": [ "36/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as \"infinity\". If it diverges to negative infinity, state your answer as \"-infinity\". If it diverges without being infinity or negative infinity, state your answer as \"divergent\". $\\int_{5}^{\\,13} \\frac{4}{\\sqrt[3]{x-5}} \\,dx$=[ANS].", "answer_v3": [ "48/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0588", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "3", "keywords": [ "improper integral' 'convergent' 'divergent' 'comparison test" ], "problem_v1": "For each of the improper integrals below, if the comparison test applies, enter either A or B followed by one letter from C to K that best applies, and if the comparison test does not apply, enter only L. For example, one possible answer is BF, and another one is L.\nHint: $0 < e^{-x}\\leq 1$ for $x\\geq 1$. [ANS] 1. $ \\int_1^\\infty \\frac{x}{\\sqrt{x^6+6}}\\,dx$ [ANS] 2. $ \\int_1^\\infty \\frac{e^{-x}}{x^2}\\,dx$ [ANS] 3. $ \\int_1^\\infty \\frac{\\cos^2(x)}{x^2+6}\\,dx$ [ANS] 4. $ \\int_1^\\infty \\frac{8+\\sin(x)}{\\sqrt{x-0.6}}\\,dx$ [ANS] 5. $ \\int_1^\\infty \\frac{1}{x^2+6}\\,dx$\nA. The integral is convergent B. The integral is divergent C. by comparison to $\\int_1^\\infty\\frac{1}{x^2-6}\\,dx$. D. by comparison to $\\int_1^\\infty\\frac{1}{x^2+6}\\,dx$. E. by comparison to $\\int_1^\\infty\\frac{\\cos^2(x)}{x^2}\\,dx$. F. by comparison to $\\int_1^\\infty\\frac{e^x}{x^2}\\,dx$. G. by comparison to $\\int_1^\\infty\\frac{-e^{-x}}{2x}\\,dx$. H. by comparison to $\\int_1^\\infty\\frac{1}{\\sqrt{x}}\\,dx$. I. by comparison to $\\int_1^\\infty\\frac{1}{\\sqrt{x^5}}\\,dx$. J. by comparison to $\\int_1^\\infty\\frac{1}{x^2}\\,dx$. K. by comparison to $\\int_1^\\infty\\frac{1}{x^3}\\,dx$. L. The comparison test does not apply.", "answer_v1": [ "AJ", "AJ", "AJ", "BH", "AJ" ], "answer_type_v1": [ "MCM", "MCM", "MCM", "MCM", "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ] ], "problem_v2": "For each of the improper integrals below, if the comparison test applies, enter either A or B followed by one letter from C to K that best applies, and if the comparison test does not apply, enter only L. For example, one possible answer is BF, and another one is L.\nHint: $0 < e^{-x}\\leq 1$ for $x\\geq 1$. [ANS] 1. $ \\int_1^\\infty \\frac{e^{-x}}{x^2}\\,dx$ [ANS] 2. $ \\int_1^\\infty \\frac{10+\\sin(x)}{\\sqrt{x-0.3}}\\,dx$ [ANS] 3. $ \\int_1^\\infty \\frac{\\cos^2(x)}{x^2+2}\\,dx$ [ANS] 4. $ \\int_1^\\infty \\frac{1}{x^2+2}\\,dx$ [ANS] 5. $ \\int_1^\\infty \\frac{x}{\\sqrt{x^6+2}}\\,dx$\nA. The integral is convergent B. The integral is divergent C. by comparison to $\\int_1^\\infty\\frac{1}{x^2-2}\\,dx$. D. by comparison to $\\int_1^\\infty\\frac{1}{x^2+2}\\,dx$. E. by comparison to $\\int_1^\\infty\\frac{\\cos^2(x)}{x^2}\\,dx$. F. by comparison to $\\int_1^\\infty\\frac{e^x}{x^2}\\,dx$. G. by comparison to $\\int_1^\\infty\\frac{-e^{-x}}{2x}\\,dx$. H. by comparison to $\\int_1^\\infty\\frac{1}{\\sqrt{x}}\\,dx$. I. by comparison to $\\int_1^\\infty\\frac{1}{\\sqrt{x^5}}\\,dx$. J. by comparison to $\\int_1^\\infty\\frac{1}{x^2}\\,dx$. K. by comparison to $\\int_1^\\infty\\frac{1}{x^3}\\,dx$. L. The comparison test does not apply.", "answer_v2": [ "AJ", "BH", "AJ", "AJ", "AJ" ], "answer_type_v2": [ "MCM", "MCM", "MCM", "MCM", "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ] ], "problem_v3": "For each of the improper integrals below, if the comparison test applies, enter either A or B followed by one letter from C to K that best applies, and if the comparison test does not apply, enter only L. For example, one possible answer is BF, and another one is L.\nHint: $0 < e^{-x}\\leq 1$ for $x\\geq 1$. [ANS] 1. $ \\int_1^\\infty \\frac{1}{x^2+3}\\,dx$ [ANS] 2. $ \\int_1^\\infty \\frac{\\cos^2(x)}{x^2+3}\\,dx$ [ANS] 3. $ \\int_1^\\infty \\frac{x}{\\sqrt{x^6+3}}\\,dx$ [ANS] 4. $ \\int_1^\\infty \\frac{8+\\sin(x)}{\\sqrt{x-0.3}}\\,dx$ [ANS] 5. $ \\int_1^\\infty \\frac{e^{-x}}{x^2}\\,dx$\nA. The integral is convergent B. The integral is divergent C. by comparison to $\\int_1^\\infty\\frac{1}{x^2-3}\\,dx$. D. by comparison to $\\int_1^\\infty\\frac{1}{x^2+3}\\,dx$. E. by comparison to $\\int_1^\\infty\\frac{\\cos^2(x)}{x^2}\\,dx$. F. by comparison to $\\int_1^\\infty\\frac{e^x}{x^2}\\,dx$. G. by comparison to $\\int_1^\\infty\\frac{-e^{-x}}{2x}\\,dx$. H. by comparison to $\\int_1^\\infty\\frac{1}{\\sqrt{x}}\\,dx$. I. by comparison to $\\int_1^\\infty\\frac{1}{\\sqrt{x^5}}\\,dx$. J. by comparison to $\\int_1^\\infty\\frac{1}{x^2}\\,dx$. K. by comparison to $\\int_1^\\infty\\frac{1}{x^3}\\,dx$. L. The comparison test does not apply.", "answer_v3": [ "AJ", "AJ", "AJ", "BH", "AJ " ], "answer_type_v3": [ "MCM", "MCM", "MCM", "MCM", "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K" ] ] }, { "id": "Calculus_-_single_variable_0589", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "3", "keywords": [ "calculus", "integrals", "improper" ], "problem_v1": "Determine whether each of the following integrals is proper, improper and convergent, or improper and divergent. [ANS] 1. $ \\int_{-\\infty}^{\\infty}\\frac{t}{t^2+9}\\,dt$\n[ANS] 2. $ \\int_{6}^{\\infty}\\frac{1}{\\sqrt{t^2-36}}\\,dt$\n[ANS] 3. $ \\int_{-9\\pi}^{35\\pi}\\sin(\\theta)\\arctan(\\theta)\\,d\\theta$\n[ANS] 4. $ \\int_{0}^{19}\\frac{1}{\\sqrt[3]{x-9}}\\,dx$\n[ANS] 5. $ \\int_{-\\pi/9}^{19\\pi/2}\\tan^2(6x)\\,dx$\n[ANS] 6. $ \\int_{1}^{\\infty}s e^{-6 s^2}\\,ds$\n[ANS] 7. $ \\int_{9}^{19}\\ln(x-9)\\,dx$\n[ANS] 8. $ \\int_{-\\infty}^{\\infty}\\sin(6 z)\\,dz$", "answer_v1": [ "Improper and divergent", "Improper and divergent", "Proper", "Improper and convergent", "Improper and divergent", "Improper and convergent", "Improper and convergent", "Improper and divergent" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ] ], "problem_v2": "Determine whether each of the following integrals is proper, improper and convergent, or improper and divergent. [ANS] 1. $ \\int_{4}^{9}\\ln(x-4)\\,dx$\n[ANS] 2. $ \\int_{-\\infty}^{\\infty}\\frac{t}{t^2+6}\\,dt$\n[ANS] 3. $ \\int_{-\\pi/4}^{9\\pi/2}\\tan^2(8x)\\,dx$\n[ANS] 4. $ \\int_{8}^{\\infty}\\frac{1}{\\sqrt{t^2-64}}\\,dt$\n[ANS] 5. $ \\int_{-6\\pi}^{20\\pi}\\sin(\\theta)\\arctan(\\theta)\\,d\\theta$\n[ANS] 6. $ \\int_{-\\infty}^{\\infty}\\sin(8 z)\\,dz$\n[ANS] 7. $ \\int_{1}^{\\infty}s e^{8 s^2}\\,ds$\n[ANS] 8. $ \\int_{0}^{9}\\frac{1}{\\sqrt[3]{x-4}}\\,dx$", "answer_v2": [ "Improper and convergent", "Improper and divergent", "Improper and divergent", "Improper and divergent", "Proper", "Improper and divergent", "Improper and divergent", "Improper and convergent" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ] ], "problem_v3": "Determine whether each of the following integrals is proper, improper and convergent, or improper and divergent. [ANS] 1. $ \\int_{-\\pi/6}^{13\\pi/2}\\tan^2(6x)\\,dx$\n[ANS] 2. $ \\int_{6}^{\\infty}\\frac{1}{\\sqrt{t^2-36}}\\,dt$\n[ANS] 3. $ \\int_{1}^{\\infty}s e^{6 s^2}\\,ds$\n[ANS] 4. $ \\int_{6}^{13}\\ln(x-6)\\,dx$\n[ANS] 5. $ \\int_{-7\\pi}^{28\\pi}\\sin(\\theta)\\arctan(\\theta)\\,d\\theta$\n[ANS] 6. $ \\int_{-\\infty}^{\\infty}\\sin(6 z)\\,dz$\n[ANS] 7. $ \\int_{-\\infty}^{\\infty}\\frac{t}{t^2+7}\\,dt$\n[ANS] 8. $ \\int_{0}^{13}\\frac{1}{\\sqrt[3]{x-6}}\\,dx$", "answer_v3": [ "Improper and divergent", "Improper and divergent", "Improper and divergent", "Improper and convergent", "Proper", "Improper and divergent", "Improper and divergent", "Improper and convergent" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ], [ "Proper", "Improper and convergent", "Improper and divergent" ] ] }, { "id": "Calculus_-_single_variable_0590", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "3", "keywords": [ "integrals", "improper" ], "problem_v1": "Find the area under the curve y=1x^{-2} from $x=8\\ $ to $\\ x=t$ and evaluate it for $t=10\\ $, $\\ t=100$. Then find the total area under this curve for $x \\geq 8$.\n(a) t=10 [ANS]\n(b) t=100 [ANS]\n(c) Total area [ANS]", "answer_v1": [ "0.025", "0.115", "0.125" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the area under the curve y=0.5x^{-1.5} from $x=5\\ $ to $\\ x=t$ and evaluate it for $t=10\\ $, $\\ t=100$. Then find the total area under this curve for $x \\geq 5$.\n(a) t=10 [ANS]\n(b) t=100 [ANS]\n(c) Total area [ANS]", "answer_v2": [ "0.13098582948312", "0.347213595499958", "0.447213595499958" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the area under the curve y=1x^{-2} from $x=6\\ $ to $\\ x=t$ and evaluate it for $t=10\\ $, $\\ t=100$. Then find the total area under this curve for $x \\geq 6$.\n(a) t=10 [ANS]\n(b) t=100 [ANS]\n(c) Total area [ANS]", "answer_v3": [ "0.0666666666666667", "0.156666666666667", "0.166666666666667" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0591", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "4", "keywords": [], "problem_v1": "Let $f(x)$ be a function that is defined and has a continuous derivative on the interval $(2, \\infty)$. Assume also that f(2)=11 |f(x)| < x^{4}+7 and \\int_{2}^{\\infty} f(x) e^{-x/7} \\,dx=2 Determine the value of \\int_{2}^{\\infty} f'(x) e^{-x/7} \\,dx [ANS]", "answer_v1": [ "-7.98053593811386" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(x)$ be a function that is defined and has a continuous derivative on the interval $(2, \\infty)$. Assume also that f(2)=-6 |f(x)| < x^{9}+2 and \\int_{2}^{\\infty} f(x) e^{-x/2} \\,dx=8 Determine the value of \\int_{2}^{\\infty} f'(x) e^{-x/2} \\,dx [ANS]", "answer_v2": [ "6.20727664702865" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(x)$ be a function that is defined and has a continuous derivative on the interval $(2, \\infty)$. Assume also that f(3)=8 |f(x)| < x^{3}+4 and \\int_{3}^{\\infty} f(x) e^{-x/4} \\,dx=2 Determine the value of \\int_{3}^{\\infty} f'(x) e^{-x/4} \\,dx [ANS]", "answer_v3": [ "-3.27893242192812" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0592", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "3", "keywords": [ "calculus", "integral", "improper", "infinite", "definite integrals" ], "problem_v1": "Calculate the integral below, if it converges. If it does not converge, enter diverges for your answer. $\\int_{2}^{\\infty} 3x^{2}e^{-x^{3}} \\,dx=$ [ANS]", "answer_v1": [ "3*e^{-8}/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the integral below, if it converges. If it does not converge, enter diverges for your answer. $\\int_{0}^{\\infty} 5xe^{-x^{2}} \\,dx=$ [ANS]", "answer_v2": [ "2.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the integral below, if it converges. If it does not converge, enter diverges for your answer. $\\int_{0}^{\\infty} 4xe^{-x^{2}} \\,dx=$ [ANS]", "answer_v3": [ "2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0593", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "2", "keywords": [ "calculus", "integral", "improper", "infinite", "definite integrals" ], "problem_v1": "Calculate the integral, if it converges. If it diverges, enter diverges for your answer. $\\int_{36}^{40}\\,{1\\over y^2-36}\\,dy=$ [ANS]", "answer_v1": [ "(1/(2*6))*(ln(40-6)-ln(40+6)-ln(36-6)+ln(36+6))" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the integral, if it converges. If it diverges, enter diverges for your answer. $\\int_{4}^{8}\\,{1\\over y^2-4}\\,dy=$ [ANS]", "answer_v2": [ "(1/(2*2))*(ln(8-2)-ln(8+2)-ln(4-2)+ln(4+2))" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the integral, if it converges. If it diverges, enter diverges for your answer. $\\int_{9}^{10}\\,{1\\over y^2-9}\\,dy=$ [ANS]", "answer_v3": [ "(1/(2*3))*(ln(10-3)-ln(10+3)-ln(9-3)+ln(9+3))" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0594", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "3", "keywords": [ "calculus", "integral", "improper", "infinite", "definite integrals" ], "problem_v1": "Find the area under the curve $y=7xe^{-x}$ for $x\\ge 2$. area=[ANS]", "answer_v1": [ "7*(2+1)*e^{-2}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the area under the curve $y=xe^{-x}$ for $x\\ge 4$. area=[ANS]", "answer_v2": [ "1*(4+1)*e^{-4}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the area under the curve $y=3xe^{-x}$ for $x\\ge 3$. area=[ANS]", "answer_v3": [ "3*(3+1)*e^{-3}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0595", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "2", "keywords": [ "calculus", "integral", "improper", "infinite", "definite integrals", "comparison test" ], "problem_v1": "For each of the following integrals, give a power or simple exponential function that if integrated on a similar infinite domain will have the same convergence or divergence behavior as the given integral, and use that to predict whether the integral converges or diverges. Note that for this problem we are not formally applying the comparison test; we are simply looking at the behavior of the integrals to build intuition.\n(To indicate convergence or divergence, enter one of the words converges or diverges in the appropriate answer blanks.)\n$\\begin{array}{ccccc}\\hline \\int_1^{\\infty} {x^{2}+6\\over x^{4}+8x^{2}+6} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\int_1^{\\infty} {x\\over x+7} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\int_1^{\\infty} {x^{3}+1\\over x^{4}+8x+6} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\int_1^{\\infty} {x+6\\over x^{4}+6x^{3}+6} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "1/(x^2)", "converges", "1", "diverges", "1/x", "diverges", "1/(x^3)", "converges" ], "answer_type_v1": [ "EX", "EX", "NV", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [ "convergence", "divergence" ], [], [ "convergence", "divergence" ], [], [ "convergence", "divergence" ], [], [ "convergence", "divergence" ] ], "problem_v2": "For each of the following integrals, give a power or simple exponential function that if integrated on a similar infinite domain will have the same convergence or divergence behavior as the given integral, and use that to predict whether the integral converges or diverges. Note that for this problem we are not formally applying the comparison test; we are simply looking at the behavior of the integrals to build intuition.\n(To indicate convergence or divergence, enter one of the words converges or diverges in the appropriate answer blanks.)\n$\\begin{array}{ccccc}\\hline \\int_1^{\\infty} {x\\over x+1} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\int_1^{\\infty} {e^{2x}\\over e^{3x}+1} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\int_1^{\\infty} {x^{2}+1\\over x^{3}+2x+1} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\int_1^{\\infty} {x\\over x^{3}+1} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "1", "diverges", "1/(e^x)", "converges", "1/x", "diverges", "1/(x^2)", "converges" ], "answer_type_v2": [ "NV", "MCS", "EX", "MCS", "EX", "MCS", "EX", "MCS" ], "options_v2": [ [], [ "convergence", "divergence" ], [], [ "convergence", "divergence" ], [], [ "convergence", "divergence" ], [], [ "convergence", "divergence" ] ], "problem_v3": "For each of the following integrals, give a power or simple exponential function that if integrated on a similar infinite domain will have the same convergence or divergence behavior as the given integral, and use that to predict whether the integral converges or diverges. Note that for this problem we are not formally applying the comparison test; we are simply looking at the behavior of the integrals to build intuition.\n(To indicate convergence or divergence, enter one of the words converges or diverges in the appropriate answer blanks.)\n$\\begin{array}{ccccc}\\hline \\int_1^{\\infty} {x+2\\over x^{3}+6x^{2}+2} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\int_1^{\\infty} {x^{2}+1\\over x^{3}+4x+2} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\int_1^{\\infty} {x\\over x^2+4x+2} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\int_1^{\\infty} {e^{2x}\\over e^{3x}+2} \\,dx: & a similar integrand is & [ANS] & so we predict the integral & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "1/(x^2)", "converges", "1/x", "diverges", "1/x", "diverges", "1/(e^x)", "converges" ], "answer_type_v3": [ "EX", "MCS", "EX", "MCS", "EX", "MCS", "EX", "MCS" ], "options_v3": [ [], [ "convergence", "divergence" ], [], [ "convergence", "divergence" ], [], [ "convergence", "divergence" ], [], [ "convergence", "divergence" ] ] }, { "id": "Calculus_-_single_variable_0596", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "3", "keywords": [ "calculus", "integral", "improper", "infinite", "definite integrals", "comparison test" ], "problem_v1": "Find the value of $a$ (to three decimal places) that makes \\int_{-\\infty}^{\\infty} a e^{-x^2/6}\\,dx=1. $a=$ [ANS]", "answer_v1": [ "1/sqrt(6*pi)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the value of $a$ (to three decimal places) that makes \\int_{-\\infty}^{\\infty} a e^{-x^2/2}\\,dx=1. $a=$ [ANS]", "answer_v2": [ "1/sqrt(2*pi)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the value of $a$ (to three decimal places) that makes \\int_{-\\infty}^{\\infty} a e^{-x^2/3}\\,dx=1. $a=$ [ANS]", "answer_v3": [ "1/sqrt(3*pi)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0597", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "3", "keywords": [ "derivative" ], "problem_v1": "Evaluate the integrals that converge, enter 'DNC' if integral Does Not Converge. $ \\int_{7}^{+\\infty} \\frac{1}{x\\sqrt{x^{2}-49}} \\;dx$ $=$ [ANS]", "answer_v1": [ "pi/14" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the integrals that converge, enter 'DNC' if integral Does Not Converge. $ \\int_{1}^{+\\infty} \\frac{1}{x\\sqrt{x^{2}-1}} \\;dx$ $=$ [ANS]", "answer_v2": [ "pi/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the integrals that converge, enter 'DNC' if integral Does Not Converge. $ \\int_{3}^{+\\infty} \\frac{1}{x\\sqrt{x^{2}-9}} \\;dx$ $=$ [ANS]", "answer_v3": [ "pi/6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0598", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "3", "keywords": [ "derivative" ], "problem_v1": "In each part, determine all values of $p$ for which the integral is improper. Enter in interval notation or \"none\" if there are no relevant values of $p$.\n(a) $ \\int_{1}^{6} \\frac{dx}{x^{p}}$ $p$ values that make integral improper [ANS]\n(b) $ \\int_{1}^{4} \\frac{dx}{x-p}$ $p$ values that make integral improper [ANS]\n(c) $ \\int_{-4}^{1} e^{-px}dx$ $p$ values that make integral improper [ANS]", "answer_v1": [ "none", "[1,4]", "none" ], "answer_type_v1": [ "OE", "INT", "OE" ], "options_v1": [ [], [], [] ], "problem_v2": "In each part, determine all values of $p$ for which the integral is improper. Enter in interval notation or \"none\" if there are no relevant values of $p$.\n(a) $ \\int_{0}^{8} \\frac{dx}{x^{p}}$ $p$ values that make integral improper [ANS]\n(b) $ \\int_{0}^{2} \\frac{dx}{x-p}$ $p$ values that make integral improper [ANS]\n(c) $ \\int_{8}^{9} e^{-px}dx$ $p$ values that make integral improper [ANS]", "answer_v2": [ "(0,infinity)", "[0,2]", "none" ], "answer_type_v2": [ "INT", "INT", "OE" ], "options_v2": [ [], [], [] ], "problem_v3": "In each part, determine all values of $p$ for which the integral is improper. Enter in interval notation or \"none\" if there are no relevant values of $p$.\n(a) $ \\int_{0}^{5} \\frac{dx}{x^{p}}$ $p$ values that make integral improper [ANS]\n(b) $ \\int_{0}^{3} \\frac{dx}{x-p}$ $p$ values that make integral improper [ANS]\n(c) $ \\int_{-6}^{7} e^{-px}dx$ $p$ values that make integral improper [ANS]", "answer_v3": [ "(0,infinity)", "[0,3]", "none" ], "answer_type_v3": [ "INT", "INT", "OE" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0599", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "3", "keywords": [ "Integral", "Improper Integral", "integrals", "improper" ], "problem_v1": "Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer-1.\n\\int_{6}^{\\infty} x e^{-4x} dx [ANS]", "answer_v1": [ "5.89864772543609E-11" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer-1.\n\\int_{3}^{\\infty} x e^{-3x} dx [ANS]", "answer_v2": [ "0.000137122004540755" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer-1.\n\\int_{3}^{\\infty} x e^{-4x} dx [ANS]", "answer_v3": [ "4.99217253707917E-06" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0600", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "3", "keywords": [ "antiderivatives" ], "problem_v1": "Find what value of $c$ does $\\int_{8}^{\\infty} \\frac {c}{x^{3}} dx=1$? Answer: [ANS]", "answer_v1": [ "128" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find what value of $c$ does $\\int_{2}^{\\infty} \\frac {c}{x^{4}} dx=1$? Answer: [ANS]", "answer_v2": [ "24" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find what value of $c$ does $\\int_{4}^{\\infty} \\frac {c}{x^{3}} dx=1$? Answer: [ANS]", "answer_v3": [ "32" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0601", "subject": "Calculus_-_single_variable", "topic": "Integrals", "subtopic": "Improper integrals", "level": "3", "keywords": [ "calculus", "integrals", "substitution" ], "problem_v1": "Find the indicated integral (if it exists) $ \\int_{-\\infty}^{\\infty} \\frac{e^{7x}}{e^{14x}+1}\\,dx$=[ANS]", "answer_v1": [ "0.224399475285714" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the indicated integral (if it exists) $ \\int_{-\\infty}^{\\infty} \\frac{e^{3x}}{e^{6x}+1}\\,dx$=[ANS]", "answer_v2": [ "0.523598775666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the indicated integral (if it exists) $ \\int_{-\\infty}^{\\infty} \\frac{e^{4x}}{e^{8x}+1}\\,dx$=[ANS]", "answer_v3": [ "0.39269908175" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0602", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric integrals", "level": "3", "keywords": [ "calculus", "integration", "integral", "trigonometry", "trigonometric", "trig", "Indefinite", "Trig Integral" ], "problem_v1": "Calculate $\\int 8 \\cos^{4} x \\, dx$.-3*R1+R3-3*R1+R3\nNote: Use C for an arbitrary constant.", "answer_v1": [ "2*[cos(x)]^3*sin(x)+3*cos(x)*sin(x)+3*x+C" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Calculate $\\int 2 \\cos^{4} x \\, dx$.-3*R1+R3-3*R1+R3\nNote: Use C for an arbitrary constant.", "answer_v2": [ "0.5*[cos(x)]^3*sin(x)+0.75*cos(x)*sin(x)+0.75*x+C" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Calculate $\\int 4 \\cos^{4} x \\, dx$.-3*R1+R3-3*R1+R3\nNote: Use C for an arbitrary constant.", "answer_v3": [ "1*[cos(x)]^3*sin(x)+1.5*cos(x)*sin(x)+1.5*x+C" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0603", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric integrals", "level": "3", "keywords": [ "calculus", "integration", "integral", "trigonometry", "trigonometric", "trig" ], "problem_v1": "Calculate $\\int{\\sin^{3}{x} \\cos^{9}{x}} \\, dx$. [ANS]", "answer_v1": [ "(-0.1)*[cos(x)]^10+0.0833333*[cos(x)]^12+C" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Calculate $\\int{\\sin^{3}{x} \\cos^{3}{x}} \\, dx$. [ANS]", "answer_v2": [ "(-0.25)*[cos(x)]^4+0.166667*[cos(x)]^6+C" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Calculate $\\int{\\sin^{3}{x} \\cos^{5}{x}} \\, dx$. [ANS]", "answer_v3": [ "(-0.166667)*[cos(x)]^6+0.125*[cos(x)]^8+C" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0604", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric integrals", "level": "3", "keywords": [ "Indefinite", "Trig Integral" ], "problem_v1": "Evaluate the indefinite integral. $\\int \\tan^{3}\\!\\left(x\\right)\\sec^{9}\\!\\left(x\\right) \\, dx$=[ANS] $+C$.", "answer_v1": [ "1/11*[sec(x)]^11-1/9*[sec(x)]^9" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Evaluate the indefinite integral. $\\int \\tan^{3}\\!\\left(x\\right)\\sec^{3}\\!\\left(x\\right) \\, dx$=[ANS] $+C$.", "answer_v2": [ "1/5*[sec(x)]^5-1/3*[sec(x)]^3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Evaluate the indefinite integral. $\\int \\tan^{3}\\!\\left(x\\right)\\sec^{5}\\!\\left(x\\right) \\, dx$=[ANS] $+C$.", "answer_v3": [ "1/7*[sec(x)]^7-1/5*[sec(x)]^5" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0605", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric integrals", "level": "3", "keywords": [ "integral' 'trig functions", "integrals", "trigonometry", "substitution", "Calculus", "Riemann Integral" ], "problem_v1": "$ \\int_0^{\\pi/6} \\sin^4(6x)\\, dx$=[ANS]", "answer_v1": [ "0.196349540875" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$ \\int_0^{\\pi/3} \\sin^4(3x)\\, dx$=[ANS]", "answer_v2": [ "0.39269908175" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$ \\int_0^{\\pi/4} \\sin^4(4x)\\, dx$=[ANS]", "answer_v3": [ "0.2945243113125" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0606", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric integrals", "level": "3", "keywords": [ "Integration", "Trigonometry", "integral' 'trig functions", "integrals", "substitution", "Calculus", "Riemann Integral" ], "problem_v1": "Evaluate the indefinite integral.\n\\int 208 \\cos^4(16x) dx [ANS] $+C$", "answer_v1": [ "(208*3*x)/8 +13*(cos(16*x))^3*sin(16*x)/4+3*13*cos(16*x)*sin(16*x)/8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Evaluate the indefinite integral.\n\\int 57 \\cos^4(3x) dx [ANS] $+C$", "answer_v2": [ "(57*3*x)/8 +19*(cos(3*x))^3*sin(3*x)/4+3*19*cos(3*x)*sin(3*x)/8" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Evaluate the indefinite integral.\n\\int 91 \\cos^4(7x) dx [ANS] $+C$", "answer_v3": [ "(91*3*x)/8 +13*(cos(7*x))^3*sin(7*x)/4+3*13*cos(7*x)*sin(7*x)/8" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0607", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Hyperbolic functions", "level": "2", "keywords": [ "calculus", "integration", "integral", "hyperbolic", "inverse hyperbolic" ], "problem_v1": "Use hyperbolic functions to calculate the integral $\\int \\frac{1}{\\sqrt{49+x^{2}}} \\, dx$.-3*R1+R3-3*R1+R3\nNote: Use C for an arbitrary constant.", "answer_v1": [ "asinh(x/7)+C" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use hyperbolic functions to calculate the integral $\\int \\frac{1}{\\sqrt{4+x^{2}}} \\, dx$.-3*R1+R3-3*R1+R3\nNote: Use C for an arbitrary constant.", "answer_v2": [ "asinh(x/2)+C" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use hyperbolic functions to calculate the integral $\\int \\frac{1}{\\sqrt{25+x^{2}}} \\, dx$.-3*R1+R3-3*R1+R3\nNote: Use C for an arbitrary constant.", "answer_v3": [ "asinh(x/5)+C" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0608", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Hyperbolic functions", "level": "2", "keywords": [ "calculus", "integration", "integral", "hyperbolic", "inverse hyperbolic" ], "problem_v1": "Calculate the integral $\\int 7\\tanh\\!\\left(x\\right)\\mathop{\\rm sech}^{2}\\!\\left(x\\right) \\, dx$. [ANS]", "answer_v1": [ "7*[tanh(x)]^2/2+C" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Calculate the integral $\\int 3\\tanh\\!\\left(x\\right)\\mathop{\\rm sech}^{2}\\!\\left(x\\right) \\, dx$. [ANS]", "answer_v2": [ "3*[tanh(x)]^2/2+C" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Calculate the integral $\\int 4\\tanh\\!\\left(x\\right)\\mathop{\\rm sech}^{2}\\!\\left(x\\right) \\, dx$. [ANS]", "answer_v3": [ "4*[tanh(x)]^2/2+C" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0609", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Hyperbolic functions", "level": "2", "keywords": [ "calculus", "integration", "integral", "hyperbolic", "inverse hyperbolic" ], "problem_v1": "Calculate the integral $\\int 7\\sinh^{-1}\\!\\left(x\\right) \\, dx$. Hint: Use Integration by Parts with $v'=1$.-3*R1+R3-3*R1+R3\nNote: Use C for an arbitrary constant.", "answer_v1": [ "7*[x*asinh(x)-sqrt(x^2+1)]+C" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Calculate the integral $\\int 3\\sinh^{-1}\\!\\left(x\\right) \\, dx$. Hint: Use Integration by Parts with $v'=1$.-3*R1+R3-3*R1+R3\nNote: Use C for an arbitrary constant.", "answer_v2": [ "3*[x*asinh(x)-sqrt(x^2+1)]+C" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Calculate the integral $\\int 4\\sinh^{-1}\\!\\left(x\\right) \\, dx$. Hint: Use Integration by Parts with $v'=1$.-3*R1+R3-3*R1+R3\nNote: Use C for an arbitrary constant.", "answer_v3": [ "4*[x*asinh(x)-sqrt(x^2+1)]+C" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0610", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Hyperbolic functions", "level": "2", "keywords": [ "calculus", "integration", "integral", "hyperbolic", "inverse hyperbolic" ], "problem_v1": "Calculate the integral $\\int^{\\frac{1}{5}}_{-\\frac{1}{5}} \\frac{1}{1-x^{2}} \\, dx$. [ANS]", "answer_v1": [ "0.405465" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the integral $\\int^{\\frac{1}{2}}_{-\\frac{1}{2}} \\frac{1}{1-x^{2}} \\, dx$. [ANS]", "answer_v2": [ "1.09861" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the integral $\\int^{\\frac{1}{3}}_{-\\frac{1}{3}} \\frac{1}{1-x^{2}} \\, dx$. [ANS]", "answer_v3": [ "0.693147" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0611", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Hyperbolic functions", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find \\int\\frac{\\cosh(8\\sqrt{z})}{ 8\\sqrt{z}}\\, dz. Answer: [ANS]+C.", "answer_v1": [ "2/(8^2)*sinh(8*sqrt(z))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find \\int\\frac{\\cosh(2\\sqrt{z})}{ 2\\sqrt{z}}\\, dz. Answer: [ANS]+C.", "answer_v2": [ "2/(2^2)*sinh(2*sqrt(z))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find \\int\\frac{\\cosh(4\\sqrt{z})}{ 4\\sqrt{z}}\\, dz. Answer: [ANS]+C.", "answer_v3": [ "2/(4^2)*sinh(4*sqrt(z))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0612", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Partial fractions", "level": "3", "keywords": [ "integral' 'partial fraction", "integrals", "partial fractions" ], "problem_v1": "Evaluate the integral.\n\\int \\frac{10x^2-48x-38}{x^3-5x^2-8x+48} dx [ANS]", "answer_v1": [ "4*ln(abs(x+3)) + 6*ln(abs(x+-4)) - -10/(x+-4)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Evaluate the integral.\n\\int \\frac{5x^2-13x-9}{x^3-3x^2+0x+4} dx [ANS]", "answer_v2": [ "1*ln(abs(x+1)) + 4*ln(abs(x+-2)) - -5/(x+-2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Evaluate the integral.\n\\int \\frac{7x^2-33x-8}{x^3-5x^2+3x+9} dx [ANS]", "answer_v3": [ "2*ln(abs(x+1)) + 5*ln(abs(x+-3)) - -11/(x+-3)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0613", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Partial fractions", "level": "2", "keywords": [ "integral' 'partial fraction" ], "problem_v1": "Write out the form of the partial fraction decomposition of the function appearing in the integral:\n\\int \\frac{5x-85}{x^2+5x-50}\\, dx Determine the numerical values of the coefficients, A and B, where A $\\leq$ B.\n\\frac{A}{denominator}+\\frac{B}{denominator} A=[ANS] B=[ANS]", "answer_v1": [ "-4", "9" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Write out the form of the partial fraction decomposition of the function appearing in the integral:\n\\int \\frac{2x-116}{x^2+4x-12}\\, dx Determine the numerical values of the coefficients, A and B, where A $\\leq$ B.\n\\frac{A}{denominator}+\\frac{B}{denominator} A=[ANS] B=[ANS]", "answer_v2": [ "-14", "16" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Write out the form of the partial fraction decomposition of the function appearing in the integral:\n\\int \\frac{2x-114}{x^2+6x-16}\\, dx Determine the numerical values of the coefficients, A and B, where A $\\leq$ B.\n\\frac{A}{denominator}+\\frac{B}{denominator} A=[ANS] B=[ANS]", "answer_v3": [ "-11", "13" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0614", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Partial fractions", "level": "3", "keywords": [ "integral' 'partial fraction" ], "problem_v1": "Consider the integral \\int \\frac{(1+9x)^{10}}{\\left(x^3-x\\right)^2 \\left(x^2-8x+7\\right)} \\,dx Enter a T or an F in each answer space below to indicate whether or not a term of the given type occurs in the general form of the complete partial fractions decomposition of the integrand. $A_1, A_2, A_3\\dots$ and $B_1, B_2, B_3,\\dots$ denote constants. [ANS] 1. $A_{3} x^3$ [ANS] 2. $\\frac{A_{15} x+B_{15}}{\\left(x^2-1\\right)^2}$ [ANS] 3. $\\frac{B_6}{x-7}$ [ANS] 4. $\\frac{B_7}{(x-1)^2}$ [ANS] 5. $A_{2}$ [ANS] 6. $\\frac{B_{11}}{(x+1)^2}$ [ANS] 7. $\\frac{B_8}{(x-1)^3}$ [ANS] 8. $\\frac{A_{10} x+B_{10}}{x^2-8x+7}$ [ANS] 9. $A_{6} x$ [ANS] 10. $\\frac{B_3}{(x+7)^2}$ [ANS] 11. $A_{4} x^2$ [ANS] 12. $\\frac{B_5}{x+7}$ [ANS] 13. $\\frac{A_9x+B_9}{\\left(x^3-x\\right)^2}$ [ANS] 14. $\\frac{A_{7}}{x}$ [ANS] 15. $\\frac{A_{8}}{x^3}$ [ANS] 16. $\\frac{B_1}{\\left(x+1\\right)^3}$ [ANS] 17. $\\frac{B_2}{x-1}$ [ANS] 18. $\\frac{A_{1}}{x^2}$ [ANS] 19. $\\frac{A_{20} x+B_{20}}{x^3-x}$ [ANS] 20. $\\frac{B_4}{x+1}$", "answer_v1": [ "F", "F", "T", "T", "T", "T", "T", "F", "T", "F", "T", "F", "F", "T", "F", "F", "T", "T", "F", "T" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the integral \\int \\frac{(1+12x)^{10}}{\\left(x^3-x\\right)^2 \\left(x^2-4x+3\\right)} \\,dx Enter a T or an F in each answer space below to indicate whether or not a term of the given type occurs in the general form of the complete partial fractions decomposition of the integrand. $A_1, A_2, A_3\\dots$ and $B_1, B_2, B_3,\\dots$ denote constants. [ANS] 1. $\\frac{B_3}{(x+3)^2}$ [ANS] 2. $\\frac{B_8}{(x-1)^3}$ [ANS] 3. $\\frac{A_{20} x+B_{20}}{x^3-x}$ [ANS] 4. $\\frac{B_7}{(x-1)^2}$ [ANS] 5. $\\frac{B_4}{x+1}$ [ANS] 6. $\\frac{A_9x+B_9}{\\left(x^3-x\\right)^2}$ [ANS] 7. $A_{3} x^3$ [ANS] 8. $\\frac{B_1}{\\left(x+1\\right)^3}$ [ANS] 9. $\\frac{A_{15} x+B_{15}}{\\left(x^2-1\\right)^2}$ [ANS] 10. $\\frac{B_{11}}{(x+1)^2}$ [ANS] 11. $\\frac{A_{8}}{x^3}$ [ANS] 12. $\\frac{B_5}{x+3}$ [ANS] 13. $\\frac{B_6}{x-3}$ [ANS] 14. $\\frac{A_{10} x+B_{10}}{x^2-4x+3}$ [ANS] 15. $A_{6} x$ [ANS] 16. $\\frac{B_2}{x-1}$ [ANS] 17. $A_{4} x^2$ [ANS] 18. $\\frac{A_{7}}{x}$ [ANS] 19. $A_{2}$ [ANS] 20. $\\frac{A_{1}}{x^2}$", "answer_v2": [ "F", "T", "F", "T", "T", "F", "F", "F", "F", "T", "F", "F", "T", "F", "T", "T", "T", "T", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the integral \\int \\frac{(1+9x)^{10}}{\\left(x^3-x\\right)^2 \\left(x^2-5x+4\\right)} \\,dx Enter a T or an F in each answer space below to indicate whether or not a term of the given type occurs in the general form of the complete partial fractions decomposition of the integrand. $A_1, A_2, A_3\\dots$ and $B_1, B_2, B_3,\\dots$ denote constants. [ANS] 1. $\\frac{B_6}{x-4}$ [ANS] 2. $A_{2}$ [ANS] 3. $\\frac{B_4}{x+1}$ [ANS] 4. $\\frac{B_8}{(x-1)^3}$ [ANS] 5. $\\frac{A_{7}}{x}$ [ANS] 6. $\\frac{A_{1}}{x^2}$ [ANS] 7. $\\frac{A_{8}}{x^3}$ [ANS] 8. $\\frac{B_3}{(x+4)^2}$ [ANS] 9. $\\frac{B_7}{(x-1)^2}$ [ANS] 10. $\\frac{B_5}{x+4}$ [ANS] 11. $\\frac{B_1}{\\left(x+1\\right)^3}$ [ANS] 12. $A_{4} x^2$ [ANS] 13. $\\frac{A_{20} x+B_{20}}{x^3-x}$ [ANS] 14. $\\frac{A_{15} x+B_{15}}{\\left(x^2-1\\right)^2}$ [ANS] 15. $\\frac{B_{11}}{(x+1)^2}$ [ANS] 16. $\\frac{B_2}{x-1}$ [ANS] 17. $\\frac{A_{10} x+B_{10}}{x^2-5x+4}$ [ANS] 18. $A_{3} x^3$ [ANS] 19. $A_{6} x$ [ANS] 20. $\\frac{A_9x+B_9}{\\left(x^3-x\\right)^2}$", "answer_v3": [ "T", "T", "T", "T", "T", "T", "F", "F", "T", "F", "F", "T", "F", "F", "T", "T", "F", "F", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0615", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Partial fractions", "level": "2", "keywords": [ "integral' 'partial fraction", "Integration", "Partial Fractions" ], "problem_v1": "Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral $ \\int \\frac{6x^3+4x^2-16x-16}{x^2-4}\\, dx$ Then the integrand decomposes into the form a x+b+\\frac{c}{x-2}+\\frac{d}{x+2} where $a$=[ANS]\n$b$=[ANS]\n$c$=[ANS]\n$d$=[ANS]\nIntegrating term by term, we obtain that $ \\int \\frac{6x^3+4x^2-16x-16}{x^2-4}\\, dx=$ [ANS] $+C$", "answer_v1": [ "6", "4", "4", "4", "6*x^2/2+4*x+4*ln(x-2)+4*ln(x+2)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral $ \\int \\frac{3x^3+5x^2-71x-135}{x^2-25}\\, dx$ Then the integrand decomposes into the form a x+b+\\frac{c}{x-5}+\\frac{d}{x+5} where $a$=[ANS]\n$b$=[ANS]\n$c$=[ANS]\n$d$=[ANS]\nIntegrating term by term, we obtain that $ \\int \\frac{3x^3+5x^2-71x-135}{x^2-25}\\, dx=$ [ANS] $+C$", "answer_v2": [ "3", "5", "1", "3", "3*x^2/2+5*x+1*ln(x-5)+3*ln(x+5)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the indefinite integral $ \\int \\frac{4x^3+4x^2-11x-18}{x^2-4}\\, dx$ Then the integrand decomposes into the form a x+b+\\frac{c}{x-2}+\\frac{d}{x+2} where $a$=[ANS]\n$b$=[ANS]\n$c$=[ANS]\n$d$=[ANS]\nIntegrating term by term, we obtain that $ \\int \\frac{4x^3+4x^2-11x-18}{x^2-4}\\, dx=$ [ANS] $+C$", "answer_v3": [ "4", "4", "2", "3", "4*x^2/2+4*x+2*ln(x-2)+3*ln(x+2)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0616", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Partial fractions", "level": "5", "keywords": [ "calculus", "integral", "antiderivatives", "indefinite integrals", "partial fractions" ], "problem_v1": "A rumor is spread in a school. For $00$, the time $t$ at which a fraction $p$ of the school population has heard the rumor is given by t(p)=\\int_{a}^{p} \\frac{b}{x(1-x)}\\,dx.\n(a) Evaluate the integral to find an explicit formula for $t(p)$. Write your answer so it has only one $\\ln$ term. $\\int_{a}^{p} \\frac{b}{x(1-x)}\\,dx=$ [ANS]\n(b) At time $t=0$, seven percent of the school population ($p=0.07$) has heard the rumor. What is $a$? $a=$ [ANS]\n(c) At time $t=1$, fifty-five percent of the school population ($p=0.55$) has heard the rumor. What is $b$? $b=$ [ANS]\n(d) At what time has ninety-three percent of the population ($p=0.93$) heard the rumor? $t=$ [ANS]", "answer_v1": [ "b*ln(abs(p*(1-a)/(a*(1-p))))", "0.07", "1/ln((0.55/(1-0.55))((1-0.07)/0.07))", "(1/ln((0.55/(1-0.55))((1-0.07)/0.07)))*ln((0.93/(1-0.93))((1-0.07)/0.07))" ], "answer_type_v1": [ "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A rumor is spread in a school. For $00$, the time $t$ at which a fraction $p$ of the school population has heard the rumor is given by t(p)=\\int_{a}^{p} \\frac{b}{x(1-x)}\\,dx.\n(a) Evaluate the integral to find an explicit formula for $t(p)$. Write your answer so it has only one $\\ln$ term. $\\int_{a}^{p} \\frac{b}{x(1-x)}\\,dx=$ [ANS]\n(b) At time $t=0$, one percent of the school population ($p=0.01$) has heard the rumor. What is $a$? $a=$ [ANS]\n(c) At time $t=1$, fifty-nine percent of the school population ($p=0.59$) has heard the rumor. What is $b$? $b=$ [ANS]\n(d) At what time has ninety percent of the population ($p=0.9$) heard the rumor? $t=$ [ANS]", "answer_v2": [ "b*ln(abs(p*(1-a)/(a*(1-p))))", "0.01", "1/ln((0.59/(1-0.59))((1-0.01)/0.01))", "(1/ln((0.59/(1-0.59))((1-0.01)/0.01)))*ln((0.9/(1-0.9))((1-0.01)/0.01))" ], "answer_type_v2": [ "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A rumor is spread in a school. For $00$, the time $t$ at which a fraction $p$ of the school population has heard the rumor is given by t(p)=\\int_{a}^{p} \\frac{b}{x(1-x)}\\,dx.\n(a) Evaluate the integral to find an explicit formula for $t(p)$. Write your answer so it has only one $\\ln$ term. $\\int_{a}^{p} \\frac{b}{x(1-x)}\\,dx=$ [ANS]\n(b) At time $t=0$, three percent of the school population ($p=0.03$) has heard the rumor. What is $a$? $a=$ [ANS]\n(c) At time $t=1$, fifty-six percent of the school population ($p=0.56$) has heard the rumor. What is $b$? $b=$ [ANS]\n(d) At what time has ninety-one percent of the population ($p=0.91$) heard the rumor? $t=$ [ANS]", "answer_v3": [ "b*ln(abs(p*(1-a)/(a*(1-p))))", "0.03", "1/ln((0.56/(1-0.56))((1-0.03)/0.03))", "(1/ln((0.56/(1-0.56))((1-0.03)/0.03)))*ln((0.91/(1-0.91))((1-0.03)/0.03))" ], "answer_type_v3": [ "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0617", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Partial fractions", "level": "2", "keywords": [ "calculus", "integral", "antiderivatives", "indefinite integrals", "partial fractions" ], "problem_v1": "Calculate the integral below by partial fractions and by using the indicated substitution. Be sure that you can show how the results you obtain are the same. \\int \\frac{2x}{x^{2}-49}\\,dx First, rewrite this with partial fractions: $\\int \\frac{2x}{x^{2}-49}\\,dx=\\int$ [ANS] $dx+\\int$ [ANS] $dx=$ [ANS]+[ANS] $+C$.\n(Note that you should not include the $+C$ in your entered answer, as it has been provided at the end of the expression.) Next, use the substitution $w=x^2-49$ to find the integral: $\\int \\frac{2x}{x^{2}-49}\\,dx=\\int$ [ANS] $dw=$ [ANS] $+C=$ [ANS] $+C$.\n(For the second answer blank, give your antiderivative in terms of the variable $w$. Again, note that you should not include the $+C$ in your answer.)", "answer_v1": [ "1/(x-7)", "1/(x+7)", "ln(|x-7|)", "ln(|x+7|)", "1/w", "ln(|w|)", "ln(|x^2-49|)" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Calculate the integral below by partial fractions and by using the indicated substitution. Be sure that you can show how the results you obtain are the same. \\int \\frac{2x}{x^{2}-1}\\,dx First, rewrite this with partial fractions: $\\int \\frac{2x}{x^{2}-1}\\,dx=\\int$ [ANS] $dx+\\int$ [ANS] $dx=$ [ANS]+[ANS] $+C$.\n(Note that you should not include the $+C$ in your entered answer, as it has been provided at the end of the expression.) Next, use the substitution $w=x^2-1$ to find the integral: $\\int \\frac{2x}{x^{2}-1}\\,dx=\\int$ [ANS] $dw=$ [ANS] $+C=$ [ANS] $+C$.\n(For the second answer blank, give your antiderivative in terms of the variable $w$. Again, note that you should not include the $+C$ in your answer.)", "answer_v2": [ "1/(x-1)", "1/(x+1)", "ln(|x-1|)", "ln(|x+1|)", "1/w", "ln(|w|)", "ln(|x^2-1|)" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Calculate the integral below by partial fractions and by using the indicated substitution. Be sure that you can show how the results you obtain are the same. \\int \\frac{2x}{x^{2}-9}\\,dx First, rewrite this with partial fractions: $\\int \\frac{2x}{x^{2}-9}\\,dx=\\int$ [ANS] $dx+\\int$ [ANS] $dx=$ [ANS]+[ANS] $+C$.\n(Note that you should not include the $+C$ in your entered answer, as it has been provided at the end of the expression.) Next, use the substitution $w=x^2-9$ to find the integral: $\\int \\frac{2x}{x^{2}-9}\\,dx=\\int$ [ANS] $dw=$ [ANS] $+C=$ [ANS] $+C$.\n(For the second answer blank, give your antiderivative in terms of the variable $w$. Again, note that you should not include the $+C$ in your answer.)", "answer_v3": [ "1/(x-3)", "1/(x+3)", "ln(|x-3|)", "ln(|x+3|)", "1/w", "ln(|w|)", "ln(|x^2-9|)" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0618", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Partial fractions", "level": "2", "keywords": [ "calculus", "integral", "antiderivatives", "indefinite integrals", "partial fractions" ], "problem_v1": "Calculate the integral: $\\int\\, \\frac{2x}{x^{2}-9x+20}\\, dx=$ [ANS]", "answer_v1": [ "-8*ln(|x-4|)+10*ln(|x-5|)+C" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Calculate the integral: $\\int\\, \\frac{3x}{x^{2}-3x+2}\\, dx=$ [ANS]", "answer_v2": [ "-3*ln(|x-1|)+6*ln(|x-2|)+C" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Calculate the integral: $\\int\\, \\frac{2x}{x^{2}-5x+6}\\, dx=$ [ANS]", "answer_v3": [ "-4*ln(|x-2|)+6*ln(|x-3|)+C" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0619", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Partial fractions", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find the coefficients of the partial fractions expansion.\n\\frac{7x^4-14x^3+84x^2-138x+187}{(x-4)(x^2+9)^2}=\\frac{{A}}{x-4}+\\frac{{B}x+{C}}{x^2+9}+\\frac{{D}x+{E}}{(x^2+9)^2}. ${A}=$ [ANS]. ${B}=$ [ANS]. ${C}=$ [ANS]. ${D}=$ [ANS]. ${E}=$ [ANS].", "answer_v1": [ "3", "4", "2", "2", "-4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Find the coefficients of the partial fractions expansion.\n\\frac{5x^4-2x^3+106x^2-38x+388}{(x-2)(x^2+16)^2}=\\frac{{A}}{x-2}+\\frac{{B}x+{C}}{x^2+16}+\\frac{{D}x+{E}}{(x^2+16)^2}. ${A}=$ [ANS]. ${B}=$ [ANS]. ${C}=$ [ANS]. ${D}=$ [ANS]. ${E}=$ [ANS].", "answer_v2": [ "2", "3", "4", "2", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Find the coefficients of the partial fractions expansion.\n\\frac{5x^4-4x^3+62x^2-43x+128}{(x-2)(x^2+9)^2}=\\frac{{A}}{x-2}+\\frac{{B}x+{C}}{x^2+9}+\\frac{{D}x+{E}}{(x^2+9)^2}. ${A}=$ [ANS]. ${B}=$ [ANS]. ${C}=$ [ANS]. ${D}=$ [ANS]. ${E}=$ [ANS].", "answer_v3": [ "2", "3", "2", "3", "-1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0620", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Partial fractions", "level": "2", "keywords": [ "calculus", "integral" ], "problem_v1": "$ \\int \\frac{1}{x^2+10x+89}\\,dx$=[ANS]+$C$", "answer_v1": [ "(1/8)*arctan((x+5)/8)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$ \\int \\frac{1}{x^2+4x+104}\\,dx$=[ANS]+$C$", "answer_v2": [ "(1/10)*arctan((x+2)/10)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$ \\int \\frac{1}{x^2+6x+73}\\,dx$=[ANS]+$C$", "answer_v3": [ "(1/8)*arctan((x+3)/8)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0621", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric substitution", "level": "4", "keywords": [ "calculus", "integration", "integral", "trigonometric substitution", "substitution", "trigonometry", "trigonometric", "trig" ], "problem_v1": "Find the volume of the solid obtained by revolving the graph of $y=7x\\sqrt{16-x^2}$ over [0,4] about the y-axis. [ANS]", "answer_v1": [ "224*pi^2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the solid obtained by revolving the graph of $y=9x\\sqrt{4-x^2}$ over [0,2] about the y-axis. [ANS]", "answer_v2": [ "18*pi^2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the solid obtained by revolving the graph of $y=8x\\sqrt{4-x^2}$ over [0,2] about the y-axis. [ANS]", "answer_v3": [ "16*pi^2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0622", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric substitution", "level": "3", "keywords": [ "integral' 'substitution' 'trig", "Calculus", "Riemann Integral" ], "problem_v1": "For each of the following integrals find an appropriate trigonometric substitution of the form $x=f(t)$ to simplify the integral.\n\\int (7x^2-6)^{3/2}\\, dx $x=$ [ANS]\n\\int \\frac{x^2}{\\sqrt{6x^2+7}}\\, dx $x=$ [ANS]\n\\int x\\sqrt{4x^2+32x+60}\\, dx $x=$ [ANS]\n\\int \\frac{x}{\\sqrt{-40-5x^2-30x}}\\, dx $x=$ [ANS]", "answer_v1": [ "sqrt(6/7)*sec(t)", "sqrt(7/6)*tan(t)", "sqrt(4/4)*sec(t)- 4", "sqrt(5/5)*sin(t)- 3" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "For each of the following integrals find an appropriate trigonometric substitution of the form $x=f(t)$ to simplify the integral.\n\\int (3x^2-8)^{3/2}\\, dx $x=$ [ANS]\n\\int \\frac{x^2}{\\sqrt{3x^2+4}}\\, dx $x=$ [ANS]\n\\int x\\sqrt{8x^2-48x+68}\\, dx $x=$ [ANS]\n\\int \\frac{x}{\\sqrt{-52-6x^2-36x}}\\, dx $x=$ [ANS]", "answer_v2": [ "sqrt(8/3)*sec(t)", "sqrt(4/3)*tan(t)", "sqrt(4/8)*sec(t)- -3", "sqrt(2/6)*sin(t)- 3" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "For each of the following integrals find an appropriate trigonometric substitution of the form $x=f(t)$ to simplify the integral.\n\\int (4x^2-6)^{3/2}\\, dx $x=$ [ANS]\n\\int \\frac{x^2}{\\sqrt{4x^2+5}}\\, dx $x=$ [ANS]\n\\int x\\sqrt{4x^2+40x+96}\\, dx $x=$ [ANS]\n\\int \\frac{x}{\\sqrt{-69-8x^2+48x}}\\, dx $x=$ [ANS]", "answer_v3": [ "sqrt(6/4)*sec(t)", "sqrt(5/4)*tan(t)", "sqrt(4/4)*sec(t)- 5", "sqrt(3/8)*sin(t)- -3" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0623", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric substitution", "level": "2", "keywords": [ "Substitution' 'Trig Substitution' 'Trigonometric Substitution" ], "problem_v1": "Match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following list. Enter the appropriate letter (A,B,C,D, or E) in each blank.\nA. $\\tan(\\theta)$ where $x=8 \\sin \\theta$ B. $\\cos(\\theta)$ where $x=8 \\sin \\theta$ C. $(1/2)\\sin(2\\theta)$ where $x=8 \\sin \\theta$ D. $\\sin(\\theta)$ where $x=8 \\tan \\theta$ E. $\\cos(\\theta)$ where $x=8 \\tan \\theta$ [ANS] 1. $ \\frac{x}{64}\\sqrt{64-x^2}$ [ANS] 2. $ \\frac{x}{\\sqrt{64+x^2}}$ [ANS] 3. $ \\frac{8}{\\sqrt{64+x^2}}$ [ANS] 4. $ \\frac{x}{\\sqrt{64-x^2}}$ [ANS] 5. $ \\frac{\\sqrt{64-x^2}}{8}$", "answer_v1": [ "C", "D", "E", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following list. Enter the appropriate letter (A,B,C,D, or E) in each blank.\nA. $\\tan(\\theta)$ where $x=2 \\sin \\theta$ B. $\\cos(\\theta)$ where $x=2 \\sin \\theta$ C. $(1/2)\\sin(2\\theta)$ where $x=2 \\sin \\theta$ D. $\\sin(\\theta)$ where $x=2 \\tan \\theta$ E. $\\cos(\\theta)$ where $x=2 \\tan \\theta$ [ANS] 1. $ \\frac{2}{\\sqrt{4+x^2}}$ [ANS] 2. $ \\frac{x}{\\sqrt{4-x^2}}$ [ANS] 3. $ \\frac{x}{4}\\sqrt{4-x^2}$ [ANS] 4. $ \\frac{x}{\\sqrt{4+x^2}}$ [ANS] 5. $ \\frac{\\sqrt{4-x^2}}{2}$", "answer_v2": [ "E", "A", "C", "D", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Match each of the trigonometric expressions below with the equivalent non-trigonometric function from the following list. Enter the appropriate letter (A,B,C,D, or E) in each blank.\nA. $\\tan(\\theta)$ where $x=4 \\sin \\theta$ B. $\\cos(\\theta)$ where $x=4 \\sin \\theta$ C. $(1/2)\\sin(2\\theta)$ where $x=4 \\sin \\theta$ D. $\\sin(\\theta)$ where $x=4 \\tan \\theta$ E. $\\cos(\\theta)$ where $x=4 \\tan \\theta$ [ANS] 1. $ \\frac{x}{\\sqrt{16+x^2}}$ [ANS] 2. $ \\frac{\\sqrt{16-x^2}}{4}$ [ANS] 3. $ \\frac{x}{16}\\sqrt{16-x^2}$ [ANS] 4. $ \\frac{x}{\\sqrt{16-x^2}}$ [ANS] 5. $ \\frac{4}{\\sqrt{16+x^2}}$", "answer_v3": [ "D", "B", "C", "A", "E" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Calculus_-_single_variable_0624", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric substitution", "level": "3", "keywords": [ "derivative" ], "problem_v1": "Evaluate the integral. $ \\int e^{x}\\sqrt{64-e^{2x}} \\;dx$ $=$ [ANS] $+C$", "answer_v1": [ "32*asin(e^x/8)+1/2*e^x*sqrt(64-e^{2*x})" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Evaluate the integral. $ \\int e^{x}\\sqrt{4-e^{2x}} \\;dx$ $=$ [ANS] $+C$", "answer_v2": [ "2*asin(e^x/2)+1/2*e^x*sqrt(4-e^{2*x})" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Evaluate the integral. $ \\int e^{x}\\sqrt{16-e^{2x}} \\;dx$ $=$ [ANS] $+C$", "answer_v3": [ "8*asin(e^x/4)+1/2*e^x*sqrt(16-e^{2*x})" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0625", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric substitution", "level": "2", "keywords": [ "Integral", "Trig Substitution", "integrals", "inverse trig functions" ], "problem_v1": "$\\int \\frac{1}{\\sqrt{25-81x^2}}dx$=[ANS]+$C$ ", "answer_v1": [ "arcsin(9*x/5)/9" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "$\\int \\frac{1}{\\sqrt{9-81x^2}}dx$=[ANS]+$C$ ", "answer_v2": [ "arcsin(9*x/3)/9" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "$\\int \\frac{1}{\\sqrt{16-49x^2}}dx$=[ANS]+$C$ ", "answer_v3": [ "arcsin(7*x/4)/7" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0626", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric substitution", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Evaluate the definite integral.\n\\int_{0}^{13 \\sin(\\frac{\\pi}{12})} \\frac{x^3}{\\sqrt{169-x^2}} dx [ANS]\n[NOTE: Remember to enter all necessary*, (, and)!! Enter arctan(x) for $\\tan^{-1} x$, sin(x) for $\\sin x$...]", "answer_v1": [ "2.5218529210912" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the definite integral.\n\\int_{0}^{19 \\sin(\\frac{\\pi}{3})} \\frac{x^3}{\\sqrt{361-x^2}} dx [ANS]\n[NOTE: Remember to enter all necessary*, (, and)!! Enter arctan(x) for $\\tan^{-1} x$, sin(x) for $\\sin x$...]", "answer_v2": [ "1428.95833333333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the definite integral.\n\\int_{0}^{13 \\sin(\\frac{\\pi}{6})} \\frac{x^3}{\\sqrt{169-x^2}} dx [ANS]\n[NOTE: Remember to enter all necessary*, (, and)!! Enter arctan(x) for $\\tan^{-1} x$, sin(x) for $\\sin x$...]", "answer_v3": [ "37.6733075808578" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0627", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Trigonometric substitution", "level": "4", "keywords": [ "integrals", "trigonometric", "substitution" ], "problem_v1": "if (navigator.appVersion.indexOf(\"MSIE\") > 0) {document.write(\"
You seem to be using Internet Explorer.
It is recommended that another browser be used to view this page.
\");} Evaluate the indefinite integral.\n\\int \\frac{\\sqrt{x^2-64}}{x}dx [ANS]\nHi Undefined_User, If you don't get this in 5 tries I'll give you a hint with an applet to help you out. Hi Undefined_User, If you don't get this in 5 tries I'll give you a hint with an applet to help you out.", "answer_v1": [ "sqrt(x^2-64)-8*atan([sqrt(x^2-64)]/8)+C" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "if (navigator.appVersion.indexOf(\"MSIE\") > 0) {document.write(\"
You seem to be using Internet Explorer.
It is recommended that another browser be used to view this page.
\");} Evaluate the indefinite integral.\n\\int \\frac{\\sqrt{x^2-4}}{x}dx [ANS]\nHi Undefined_User, If you don't get this in 5 tries I'll give you a hint with an applet to help you out. Hi Undefined_User, If you don't get this in 5 tries I'll give you a hint with an applet to help you out.", "answer_v2": [ "sqrt(x^2-4)-2*atan([sqrt(x^2-4)]/2)+C" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "if (navigator.appVersion.indexOf(\"MSIE\") > 0) {document.write(\"
You seem to be using Internet Explorer.
It is recommended that another browser be used to view this page.
\");} Evaluate the indefinite integral.\n\\int \\frac{\\sqrt{x^2-16}}{x}dx [ANS]\nHi Undefined_User, If you don't get this in 5 tries I'll give you a hint with an applet to help you out. Hi Undefined_User, If you don't get this in 5 tries I'll give you a hint with an applet to help you out.", "answer_v3": [ "sqrt(x^2-16)-4*atan([sqrt(x^2-16)]/4)+C" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0628", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Tables of integrals", "level": "2", "keywords": [ "calculus", "integral", "antiderivatives", "indefinite integrals", "tables" ], "problem_v1": "Antidifferentiate using the table of integrals. You may need to transform the integrand first. $\\int \\sin 7 t \\cos 6 t\\,dt=$ [ANS]", "answer_v1": [ "1/(36-49)*[6*sin(7*t)*sin(6*t)+7*cos(7*t)*cos(6*t)]+C" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Antidifferentiate using the table of integrals. You may need to transform the integrand first. $\\int \\sin 2 t \\cos 8 t\\,dt=$ [ANS]", "answer_v2": [ "1/(64-4)*[8*sin(2*t)*sin(8*t)+2*cos(2*t)*cos(8*t)]+C" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Antidifferentiate using the table of integrals. You may need to transform the integrand first. $\\int \\sin 4 t \\cos 6 t\\,dt=$ [ANS]", "answer_v3": [ "1/(36-16)*[6*sin(4*t)*sin(6*t)+4*cos(4*t)*cos(6*t)]+C" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0629", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Tables of integrals", "level": "2", "keywords": [ "calculus", "integral", "antiderivatives", "indefinite integrals", "tables" ], "problem_v1": "Antidifferentiate using a table of integrals. You may need to transform the integrand first. $ \\int\\frac {dx}{\\sqrt{64-16x^2}}=$ [ANS]", "answer_v1": [ "0.25*asin(4*x/8)+C" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Antidifferentiate using a table of integrals. You may need to transform the integrand first. $ \\int\\frac {dx}{\\sqrt{4-25x^2}}=$ [ANS]", "answer_v2": [ "0.2*asin(5*x/2)+C" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Antidifferentiate using a table of integrals. You may need to transform the integrand first. $ \\int\\frac {dx}{\\sqrt{16-16x^2}}=$ [ANS]", "answer_v3": [ "0.25*asin(4*x/4)+C" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0630", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Computer algebra system", "level": "2", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "properties of integrals", "area" ], "problem_v1": "Using a calculator, find the area of the region under $y=8\\ln(5x)$ and above $y=8$ for $3\\le x\\le 7$. area=[ANS]", "answer_v1": [ "70.1062866169541" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Using a calculator, find the area of the region under $y=3\\ln(7x)$ and above $y=7$ for $2\\le x\\le 5$. area=[ANS]", "answer_v2": [ "7.49587694464965" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Using a calculator, find the area of the region under $y=5\\ln(5x)$ and above $y=4$ for $2\\le x\\le 5$. area=[ANS]", "answer_v3": [ "30.4460446917646" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0631", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Mixed techniques", "level": "4", "keywords": [ "integrals", "integration by parts", "Integration", "Parts", "Trig Substitution" ], "problem_v1": "Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the definite integral $ \\int_0^{1/7} x\\sin^{-1}(7x)\\, dx$ The first step in evaluating this integral is to apply integration by parts: \\int u\\, dv=uv-\\int v\\, du where $u$=[ANS]\nand $dv=h(x)\\,dx$ where $h(x)$=[ANS]\nNote: Use $\\arcsin(x)$ for $\\sin^{-1}(x)$. After integrating by parts, we obtain the integral $ \\int_0^{1/7} v\\,du=\\int_0^{1/7} f(x)\\,dx$ on the right hand side where $f(x)$=[ANS]\nThe most appropriate substitution to simplify this integral is $x=g(t)$ where $g(t)$=[ANS]\nNote: We are using $t$ as variable for angles instead of $\\theta$, since there is no standard way to type $\\theta$ on a computer keyboard.\nAfter making this substitution and simplifying (using trig identities), we obtain the integral $ \\int_a^b k(t)\\,dt$ where $k(t)$=[ANS]\n$a$=[ANS]\n$b$=[ANS]\nAfter evaluating this integral and plugging back into the integration by parts formula we obtain: $ \\int_0^{1/7} x\\sin^{-1}(7x)\\, dx$=[ANS]", "answer_v1": [ "asin(7*x)", "x", "7*x^2/(2*sqrt(1-7^2*x^2))", "[sin(t)]/7", "[sin(t)]^2/98", "0", "1.5708", "0.0080142669744898" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the definite integral $ \\int_0^{1/3} x\\sin^{-1}(3x)\\, dx$ The first step in evaluating this integral is to apply integration by parts: \\int u\\, dv=uv-\\int v\\, du where $u$=[ANS]\nand $dv=h(x)\\,dx$ where $h(x)$=[ANS]\nNote: Use $\\arcsin(x)$ for $\\sin^{-1}(x)$. After integrating by parts, we obtain the integral $ \\int_0^{1/3} v\\,du=\\int_0^{1/3} f(x)\\,dx$ on the right hand side where $f(x)$=[ANS]\nThe most appropriate substitution to simplify this integral is $x=g(t)$ where $g(t)$=[ANS]\nNote: We are using $t$ as variable for angles instead of $\\theta$, since there is no standard way to type $\\theta$ on a computer keyboard.\nAfter making this substitution and simplifying (using trig identities), we obtain the integral $ \\int_a^b k(t)\\,dt$ where $k(t)$=[ANS]\n$a$=[ANS]\n$b$=[ANS]\nAfter evaluating this integral and plugging back into the integration by parts formula we obtain: $ \\int_0^{1/3} x\\sin^{-1}(3x)\\, dx$=[ANS]", "answer_v2": [ "asin(3*x)", "x", "3*x^2/(2*sqrt(1-3^2*x^2))", "[sin(t)]/3", "[sin(t)]^2/18", "0", "1.5708", "0.0436332313055556" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Consider the definite integral $ \\int_0^{1/4} x\\sin^{-1}(4x)\\, dx$ The first step in evaluating this integral is to apply integration by parts: \\int u\\, dv=uv-\\int v\\, du where $u$=[ANS]\nand $dv=h(x)\\,dx$ where $h(x)$=[ANS]\nNote: Use $\\arcsin(x)$ for $\\sin^{-1}(x)$. After integrating by parts, we obtain the integral $ \\int_0^{1/4} v\\,du=\\int_0^{1/4} f(x)\\,dx$ on the right hand side where $f(x)$=[ANS]\nThe most appropriate substitution to simplify this integral is $x=g(t)$ where $g(t)$=[ANS]\nNote: We are using $t$ as variable for angles instead of $\\theta$, since there is no standard way to type $\\theta$ on a computer keyboard.\nAfter making this substitution and simplifying (using trig identities), we obtain the integral $ \\int_a^b k(t)\\,dt$ where $k(t)$=[ANS]\n$a$=[ANS]\n$b$=[ANS]\nAfter evaluating this integral and plugging back into the integration by parts formula we obtain: $ \\int_0^{1/4} x\\sin^{-1}(4x)\\, dx$=[ANS]", "answer_v3": [ "asin(4*x)", "x", "4*x^2/(2*sqrt(1-4^2*x^2))", "[sin(t)]/4", "[sin(t)]^2/32", "0", "1.5708", "0.024543692609375" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0632", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Mixed techniques", "level": "3", "keywords": [ "calculus", "integral", "integration by parts", "indefinite integrals", "antiderivatives" ], "problem_v1": "For each of the following integrals, indicate whether integration by substitution or integration by parts is more appropriate, or if neither method is appropriate. Do not evaluate the integrals. 1. $\\int\\,x\\sin x\\,dx$ [ANS] A. substitution B. integration by parts C. neither\n2. $\\int\\,{x^{3}\\over 1+x^{4}}\\,dx$ [ANS] A. integration by parts B. neither C. substitution\n3. $\\int\\,x^{3} e^{x^{4}}\\,dx$ [ANS] A. integration by parts B. substitution C. neither\n4. $\\int\\,x^{3} \\cos(x^{4})\\,dx$ [ANS] A. integration by parts B. substitution C. neither\n5. $\\int\\,{1\\over\\sqrt{8x+1}}\\,dx$ [ANS] A. neither B. integration by parts C. substitution\n(Note that because this is multiple choice, you will not be able to see which parts of the problem you got correct.) (Note that because this is multiple choice, you will not be able to see which parts of the problem you got correct.)", "answer_v1": [ "B", "C", "B", "B", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "For each of the following integrals, indicate whether integration by substitution or integration by parts is more appropriate, or if neither method is appropriate. Do not evaluate the integrals. 1. $\\int\\,x\\sin x\\,dx$ [ANS] A. neither B. substitution C. integration by parts\n2. $\\int\\,{x^{4}\\over 1+x^{5}}\\,dx$ [ANS] A. integration by parts B. substitution C. neither\n3. $\\int\\,x^{4} e^{x^{5}}\\,dx$ [ANS] A. neither B. substitution C. integration by parts\n4. $\\int\\,x^{4} \\cos(x^{5})\\,dx$ [ANS] A. neither B. integration by parts C. substitution\n5. $\\int\\,{1\\over\\sqrt{2x+1}}\\,dx$ [ANS] A. integration by parts B. substitution C. neither\n(Note that because this is multiple choice, you will not be able to see which parts of the problem you got correct.) (Note that because this is multiple choice, you will not be able to see which parts of the problem you got correct.)", "answer_v2": [ "C", "B", "B", "C", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "For each of the following integrals, indicate whether integration by substitution or integration by parts is more appropriate, or if neither method is appropriate. Do not evaluate the integrals. 1. $\\int\\,x\\sin x\\,dx$ [ANS] A. substitution B. integration by parts C. neither\n2. $\\int\\,{x^{3}\\over 1+x^{4}}\\,dx$ [ANS] A. neither B. substitution C. integration by parts\n3. $\\int\\,x^{3} e^{x^{4}}\\,dx$ [ANS] A. neither B. integration by parts C. substitution\n4. $\\int\\,x^{3} \\cos(x^{4})\\,dx$ [ANS] A. neither B. integration by parts C. substitution\n5. $\\int\\,{1\\over\\sqrt{4x+1}}\\,dx$ [ANS] A. integration by parts B. substitution C. neither\n(Note that because this is multiple choice, you will not be able to see which parts of the problem you got correct.) (Note that because this is multiple choice, you will not be able to see which parts of the problem you got correct.)", "answer_v3": [ "B", "B", "C", "C", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Calculus_-_single_variable_0633", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Mixed techniques", "level": "4", "keywords": [ "calculus", "integral", "integration by parts", "indefinite integrals", "antiderivatives" ], "problem_v1": "Find approximately, by using a calculator or other method to estimate the area, $\\int_0^1\\,u^{5}\\,\\arcsin(u^{6})\\,du \\approx$ [ANS]\nThen find the answer exactly, by applying the Fundamental Theorem of Calculus: if $F(u)$ is the antiderivative of $u^{5}\\,\\arcsin(u^{6})$ having no added constant, then\n\\int_0^1\\,u^{5}\\,\\arcsin(u^{6})\\,du=F(1)-F(0), where $F(1)=$ [ANS]\nand $F(0)=$ [ANS].", "answer_v1": [ "0.09513", "(1/6)*(pi/2)", "(1/6)" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find approximately, by using a calculator or other method to estimate the area, $\\int_0^1\\,u\\,\\arcsin(u^{2})\\,du \\approx$ [ANS]\nThen find the answer exactly, by applying the Fundamental Theorem of Calculus: if $F(u)$ is the antiderivative of $u\\,\\arcsin(u^{2})$ having no added constant, then\n\\int_0^1\\,u\\,\\arcsin(u^{2})\\,du=F(1)-F(0), where $F(1)=$ [ANS]\nand $F(0)=$ [ANS].", "answer_v2": [ "0.2854", "(1/2)*(pi/2)", "(1/2)" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find approximately, by using a calculator or other method to estimate the area, $\\int_0^1\\,u^{2}\\,\\arcsin(u^{3})\\,du \\approx$ [ANS]\nThen find the answer exactly, by applying the Fundamental Theorem of Calculus: if $F(u)$ is the antiderivative of $u^{2}\\,\\arcsin(u^{3})$ having no added constant, then\n\\int_0^1\\,u^{2}\\,\\arcsin(u^{3})\\,du=F(1)-F(0), where $F(1)=$ [ANS]\nand $F(0)=$ [ANS].", "answer_v3": [ "0.19027", "(1/3)*(pi/2)", "(1/3)" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0634", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Mixed techniques", "level": "3", "keywords": [ "calculus", "integral", "indefinite integrals", "substitution", "antiderivatives" ], "problem_v1": "For each of the following, determine if substitution can be used to evaluate the integral. If so, fill in the substitution variable $w$ and the value of the integral; if not, enter na for both $w$ and the antiderivative.\n$\\begin{array}{ccc}\\hline integral & evaluates to & w=\\\\ \\hline \\int\\,{\\sin(x)\\over 7+\\cos(x)}\\,dx=& [ANS] & [ANS] \\\\ \\hline \\int\\,x^{4} \\cos(x^{3})\\,dx=& [ANS] & [ANS] \\\\ \\hline \\int\\,{x^{4}\\over 7+x^{4}}\\,dx=& [ANS] & [ANS] \\\\ \\hline \\int\\,x^{4}\\,e^{x^{3}}\\,dx=& [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "-1*ln(|7+cos(x)|)+C", "7 + cos(x)", "na", "na", "na", "na", "na", "na" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "For each of the following, determine if substitution can be used to evaluate the integral. If so, fill in the substitution variable $w$ and the value of the integral; if not, enter na for both $w$ and the antiderivative.\n$\\begin{array}{ccc}\\hline integral & evaluates to & w=\\\\ \\hline \\int\\,{x^{5}\\over 1+x^{5}}\\,dx=& [ANS] & [ANS] \\\\ \\hline \\int\\,{\\cos(x)\\over 1+\\cos(x)}\\,dx=& [ANS] & [ANS] \\\\ \\hline \\int\\,x^{5} \\cos(x^{4})\\,dx=& [ANS] & [ANS] \\\\ \\hline \\int\\,x^{4} \\cos(x^{5})\\,dx=& [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "na", "na", "na", "na", "na", "na", "0.2*sin(x^5)+C", "x^5" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "For each of the following, determine if substitution can be used to evaluate the integral. If so, fill in the substitution variable $w$ and the value of the integral; if not, enter na for both $w$ and the antiderivative.\n$\\begin{array}{ccc}\\hline integral & evaluates to & w=\\\\ \\hline \\int\\,x^{3}\\,e^{x^{4}}\\,dx=& [ANS] & [ANS] \\\\ \\hline \\int\\,x^{4} \\sin(x^{3})\\,dx=& [ANS] & [ANS] \\\\ \\hline \\int\\,{x^{4}\\over 3+x^{4}}\\,dx=& [ANS] & [ANS] \\\\ \\hline \\int\\,{\\sin(x)\\over 3+\\sin(x)}\\,dx=& [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "0.25*e^{x^4}+C", "x^4", "na", "na", "na", "na", "na", "na" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0635", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Mixed techniques", "level": "3", "keywords": [ "integrals", "integration by parts", "substitution method" ], "problem_v1": "First make a substitution and then use integration by parts to evaluate the integral.\n\\int x^{11} \\cos (x^6) dx Answer: [ANS] $+$ $C$", "answer_v1": [ "1/6 * x^6 * sin(x^6) + 1/6 * cos(x^6)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "First make a substitution and then use integration by parts to evaluate the integral.\n\\int x^{3} \\cos (x^2) dx Answer: [ANS] $+$ $C$", "answer_v2": [ "1/2 * x^2 * sin(x^2) + 1/2 * cos(x^2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "First make a substitution and then use integration by parts to evaluate the integral.\n\\int x^{5} \\cos (x^3) dx Answer: [ANS] $+$ $C$", "answer_v3": [ "1/3 * x^3 * sin(x^3) + 1/3 * cos(x^3)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0636", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "3", "keywords": [], "problem_v1": "\\int_{1}^{13} 6\\ln x \\, dx a) Approximate the definite integral with the Trapezoid Rule and $n=6$. [ANS]\nb) Approximate the definite integral with Simpson's Rule and $n=6$. [ANS]", "answer_v1": [ "126.379", "127.792" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "\\int_{1}^{4} 9\\ln x \\, dx a) Approximate the definite integral with the Trapezoid Rule and $n=6$. [ANS]\nb) Approximate the definite integral with Simpson's Rule and $n=6$. [ANS]", "answer_v2": [ "22.7674", "22.9018" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "\\int_{1}^{7} 6\\ln x \\, dx a) Approximate the definite integral with the Trapezoid Rule and $n=6$. [ANS]\nb) Approximate the definite integral with Simpson's Rule and $n=6$. [ANS]", "answer_v3": [ "45.3132", "45.6936" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0637", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "calculus", "integrals", "integration", "riemann sums", "partial sums" ], "problem_v1": "Calculate $M_{6}$ for $f(x)=5\\cdot\\ln(x^{3})$ over $[1,2]$. $M_{6}=$ [ANS]", "answer_v1": [ "5.80307" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate $M_{6}$ for $f(x)=7\\cdot\\ln(x^{2})$ over $[1,2]$. $M_{6}=$ [ANS]", "answer_v2": [ "5.4162" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate $M_{6}$ for $f(x)=6\\cdot\\ln(x^{2})$ over $[1,2]$. $M_{6}=$ [ANS]", "answer_v3": [ "4.64246" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0638", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "calculus", "integrals", "integration", "riemann sums", "partial sums" ], "problem_v1": "Calculate $R_{8}$ for $f(x)=9-x$ over $[3,5]$. $R_{8}=$ [ANS]", "answer_v1": [ "9.75" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate $R_{8}$ for $f(x)=6-x$ over $[3,5]$. $R_{8}=$ [ANS]", "answer_v2": [ "3.75" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate $R_{8}$ for $f(x)=7-x$ over $[3,5]$. $R_{8}=$ [ANS]", "answer_v3": [ "5.75" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0639", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "calculus", "integrals", "integration", "riemann sums", "partial sums" ], "problem_v1": "Calculate $L_{4}$ for $f(x)=60 \\cos\\left(\\frac{x}{3}\\right)$ over $[\\frac{3 \\pi}{4},\\frac{3 \\pi}{2}]$. $L_{4}=$ [ANS]", "answer_v1": [ "65.0469" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate $L_{4}$ for $f(x)=94 \\cos\\left(\\frac{x}{2}\\right)$ over $[\\frac{2 \\pi}{4},\\frac{2 \\pi}{2}]$. $L_{4}=$ [ANS]", "answer_v2": [ "67.9379" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate $L_{4}$ for $f(x)=62 \\cos\\left(\\frac{x}{2}\\right)$ over $[\\frac{2 \\pi}{4},\\frac{2 \\pi}{2}]$. $L_{4}=$ [ANS]", "answer_v3": [ "44.8101" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0640", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "calculus", "integration", "integral", "numerical", "numerical integration" ], "problem_v1": "Calculate the integral approximation $S_{10}$ for $\\int^{7}_{4} \\frac{1}{x^{5}+1} \\, dx$. $S_{10}$=[ANS]", "answer_v1": [ "0.000872153639" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the integral approximation $S_{6}$ for $\\int^{10}_{1} \\frac{1}{x^{3}+1} \\, dx$. $S_{6}$=[ANS]", "answer_v2": [ "0.404292432" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the integral approximation $S_{8}$ for $\\int^{7}_{2} \\frac{1}{x^{4}+1} \\, dx$. $S_{8}$=[ANS]", "answer_v3": [ "0.0400166769" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0641", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "calculus", "integration", "integral", "numerical", "numerical integration" ], "problem_v1": "Calculate the integral approximations $T_{8}$ and $M_{8}$ for $\\int^{8}_{4} x^{4}\\, dx$. Your answers must be accurate to 8 decimal places.\n$T_{8}$=[ANS]\n$M_{8}$=[ANS]", "answer_v1": [ "6386.125", "6330.14062" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Calculate the integral approximations $T_{6}$ and $M_{6}$ for $\\int^{10}_{1} x^{2}\\, dx$. Your answers must be accurate to 8 decimal places.\n$T_{6}$=[ANS]\n$M_{6}$=[ANS]", "answer_v2": [ "336.375", "331.3125" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Calculate the integral approximations $T_{6}$ and $M_{6}$ for $\\int^{7}_{2} x^{3}\\, dx$. Your answers must be accurate to 8 decimal places.\n$T_{6}$=[ANS]\n$M_{6}$=[ANS]", "answer_v3": [ "604.0625", "592.34375" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0642", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "3", "keywords": [ "calculus", "integration", "integral", "numerical", "numerical integration" ], "problem_v1": "Use the Error Bound to find a value of N for which $Error(S_N) \\le 1\\times 10^{-9}$. $\\int^{7}_{1}{5x^{1.5}} \\, dx$. $N \\ge$ [ANS]", "answer_v1": [ "592" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the Error Bound to find a value of N for which $Error(S_N) \\le 1\\times 10^{-9}$. $\\int^{7}_{1}{2x^{1.5}} \\, dx$. $N \\ge$ [ANS]", "answer_v2": [ "470" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the Error Bound to find a value of N for which $Error(S_N) \\le 1\\times 10^{-9}$. $\\int^{7}_{1}{3x^{1.5}} \\, dx$. $N \\ge$ [ANS]", "answer_v3": [ "520" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0643", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "3", "keywords": [ "calculus", "integration", "integral", "numerical", "numerical integration", "disk", "disk method" ], "problem_v1": "Calculate the approximation to the volume of the solid obtained by rotating y=$\\cos^{4}\\!\\left(x\\right)$ about the x-axis over the interval $\\left[0, \\frac{\\pi}{2} \\right]$. Use the midpoint method, with N=8. $V \\approx$ [ANS]", "answer_v1": [ "1.34936" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the approximation to the volume of the solid obtained by rotating y=$\\cos\\!\\left(x\\right)$ about the x-axis over the interval $\\left[0, \\frac{\\pi}{2} \\right]$. Use the midpoint method, with N=8. $V \\approx$ [ANS]", "answer_v2": [ "2.4674" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the approximation to the volume of the solid obtained by rotating y=$\\cos^{2}\\!\\left(x\\right)$ about the x-axis over the interval $\\left[0, \\frac{\\pi}{2} \\right]$. Use the midpoint method, with N=8. $V \\approx$ [ANS]", "answer_v3": [ "1.85055" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0644", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "3", "keywords": [ "calculus", "integration", "integral", "numerical", "numerical integration" ], "problem_v1": "Consider the integral approximation $M_{20}$ of $\\int^{\\frac{\\pi}{16}}_{0}{\\cos\\!\\left(x\\right)} \\, dx$. Does $M_{20}$ overestimate or underestimate the exact value? [ANS] A. overestimates B. underestimates\nFind the error bound for $M_{20}$ without calculating $M_N$ using the result that Error(M_{N}) \\le \\frac{M(b-a)^3}{24N^2}, where $M$ is the least upper bound for all absolute values of the second derivatives of the function $\\cos\\!\\left(x\\right)$ on the interval $[a,b]$. $Error(M_{20}) \\le$ [ANS]", "answer_v1": [ "A", "7.8853E-7" ], "answer_type_v1": [ "MCS", "NV" ], "options_v1": [ [ "A", "B" ], [] ], "problem_v2": "Consider the integral approximation $M_{20}$ of $\\int^{\\frac{\\pi}{4}}_{0}{\\cos\\!\\left(x\\right)} \\, dx$. Does $M_{20}$ overestimate or underestimate the exact value? [ANS] A. underestimates B. overestimates\nFind the error bound for $M_{20}$ without calculating $M_N$ using the result that Error(M_{N}) \\le \\frac{M(b-a)^3}{24N^2}, where $M$ is the least upper bound for all absolute values of the second derivatives of the function $\\cos\\!\\left(x\\right)$ on the interval $[a,b]$. $Error(M_{20}) \\le$ [ANS]", "answer_v2": [ "B", "5.04659E-5" ], "answer_type_v2": [ "MCS", "NV" ], "options_v2": [ [ "A", "B" ], [] ], "problem_v3": "Consider the integral approximation $M_{20}$ of $\\int^{\\frac{\\pi}{5}}_{0}{\\cos\\!\\left(x\\right)} \\, dx$. Does $M_{20}$ overestimate or underestimate the exact value? [ANS] A. underestimates B. overestimates\nFind the error bound for $M_{20}$ without calculating $M_N$ using the result that Error(M_{N}) \\le \\frac{M(b-a)^3}{24N^2}, where $M$ is the least upper bound for all absolute values of the second derivatives of the function $\\cos\\!\\left(x\\right)$ on the interval $[a,b]$. $Error(M_{20}) \\le$ [ANS]", "answer_v3": [ "B", "2.58386E-5" ], "answer_type_v3": [ "MCS", "NV" ], "options_v3": [ [ "A", "B" ], [] ] }, { "id": "Calculus_-_single_variable_0645", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "calculus", "integration", "integral", "numerical", "numerical integration" ], "problem_v1": "Calculate the integral approximations $T_{6}$ and $M_{6}$ for $\\int^{\\frac{\\pi}{9}}_{0} \\sec^{3} x \\, dx$. $T_{6}$=-3*R1+R3-3*R1+R3\n$M_{6}$=\u21d2 \u21d2", "answer_v1": [ "0.372224643", "0.371668439" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Calculate the integral approximations $T_{4}$ and $M_{4}$ for $\\int^{\\frac{\\pi}{3}}_{0} \\sec^{4} x \\, dx$. $T_{4}$=-3*R1+R3-3*R1+R3\n$M_{4}$=\u21d2 \u21d2", "answer_v2": [ "4.03865509", "3.19739442" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Calculate the integral approximations $T_{4}$ and $M_{4}$ for $\\int^{\\frac{\\pi}{5}}_{0} \\sec^{3} x \\, dx$. $T_{4}$=-3*R1+R3-3*R1+R3\n$M_{4}$=\u21d2 \u21d2", "answer_v3": [ "0.794556487", "0.781997552" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0646", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "calculus", "integration", "riemann sums" ], "problem_v1": "Estimate $\\int_{3}^{\\,5} 3 \\sqrt{x} \\,dx$ using left endpoints for $n=4$ approximating rectangles. $\\int_{3}^{\\,5} 3 \\sqrt{x} \\,dx$ is approximately [ANS]", "answer_v1": [ "11.5863" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Estimate $\\int_{1}^{\\,3.5} 2 \\sqrt{x} \\,dx$ using left endpoints for $n=5$ approximating rectangles. $\\int_{1}^{\\,3.5} 2 \\sqrt{x} \\,dx$ is approximately [ANS]", "answer_v2": [ "6.95215" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Estimate $\\int_{1}^{\\,3} 2 \\sqrt{x} \\,dx$ using left endpoints for $n=4$ approximating rectangles. $\\int_{1}^{\\,3} 2 \\sqrt{x} \\,dx$ is approximately [ANS]", "answer_v3": [ "5.2201" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0647", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "integral' 'summation' 'area' 'riemann", "Definite", "Integral", "Approximate", "Midpoint Rule", "integrals", "Riemann Sums", "integrals' 'riemann sums" ], "problem_v1": "Use the Midpoint Rule to approximate \\int_{-1.5}^{5.5} x^3 dx with $n=7$. [ANS]", "answer_v1": [ "224" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the Midpoint Rule to approximate \\int_{-2.5}^{3.5} x^3 dx with $n=6$. [ANS]", "answer_v2": [ "27" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the Midpoint Rule to approximate \\int_{-1.5}^{3.5} x^3 dx with $n=5$. [ANS]", "answer_v3": [ "35" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0648", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "3", "keywords": [ "integrals", "approximation", "calculus", "Riemann Integral" ], "problem_v1": "Approximate the following integral using the indicated methods. Round your answers to six decimal places.\n\\int_{0}^{1} e^{-5x^2} dx\n(a) Trapezoidal Rule with 4 subintervals [ANS]\n(b) Midpoint Rule with 4 subintervals [ANS]\n(c) Simpson's Rule with 4 subintervals [ANS]", "answer_v1": [ "0.395386016800421", "0.395866307143549", "0.395535727340605" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Approximate the following integral using the indicated methods. Round your answers to six decimal places.\n\\int_{0}^{1} e^{-2x^2} dx\n(a) Trapezoidal Rule with 4 subintervals [ANS]\n(b) Midpoint Rule with 4 subintervals [ANS]\n(c) Simpson's Rule with 4 subintervals [ANS]", "answer_v2": [ "0.595336917818471", "0.599542841266713", "0.598082840202805" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Approximate the following integral using the indicated methods. Round your answers to six decimal places.\n\\int_{0}^{1} e^{-3x^2} dx\n(a) Trapezoidal Rule with 4 subintervals [ANS]\n(b) Midpoint Rule with 4 subintervals [ANS]\n(c) Simpson's Rule with 4 subintervals [ANS]", "answer_v3": [ "0.502817651253163", "0.505095217095571", "0.504213520516726" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0649", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "3", "keywords": [], "problem_v1": "Estimate $\\int_{1.5}^{10.25} \\left|7-x\\right| dx$ using $5$ divisions and Left=[ANS] $\\Big($ [ANS] [ANS] [ANS] [ANS] [ANS] $\\Big)=$ [ANS]\nRight=[ANS] $\\Big($ [ANS] [ANS] [ANS] [ANS] [ANS] $\\Big)=$ [ANS]\nMidpoint=[ANS] $\\Big($ [ANS] [ANS] [ANS] [ANS] [ANS] $\\Big)=$ [ANS]\nTrapezoid=[ANS]\nSimpson's=[ANS]", "answer_v1": [ "1.75", "5.5", "3.75", "2", "0.25", "1.5", "22.75", "1.75", "3.75", "2", "0.25", "1.5", "3.25", "18.8125", "1.75", "4.625", "2.875", "1.125", "0.625", "2.375", "20.3438", "20.7812", "20.4896" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Estimate $\\int_{7}^{11} \\left|1-x\\right| dx$ using $4$ divisions and Left=[ANS] $\\Big($ [ANS] [ANS] [ANS] [ANS] $\\Big)=$ [ANS]\nRight=[ANS] $\\Big($ [ANS] [ANS] [ANS] [ANS] $\\Big)=$ [ANS]\nMidpoint=[ANS] $\\Big($ [ANS] [ANS] [ANS] [ANS] $\\Big)=$ [ANS]\nTrapezoid=[ANS]\nSimpson's=[ANS]", "answer_v2": [ "1", "6", "7", "8", "9", "30", "1", "7", "8", "9", "10", "34", "1", "6.5", "7.5", "8.5", "9.5", "32", "32", "32" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Estimate $\\int_{1.5}^{6.5} \\left|3-x\\right| dx$ using $4$ divisions and Left=[ANS] $\\Big($ [ANS] [ANS] [ANS] [ANS] $\\Big)=$ [ANS]\nRight=[ANS] $\\Big($ [ANS] [ANS] [ANS] [ANS] $\\Big)=$ [ANS]\nMidpoint=[ANS] $\\Big($ [ANS] [ANS] [ANS] [ANS] $\\Big)=$ [ANS]\nTrapezoid=[ANS]\nSimpson's=[ANS]", "answer_v3": [ "1.25", "1.5", "0.25", "1", "2.25", "6.25", "1.25", "0.25", "1", "2.25", "3.5", "8.75", "1.25", "0.875", "0.375", "1.625", "2.875", "7.1875", "7.5", "7.29167" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0650", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "exponentials", "logarithms", "integrals" ], "problem_v1": "This problem is a reprise of problem 5 with 1.1 replaced by 1.001 Compute an approximation to \\int_1^{8}\\frac{1}{x}\\,dx\\,, which gives the area under $y=\\frac{1}{x}$ for $1\\le x\\le 8$, using a modified Riemann sum with the (NOT equally spaced) partition 1,1.001,1.001^2,1.001^3,\\dots,1.001^N,8 and left hand endpoints EXCEPT neglecting the area of the last rectangle. Here $N$ denotes the largest possible power which fits in the interval $1\\le x\\le 8$. Please note that the problem is NOT asking for the value of $\\int_1^{8}\\frac{1}{x}\\,dx$. Rather it is asking for the EXACT values of the areas of approximating rectangles and for the EXACT value of the sum of the areas of the rectangles. Calculator approximations (no matter how accurate) will NOT be accepted. Do the calculations by hand using fractions (until you notice the pattern in the areas). The number of approximating rectangles is: $N$=[ANS]\nThe area of the first rectangle=[ANS]\nThe area of the second rectangle=[ANS]\nThe area of the third rectangle=[ANS]\nThe sum of the areas of the $N$ rectangles=[ANS]", "answer_v1": [ "2080", "0.001", "0.001", "0.001", "2.08" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "This problem is a reprise of problem 5 with 1.1 replaced by 1.001 Compute an approximation to \\int_1^{2}\\frac{1}{x}\\,dx\\,, which gives the area under $y=\\frac{1}{x}$ for $1\\le x\\le 2$, using a modified Riemann sum with the (NOT equally spaced) partition 1,1.001,1.001^2,1.001^3,\\dots,1.001^N,2 and left hand endpoints EXCEPT neglecting the area of the last rectangle. Here $N$ denotes the largest possible power which fits in the interval $1\\le x\\le 2$. Please note that the problem is NOT asking for the value of $\\int_1^{2}\\frac{1}{x}\\,dx$. Rather it is asking for the EXACT values of the areas of approximating rectangles and for the EXACT value of the sum of the areas of the rectangles. Calculator approximations (no matter how accurate) will NOT be accepted. Do the calculations by hand using fractions (until you notice the pattern in the areas). The number of approximating rectangles is: $N$=[ANS]\nThe area of the first rectangle=[ANS]\nThe area of the second rectangle=[ANS]\nThe area of the third rectangle=[ANS]\nThe sum of the areas of the $N$ rectangles=[ANS]", "answer_v2": [ "693", "0.001", "0.001", "0.001", "0.693" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "This problem is a reprise of problem 5 with 1.1 replaced by 1.001 Compute an approximation to \\int_1^{4}\\frac{1}{x}\\,dx\\,, which gives the area under $y=\\frac{1}{x}$ for $1\\le x\\le 4$, using a modified Riemann sum with the (NOT equally spaced) partition 1,1.001,1.001^2,1.001^3,\\dots,1.001^N,4 and left hand endpoints EXCEPT neglecting the area of the last rectangle. Here $N$ denotes the largest possible power which fits in the interval $1\\le x\\le 4$. Please note that the problem is NOT asking for the value of $\\int_1^{4}\\frac{1}{x}\\,dx$. Rather it is asking for the EXACT values of the areas of approximating rectangles and for the EXACT value of the sum of the areas of the rectangles. Calculator approximations (no matter how accurate) will NOT be accepted. Do the calculations by hand using fractions (until you notice the pattern in the areas). The number of approximating rectangles is: $N$=[ANS]\nThe area of the first rectangle=[ANS]\nThe area of the second rectangle=[ANS]\nThe area of the third rectangle=[ANS]\nThe sum of the areas of the $N$ rectangles=[ANS]", "answer_v3": [ "1386", "0.001", "0.001", "0.001", "1.386" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0651", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "calculus", "integral", "definite integrals", "distance" ], "problem_v1": "In this problem, use the general expressions for left and right sums, \\mbox{left-hand sum}=f(t_0)\\Delta t+f(t_1)\\Delta t+\\cdots+f(t_{n-1})\\Delta t and \\mbox{right-hand sum}=f(t_1)\\Delta t+f(t_2)\\Delta t+\\cdots+f(t_{n})\\Delta t, and the following table:\n$\\begin{array}{cccccc}\\hline t & 0 & 5 & 10 & 15 & 20 \\\\ \\hline f(t) & 32 & 29 & 26 & 24 & 22 \\\\ \\hline \\end{array}$\nA. If we use $n=4$ subdivisions, fill in the values: $\\Delta t=$ [ANS]\n$t_0=$ [ANS] ; $t_1=$ [ANS] ; $t_2=$ [ANS] ; $t_3=$ [ANS] ; $t_4=$ [ANS]\n$f(t_0)=$ [ANS] ; $f(t_1)=$ [ANS] ; $f(t_2)=$ [ANS] ; $f(t_3)=$ [ANS] ; $f(t_4)=$ [ANS]\nB. Find the left and right sums using $n=4$ left sum=[ANS]\nright sum=[ANS]\nC. If we use $n=2$ subdivisions, fill in the values: $\\Delta t=$ [ANS]\n$t_0=$ [ANS] ; $t_1=$ [ANS] ; $t_2=$ [ANS]\n$f(t_0)=$ [ANS] ; $f(t_1)=$ [ANS] ; $f(t_2)=$ [ANS]\nD. Find the left and right sums using $n=2$ left sum=[ANS]\nright sum=[ANS]", "answer_v1": [ "5", "0", "5", "10", "15", "20", "32", "29", "26", "24", "22", "555", "505", "2*5", "0", "10", "20", "32", "26", "22", "580", "480" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "In this problem, use the general expressions for left and right sums, \\mbox{left-hand sum}=f(t_0)\\Delta t+f(t_1)\\Delta t+\\cdots+f(t_{n-1})\\Delta t and \\mbox{right-hand sum}=f(t_1)\\Delta t+f(t_2)\\Delta t+\\cdots+f(t_{n})\\Delta t, and the following table:\n$\\begin{array}{cccccc}\\hline t & 0 & 2 & 4 & 6 & 8 \\\\ \\hline f(t) & 39 & 38 & 36 & 32 & 30 \\\\ \\hline \\end{array}$\nA. If we use $n=4$ subdivisions, fill in the values: $\\Delta t=$ [ANS]\n$t_0=$ [ANS] ; $t_1=$ [ANS] ; $t_2=$ [ANS] ; $t_3=$ [ANS] ; $t_4=$ [ANS]\n$f(t_0)=$ [ANS] ; $f(t_1)=$ [ANS] ; $f(t_2)=$ [ANS] ; $f(t_3)=$ [ANS] ; $f(t_4)=$ [ANS]\nB. Find the left and right sums using $n=4$ left sum=[ANS]\nright sum=[ANS]\nC. If we use $n=2$ subdivisions, fill in the values: $\\Delta t=$ [ANS]\n$t_0=$ [ANS] ; $t_1=$ [ANS] ; $t_2=$ [ANS]\n$f(t_0)=$ [ANS] ; $f(t_1)=$ [ANS] ; $f(t_2)=$ [ANS]\nD. Find the left and right sums using $n=2$ left sum=[ANS]\nright sum=[ANS]", "answer_v2": [ "2", "0", "2", "4", "6", "8", "39", "38", "36", "32", "30", "290", "272", "2*2", "0", "4", "8", "39", "36", "30", "300", "264" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "In this problem, use the general expressions for left and right sums, \\mbox{left-hand sum}=f(t_0)\\Delta t+f(t_1)\\Delta t+\\cdots+f(t_{n-1})\\Delta t and \\mbox{right-hand sum}=f(t_1)\\Delta t+f(t_2)\\Delta t+\\cdots+f(t_{n})\\Delta t, and the following table:\n$\\begin{array}{cccccc}\\hline t & 0 & 3 & 6 & 9 & 12 \\\\ \\hline f(t) & 32 & 30 & 27 & 26 & 24 \\\\ \\hline \\end{array}$\nA. If we use $n=4$ subdivisions, fill in the values: $\\Delta t=$ [ANS]\n$t_0=$ [ANS] ; $t_1=$ [ANS] ; $t_2=$ [ANS] ; $t_3=$ [ANS] ; $t_4=$ [ANS]\n$f(t_0)=$ [ANS] ; $f(t_1)=$ [ANS] ; $f(t_2)=$ [ANS] ; $f(t_3)=$ [ANS] ; $f(t_4)=$ [ANS]\nB. Find the left and right sums using $n=4$ left sum=[ANS]\nright sum=[ANS]\nC. If we use $n=2$ subdivisions, fill in the values: $\\Delta t=$ [ANS]\n$t_0=$ [ANS] ; $t_1=$ [ANS] ; $t_2=$ [ANS]\n$f(t_0)=$ [ANS] ; $f(t_1)=$ [ANS] ; $f(t_2)=$ [ANS]\nD. Find the left and right sums using $n=2$ left sum=[ANS]\nright sum=[ANS]", "answer_v3": [ "3", "0", "3", "6", "9", "12", "32", "30", "27", "26", "24", "345", "321", "2*3", "0", "6", "12", "32", "27", "24", "354", "306" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0652", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "5", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "area", "properties of integrals" ], "problem_v1": "Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, made at the start of each month, show the rate at which pollutants are escaping (in tons/month) in the gas:\n$\\begin{array}{cccccccc}\\hline Time (months) & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline Rate & 7 & 10 & 15 & 22 & 30 & 39 & 50 \\\\ \\hline \\end{array}$\nA. Make an overestimate and an underestimate of the total quantity of pollutants that escape during the first month. overestimate=[ANS] tons underestimate=[ANS] tons B. Make an overestimate and an underestimate of the total quantity of pollutants that escape for the whole six months for which we have data. overestimate=[ANS]\nunderestimate=[ANS]\nC. How often would measurements have to be made to find an overestimate and an underestimate (for the quantity of pollutants that escaped) during the first six months which differ by exactly 1 ton from each other? [ANS] times a month.", "answer_v1": [ "10", "7", "166", "123", "43" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, made at the start of each month, show the rate at which pollutants are escaping (in tons/month) in the gas:\n$\\begin{array}{cccccccc}\\hline Time (months) & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline Rate & 2 & 6 & 10 & 15 & 23 & 32 & 41 \\\\ \\hline \\end{array}$\nA. Make an overestimate and an underestimate of the total quantity of pollutants that escape during the first month. overestimate=[ANS] tons underestimate=[ANS] tons B. Make an overestimate and an underestimate of the total quantity of pollutants that escape for the whole six months for which we have data. overestimate=[ANS]\nunderestimate=[ANS]\nC. How often would measurements have to be made to find an overestimate and an underestimate (for the quantity of pollutants that escaped) during the first six months which differ by exactly 1 ton from each other? [ANS] times a month.", "answer_v2": [ "6", "2", "127", "88", "39" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, made at the start of each month, show the rate at which pollutants are escaping (in tons/month) in the gas:\n$\\begin{array}{cccccccc}\\hline Time (months) & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline Rate & 4 & 7 & 11 & 17 & 23 & 30 & 40 \\\\ \\hline \\end{array}$\nA. Make an overestimate and an underestimate of the total quantity of pollutants that escape during the first month. overestimate=[ANS] tons underestimate=[ANS] tons B. Make an overestimate and an underestimate of the total quantity of pollutants that escape for the whole six months for which we have data. overestimate=[ANS]\nunderestimate=[ANS]\nC. How often would measurements have to be made to find an overestimate and an underestimate (for the quantity of pollutants that escaped) during the first six months which differ by exactly 1 ton from each other? [ANS] times a month.", "answer_v3": [ "7", "4", "128", "92", "36" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0653", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "3", "keywords": [ "calculus", "integral", "approximation", "definite integrals", "simpson", "errors" ], "problem_v1": "Note: for this problem, because later answers depend on earlier ones, you must enter answers for all answer blanks for the problem to be correctly graded. If you would like to get feedback before you completed all computations, enter a \"1\" for each answer you did not yet compute and then submit the problem. (But note that this will, obviously, result in a problem submission.) Note: for this problem, because later answers depend on earlier ones, you must enter answers for all answer blanks for the problem to be correctly graded. If you would like to get feedback before you completed all computations, enter a \"1\" for each answer you did not yet compute and then submit the problem. (But note that this will, obviously, result in a problem submission.)\n(a) What is the exact value of $\\int_{0}^{6}\\,e^x\\,dx$? $\\int_{0}^{6}\\,e^x\\,dx=$ [ANS]\n(b) Find LEFT(2), RIGHT(2), TRAP(2), MID(2), and SIMP(2); compute the error for each.\n$\\begin{array}{cccccc}\\hline & LEFT(2) & RIGHT(2) & TRAP(2) & MID(2) & SIMP(2) \\\\ \\hline value & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline error & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(c) Repeat part (b) with $n=4$ (instead of $n=2$).\n$\\begin{array}{cccccc}\\hline & LEFT(4) & RIGHT(4) & TRAP(4) & MID(4) & SIMP(4) \\\\ \\hline value & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline error & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(d) For each rule in part (b), as $n$ goes from $n=2$ to $n=4$, does the error go down approximately as you would expect? Explain by calculating the ratios of the errors: Error LEFT(2)/Error LEFT(4)=[ANS]\nError RIGHT(2)/Error RIGHT(4)=[ANS]\nError TRAP(2)/Error TRAP(4)=[ANS]\nError MID(2)/Error MID(4)=[ANS]\nError SIMP(2)/Error SIMP(4)=[ANS]\n(Be sure that you can explain in words why these do (or don't) make sense.) (Be sure that you can explain in words why these do (or don't) make sense.)", "answer_v1": [ "e^6-1", "3*(1+e^3)", "3*(e^3+e^6)", "1.5*(1+2*e^3+e^6)", "3*[e^1.5+e^{4.5}]", "(2*283.496+666.9)/3", "402.429-63.2566", "402.429-1270.54", "402.429-666.9", "402.429-283.496", "402.429-411.297", "1.5*[1+e^1.5+e^3+e^{4.5}]", "1.5*[e^1.5+e^3+e^{4.5}+e^6]", "0.75*[1+2*e^1.5+2*e^3+2*e^{4.5}+e^6]", "1.5*[e^0.75+e^{2.25}+e^{3.75}+e^{5.25}]", "(2*367.038+475.198)/3", "402.429-173.377", "402.429-777.02", "402.429-475.198", "402.429-367.038", "402.429-403.091", "339.172/229.052", "-868.111/-374.591", "-264.471/-72.769", "118.933/35.391", "-8.868/-0.662" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Note: for this problem, because later answers depend on earlier ones, you must enter answers for all answer blanks for the problem to be correctly graded. If you would like to get feedback before you completed all computations, enter a \"1\" for each answer you did not yet compute and then submit the problem. (But note that this will, obviously, result in a problem submission.) Note: for this problem, because later answers depend on earlier ones, you must enter answers for all answer blanks for the problem to be correctly graded. If you would like to get feedback before you completed all computations, enter a \"1\" for each answer you did not yet compute and then submit the problem. (But note that this will, obviously, result in a problem submission.)\n(a) What is the exact value of $\\int_{0}^{3}\\,e^x\\,dx$? $\\int_{0}^{3}\\,e^x\\,dx=$ [ANS]\n(b) Find LEFT(2), RIGHT(2), TRAP(2), MID(2), and SIMP(2); compute the error for each.\n$\\begin{array}{cccccc}\\hline & LEFT(2) & RIGHT(2) & TRAP(2) & MID(2) & SIMP(2) \\\\ \\hline value & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline error & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(c) Repeat part (b) with $n=4$ (instead of $n=2$).\n$\\begin{array}{cccccc}\\hline & LEFT(4) & RIGHT(4) & TRAP(4) & MID(4) & SIMP(4) \\\\ \\hline value & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline error & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(d) For each rule in part (b), as $n$ goes from $n=2$ to $n=4$, does the error go down approximately as you would expect? Explain by calculating the ratios of the errors: Error LEFT(2)/Error LEFT(4)=[ANS]\nError RIGHT(2)/Error RIGHT(4)=[ANS]\nError TRAP(2)/Error TRAP(4)=[ANS]\nError MID(2)/Error MID(4)=[ANS]\nError SIMP(2)/Error SIMP(4)=[ANS]\n(Be sure that you can explain in words why these do (or don't) make sense.) (Be sure that you can explain in words why these do (or don't) make sense.)", "answer_v2": [ "e^3-1", "1.5*(1+e^1.5)", "1.5*(e^1.5+e^3)", "0.75*(1+2*e^1.5+e^3)", "1.5*[e^0.75+e^{2.25}]", "(2*17.4071+22.5367)/3", "19.0855-8.22253", "19.0855-36.8508", "19.0855-22.5367", "19.0855-17.4071", "19.0855-19.117", "0.75*[1+e^0.75+e^1.5+e^{2.25}]", "0.75*[e^0.75+e^1.5+e^{2.25}+e^3]", "0.375*[1+2*e^0.75+2*e^1.5+2*e^{2.25}+e^3]", "0.75*[e^0.375+e^{1.125}+e^{1.875}+e^{2.625}]", "(2*18.6455+19.9719)/3", "19.0855-12.8148", "19.0855-27.129", "19.0855-19.9719", "19.0855-18.6455", "19.0855-19.0876", "10.863/6.2707", "-17.7653/-8.0435", "-3.4512/-0.8864", "1.6784/0.44", "-0.0315/-0.0021" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Note: for this problem, because later answers depend on earlier ones, you must enter answers for all answer blanks for the problem to be correctly graded. If you would like to get feedback before you completed all computations, enter a \"1\" for each answer you did not yet compute and then submit the problem. (But note that this will, obviously, result in a problem submission.) Note: for this problem, because later answers depend on earlier ones, you must enter answers for all answer blanks for the problem to be correctly graded. If you would like to get feedback before you completed all computations, enter a \"1\" for each answer you did not yet compute and then submit the problem. (But note that this will, obviously, result in a problem submission.)\n(a) What is the exact value of $\\int_{0}^{4}\\,e^x\\,dx$? $\\int_{0}^{4}\\,e^x\\,dx=$ [ANS]\n(b) Find LEFT(2), RIGHT(2), TRAP(2), MID(2), and SIMP(2); compute the error for each.\n$\\begin{array}{cccccc}\\hline & LEFT(2) & RIGHT(2) & TRAP(2) & MID(2) & SIMP(2) \\\\ \\hline value & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline error & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(c) Repeat part (b) with $n=4$ (instead of $n=2$).\n$\\begin{array}{cccccc}\\hline & LEFT(4) & RIGHT(4) & TRAP(4) & MID(4) & SIMP(4) \\\\ \\hline value & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline error & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n(d) For each rule in part (b), as $n$ goes from $n=2$ to $n=4$, does the error go down approximately as you would expect? Explain by calculating the ratios of the errors: Error LEFT(2)/Error LEFT(4)=[ANS]\nError RIGHT(2)/Error RIGHT(4)=[ANS]\nError TRAP(2)/Error TRAP(4)=[ANS]\nError MID(2)/Error MID(4)=[ANS]\nError SIMP(2)/Error SIMP(4)=[ANS]\n(Be sure that you can explain in words why these do (or don't) make sense.) (Be sure that you can explain in words why these do (or don't) make sense.)", "answer_v3": [ "e^4-1", "2*(1+e^2)", "2*(e^2+e^4)", "1*(1+2*e^2+e^4)", "2*[e^1+e^3]", "(2*45.6076+70.3763)/3", "53.5982-16.7781", "53.5982-123.974", "53.5982-70.3763", "53.5982-45.6076", "53.5982-53.8638", "1*[1+e^1+e^2+e^3]", "1*[e^1+e^2+e^3+e^4]", "0.5*[1+2*e^1+2*e^2+2*e^3+e^4]", "1*[e^0.5+e^{1.5}+e^{2.5}+e^{3.5}]", "(2*51.4284+57.9919)/3", "53.5982-31.1929", "53.5982-84.791", "53.5982-57.9919", "53.5982-51.4284", "53.5982-53.6162", "36.8201/22.4053", "-70.3758/-31.1928", "-16.7781/-4.3937", "7.9906/2.1698", "-0.2656/-0.018" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0654", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Determine an $n$ so that the trapezoidal rule will approximate the integral \\int_{5}^{11} \\frac{7 \\, dx}{4+x} with an error $E_{n}$ satisfying $|E_{n}| \\le 0.0002$.\nThe theoretical error bound for the Trapezoid rule is given by E_{n}=-\\dfrac{(b-a)^{3}}{12 n^{2}} f''(c) where $c$ is some point between $a$ and $b$. It predicts that the desired accuracy will be achieved if the number of terms $n$ is at least [ANS].", "answer_v1": [ "42.5739709641549" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine an $n$ so that the trapezoidal rule will approximate the integral \\int_{7}^{10} \\frac{4 \\, dx}{9+x} with an error $E_{n}$ satisfying $|E_{n}| \\le 0.0002$.\nThe theoretical error bound for the Trapezoid rule is given by E_{n}=-\\dfrac{(b-a)^{3}}{12 n^{2}} f''(c) where $c$ is some point between $a$ and $b$. It predicts that the desired accuracy will be achieved if the number of terms $n$ is at least [ANS].", "answer_v2": [ "5.6875" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine an $n$ so that the trapezoidal rule will approximate the integral \\int_{6}^{10} \\frac{6 \\, dx}{3+x} with an error $E_{n}$ satisfying $|E_{n}| \\le 0.0003$.\nThe theoretical error bound for the Trapezoid rule is given by E_{n}=-\\dfrac{(b-a)^{3}}{12 n^{2}} f''(c) where $c$ is some point between $a$ and $b$. It predicts that the desired accuracy will be achieved if the number of terms $n$ is at least [ANS].", "answer_v3": [ "18.1066746426556" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0655", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "integrals", "approximation", "Simpson" ], "problem_v1": "Use Simpson's Rule and the Trapezoid Rule to estimate the value of the integral $ \\int_{-2}^{4} (x^{2}-x^{3}+x+2) \\,dx$. In both cases, use $n=2$ subdivisions. Simpson's Rule approximation $S_2=$ [ANS]\nTrapezoid Rule approximation $T_2=$ [ANS]\nHint: $f(-2)=12$, $f(1)=3$, and $f(4)=-42$ for the integrand $f$. Note: Simpson's rule with $n=2$ (or larger) gives the exact value of the integral of a cubic function.", "answer_v1": [ "-18", "-36" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Use Simpson's Rule and the Trapezoid Rule to estimate the value of the integral $ \\int_{-4}^{0} (3x-\\left(3x^{3}+2x^{2}\\right)-3) \\,dx$. In both cases, use $n=2$ subdivisions. Simpson's Rule approximation $S_2=$ [ANS]\nTrapezoid Rule approximation $T_2=$ [ANS]\nHint: $f(-4)=145$, $f(-2)=7$, and $f(0)=-3$ for the integrand $f$. Note: Simpson's rule with $n=2$ (or larger) gives the exact value of the integral of a cubic function.", "answer_v2": [ "113.333333333333", "156" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Use Simpson's Rule and the Trapezoid Rule to estimate the value of the integral $ \\int_{0}^{6} (x-\\left(x^{3}+2x^{2}\\right)-1) \\,dx$. In both cases, use $n=2$ subdivisions. Simpson's Rule approximation $S_2=$ [ANS]\nTrapezoid Rule approximation $T_2=$ [ANS]\nHint: $f(0)=-1$, $f(3)=-43$, and $f(6)=-283$ for the integrand $f$. Note: Simpson's rule with $n=2$ (or larger) gives the exact value of the integral of a cubic function.", "answer_v3": [ "-456", "-555" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0656", "subject": "Calculus_-_single_variable", "topic": "Techniques of integration", "subtopic": "Approximation", "level": "2", "keywords": [ "integrals", "approximation", "Simpson" ], "problem_v1": "Use Simpson's Rule and all the data in the following table to estimate the value of $\\int_{9}^{15}y\\,dx$.\n$\\begin{array}{cccccccc}\\hline x & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\\\ \\hline y & 5 & 2 & 2 & 4 &-4 &-3 & 1 \\\\ \\hline \\end{array}$\nAnswer: [ANS]", "answer_v1": [ "(1/3)*(5 + 4*2 + 2*2 + 4*4 - 2*4 - 4*3 + 1)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Simpson's Rule and all the data in the following table to estimate the value of $\\int_{-3}^{3}y\\,dx$.\n$\\begin{array}{cccccccc}\\hline x &-3 &-2 &-1 & 0 & 1 & 2 & 3 \\\\ \\hline y &-8 & 8 &-7 &-3 & 8 &-3 &-6 \\\\ \\hline \\end{array}$\nAnswer: [ANS]", "answer_v2": [ "(1/3)*(-8 + 4*8 + 2*-7 + 4*-3 + 2*8 + 4*-3 + -6)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Simpson's Rule and all the data in the following table to estimate the value of $\\int_{25}^{31}y\\,dx$.\n$\\begin{array}{cccccccc}\\hline x & 25 & 26 & 27 & 28 & 29 & 30 & 31 \\\\ \\hline y &-4 & 2 &-4 & 1 &-6 &-3 & 6 \\\\ \\hline \\end{array}$\nAnswer: [ANS]", "answer_v3": [ "(1/3)*(-4 + 4*2 + 2*-4 + 4*1 + 2*-6 + 4*-3 + 6)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0657", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "4", "keywords": [ "integrals", "average value of a function" ], "problem_v1": "(a) Find the average value of $f(x)=25-x^2$ on the interval $[0,4]$.\nAnswer: [ANS]\n(b) Find a value $c$ in the interval $[0,4]$ such that $f(c)$ is equal to the average value.\nAnswer: [ANS]", "answer_v1": [ "19.6666666666667", "2.3094010767585" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) Find the average value of $f(x)=25-x^2$ on the interval $[0,1]$.\nAnswer: [ANS]\n(b) Find a value $c$ in the interval $[0,1]$ such that $f(c)$ is equal to the average value.\nAnswer: [ANS]", "answer_v2": [ "24.6666666666667", "0.577350269189625" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) Find the average value of $f(x)=25-x^2$ on the interval $[0,2]$.\nAnswer: [ANS]\n(b) Find a value $c$ in the interval $[0,2]$ such that $f(c)$ is equal to the average value.\nAnswer: [ANS]", "answer_v3": [ "23.6666666666667", "1.15470053837925" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0658", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "3", "keywords": [ "integrals", "average value of a function" ], "problem_v1": "Find the average value of $f(x)=\\cos^4x \\sin x$ on the interval $[0,7]$\nAnswer: [ANS]", "answer_v1": [ "(1-((cos(7)))^5)/(5*7)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the average value of $f(x)=\\cos^4x \\sin x$ on the interval $[0,1]$\nAnswer: [ANS]", "answer_v2": [ "(1-((cos(1)))^5)/(5*1)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the average value of $f(x)=\\cos^4x \\sin x$ on the interval $[0,3]$\nAnswer: [ANS]", "answer_v3": [ "(1-((cos(3)))^5)/(5*3)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0659", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "3", "keywords": [], "problem_v1": "Find the average value of the function $f(x)=6 \\sin x$ on the given intervals. a) Average value on $[0, \\pi/2]$: [ANS]\nb) Average value on $[0, \\pi]$: [ANS]\nc) Average value on $[0, 2\\pi]$: [ANS]", "answer_v1": [ "3.81972", "3.81972", "0" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the average value of the function $f(x)=2 \\sin x$ on the given intervals. a) Average value on $[0, \\pi/2]$: [ANS]\nb) Average value on $[0, \\pi]$: [ANS]\nc) Average value on $[0, 2\\pi]$: [ANS]", "answer_v2": [ "1.27324", "1.27324", "0" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the average value of the function $f(x)=3 \\sin x$ on the given intervals. a) Average value on $[0, \\pi/2]$: [ANS]\nb) Average value on $[0, \\pi]$: [ANS]\nc) Average value on $[0, 2\\pi]$: [ANS]", "answer_v3": [ "1.90986", "1.90986", "0" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0660", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "4", "keywords": [ "calculus", "integrals", "integration", "volume" ], "problem_v1": "Let M be the average value of $f(x)=x^{8}$ on $[0,6]$. Find a value of $c$ in $[0,6]$ such that $f(c)=M$. $c=$ [ANS]", "answer_v1": [ "4.55901411390956" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let M be the average value of $f(x)=x^{2}$ on $[0,10]$. Find a value of $c$ in $[0,10]$ such that $f(c)=M$. $c=$ [ANS]", "answer_v2": [ "5.77350269189626" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let M be the average value of $f(x)=x^{4}$ on $[0,7]$. Find a value of $c$ in $[0,7]$ such that $f(c)=M$. $c=$ [ANS]", "answer_v3": [ "4.68118213483495" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0661", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "5", "keywords": [ "integration", "average value of a function", "calculus", "integrals", "average value", "function", "temperature", "\"Trigonometry" ], "problem_v1": "In a certain city the temperature (in degrees Fahrenheit) $t$ hours after 9am was approximated by the function T(t)=70+14 \\sin \\left(\\frac{\\pi t}{12} \\right) Determine the temperature at 9 am. [ANS]\nDetermine the temperature at 3 pm. [ANS]\nFind the average temperature during the period from 9 am to 9 pm. [ANS]", "answer_v1": [ "70", "84", "78.9126768131461" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "In a certain city the temperature (in degrees Fahrenheit) $t$ hours after 9am was approximated by the function T(t)=30+19 \\sin \\left(\\frac{\\pi t}{12} \\right) Determine the temperature at 9 am. [ANS]\nDetermine the temperature at 3 pm. [ANS]\nFind the average temperature during the period from 9 am to 9 pm. [ANS]", "answer_v2": [ "30", "49", "42.095775674984" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "In a certain city the temperature (in degrees Fahrenheit) $t$ hours after 9am was approximated by the function T(t)=40+14 \\sin \\left(\\frac{\\pi t}{12} \\right) Determine the temperature at 9 am. [ANS]\nDetermine the temperature at 3 pm. [ANS]\nFind the average temperature during the period from 9 am to 9 pm. [ANS]", "answer_v3": [ "40", "54", "48.9126768131461" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0662", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "3", "keywords": [ "calculus", "integral", "antiderivatives" ], "problem_v1": "The average value of the function $v(x)=4x$ on the interval $[1,c]$ is equal to 7. Find $c$ if $c > 1$. $c=$ [ANS]", "answer_v1": [ "-1+2*7/4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The average value of the function $v(x)=2x$ on the interval $[1,c]$ is equal to 7. Find $c$ if $c > 1$. $c=$ [ANS]", "answer_v2": [ "-1+2*7/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The average value of the function $v(x)=2x$ on the interval $[1,c]$ is equal to 6. Find $c$ if $c > 1$. $c=$ [ANS]", "answer_v3": [ "-1+2*6/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0663", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "5", "keywords": [ "calculus", "integral", "fundamental theorem of calculus", "area", "properties of integrals" ], "problem_v1": "The number of hours, $H$, of daylight in Madrid as a function of date is approximated by the formula H=12+2.4 \\sin\\left(0.0172(t-80)\\right), where $t$ is the number of days since the start of the year. (We can think of $t=0$ as the stroke of midnight on Dec. 31/Jan 1; thus, January falls between $t=0$ and $t=31$, February falls between $t=31$ and $t=59$, etc.). Find the average number of hours of daylight in Madrid (assuming in each case that it is not a leap year): A. in January: average hours=[ANS]\nB. in July: average hours=[ANS]\nC. over a year: average hours=[ANS]\nBe sure that you give a reasonably accurate estimate for parts A and B, and that you can explain why the relative magnitudes of your answers to parts A, B, and C are reasonable. Be sure that you give a reasonably accurate estimate for parts A and B, and that you can explain why the relative magnitudes of your answers to parts A, B, and C are reasonable.", "answer_v1": [ "9.87633", "14.1528", "12" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The number of hours, $H$, of daylight in Madrid as a function of date is approximated by the formula H=12+2.4 \\sin\\left(0.0172(t-80)\\right), where $t$ is the number of days since the start of the year. (We can think of $t=0$ as the stroke of midnight on Dec. 31/Jan 1; thus, January falls between $t=0$ and $t=31$, February falls between $t=31$ and $t=59$, etc.). Find the average number of hours of daylight in Madrid (assuming in each case that it is not a leap year): A. in October: average hours=[ANS]\nB. in August: average hours=[ANS]\nC. over a year: average hours=[ANS]\nBe sure that you give a reasonably accurate estimate for parts A and B, and that you can explain why the relative magnitudes of your answers to parts A, B, and C are reasonable. Be sure that you give a reasonably accurate estimate for parts A and B, and that you can explain why the relative magnitudes of your answers to parts A, B, and C are reasonable.", "answer_v2": [ "10.9799", "13.3481", "12" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The number of hours, $H$, of daylight in Madrid as a function of date is approximated by the formula H=12+2.4 \\sin\\left(0.0172(t-80)\\right), where $t$ is the number of days since the start of the year. (We can think of $t=0$ as the stroke of midnight on Dec. 31/Jan 1; thus, January falls between $t=0$ and $t=31$, February falls between $t=31$ and $t=59$, etc.). Find the average number of hours of daylight in Madrid (assuming in each case that it is not a leap year): A. in November: average hours=[ANS]\nB. in July: average hours=[ANS]\nC. over a year: average hours=[ANS]\nBe sure that you give a reasonably accurate estimate for parts A and B, and that you can explain why the relative magnitudes of your answers to parts A, B, and C are reasonable. Be sure that you give a reasonably accurate estimate for parts A and B, and that you can explain why the relative magnitudes of your answers to parts A, B, and C are reasonable.", "answer_v3": [ "10.0432", "14.1528", "12" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0664", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "2", "keywords": [ "derivative" ], "problem_v1": "(a) Suppose that the acceleration function of a particle moving along a coordinate line is $a(t)=t+4$. Find the average acceleration of the particle over the time interval $0 \\leq t \\leq 6$ by integrating. $a_{\\mathrm{ave}}$ $=$ [ANS]\n(b) Suppose that the velocity function of a particle moving along a coordinate line is $v(t)=2\\cos\\!\\left(t\\right)$. Find the average acceleration of the particle over the time interval $0 \\leq t \\leq \\frac{\\pi}{4}$ algebraically. $a_{\\mathrm{ave}}$ $=$ [ANS]", "answer_v1": [ "7", "8/pi*(1/[sqrt(2)]-1)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) Suppose that the acceleration function of a particle moving along a coordinate line is $a(t)=t-7$. Find the average acceleration of the particle over the time interval $0 \\leq t \\leq 10$ by integrating. $a_{\\mathrm{ave}}$ $=$ [ANS]\n(b) Suppose that the velocity function of a particle moving along a coordinate line is $v(t)=-6\\cos\\!\\left(t\\right)$. Find the average acceleration of the particle over the time interval $0 \\leq t \\leq \\frac{\\pi}{4}$ algebraically. $a_{\\mathrm{ave}}$ $=$ [ANS]", "answer_v2": [ "-2", "24/pi*(1-1/[sqrt(2)])" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) Suppose that the acceleration function of a particle moving along a coordinate line is $a(t)=t-3$. Find the average acceleration of the particle over the time interval $0 \\leq t \\leq 7$ by integrating. $a_{\\mathrm{ave}}$ $=$ [ANS]\n(b) Suppose that the velocity function of a particle moving along a coordinate line is $v(t)=-4\\cos\\!\\left(t\\right)$. Find the average acceleration of the particle over the time interval $0 \\leq t \\leq \\frac{\\pi}{4}$ algebraically. $a_{\\mathrm{ave}}$ $=$ [ANS]", "answer_v3": [ "1/2", "16/pi*(1-1/[sqrt(2)])" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0665", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "2", "keywords": [ "derivative" ], "problem_v1": "(a) The temperature of a $20$ m long metal bar is $18^\u25e6$ C at one end and $31^\u25e6$ C at the other end. Assuming that the temperature increases linearly from the cooler end to the hotter end, what is the average temperature of the bar? $T_{\\mathrm{ave}}$ $=$ [ANS] $^\u25e6$ C (b) Explain why there must be a point on the bar where the temperature is the same as the average, and find it. [Let $x$ be the distance along the bar from zero.] $x$ $=$ [ANS] m along the bar.", "answer_v1": [ "(31+18)/2", "20/2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) The temperature of a $8$ m long metal bar is $10^\u25e6$ C at one end and $35^\u25e6$ C at the other end. Assuming that the temperature increases linearly from the cooler end to the hotter end, what is the average temperature of the bar? $T_{\\mathrm{ave}}$ $=$ [ANS] $^\u25e6$ C (b) Explain why there must be a point on the bar where the temperature is the same as the average, and find it. [Let $x$ be the distance along the bar from zero.] $x$ $=$ [ANS] m along the bar.", "answer_v2": [ "(35+10)/2", "8/2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) The temperature of a $11$ m long metal bar is $13^\u25e6$ C at one end and $31^\u25e6$ C at the other end. Assuming that the temperature increases linearly from the cooler end to the hotter end, what is the average temperature of the bar? $T_{\\mathrm{ave}}$ $=$ [ANS] $^\u25e6$ C (b) Explain why there must be a point on the bar where the temperature is the same as the average, and find it. [Let $x$ be the distance along the bar from zero.] $x$ $=$ [ANS] m along the bar.", "answer_v3": [ "(31+13)/2", "11/2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0666", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "2", "keywords": [ "derivative" ], "problem_v1": "Find the average value of the function $f(x)=\\sqrt[3]{x}$ on the interval $[1,27]$. $f_{\\mathrm{ave}}$ $=$ [ANS]", "answer_v1": [ "2.30769" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the average value of the function $f(x)=\\sqrt[3]{x}$ on the interval $[-8,27]$. $f_{\\mathrm{ave}}$ $=$ [ANS]", "answer_v2": [ "1.39286" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the average value of the function $f(x)=\\sqrt[3]{x}$ on the interval $[-1,8]$. $f_{\\mathrm{ave}}$ $=$ [ANS]", "answer_v3": [ "1.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0667", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "2", "keywords": [ "derivative" ], "problem_v1": "Suppose that the value of a yacht in dollars after t years of use is $V (t)=325000e^{-0.18t}$. What is the average value of the yacht over its first 14 years of use? $V_{\\mathrm{ave}}$ $=$ \\$ [ANS]", "answer_v1": [ "128968*[1-e^{-2.52}]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that the value of a yacht in dollars after t years of use is $V (t)=125000e^{-0.12t}$. What is the average value of the yacht over its first 7 years of use? $V_{\\mathrm{ave}}$ $=$ \\$ [ANS]", "answer_v2": [ "148810*[1-e^{-0.84}]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that the value of a yacht in dollars after t years of use is $V (t)=200000e^{-0.18t}$. What is the average value of the yacht over its first 9 years of use? $V_{\\mathrm{ave}}$ $=$ \\$ [ANS]", "answer_v3": [ "123457*[1-e^{-1.62}]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0668", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "2", "keywords": [ "derivative" ], "problem_v1": "Find the average value of the function $f(x)=7e^{x}$ on the interval $[-2,\\ln{7}]$. $f_{\\mathrm{ave}}$ $=$ [ANS]", "answer_v1": [ "7/[ln(7)+2]*[7-e^{-2}]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the average value of the function $f(x)=3e^{x}$ on the interval $[-8,\\ln{10}]$. $f_{\\mathrm{ave}}$ $=$ [ANS]", "answer_v2": [ "3/[ln(10)+8]*[10-e^{-8}]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the average value of the function $f(x)=4e^{x}$ on the interval $[-6,\\ln{7}]$. $f_{\\mathrm{ave}}$ $=$ [ANS]", "answer_v3": [ "4/[ln(7)+6]*[7-e^{-6}]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0669", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "2", "keywords": [ "antiderivatives" ], "problem_v1": "The cost function for a product is $C(x)=0.7x^{2}+160x+160$. Find average cost over $[0,750]$. Answer: [ANS]", "answer_v1": [ "191410" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The cost function for a product is $C(x)=0.2x^{2}+200x+110$. Find average cost over $[0,400]$. Answer: [ANS]", "answer_v2": [ "50776.6666666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The cost function for a product is $C(x)=0.4x^{2}+160x+130$. Find average cost over $[0,600]$. Answer: [ANS]", "answer_v3": [ "96130" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0670", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "5", "keywords": [ "calculus", "integrals", "average value", "function", "volumes" ], "problem_v1": "A solid lies between two parallel planes $5$ feet apart and has a volume of $45$ cubic feet. What is the average area of cross-sections of the solid by planes that lie between the given ones? [ANS]", "answer_v1": [ "9" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A solid lies between two parallel planes $7$ feet apart and has a volume of $31$ cubic feet. What is the average area of cross-sections of the solid by planes that lie between the given ones? [ANS]", "answer_v2": [ "4.42857142857143" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A solid lies between two parallel planes $5$ feet apart and has a volume of $36$ cubic feet. What is the average area of cross-sections of the solid by planes that lie between the given ones? [ANS]", "answer_v3": [ "7.2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0671", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Average value of a function", "level": "3", "keywords": [ "calculus", "integrals", "average value", "function", "integrals", "theory", "Integral", "Mean Value" ], "problem_v1": "Find the mean value of the function $f(x)=|8-x|$ on the closed interval $[6, 10]$. mean value=[ANS]", "answer_v1": [ "1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the mean value of the function $f(x)=|6-x|$ on the closed interval $[3, 7]$. mean value=[ANS]", "answer_v2": [ "1.25" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the mean value of the function $f(x)=|6-x|$ on the closed interval $[4, 7]$. mean value=[ANS]", "answer_v3": [ "0.833333333333333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0672", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Areas between curves", "level": "4", "keywords": [ "integrals", "area between curvers" ], "problem_v1": "Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y$. Then find the area of the region.\ny=6x, y=5x^2 Answer: [ANS]", "answer_v1": [ "(6^3)/(6*(5^2))" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y$. Then find the area of the region.\ny=2x, y=7x^2 Answer: [ANS]", "answer_v2": [ "(2^3)/(6*(7^2))" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $x$ or $y$. Then find the area of the region.\ny=3x, y=5x^2 Answer: [ANS]", "answer_v3": [ "(3^3)/(6*(5^2))" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0673", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Areas between curves", "level": "3", "keywords": [ "integrals", "area between curvers" ], "problem_v1": "Find the area of the region bounded by $y=8 \\cos x$, $y=13 \\sec^2(x)$, $x=-\\pi/4$, and $x=\\pi/4$.\nAnswer: [ANS]", "answer_v1": [ "2*13-8 sqrt(2)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of the region bounded by $y=2 \\cos x$, $y=11 \\sec^2(x)$, $x=-\\pi/4$, and $x=\\pi/4$.\nAnswer: [ANS]", "answer_v2": [ "2*11-2 sqrt(2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of the region bounded by $y=4 \\cos x$, $y=10 \\sec^2(x)$, $x=-\\pi/4$, and $x=\\pi/4$.\nAnswer: [ANS]", "answer_v3": [ "2*10-4 sqrt(2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0674", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Areas between curves", "level": "3", "keywords": [ "calculus", "integrals", "integration", "area between curves" ], "problem_v1": "Find the area between $y=e^x$ and $y=e^{5x}$ over $[0,1]$. $A=$ [ANS]", "answer_v1": [ "27.7643499920563" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area between $y=e^x$ and $y=e^{2x}$ over $[0,1]$. $A=$ [ANS]", "answer_v2": [ "1.47624622100628" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area between $y=e^x$ and $y=e^{3x}$ over $[0,1]$. $A=$ [ANS]", "answer_v3": [ "4.64356381260351" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0675", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Areas between curves", "level": "4", "keywords": [ "calculus", "integration", "area", "area between curves", "integration", "area between curves", "area", "integrals" ], "problem_v1": "Find the area of the region that is enclosed between $y=11x^{2}-x^{3}+x$ and $y=x^{2}+25x$. The area is [ANS].", "answer_v1": [ "49.3333" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of the region that is enclosed between $y=6x^{2}-x^{3}+x$ and $y=x^{2}+5x$. The area is [ANS].", "answer_v2": [ "11.8333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of the region that is enclosed between $y=7x^{2}-x^{3}+x$ and $y=x^{2}+9x$. The area is [ANS].", "answer_v3": [ "8" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0676", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Areas between curves", "level": "4", "keywords": [ "Integral", "Area", "Between Curve", "integrals", "area between curves" ], "problem_v1": "Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Find the area bounded by the two curves: x=100000\\left(11 \\sqrt{y}-1\\right) x=100000\\left(\\frac{11 \\sqrt{y}-1}{8 \\sqrt{y}}\\right) The appropriate definite integral for computing this area has integrand [ANS] ; lower limit of integration=[ANS] ; and upper limit of integration=[ANS]\nThis definite integral has value=[ANS]\nThis is the area of the region enclosed by the two curves.", "answer_v1": [ "100000*(11*sqrt(y)-1)*(1/(8*sqrt(y))-1)", "0.00826446280991736", "0.015625", "14.5273760330579" ], "answer_type_v1": [ "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Find the area bounded by the two curves: x=100000\\left(8 \\sqrt{y}-1\\right) x=100000\\left(\\frac{8 \\sqrt{y}-1}{4 \\sqrt{y}}\\right) The appropriate definite integral for computing this area has integrand [ANS] ; lower limit of integration=[ANS] ; and upper limit of integration=[ANS]\nThis definite integral has value=[ANS]\nThis is the area of the region enclosed by the two curves.", "answer_v2": [ "100000*(8*sqrt(y)-1)*(1/(4*sqrt(y))-1)", "0.015625", "0.0625", "520.833333333333" ], "answer_type_v2": [ "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Note: You can get full credit for this problem by just answering the last question correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit. Find the area bounded by the two curves: x=100000\\left(8 \\sqrt{y}-1\\right) x=100000\\left(\\frac{8 \\sqrt{y}-1}{5 \\sqrt{y}}\\right) The appropriate definite integral for computing this area has integrand [ANS] ; lower limit of integration=[ANS] ; and upper limit of integration=[ANS]\nThis definite integral has value=[ANS]\nThis is the area of the region enclosed by the two curves.", "answer_v3": [ "100000*(8*sqrt(y)-1)*(1/(5*sqrt(y))-1)", "0.015625", "0.04", "112.5" ], "answer_type_v3": [ "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0677", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Areas between curves", "level": "4", "keywords": [ "integrals", "area between curves" ], "problem_v1": "Consider the area between the graphs $x+2 y=28$ and $x+7=y^2$. This area can be computed in two different ways using integrals\nFirst of all it can be computed as a sum of two integrals \\int_a^b f(x)\\,dx+\\int_b^c g(x)\\,dx where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $f(x)=$ [ANS]\n$g(x)=$ [ANS]\nAlternatively this area can be computed as a single integral \\int_{\\alpha}^{\\beta} h(y)\\,dy where $\\alpha=$ [ANS], $\\beta=$ [ANS] and $h(y)=$ [ANS]\nEither way we find that the area is [ANS].", "answer_v1": [ "-7", "18", "42", "2*sqrt(x+7)", " (28 - x)/2 + sqrt(x+7)", "-7", "5", "28 - 2*y - y^2 + 7", "288" ], "answer_type_v1": [ "NV", "NV", "NV", "EX", "EX", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the area between the graphs $x+6 y=12$ and $x+4=y^2$. This area can be computed in two different ways using integrals\nFirst of all it can be computed as a sum of two integrals \\int_a^b f(x)\\,dx+\\int_b^c g(x)\\,dx where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $f(x)=$ [ANS]\n$g(x)=$ [ANS]\nAlternatively this area can be computed as a single integral \\int_{\\alpha}^{\\beta} h(y)\\,dy where $\\alpha=$ [ANS], $\\beta=$ [ANS] and $h(y)=$ [ANS]\nEither way we find that the area is [ANS].", "answer_v2": [ "-4", "0", "60", "2*sqrt(x+4)", " (12 - x)/6 + sqrt(x+4)", "-8", "2", "12 - 6*y - y^2 + 4", "166.666666666667" ], "answer_type_v2": [ "NV", "NV", "NV", "EX", "EX", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the area between the graphs $x+4 y=16$ and $x+5=y^2$. This area can be computed in two different ways using integrals\nFirst of all it can be computed as a sum of two integrals \\int_a^b f(x)\\,dx+\\int_b^c g(x)\\,dx where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $f(x)=$ [ANS]\n$g(x)=$ [ANS]\nAlternatively this area can be computed as a single integral \\int_{\\alpha}^{\\beta} h(y)\\,dy where $\\alpha=$ [ANS], $\\beta=$ [ANS] and $h(y)=$ [ANS]\nEither way we find that the area is [ANS].", "answer_v3": [ "-5", "4", "44", "2*sqrt(x+5)", " (16 - x)/4 + sqrt(x+5)", "-7", "3", "16 - 4*y - y^2 + 5", "166.666666666667" ], "answer_type_v3": [ "NV", "NV", "NV", "EX", "EX", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0678", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Areas between curves", "level": "4", "keywords": [ "integrals", "area between curves", "Area", "Between Curves", "Integral" ], "problem_v1": "Find the area between the curves: $y=x^3-13x^2+40x$ and $y=-x^3+13x^2-40x$ [ANS]", "answer_v1": [ "287.666666666667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area between the curves: $y=x^3-12x^2+20x$ and $y=-x^3+12x^2-20x$ [ANS]", "answer_v2": [ "1048" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area between the curves: $y=x^3-12x^2+27x$ and $y=-x^3+12x^2-27x$ [ANS]", "answer_v3": [ "499.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0679", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Areas between curves", "level": "2", "keywords": [ "calculus", "definite integrals", "fundamental theorem of calculus" ], "problem_v1": "Find the area under $y=5\\sin\\!\\left(x\\right)$ and above $y=5\\cos\\!\\left(x\\right)$ for $\\frac{\\pi2 \\le x \\le \\frac}{3\\pi}2$. (Note that this area may not be defined over the entire interval.) (Note that this area may not be defined over the entire interval.) area=[ANS]", "answer_v1": [ "5*[1+sqrt(2)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area under $y=2\\sin\\!\\left(x\\right)$ and above $y=2\\cos\\!\\left(x\\right)$ for $0 \\le x \\le \\pi$. (Note that this area may not be defined over the entire interval.) (Note that this area may not be defined over the entire interval.) area=[ANS]", "answer_v2": [ "2*[1+sqrt(2)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area under $y=3\\sin\\!\\left(x\\right)$ and above $y=3\\cos\\!\\left(x\\right)$ for $0 \\le x \\le \\pi$. (Note that this area may not be defined over the entire interval.) (Note that this area may not be defined over the entire interval.) area=[ANS]", "answer_v3": [ "3*[1+sqrt(2)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0681", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Areas between curves", "level": "3", "keywords": [ "antiderivatives" ], "problem_v1": "Find area enclosed by $f(x)=\\sqrt {x+41}$ and $g(x)=\\frac{1}{21} x+\\frac {139}{21}$. Answer: [ANS]", "answer_v1": [ "57.1666666666665" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find area enclosed by $f(x)=\\sqrt {x+98}$ and $g(x)=\\frac{1}{23} x+\\frac {228}{23}$. Answer: [ANS]", "answer_v2": [ "4.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find area enclosed by $f(x)=\\sqrt {x+45}$ and $g(x)=\\frac{1}{18} x+\\frac {122}{18}$. Answer: [ANS]", "answer_v3": [ "10.6666666666666" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0682", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Areas between curves", "level": "4", "keywords": [ "calculus", "integration", "area", "area between curves" ], "problem_v1": "More on Areas. Farmer Jones, and his wife, Dr. Jones, both mathematicians, decide to build a fence in their field to keep the sheep safe. Being mathematicians, they decide that the fences are to be in the shape of the parabolas $y=9x^2$ and $y=x^2+8$. What is the area of the enclosed region? [ANS]", "answer_v1": [ "10.6666666666667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "More on Areas. Farmer Jones, and his wife, Dr. Jones, both mathematicians, decide to build a fence in their field to keep the sheep safe. Being mathematicians, they decide that the fences are to be in the shape of the parabolas $y=2x^2$ and $y=x^2+12$. What is the area of the enclosed region? [ANS]", "answer_v2": [ "55.4256258422041" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "More on Areas. Farmer Jones, and his wife, Dr. Jones, both mathematicians, decide to build a fence in their field to keep the sheep safe. Being mathematicians, they decide that the fences are to be in the shape of the parabolas $y=5x^2$ and $y=x^2+8$. What is the area of the enclosed region? [ANS]", "answer_v3": [ "15.084944665313" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0683", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Areas between curves", "level": "4", "keywords": [ "calculus", "parametric", "area", "integration" ], "problem_v1": "There is a line through the origin that divides the region bounded by the parabola $y=7x-6x^2$ and the x-axis into two regions with equal area. What is the slope of that line? [ANS]\nHint: Draw a picture, write the area between the parabola and the line in terms of the slope of the line, and solve for the slope.", "answer_v1": [ "1.4440963181113" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "There is a line through the origin that divides the region bounded by the parabola $y=2x-8x^2$ and the x-axis into two regions with equal area. What is the slope of that line? [ANS]\nHint: Draw a picture, write the area between the parabola and the line in terms of the slope of the line, and solve for the slope.", "answer_v2": [ "0.4125989480318" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "There is a line through the origin that divides the region bounded by the parabola $y=4x-6x^2$ and the x-axis into two regions with equal area. What is the slope of that line? [ANS]\nHint: Draw a picture, write the area between the parabola and the line in terms of the slope of the line, and solve for the slope.", "answer_v3": [ "0.825197896063601" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0684", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by slices", "level": "5", "keywords": [ "calculus", "integrals", "integration", "volume" ], "problem_v1": "Find the volume of the solid whose base is the circle $x^2+y^2=64$ and the cross sections perpendicular to the $x$-axis are triangles whose height and base are equal. Find the area of the vertical cross section $A$ at the level $x=5$. $A=$ [ANS]\n$V=$ [ANS]", "answer_v1": [ "78", "1365.33333333333" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the volume of the solid whose base is the circle $x^2+y^2=9$ and the cross sections perpendicular to the $x$-axis are triangles whose height and base are equal. Find the area of the vertical cross section $A$ at the level $x=2$. $A=$ [ANS]\n$V=$ [ANS]", "answer_v2": [ "10", "72" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the volume of the solid whose base is the circle $x^2+y^2=25$ and the cross sections perpendicular to the $x$-axis are triangles whose height and base are equal. Find the area of the vertical cross section $A$ at the level $x=3$. $A=$ [ANS]\n$V=$ [ANS]", "answer_v3": [ "32", "333.333333333333" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0685", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by slices", "level": "5", "keywords": [ "Integration", "Volume" ], "problem_v1": "Find the volume of a right circular cone with height $24$ and base radius $8$. $V=$ [ANS]", "answer_v1": [ "1608.49543863797" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of a right circular cone with height $8$ and base radius $2$. $V=$ [ANS]", "answer_v2": [ "33.5103216382911" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of a right circular cone with height $12$ and base radius $4$. $V=$ [ANS]", "answer_v3": [ "201.061929829747" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0686", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by slices", "level": "5", "keywords": [ "integral' 'volume" ], "problem_v1": "The base of a certain solid is the triangle with vertices at $(-12,6)$, $(6,6)$, and the origin. Cross-sections perpendicular to the $y$-axis are squares. Then the volume of the solid is [ANS].", "answer_v1": [ "648" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The base of a certain solid is the triangle with vertices at $(-4,2)$, $(2,2)$, and the origin. Cross-sections perpendicular to the $y$-axis are squares. Then the volume of the solid is [ANS].", "answer_v2": [ "24" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The base of a certain solid is the triangle with vertices at $(-6,3)$, $(3,3)$, and the origin. Cross-sections perpendicular to the $y$-axis are squares. Then the volume of the solid is [ANS].", "answer_v3": [ "81" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0687", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by slices", "level": "5", "keywords": [ "integral' 'volume", "Integration", "Volume", "calculus", "integrals", "volumes" ], "problem_v1": "You wake up one morning, and find yourself wearing a toga and scarab ring. Always a logical person, you conclude that you must have become an Egyptian pharoah. You decide to honor yourself with a pyramid of your own design. You decide it should have height $h=4010$ and a square base with side $s=1870$ To impress your Egyptian subjects, find the volume of the pyramid. [ANS]", "answer_v1": [ "4674189666.66667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You wake up one morning, and find yourself wearing a toga and scarab ring. Always a logical person, you conclude that you must have become an Egyptian pharoah. You decide to honor yourself with a pyramid of your own design. You decide it should have height $h=1330$ and a square base with side $s=2400$ To impress your Egyptian subjects, find the volume of the pyramid. [ANS]", "answer_v2": [ "2553600000" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You wake up one morning, and find yourself wearing a toga and scarab ring. Always a logical person, you conclude that you must have become an Egyptian pharoah. You decide to honor yourself with a pyramid of your own design. You decide it should have height $h=2250$ and a square base with side $s=1910$ To impress your Egyptian subjects, find the volume of the pyramid. [ANS]", "answer_v3": [ "2736075000" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0688", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by slices", "level": "5", "keywords": [ "integral' 'volume", "integrals", "volume" ], "problem_v1": "A soda glass has the shape of the surface generated by revolving the graph of $y=7x^2$ for $0\\le x\\le 1$ about the $y$-axis. Soda is extracted from the glass through a straw at the rate of $1/2$ cubic inch per second. How fast is the soda level in the glass dropping when the level is 4 inches? (Answer should be implicitly in units of inches per second. Do not put units in your answer. Also your answer should be positive, since we are asking for the rate at which the level DROPS rather than rises.) answer: [ANS]", "answer_v1": [ "0.278521150410817" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A soda glass has the shape of the surface generated by revolving the graph of $y=5x^2$ for $0\\le x\\le 1$ about the $y$-axis. Soda is extracted from the glass through a straw at the rate of $1/2$ cubic inch per second. How fast is the soda level in the glass dropping when the level is 3 inches? (Answer should be implicitly in units of inches per second. Do not put units in your answer. Also your answer should be positive, since we are asking for the rate at which the level DROPS rather than rises.) answer: [ANS]", "answer_v2": [ "0.265258238486492" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A soda glass has the shape of the surface generated by revolving the graph of $y=6x^2$ for $0\\le x\\le 1$ about the $y$-axis. Soda is extracted from the glass through a straw at the rate of $1/2$ cubic inch per second. How fast is the soda level in the glass dropping when the level is 3 inches? (Answer should be implicitly in units of inches per second. Do not put units in your answer. Also your answer should be positive, since we are asking for the rate at which the level DROPS rather than rises.) answer: [ANS]", "answer_v3": [ "0.318309886183791" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0689", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by slices", "level": "4", "keywords": [ "integral' 'volume" ], "problem_v1": "Find the volume of a pyramid with height 27 and rectangular base with dimensions 4 and 10. [ANS]", "answer_v1": [ "360" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of a pyramid with height 16 and rectangular base with dimensions 6 and 7. [ANS]", "answer_v2": [ "224" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of a pyramid with height 20 and rectangular base with dimensions 5 and 8. [ANS]", "answer_v3": [ "266.666666666667" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0690", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by slices", "level": "5", "keywords": [ "calculus", "integrals", "volumes" ], "problem_v1": "Find the volume of a cap of a sphere with radius $r=75$ and height $h=38$. Volume=[ANS]", "answer_v1": [ "282772.660354515" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of a cap of a sphere with radius $r=94$ and height $h=5$. Volume=[ANS]", "answer_v2": [ "7251.84304203644" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of a cap of a sphere with radius $r=67$ and height $h=16$. Volume=[ANS]", "answer_v3": [ "49595.2760246709" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0691", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by slices", "level": "5", "keywords": [ "calculus", "integrals", "volumes" ], "problem_v1": "Find the volume of the frustum of a pyramid with a square base of side length $b=37$, square top of side $a=15$, and height $h=113$. Volume=[ANS]", "answer_v1": [ "80945.6666666667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the frustum of a pyramid with a square base of side length $b=28$, square top of side $a=24$, and height $h=13$. Volume=[ANS]", "answer_v2": [ "8805.33333333333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the frustum of a pyramid with a square base of side length $b=26$, square top of side $a=16$, and height $h=48$. Volume=[ANS]", "answer_v3": [ "21568" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0692", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by slices", "level": "5", "keywords": [ "integrals", "volume" ], "problem_v1": "As viewed from above, a swimming pool has the shape of the ellipse \\frac{x^2}{4900}+\\frac{y^2}{2500}=1, where $x$ and $y$ are measured in feet. The cross sections perpendicular to the $x$-axis are squares. Find the total volume of the pool. $V$=[ANS] cubic feet", "answer_v1": [ "933333.333333333" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "As viewed from above, a swimming pool has the shape of the ellipse \\frac{x^2}{2500}+\\frac{y^2}{400}=1, where $x$ and $y$ are measured in feet. The cross sections perpendicular to the $x$-axis are squares. Find the total volume of the pool. $V$=[ANS] cubic feet", "answer_v2": [ "106666.666666667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "As viewed from above, a swimming pool has the shape of the ellipse \\frac{x^2}{2500}+\\frac{y^2}{900}=1, where $x$ and $y$ are measured in feet. The cross sections perpendicular to the $x$-axis are squares. Find the total volume of the pool. $V$=[ANS] cubic feet", "answer_v3": [ "240000" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0693", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by slices", "level": "5", "keywords": [ "Integral", "Volume" ], "problem_v1": "A ball of radius $16$ has a round hole of radius $7$ drilled through its center. Find the volume of the resulting solid.\nAnswer: [ANS]", "answer_v1": [ "(4/3)*pi*(sqrt(16*16-7*7))^3" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A ball of radius $10$ has a round hole of radius $9$ drilled through its center. Find the volume of the resulting solid.\nAnswer: [ANS]", "answer_v2": [ "(4/3)*pi*(sqrt(10*10-9*9))^3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A ball of radius $12$ has a round hole of radius $7$ drilled through its center. Find the volume of the resulting solid.\nAnswer: [ANS]", "answer_v3": [ "(4/3)*pi*(sqrt(12*12-7*7))^3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0694", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by disks", "level": "4", "keywords": [ "calculus", "integration", "integrals", "volumes of revolution", "revolution", "disk method" ], "problem_v1": "Find the volume of the solid obtained by rotating the region under the graph of the function $f(x)=5x-x^2$ about the $x$-axis over the interval $[0,5]$. $V=$ [ANS]", "answer_v1": [ "327.249234748937" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the solid obtained by rotating the region under the graph of the function $f(x)=2x-x^2$ about the $x$-axis over the interval $[0,2]$. $V=$ [ANS]", "answer_v2": [ "3.35103216382911" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the solid obtained by rotating the region under the graph of the function $f(x)=3x-x^2$ about the $x$-axis over the interval $[0,3]$. $V=$ [ANS]", "answer_v3": [ "25.4469004940773" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0695", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by disks", "level": "4", "keywords": [ "integrals", "volume", "washers", "disks" ], "problem_v1": "The volume of the solid obtained by rotating the region enclosed by y=e^{4x}+4, \\quad y=0, \\quad x=0, \\quad x=0.8 about the x-axis can be computed using the method of disks or washers via an integral\n$ V=\\int_a^b$ [ANS] with limits of integration $a=$ and $b=$ .", "answer_v1": [ "pi*(e^{4*x}+4)^2", "dx", "0", "0.8" ], "answer_type_v1": [ "EX", "MCS", "NV", "NV" ], "options_v1": [ [], [ "dx", "dy" ], [], [] ], "problem_v2": "The volume of the solid obtained by rotating the region enclosed by y=e^{5x}+1, \\quad y=0, \\quad x=0, \\quad x=0.1 about the x-axis can be computed using the method of disks or washers via an integral\n$ V=\\int_a^b$ [ANS] with limits of integration $a=$ and $b=$ .", "answer_v2": [ "pi*(e^{5*x}+1)^2", "dx", "0", "0.1" ], "answer_type_v2": [ "EX", "MCS", "NV", "NV" ], "options_v2": [ [], [ "dx", "dy" ], [], [] ], "problem_v3": "The volume of the solid obtained by rotating the region enclosed by y=e^{4x}+2, \\quad y=0, \\quad x=0, \\quad x=0.4 about the x-axis can be computed using the method of disks or washers via an integral\n$ V=\\int_a^b$ [ANS] with limits of integration $a=$ and $b=$ .", "answer_v3": [ "pi*(e^{4*x}+2)^2", "dx", "0", "0.4" ], "answer_type_v3": [ "EX", "MCS", "NV", "NV" ], "options_v3": [ [], [ "dx", "dy" ], [], [] ] }, { "id": "Calculus_-_single_variable_0696", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by disks", "level": "5", "keywords": [ "integral' 'volume", "Integration", "Volume" ], "problem_v1": "As a hardworking student, plagued by too much homework, you spend all night doing math homework. By 6am, you imagine yourself to be a region bounded by $y=9x^2$ $x=0$ $x=3$ $y=0$ As you grow more and more tired, the world begins to spin around you. However, according to Newton, there is no difference between the world spinning around you, and you spinning around the world. Unfortunately, you are so tired that you think the world is the x-axis. What is the volume of the solid you (the region) create by spinning about the x-axis? [ANS]", "answer_v1": [ "12367.1936401216" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "As a hardworking student, plagued by too much homework, you spend all night doing math homework. By 6am, you imagine yourself to be a region bounded by $y=2x^2$ $x=0$ $x=4$ $y=0$ As you grow more and more tired, the world begins to spin around you. However, according to Newton, there is no difference between the world spinning around you, and you spinning around the world. Unfortunately, you are so tired that you think the world is the x-axis. What is the volume of the solid you (the region) create by spinning about the x-axis? [ANS]", "answer_v2": [ "2573.59270182076" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "As a hardworking student, plagued by too much homework, you spend all night doing math homework. By 6am, you imagine yourself to be a region bounded by $y=5x^2$ $x=0$ $x=3$ $y=0$ As you grow more and more tired, the world begins to spin around you. However, according to Newton, there is no difference between the world spinning around you, and you spinning around the world. Unfortunately, you are so tired that you think the world is the x-axis. What is the volume of the solid you (the region) create by spinning about the x-axis? [ANS]", "answer_v3": [ "3817.0350741116" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0697", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by disks", "level": "4", "keywords": [], "problem_v1": "Find the volume of the solid that results when the region bounded by $y=\\sqrt{x}\\;$, $y=0$ and $x=49$ is revolved about the line $x=49$.\nVolume $=$ [ANS]", "answer_v1": [ "134456/15*pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the solid that results when the region bounded by $y=\\sqrt{x}\\;$, $y=0$ and $x=1$ is revolved about the line $x=1$.\nVolume $=$ [ANS]", "answer_v2": [ "8/15*pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the solid that results when the region bounded by $y=\\sqrt{x}\\;$, $y=0$ and $x=9$ is revolved about the line $x=9$.\nVolume $=$ [ANS]", "answer_v3": [ "648/5*pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0698", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by disks", "level": "5", "keywords": [ "calculus", "volume" ], "problem_v1": "Find the volume of a frustum of a right circular cone with height 25, lower base radius 27 and top radius 13. Volume=[ANS]", "answer_v1": [ "32698.7435361138" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of a frustum of a right circular cone with height 10, lower base radius 26 and top radius 19. Volume=[ANS]", "answer_v2": [ "16032.5945088199" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of a frustum of a right circular cone with height 15, lower base radius 22 and top radius 13. Volume=[ANS]", "answer_v3": [ "14749.7775086041" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0699", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by washers", "level": "4", "keywords": [ "integral' 'volume' 'rotation", "Integral", "Volume", "calculus", "integrals", "volumes" ], "problem_v1": "Find the volume formed by rotating the region enclosed by: $x=9 y$ and $y^3=x$ with $y \\geq 0$ about the $y$-axis [ANS]", "answer_v1": [ "1308.69773969541" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume formed by rotating the region enclosed by: $x=2 y$ and $y^3=x$ with $y \\geq 0$ about the $y$-axis [ANS]", "answer_v2": [ "6.77010733433656" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume formed by rotating the region enclosed by: $x=5 y$ and $y^3=x$ with $y \\geq 0$ about the $y$-axis [ANS]", "answer_v3": [ "167.257493596208" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0700", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by cylindrical shells", "level": "4", "keywords": [ "integration", "cylindrical shells" ], "problem_v1": "Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the $y$-axis.\n$y=\\frac{1}{x},\\; x=2,\\; x=6,\\;y=0$ Volume $=$ [ANS]", "answer_v1": [ "8*pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the $y$-axis.\n$y=\\frac{1}{x},\\; x=1,\\; x=8,\\;y=0$ Volume $=$ [ANS]", "answer_v2": [ "14*pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the $y$-axis.\n$y=\\frac{1}{x},\\; x=1,\\; x=6,\\;y=0$ Volume $=$ [ANS]", "answer_v3": [ "10*pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0701", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Volumes by cylindrical shells", "level": "5", "keywords": [ "Solids of Revolution", "Flash applets", "NSF-0941388" ], "problem_v1": "ww_applet_list[\"solidsWW\"].visible=1;\nif (navigator.appVersion.indexOf(\"MSIE\") > 0) {document.write(\"
You seem to be using Internet Explorer.
It is recommended that another browser be used to view this page.
\");}\nFind the volume of the figure shown. The cross-section of the figure is a regular 6-sided polygon. The area of the polygon can be computed as a function of the length of a line segment from the center of the 6-sided polygon to the midpoint of one of its sides and is given by $6x^2\\tan\\left(\\frac{\\pi}{6}\\right)$ where $x$ is the length of the bisector of one of the sides (shown in black on the cross-section graph). A formula similar to the cylindrical shells formula will then provide the volume of the figure. Simply replace $\\pi$ in the formula V=2\\pi\\int x f(x) dx with $6 \\tan\\left(\\frac{\\pi}{6}\\right)$ to find the volume of the solid shown where for this solid f(x)=\\begin{cases}x&x\\le 10\\\\ 20-x&10 0) {document.write(\"
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It is recommended that another browser be used to view this page.
\");}\nFind the volume of the figure shown. The cross-section of the figure is a regular 8-sided polygon. The area of the polygon can be computed as a function of the length of a line segment from the center of the 8-sided polygon to the midpoint of one of its sides and is given by $8x^2\\tan\\left(\\frac{\\pi}{8}\\right)$ where $x$ is the length of the bisector of one of the sides (shown in black on the cross-section graph). A formula similar to the cylindrical shells formula will then provide the volume of the figure. Simply replace $\\pi$ in the formula V=2\\pi\\int x f(x) dx with $8 \\tan\\left(\\frac{\\pi}{8}\\right)$ to find the volume of the solid shown where for this solid f(x)=\\begin{cases}x&x\\le 4\\\\ 8-x&4 0) {document.write(\"
You seem to be using Internet Explorer.
It is recommended that another browser be used to view this page.
\");}\nFind the volume of the figure shown. The cross-section of the figure is a regular 6-sided polygon. The area of the polygon can be computed as a function of the length of a line segment from the center of the 6-sided polygon to the midpoint of one of its sides and is given by $6x^2\\tan\\left(\\frac{\\pi}{6}\\right)$ where $x$ is the length of the bisector of one of the sides (shown in black on the cross-section graph). A formula similar to the cylindrical shells formula will then provide the volume of the figure. Simply replace $\\pi$ in the formula V=2\\pi\\int x f(x) dx with $6 \\tan\\left(\\frac{\\pi}{6}\\right)$ to find the volume of the solid shown where for this solid f(x)=\\begin{cases}x&x\\le 6\\\\ 12-x&6 0$, in years, the company's extraction plan is a linear declining function of time as follows: q (t)=a-b t, where $q(t)$ is the rate of extraction of oil in millions of barrels per year at time $t$ and $b=0.15$ and $a=16$.\n(a) How long does it take to exhaust the entire reserve? time=[ANS] years\n(b) The oil price is a constant 45 dollars per barrel, the extraction cost per barrel is a constant 12 dollars, and the market interest rate is 8 percent per year, compounded continuously. What is the present value of the company's profit? value=[ANS] millions of dollars", "answer_v1": [ "16/0.15-1/0.15*sqrt(16*16-2*0.15*180)", "(-12+45)*[-0.15+16*0.08+e^{-11.9155*0.08}*(0.15-16*0.08+0.15*11.9155*0.08)]/(0.08^2)" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "An oil company discovered an oil reserve of 100 million barrels. For time $t > 0$, in years, the company's extraction plan is a linear declining function of time as follows: q (t)=a-b t, where $q(t)$ is the rate of extraction of oil in millions of barrels per year at time $t$ and $b=0.2$ and $a=10$.\n(a) How long does it take to exhaust the entire reserve? time=[ANS] years\n(b) The oil price is a constant 35 dollars per barrel, the extraction cost per barrel is a constant 20 dollars, and the market interest rate is 8 percent per year, compounded continuously. What is the present value of the company's profit? value=[ANS] millions of dollars", "answer_v2": [ "10/0.2-1/0.2*sqrt(10*10-2*0.2*100)", "(-20+35)*[-0.2+10*0.08+e^{-11.2702*0.08}*(0.2-10*0.08+0.2*11.2702*0.08)]/(0.08^2)" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "An oil company discovered an oil reserve of 130 million barrels. For time $t > 0$, in years, the company's extraction plan is a linear declining function of time as follows: q (t)=a-b t, where $q(t)$ is the rate of extraction of oil in millions of barrels per year at time $t$ and $b=0.15$ and $a=12$.\n(a) How long does it take to exhaust the entire reserve? time=[ANS] years\n(b) The oil price is a constant 40 dollars per barrel, the extraction cost per barrel is a constant 12 dollars, and the market interest rate is 9 percent per year, compounded continuously. What is the present value of the company's profit? value=[ANS] millions of dollars", "answer_v3": [ "12/0.15-1/0.15*sqrt(12*12-2*0.15*130)", "(-12+40)*[-0.15+12*0.09+e^{-11.687*0.09}*(0.15-12*0.09+0.15*11.687*0.09)]/(0.09^2)" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0798", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Economics", "level": "", "keywords": [ "calculus", "integral", "economics", "present and future value" ], "problem_v1": "A bank account earns 11 percent interest compounded continuously. At what (constant, continuous) rate must a parent deposit money into such an account in order to save 200000 dollars in 14 years for a child's college expenses? rate=[ANS] (dollars/year) If the parent decides instead to deposit a lump sum now in order to attain the goal of 200000 dollars in 14 years, how much must be deposited now? amount=[ANS] (dollars)", "answer_v1": [ " 200000.00* 0.11/[e^{ 0.11* 14.00}- 1.00]", " 200000.00/[e^{ 0.11* 14.00}]" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A bank account earns 5 percent interest compounded continuously. At what (constant, continuous) rate must a parent deposit money into such an account in order to save 250000 dollars in 9 years for a child's college expenses? rate=[ANS] (dollars/year) If the parent decides instead to deposit a lump sum now in order to attain the goal of 250000 dollars in 9 years, how much must be deposited now? amount=[ANS] (dollars)", "answer_v2": [ " 250000.00* 0.05/[e^{ 0.05* 9.00}- 1.00]", " 250000.00/[e^{ 0.05* 9.00}]" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A bank account earns 7 percent interest compounded continuously. At what (constant, continuous) rate must a parent deposit money into such an account in order to save 200000 dollars in 11 years for a child's college expenses? rate=[ANS] (dollars/year) If the parent decides instead to deposit a lump sum now in order to attain the goal of 200000 dollars in 11 years, how much must be deposited now? amount=[ANS] (dollars)", "answer_v3": [ " 200000.00* 0.07/[e^{ 0.07* 11.00}- 1.00]", " 200000.00/[e^{ 0.07* 11.00}]" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0799", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Economics", "level": "", "keywords": [ "calculus", "integral", "economics", "present and future value" ], "problem_v1": "Find the present and future values of an income stream of 3500 dollars a year, for a period of 5 years, if the continuous interest rate is 7 percent. present value=[ANS] dollars future value=[ANS] dollars (In either case, specify your answer to the nearest cent (i.e., 0.01 dollar).) (In either case, specify your answer to the nearest cent (i.e., 0.01 dollar).)", "answer_v1": [ " 20953.38/[e^{ 0.07* 5.00}]", " 3500.00/ 0.07*[e^{ 0.07* 5.00}- 1.00]" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the present and future values of an income stream of 1000 dollars a year, for a period of 5 years, if the continuous interest rate is 9 percent. present value=[ANS] dollars future value=[ANS] dollars (In either case, specify your answer to the nearest cent (i.e., 0.01 dollar).) (In either case, specify your answer to the nearest cent (i.e., 0.01 dollar).)", "answer_v2": [ " 6314.58/[e^{ 0.09* 5.00}]", " 1000.00/ 0.09*[e^{ 0.09* 5.00}- 1.00]" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the present and future values of an income stream of 2000 dollars a year, for a period of 5 years, if the continuous interest rate is 7 percent. present value=[ANS] dollars future value=[ANS] dollars (In either case, specify your answer to the nearest cent (i.e., 0.01 dollar).) (In either case, specify your answer to the nearest cent (i.e., 0.01 dollar).)", "answer_v3": [ " 11973.36/[e^{ 0.07* 5.00}]", " 2000.00/ 0.07*[e^{ 0.07* 5.00}- 1.00]" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0800", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Economics", "level": "", "keywords": [ "calculus", "integral", "economics", "present and future value" ], "problem_v1": "Sales of Version 6.0 of a computer software package start out high and decrease exponentially. At time $t$, in years, the sales are $s(t)=45 e^{-t}$ thousands of dollars per year. After 4 years, Version 7.0 of the software is released and replaces Version 6.0. Assume that all income from software sales is immediately invested in government bonds which pay interest at a 7 percent rate compounded continuously, calculate the total value of sales of Version 6.0 over the four year period. value=[ANS] thousand dollars", "answer_v1": [ "45*e^{0.07*4}/(0.07+1)*(1-e^{-(1+0.07)*4})" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sales of Version 3.0 of a computer software package start out high and decrease exponentially. At time $t$, in years, the sales are $s(t)=60 e^{-t}$ thousands of dollars per year. After 3 years, Version 4.0 of the software is released and replaces Version 3.0. Assume that all income from software sales is immediately invested in government bonds which pay interest at a 4 percent rate compounded continuously, calculate the total value of sales of Version 3.0 over the three year period. value=[ANS] thousand dollars", "answer_v2": [ "60*e^{0.04*3}/(0.04+1)*(1-e^{-(1+0.04)*3})" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sales of Version 4.0 of a computer software package start out high and decrease exponentially. At time $t$, in years, the sales are $s(t)=45 e^{-t}$ thousands of dollars per year. After 3 years, Version 5.0 of the software is released and replaces Version 4.0. Assume that all income from software sales is immediately invested in government bonds which pay interest at a 5 percent rate compounded continuously, calculate the total value of sales of Version 4.0 over the three year period. value=[ANS] thousand dollars", "answer_v3": [ "45*e^{0.05*3}/(0.05+1)*(1-e^{-(1+0.05)*3})" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0801", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Economics", "level": "", "keywords": [ "calculus", "integral", "economics", "present and future value" ], "problem_v1": "In May 1991, Car and Driver described a Jaguar that sold for 980,000 dollars. Suppose that at that price only 55 have been sold. If it is estimated that 350 could have been sold if the price had been 560,000 dollars. Assuming that the demand curve is a straight line, and that 560,000 dollars and 350 are the equilibrium price and quantity, find the consumer surplus at the equilibrium price. surplus=[ANS] thousands of dollars", "answer_v1": [ "1/2*(980-560)*350/(350-55)*350" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In May 1991, Car and Driver described a Jaguar that sold for 980,000 dollars. Suppose that at that price only 40 have been sold. If it is estimated that 400 could have been sold if the price had been 510,000 dollars. Assuming that the demand curve is a straight line, and that 510,000 dollars and 400 are the equilibrium price and quantity, find the consumer surplus at the equilibrium price. surplus=[ANS] thousands of dollars", "answer_v2": [ "1/2*(980-510)*400/(400-40)*400" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In May 1991, Car and Driver described a Jaguar that sold for 980,000 dollars. Suppose that at that price only 45 have been sold. If it is estimated that 375 could have been sold if the price had been 530,000 dollars. Assuming that the demand curve is a straight line, and that 530,000 dollars and 375 are the equilibrium price and quantity, find the consumer surplus at the equilibrium price. surplus=[ANS] thousands of dollars", "answer_v3": [ "1/2*(980-530)*375/(375-45)*375" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0802", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Economics", "level": "4", "keywords": [ "derivatives" ], "problem_v1": "Find producer's surplus at the market equilibrium point if supply function is $p=0.7x+13$ and the demand function is $p=\\frac{409.2}{x+14}$. Answer: [ANS]", "answer_v1": [ "22.4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find producer's surplus at the market equilibrium point if supply function is $p=0.2x+19$ and the demand function is $p=\\frac{200}{x+5}$. Answer: [ANS]", "answer_v2": [ "2.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find producer's surplus at the market equilibrium point if supply function is $p=0.4x+13$ and the demand function is $p=\\frac{215.6}{x+8}$. Answer: [ANS]", "answer_v3": [ "7.2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0803", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Biology", "level": "5", "keywords": [ "calculus", "integrals", "integration", "exponential decay/growth" ], "problem_v1": "The population of a city is P(t)=8e^{0.12t} (in millions), where t is measured in years.\n(a) Calculate the doubling time of the population. (b) How long does it take for the population to triple in size? (c) How long does it take for the population to quadruple in size?\n(a) [ANS]\n(b) [ANS]\n(c) [ANS]", "answer_v1": [ "5.77623", "9.1551", "11.5525" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The population of a city is P(t)=2e^{0.19t} (in millions), where t is measured in years.\n(a) Calculate the doubling time of the population. (b) How long does it take for the population to triple in size? (c) How long does it take for the population to quadruple in size?\n(a) [ANS]\n(b) [ANS]\n(c) [ANS]", "answer_v2": [ "3.64814", "5.78217", "7.29629" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The population of a city is P(t)=4e^{0.13t} (in millions), where t is measured in years.\n(a) Calculate the doubling time of the population. (b) How long does it take for the population to triple in size? (c) How long does it take for the population to quadruple in size?\n(a) [ANS]\n(b) [ANS]\n(c) [ANS]", "answer_v3": [ "5.3319", "8.45086", "10.6638" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0804", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Biology", "level": "5", "keywords": [ "calculus", "integrals", "integration", "exponential decay/growth" ], "problem_v1": "A certain bacteria population P obeys the exponential growth law P(t)=3000e^{0.5t} (t in hours)\n(a) How many bacteria are present initially? (b) At what time will there be 10000 bacteria?\n(a) [ANS]\n(b) [ANS]", "answer_v1": [ "3000", "2.41" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A certain bacteria population P obeys the exponential growth law P(t)=500e^{2.9t} (t in hours)\n(a) How many bacteria are present initially? (b) At what time will there be 10000 bacteria?\n(a) [ANS]\n(b) [ANS]", "answer_v2": [ "500", "1.03" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A certain bacteria population P obeys the exponential growth law P(t)=1000e^{3.5t} (t in hours)\n(a) How many bacteria are present initially? (b) At what time will there be 10000 bacteria?\n(a) [ANS]\n(b) [ANS]", "answer_v3": [ "1000", "0.66" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0805", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Biology", "level": "5", "keywords": [ "calculus", "integrals", "integration", "net change", "total change" ], "problem_v1": "A population of insects increases at a rate of $270+12 t+1.2 t^2$ insects per day. Find the insect population after 5 days, assuming that there are 40 insects at $t=0$. Round your answer to the nearest whole number. Answer: [ANS] insects", "answer_v1": [ "1590" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A population of insects increases at a rate of $160+16 t+0.3 t^2$ insects per day. Find the insect population after 4 days, assuming that there are 80 insects at $t=0$. Round your answer to the nearest whole number. Answer: [ANS] insects", "answer_v2": [ "854" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A population of insects increases at a rate of $200+12 t+0.6 t^2$ insects per day. Find the insect population after 5 days, assuming that there are 40 insects at $t=0$. Round your answer to the nearest whole number. Answer: [ANS] insects", "answer_v3": [ "1215" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0806", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Biology", "level": "2", "keywords": [ "calculus", "indefinite integrals", "net change theorem" ], "problem_v1": "A population of cattle is increasing at a rate of $700+60 \\, t$ per year, where $t$ is measured in years. By how much does the population increase between the 6th and the 11th years? Total Increase=[ANS]", "answer_v1": [ "6050" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A population of cattle is increasing at a rate of $100+90 \\, t$ per year, where $t$ is measured in years. By how much does the population increase between the 2nd and the 5th years? Total Increase=[ANS]", "answer_v2": [ "1245" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A population of cattle is increasing at a rate of $300+60 \\, t$ per year, where $t$ is measured in years. By how much does the population increase between the 3rd and the 7th years? Total Increase=[ANS]", "answer_v3": [ "2400" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0807", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Probability and statistics", "level": "3", "keywords": [ "density function' 'integral" ], "problem_v1": "Find the value of $ C$ so that the function f(x)=\\begin{cases} Cx^{8} & \\text{if} \\;\\; 0 \\leq x \\leq 7 \\\\ 0 & \\text{otherwise} \\end{cases} is a density function. [ANS]", "answer_v1": [ "2.2302839000241E-07" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the value of $ C$ so that the function f(x)=\\begin{cases} Cx^{1.5} & \\text{if} \\;\\; 0 \\leq x \\leq 10 \\\\ 0 & \\text{otherwise} \\end{cases} is a density function. [ANS]", "answer_v2": [ "0.00790569415042095" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the value of $ C$ so that the function f(x)=\\begin{cases} Cx^{3.5} & \\text{if} \\;\\; 0 \\leq x \\leq 7 \\\\ 0 & \\text{otherwise} \\end{cases} is a density function. [ANS]", "answer_v3": [ "0.000708388225131829" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0808", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Probability and statistics", "level": "2", "keywords": [ "density function' 'integral" ], "problem_v1": "At a restaurant, the density function for the time a customer has to wait before being seated is given by f(t)=\\begin{cases} 0 & \\text{if} \\;\\; t<0\\\\ 3 e^{-3 t} & \\text{if} \\;\\; t\\geq 0. \\end{cases} Find the probability that a customer will have to wait at least $6$ minutes for a table. [ANS]", "answer_v1": [ "1.52299797447126E-08" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "At a restaurant, the density function for the time a customer has to wait before being seated is given by f(t)=\\begin{cases} 0 & \\text{if} \\;\\; t<0\\\\ 1 e^{-1 t} & \\text{if} \\;\\; t\\geq 0. \\end{cases} Find the probability that a customer will have to wait at least $7$ minutes for a table. [ANS]", "answer_v2": [ "0.000911881965554517" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "At a restaurant, the density function for the time a customer has to wait before being seated is given by f(t)=\\begin{cases} 0 & \\text{if} \\;\\; t<0\\\\ 1 e^{-1 t} & \\text{if} \\;\\; t\\geq 0. \\end{cases} Find the probability that a customer will have to wait at least $6$ minutes for a table. [ANS]", "answer_v3": [ "0.00247875217666636" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0809", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Probability and statistics", "level": "2", "keywords": [ "density function' 'integral" ], "problem_v1": "Suppose that a density function is given by the formula f(x)=\\begin{cases} {\\textstyle\\frac{1}{72}} x & \\text{if} \\;\\; 0 \\leq x \\leq 8\\\\ {\\textstyle\\frac{1}{5}}-{\\textstyle\\frac{1}{90}} x & \\text{if} \\;\\; 8 \\leq x \\leq 18 \\\\ 0 & \\text{otherwise.} \\end{cases} Find the probability that $x$ is between $5$ and $11$. [ANS]", "answer_v1": [ "133/240" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that a density function is given by the formula f(x)=\\begin{cases} {\\textstyle\\frac{2}{75}} x & \\text{if} \\;\\; 0 \\leq x \\leq 5\\\\ {\\textstyle\\frac{1}{5}}-{\\textstyle\\frac{1}{75}} x & \\text{if} \\;\\; 5 \\leq x \\leq 15 \\\\ 0 & \\text{otherwise.} \\end{cases} Find the probability that $x$ is between $4$ and $10$. [ANS]", "answer_v2": [ "31/50" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that a density function is given by the formula f(x)=\\begin{cases} {\\textstyle\\frac{1}{40}} x & \\text{if} \\;\\; 0 \\leq x \\leq 5\\\\ {\\textstyle\\frac{2}{11}}-{\\textstyle\\frac{1}{88}} x & \\text{if} \\;\\; 5 \\leq x \\leq 16 \\\\ 0 & \\text{otherwise.} \\end{cases} Find the probability that $x$ is between $4$ and $9$. [ANS]", "answer_v3": [ "459/880" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0810", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Probability and statistics", "level": "5", "keywords": [ "density function' 'integral" ], "problem_v1": "At a particular fast-food restaurant, food is often made in advance, and then it is placed under a heater until it is sold. The distribution function for the number of minutes that passes between when a cheeseburger is made and when it is sold is given by f(t)=\\begin{cases} \\frac{2}{17}-\\frac{2}{289}t & \\text{if} \\;\\; 0\\leq t \\leq 17\\\\ 0 & \\text{otherwise.} \\end{cases} Find the median number of minutes that a cheeseburger sits under the heater. [ANS]", "answer_v1": [ "4.97918471982869" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "At a particular fast-food restaurant, food is often made in advance, and then it is placed under a heater until it is sold. The distribution function for the number of minutes that passes between when a cheeseburger is made and when it is sold is given by f(t)=\\begin{cases} \\frac{2}{8}-\\frac{2}{64}t & \\text{if} \\;\\; 0\\leq t \\leq 8\\\\ 0 & \\text{otherwise.} \\end{cases} Find the median number of minutes that a cheeseburger sits under the heater. [ANS]", "answer_v2": [ "2.34314575050762" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "At a particular fast-food restaurant, food is often made in advance, and then it is placed under a heater until it is sold. The distribution function for the number of minutes that passes between when a cheeseburger is made and when it is sold is given by f(t)=\\begin{cases} \\frac{2}{11}-\\frac{2}{121}t & \\text{if} \\;\\; 0\\leq t \\leq 11\\\\ 0 & \\text{otherwise.} \\end{cases} Find the median number of minutes that a cheeseburger sits under the heater. [ANS]", "answer_v3": [ "3.22182540694798" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0811", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Probability and statistics", "level": "5", "keywords": [ "calculus", "integral", "probability distributions", "probability", "mean", "median" ], "problem_v1": "Consider a group of people who have received treatment for a disease such as cancer. Let $t$ be the survival time, the number of years a person lives after receiving treatment. The density function giving the distribution of $t$ is $p(t)=Ce^{-Ct}$ for some positive constant $C$, and the cumulative distribution function is $P(t)=\\int_0^t p(x)\\,dx$. Think carefully about what the practical meaning of $P(t)$ is, being sure that you can put it into words.\n(a) The survival function, $S(t)$, is the probability that a randomly selected person survives for at least $t$ years. Find a formula for $S(t)$. $S(t)=$ [ANS]\n(b) Suppose that a patient has a 80 percent chance of surviving at least 4 years. Find $C$. $C=$ [ANS]\n(c) Using the value of $C$ you found in (b), find each of the following: the probability that the patient survives up to (that is, less than or equal to) 5 years: [ANS]\nthe mean survival time for patients with this survival function, in years: [ANS]", "answer_v1": [ "e^{-C*t}", "-(1/4)*ln(80/100)", "1-(80/100)^{1.25}", "-4/[ln(80/100)]" ], "answer_type_v1": [ "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider a group of people who have received treatment for a disease such as cancer. Let $t$ be the survival time, the number of years a person lives after receiving treatment. The density function giving the distribution of $t$ is $p(t)=Ce^{-Ct}$ for some positive constant $C$, and the cumulative distribution function is $P(t)=\\int_0^t p(x)\\,dx$. Think carefully about what the practical meaning of $P(t)$ is, being sure that you can put it into words.\n(a) The survival function, $S(t)$, is the probability that a randomly selected person survives for at least $t$ years. Find a formula for $S(t)$. $S(t)=$ [ANS]\n(b) Suppose that a patient has a 55 percent chance of surviving at least 5 years. Find $C$. $C=$ [ANS]\n(c) Using the value of $C$ you found in (b), find each of the following: the probability that the patient survives up to (that is, less than or equal to) 4 years: [ANS]\nthe mean survival time for patients with this survival function, in years: [ANS]", "answer_v2": [ "e^{-C*t}", "-(1/5)*ln(55/100)", "1-(55/100)^{0.8}", "-5/[ln(55/100)]" ], "answer_type_v2": [ "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider a group of people who have received treatment for a disease such as cancer. Let $t$ be the survival time, the number of years a person lives after receiving treatment. The density function giving the distribution of $t$ is $p(t)=Ce^{-Ct}$ for some positive constant $C$, and the cumulative distribution function is $P(t)=\\int_0^t p(x)\\,dx$. Think carefully about what the practical meaning of $P(t)$ is, being sure that you can put it into words.\n(a) The survival function, $S(t)$, is the probability that a randomly selected person survives for at least $t$ years. Find a formula for $S(t)$. $S(t)=$ [ANS]\n(b) Suppose that a patient has a 65 percent chance of surviving at least 4 years. Find $C$. $C=$ [ANS]\n(c) Using the value of $C$ you found in (b), find each of the following: the probability that the patient survives up to (that is, less than or equal to) 5 years: [ANS]\nthe mean survival time for patients with this survival function, in years: [ANS]", "answer_v3": [ "e^{-C*t}", "-(1/4)*ln(65/100)", "1-(65/100)^{1.25}", "-4/[ln(65/100)]" ], "answer_type_v3": [ "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0812", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Probability and statistics", "level": "5", "keywords": [ "calculus", "integral", "probability distributions", "probability", "mean", "median" ], "problem_v1": "The distribution of IQ scores can be modeled by a normal distribution with mean 100 and standard deviation 15.\n(a) Let $x$ be a person's IQ score. Write the formula for the density function of IQ scores. $p(x)=$ [ANS]\n(b) Estimate the fraction of the population with IQ between 125 and 130. fraction=[ANS]", "answer_v1": [ "1/[15*sqrt(2*pi)]*e^{-(1/2)*[(x-100)/15]^2}", "0.0250402" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The distribution of IQ scores can be modeled by a normal distribution with mean 100 and standard deviation 15.\n(a) Let $x$ be a person's IQ score. Write the formula for the density function of IQ scores. $p(x)=$ [ANS]\n(b) Estimate the fraction of the population with IQ between 80 and 85. fraction=[ANS]", "answer_v2": [ "1/[15*sqrt(2*pi)]*e^{-(1/2)*[(x-100)/15]^2}", "0.067444" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The distribution of IQ scores can be modeled by a normal distribution with mean 100 and standard deviation 15.\n(a) Let $x$ be a person's IQ score. Write the formula for the density function of IQ scores. $p(x)=$ [ANS]\n(b) Estimate the fraction of the population with IQ between 75 and 80. fraction=[ANS]", "answer_v3": [ "1/[15*sqrt(2*pi)]*e^{-(1/2)*[(x-100)/15]^2}", "0.0434209" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0813", "subject": "Calculus_-_single_variable", "topic": "Applications of integration", "subtopic": "Probability and statistics", "level": "5", "keywords": [ "calculus", "integral", "probability distributions", "probability" ], "problem_v1": "Suppose that $p(x)$ is the density function for heights of American men, in inches, and suppose that $p(69)=0.21$. Think carefully about what the meaning of this mathematical statement is.\n(a) Approximately what percent of American men are between 68.7 and 69.3 inches tall? Approximately [ANS] percent. (b) Suppose that the average height of American men is 69 inches. Would you expect $p(77) > p(69)$ or $p(77) < p(69)$? [ANS] A. $p(77)>p(69)$ B. $p(77) p(69)$ or $p(74) < p(69)$? [ANS] A. $p(74)>p(69)$ B. $p(74) p(69)$ or $p(75) < p(69)$? [ANS] A. $p(75)>p(69)$ B. $p(75)M$. (Note that we are using $t$ instead of $\\epsilon$ in the definition in order to allow you to enter your answer more easily). $M=$ [ANS] (Enter your answer as a function of $t$)", "answer_v1": [ "1995", "199995", "2/t-5" ], "answer_type_v1": [ "NV", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $a_n=\\frac{n}{n+3}$. Find the smallest number $M$ such that:\n(a) $|a_n-1|\\le 0.001$ for $n\\ge M$ $M=$ [ANS]\n(b) $|a_n-1|\\le 0.00001$ for $n\\ge M$ $M=$ [ANS]\n(c) Now use the limit definition to prove that $\\lim_{n\\to\\infty}a_n=1$. That is, find the smallest value of $M$ (in terms of $t$) such that $\\left|a_n-1\\right|M$. (Note that we are using $t$ instead of $\\epsilon$ in the definition in order to allow you to enter your answer more easily). $M=$ [ANS] (Enter your answer as a function of $t$)", "answer_v2": [ "2997", "299997", "3/t-3" ], "answer_type_v2": [ "NV", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $a_n=\\frac{n+1}{n+3}$. Find the smallest number $M$ such that:\n(a) $|a_n-1|\\le 0.001$ for $n\\ge M$ $M=$ [ANS]\n(b) $|a_n-1|\\le 0.00001$ for $n\\ge M$ $M=$ [ANS]\n(c) Now use the limit definition to prove that $\\lim_{n\\to\\infty}a_n=1$. That is, find the smallest value of $M$ (in terms of $t$) such that $\\left|a_n-1\\right|M$. (Note that we are using $t$ instead of $\\epsilon$ in the definition in order to allow you to enter your answer more easily). $M=$ [ANS] (Enter your answer as a function of $t$)", "answer_v3": [ "1997", "199997", "2/t-3" ], "answer_type_v3": [ "NV", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0821", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Limit of a sequence", "level": "2", "keywords": [ "sequence", "limit", "convergent", "divergent", "Sequences ", "Limits" ], "problem_v1": "Match each sequence below to statement that BEST fits it.\nSTATEMENTS\nZ. The sequence converges to zero; I. The sequence diverges to positive infinity; F. The sequence has a finite non-zero limit; D. The sequence diverges, but not to infinity. SEQUENCES [ANS] 1. $\\frac{(-5)^n}{n!}$ [ANS] 2. $\\left(5n^{2n} \\right) ^{1/n}$ [ANS] 3. $\\left(-1 \\right) ^{-n} \\frac{2n}{\\ln(n)}$ [ANS] 4. $\\frac{100n^2+1}{3n!}$ [ANS] 5. $\\frac{5^{n}}{n!}$ [ANS] 6. $\\sqrt{n^{2}+4n}-\\sqrt{n^{2}}$ [ANS] 7. $\\cos ^{2}(n)+\\sin ^{2}(n)$ [ANS] 8. $\\left(\\frac{e}{10} \\right) ^{n}$", "answer_v1": [ "Z", "I", "D", "Z", "Z", "F", "F", "Z" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ] ], "problem_v2": "Match each sequence below to statement that BEST fits it.\nSTATEMENTS\nZ. The sequence converges to zero; I. The sequence diverges to positive infinity; F. The sequence has a finite non-zero limit; D. The sequence diverges, but not to infinity. SEQUENCES [ANS] 1. $\\sqrt{n^{2}+4n}-\\sqrt{n^{2}}$ [ANS] 2. $\\cos ^{2}(n)+\\sin ^{2}(n)$ [ANS] 3. $\\frac{5^{n}}{n!}$ [ANS] 4. $\\left(-1 \\right) ^{-n} \\frac{2n}{\\ln(n)}$ [ANS] 5. $\\frac{(-5)^n}{n!}$ [ANS] 6. $\\left(\\frac{e}{10} \\right) ^{n}$ [ANS] 7. $\\left(5n^{2n} \\right) ^{1/n}$ [ANS] 8. $\\frac{100n^2+1}{3n!}$", "answer_v2": [ "F", "F", "Z", "D", "Z", "Z", "I", "Z" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ] ], "problem_v3": "Match each sequence below to statement that BEST fits it.\nSTATEMENTS\nZ. The sequence converges to zero; I. The sequence diverges to positive infinity; F. The sequence has a finite non-zero limit; D. The sequence diverges, but not to infinity. SEQUENCES [ANS] 1. $\\left(\\frac{e}{10} \\right) ^{n}$ [ANS] 2. $\\frac{100n^2+1}{3n!}$ [ANS] 3. $\\frac{5^{n}}{n!}$ [ANS] 4. $\\left(5n^{2n} \\right) ^{1/n}$ [ANS] 5. $\\sqrt{n^{2}+4n}-\\sqrt{n^{2}}$ [ANS] 6. $\\frac{(-5)^n}{n!}$ [ANS] 7. $\\cos ^{2}(n)+\\sin ^{2}(n)$ [ANS] 8. $\\left(-1 \\right) ^{-n} \\frac{2n}{\\ln(n)}$", "answer_v3": [ "Z", "Z", "Z", "I", "F", "Z", "F", "D" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ], [ "Z", "I", "F", "D" ] ] }, { "id": "Calculus_-_single_variable_0822", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Limit of a sequence", "level": "2", "keywords": [ "sequence", "limit", "convergent", "divergent", "Sequences", "convergence" ], "problem_v1": "If a sequence $c_1, c_2, c_3,...$ has limit K then the sequence $e^{c_1}, e^{c_2}, e^{c_3},...$ has limit $e^K.$ Use this fact together with l'Hopital's rule to compute the limit of the sequence given by $b_n=(1+\\frac{4.8}{n})^n.$ [ANS]", "answer_v1": [ "121.510417518735" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If a sequence $c_1, c_2, c_3,...$ has limit K then the sequence $e^{c_1}, e^{c_2}, e^{c_3},...$ has limit $e^K.$ Use this fact together with l'Hopital's rule to compute the limit of the sequence given by $b_n=(1+\\frac{1.4}{n})^n.$ [ANS]", "answer_v2": [ "4.05519996684467" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If a sequence $c_1, c_2, c_3,...$ has limit K then the sequence $e^{c_1}, e^{c_2}, e^{c_3},...$ has limit $e^K.$ Use this fact together with l'Hopital's rule to compute the limit of the sequence given by $b_n=(1+\\frac{2.5}{n})^n.$ [ANS]", "answer_v3": [ "12.1824939607035" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0823", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Limit of a sequence", "level": "2", "keywords": [ "Calculus" ], "problem_v1": "Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several examples, especially examples which might show that the statement is false!. You must get all of the answers correct to receive credit. [ANS] 1. Every sequence which is convergent must be bounded. [ANS] 2. The sequence 1, 2, 3, 4,... has a finite accumulation point. [ANS] 3. The sequence 1,-1, 2,-1, 3,-1, 4,-1, 5... has a finite accumulation point. [ANS] 4. The sequence 1,-1, 1,-1, 1,-1,... does not have a convergent subsequence.", "answer_v1": [ "T", "F", "T", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several examples, especially examples which might show that the statement is false!. You must get all of the answers correct to receive credit. [ANS] 1. Every bounded sequence converges to a limit point. [ANS] 2. Every sequence which converges is either an increasing sequence or a decreasing sequence. [ANS] 3. Every bounded sequence has a subsequence which converges to a limit point. [ANS] 4. The sequence 1, 2, 3, 4,... has no finite limit.", "answer_v2": [ "F", "F", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false. A good technique is to think of several examples, especially examples which might show that the statement is false!. You must get all of the answers correct to receive credit. [ANS] 1. The sequence of rational numbers 3.1, 3.14, 3.141, 3.14159,... which approximates the ratio of the circumference of a circle and its diameter, has a rational number as its limit point. [ANS] 2. The sequence 1,-1, 2,-1, 3,-1, 4,-1, 5... has a finite accumulation point. [ANS] 3. Every bounded sequence has a subsequence which converges to a limit point. [ANS] 4. The sequence 1, 2, 3, 4,... has a finite accumulation point.", "answer_v3": [ "F", "T", "T", "F" ], "answer_type_v3": [ "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0824", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Limit of a sequence", "level": "3", "keywords": [ "monotonic' 'sequence" ], "problem_v1": "We want to determine if the sequence $\\frac{10^{n}}{5^{n^{2}}}$ is monotonic. Using the ratio test we get that $\\frac{s_{n+1}}{s_{n}}=$ [ANS] [ANS]: >> 1 Hence the sequence is [ANS]", "answer_v1": [ "10/[5^{2*n+1}]", "<", "monotone decreasing" ], "answer_type_v1": [ "EX", "MCS", "MCS" ], "options_v1": [ [], [ "=", "<" ], [ "monotone increasing", "monotone nonincreasing", "monotone nondecreasing", "cannot be determined" ] ], "problem_v2": "We want to determine if the sequence $\\frac{4^{n}}{2^{n^{2}}}$ is monotonic. Using the ratio test we get that $\\frac{s_{n+1}}{s_{n}}=$ [ANS] [ANS]: >> 1 Hence the sequence is [ANS]", "answer_v2": [ "4/[2^{2*n+1}]", "<", "monotone decreasing" ], "answer_type_v2": [ "EX", "MCS", "MCS" ], "options_v2": [ [], [ "=", "<" ], [ "monotone increasing", "monotone nonincreasing", "monotone nondecreasing", "cannot be determined" ] ], "problem_v3": "We want to determine if the sequence $\\frac{6^{n}}{3^{n^{2}}}$ is monotonic. Using the ratio test we get that $\\frac{s_{n+1}}{s_{n}}=$ [ANS] [ANS]: >> 1 Hence the sequence is [ANS]", "answer_v3": [ "6/[3^{2*n+1}]", "<", "monotone decreasing" ], "answer_type_v3": [ "EX", "MCS", "MCS" ], "options_v3": [ [], [ "=", "<" ], [ "monotone increasing", "monotone nonincreasing", "monotone nondecreasing", "cannot be determined" ] ] }, { "id": "Calculus_-_single_variable_0825", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Limit of a sequence", "level": "2", "keywords": [], "problem_v1": "Write out the first five terms of the sequence with, $\\left[{\\left(1-\\frac{5}{n+6}\\right)^{n}}\\right]_{n=1}^{\\infty}$, determine whether the sequence converges, and if so find its limit. Enter the following information for $a_n=\\left(1-\\frac{5}{n+6}\\right)^{n}$. $a_1=$ [ANS]\n$a_2=$ [ANS]\n$a_3=$ [ANS]\n$a_4=$ [ANS]\n$a_5=$ [ANS]\n$ \\lim_{n\\to\\infty} \\left(1-\\frac{5}{n+6}\\right)^{n}=$ [ANS]\n(Enter DNE if limit Does Not Exist.) Does the sequence converge [ANS] (Enter \"yes\" or \"no\").", "answer_v1": [ "[1-5/(1+6)]^1", "[1-5/(2+6)]^2", "[1-5/(3+6)]^3", "[1-5/(4+6)]^4", "[1-5/(5+6)]^5", "e^{-5}", "yes" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Write out the first five terms of the sequence with, $\\left[{\\left(1-\\frac{8}{n}\\right)^{n}}\\right]_{n=1}^{\\infty}$, determine whether the sequence converges, and if so find its limit. Enter the following information for $a_n=\\left(1-\\frac{8}{n}\\right)^{n}$. $a_1=$ [ANS]\n$a_2=$ [ANS]\n$a_3=$ [ANS]\n$a_4=$ [ANS]\n$a_5=$ [ANS]\n$ \\lim_{n\\to\\infty} \\left(1-\\frac{8}{n}\\right)^{n}=$ [ANS]\n(Enter DNE if limit Does Not Exist.) Does the sequence converge [ANS] (Enter \"yes\" or \"no\").", "answer_v2": [ "[1-8/(1+0)]^1", "[1-8/(2+0)]^2", "[1-8/(3+0)]^3", "[1-8/(4+0)]^4", "[1-8/(5+0)]^5", "e^{-8}", "yes" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Write out the first five terms of the sequence with, $\\left[{\\left(1-\\frac{5}{n+2}\\right)^{n}}\\right]_{n=1}^{\\infty}$, determine whether the sequence converges, and if so find its limit. Enter the following information for $a_n=\\left(1-\\frac{5}{n+2}\\right)^{n}$. $a_1=$ [ANS]\n$a_2=$ [ANS]\n$a_3=$ [ANS]\n$a_4=$ [ANS]\n$a_5=$ [ANS]\n$ \\lim_{n\\to\\infty} \\left(1-\\frac{5}{n+2}\\right)^{n}=$ [ANS]\n(Enter DNE if limit Does Not Exist.) Does the sequence converge [ANS] (Enter \"yes\" or \"no\").", "answer_v3": [ "[1-5/(1+2)]^1", "[1-5/(2+2)]^2", "[1-5/(3+2)]^3", "[1-5/(4+2)]^4", "[1-5/(5+2)]^5", "e^{-5}", "yes" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "TF" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0827", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Limit of a sequence", "level": "2", "keywords": [ "calculus", "sequence", "limit", "convergent", "divergent", "Sequences", "convergence" ], "problem_v1": "Find the limit of the sequence whose terms are given by $a_n=(n^2)(1-\\cos (\\frac{4.8}{n})).$ [ANS]", "answer_v1": [ "11.52" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the limit of the sequence whose terms are given by $a_n=(n^2)(1-\\cos (\\frac{1.4}{n})).$ [ANS]", "answer_v2": [ "0.98" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the limit of the sequence whose terms are given by $a_n=(n^2)(1-\\cos (\\frac{2.5}{n})).$ [ANS]", "answer_v3": [ "3.125" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0828", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Limit of a sequence", "level": "2", "keywords": [ "Sequences", "limits" ], "problem_v1": "Determine whether the sequences are increasing, decreasing, or not monotonic. If increasing, enter I as your answer. If decreasing, enter D as your answer. If not monotonic, enter N as your answer. [ANS] 1. $a_n=\\frac{1}{5 n+8}$ [ANS] 2. $a_n=\\frac{\\sqrt{n+5}}{8 n+5}$ [ANS] 3. $a_n=\\frac{\\cos n}{5^n}$ [ANS] 4. $a_n=\\frac{n-5}{n+5}$", "answer_v1": [ "D", "D", "N", "I" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [ "D", "N", "I" ], [ "D", "N", "I" ], [ "D", "N", "I" ], [ "D", "N", "I" ] ], "problem_v2": "Determine whether the sequences are increasing, decreasing, or not monotonic. If increasing, enter I as your answer. If decreasing, enter D as your answer. If not monotonic, enter N as your answer. [ANS] 1. $a_n=\\frac{\\cos n}{2^n}$ [ANS] 2. $a_n=\\frac{1}{2 n+9}$ [ANS] 3. $a_n=\\frac{\\sqrt{n+2}}{9 n+2}$ [ANS] 4. $a_n=\\frac{n-2}{n+2}$", "answer_v2": [ "N", "D", "D", "I" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [ "D", "N", "I" ], [ "D", "N", "I" ], [ "D", "N", "I" ], [ "D", "N", "I" ] ], "problem_v3": "Determine whether the sequences are increasing, decreasing, or not monotonic. If increasing, enter I as your answer. If decreasing, enter D as your answer. If not monotonic, enter N as your answer. [ANS] 1. $a_n=\\frac{n-3}{n+3}$ [ANS] 2. $a_n=\\frac{1}{3 n+8}$ [ANS] 3. $a_n=\\frac{\\cos n}{3^n}$ [ANS] 4. $a_n=\\frac{\\sqrt{n+3}}{8 n+3}$", "answer_v3": [ "I", "D", "N", "D" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [ "D", "N", "I" ], [ "D", "N", "I" ], [ "D", "N", "I" ], [ "D", "N", "I" ] ] }, { "id": "Calculus_-_single_variable_0829", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Series notation", "level": "3", "keywords": [ "Series", "reindex" ], "problem_v1": "Reindex the series to start at $k=0$ $y=\\sum_{k=7}^\\infty \\left(k+1\\right)x^{k+3}=\\sum_{k=0}^\\infty$ [ANS]", "answer_v1": [ "(k+7+1)*x^{k+7+3}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Reindex the series to start at $k=0$ $y=\\sum_{k=2}^\\infty \\left(k+1\\right)x^{k+3}=\\sum_{k=0}^\\infty$ [ANS]", "answer_v2": [ "(k+2+1)*x^{k+2+3}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Reindex the series to start at $k=0$ $y=\\sum_{k=4}^\\infty \\left(k+1\\right)x^{k+3}=\\sum_{k=0}^\\infty$ [ANS]", "answer_v3": [ "(k+4+1)*x^{k+4+3}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0830", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Series notation", "level": "3", "keywords": [ "calculus", "integral", "series", "power series", "interval of convergence", "radius of convergence" ], "problem_v1": "Consider the series \\sum_{n=n_0}^{\\infty}\\,a_n=(x-8)^{3}+\\frac{(x-8)^{6}}{4\\cdot 2!}+\\frac{(x-8)^{9}}{16\\cdot 3!}+\\frac{(x-8)^{12}}{64\\cdot 4!}+\\cdots Find an expression for $a_n$. $a_n=$ [ANS]\nIn the summation formula $n$ starts at $n=n_0$. What is your starting index $n_0$? $n_0=$ [ANS]\n(Note that because the validity of either of your answers depends on the other, if you enter only one, both will be marked wrong.) (Note that because the validity of either of your answers depends on the other, if you enter only one, both will be marked wrong.)", "answer_v1": [ "(x-8)^{3*n}/[4^{n-1}*n!]", "1" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider the series \\sum_{n=n_0}^{\\infty}\\,a_n=(x-5)^{3}+\\frac{(x-5)^{6}}{2\\cdot 2!}+\\frac{(x-5)^{9}}{4\\cdot 3!}+\\frac{(x-5)^{12}}{8\\cdot 4!}+\\cdots Find an expression for $a_n$. $a_n=$ [ANS]\nIn the summation formula $n$ starts at $n=n_0$. What is your starting index $n_0$? $n_0=$ [ANS]\n(Note that because the validity of either of your answers depends on the other, if you enter only one, both will be marked wrong.) (Note that because the validity of either of your answers depends on the other, if you enter only one, both will be marked wrong.)", "answer_v2": [ "(x-5)^{3*n}/[2^{n-1}*n!]", "1" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider the series \\sum_{n=n_0}^{\\infty}\\,a_n=(x-6)^{3}+\\frac{(x-6)^{6}}{3\\cdot 2!}+\\frac{(x-6)^{9}}{9\\cdot 3!}+\\frac{(x-6)^{12}}{27\\cdot 4!}+\\cdots Find an expression for $a_n$. $a_n=$ [ANS]\nIn the summation formula $n$ starts at $n=n_0$. What is your starting index $n_0$? $n_0=$ [ANS]\n(Note that because the validity of either of your answers depends on the other, if you enter only one, both will be marked wrong.) (Note that because the validity of either of your answers depends on the other, if you enter only one, both will be marked wrong.)", "answer_v3": [ "(x-6)^{3*n}/[3^{n-1}*n!]", "1" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0831", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Partial sums", "level": "3", "keywords": [ "calculus", "derivatives", "slope" ], "problem_v1": "Write $S=\\sum\\limits_{n=8}^\\infty \\frac{1}{n(n-1)}$ as a telescoping series and find its sum. $S_N$=[ANS]\n$S$=[ANS]", "answer_v1": [ "0.142857-1/N", "0.142857" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Write $S=\\sum\\limits_{n=4}^\\infty \\frac{1}{n(n-1)}$ as a telescoping series and find its sum. $S_N$=[ANS]\n$S$=[ANS]", "answer_v2": [ "0.333333-1/N", "0.333333" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Write $S=\\sum\\limits_{n=5}^\\infty \\frac{1}{n(n-1)}$ as a telescoping series and find its sum. $S_N$=[ANS]\n$S$=[ANS]", "answer_v3": [ "0.25-1/N", "0.25" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0832", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Partial sums", "level": "2", "keywords": [ "Series", "Sequence", "Partial Sum", "calculus", "convergent", "divergent", "limit" ], "problem_v1": "Let $r=\\frac{17}{28}.$ For both of the following answer blanks, decide whether the given sequence or series is convergent or divergent. If convergent, enter the limit (for a sequence) or the sum (for a series). If divergent, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, or DIV otherwise.\nA. Consider the sequence $\\lbrace n r^n \\rbrace$.\n$ \\lim_{n \\rightarrow \\infty} n r^n=$ [ANS]\nB. Take my word for it that it can be shown that\n\\sum_{i=1}^{n} i r^i=\\frac{n r^{n+2}-(n+1)r^{n+1}+r}{(1-r)^2}.\nNow consider the series $ \\sum_{n=1}^{\\infty} \\, n r^n$.\n$ \\sum_{n=1}^{\\infty} \\, n r^n=$ [ANS]", "answer_v1": [ "0", "3.93388429752066" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $r=\\frac{5}{20}.$ For both of the following answer blanks, decide whether the given sequence or series is convergent or divergent. If convergent, enter the limit (for a sequence) or the sum (for a series). If divergent, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, or DIV otherwise.\nA. Consider the sequence $\\lbrace n r^n \\rbrace$.\n$ \\lim_{n \\rightarrow \\infty} n r^n=$ [ANS]\nB. Take my word for it that it can be shown that\n\\sum_{i=1}^{n} i r^i=\\frac{n r^{n+2}-(n+1)r^{n+1}+r}{(1-r)^2}.\nNow consider the series $ \\sum_{n=1}^{\\infty} \\, n r^n$.\n$ \\sum_{n=1}^{\\infty} \\, n r^n=$ [ANS]", "answer_v2": [ "0", "0.444444444444444" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $r=\\frac{12}{23}.$ For both of the following answer blanks, decide whether the given sequence or series is convergent or divergent. If convergent, enter the limit (for a sequence) or the sum (for a series). If divergent, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, or DIV otherwise.\nA. Consider the sequence $\\lbrace n r^n \\rbrace$.\n$ \\lim_{n \\rightarrow \\infty} n r^n=$ [ANS]\nB. Take my word for it that it can be shown that\n\\sum_{i=1}^{n} i r^i=\\frac{n r^{n+2}-(n+1)r^{n+1}+r}{(1-r)^2}.\nNow consider the series $ \\sum_{n=1}^{\\infty} \\, n r^n$.\n$ \\sum_{n=1}^{\\infty} \\, n r^n=$ [ANS]", "answer_v3": [ "0", "2.28099173553719" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0833", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Partial sums", "level": "2", "keywords": [ "Series", "Converge", "Sequences", "convergence" ], "problem_v1": "Let $a_n$ be the n th digit after the decimal point in $8 \\pi+6 e$. Evaluate\n\\sum_{n=1}^\\infty a_n (.1)^n. [ANS]", "answer_v1": [ "0.44243219947262" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $a_n$ be the n th digit after the decimal point in $2 \\pi+9 e$. Evaluate\n\\sum_{n=1}^\\infty a_n (.1)^n. [ANS]", "answer_v2": [ "0.747721763310992" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $a_n$ be the n th digit after the decimal point in $4 \\pi+6 e$. Evaluate\n\\sum_{n=1}^\\infty a_n (.1)^n. [ANS]", "answer_v3": [ "0.876061585113444" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0834", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Partial sums", "level": "2", "keywords": [ "Series", "Partial Sum", "Series", "Summation" ], "problem_v1": "Let s_k=\\sum_{n=1}^k n(.1)^n Find $s_{8}.$ $s_{8}=$ [ANS]", "answer_v1": [ "0.12345678" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let s_k=\\sum_{n=1}^k n(.1)^n Find $s_{3}.$ $s_{3}=$ [ANS]", "answer_v2": [ "0.123" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let s_k=\\sum_{n=1}^k n(.1)^n Find $s_{5}.$ $s_{5}=$ [ANS]", "answer_v3": [ "0.12345" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0835", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Partial sums", "level": "2", "keywords": [ "Series", "Converge", "Diverge", "Telescope", "Exponential", "Series", "Summation" ], "problem_v1": "If the following series converges, compute its sum. Otherwise, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, and DIV otherwise.\n\\sum_{n=1}^\\infty(e^{-8 n}-e^{-8 (n+1)}) [ANS]", "answer_v1": [ "0.000335462627902512" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If the following series converges, compute its sum. Otherwise, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, and DIV otherwise.\n\\sum_{n=1}^\\infty(e^{-1 n}-e^{-1 (n+1)}) [ANS]", "answer_v2": [ "0.367879441171442" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If the following series converges, compute its sum. Otherwise, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, and DIV otherwise.\n\\sum_{n=1}^\\infty(e^{-4 n}-e^{-4 (n+1)}) [ANS]", "answer_v3": [ "0.0183156388887342" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0836", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Partial sums", "level": "2", "keywords": [ "Series", "Converge", "Diverge", "Telescope", "Partial Fractions", "Series", "Summation", "Divergent", "Convergent", "calculus", "series" ], "problem_v1": "Determine the sum of the series \\sum _ {n=1} ^ \\infty \\frac{8}{n\\!\\left(n+2\\right)} if possible. (If the series diverges, enter 'infinity', '-infinity' or 'dne' as appropriate.)\nAnswer: [ANS]\n(Hint: try breaking the summands up partial fractions-style.)", "answer_v1": [ "6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the sum of the series \\sum _ {n=1} ^ \\infty \\frac{1}{n\\!\\left(n+2\\right)} if possible. (If the series diverges, enter 'infinity', '-infinity' or 'dne' as appropriate.)\nAnswer: [ANS]\n(Hint: try breaking the summands up partial fractions-style.)", "answer_v2": [ "3/4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the sum of the series \\sum _ {n=1} ^ \\infty \\frac{4}{n\\!\\left(n+2\\right)} if possible. (If the series diverges, enter 'infinity', '-infinity' or 'dne' as appropriate.)\nAnswer: [ANS]\n(Hint: try breaking the summands up partial fractions-style.)", "answer_v3": [ "3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0837", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Partial sums", "level": "3", "keywords": [ "Series", "Converge", "Diverge", "Telescope", "Exponential", "Trigonometry", "Series", "Summation", "calculus", "convergent", "divergent" ], "problem_v1": "Decide whether each of the following series converges. If a given series converges, compute its sum. Otherwise, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, and DIV otherwise. [ANS] 1. \\sum_{n=1}^\\infty\\left(\\sin(10 n)-\\sin(10 (n+1))\\right) [ANS] 2. \\sum_{n=1}^\\infty\\left(e^{8 n}-e^{8 (n+1)}\\right) [ANS] 3. \\sum_{n=1}^\\infty\\left(\\sin\\left(\\frac{10}{n}\\right)-\\sin\\left(\\frac{10}{n+1}\\right)\\right)", "answer_v1": [ "DIV", "MINF", "-0.54402111088937" ], "answer_type_v1": [ "MCS", "MCS", "NV" ], "options_v1": [ [ "DIV", "INF", "MINF" ], [ "DIV", "INF", "MINF" ], [] ], "problem_v2": "Decide whether each of the following series converges. If a given series converges, compute its sum. Otherwise, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, and DIV otherwise. [ANS] 1. \\sum_{n=1}^\\infty\\left(\\sin\\left(\\frac{-6}{n}\\right)-\\sin\\left(\\frac{-6}{n+1}\\right)\\right) [ANS] 2. \\sum_{n=1}^\\infty\\left(\\sin(-6 n)-\\sin(-6 (n+1))\\right) [ANS] 3. \\sum_{n=1}^\\infty\\left(e^{12 n}-e^{12 (n+1)}\\right)", "answer_v2": [ "0.279415498198926", "DIV", "MINF" ], "answer_type_v2": [ "NV", "MCS", "MCS" ], "options_v2": [ [], [ "DIV", "INF", "MINF" ], [ "DIV", "INF", "MINF" ] ], "problem_v3": "Decide whether each of the following series converges. If a given series converges, compute its sum. Otherwise, enter INF if it diverges to infinity, MINF if it diverges to minus infinity, and DIV otherwise. [ANS] 1. \\sum_{n=1}^\\infty\\left(\\sin\\left(\\frac{-2}{n}\\right)-\\sin\\left(\\frac{-2}{n+1}\\right)\\right) [ANS] 2. \\sum_{n=1}^\\infty\\left(e^{9 n}-e^{9 (n+1)}\\right) [ANS] 3. \\sum_{n=1}^\\infty\\left(\\sin(-2 n)-\\sin(-2 (n+1))\\right)", "answer_v3": [ "-0.909297426825682", "MINF", "DIV" ], "answer_type_v3": [ "NV", "MCS", "MCS" ], "options_v3": [ [], [ "DIV", "INF", "MINF" ], [ "DIV", "INF", "MINF" ] ] }, { "id": "Calculus_-_single_variable_0838", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Partial sums", "level": "2", "keywords": [ "Series", "Converge", "Diverge", "Telescope", "Trigonometry", "Series", "Summation", "calculus" ], "problem_v1": "Determine the sum of the following series.\n\\sum_{n=1}^\\infty \\left(\\sin \\left(\\frac{5}{n} \\right)-\\sin \\left(\\frac{5}{n+1}\\right)\\right) [ANS]", "answer_v1": [ "-0.958924274663138" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the sum of the following series.\n\\sum_{n=1}^\\infty \\left(\\sin \\left(\\frac{-9}{n} \\right)-\\sin \\left(\\frac{-9}{n+1}\\right)\\right) [ANS]", "answer_v2": [ "-0.412118485241757" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the sum of the following series.\n\\sum_{n=1}^\\infty \\left(\\sin \\left(\\frac{-4}{n} \\right)-\\sin \\left(\\frac{-4}{n+1}\\right)\\right) [ANS]", "answer_v3": [ "0.756802495307928" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0839", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Partial sums", "level": "3", "keywords": [], "problem_v1": "Determine whether the series converges, and if so find its sum. $ \\sum_{k=1}^{\\infty} \\frac{1}{25k^{2}-5k-6}=$ [ANS]\n(Enter DNE if the sum does not exist.)", "answer_v1": [ "1/5/(5-3)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine whether the series converges, and if so find its sum. $ \\sum_{k=1}^{\\infty} \\frac{1}{4k^{2}-1}=$ [ANS]\n(Enter DNE if the sum does not exist.)", "answer_v2": [ "1/2/(2-1)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine whether the series converges, and if so find its sum. $ \\sum_{k=1}^{\\infty} \\frac{1}{9k^{2}-3k-2}=$ [ANS]\n(Enter DNE if the sum does not exist.)", "answer_v3": [ "1/3/(3-2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0840", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Partial sums", "level": "3", "keywords": [], "problem_v1": "Consider the series ${\\sum_{n=0}^{\\infty} {5e^{-n}}}$.\nThe general formula for the sum of the first $n$ terms is $S_n=$ [ANS]. Your answer should be in terms of $n$.\nThe sum of a series is defined as the limit of the sequence of partial sums, which means ${\\sum_{n=0}^{\\infty} {5e^{-n}}=\\lim_{n \\to \\infty} \\bigg(}$ [ANS] ${\\bigg)=}$ [ANS].\nSelect all true statements (there may be more than one correct answer): [ANS] A. The series converges. B. The series is a p-series. C. The series is a telescoping series (i.e., it is like a collapsible telescope). D. The series is a geometric series.", "answer_v1": [ "5*[1-(1/e)^{n+1}]/(1-1/e)", "5*[1-(1/e)^{n+1}]/(1-1/e)", "7.90988", "AD" ], "answer_type_v1": [ "EX", "EX", "NV", "MCM" ], "options_v1": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v2": "Consider the series ${\\sum_{n=0}^{\\infty} {2e^{-n}}}$.\nThe general formula for the sum of the first $n$ terms is $S_n=$ [ANS]. Your answer should be in terms of $n$.\nThe sum of a series is defined as the limit of the sequence of partial sums, which means ${\\sum_{n=0}^{\\infty} {2e^{-n}}=\\lim_{n \\to \\infty} \\bigg(}$ [ANS] ${\\bigg)=}$ [ANS].\nSelect all true statements (there may be more than one correct answer): [ANS] A. The series is a p-series. B. The series is a telescoping series (i.e., it is like a collapsible telescope). C. The series converges. D. The series is a geometric series.", "answer_v2": [ "2*[1-(1/e)^{n+1}]/(1-1/e)", "2*[1-(1/e)^{n+1}]/(1-1/e)", "3.16395", "CD" ], "answer_type_v2": [ "EX", "EX", "NV", "MCM" ], "options_v2": [ [], [], [], [ "A", "B", "C", "D" ] ], "problem_v3": "Consider the series ${\\sum_{n=0}^{\\infty} {3e^{-n}}}$.\nThe general formula for the sum of the first $n$ terms is $S_n=$ [ANS]. Your answer should be in terms of $n$.\nThe sum of a series is defined as the limit of the sequence of partial sums, which means ${\\sum_{n=0}^{\\infty} {3e^{-n}}=\\lim_{n \\to \\infty} \\bigg(}$ [ANS] ${\\bigg)=}$ [ANS].\nSelect all true statements (there may be more than one correct answer): [ANS] A. The series is a geometric series. B. The series is a p-series. C. The series converges. D. The series is a telescoping series (i.e., it is like a collapsible telescope).", "answer_v3": [ "3*[1-(1/e)^{n+1}]/(1-1/e)", "3*[1-(1/e)^{n+1}]/(1-1/e)", "4.74593", "AC" ], "answer_type_v3": [ "EX", "EX", "NV", "MCM" ], "options_v3": [ [], [], [], [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_single_variable_0841", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor polynomials", "level": "3", "keywords": [ "calculus", "taylor series", "taylor polynomials", "partial sums" ], "problem_v1": "Compute $T_2(x)$ at $x=0.8$ for $y=e^{x}$ and use a calculator to compute the error $|e^{x}-T_2(x)|$ at $x=0.3$. $T_2(x)=$ [ANS]\n$|e^{x}-T_2(x)|$=[ANS]", "answer_v1": [ "e^0.8+e^0.8*(x-0.8)+e^0.8/2*(x-0.8)^2", "0.0411043" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Compute $T_2(x)$ at $x=0$ for $y=e^{x}$ and use a calculator to compute the error $|e^{x}-T_2(x)|$ at $x=1.3$. $T_2(x)=$ [ANS]\n$|e^{x}-T_2(x)|$=[ANS]", "answer_v2": [ "1+x+0.5*x^2", "0.524297" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Compute $T_2(x)$ at $x=0.3$ for $y=e^{x}$ and use a calculator to compute the error $|e^{x}-T_2(x)|$ at $x=0.3$. $T_2(x)=$ [ANS]\n$|e^{x}-T_2(x)|$=[ANS]", "answer_v3": [ "e^0.3+e^0.3*(x-0.3)+e^0.3/2*(x-0.3)^2", "0" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0842", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor polynomials", "level": "2", "keywords": [ "calculus", "taylor series", "taylor polynomials", "partial sums" ], "problem_v1": "Calculate the Taylor polynomials $T_2(x)$ and $T_3(x)$ centered at $x=8$ for $f(x)=e^{-x}+e^{-2x}$.\n$T_2(x)$ must be of the form A+B(x-8)+C(x-8)^2 where $A$=: [ANS]\n$B$=: [ANS] and $C$=: [ANS]\n$T_3(x)$ must be of the form D+E(x-8)+F(x-8)^2+G(x-8)^3 where $D$=: [ANS]\n$E$=: [ANS]\n$F$=: [ANS] and $G$=: [ANS]", "answer_v1": [ "e^{-8}+e^{-16}", "-[e^{-8}+2*e^{-16}]", "[e^{-8}+4*e^{-16}]/2", "e^{-8}+e^{-16}", "-[e^{-8}+2*e^{-16}]", "[e^{-8}+4*e^{-16}]/2", "-[e^{-8}+8*e^{-16}]/6" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Calculate the Taylor polynomials $T_2(x)$ and $T_3(x)$ centered at $x=0$ for $f(x)=e^{-x}+e^{-2x}$.\n$T_2(x)$ must be of the form A+B(x-0)+C(x-0)^2 where $A$=: [ANS]\n$B$=: [ANS] and $C$=: [ANS]\n$T_3(x)$ must be of the form D+E(x-0)+F(x-0)^2+G(x-0)^3 where $D$=: [ANS]\n$E$=: [ANS]\n$F$=: [ANS] and $G$=: [ANS]", "answer_v2": [ "2", "-3", "2.5", "2", "-3", "2.5", "-1.5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Calculate the Taylor polynomials $T_2(x)$ and $T_3(x)$ centered at $x=3$ for $f(x)=e^{-x}+e^{-2x}$.\n$T_2(x)$ must be of the form A+B(x-3)+C(x-3)^2 where $A$=: [ANS]\n$B$=: [ANS] and $C$=: [ANS]\n$T_3(x)$ must be of the form D+E(x-3)+F(x-3)^2+G(x-3)^3 where $D$=: [ANS]\n$E$=: [ANS]\n$F$=: [ANS] and $G$=: [ANS]", "answer_v3": [ "e^{-3}+e^{-6}", "-[e^{-3}+2*e^{-6}]", "[e^{-3}+4*e^{-6}]/2", "e^{-3}+e^{-6}", "-[e^{-3}+2*e^{-6}]", "[e^{-3}+4*e^{-6}]/2", "-[e^{-3}+8*e^{-6}]/6" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0843", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor polynomials", "level": "4", "keywords": [ "Power Series", "Taylor Series" ], "problem_v1": "Find the degree 3 Taylor polynomial $T_3(x)$ centered at $a=6$ of the function $f(x)=(5x-14)^{5/4}$.\n$T_3(x)=$ [ANS]\n[ANS] The function $f(x)=(5x-14)^{5/4}$ equals its third degree Taylor polynomial $T_3(x)$ centered at $a=6$. Hint: Graph both of them. If it looks like they are equal, then do the algebra.", "answer_v1": [ "2^5 + 5 * 2 * 5 / 4 * (x-6) + 0.9765625/2 * (x-6)^2 + -0.2288818359375/6 * (x-6)^3", "FALSE" ], "answer_type_v1": [ "EX", "MCS" ], "options_v1": [ [], [ "True", "False", "Cannot be determined" ] ], "problem_v2": "Find the degree 3 Taylor polynomial $T_3(x)$ centered at $a=2$ of the function $f(x)=(-7x+78)^{4/3}$.\n$T_3(x)=$ [ANS]\n[ANS] The function $f(x)=(-7x+78)^{4/3}$ equals its third degree Taylor polynomial $T_3(x)$ centered at $a=2$. Hint: Graph both of them. If it looks like they are equal, then do the algebra.", "answer_v2": [ "4^4 + 4 * 4 * -7 / 3 * (x-2) + 1.36111111111111/2 * (x-2)^2 + 0.0992476851851852/6 * (x-2)^3", "FALSE" ], "answer_type_v2": [ "EX", "MCS" ], "options_v2": [ [], [ "True", "False", "Cannot be determined" ] ], "problem_v3": "Find the degree 3 Taylor polynomial $T_3(x)$ centered at $a=2$ of the function $f(x)=(-5x+18)^{4/3}$.\n$T_3(x)=$ [ANS]\n[ANS] The function $f(x)=(-5x+18)^{4/3}$ equals its third degree Taylor polynomial $T_3(x)$ centered at $a=2$. Hint: Graph both of them. If it looks like they are equal, then do the algebra.", "answer_v3": [ "2^4 + 4 * 2 * -5 / 3 * (x-2) + 2.77777777777778/2 * (x-2)^2 + 1.15740740740741/6 * (x-2)^3", "FALSE" ], "answer_type_v3": [ "EX", "MCS" ], "options_v3": [ [], [ "True", "False", "Cannot be determined" ] ] }, { "id": "Calculus_-_single_variable_0844", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor polynomials", "level": "2", "keywords": [ "Series", "Taylor", "calculus", "taylor polynomial", "arctan", "Taylor Series" ], "problem_v1": "Find $T_{4}(x)$: the Taylor polynomial of degree 4 of the function $f(x)=\\arctan(12x)$ at $a=0$. (You need to enter a function.) $T_{4}(x)=$ [ANS]", "answer_v1": [ "(12*x-(12^3)*x^3/3)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find $T_{4}(x)$: the Taylor polynomial of degree 4 of the function $f(x)=\\arctan(3x)$ at $a=0$. (You need to enter a function.) $T_{4}(x)=$ [ANS]", "answer_v2": [ "(3*x-(3^3)*x^3/3)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find $T_{4}(x)$: the Taylor polynomial of degree 4 of the function $f(x)=\\arctan(6x)$ at $a=0$. (You need to enter a function.) $T_{4}(x)=$ [ANS]", "answer_v3": [ "(6*x-(6^3)*x^3/3)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0845", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor polynomials", "level": "3", "keywords": [ "calculus", "taylor series", "error", "polynomials" ], "problem_v1": "Imagine that you need to compute $e^{0.4}$ but you have no calculator or other aid to enable you to compute it exactly, only paper and pencil. You decide to use a third-degree Taylor polynomial expanded around $x=0$. Use the fact that $e^{0.4} 2$, $ \\frac{\\ln (n)}{n^2} > \\frac{1}{n^2}$, and the series $ \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{\\ln (n)}{n^2}$ converges. [ANS] 2. For all $n > 1$, $ \\frac{n}{6-n^3} < \\frac{1}{n^2}$, and the series $ \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{n}{6-n^3}$ converges. [ANS] 3. For all $n > 2$, $ \\frac{\\sqrt{n+1}}{n} > \\frac{1}{n}$, and the series $ \\sum \\frac{1}{n}$ diverges, so by the Comparison Test, the series $ \\sum \\frac{\\sqrt{n+1}}{n}$ diverges. [ANS] 4. For all $n > 2$, $ \\frac{n}{n^3-6} < \\frac{2}{n^2}$, and the series $ 2 \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{n}{n^3-6}$ converges. [ANS] 5. For all $n > 1$, $ \\frac{\\sin^2 (n)}{n^2} < \\frac{1}{n^2}$, and the series $ \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{\\sin^2 (n)}{n^2}$ converges. [ANS] 6. For all $n > 2$, $ \\frac{1}{n^2-5} < \\frac{1}{n^2}$, and the series $ \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{1}{n^2-5}$ converges.", "answer_v1": [ "I", "I", "C", "C", "C", "I" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "C", "I" ], [ "C", "I" ], [ "C", "I" ], [ "C", "I" ], [ "C", "I" ], [ "C", "I" ] ], "problem_v2": "Each of the following statements is an attempt to show that a given series is convergent or divergent not using the Comparison Test (NOT the Limit Comparison Test.) For each statement, enter C (for \"correct\") if the argument is valid, or enter I (for \"incorrect\") if any part of the argument is flawed. (Note: if the conclusion is true but the argument that led to it was wrong, you must enter I.) [ANS] 1. For all $n > 1$, $ \\frac{n}{1-n^3} < \\frac{1}{n^2}$, and the series $ \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{n}{1-n^3}$ converges. [ANS] 2. For all $n > 1$, $ \\frac{\\arctan (n)}{n^3} < \\frac{\\pi}{2n^3}$, and the series $ \\frac{\\pi}{2} \\sum \\frac{1}{n^3}$ converges, so by the Comparison Test, the series $ \\sum \\frac{\\arctan (n)}{n^3}$ converges. [ANS] 3. For all $n > 2$, $ \\frac{\\sqrt{n+1}}{n} > \\frac{1}{n}$, and the series $ \\sum \\frac{1}{n}$ diverges, so by the Comparison Test, the series $ \\sum \\frac{\\sqrt{n+1}}{n}$ diverges. [ANS] 4. For all $n > 1$, $ \\frac{1}{n \\ln (n)} < \\frac{2}{n}$, and the series $ 2 \\sum \\frac{1}{n}$ diverges, so by the Comparison Test, the series $ \\sum \\frac{1}{n \\ln (n)}$ diverges. [ANS] 5. For all $n > 2$, $ \\frac{n}{n^3-2} < \\frac{2}{n^2}$, and the series $ 2 \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{n}{n^3-2}$ converges. [ANS] 6. For all $n > 2$, $ \\frac{\\ln (n)}{n^2} > \\frac{1}{n^2}$, and the series $ \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{\\ln (n)}{n^2}$ converges.", "answer_v2": [ "I", "C", "C", "I", "C", "I" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "C", "I" ], [ "C", "I" ], [ "C", "I" ], [ "C", "I" ], [ "C", "I" ], [ "C", "I" ] ], "problem_v3": "Each of the following statements is an attempt to show that a given series is convergent or divergent not using the Comparison Test (NOT the Limit Comparison Test.) For each statement, enter C (for \"correct\") if the argument is valid, or enter I (for \"incorrect\") if any part of the argument is flawed. (Note: if the conclusion is true but the argument that led to it was wrong, you must enter I.) [ANS] 1. For all $n > 2$, $ \\frac{n}{n^3-3} < \\frac{2}{n^2}$, and the series $ 2 \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{n}{n^3-3}$ converges. [ANS] 2. For all $n > 2$, $ \\frac{\\sqrt{n+1}}{n} > \\frac{1}{n}$, and the series $ \\sum \\frac{1}{n}$ diverges, so by the Comparison Test, the series $ \\sum \\frac{\\sqrt{n+1}}{n}$ diverges. [ANS] 3. For all $n > 2$, $ \\frac{1}{n^2-5} < \\frac{1}{n^2}$, and the series $ \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{1}{n^2-5}$ converges. [ANS] 4. For all $n > 1$, $ \\frac{\\arctan (n)}{n^3} < \\frac{\\pi}{2n^3}$, and the series $ \\frac{\\pi}{2} \\sum \\frac{1}{n^3}$ converges, so by the Comparison Test, the series $ \\sum \\frac{\\arctan (n)}{n^3}$ converges. [ANS] 5. For all $n > 1$, $ \\frac{\\sin^2 (n)}{n^2} < \\frac{1}{n^2}$, and the series $ \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{\\sin^2 (n)}{n^2}$ converges. [ANS] 6. For all $n > 2$, $ \\frac{\\ln (n)}{n^2} > \\frac{1}{n^2}$, and the series $ \\sum \\frac{1}{n^2}$ converges, so by the Comparison Test, the series $ \\sum \\frac{\\ln (n)}{n^2}$ converges.", "answer_v3": [ "C", "C", "I", "C", "C", "I" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "C", "I" ], [ "C", "I" ], [ "C", "I" ], [ "C", "I" ], [ "C", "I" ], [ "C", "I" ] ] }, { "id": "Calculus_-_single_variable_0898", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Comparison tests", "level": "3", "keywords": [ "calculus", "integral", "series", "convergence", "ratio test", "comparison test", "limit comparison test", "alternating series" ], "problem_v1": "For each of the following, carefully determine whether the series converges or not.\n(a) $\\sum\\limits_{n=1}^{\\infty}\\,{6^n\\over(2n)!}$ [ANS] A. converges B. diverges\n(b) $\\sum\\limits_{n=1}^{\\infty}\\,{(-1)^{n-1} n^5\\over 6^n}$ [ANS] A. converges B. diverges\n(c) $\\sum\\limits_{n=2}^{\\infty}\\,{6\\over\\ln n^5}$ [ANS] A. converges B. diverges\n(d) $\\sum\\limits_{n=1}^{\\infty}\\,{n(n+3)\\over\\sqrt{n^3+6 n^2}}$ [ANS] A. converges B. diverges", "answer_v1": [ "A", "A", "B", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v2": "For each of the following, carefully determine whether the series converges or not.\n(a) $\\sum\\limits_{n=1}^{\\infty}\\,{(n-1)!\\over n^2}$ [ANS] A. converges B. diverges\n(b) $\\sum\\limits_{n=1}^{\\infty}\\,{n(n+5)\\over\\sqrt{n^3+3 n^2}}$ [ANS] A. converges B. diverges\n(c) $\\sum\\limits_{n=1}^{\\infty}\\,{(-1)^{n-1} 3^n\\over n^2}$ [ANS] A. converges B. diverges\n(d) $\\sum\\limits_{n=1}^{\\infty}\\,{3+\\sin(n)\\over n^2+5}$ [ANS] A. converges B. diverges", "answer_v2": [ "B", "B", "B", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ], "problem_v3": "For each of the following, carefully determine whether the series converges or not.\n(a) $\\sum\\limits_{n=1}^{\\infty}\\,{4+\\sin(n)\\over n^3+4}$ [ANS] A. converges B. diverges\n(b) $\\sum\\limits_{n=1}^{\\infty}\\,{(2n)!\\over 4^n}$ [ANS] A. converges B. diverges\n(c) $\\sum\\limits_{n=1}^{\\infty}\\,{(-1)^{n-1} n^3\\over 4^n}$ [ANS] A. converges B. diverges\n(d) $\\sum\\limits_{n=1}^{\\infty}\\,{n(n+4)\\over\\sqrt{n^3+4 n^2}}$ [ANS] A. converges B. diverges", "answer_v3": [ "A", "B", "A", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B" ], [ "A", "B" ], [ "A", "B" ], [ "A", "B" ] ] }, { "id": "Calculus_-_single_variable_0899", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Comparison tests", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Match each of the following with the correct statement. C stands for Convergent, D stands for Divergent. [ANS] 1. $ \\sum_{n=1}^\\infty \\frac{1}{5+\\root 6 \\of {n^{4}}}$ [ANS] 2. $ \\sum_{n=1}^\\infty \\frac{6+4 ^n}{7+5 ^n}$ [ANS] 3. $ \\sum_{n=1}^\\infty \\frac{\\ln(n)}{7 n}$ [ANS] 4. $ \\sum_{n=1}^\\infty \\frac{6}{n(n+5)}$ [ANS] 5. $ \\sum_{n=2}^\\infty \\frac{6}{n^{7}-16}$", "answer_v1": [ "D", "C", "D", "C", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "C", "D" ], [ "C", "D" ], [ "C", "D" ], [ "C", "D" ], [ "C", "D" ] ], "problem_v2": "Match each of the following with the correct statement. C stands for Convergent, D stands for Divergent. [ANS] 1. $ \\sum_{n=1}^\\infty \\frac{\\ln(n)}{3 n}$ [ANS] 2. $ \\sum_{n=3}^\\infty \\frac{2}{n^{3}-16}$ [ANS] 3. $ \\sum_{n=1}^\\infty \\frac{2}{n(n+7)}$ [ANS] 4. $ \\sum_{n=1}^\\infty \\frac{1}{2+\\root 4 \\of {n^{7}}}$ [ANS] 5. $ \\sum_{n=1}^\\infty \\frac{4+6 ^n}{7+1 ^n}$", "answer_v2": [ "D", "C", "C", "C", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "C", "D" ], [ "C", "D" ], [ "C", "D" ], [ "C", "D" ], [ "C", "D" ] ], "problem_v3": "Match each of the following with the correct statement. C stands for Convergent, D stands for Divergent. [ANS] 1. $ \\sum_{n=1}^\\infty \\frac{1}{3+\\root 5 \\of {n^{4}}}$ [ANS] 2. $ \\sum_{n=1}^\\infty \\frac{3}{n(n+5)}$ [ANS] 3. $ \\sum_{n=1}^\\infty \\frac{10+9 ^n}{3+3 ^n}$ [ANS] 4. $ \\sum_{n=2}^\\infty \\frac{3}{n^{9}-25}$ [ANS] 5. $ \\sum_{n=1}^\\infty \\frac{\\ln(n)}{9 n}$", "answer_v3": [ "D", "C", "D", "C", "D" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "C", "D" ], [ "C", "D" ], [ "C", "D" ], [ "C", "D" ], [ "C", "D" ] ] }, { "id": "Calculus_-_single_variable_0900", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Comparison tests", "level": "3", "keywords": [ "series", "limit comparison test" ], "problem_v1": "Use the limit comparison test to determine whether $ \\sum_{n=14}^{\\infty} a_n=\\sum_{n=14}^{\\infty} \\frac{8 n^3-6 n^2+14}{4+4 n^4}$ converges or diverges.\n(a) Choose a series $ \\sum_{n=14}^\\infty b_n$ with terms of the form $ b_n=\\frac{1}{n^p}$ and apply the limit comparison test. Write your answer as a fully simplified fraction. For $n \\geq 14$, $\\begin{array}{cccc}\\hline & \\lim_{n \\to \\infty} \\frac{a_{n}}{b_{n}}=\\lim_{n \\to \\infty} & & [ANS] [ANS] \\\\ \\hline \\end{array}$\n(b) Evaluate the limit in the previous part. Enter $\\infty$ as infinity and $-\\infty$ as-infinity.-infinity. If the limit does not exist, enter DNE. DNE. $ \\lim_{n\\to\\infty} \\frac{a_{n}}{b_{n}} \\,$=[ANS]\n(c) By the limit comparison test, does the series converge, diverge, or is the test inconclusive? [ANS]", "answer_v1": [ "8*n^4-6*n^3+14*n", "4*n^4+4", "2", "Diverges" ], "answer_type_v1": [ "EX", "EX", "NV", "MCS" ], "options_v1": [ [], [], [], [ "Converges", "Diverges", "Inconclusive" ] ], "problem_v2": "Use the limit comparison test to determine whether $ \\sum_{n=7}^{\\infty} a_n=\\sum_{n=7}^{\\infty} \\frac{5 n^3-9 n^2+7}{9+3 n^4}$ converges or diverges.\n(a) Choose a series $ \\sum_{n=7}^\\infty b_n$ with terms of the form $ b_n=\\frac{1}{n^p}$ and apply the limit comparison test. Write your answer as a fully simplified fraction. For $n \\geq 7$, $\\begin{array}{cccc}\\hline & \\lim_{n \\to \\infty} \\frac{a_{n}}{b_{n}}=\\lim_{n \\to \\infty} & & [ANS] [ANS] \\\\ \\hline \\end{array}$\n(b) Evaluate the limit in the previous part. Enter $\\infty$ as infinity and $-\\infty$ as-infinity.-infinity. If the limit does not exist, enter DNE. DNE. $ \\lim_{n\\to\\infty} \\frac{a_{n}}{b_{n}} \\,$=[ANS]\n(c) By the limit comparison test, does the series converge, diverge, or is the test inconclusive? [ANS]", "answer_v2": [ "5*n^4-9*n^3+7*n", "3*n^4+9", "1.66667", "Diverges" ], "answer_type_v2": [ "EX", "EX", "NV", "MCS" ], "options_v2": [ [], [], [], [ "Converges", "Diverges", "Inconclusive" ] ], "problem_v3": "Use the limit comparison test to determine whether $ \\sum_{n=9}^{\\infty} a_n=\\sum_{n=9}^{\\infty} \\frac{6 n^3-6 n^2+9}{3+3 n^4}$ converges or diverges.\n(a) Choose a series $ \\sum_{n=9}^\\infty b_n$ with terms of the form $ b_n=\\frac{1}{n^p}$ and apply the limit comparison test. Write your answer as a fully simplified fraction. For $n \\geq 9$, $\\begin{array}{cccc}\\hline & \\lim_{n \\to \\infty} \\frac{a_{n}}{b_{n}}=\\lim_{n \\to \\infty} & & [ANS] [ANS] \\\\ \\hline \\end{array}$\n(b) Evaluate the limit in the previous part. Enter $\\infty$ as infinity and $-\\infty$ as-infinity.-infinity. If the limit does not exist, enter DNE. DNE. $ \\lim_{n\\to\\infty} \\frac{a_{n}}{b_{n}} \\,$=[ANS]\n(c) By the limit comparison test, does the series converge, diverge, or is the test inconclusive? [ANS]", "answer_v3": [ "6*n^4-6*n^3+9*n", "3*n^4+3", "2", "Diverges" ], "answer_type_v3": [ "EX", "EX", "NV", "MCS" ], "options_v3": [ [], [], [], [ "Converges", "Diverges", "Inconclusive" ] ] }, { "id": "Calculus_-_single_variable_0901", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Integral test", "level": "2", "keywords": [ "calculus", "infinite series", "series", "converge", "convergence", "comparison test", "integral test", "limit" ], "problem_v1": "Use the Integral Test to determine whether the infinite series is convergent. \\sum_{n=5}^{\\infty} 12 ne^{-n^2} Fill in the corresponding integrand and the value of the improper integral. Enter inf for $\\infty$,-inf for $-\\infty$, and DNE if the limit does not exist. Compare with $\\int_{5}^{\\infty}$ [ANS] $dx$=[ANS]\nBy the Integral Test, the infinite series $ \\sum_{n=5}^{\\infty} 12 ne^{-n^2}$ [ANS] A. converges B. diverges", "answer_v1": [ "12*x*e^{-x^2}", "8.33277E-11", "A" ], "answer_type_v1": [ "EX", "NV", "MCS" ], "options_v1": [ [], [], [ "A", "B" ] ], "problem_v2": "Use the Integral Test to determine whether the infinite series is convergent. \\sum_{n=1}^{\\infty} 20 ne^{-n^2} Fill in the corresponding integrand and the value of the improper integral. Enter inf for $\\infty$,-inf for $-\\infty$, and DNE if the limit does not exist. Compare with $\\int_{1}^{\\infty}$ [ANS] $dx$=[ANS]\nBy the Integral Test, the infinite series $ \\sum_{n=1}^{\\infty} 20 ne^{-n^2}$ [ANS] A. converges B. diverges", "answer_v2": [ "20*x*e^{-x^2}", "3.67879", "A" ], "answer_type_v2": [ "EX", "NV", "MCS" ], "options_v2": [ [], [], [ "A", "B" ] ], "problem_v3": "Use the Integral Test to determine whether the infinite series is convergent. \\sum_{n=2}^{\\infty} 14 ne^{-n^2} Fill in the corresponding integrand and the value of the improper integral. Enter inf for $\\infty$,-inf for $-\\infty$, and DNE if the limit does not exist. Compare with $\\int_{2}^{\\infty}$ [ANS] $dx$=[ANS]\nBy the Integral Test, the infinite series $ \\sum_{n=2}^{\\infty} 14 ne^{-n^2}$ [ANS] A. converges B. diverges", "answer_v3": [ "14*x*e^{-x^2}", "0.128209", "A" ], "answer_type_v3": [ "EX", "NV", "MCS" ], "options_v3": [ [], [], [ "A", "B" ] ] }, { "id": "Calculus_-_single_variable_0902", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Integral test", "level": "3", "keywords": [ "series", "divergent", "convergent", "integral test" ], "problem_v1": "A. Suppose that f(x) is a function that is positive and decreasing. Recall that by the integral test: $ \\int_p^{\\infty} f(x)dx \\leq \\sum_{n=p} ^{\\infty} f(n).$ Recall that $ e=\\sum_{n=0}^{\\infty} \\frac{1}{n!}.$ Suppose that for each positive integer k, $f(k)=\\frac{1}{k!}$. Find an upper bound B for $ \\int_2^{\\infty} f(x)dx.$ B=[ANS]\nB. A function is given by h(k)= \\int_0^{\\infty} x^k e^{-x} dx.. Its values may be found in tables. Make the change of variables $y=x \\ln(4)$ to express $ I=\\int_0^{\\infty} x^{5} 4^{-x}dx$ as a constant C times $h(5).$ Find C. C=[ANS]\nC. Let $g(x)=x^5 4 ^{-x}.$ Find the smallest number M such that the function g is decreasing for all $x > M.$ C. M=[ANS]\nD. Does $ \\sum_{n=1}^{\\infty} n^{5}4 ^{-n}$ converge or diverge? Answer with one letter, C or D. [ANS]", "answer_v1": [ "0.718281828459045", "0.140885812113652", "3.60673760222241", "C" ], "answer_type_v1": [ "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A. Suppose that f(x) is a function that is positive and decreasing. Recall that by the integral test: $ \\int_p^{\\infty} f(x)dx \\leq \\sum_{n=p} ^{\\infty} f(n).$ Recall that $ e=\\sum_{n=0}^{\\infty} \\frac{1}{n!}.$ Suppose that for each positive integer k, $f(k)=\\frac{1}{k!}$. Find an upper bound B for $ \\int_2^{\\infty} f(x)dx.$ B=[ANS]\nB. A function is given by h(k)= \\int_0^{\\infty} x^k e^{-x} dx.. Its values may be found in tables. Make the change of variables $y=x \\ln(5)$ to express $ I=\\int_0^{\\infty} x^{2} 5^{-x}dx$ as a constant C times $h(2).$ Find C. C=[ANS]\nC. Let $g(x)=x^2 5 ^{-x}.$ Find the smallest number M such that the function g is decreasing for all $x > M.$ C. M=[ANS]\nD. Does $ \\sum_{n=1}^{\\infty} n^{2}5 ^{-n}$ converge or diverge? Answer with one letter, C or D. [ANS]", "answer_v2": [ "0.718281828459045", "0.239870763526583", "1.24266986911922", "C" ], "answer_type_v2": [ "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A. Suppose that f(x) is a function that is positive and decreasing. Recall that by the integral test: $ \\int_p^{\\infty} f(x)dx \\leq \\sum_{n=p} ^{\\infty} f(n).$ Recall that $ e=\\sum_{n=0}^{\\infty} \\frac{1}{n!}.$ Suppose that for each positive integer k, $f(k)=\\frac{1}{k!}$. Find an upper bound B for $ \\int_2^{\\infty} f(x)dx.$ B=[ANS]\nB. A function is given by h(k)= \\int_0^{\\infty} x^k e^{-x} dx.. Its values may be found in tables. Make the change of variables $y=x \\ln(4)$ to express $ I=\\int_0^{\\infty} x^{3} 4^{-x}dx$ as a constant C times $h(3).$ Find C. C=[ANS]\nC. Let $g(x)=x^3 4 ^{-x}.$ Find the smallest number M such that the function g is decreasing for all $x > M.$ C. M=[ANS]\nD. Does $ \\sum_{n=1}^{\\infty} n^{3}4 ^{-n}$ converge or diverge? Answer with one letter, C or D. [ANS]", "answer_v3": [ "0.718281828459045", "0.27075605219327", "2.16404256133345", "C" ], "answer_type_v3": [ "NV", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0903", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Integral test", "level": "2", "keywords": [ "Series", "Converge", "Diverge", "Integral Test", "calculus", "convergence", "integral test", "convergent", "divergent", "Series", "Integral test", "Convergence" ], "problem_v1": "Find the value of the integral \\int_2^{\\infty} \\frac{dx}{8x (\\ln(6x))^2}. [ANS]\nDetermine whether the series\n\\sum_{n=2}^\\infty \\frac{1}{8 n (\\ln(6 n))^2} is convergent. Enter C if the series is convergent, D if the series is divergent. [ANS]", "answer_v1": [ "0.0503037005477306", "C" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find the value of the integral \\int_2^{\\infty} \\frac{dx}{2x (\\ln(9x))^2}. [ANS]\nDetermine whether the series\n\\sum_{n=2}^\\infty \\frac{1}{2 n (\\ln(9 n))^2} is convergent. Enter C if the series is convergent, D if the series is divergent. [ANS]", "answer_v2": [ "0.172988128130597", "C" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find the value of the integral \\int_2^{\\infty} \\frac{dx}{4x (\\ln(6x))^2}. [ANS]\nDetermine whether the series\n\\sum_{n=2}^\\infty \\frac{1}{4 n (\\ln(6 n))^2} is convergent. Enter C if the series is convergent, D if the series is divergent. [ANS]", "answer_v3": [ "0.100607401095461", "C" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0904", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Integral test", "level": "2", "keywords": [ "calculus", "integral", "series", "convergence", "integral test" ], "problem_v1": "For each of the following series, indicate whether the integral test can be used to determine its convergence or not, and if not, why. A. $\\sum\\limits_{n=1}^{\\infty} (n-6)^7\\,6^{-n}$ Can the integral test be used to test convergence? [ANS] A. no, because the terms in the series do not decrease in magnitude B. no, because the terms in the series are not all positive for $n\\ge c$, for some $c>0$ C. no, because the series is not a geometric series D. no, because the terms in the series are not recursively defined E. no, because the function $f(x)$ (where $a_n=f(n)$) is not defined for all $x$ F. yes\nB. $\\sum\\limits_{n=1}^{\\infty} \\frac{6}{n(|\\cos(n)|+2)}$ Can the integral test be used to test convergence? [ANS] A. no, because the terms in the series do not decrease in magnitude B. no, because the terms in the series are not all positive for $n\\ge c$, for some $c>0$ C. no, because the series is not a geometric series D. no, because the terms in the series are not recursively defined E. no, because the function $f(x)$ (where $a_n=f(n)$) is not defined for all $x$ F. yes\nC. $\\sum\\limits_{n=1}^{\\infty} \\frac{\\cos(n)}{n^{7}}$ Can the integral test be used to test convergence? [ANS] A. no, because the terms in the series do not decrease in magnitude B. no, because the terms in the series are not all positive for $n\\ge c$, for some $c>0$ C. no, because the series is not a geometric series D. no, because the terms in the series are not recursively defined E. no, because the function $f(x)$ (where $a_n=f(n)$) is not defined for all $x$ F. yes", "answer_v1": [ "F", "A", "B" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "For each of the following series, indicate whether the integral test can be used to determine its convergence or not, and if not, why. A. $\\sum\\limits_{n=1}^{\\infty} {n\\over n^2-7.8}$ Can the integral test be used to test convergence? [ANS] A. no, because the terms in the series do not decrease in magnitude B. no, because the terms in the series are not all positive for $n\\ge c$, for some $c>0$ C. no, because the series is not a geometric series D. no, because the terms in the series are not recursively defined E. no, because the function $f(x)$ (where $a_n=f(n)$) is not defined for all $x$ F. yes\nB. $\\sum\\limits_{n=1}^{\\infty} \\frac{8}{n^{2}\\,(\\cos(\\pi n)+\\sin(\\pi n)+2)}$ Can the integral test be used to test convergence? [ANS] A. no, because the terms in the series do not decrease in magnitude B. no, because the terms in the series are not all positive for $n\\ge c$, for some $c>0$ C. no, because the series is not a geometric series D. no, because the terms in the series are not recursively defined E. no, because the function $f(x)$ (where $a_n=f(n)$) is not defined for all $x$ F. yes\nC. $\\sum\\limits_{n=1}^{\\infty} {\\sin(8 n)+2}$ Can the integral test be used to test convergence? [ANS] A. no, because the terms in the series do not decrease in magnitude B. no, because the terms in the series are not all positive for $n\\ge c$, for some $c>0$ C. no, because the series is not a geometric series D. no, because the terms in the series are not recursively defined E. no, because the function $f(x)$ (where $a_n=f(n)$) is not defined for all $x$ F. yes", "answer_v2": [ "F", "A", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "For each of the following series, indicate whether the integral test can be used to determine its convergence or not, and if not, why. A. $\\sum\\limits_{n=1}^{\\infty} \\frac{6}{n(\\cos(n)+\\sin(n)+2)}$ Can the integral test be used to test convergence? [ANS] A. no, because the terms in the series do not decrease in magnitude B. no, because the terms in the series are not all positive for $n\\ge c$, for some $c>0$ C. no, because the series is not a geometric series D. no, because the terms in the series are not recursively defined E. no, because the function $f(x)$ (where $a_n=f(n)$) is not defined for all $x$ F. yes\nB. $\\sum\\limits_{n=1}^{\\infty} (n-6)^4\\,6^{-n}$ Can the integral test be used to test convergence? [ANS] A. no, because the terms in the series do not decrease in magnitude B. no, because the terms in the series are not all positive for $n\\ge c$, for some $c>0$ C. no, because the series is not a geometric series D. no, because the terms in the series are not recursively defined E. no, because the function $f(x)$ (where $a_n=f(n)$) is not defined for all $x$ F. yes\nC. $\\sum\\limits_{n=1}^{\\infty} \\frac{6}{n(|\\cos(n)|+2)}$ Can the integral test be used to test convergence? [ANS] A. no, because the terms in the series do not decrease in magnitude B. no, because the terms in the series are not all positive for $n\\ge c$, for some $c>0$ C. no, because the series is not a geometric series D. no, because the terms in the series are not recursively defined E. no, because the function $f(x)$ (where $a_n=f(n)$) is not defined for all $x$ F. yes", "answer_v3": [ "A", "F", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Calculus_-_single_variable_0905", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Integral test", "level": "3", "keywords": [ "Series' 'Sequences' 'summation" ], "problem_v1": "$(a)$ Compute $s_{5}$ of $ \\sum_{n=1}^\\infty\\frac{8}{6 n^6}$ [ANS]\n$(b)$ Estimate the error in using $s_{5}$ as an approximation of the sum of the series. (I.e. use $ \\int_{5}^{\\infty} f(x)dx \\geq r_{5}$) [ANS]\n$(c)$ Use n=5 and s_n+\\int_{n+1}^{\\infty} f(x)dx \\leq s \\leq s_n+\\int_{n}^{\\infty}f(x)dx to find a better estimate of the sum. [ANS] $\\leq s \\leq$ [ANS]", "answer_v1": [ "1.35640651031664", "8.53333333333333E-05", "1.35644080386946", "1.35649184364998" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "$(a)$ Compute $s_{4}$ of $ \\sum_{n=1}^\\infty\\frac{1}{10 n^3}$ [ANS]\n$(b)$ Estimate the error in using $s_{4}$ as an approximation of the sum of the series. (I.e. use $ \\int_{4}^{\\infty} f(x)dx \\geq r_{4}$) [ANS]\n$(c)$ Use n=4 and s_n+\\int_{n+1}^{\\infty} f(x)dx \\leq s \\leq s_n+\\int_{n}^{\\infty}f(x)dx to find a better estimate of the sum. [ANS] $\\leq s \\leq$ [ANS]", "answer_v2": [ "0.117766203703704", "0.003125", "0.119766203703704", "0.120891203703704" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "$(a)$ Compute $s_{5}$ of $ \\sum_{n=1}^\\infty\\frac{4}{7 n^4}$ [ANS]\n$(b)$ Estimate the error in using $s_{5}$ as an approximation of the sum of the series. (I.e. use $ \\int_{5}^{\\infty} f(x)dx \\geq r_{5}$) [ANS]\n$(c)$ Use n=5 and s_n+\\int_{n+1}^{\\infty} f(x)dx \\leq s \\leq s_n+\\int_{n}^{\\infty}f(x)dx to find a better estimate of the sum. [ANS] $\\leq s \\leq$ [ANS]", "answer_v3": [ "0.617343959435626", "0.00152380952380952", "0.618225793650794", "0.618867768959436" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0906", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Integral test", "level": "3", "keywords": [], "problem_v1": "Suppose $a(x)$ is a continuous, positive, decreasing function for $x$ in the interval $\\lbrack 1, \\infty)$, and $\\lbrace a_n \\rbrace$ is the sequence defined by $a_n=a(n)$ for every natural number $n$.\nIf ${\\sum_{n=1}^{\\infty} a_n}$ diverges, then ${\\int_{1}^{\\infty} a(x) \\, dx}$ [ANS] because [ANS] A. $ \\sum_{n=1}^{\\infty} a_n < a_1+\\int_{1}^{\\infty} a(x) \\, dx$. B. $ \\int_{0}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$. C. $ \\sum_{n=1}^{\\infty} a_n < \\int_{1}^{\\infty} a(x) \\, dx$. D. $ \\int_{1}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$.\nIf ${\\sum_{n=1}^{\\infty} a_n}$ converges, then ${\\int_{1}^{\\infty} a(x) \\, dx}$ [ANS] because [ANS] A. $ \\int_{1}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$. B. $ \\int_{0}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$. C. $ \\sum_{n=1}^{\\infty} a_n < \\int_{1}^{\\infty} a(x) \\, dx$. D. $ \\sum_{n=1}^{\\infty} a_n < a_1+\\int_{1}^{\\infty} a(x) \\, dx$.", "answer_v1": [ "diverges", "A", "converges", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "converges", "diverges", "cannot be determined" ], [ "A", "B", "C", "D" ], [ "converges", "diverges", "cannot be determined" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Suppose $a(x)$ is a continuous, positive, decreasing function for $x$ in the interval $\\lbrack 1, \\infty)$, and $\\lbrace a_n \\rbrace$ is the sequence defined by $a_n=a(n)$ for every natural number $n$.\nIf ${\\sum_{n=1}^{\\infty} a_n}$ diverges, then ${\\int_{1}^{\\infty} a(x) \\, dx}$ [ANS] because [ANS] A. $ \\sum_{n=1}^{\\infty} a_n < a_1+\\int_{1}^{\\infty} a(x) \\, dx$. B. $ \\int_{0}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$. C. $ \\sum_{n=1}^{\\infty} a_n < \\int_{1}^{\\infty} a(x) \\, dx$. D. $ \\int_{1}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$.\nIf ${\\sum_{n=1}^{\\infty} a_n}$ converges, then ${\\int_{1}^{\\infty} a(x) \\, dx}$ [ANS] because [ANS] A. $ \\sum_{n=1}^{\\infty} a_n < \\int_{1}^{\\infty} a(x) \\, dx$. B. $ \\int_{0}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$. C. $ \\sum_{n=1}^{\\infty} a_n < a_1+\\int_{1}^{\\infty} a(x) \\, dx$. D. $ \\int_{1}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$.", "answer_v2": [ "diverges", "A", "converges", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "converges", "diverges", "cannot be determined" ], [ "A", "B", "C", "D" ], [ "converges", "diverges", "cannot be determined" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Suppose $a(x)$ is a continuous, positive, decreasing function for $x$ in the interval $\\lbrack 1, \\infty)$, and $\\lbrace a_n \\rbrace$ is the sequence defined by $a_n=a(n)$ for every natural number $n$.\nIf ${\\sum_{n=1}^{\\infty} a_n}$ diverges, then ${\\int_{1}^{\\infty} a(x) \\, dx}$ [ANS] because [ANS] A. $ \\sum_{n=1}^{\\infty} a_n < a_1+\\int_{1}^{\\infty} a(x) \\, dx$. B. $ \\sum_{n=1}^{\\infty} a_n < \\int_{1}^{\\infty} a(x) \\, dx$. C. $ \\int_{0}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$. D. $ \\int_{1}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$.\nIf ${\\sum_{n=1}^{\\infty} a_n}$ converges, then ${\\int_{1}^{\\infty} a(x) \\, dx}$ [ANS] because [ANS] A. $ \\sum_{n=1}^{\\infty} a_n < \\int_{1}^{\\infty} a(x) \\, dx$. B. $ \\int_{1}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$. C. $ \\int_{0}^{\\infty} a(x) \\, dx < \\sum_{n=1}^{\\infty} a_n$. D. $ \\sum_{n=1}^{\\infty} a_n < a_1+\\int_{1}^{\\infty} a(x) \\, dx$.", "answer_v3": [ "diverges", "A", "converges", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "converges", "diverges", "cannot be determined" ], [ "A", "B", "C", "D" ], [ "converges", "diverges", "cannot be determined" ], [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_single_variable_0907", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Integral test", "level": "3", "keywords": [ "Series", "Summation" ], "problem_v1": "The series \\sum_{n=2}^\\infty \\frac{1}{n(\\ln n)^6} is convergent. (A). According to the Remainder Estimate for the Integral Test, the error in the approximation $s\\approx s_n$ (where $s$ is the value of the infinite sum and $s_n$ is the $n$-th partial sum) is $|s-s_n| \\le$ [ANS]\n(B). Find the smallest value of $n$ such that this upper bound is less than 0.006. $n=$ [ANS]", "answer_v1": [ "1/(5*[ln(n)]^5)", "8" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The series \\sum_{n=2}^\\infty \\frac{1}{n(\\ln n)^3} is convergent. (A). According to the Remainder Estimate for the Integral Test, the error in the approximation $s\\approx s_n$ (where $s$ is the value of the infinite sum and $s_n$ is the $n$-th partial sum) is $|s-s_n| \\le$ [ANS]\n(B). Find the smallest value of $n$ such that this upper bound is less than 0.09. $n=$ [ANS]", "answer_v2": [ "1/(2*[ln(n)]^2)", "11" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The series \\sum_{n=2}^\\infty \\frac{1}{n(\\ln n)^4} is convergent. (A). According to the Remainder Estimate for the Integral Test, the error in the approximation $s\\approx s_n$ (where $s$ is the value of the infinite sum and $s_n$ is the $n$-th partial sum) is $|s-s_n| \\le$ [ANS]\n(B). Find the smallest value of $n$ such that this upper bound is less than 0.06. $n=$ [ANS]", "answer_v3": [ "1/(3*[ln(n)]^3)", "6" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0908", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Root test", "level": "2", "keywords": [ "calculus", "series", "sequences", "convergence", "root test" ], "problem_v1": "Use the Root Test to determine the convergence or divergence of the given series or state that the Root Test is inconclusive. \\sum\\limits_{n=1}^{\\infty} \\left(1+\\frac{1}{8 n}\\right)^{-n^2} $L=\\lim\\limits_{n \\to \\infty} \\sqrt[n]{\\left| a_n \\right|}=$ [ANS] (Enter 'inf' for $\\infty$.) $\\sum_{n=4}^{\\infty} \\left(1+\\frac{1}{8 n}\\right)^{-n^2}$ is: [ANS] A. convergent B. divergent C. The Root Test is inconclusive", "answer_v1": [ "0.882497", "A" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "A", "B", "C" ] ], "problem_v2": "Use the Root Test to determine the convergence or divergence of the given series or state that the Root Test is inconclusive. \\sum\\limits_{n=1}^{\\infty} \\left(1+\\frac{1}{2 n}\\right)^{-n^2} $L=\\lim\\limits_{n \\to \\infty} \\sqrt[n]{\\left| a_n \\right|}=$ [ANS] (Enter 'inf' for $\\infty$.) $\\sum_{n=4}^{\\infty} \\left(1+\\frac{1}{2 n}\\right)^{-n^2}$ is: [ANS] A. convergent B. divergent C. The Root Test is inconclusive", "answer_v2": [ "0.606531", "A" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "A", "B", "C" ] ], "problem_v3": "Use the Root Test to determine the convergence or divergence of the given series or state that the Root Test is inconclusive. \\sum\\limits_{n=1}^{\\infty} \\left(1+\\frac{1}{4 n}\\right)^{-n^2} $L=\\lim\\limits_{n \\to \\infty} \\sqrt[n]{\\left| a_n \\right|}=$ [ANS] (Enter 'inf' for $\\infty$.) $\\sum_{n=4}^{\\infty} \\left(1+\\frac{1}{4 n}\\right)^{-n^2}$ is: [ANS] A. convergent B. divergent C. The Root Test is inconclusive", "answer_v3": [ "0.778801", "A" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "A", "B", "C" ] ] }, { "id": "Calculus_-_single_variable_0909", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Alternating series test", "level": "2", "keywords": [], "problem_v1": "Approximate the value of the series to within an error of at most $10^{-4}$. \\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{(n+76)(n+77)} According to Equation (2): \\left| S_N-S \\right|\\le a_{N+1} what is the smallest value of $N$ that approximates $S$ to within an error of at most $10^{-4}$? $N=$ [ANS]\n$S\\approx$ [ANS]", "answer_v1": [ "23", "0.000134322" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Approximate the value of the series to within an error of at most $10^{-3}$. \\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{(n+10)(n+2)} According to Equation (2): \\left| S_N-S \\right|\\le a_{N+1} what is the smallest value of $N$ that approximates $S$ to within an error of at most $10^{-3}$? $N=$ [ANS]\n$S\\approx$ [ANS]", "answer_v2": [ "25", "0.0187161" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Approximate the value of the series to within an error of at most $10^{-3}$. \\sum_{n=1}^{\\infty} \\frac{(-1)^{n+1}}{(n+7)(n+3)} According to Equation (2): \\left| S_N-S \\right|\\le a_{N+1} what is the smallest value of $N$ that approximates $S$ to within an error of at most $10^{-3}$? $N=$ [ANS]\n$S\\approx$ [ANS]", "answer_v3": [ "26", "0.0179468" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0910", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Alternating series test", "level": "2", "keywords": [ "series", "divergent", "convergent", "alternating series", "approximation", "alternating", "series" ], "problem_v1": "For the following alternating series, $ \\sum_{n=1}^\\infty a_n=1-\\frac{1}{10}+\\frac{1}{100}-\\frac{1}{1000}+...$ how many terms do you have to go for your approximation (your partial sum) to be within 1e-08 from the convergent value of that series? [ANS]", "answer_v1": [ "8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "For the following alternating series, $ \\sum_{n=1}^\\infty a_n=1-\\frac{1}{10}+\\frac{1}{100}-\\frac{1}{1000}+...$ how many terms do you have to go for your approximation (your partial sum) to be within 0.01 from the convergent value of that series? [ANS]", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "For the following alternating series, $ \\sum_{n=1}^\\infty a_n=1-\\frac{1}{10}+\\frac{1}{100}-\\frac{1}{1000}+...$ how many terms do you have to go for your approximation (your partial sum) to be within 0.0001 from the convergent value of that series? [ANS]", "answer_v3": [ "4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0911", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Absolute and conditional convergence", "level": "3", "keywords": [ "series", "divergent", "convergent", "absolute convergence", "ratio test", "conditionally convergent", "calculus" ], "problem_v1": "Consider the series $ \\sum_{n=1}^{\\infty} a_n$ where a_n=\\frac{(7 n^2+5) 8^{n+2}}{7^{n}} In this problem you must attempt to use the Ratio Test to decide whether the series converges.\nCompute L=\\lim_{n\\rightarrow\\infty} \\left| \\frac{a_{n+1}}{a_n} \\right| Enter the numerical value of the limit $L$ if it converges, INF if the limit for $L$ diverges to infinity, MINF if it diverges to negative infinity, or DIV if it diverges but not to infinity or negative infinity. $L=$ [ANS]\nWhich of the following statements is true? A. The Ratio Test says that the series converges absolutely. B. The Ratio Test says that the series diverges. C. The Ratio Test says that the series converges conditionally. D. The Ratio Test is inconclusive, but the series converges absolutely by another test or tests. E. The Ratio Test is inconclusive, but the series diverges by another test or tests. F. The Ratio Test is inconclusive, but the series converges conditionally by another test or tests. Enter the letter for your choice here: [ANS]", "answer_v1": [ "1.14285714285714", "B" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Consider the series $ \\sum_{n=1}^{\\infty} a_n$ where a_n=\\frac{(-5)^{n}}{(2 n+7) 8^{n+2}} In this problem you must attempt to use the Ratio Test to decide whether the series converges.\nCompute L=\\lim_{n\\rightarrow\\infty} \\left| \\frac{a_{n+1}}{a_n} \\right| Enter the numerical value of the limit $L$ if it converges, INF if the limit for $L$ diverges to infinity, MINF if it diverges to negative infinity, or DIV if it diverges but not to infinity or negative infinity. $L=$ [ANS]\nWhich of the following statements is true? A. The Ratio Test says that the series converges absolutely. B. The Ratio Test says that the series diverges. C. The Ratio Test says that the series converges conditionally. D. The Ratio Test is inconclusive, but the series converges absolutely by another test or tests. E. The Ratio Test is inconclusive, but the series diverges by another test or tests. F. The Ratio Test is inconclusive, but the series converges conditionally by another test or tests. Enter the letter for your choice here: [ANS]", "answer_v2": [ "0.625", "A" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Consider the series $ \\sum_{n=1}^{\\infty} a_n$ where a_n=\\frac{6^{n+2}}{(4 n^2+5) 4^{n}} In this problem you must attempt to use the Ratio Test to decide whether the series converges.\nCompute L=\\lim_{n\\rightarrow\\infty} \\left| \\frac{a_{n+1}}{a_n} \\right| Enter the numerical value of the limit $L$ if it converges, INF if the limit for $L$ diverges to infinity, MINF if it diverges to negative infinity, or DIV if it diverges but not to infinity or negative infinity. $L=$ [ANS]\nWhich of the following statements is true? A. The Ratio Test says that the series converges absolutely. B. The Ratio Test says that the series diverges. C. The Ratio Test says that the series converges conditionally. D. The Ratio Test is inconclusive, but the series converges absolutely by another test or tests. E. The Ratio Test is inconclusive, but the series diverges by another test or tests. F. The Ratio Test is inconclusive, but the series converges conditionally by another test or tests. Enter the letter for your choice here: [ANS]", "answer_v3": [ "1.5", "B" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Calculus_-_single_variable_0912", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Strategy for testing series", "level": "3", "keywords": [ "series", "divergent", "convergent", "comparison test", "absolute convergence", "alternating series", "geometric series" ], "problem_v1": "Select the FIRST correct reason why the given series converges.\nA. Convergent geometric series B. Convergent \\(p\\) series C. Comparison (or Limit Comparison) with a geometric or \\(p\\) series D. Converges by alternating series test [ANS] 1. $ \\sum_{n=1}^\\infty \\frac{(n+1)(15)^n}{4^{2n}}$ [ANS] 2. $ \\sum_{n=1}^\\infty \\frac{5 (7)^n}{10^{2n}}$ [ANS] 3. $ \\sum_{n=1}^\\infty \\frac{(-1)^n}{5 n+6}$ [ANS] 4. $ \\sum_{n=1}^\\infty (-1)^n \\frac{\\sqrt{n}}{n+6}$ [ANS] 5. $ \\sum_{n=1}^\\infty \\frac{\\sin^2 (3 n)}{n^2}$ [ANS] 6. $ \\sum_{n=1}^\\infty \\frac{n^2+\\sqrt{n}}{n^4-6}$", "answer_v1": [ "C", "A", "D", "D", "C", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Select the FIRST correct reason why the given series converges.\nA. Convergent geometric series B. Convergent \\(p\\) series C. Comparison (or Limit Comparison) with a geometric or \\(p\\) series D. Converges by alternating series test [ANS] 1. $ \\sum_{n=1}^\\infty \\frac{\\cos(n \\pi)}{\\ln(2 n)}$ [ANS] 2. $ \\sum_{n=1}^\\infty \\frac{2 (5)^n}{9^{2n}}$ [ANS] 3. $ \\sum_{n=1}^\\infty \\frac{(n+1)(15)^n}{4^{2n}}$ [ANS] 4. $ \\sum_{n=1}^\\infty \\frac{(-1)^n \\ln(e^n)}{n^{5} \\cos(n \\pi)}$ [ANS] 5. $ \\sum_{n=1}^\\infty \\frac{(-1)^n}{2 n+4}$ [ANS] 6. $ \\sum_{n=1}^\\infty \\frac{\\sin^2 (7 n)}{n^2}$", "answer_v2": [ "D", "A", "C", "B", "D", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Select the FIRST correct reason why the given series converges.\nA. Convergent geometric series B. Convergent \\(p\\) series C. Comparison (or Limit Comparison) with a geometric or \\(p\\) series D. Converges by alternating series test [ANS] 1. $ \\sum_{n=1}^\\infty \\frac{(-1)^n}{3 n+5}$ [ANS] 2. $ \\sum_{n=1}^\\infty \\frac{(n+1)(24)^n}{5^{2n}}$ [ANS] 3. $ \\sum_{n=1}^\\infty \\frac{\\sin^2 (3 n)}{n^2}$ [ANS] 4. $ \\sum_{n=1}^\\infty \\frac{\\cos(n \\pi)}{\\ln(3 n)}$ [ANS] 5. $ \\sum_{n=1}^\\infty \\frac{n^2+\\sqrt{n}}{n^4-9}$ [ANS] 6. $ \\sum_{n=1}^\\infty \\frac{(-1)^n \\ln(e^n)}{n^{9} \\cos(n \\pi)}$", "answer_v3": [ "D", "C", "C", "D", "C", "B" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_single_variable_0913", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Interval of convergence of a power series", "level": "3", "keywords": [ "calculus", "series", "sequences", "power series", "convergence", "radius of convergence", "interval of convergence" ], "problem_v1": "Use the Ratio Test to determine the radius of convergence of the following series:\n\\sum_{n=0}^{\\infty} \\frac{x^n}{23^n} $R=$ [ANS]", "answer_v1": [ "23" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the Ratio Test to determine the radius of convergence of the following series:\n\\sum_{n=0}^{\\infty} \\frac{x^n}{4^n} $R=$ [ANS]", "answer_v2": [ "4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the Ratio Test to determine the radius of convergence of the following series:\n\\sum_{n=0}^{\\infty} \\frac{x^n}{10^n} $R=$ [ANS]", "answer_v3": [ "10" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0914", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Interval of convergence of a power series", "level": "2", "keywords": [], "problem_v1": "Find the Maclaurin series and corresponding interval of convergence of the following function.\nf(x)=\\frac{1-\\cos (x^{8})}{x^{3}} after removing the removable discontinuity at $x=0$. $f(x)=\\sum\\limits_{n=1}^{\\infty}$ [ANS]\nThe interval of convergence for this power series is: [ANS]", "answer_v1": [ "(-1)^{n+1}*x^{16*n-3}/[(2*n)!]", "(-infinity,infinity)" ], "answer_type_v1": [ "EX", "INT" ], "options_v1": [ [], [] ], "problem_v2": "Find the Maclaurin series and corresponding interval of convergence of the following function.\nf(x)=\\frac{1-\\cos (x^{2})}{x^{4}} after removing the removable discontinuity at $x=0$. $f(x)=\\sum\\limits_{n=1}^{\\infty}$ [ANS]\nThe interval of convergence for this power series is: [ANS]", "answer_v2": [ "(-1)^{n+1}*x^{4*n-4}/[(2*n)!]", "(-infinity,infinity)" ], "answer_type_v2": [ "EX", "INT" ], "options_v2": [ [], [] ], "problem_v3": "Find the Maclaurin series and corresponding interval of convergence of the following function.\nf(x)=\\frac{1-\\cos (x^{4})}{x^{3}} after removing the removable discontinuity at $x=0$. $f(x)=\\sum\\limits_{n=1}^{\\infty}$ [ANS]\nThe interval of convergence for this power series is: [ANS]", "answer_v3": [ "(-1)^{n+1}*x^{8*n-3}/[(2*n)!]", "(-infinity,infinity)" ], "answer_type_v3": [ "EX", "INT" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0915", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Interval of convergence of a power series", "level": "2", "keywords": [], "problem_v1": "Find the Taylor series, centered at $c=6$, for the function f(x)=\\frac{1}{x}. $ f(x)=\\sum_{n=0}^{\\infty}$ [ANS]. The interval of convergence is: [ANS].", "answer_v1": [ "(-1)^n/[6^{n+1}]*(x-6)^n", "(0,12)" ], "answer_type_v1": [ "EX", "INT" ], "options_v1": [ [], [] ], "problem_v2": "Find the Taylor series, centered at $c=2$, for the function f(x)=\\frac{1}{x}. $ f(x)=\\sum_{n=0}^{\\infty}$ [ANS]. The interval of convergence is: [ANS].", "answer_v2": [ "(-1)^n/[2^{n+1}]*(x-2)^n", "(0,4)" ], "answer_type_v2": [ "EX", "INT" ], "options_v2": [ [], [] ], "problem_v3": "Find the Taylor series, centered at $c=3$, for the function f(x)=\\frac{1}{x}. $ f(x)=\\sum_{n=0}^{\\infty}$ [ANS]. The interval of convergence is: [ANS].", "answer_v3": [ "(-1)^n/[3^{n+1}]*(x-3)^n", "(0,6)" ], "answer_type_v3": [ "EX", "INT" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0916", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Interval of convergence of a power series", "level": "4", "keywords": [ "Series", "Power", "Radius", "Convergence", "Coefficient" ], "problem_v1": "The function $f(x)=8x \\arctan (6x)$ is represented as a power series $ f(x)=\\sum_{n=0}^\\infty c_n x^n.$ Find the first few coefficients in the power series. $c_0=$ [ANS]\n$c_1=$ [ANS]\n$c_2=$ [ANS]\n$c_3=$ [ANS]\n$c_4=$ [ANS]\nFind the radius of convergence $R$ of the series. $R=$ [ANS]", "answer_v1": [ "0", "0", "48", "0", "-576", "0.166666666666667" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The function $f(x)=2x \\arctan (8x)$ is represented as a power series $ f(x)=\\sum_{n=0}^\\infty c_n x^n.$ Find the first few coefficients in the power series. $c_0=$ [ANS]\n$c_1=$ [ANS]\n$c_2=$ [ANS]\n$c_3=$ [ANS]\n$c_4=$ [ANS]\nFind the radius of convergence $R$ of the series. $R=$ [ANS]", "answer_v2": [ "0", "0", "16", "0", "-341.333333333333", "0.125" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The function $f(x)=4x \\arctan (6x)$ is represented as a power series $ f(x)=\\sum_{n=0}^\\infty c_n x^n.$ Find the first few coefficients in the power series. $c_0=$ [ANS]\n$c_1=$ [ANS]\n$c_2=$ [ANS]\n$c_3=$ [ANS]\n$c_4=$ [ANS]\nFind the radius of convergence $R$ of the series. $R=$ [ANS]", "answer_v3": [ "0", "0", "24", "0", "-288", "0.166666666666667" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0917", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Interval of convergence of a power series", "level": "3", "keywords": [ "Series", "Power", "Interval", "Convergence", "Power Series" ], "problem_v1": "Find all the values of x such that the given series would converge. \\sum_{n=1}^\\infty \\frac{x^n}{(7)^n (\\sqrt{n}+9)} The series is convergent from $x=$ [ANS], left end included (enter Y or N): [ANS]\nto $x=$ [ANS], right end included (enter Y or N): [ANS]", "answer_v1": [ "-7", "Y", "7", "N" ], "answer_type_v1": [ "NV", "TF", "NV", "TF" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find all the values of x such that the given series would converge. \\sum_{n=1}^\\infty \\frac{x^n}{(11)^n (\\sqrt{n}+2)} The series is convergent from $x=$ [ANS], left end included (enter Y or N): [ANS]\nto $x=$ [ANS], right end included (enter Y or N): [ANS]", "answer_v2": [ "-11", "Y", "11", "N" ], "answer_type_v2": [ "NV", "TF", "NV", "TF" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find all the values of x such that the given series would converge. \\sum_{n=1}^\\infty \\frac{x^n}{(8)^n (\\sqrt{n}+5)} The series is convergent from $x=$ [ANS], left end included (enter Y or N): [ANS]\nto $x=$ [ANS], right end included (enter Y or N): [ANS]", "answer_v3": [ "-8", "Y", "8", "N" ], "answer_type_v3": [ "NV", "TF", "NV", "TF" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0918", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Interval of convergence of a power series", "level": "3", "keywords": [ "Series", "Power", "Radius", "Convergence", "Double Factorial" ], "problem_v1": "Define the double factorial of $n$, denoted $n!!$, as follows: n!!=\\begin{cases} 1 \\cdot 3 \\cdot 5 \\cdot \\cdot \\cdot \\cdot (n-2) \\cdot n & \\text{if} n \\text{is odd} \\\\ 2 \\cdot 4 \\cdot 6 \\cdot \\cdot \\cdot \\cdot (n-2) \\cdot n & \\text{if} n \\text{is even} \\end{cases} where $(-1)!!=0!!=1!!=1$, $2!!=2$, and $3!!=3$. Thus, if $n$ is even, then $n!!$ is the product of all the even integers, between $1$ and $n$ and, if $n$ is odd, then $n!!$ is the product of all the odd integers, between $1$ and $n$. Find the radius of convergence for the given power series. \\sum_{n=1}^\\infty \\frac {8^n \\cdot n! \\cdot (4 n+7)! \\cdot (2n)!!} {6^n \\cdot [(n+3)!]^4 \\cdot (4n-2)!!} (6x-2)^n The radius of convergence, $R=$ [ANS].", "answer_v1": [ "0.00390625" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Define the double factorial of $n$, denoted $n!!$, as follows: n!!=\\begin{cases} 1 \\cdot 3 \\cdot 5 \\cdot \\cdot \\cdot \\cdot (n-2) \\cdot n & \\text{if} n \\text{is odd} \\\\ 2 \\cdot 4 \\cdot 6 \\cdot \\cdot \\cdot \\cdot (n-2) \\cdot n & \\text{if} n \\text{is even} \\end{cases} where $(-1)!!=0!!=1!!=1$, $2!!=2$, and $3!!=3$. Thus, if $n$ is even, then $n!!$ is the product of all the even integers, between $1$ and $n$ and, if $n$ is odd, then $n!!$ is the product of all the odd integers, between $1$ and $n$. Find the radius of convergence for the given power series. \\sum_{n=1}^\\infty \\frac {2^n \\cdot n! \\cdot (2 n+4)! \\cdot (2n)!!} {9^n \\cdot [(n+9)!]^2 \\cdot (4n-2)!!} (-3x+1)^n The radius of convergence, $R=$ [ANS].", "answer_v2": [ "3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Define the double factorial of $n$, denoted $n!!$, as follows: n!!=\\begin{cases} 1 \\cdot 3 \\cdot 5 \\cdot \\cdot \\cdot \\cdot (n-2) \\cdot n & \\text{if} n \\text{is odd} \\\\ 2 \\cdot 4 \\cdot 6 \\cdot \\cdot \\cdot \\cdot (n-2) \\cdot n & \\text{if} n \\text{is even} \\end{cases} where $(-1)!!=0!!=1!!=1$, $2!!=2$, and $3!!=3$. Thus, if $n$ is even, then $n!!$ is the product of all the even integers, between $1$ and $n$ and, if $n$ is odd, then $n!!$ is the product of all the odd integers, between $1$ and $n$. Find the radius of convergence for the given power series. \\sum_{n=1}^\\infty \\frac {4^n \\cdot n! \\cdot (3 n+5)! \\cdot (2n)!!} {6^n \\cdot [(n+2)!]^3 \\cdot (4n-2)!!} (8x+7)^n The radius of convergence, $R=$ [ANS].", "answer_v3": [ "0.0555555555555556" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0919", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Interval of convergence of a power series", "level": "3", "keywords": [ "calculus", "integral", "taylor series", "series" ], "problem_v1": "Consider the functions $y=\\cos(x)$ and $y=e^{-x^2}$. A. Write the Taylor expansions for the two functions about $x=0$. What is similar about the two series? What is different?\nLooking at the series, which function do you predict will be greater over the interval (-1,1)? [ANS] A. $\\cos(x)$ B. $e^{-x^2}$\n(Graph the functions to verify that your answer is correct!) (Graph the functions to verify that your answer is correct!) B. Are these functions even or odd? [ANS] A. Even B. Odd\nC. Find the radii of convergence for your two series. For $\\cos(x)$, the radius of convergence is [ANS]\nFor $e^{-x^2}$, the radius of convergence is [ANS]\n(Enter (Enter infinity if the radius of convergence is infinite.) if the radius of convergence is infinite.) Looking at the relative sizes of the successive terms in your series, note how the radii of convergence you found make sense.", "answer_v1": [ "A", "A", "infinity", "infinity" ], "answer_type_v1": [ "MCS", "MCS", "NV", "NV" ], "options_v1": [ [ "A", "B" ], [ "A", "B" ], [], [] ], "problem_v2": "Consider the functions $y=1-\\sin(x^2)$ and $y=2-\\sqrt{1+x^2}$. A. Write the Taylor expansions for the two functions about $x=0$. What is similar about the two series? What is different?\nLooking at the series, which function do you predict will be greater over the interval (-1,1)? [ANS] A. $1-\\sin(x^2)$ B. $2-\\sqrt{1+x^2}$\n(Graph the functions to verify that your answer is correct!) (Graph the functions to verify that your answer is correct!) B. Are these functions even or odd? [ANS] A. Even B. Odd\nC. Find the radii of convergence for your two series. For $1-\\sin(x^2)$, the radius of convergence is [ANS]\nFor $2-\\sqrt{1+x^2}$, the radius of convergence is [ANS]\n(Enter (Enter infinity if the radius of convergence is infinite.) if the radius of convergence is infinite.) Looking at the relative sizes of the successive terms in your series, note how the radii of convergence you found make sense.", "answer_v2": [ "B", "A", "infinity", "1" ], "answer_type_v2": [ "MCS", "MCS", "NV", "NV" ], "options_v2": [ [ "A", "B" ], [ "A", "B" ], [], [] ], "problem_v3": "Consider the functions $y=1-\\ln(1+x^2)$ and $y=\\cos(x)$. A. Write the Taylor expansions for the two functions about $x=0$. What is similar about the two series? What is different?\nLooking at the series, which function do you predict will be greater over the interval (-1,1)? [ANS] A. $1-\\ln(1+x^2)$ B. $\\cos(x)$\n(Graph the functions to verify that your answer is correct!) (Graph the functions to verify that your answer is correct!) B. Are these functions even or odd? [ANS] A. Even B. Odd\nC. Find the radii of convergence for your two series. For $1-\\ln(1+x^2)$, the radius of convergence is [ANS]\nFor $\\cos(x)$, the radius of convergence is [ANS]\n(Enter (Enter infinity if the radius of convergence is infinite.) if the radius of convergence is infinite.) Looking at the relative sizes of the successive terms in your series, note how the radii of convergence you found make sense.", "answer_v3": [ "B", "A", "1", "infinity" ], "answer_type_v3": [ "MCS", "MCS", "NV", "NV" ], "options_v3": [ [ "A", "B" ], [ "A", "B" ], [], [] ] }, { "id": "Calculus_-_single_variable_0920", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Interval of convergence of a power series", "level": "3", "keywords": [ "calculus", "integral", "series", "power series", "interval of convergence", "radius of convergence" ], "problem_v1": "Find the interval of convergence for the power series \\sum_{n=2}^{\\infty} {n(x-6)^n\\over 4^{2n}} interval of convergence=[ANS]\n(Enter your answer as an interval: thus, if the interval of convergence were $-3 < x\\le 5$, you would enter (-3,5]. Use Inf for any endpoint at infinity.)", "answer_v1": [ "(-10,22)" ], "answer_type_v1": [ "INT" ], "options_v1": [ [] ], "problem_v2": "Find the interval of convergence for the power series \\sum_{n=2}^{\\infty} {(x-1)^n\\over 5 n} interval of convergence=[ANS]\n(Enter your answer as an interval: thus, if the interval of convergence were $-3 < x\\le 5$, you would enter (-3,5]. Use Inf for any endpoint at infinity.)", "answer_v2": [ "[0,2)" ], "answer_type_v2": [ "INT" ], "options_v2": [ [] ], "problem_v3": "Find the interval of convergence for the power series \\sum_{n=2}^{\\infty} {(x-3)^n\\over 4 n} interval of convergence=[ANS]\n(Enter your answer as an interval: thus, if the interval of convergence were $-3 < x\\le 5$, you would enter (-3,5]. Use Inf for any endpoint at infinity.)", "answer_v3": [ "[2,4)" ], "answer_type_v3": [ "INT" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0921", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Maclaurin series", "level": "1", "keywords": [], "problem_v1": "Write out the first four terms of the Maclaurin series of $f(x)$ if f(0)=8, \\qquad f'(0)=3, \\qquad f''(0)=4, \\qquad f'''(0)=7 $f(x)$=[ANS] $+\\cdots$", "answer_v1": [ "8+3*x+4*x^2/2+7*x^3/6" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write out the first four terms of the Maclaurin series of $f(x)$ if f(0)=-13, \\qquad f'(0)=13, \\qquad f''(0)=-11, \\qquad f'''(0)=-5 $f(x)$=[ANS] $+\\cdots$", "answer_v2": [ "13*x-13-11*x^2/2-5*x^3/6" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write out the first four terms of the Maclaurin series of $f(x)$ if f(0)=-6, \\qquad f'(0)=3, \\qquad f''(0)=-7, \\qquad f'''(0)=2 $f(x)$=[ANS] $+\\cdots$", "answer_v3": [ "3*x-6-7*x^2/2+2*x^3/6" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0922", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Maclaurin series", "level": "2", "keywords": [ "Series", "MacLaurin" ], "problem_v1": "Match the series with the right expression. (Use the Maclaurin series.) [ANS] 1. $ \\sum_{n=0}^\\infty \\frac{(-1)^n 3^{-(2n+1)}}{2n+1}$ [ANS] 2. $ \\sum_{n=0}^\\infty \\frac{3^n}{n!}$ [ANS] 3. $ \\sum_{n=0}^\\infty \\frac{(-1)^n 3^{2n}}{(2n)!}$ [ANS] 4. $ \\sum_{n=0}^\\infty (-1)^n \\frac{3^{2n+1}}{(2n+1)!}$\nA. $e^3$ B. $\\arctan(1/3)$ C. $\\sin(3)$ D. $\\cos(3)$", "answer_v1": [ "B", "A", "D", "C" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v2": "Match the series with the right expression. (Use the Maclaurin series.) [ANS] 1. $ \\sum_{n=0}^\\infty (-1)^n \\frac{3^{2n+1}}{(2n+1)!}$ [ANS] 2. $ \\sum_{n=0}^\\infty \\frac{(-1)^n 3^{-(2n+1)}}{2n+1}$ [ANS] 3. $ \\sum_{n=0}^\\infty \\frac{3^n}{n!}$ [ANS] 4. $ \\sum_{n=0}^\\infty \\frac{(-1)^n 3^{2n}}{(2n)!}$\nA. $\\cos(3)$ B. $\\sin(3)$ C. $\\arctan(1/3)$ D. $e^3$", "answer_v2": [ "B", "C", "D", "A" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ], "problem_v3": "Match the series with the right expression. (Use the Maclaurin series.) [ANS] 1. $ \\sum_{n=0}^\\infty \\frac{3^n}{n!}$ [ANS] 2. $ \\sum_{n=0}^\\infty \\frac{(-1)^n 3^{2n}}{(2n)!}$ [ANS] 3. $ \\sum_{n=0}^\\infty (-1)^n \\frac{3^{2n+1}}{(2n+1)!}$ [ANS] 4. $ \\sum_{n=0}^\\infty \\frac{(-1)^n 3^{-(2n+1)}}{2n+1}$\nA. $e^3$ B. $\\sin(3)$ C. $\\arctan(1/3)$ D. $\\cos(3)$", "answer_v3": [ "A", "D", "B", "C" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ], [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_single_variable_0923", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Maclaurin series", "level": "4", "keywords": [ "approximation", "estimation theorem" ], "problem_v1": "Find an upper bound on the error that can result if $\\cos (x)$ is approximated by $1-(x^2/2!)+(x^4/4!)$ over the interval $[-0.8, 0.8]$. upper bound: [ANS] Enter answer to six decimal places.", "answer_v1": [ "0.000364" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find an upper bound on the error that can result if $\\cos (x)$ is approximated by $1-(x^2/2!)+(x^4/4!)$ over the interval $[-0.3, 0.3]$. upper bound: [ANS] Enter answer to six decimal places.", "answer_v2": [ "1E-6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find an upper bound on the error that can result if $\\cos (x)$ is approximated by $1-(x^2/2!)+(x^4/4!)$ over the interval $[-0.5, 0.5]$. upper bound: [ANS] Enter answer to six decimal places.", "answer_v3": [ "2.2E-5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0924", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Maclaurin series", "level": "2", "keywords": [ "approximation", "estimation theorem" ], "problem_v1": "Approximate $\\tan^{-1}(0.21)$ to three decimal-place accuracy using the MacLaurin series for $\\tan^{-1}\\!\\left(x\\right)$. $\\tan^{-1}(0.21)\\approx$ [ANS]", "answer_v1": [ "0.207" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Approximate $\\tan^{-1}(0.15)$ to three decimal-place accuracy using the MacLaurin series for $\\tan^{-1}\\!\\left(x\\right)$. $\\tan^{-1}(0.15)\\approx$ [ANS]", "answer_v2": [ "0.149" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Approximate $\\tan^{-1}(0.17)$ to three decimal-place accuracy using the MacLaurin series for $\\tan^{-1}\\!\\left(x\\right)$. $\\tan^{-1}(0.17)\\approx$ [ANS]", "answer_v3": [ "0.168" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0925", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Maclaurin series", "level": "3", "keywords": [ "approximation", "estimation theorem" ], "problem_v1": "Use the following binomial series formula (1+x)^m=1+m x+\\frac{m(m-1)}{2!}x^2+\\cdot\\cdot\\cdot+\\frac{m(m-1)\\cdots(m-k+1)}{k!}x^k+\\cdot\\cdot\\cdot to obtain the MacLaurin series for\n(a) $\\frac{1}{\\left(1+x\\right)^{9}}= \\sum_{k=0}^{\\infty}$ [ANS]\n(b) $\\sqrt[6]{1+x}=$ [ANS] $+\\cdot\\cdot\\cdot$. Enter first 4 terms only.", "answer_v1": [ "(-1)^k*(k+8)!/(8!*k!)*x^k", "1+1/6*x+-5/(6^2*2!)*x^2+-5*-11/(6^3*3!)*x^3" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Use the following binomial series formula (1+x)^m=1+m x+\\frac{m(m-1)}{2!}x^2+\\cdot\\cdot\\cdot+\\frac{m(m-1)\\cdots(m-k+1)}{k!}x^k+\\cdot\\cdot\\cdot to obtain the MacLaurin series for\n(a) $\\frac{1}{\\left(1+x\\right)^{3}}= \\sum_{k=0}^{\\infty}$ [ANS]\n(b) $\\sqrt[7]{1+x}=$ [ANS] $+\\cdot\\cdot\\cdot$. Enter first 4 terms only.", "answer_v2": [ "(-1)^k*(k+2)!/(2!*k!)*x^k", "1+1/7*x+-6/(7^2*2!)*x^2+-6*-13/(7^3*3!)*x^3" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Use the following binomial series formula (1+x)^m=1+m x+\\frac{m(m-1)}{2!}x^2+\\cdot\\cdot\\cdot+\\frac{m(m-1)\\cdots(m-k+1)}{k!}x^k+\\cdot\\cdot\\cdot to obtain the MacLaurin series for\n(a) $\\frac{1}{\\left(1+x\\right)^{5}}= \\sum_{k=0}^{\\infty}$ [ANS]\n(b) $\\sqrt[6]{1+x}=$ [ANS] $+\\cdot\\cdot\\cdot$. Enter first 4 terms only.", "answer_v3": [ "(-1)^k*(k+4)!/(4!*k!)*x^k", "1+1/6*x+-5/(6^2*2!)*x^2+-5*-11/(6^3*3!)*x^3" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0926", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Maclaurin series", "level": "3", "keywords": [ "linear approximation", "differentials" ], "problem_v1": "Use sigma notation to write the Maclaurin series for the function. $\\ln\\!\\left(8+x\\right)$. Maclaurin series [ANS] $+\\sum_{k=1}^{\\infty}$ [ANS]\n(Note first term separate and summation from $k=1$)", "answer_v1": [ "ln(8)", "(-1)^{k-1}*x^k/(8^k*k)" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Use sigma notation to write the Maclaurin series for the function. $\\ln\\!\\left(2+x\\right)$. Maclaurin series [ANS] $+\\sum_{k=1}^{\\infty}$ [ANS]\n(Note first term separate and summation from $k=1$)", "answer_v2": [ "ln(2)", "(-1)^{k-1}*x^k/(2^k*k)" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Use sigma notation to write the Maclaurin series for the function. $\\ln\\!\\left(4+x\\right)$. Maclaurin series [ANS] $+\\sum_{k=1}^{\\infty}$ [ANS]\n(Note first term separate and summation from $k=1$)", "answer_v3": [ "ln(4)", "(-1)^{k-1}*x^k/(4^k*k)" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0927", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor series", "level": "2", "keywords": [ "Taylor", "Series", "Coefficient", "Power Series", "Taylor Series", "Series", "Taylor" ], "problem_v1": "Write the Taylor series for $f(x)=e^{x}$ about $x=2$ as $ \\sum_{n=0}^\\infty c_n(x-2)^n.$ Find the first five coefficients.\n$\\begin{array}{ccc}\\hline c_0 &=& [ANS] \\\\ \\hline c_1 &=& [ANS] \\\\ \\hline c_2 &=& [ANS] \\\\ \\hline c_3 &=& [ANS] \\\\ \\hline c_4 &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "7.38906", "7.38906", "3.69453", "1.23151", "0.307877" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Write the Taylor series for $f(x)=e^{x}$ about $x=-4$ as $ \\sum_{n=0}^\\infty c_n(x+4)^n.$ Find the first five coefficients.\n$\\begin{array}{ccc}\\hline c_0 &=& [ANS] \\\\ \\hline c_1 &=& [ANS] \\\\ \\hline c_2 &=& [ANS] \\\\ \\hline c_3 &=& [ANS] \\\\ \\hline c_4 &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "0.0183156", "0.0183156", "0.00915782", "0.00305261", "0.000763152" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Write the Taylor series for $f(x)=e^{x}$ about $x=-2$ as $ \\sum_{n=0}^\\infty c_n(x+2)^n.$ Find the first five coefficients.\n$\\begin{array}{ccc}\\hline c_0 &=& [ANS] \\\\ \\hline c_1 &=& [ANS] \\\\ \\hline c_2 &=& [ANS] \\\\ \\hline c_3 &=& [ANS] \\\\ \\hline c_4 &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "0.135335", "0.135335", "0.0676676", "0.0225559", "0.00563897" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0928", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor series", "level": "3", "keywords": [ "taylor", "series" ], "problem_v1": "Let $f(x)= \\frac{7+6x}{x}$.\nCompute\n$\\begin{array}{ccccccccccccc}\\hline f(x) & &=& & [ANS] & & \\hskip 30pt & & f(1) & &=& & [ANS] \\\\ \\hline f^\\prime(x) & &=& & [ANS] & & \\hskip 30pt & & f^\\prime(1) & &=& & [ANS] \\\\ \\hline f^{\\prime\\prime}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{\\prime\\prime}(1) & &=& & [ANS] \\\\ \\hline f^{\\prime\\prime\\prime}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{\\prime\\prime\\prime}(1) & &=& & [ANS] \\\\ \\hline f^{(iv)}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{(iv)}(1) & &=& & [ANS] \\\\ \\hline f^{(v)}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{(v)}(1) & &=& & [ANS] \\\\ \\hline \\end{array}$\nWe see that the first term does not fit a pattern, but we also see that $f^{(k)}(1)$=[ANS] for $k\\geq 1$. Hence we see that the Taylor series for $f$ centered at $1$ is given by\n$f(x)=13+ \\sum_{k=1}^\\infty$ [ANS] $(x-1)^k$.", "answer_v1": [ "(7+6*x)/x", "13", "[6*x-(7+6*x)]/(x^2)", "-7", "-([6*x-(7+6*x)]*2*x/[(x^2)^2])", "14", "-[(2*[6*x-(7+6*x)]*(x^2)^2-[6*x-(7+6*x)]*2*x*2*x^2*2*x)/([(x^2)^2]^2)]", "-42", "-([(2*[6*x-(7+6*x)]*2*x^2*2*x-(2*[6*x-(7+6*x)]*2*x^2*2*x+[6*x-(7+6*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*[(x^2)^2]^2-(2*[6*x-(7+6*x)]*(x^2)^2-[6*x-(7+6*x)]*2*x*2*x^2*2*x)*2*(x^2)^2*2*x^2*2*x]/[([(x^2)^2]^2)^2])", "168", "-[([(2*[6*x-(7+6*x)]*(2*2*x*2*x+2*2*x^2)-(2*[6*x-(7+6*x)]*(2*2*x*2*x+2*2*x^2)+2*[6*x-(7+6*x)]*(2*2*x*2*x+2*2*x^2)+[6*x-(7+6*x)]*2*x*(4*2*x+2*2*2*x+2*2*2*x)))*[(x^2)^2]^2+(2*[6*x-(7+6*x)]*2*x^2*2*x-(2*[6*x-(7+6*x)]*2*x^2*2*x+[6*x-(7+6*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*2*(x^2)^2*2*x^2*2*x-[(2*[6*x-(7+6*x)]*2*x^2*2*x-(2*[6*x-(7+6*x)]*2*x^2*2*x+[6*x-(7+6*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*2*(x^2)^2*2*x^2*2*x+(2*[6*x-(7+6*x)]*(x^2)^2-[6*x-(7+6*x)]*2*x*2*x^2*2*x)*[2*2*x^2*2*x*2*x^2*2*x+2*(x^2)^2*(2*2*x*2*x+2*2*x^2)]]]*([(x^2)^2]^2)^2-[(2*[6*x-(7+6*x)]*2*x^2*2*x-(2*[6*x-(7+6*x)]*2*x^2*2*x+[6*x-(7+6*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*[(x^2)^2]^2-(2*[6*x-(7+6*x)]*(x^2)^2-[6*x-(7+6*x)]*2*x*2*x^2*2*x)*2*(x^2)^2*2*x^2*2*x]*2*[(x^2)^2]^2*2*(x^2)^2*2*x^2*2*x)/([([(x^2)^2]^2)^2]^2)]", "-840", "(-1)^k*(k!)*7", "(-1)^k*7" ], "answer_type_v1": [ "EX", "NV", "EX", "NV", "EX", "NV", "EX", "NV", "EX", "NV", "EX", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $f(x)= \\frac{2+8x}{x}$.\nCompute\n$\\begin{array}{ccccccccccccc}\\hline f(x) & &=& & [ANS] & & \\hskip 30pt & & f(1) & &=& & [ANS] \\\\ \\hline f^\\prime(x) & &=& & [ANS] & & \\hskip 30pt & & f^\\prime(1) & &=& & [ANS] \\\\ \\hline f^{\\prime\\prime}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{\\prime\\prime}(1) & &=& & [ANS] \\\\ \\hline f^{\\prime\\prime\\prime}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{\\prime\\prime\\prime}(1) & &=& & [ANS] \\\\ \\hline f^{(iv)}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{(iv)}(1) & &=& & [ANS] \\\\ \\hline f^{(v)}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{(v)}(1) & &=& & [ANS] \\\\ \\hline \\end{array}$\nWe see that the first term does not fit a pattern, but we also see that $f^{(k)}(1)$=[ANS] for $k\\geq 1$. Hence we see that the Taylor series for $f$ centered at $1$ is given by\n$f(x)=10+ \\sum_{k=1}^\\infty$ [ANS] $(x-1)^k$.", "answer_v2": [ "(2+8*x)/x", "10", "[8*x-(2+8*x)]/(x^2)", "-2", "-([8*x-(2+8*x)]*2*x/[(x^2)^2])", "4", "-[(2*[8*x-(2+8*x)]*(x^2)^2-[8*x-(2+8*x)]*2*x*2*x^2*2*x)/([(x^2)^2]^2)]", "-12", "-([(2*[8*x-(2+8*x)]*2*x^2*2*x-(2*[8*x-(2+8*x)]*2*x^2*2*x+[8*x-(2+8*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*[(x^2)^2]^2-(2*[8*x-(2+8*x)]*(x^2)^2-[8*x-(2+8*x)]*2*x*2*x^2*2*x)*2*(x^2)^2*2*x^2*2*x]/[([(x^2)^2]^2)^2])", "48", "-[([(2*[8*x-(2+8*x)]*(2*2*x*2*x+2*2*x^2)-(2*[8*x-(2+8*x)]*(2*2*x*2*x+2*2*x^2)+2*[8*x-(2+8*x)]*(2*2*x*2*x+2*2*x^2)+[8*x-(2+8*x)]*2*x*(4*2*x+2*2*2*x+2*2*2*x)))*[(x^2)^2]^2+(2*[8*x-(2+8*x)]*2*x^2*2*x-(2*[8*x-(2+8*x)]*2*x^2*2*x+[8*x-(2+8*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*2*(x^2)^2*2*x^2*2*x-[(2*[8*x-(2+8*x)]*2*x^2*2*x-(2*[8*x-(2+8*x)]*2*x^2*2*x+[8*x-(2+8*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*2*(x^2)^2*2*x^2*2*x+(2*[8*x-(2+8*x)]*(x^2)^2-[8*x-(2+8*x)]*2*x*2*x^2*2*x)*[2*2*x^2*2*x*2*x^2*2*x+2*(x^2)^2*(2*2*x*2*x+2*2*x^2)]]]*([(x^2)^2]^2)^2-[(2*[8*x-(2+8*x)]*2*x^2*2*x-(2*[8*x-(2+8*x)]*2*x^2*2*x+[8*x-(2+8*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*[(x^2)^2]^2-(2*[8*x-(2+8*x)]*(x^2)^2-[8*x-(2+8*x)]*2*x*2*x^2*2*x)*2*(x^2)^2*2*x^2*2*x]*2*[(x^2)^2]^2*2*(x^2)^2*2*x^2*2*x)/([([(x^2)^2]^2)^2]^2)]", "-240", "(-1)^k*(k!)*2", "(-1)^k*2" ], "answer_type_v2": [ "EX", "NV", "EX", "NV", "EX", "NV", "EX", "NV", "EX", "NV", "EX", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $f(x)= \\frac{4+6x}{x}$.\nCompute\n$\\begin{array}{ccccccccccccc}\\hline f(x) & &=& & [ANS] & & \\hskip 30pt & & f(1) & &=& & [ANS] \\\\ \\hline f^\\prime(x) & &=& & [ANS] & & \\hskip 30pt & & f^\\prime(1) & &=& & [ANS] \\\\ \\hline f^{\\prime\\prime}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{\\prime\\prime}(1) & &=& & [ANS] \\\\ \\hline f^{\\prime\\prime\\prime}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{\\prime\\prime\\prime}(1) & &=& & [ANS] \\\\ \\hline f^{(iv)}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{(iv)}(1) & &=& & [ANS] \\\\ \\hline f^{(v)}(x) & &=& & [ANS] & & \\hskip 30pt & & f^{(v)}(1) & &=& & [ANS] \\\\ \\hline \\end{array}$\nWe see that the first term does not fit a pattern, but we also see that $f^{(k)}(1)$=[ANS] for $k\\geq 1$. Hence we see that the Taylor series for $f$ centered at $1$ is given by\n$f(x)=10+ \\sum_{k=1}^\\infty$ [ANS] $(x-1)^k$.", "answer_v3": [ "(4+6*x)/x", "10", "[6*x-(4+6*x)]/(x^2)", "-4", "-([6*x-(4+6*x)]*2*x/[(x^2)^2])", "8", "-[(2*[6*x-(4+6*x)]*(x^2)^2-[6*x-(4+6*x)]*2*x*2*x^2*2*x)/([(x^2)^2]^2)]", "-24", "-([(2*[6*x-(4+6*x)]*2*x^2*2*x-(2*[6*x-(4+6*x)]*2*x^2*2*x+[6*x-(4+6*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*[(x^2)^2]^2-(2*[6*x-(4+6*x)]*(x^2)^2-[6*x-(4+6*x)]*2*x*2*x^2*2*x)*2*(x^2)^2*2*x^2*2*x]/[([(x^2)^2]^2)^2])", "96", "-[([(2*[6*x-(4+6*x)]*(2*2*x*2*x+2*2*x^2)-(2*[6*x-(4+6*x)]*(2*2*x*2*x+2*2*x^2)+2*[6*x-(4+6*x)]*(2*2*x*2*x+2*2*x^2)+[6*x-(4+6*x)]*2*x*(4*2*x+2*2*2*x+2*2*2*x)))*[(x^2)^2]^2+(2*[6*x-(4+6*x)]*2*x^2*2*x-(2*[6*x-(4+6*x)]*2*x^2*2*x+[6*x-(4+6*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*2*(x^2)^2*2*x^2*2*x-[(2*[6*x-(4+6*x)]*2*x^2*2*x-(2*[6*x-(4+6*x)]*2*x^2*2*x+[6*x-(4+6*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*2*(x^2)^2*2*x^2*2*x+(2*[6*x-(4+6*x)]*(x^2)^2-[6*x-(4+6*x)]*2*x*2*x^2*2*x)*[2*2*x^2*2*x*2*x^2*2*x+2*(x^2)^2*(2*2*x*2*x+2*2*x^2)]]]*([(x^2)^2]^2)^2-[(2*[6*x-(4+6*x)]*2*x^2*2*x-(2*[6*x-(4+6*x)]*2*x^2*2*x+[6*x-(4+6*x)]*2*x*(2*2*x*2*x+2*2*x^2)))*[(x^2)^2]^2-(2*[6*x-(4+6*x)]*(x^2)^2-[6*x-(4+6*x)]*2*x*2*x^2*2*x)*2*(x^2)^2*2*x^2*2*x]*2*[(x^2)^2]^2*2*(x^2)^2*2*x^2*2*x)/([([(x^2)^2]^2)^2]^2)]", "-480", "(-1)^k*(k!)*4", "(-1)^k*4" ], "answer_type_v3": [ "EX", "NV", "EX", "NV", "EX", "NV", "EX", "NV", "EX", "NV", "EX", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0929", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor series", "level": "2", "keywords": [ "calculus", "integral", "taylor series", "series" ], "problem_v1": "Find the first four terms of the Taylor series for the function $4 \\sin(x)$ about the point $a=\\pi/6$. (Your answers should include the variable x when appropriate.) degree 0 term=[ANS]\ndegree 1 term=[ANS]\ndegree 2 term=[ANS]\ndegree 3 term=[ANS]", "answer_v1": [ "4/2", "sqrt(3)*4*(x-pi/6)/2", "-4*(x-pi/6)^2/4", "-4*(x-pi/6)^3/(4*sqrt(3))" ], "answer_type_v1": [ "NV", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the first four terms of the Taylor series for the function $\\cos(x)$ about the point $a=-\\pi/4$. (Your answers should include the variable x when appropriate.) degree 0 term=[ANS]\ndegree 1 term=[ANS]\ndegree 2 term=[ANS]\ndegree 3 term=[ANS]", "answer_v2": [ "1/sqrt(2)", "1*(x+pi/4)/sqrt(2)", "-1*(x+pi/4)^2/(2*sqrt(2))", "-1*(x+pi/4)^3/(6*sqrt(2))" ], "answer_type_v2": [ "NV", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the first four terms of the Taylor series for the function $2 \\sin(x)$ about the point $a=\\pi/4$. (Your answers should include the variable x when appropriate.) degree 0 term=[ANS]\ndegree 1 term=[ANS]\ndegree 2 term=[ANS]\ndegree 3 term=[ANS]", "answer_v3": [ "2/sqrt(2)", "2*(x-pi/4)/sqrt(2)", "-2*(x-pi/4)^2/(2*sqrt(2))", "-2*(x-pi/4)^3/(6*sqrt(2))" ], "answer_type_v3": [ "NV", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0930", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor series", "level": "3", "keywords": [ "taylor series" ], "problem_v1": "By recognizing the series-((1/2))+\\frac{((1/2))^2}2-\\frac{((1/2))^3}3+\\cdots+(-1)^{n}\\frac{((1/2))^n}n+\\cdots as a Taylor series evaluated at a particular value of $x$, find the sum of the convergent series. sum=[ANS]", "answer_v1": [ "ln(1/(1+0.5))" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "By recognizing the series-((1/4))-\\frac{((1/4))^2}2-\\frac{((1/4))^3}3-\\cdots-\\frac{((1/4))^n}n-\\cdots as a Taylor series evaluated at a particular value of $x$, find the sum of the convergent series. sum=[ANS]", "answer_v2": [ "ln(1-0.25)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "By recognizing the series ((1/2))+\\frac{((1/2))^2}2+\\frac{((1/2))^3}3+\\cdots+\\frac{((1/2))^n}n+\\cdots as a Taylor series evaluated at a particular value of $x$, find the sum of the convergent series. sum=[ANS]", "answer_v3": [ "ln(1/(1-0.5))" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0931", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor series", "level": "3", "keywords": [ "calculus", "integral", "taylor series", "series" ], "problem_v1": "For each of the following, solve exactly for the variable $x$.\n(a) $1-\\frac{x^2}{2!}+\\frac{x^4}{4!}-\\cdots=0.3$ $x=$ [ANS]\n(b) $1+x+\\frac{x^2}{2!}+\\frac{x^3}{3!}+\\cdots=3$ $x=$ [ANS]", "answer_v1": [ "acos(0.3)", "ln(3)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "For each of the following, solve exactly for the variable $x$.\n(a) $1+x+x^2+x^3+\\cdots=7$ $x=$ [ANS]\n(b) $1-\\frac{x^2}{2!}+\\frac{x^4}{4!}-\\cdots=0.3$ $x=$ [ANS]", "answer_v2": [ "(7-1)/(1*7)", "acos(0.3)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "For each of the following, solve exactly for the variable $x$.\n(a) $x-\\frac{x^3}{3!}+\\frac{x^5}{5!}-\\cdots=0.2$ $x=$ [ANS]\n(b) $1-\\frac{x^2}{2!}+\\frac{x^4}{4!}-\\cdots=0.4$ $x=$ [ANS]", "answer_v3": [ "asin(0.2)", "acos(0.4)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0932", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor series", "level": "2", "keywords": [ "calculus", "integral", "taylor series", "series" ], "problem_v1": "Find the first four nonzero terms of the Taylor series for the function f(x)=\\sin(-7x) about $x=2$. $f(x)\\vert_{x \\approx 2} \\approx$ [ANS]", "answer_v1": [ "-[sin(14)]-7*cos(14)*(x-2)+49/2*sin(14)*(x-2)^2+343/6*cos(14)*(x-2)^3" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the first four nonzero terms of the Taylor series for the function f(x)=\\sin(-2x) about $x=3$. $f(x)\\vert_{x \\approx 3} \\approx$ [ANS]", "answer_v2": [ "-[sin(6)]-2*cos(6)*(x-3)+4/2*sin(6)*(x-3)^2+8/6*cos(6)*(x-3)^3" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the first four nonzero terms of the Taylor series for the function f(x)=\\sin(-4x) about $x=2$. $f(x)\\vert_{x \\approx 2} \\approx$ [ANS]", "answer_v3": [ "-[sin(8)]-4*cos(8)*(x-2)+16/2*sin(8)*(x-2)^2+64/6*cos(8)*(x-2)^3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0933", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor series", "level": "3", "keywords": [ "linear approximation", "differentials" ], "problem_v1": "Use sigma notation to write the Taylor series about $x=x_0$ for the function. $\\cos\\!\\left(2x\\right),\\;x_0=\\frac{\\pi}{4}$. Taylor series $=\\sum_{k=0}^{\\infty}$ [ANS]", "answer_v1": [ "(-1)^{k+1}*2^{2*k+1}*(x-pi/4)^{2*k+1}/(2*k+1)!" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use sigma notation to write the Taylor series about $x=x_0$ for the function. $\\cos\\!\\left(-5x\\right),\\;x_0=-\\frac{\\pi}{10}$. Taylor series $=\\sum_{k=0}^{\\infty}$ [ANS]", "answer_v2": [ "(-1)^{k+1}*(-5)^{2*k+1}*[x+(pi/10)]^{2*k+1}/(2*k+1)!" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use sigma notation to write the Taylor series about $x=x_0$ for the function. $\\cos\\!\\left(-2x\\right),\\;x_0=-\\frac{\\pi}{4}$. Taylor series $=\\sum_{k=0}^{\\infty}$ [ANS]", "answer_v3": [ "(-1)^{k+1}*(-2)^{2*k+1}*[x+(pi/4)]^{2*k+1}/(2*k+1)!" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0934", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Taylor series", "level": "3", "keywords": [ "calculus", "taylor series", "interval of convergence" ], "problem_v1": "Find the first three nonzero terms of the Taylor series for the function $f(x)=\\sqrt{8x-x^{2}}$ about the point $a=4$. (Your answers should include the variable x when appropriate.)\n$\\sqrt{8x-x^{2}}=$ [ANS]+[ANS]+[ANS]+...", "answer_v1": [ "4", "-1/8*(x-4)^2", "-1/512*(x-4)^4" ], "answer_type_v1": [ "NV", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the first three nonzero terms of the Taylor series for the function $f(x)=\\sqrt{2x-x^{2}}$ about the point $a=1$. (Your answers should include the variable x when appropriate.)\n$\\sqrt{2x-x^{2}}=$ [ANS]+[ANS]+[ANS]+...", "answer_v2": [ "1", "-1/2*(x-1)^2", "-1/8*(x-1)^4" ], "answer_type_v2": [ "NV", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the first three nonzero terms of the Taylor series for the function $f(x)=\\sqrt{4x-x^{2}}$ about the point $a=2$. (Your answers should include the variable x when appropriate.)\n$\\sqrt{4x-x^{2}}=$ [ANS]+[ANS]+[ANS]+...", "answer_v3": [ "2", "-1/4*(x-2)^2", "-1/64*(x-2)^4" ], "answer_type_v3": [ "NV", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0935", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Power series", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Suppose that $f(x)$ and $g(x)$ are given by the power series $f(x)=6+5x+5x^2+4x^3+\\cdots$ and $g(x)=3+4x+4x^2+3x^3+\\cdots.$ By multiplying power series, find the first few terms of the series for the product $h(x)=f(x)\\cdot g(x)=c_0+c_1x+c_2x^2+c_3x^3+\\cdots.$\n$c_0$=[ANS]\n$c_1$=[ANS]\n$c_2$=[ANS]\n$c_3$=[ANS]", "answer_v1": [ "18", "39", "59", "70" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose that $f(x)$ and $g(x)$ are given by the power series $f(x)=2+7x+2x^2+3x^3+\\cdots$ and $g(x)=7+4x+2x^2+2x^3+\\cdots.$ By multiplying power series, find the first few terms of the series for the product $h(x)=f(x)\\cdot g(x)=c_0+c_1x+c_2x^2+c_3x^3+\\cdots.$\n$c_0$=[ANS]\n$c_1$=[ANS]\n$c_2$=[ANS]\n$c_3$=[ANS]", "answer_v2": [ "14", "57", "46", "47" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose that $f(x)$ and $g(x)$ are given by the power series $f(x)=3+5x+3x^2+4x^3+\\cdots$ and $g(x)=3+4x+5x^2+4x^3+\\cdots.$ By multiplying power series, find the first few terms of the series for the product $h(x)=f(x)\\cdot g(x)=c_0+c_1x+c_2x^2+c_3x^3+\\cdots.$\n$c_0$=[ANS]\n$c_1$=[ANS]\n$c_2$=[ANS]\n$c_3$=[ANS]", "answer_v3": [ "9", "27", "44", "61" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0936", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Representations of functions as series", "level": "3", "keywords": [ "taylor series" ], "problem_v1": "Expand the quantity \\sqrt[5]{P+t} about 0 in terms of $\\frac tP$ Give four nonzero terms. $\\sqrt[5]{P+t} \\approx$ [ANS]", "answer_v1": [ "P^{0.2}*[1+1/5*t/P+1/5*(1/5-1)/2!*(t/P)^2+1/5*(1/5-1)*(1/5-2)/3!*(t/P)^3]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Expand the quantity \\sqrt[2]{P+t} about 0 in terms of $\\frac tP$ Give four nonzero terms. $\\sqrt[2]{P+t} \\approx$ [ANS]", "answer_v2": [ "P^{0.5}*[1+1/2*t/P+1/2*(1/2-1)/2!*(t/P)^2+1/2*(1/2-1)*(1/2-2)/3!*(t/P)^3]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Expand the quantity \\sqrt[3]{P+t} about 0 in terms of $\\frac tP$ Give four nonzero terms. $\\sqrt[3]{P+t} \\approx$ [ANS]", "answer_v3": [ "P^{\\frac{1}{3}}*[1+1/3*t/P+1/3*(1/3-1)/2!*(t/P)^2+1/3*(1/3-1)*(1/3-2)/3!*(t/P)^3]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0937", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Representations of functions as series", "level": "3", "keywords": [ "calculus", "integral", "taylor series", "series" ], "problem_v1": "Find the Taylor series about 0 for each of the functions below. Give the first three non-zero terms for each. A. $2\\cos(x)+x^2=$ [ANS] $+$ [ANS] $+$ [ANS] $+\\cdots$ B. ${1\\over\\sqrt{1+x^{3}}}=$ [ANS] $+$ [ANS] $+$ [ANS] $+\\cdots$ For each of these series, also be sure that you can find the general term in the series! For each of these series, also be sure that you can find the general term in the series!", "answer_v1": [ "2", "2*x^4/(4!)", "-2*x^6/(6!)", "1", "-x^3/2", "3*x^6/8" ], "answer_type_v1": [ "NV", "EX", "EX", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find the Taylor series about 0 for each of the functions below. Give the first three non-zero terms for each. A. $(1+2x)^{4}=$ [ANS] $+$ [ANS] $+$ [ANS] $+\\cdots$ B. $2\\cos(x)+x^2=$ [ANS] $+$ [ANS] $+$ [ANS] $+\\cdots$ For each of these series, also be sure that you can find the general term in the series! For each of these series, also be sure that you can find the general term in the series!", "answer_v2": [ "1", "8*x", "24*x^2", "2", "2*x^4/(4!)", "-2*x^6/(6!)" ], "answer_type_v2": [ "NV", "EX", "EX", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find the Taylor series about 0 for each of the functions below. Give the first three non-zero terms for each. A. $x^{3} \\sin(x^2)-x^{5}=$ [ANS] $+$ [ANS] $+$ [ANS] $+\\cdots$ B. $2\\cos(x)+x^2=$ [ANS] $+$ [ANS] $+$ [ANS] $+\\cdots$ For each of these series, also be sure that you can find the general term in the series! For each of these series, also be sure that you can find the general term in the series!", "answer_v3": [ "-x^9/(3!)", "x^13/(5!)", "-x^17/(7!)", "2", "2*x^4/(4!)", "-2*x^6/(6!)" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0938", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Representations of functions as series", "level": "3", "keywords": [ "Power Series' 'integral", "calculus", "power series", "integral", "estimate", "arctan", "Series", "Power", "Radius", "Convergence", "Coefficient", "Estimate", "Error", "Power Series" ], "problem_v1": "(a) Evaluate the integral $ \\int_{0}^{2} \\frac{40}{x^2+4} dx$. Your answer should be in the form $k\\pi$, where $k$ is an integer. What is the value of $k$? (Hint: $\\frac{d \\arctan(x)}{dx}=\\frac{1}{x^2+1}$) $k=$ [ANS]\n(b) Now, lets evaluate the same integral using power series. First, find the power series for the function $f(x)=\\frac{40}{x^2+4}$. Then, integrate it from 0 to 2, and call it S. S should be an infinite series $\\sum_{n=0}^\\infty a_n$. What are the first few terms of S? $a_0=$ [ANS]\n$a_1=$ [ANS]\n$a_2=$ [ANS]\n$a_3=$ [ANS]\n$a_4=$ [ANS]\n(c) The answer in part (a) equals the sum of the infinite series in part (b) (why?). Hence, if you divide your infinite series from (b) by $k$ (the answer to\n(a)), you have found an estimate for the value of $\\pi$ in terms of an infinite series. Approximate the value of $\\pi$ by the first 5 terms. [ANS]. (d) What is an upper bound for your error of your estimate if you use the first 10 terms? (Use the alternating series estimation.) [ANS].", "answer_v1": [ "5", "20", "-6.66666666666667", "4", "-2.85714285714286", "2.22222222222222", "3.33968253968254", "0.19047619047619" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "(a) Evaluate the integral $ \\int_{0}^{2} \\frac{16}{x^2+4} dx$. Your answer should be in the form $k\\pi$, where $k$ is an integer. What is the value of $k$? (Hint: $\\frac{d \\arctan(x)}{dx}=\\frac{1}{x^2+1}$) $k=$ [ANS]\n(b) Now, lets evaluate the same integral using power series. First, find the power series for the function $f(x)=\\frac{16}{x^2+4}$. Then, integrate it from 0 to 2, and call it S. S should be an infinite series $\\sum_{n=0}^\\infty a_n$. What are the first few terms of S? $a_0=$ [ANS]\n$a_1=$ [ANS]\n$a_2=$ [ANS]\n$a_3=$ [ANS]\n$a_4=$ [ANS]\n(c) The answer in part (a) equals the sum of the infinite series in part (b) (why?). Hence, if you divide your infinite series from (b) by $k$ (the answer to\n(a)), you have found an estimate for the value of $\\pi$ in terms of an infinite series. Approximate the value of $\\pi$ by the first 5 terms. [ANS]. (d) What is an upper bound for your error of your estimate if you use the first 12 terms? (Use the alternating series estimation.) [ANS].", "answer_v2": [ "2", "8", "-2.66666666666667", "1.6", "-1.14285714285714", "0.888888888888889", "3.33968253968254", "0.16" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "(a) Evaluate the integral $ \\int_{0}^{2} \\frac{24}{x^2+4} dx$. Your answer should be in the form $k\\pi$, where $k$ is an integer. What is the value of $k$? (Hint: $\\frac{d \\arctan(x)}{dx}=\\frac{1}{x^2+1}$) $k=$ [ANS]\n(b) Now, lets evaluate the same integral using power series. First, find the power series for the function $f(x)=\\frac{24}{x^2+4}$. Then, integrate it from 0 to 2, and call it S. S should be an infinite series $\\sum_{n=0}^\\infty a_n$. What are the first few terms of S? $a_0=$ [ANS]\n$a_1=$ [ANS]\n$a_2=$ [ANS]\n$a_3=$ [ANS]\n$a_4=$ [ANS]\n(c) The answer in part (a) equals the sum of the infinite series in part (b) (why?). Hence, if you divide your infinite series from (b) by $k$ (the answer to\n(a)), you have found an estimate for the value of $\\pi$ in terms of an infinite series. Approximate the value of $\\pi$ by the first 5 terms. [ANS]. (d) What is an upper bound for your error of your estimate if you use the first 10 terms? (Use the alternating series estimation.) [ANS].", "answer_v3": [ "3", "12", "-4", "2.4", "-1.71428571428571", "1.33333333333333", "3.33968253968254", "0.19047619047619" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0939", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Applications of Taylor polynomials", "level": "3", "keywords": [ "Power", "Series", "Limit", "calculus", "taylor series", "maclaurin series", "Series", "Taylor", "Taylor Series", "cos" ], "problem_v1": "Evaluate \\lim_{x \\to 0} \\frac{e^{-3x^3}-1+3x^3-\\frac{9}{2}x^6}{12x^9}. Limit=[ANS]\nHint: Use a power series expansion.", "answer_v1": [ "-0.375" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate \\lim_{x \\to 0} \\frac{e^{-3x^3}-1+3x^3-\\frac{9}{2}x^6}{3x^9}. Limit=[ANS]\nHint: Use a power series expansion.", "answer_v2": [ "-1.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate \\lim_{x \\to 0} \\frac{e^{-3x^3}-1+3x^3-\\frac{9}{2}x^6}{6x^9}. Limit=[ANS]\nHint: Use a power series expansion.", "answer_v3": [ "-0.75" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0940", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Applications of Taylor polynomials", "level": "3", "keywords": [ "Series", "Maclaurin", "Derivative", "calculus", "Coefficient", "taylor series", "derivative", "maclaurin series" ], "problem_v1": "Let $ f(x)=\\frac{\\cos\\left(5x^{2} \\right)-1}{x^{3}}$. Evaluate the $9^{\\rm th}$ derivative of $f$ at $x=0$. $ f^{(9)}(0)=$ [ANS]\nHint: Build a Maclaurin series for $f(x)$ from the series for $\\cos (x)$.", "answer_v1": [ "-7.875E+6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $ f(x)=\\frac{\\cos\\left(2x^{2} \\right)-1}{x^{2}}$. Evaluate the $10^{\\rm th}$ derivative of $f$ at $x=0$. $ f^{(10)}(0)=$ [ANS]\nHint: Build a Maclaurin series for $f(x)$ from the series for $\\cos (x)$.", "answer_v2": [ "-322560" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $ f(x)=\\frac{\\cos\\left(3x^{2} \\right)-1}{x^{2}}$. Evaluate the $10^{\\rm th}$ derivative of $f$ at $x=0$. $ f^{(10)}(0)=$ [ANS]\nHint: Build a Maclaurin series for $f(x)$ from the series for $\\cos (x)$.", "answer_v3": [ "-3.67416E+6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0941", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Applications of Taylor polynomials", "level": "3", "keywords": [ "Series", "Taylor", "Integral", "Taylor Series", "Integrals", "calculus", "maclaurin series", "taylor series", "estimate", "integral", "MacLaurin" ], "problem_v1": "Let $ F(x)=\\int_0^{x} e^{-4 t^4} \\ dt$. Find the MacLaurin polynomial of degree 5 for $F(x)$. [ANS]\nUse this polynomial to estimate the value of $ \\int_0^{0.18} e^{-4x^4} \\ dx$. [ANS]", "answer_v1": [ "x - 4 * x^5 / 5", "0.17984883456" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $ F(x)=\\int_0^{x} e^{-5 t^4} \\ dt$. Find the MacLaurin polynomial of degree 5 for $F(x)$. [ANS]\nUse this polynomial to estimate the value of $ \\int_0^{0.1} e^{-5x^4} \\ dx$. [ANS]", "answer_v2": [ "x - 5 * x^5 / 5", "0.09999" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $ F(x)=\\int_0^{x} e^{-4 t^4} \\ dt$. Find the MacLaurin polynomial of degree 5 for $F(x)$. [ANS]\nUse this polynomial to estimate the value of $ \\int_0^{0.13} e^{-4x^4} \\ dx$. [ANS]", "answer_v3": [ "x - 4 * x^5 / 5", "0.12997029656" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0942", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Applications of Taylor polynomials", "level": "2", "keywords": [ "calculus", "integral", "taylor series", "series" ], "problem_v1": "For values of $y$ near 0, put the following functions in increasing order, by using their Taylor expansions.\n(a) $\\sin(y^2)$ (b) $1-{1\\over1+y^2}$ (c) $\\ln(1+y^2)$ [ANS] $<$ [ANS] $<$ [ANS]\n(Fill in the functions, as appropriate, in the answer blanks.) (Fill in the functions, as appropriate, in the answer blanks.)", "answer_v1": [ "1 - 1/(1+y^2)", "ln(1 + y^2)", "sin(y^2)" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "For values of $y$ near 0, put the following functions in increasing order, by using their Taylor expansions.\n(a) $\\sqrt{1+y^2}-1$ (b) ${1\\over1-y^2}-1$ (c) $1-\\cos(y)$ [ANS] $<$ [ANS] $<$ [ANS]\n(Fill in the functions, as appropriate, in the answer blanks.) (Fill in the functions, as appropriate, in the answer blanks.)", "answer_v2": [ "sqrt(1+y^2) - 1", "1 - cos(y)", "(1/(1 - y^2)) - 1" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "For values of $y$ near 0, put the following functions in increasing order, by using their Taylor expansions.\n(a) $y^2 e^{-y^2}$ (b) $\\ln(1+y^2)$ (c) $1-\\cos(y)$ [ANS] $<$ [ANS] $<$ [ANS]\n(Fill in the functions, as appropriate, in the answer blanks.) (Fill in the functions, as appropriate, in the answer blanks.)", "answer_v3": [ "1 - cos(y)", "y^2*e^{-y^2}", "ln(1 + y^2)" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0943", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Applications of Taylor polynomials", "level": "2", "keywords": [ "calculus", "integral", "taylor series", "polynomials" ], "problem_v1": "Suppose $g$ is a function which has continuous derivatives, and that $g(7)=1, g'(7)=1$, $g''(7)=2$, $g'''(7)=-2$.\n(a) What is the Taylor polynomial of degree 2 for $g$ near $7$? $P_2(x)=$ [ANS]\n(b) What is the Taylor polynomial of degree 3 for $g$ near $7$? $P_3(x)=$ [ANS]\n(c) Use the two polynomials that you found in parts\n(a) and (b) to approximate $g(6.9)$. With $P_2$, $g(6.9)\\approx$ [ANS]\nWith $P_3$, $g(6.9)\\approx$ [ANS]", "answer_v1": [ "1+1*(x-7)+1/2!*2*(x-7)^2", "1+1*(x-7)+1/2!*2*(x-7)^2+1/3!*-2*(x-7)^3", "0.91", "0.910333" ], "answer_type_v1": [ "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose $g$ is a function which has continuous derivatives, and that $g(1)=5, g'(1)=-4$, $g''(1)=-2$, $g'''(1)=5$.\n(a) What is the Taylor polynomial of degree 2 for $g$ near $1$? $P_2(x)=$ [ANS]\n(b) What is the Taylor polynomial of degree 3 for $g$ near $1$? $P_3(x)=$ [ANS]\n(c) Use the two polynomials that you found in parts\n(a) and (b) to approximate $g(0.9)$. With $P_2$, $g(0.9)\\approx$ [ANS]\nWith $P_3$, $g(0.9)\\approx$ [ANS]", "answer_v2": [ "5+-4*(x-1)+1/2!*-2*(x-1)^2", "5+-4*(x-1)+1/2!*-2*(x-1)^2+1/3!*5*(x-1)^3", "5.39", "5.38917" ], "answer_type_v2": [ "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose $g$ is a function which has continuous derivatives, and that $g(3)=1, g'(3)=-2$, $g''(3)=1$, $g'''(3)=-3$.\n(a) What is the Taylor polynomial of degree 2 for $g$ near $3$? $P_2(x)=$ [ANS]\n(b) What is the Taylor polynomial of degree 3 for $g$ near $3$? $P_3(x)=$ [ANS]\n(c) Use the two polynomials that you found in parts\n(a) and (b) to approximate $g(2.9)$. With $P_2$, $g(2.9)\\approx$ [ANS]\nWith $P_3$, $g(2.9)\\approx$ [ANS]", "answer_v3": [ "1+-2*(x-3)+1/2!*1*(x-3)^2", "1+-2*(x-3)+1/2!*1*(x-3)^2+1/3!*-3*(x-3)^3", "1.205", "1.2055" ], "answer_type_v3": [ "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0944", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Applications of Taylor polynomials", "level": "2", "keywords": [ "Power Series" ], "problem_v1": "The Taylor series of $f$, centered at $a$, is $f(x)=f(a)+f'(a) (x-a)+\\frac{12 f''(a) (x-a)^2+\\cdots$ In this problem you will use this formula with $f(x)=\\sqrt}{x}$ and $a=$ [ANS] to approximate $\\sqrt{10010}$. This is the best value of $a$ to use since it is close to 10010, and you can compute $f(a)$ without a calculator. In fact, you can easily compute $f'(a)$ and $f''(a)$ without a calculator, too. Try to do this whole problem without a calculator. The first approximation is $\\sqrt{10010} \\approx T_1(10010)=f(a)+f'(a) (x-a)=$ [ANS]\nYou might remember this as the linear approximation from Calculus I. The quadratic approximation to $f$ near $x=a$ gives $\\sqrt{10010} \\approx T_2(10010)=f(a)+f'(a) (x-a)+\\frac{12 f''(a) (x-a)^2=$ [ANS] The error (positive or negative) of the first approximation is $T_1(10010)-\\sqrt}{10010}=$ [ANS]\nThe error (positive or negative) of the second approximation is $T_2(10010)-\\sqrt{10010}=$ [ANS]", "answer_v1": [ "10000", "100.05", "100.0499875", "1.24937538998893E-05", "-6.24609697297274E-9" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "The Taylor series of $f$, centered at $a$, is $f(x)=f(a)+f'(a) (x-a)+\\frac{12 f''(a) (x-a)^2+\\cdots$ In this problem you will use this formula with $f(x)=\\sqrt}{x}$ and $a=$ [ANS] to approximate $\\sqrt{105}$. This is the best value of $a$ to use since it is close to 105, and you can compute $f(a)$ without a calculator. In fact, you can easily compute $f'(a)$ and $f''(a)$ without a calculator, too. Try to do this whole problem without a calculator. The first approximation is $\\sqrt{105} \\approx T_1(105)=f(a)+f'(a) (x-a)=$ [ANS]\nYou might remember this as the linear approximation from Calculus I. The quadratic approximation to $f$ near $x=a$ gives $\\sqrt{105} \\approx T_2(105)=f(a)+f'(a) (x-a)+\\frac{12 f''(a) (x-a)^2=$ [ANS] The error (positive or negative) of the first approximation is $T_1(105)-\\sqrt}{105}=$ [ANS]\nThe error (positive or negative) of the second approximation is $T_2(105)-\\sqrt{105}=$ [ANS]", "answer_v2": [ "100", "10.25", "10.246875", "0.00304923404040203", "-7.57659595986837E-5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "The Taylor series of $f$, centered at $a$, is $f(x)=f(a)+f'(a) (x-a)+\\frac{12 f''(a) (x-a)^2+\\cdots$ In this problem you will use this formula with $f(x)=\\sqrt}{x}$ and $a=$ [ANS] to approximate $\\sqrt{101}$. This is the best value of $a$ to use since it is close to 101, and you can compute $f(a)$ without a calculator. In fact, you can easily compute $f'(a)$ and $f''(a)$ without a calculator, too. Try to do this whole problem without a calculator. The first approximation is $\\sqrt{101} \\approx T_1(101)=f(a)+f'(a) (x-a)=$ [ANS]\nYou might remember this as the linear approximation from Calculus I. The quadratic approximation to $f$ near $x=a$ gives $\\sqrt{101} \\approx T_2(101)=f(a)+f'(a) (x-a)+\\frac{12 f''(a) (x-a)^2=$ [ANS] The error (positive or negative) of the first approximation is $T_1(101)-\\sqrt}{101}=$ [ANS]\nThe error (positive or negative) of the second approximation is $T_2(101)-\\sqrt{101}=$ [ANS]", "answer_v3": [ "100", "10.05", "10.049875", "0.000124378879110765", "-6.21120889832127E-7" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0945", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "3", "keywords": [ "sine series", "cosine series", "Fourier series", "odd periodic extension", "even periodic extension" ], "problem_v1": "Take the function $f(t)=8 t^{3}(t-4)$ defined on $[0,4]$. Let $F_{\\text{odd}}$ and $F_{\\text{even}}$ be the odd and the even periodic extensions.\nCompute: $F_{\\text{odd}}(0.1)={}$ [ANS]\n$F_{\\text{odd}}(-0.5)={}$ [ANS]\n$F_{\\text{odd}}(6)={}$ [ANS]\n$F_{\\text{odd}}(-6)={}$ [ANS]\n$F_{\\text{even}}(0.1)={}$ [ANS]\n$F_{\\text{even}}(-0.5)={}$ [ANS]\n$F_{\\text{even}}(6)={}$ [ANS]\n$F_{\\text{even}}(-6)={}$ [ANS]", "answer_v1": [ "8*0.1^3*(0.1-4)", "3.5", "128", "8*2^3*(2-4)", "8*0.1^3*(0.1-4)", "8*0.5^3*(0.5-4)", "8*2^3*(2-4)", "8*2^3*(2-4)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Take the function $f(t)=2 t^{4}(t-2)$ defined on $[0,2]$. Let $F_{\\text{odd}}$ and $F_{\\text{even}}$ be the odd and the even periodic extensions.\nCompute: $F_{\\text{odd}}(0.1)={}$ [ANS]\n$F_{\\text{odd}}(-0.5)={}$ [ANS]\n$F_{\\text{odd}}(3)={}$ [ANS]\n$F_{\\text{odd}}(-3)={}$ [ANS]\n$F_{\\text{even}}(0.1)={}$ [ANS]\n$F_{\\text{even}}(-0.5)={}$ [ANS]\n$F_{\\text{even}}(3)={}$ [ANS]\n$F_{\\text{even}}(-3)={}$ [ANS]", "answer_v2": [ "2*0.1^4*(0.1-2)", "0.1875", "2", "2*1^4*(1-2)", "2*0.1^4*(0.1-2)", "2*0.5^4*(0.5-2)", "2*1^4*(1-2)", "2*1^4*(1-2)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Take the function $f(t)=4 t^{3}(t-3)$ defined on $[0,3]$. Let $F_{\\text{odd}}$ and $F_{\\text{even}}$ be the odd and the even periodic extensions.\nCompute: $F_{\\text{odd}}(0.1)={}$ [ANS]\n$F_{\\text{odd}}(-0.5)={}$ [ANS]\n$F_{\\text{odd}}(4.5)={}$ [ANS]\n$F_{\\text{odd}}(-4.5)={}$ [ANS]\n$F_{\\text{even}}(0.1)={}$ [ANS]\n$F_{\\text{even}}(-0.5)={}$ [ANS]\n$F_{\\text{even}}(4.5)={}$ [ANS]\n$F_{\\text{even}}(-4.5)={}$ [ANS]", "answer_v3": [ "4*0.1^3*(0.1-3)", "1.25", "20.25", "4*1.5^3*(1.5-3)", "4*0.1^3*(0.1-3)", "4*0.5^3*(0.5-3)", "4*1.5^3*(1.5-3)", "4*1.5^3*(1.5-3)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0946", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "2", "keywords": [ "Fourier series" ], "problem_v1": "Compute the coefficients of the Fourier series for the 2-periodic function $f(t)=8+6 \\cos(2\\pi t)+6 \\sin(3\\pi t)$.\nWe are using the convention that the constant term is $\\frac{a_0}{2}$.\n$a_0={}$ [ANS]\n$a_1={}$ [ANS]\n$a_2={}$ [ANS]\n$a_3={}$ [ANS]\n$a_4={}$ [ANS]\n$b_1={}$ [ANS]\n$b_2={}$ [ANS]\n$b_3={}$ [ANS]\n$b_4={}$ [ANS]", "answer_v1": [ "8*2", "0", "6", "0", "0", "0", "0", "6", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Compute the coefficients of the Fourier series for the 2-periodic function $f(t)=2+9 \\cos(2\\pi t)+3 \\sin(3\\pi t)$.\nWe are using the convention that the constant term is $\\frac{a_0}{2}$.\n$a_0={}$ [ANS]\n$a_1={}$ [ANS]\n$a_2={}$ [ANS]\n$a_3={}$ [ANS]\n$a_4={}$ [ANS]\n$b_1={}$ [ANS]\n$b_2={}$ [ANS]\n$b_3={}$ [ANS]\n$b_4={}$ [ANS]", "answer_v2": [ "2*2", "0", "9", "0", "0", "0", "0", "3", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Compute the coefficients of the Fourier series for the 2-periodic function $f(t)=4+6 \\cos(2\\pi t)+4 \\sin(3\\pi t)$.\nWe are using the convention that the constant term is $\\frac{a_0}{2}$.\n$a_0={}$ [ANS]\n$a_1={}$ [ANS]\n$a_2={}$ [ANS]\n$a_3={}$ [ANS]\n$a_4={}$ [ANS]\n$b_1={}$ [ANS]\n$b_2={}$ [ANS]\n$b_3={}$ [ANS]\n$b_4={}$ [ANS]", "answer_v3": [ "4*2", "0", "6", "0", "0", "0", "0", "4", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0947", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "3", "keywords": [ "Fourier series" ], "problem_v1": "Take the function defined by the series $f(t)=\\sum_{n=1}^\\infty \\frac{8}{n^5} \\sin(n \\pi t)$.\nFind the series of the derivative $f'(t)=\\frac{a_0}{2}+\\sum_{n=1}^\\infty a_n \\cos(n \\pi t)+b_n \\sin(n \\pi t)$. $a_0={}$ [ANS]\n$a_n={}$ [ANS]\n$b_n={}$ [ANS]", "answer_v1": [ "0", "pi*8/(n^4)", "0" ], "answer_type_v1": [ "NV", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Take the function defined by the series $f(t)=\\sum_{n=1}^\\infty \\frac{2}{n^5} \\sin(n \\pi t)$.\nFind the series of the derivative $f'(t)=\\frac{a_0}{2}+\\sum_{n=1}^\\infty a_n \\cos(n \\pi t)+b_n \\sin(n \\pi t)$. $a_0={}$ [ANS]\n$a_n={}$ [ANS]\n$b_n={}$ [ANS]", "answer_v2": [ "0", "pi*2/(n^4)", "0" ], "answer_type_v2": [ "NV", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Take the function defined by the series $f(t)=\\sum_{n=1}^\\infty \\frac{4}{n^5} \\sin(n \\pi t)$.\nFind the series of the derivative $f'(t)=\\frac{a_0}{2}+\\sum_{n=1}^\\infty a_n \\cos(n \\pi t)+b_n \\sin(n \\pi t)$. $a_0={}$ [ANS]\n$a_n={}$ [ANS]\n$b_n={}$ [ANS]", "answer_v3": [ "0", "pi*4/(n^4)", "0" ], "answer_type_v3": [ "NV", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0948", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "3", "keywords": [], "problem_v1": "If $f$ is the Fourier series of $g(x)=5x+7$ for $-8 < x < 8$, then $f(x)=$ [ANS] $+\\sum_{n=1}^{\\infty} \\Bigg[\\Big($ [ANS] $\\Big)\\cos\\Big(\\frac{n \\pi}{8}x\\Big)+\\Big($ [ANS] $\\Big)\\sin\\Big(\\frac{n \\pi}{8}x\\Big) \\Bigg]$ What does $f(-8)$ equal? $\\ \\ \\ \\ f(-8)=$ [ANS]\nWhat does $f(-4)$ equal? $\\ \\ \\ \\ f(-4)=$ [ANS]\nWhat does $f(0)$ equal? $\\ \\ \\ \\ f(0)=$ [ANS]\nWhat does $f(6)$ equal? $\\ \\ \\ \\ f(6)=$ [ANS]\nWhat does $f(8)$ equal? $\\ \\ \\ \\ f(8)=$ [ANS]", "answer_v1": [ "7", "0", "-2*5*8*cos(n*3.14159)/(n*3.14159)", "7", "-13", "7", "37", "7" ], "answer_type_v1": [ "NV", "NV", "EX", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "If $f$ is the Fourier series of $g(x)=5x+7$ for $-8 < x < 8$, then $f(x)=$ [ANS] $+\\sum_{n=1}^{\\infty} \\Bigg[\\Big($ [ANS] $\\Big)\\cos\\Big(\\frac{n \\pi}{8}x\\Big)+\\Big($ [ANS] $\\Big)\\sin\\Big(\\frac{n \\pi}{8}x\\Big) \\Bigg]$ What does $f(-8)$ equal? $\\ \\ \\ \\ f(-8)=$ [ANS]\nWhat does $f(-4)$ equal? $\\ \\ \\ \\ f(-4)=$ [ANS]\nWhat does $f(0)$ equal? $\\ \\ \\ \\ f(0)=$ [ANS]\nWhat does $f(1)$ equal? $\\ \\ \\ \\ f(1)=$ [ANS]\nWhat does $f(8)$ equal? $\\ \\ \\ \\ f(8)=$ [ANS]", "answer_v2": [ "7", "0", "-2*5*8*cos(n*3.14159)/(n*3.14159)", "7", "-13", "7", "12", "7" ], "answer_type_v2": [ "NV", "NV", "EX", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "If $f$ is the Fourier series of $g(x)=5x+7$ for $-8 < x < 8$, then $f(x)=$ [ANS] $+\\sum_{n=1}^{\\infty} \\Bigg[\\Big($ [ANS] $\\Big)\\cos\\Big(\\frac{n \\pi}{8}x\\Big)+\\Big($ [ANS] $\\Big)\\sin\\Big(\\frac{n \\pi}{8}x\\Big) \\Bigg]$ What does $f(-8)$ equal? $\\ \\ \\ \\ f(-8)=$ [ANS]\nWhat does $f(-4)$ equal? $\\ \\ \\ \\ f(-4)=$ [ANS]\nWhat does $f(0)$ equal? $\\ \\ \\ \\ f(0)=$ [ANS]\nWhat does $f(3)$ equal? $\\ \\ \\ \\ f(3)=$ [ANS]\nWhat does $f(8)$ equal? $\\ \\ \\ \\ f(8)=$ [ANS]", "answer_v3": [ "7", "0", "-2*5*8*cos(n*3.14159)/(n*3.14159)", "7", "-13", "7", "22", "7" ], "answer_type_v3": [ "NV", "NV", "EX", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0949", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "5", "keywords": [ "series", "fourier", "Fourier" ], "problem_v1": "Compute the Fourier series of the function $f(x)=8\\sin\\!\\left(4x\\right)$ on $[-\\pi,\\pi]$ using the trig identities\n$ \\sin(u)\\sin(v)=\\frac{1}{2}\\left[\\cos(u-v)-\\cos(u+v)\\right]$\n$ \\sin(u)\\cos(v)=\\frac{1}{2}\\left[\\sin(u+v)+\\sin(u-v)\\right]$\n$\\begin{array}{ccccccccccc}\\hline a_0 & &=& & [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline a_n & &=& & [ANS] \\int [ANS] & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$ \\mbox{Apply trig identity}$\n$\\begin{array}{ccccccccccccccc}\\hline \\hskip 10pt & &=\\frac{8}{2\\pi}\\Bigg[& & [ANS] \\int [ANS] & & [ANS] & & dx+& & [ANS] \\int [ANS] & & [ANS] & & dx \\Bigg] \\\\ \\hline \\end{array}$\n$\\hskip 15pt=$ [ANS]\n$\\begin{array}{ccccccccc}\\hline b_n & &=& & [ANS] \\int [ANS] & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$ \\mbox{Apply trig identity}$\n$\\begin{array}{ccccccccccccccc}\\hline \\hskip 10pt & &=\\frac{8}{2\\pi}\\Bigg[& & [ANS] \\int [ANS] & & [ANS] & & dx-& & [ANS] \\int [ANS] & & [ANS] & & dx \\Bigg] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline \\hskip 10pt & &=& & [ANS] n=4 [ANS] n\\neq4 \\\\ \\hline \\end{array}$", "answer_v1": [ "-3.14159", "pi", "0.31831*8*sin(4*x)", "0", "-3.14159", "pi", "0.31831*8*sin(4*x)*cos(n*x)", "-3.14159", "pi", "sin((4+n)*x)", "-3.14159", "pi", "sin((4-n)*x)", "0", "-3.14159", "pi", "0.31831*8*sin(4*x)*sin(n*x)", "-3.14159", "pi", "cos((4-n)*x)", "-3.14159", "pi", "cos((4+n)*x)", "8", "0" ], "answer_type_v1": [ "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Compute the Fourier series of the function $f(x)=2\\sin\\!\\left(5x\\right)$ on $[-\\pi,\\pi]$ using the trig identities\n$ \\sin(u)\\sin(v)=\\frac{1}{2}\\left[\\cos(u-v)-\\cos(u+v)\\right]$\n$ \\sin(u)\\cos(v)=\\frac{1}{2}\\left[\\sin(u+v)+\\sin(u-v)\\right]$\n$\\begin{array}{ccccccccccc}\\hline a_0 & &=& & [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline a_n & &=& & [ANS] \\int [ANS] & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$ \\mbox{Apply trig identity}$\n$\\begin{array}{ccccccccccccccc}\\hline \\hskip 10pt & &=\\frac{2}{2\\pi}\\Bigg[& & [ANS] \\int [ANS] & & [ANS] & & dx+& & [ANS] \\int [ANS] & & [ANS] & & dx \\Bigg] \\\\ \\hline \\end{array}$\n$\\hskip 15pt=$ [ANS]\n$\\begin{array}{ccccccccc}\\hline b_n & &=& & [ANS] \\int [ANS] & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$ \\mbox{Apply trig identity}$\n$\\begin{array}{ccccccccccccccc}\\hline \\hskip 10pt & &=\\frac{2}{2\\pi}\\Bigg[& & [ANS] \\int [ANS] & & [ANS] & & dx-& & [ANS] \\int [ANS] & & [ANS] & & dx \\Bigg] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline \\hskip 10pt & &=& & [ANS] n=5 [ANS] n\\neq5 \\\\ \\hline \\end{array}$", "answer_v2": [ "-3.14159", "pi", "0.31831*2*sin(5*x)", "0", "-3.14159", "pi", "0.31831*2*sin(5*x)*cos(n*x)", "-3.14159", "pi", "sin((5+n)*x)", "-3.14159", "pi", "sin((5-n)*x)", "0", "-3.14159", "pi", "0.31831*2*sin(5*x)*sin(n*x)", "-3.14159", "pi", "cos((5-n)*x)", "-3.14159", "pi", "cos((5+n)*x)", "2", "0" ], "answer_type_v2": [ "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Compute the Fourier series of the function $f(x)=4\\sin\\!\\left(4x\\right)$ on $[-\\pi,\\pi]$ using the trig identities\n$ \\sin(u)\\sin(v)=\\frac{1}{2}\\left[\\cos(u-v)-\\cos(u+v)\\right]$\n$ \\sin(u)\\cos(v)=\\frac{1}{2}\\left[\\sin(u+v)+\\sin(u-v)\\right]$\n$\\begin{array}{ccccccccccc}\\hline a_0 & &=& & [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccc}\\hline a_n & &=& & [ANS] \\int [ANS] & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$ \\mbox{Apply trig identity}$\n$\\begin{array}{ccccccccccccccc}\\hline \\hskip 10pt & &=\\frac{4}{2\\pi}\\Bigg[& & [ANS] \\int [ANS] & & [ANS] & & dx+& & [ANS] \\int [ANS] & & [ANS] & & dx \\Bigg] \\\\ \\hline \\end{array}$\n$\\hskip 15pt=$ [ANS]\n$\\begin{array}{ccccccccc}\\hline b_n & &=& & [ANS] \\int [ANS] & & [ANS] & & dx \\\\ \\hline \\end{array}$\n$ \\mbox{Apply trig identity}$\n$\\begin{array}{ccccccccccccccc}\\hline \\hskip 10pt & &=\\frac{4}{2\\pi}\\Bigg[& & [ANS] \\int [ANS] & & [ANS] & & dx-& & [ANS] \\int [ANS] & & [ANS] & & dx \\Bigg] \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccc}\\hline \\hskip 10pt & &=& & [ANS] n=4 [ANS] n\\neq4 \\\\ \\hline \\end{array}$", "answer_v3": [ "-3.14159", "pi", "0.31831*4*sin(4*x)", "0", "-3.14159", "pi", "0.31831*4*sin(4*x)*cos(n*x)", "-3.14159", "pi", "sin((4+n)*x)", "-3.14159", "pi", "sin((4-n)*x)", "0", "-3.14159", "pi", "0.31831*4*sin(4*x)*sin(n*x)", "-3.14159", "pi", "cos((4-n)*x)", "-3.14159", "pi", "cos((4+n)*x)", "4", "0" ], "answer_type_v3": [ "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0950", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "3", "keywords": [ "series", "fourier", "Fourier" ], "problem_v1": "Note: The formulas for the Fourier transform on half intervals are often given in the form $\\frac{2}{L}\\int_{0}^Lf(x)cos\\left(\\frac{n\\pi x}{L}\\right) dx$, with the $\\frac{2}{L}$ outside the integral. Computing these integrals will often involve u-substitutions, integration by parts, and other integration techniques that will produce all kinds of constants. For example the formula for the cosine coefficient would be $\\int_{0}^L\\frac{2}{L}f(x)cos\\left(\\frac{n\\pi x}{L}\\right) dx$. When performing an integration by parts, all constants are included in the u term.\nFor the function $f(x)=\\cases{1 &-8 < x < 0\\\\-1 & 0 < x < 8\\\\}$ on the interval $[-8,8]$ we need only compute one set of the Fourier coefficients:\n$\\begin{array}{ccccccc}\\hline [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "0", "8", "0.25*-1*sin(n*pi*x/8)", "2/(n*pi)*[(-1)^n-1]" ], "answer_type_v1": [ "NV", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Note: The formulas for the Fourier transform on half intervals are often given in the form $\\frac{2}{L}\\int_{0}^Lf(x)cos\\left(\\frac{n\\pi x}{L}\\right) dx$, with the $\\frac{2}{L}$ outside the integral. Computing these integrals will often involve u-substitutions, integration by parts, and other integration techniques that will produce all kinds of constants. For example the formula for the cosine coefficient would be $\\int_{0}^L\\frac{2}{L}f(x)cos\\left(\\frac{n\\pi x}{L}\\right) dx$. When performing an integration by parts, all constants are included in the u term.\nFor the function $f(x)=\\cases{1 &-2 < x < 0\\\\-1 & 0 < x < 2\\\\}$ on the interval $[-2,2]$ we need only compute one set of the Fourier coefficients:\n$\\begin{array}{ccccccc}\\hline [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "0", "2", "1*-1*sin(n*pi*x/2)", "2/(n*pi)*[(-1)^n-1]" ], "answer_type_v2": [ "NV", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Note: The formulas for the Fourier transform on half intervals are often given in the form $\\frac{2}{L}\\int_{0}^Lf(x)cos\\left(\\frac{n\\pi x}{L}\\right) dx$, with the $\\frac{2}{L}$ outside the integral. Computing these integrals will often involve u-substitutions, integration by parts, and other integration techniques that will produce all kinds of constants. For example the formula for the cosine coefficient would be $\\int_{0}^L\\frac{2}{L}f(x)cos\\left(\\frac{n\\pi x}{L}\\right) dx$. When performing an integration by parts, all constants are included in the u term.\nFor the function $f(x)=\\cases{1 &-4 < x < 0\\\\-1 & 0 < x < 4\\\\}$ on the interval $[-4,4]$ we need only compute one set of the Fourier coefficients:\n$\\begin{array}{ccccccc}\\hline [ANS] \\int [ANS] & & [ANS] & & dx=& & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "0", "4", "0.5*-1*sin(n*pi*x/4)", "2/(n*pi)*[(-1)^n-1]" ], "answer_type_v3": [ "NV", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0951", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "3", "keywords": [ "calculus", "integral", "fourier", "series", "approximation" ], "problem_v1": "Construct the first three Fourier approximations to the square wave function f(x)=\\begin{cases}1 &-\\pi \\le x <-\\pi/2\\cr-1 &-\\pi/2 \\le x < \\pi/2\\cr 1 & \\pi/2 \\le x < \\pi\\end{cases} $F_1(x)=$ [ANS]\n$F_2(x)=$ [ANS]\n$F_3(x)=$ [ANS]\nUsing a calculator, graph the function and the first three Fourier approximations to see how the approximation matches the function $f(x)$.", "answer_v1": [ "0-4/pi*cos(x)", "0-4/pi*cos(x)+0*cos(2*x)", "0-4/pi*cos(x)+0*cos(2*x)+4/(3*pi)*cos(3*x)" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Construct the first three Fourier approximations to the square wave function f(x)=\\begin{cases}-1 &-\\pi \\le x < 0\\cr 1 & 0 \\le x < \\pi\\end{cases} $F_1(x)=$ [ANS]\n$F_2(x)=$ [ANS]\n$F_3(x)=$ [ANS]\nUsing a calculator, graph the function and the first three Fourier approximations to see how the approximation matches the function $f(x)$.", "answer_v2": [ "0+4/pi*sin(x)", "0+4/pi*sin(x)+0*sin(2*x)", "0+4/pi*sin(x)+0*sin(2*x)+4/(3*pi)*sin(3*x)" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Construct the first three Fourier approximations to the square wave function f(x)=\\begin{cases}1 &-\\pi \\le x < 0\\cr-1 & 0 \\le x < \\pi\\end{cases} $F_1(x)=$ [ANS]\n$F_2(x)=$ [ANS]\n$F_3(x)=$ [ANS]\nUsing a calculator, graph the function and the first three Fourier approximations to see how the approximation matches the function $f(x)$.", "answer_v3": [ "0-4/pi*sin(x)", "0-4/pi*sin(x)+0*sin(2*x)", "0-4/pi*sin(x)+0*sin(2*x)-4/(3*pi)*sin(3*x)" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0952", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "2", "keywords": [ "calculus", "integral", "fourier", "series", "approximation" ], "problem_v1": "Suppose you're given the following Fourier coefficients for a function on the interval $[-\\pi,\\pi]$: $ a_0=\\frac{2}{2\\pi}$, $ a_1=\\frac{1}{\\pi}$, $ a_2=\\frac{-1}{2 \\pi}$, $ a_3=\\frac{1}{3 \\pi}$, $ b_1=\\frac{2}{\\pi}$, $ b_2=\\frac{-1}{2 \\pi}$, $ b_3=\\frac{-1}{3 \\pi}$. Find the following Fourier approximations to the Fourier series $ a_0+\\sum_{n=1}^{\\infty} (a_n\\cos(nx)+b_n\\sin(nx))$.\n$F_0(x)=$ [ANS]\n$F_1(x)=$ [ANS]\n$F_2(x)=$ R2 R2\n$F_3(x)=$", "answer_v1": [ "2/(2*pi)", "2/(2*pi)+1/pi*cos(x)+2/pi*sin(x)", "2/(2*pi)+1/pi*cos(x)+2/pi*sin(x)+-1/(2*pi)*cos(2*x)+-1/(2*pi)*sin(2*x)", "2/(2*pi)+1/pi*cos(x)+2/pi*sin(x)+-1/(2*pi)*cos(2*x)+-1/(2*pi)*sin(2*x)+1/(3*pi)*cos(3*x)+-1/(3*pi)*sin(3*x)" ], "answer_type_v1": [ "NV", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose you're given the following Fourier coefficients for a function on the interval $[-\\pi,\\pi]$: $ a_0=\\frac{-3}{2\\pi}$, $ a_1=\\frac{-2}{\\pi}$, $ a_2=\\frac{3}{2 \\pi}$, $ a_3=\\frac{-2}{3 \\pi}$, $ b_1=\\frac{-1}{\\pi}$, $ b_2=\\frac{-1}{2 \\pi}$, $ b_3=\\frac{-1}{3 \\pi}$. Find the following Fourier approximations to the Fourier series $ a_0+\\sum_{n=1}^{\\infty} (a_n\\cos(nx)+b_n\\sin(nx))$.\n$F_0(x)=$ [ANS]\n$F_1(x)=$ [ANS]\n$F_2(x)=$ R2 R2\n$F_3(x)=$", "answer_v2": [ "-3/(2*pi)", "-3/(2*pi)+-2/pi*cos(x)+-1/pi*sin(x)", "-3/(2*pi)+-2/pi*cos(x)+-1/pi*sin(x)+3/(2*pi)*cos(2*x)+-1/(2*pi)*sin(2*x)", "-3/(2*pi)+-2/pi*cos(x)+-1/pi*sin(x)+3/(2*pi)*cos(2*x)+-1/(2*pi)*sin(2*x)+-2/(3*pi)*cos(3*x)+-1/(3*pi)*sin(3*x)" ], "answer_type_v2": [ "NV", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose you're given the following Fourier coefficients for a function on the interval $[-\\pi,\\pi]$: $ a_0=\\frac{-1}{2\\pi}$, $ a_1=\\frac{-2}{\\pi}$, $ a_2=\\frac{-1}{2 \\pi}$, $ a_3=\\frac{3}{3 \\pi}$, $ b_1=\\frac{-2}{\\pi}$, $ b_2=\\frac{2}{2 \\pi}$, $ b_3=\\frac{3}{3 \\pi}$. Find the following Fourier approximations to the Fourier series $ a_0+\\sum_{n=1}^{\\infty} (a_n\\cos(nx)+b_n\\sin(nx))$.\n$F_0(x)=$ [ANS]\n$F_1(x)=$ [ANS]\n$F_2(x)=$ R2 R2\n$F_3(x)=$", "answer_v3": [ "-1/(2*pi)", "-1/(2*pi)+-2/pi*cos(x)+-2/pi*sin(x)", "-1/(2*pi)+-2/pi*cos(x)+-2/pi*sin(x)+-1/(2*pi)*cos(2*x)+2/(2*pi)*sin(2*x)", "-1/(2*pi)+-2/pi*cos(x)+-2/pi*sin(x)+-1/(2*pi)*cos(2*x)+2/(2*pi)*sin(2*x)+3/(3*pi)*cos(3*x)+3/(3*pi)*sin(3*x)" ], "answer_type_v3": [ "NV", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0953", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "2", "keywords": [ "integral", "fourier", "series" ], "problem_v1": "Suppose that $f(x)$ is periodic with period $[-\\pi,\\,\\pi)$ and has the following complex Fourier coefficients: $\\ldots \\quad c_0=3,\\quad c_1=1+i,\\quad c_2=2-2i,\\quad c_3=-2+i,\\quad \\ldots$ (A) Compute the following complex Fourier coefficients. $c_{-3}=$ [ANS], $c_{-2}=$ [ANS], $c_{-1}=$ [ANS]\n(B) Compute the real Fourier coefficients. (Remember that $e^{i\\,kx}=\\cos(k\\,x)+i\\,\\sin(k\\,x)$.) $a_0=$ [ANS] $,\\ a_1=$ [ANS] $,\\ a_2=$ [ANS] $,\\ a_3=$ [ANS] $,\\ \\ldots$ $\\phantom{b_0=X,}\\qquad\\qquad b_1=$ [ANS] $,\\ b_2=$ [ANS] $,\\ b_3=$ [ANS] $,\\ \\ldots$ (C) Compute the complex Fourier coefficients of the following. (i) The derivative $f'(x)$. $c_{0}=$ [ANS], $c_{1}=$ [ANS], $c_{2}=$ [ANS], $c_{3}=$ [ANS]\n(ii) The shifted function $ f\\bigl(x+\\frac{\\pi}{4}\\bigr)$ $c_{0}=$ [ANS], $c_{1}=$ [ANS], $c_{2}=$ [ANS], $c_{3}=$ [ANS]\n(iii) The function $f(3x)$. $c_{0}=$ [ANS], $c_{1}=$ [ANS], $c_{2}=$ [ANS], $c_{3}=$ [ANS]", "answer_v1": [ "-2-i", "2+2i", "1-i", "3", "2", "4", "-4", "-2", "4", "-2", "0", "-1+i", "4+4i", "-3-6i", "3", "1.41421i", "2+2i", "0.707107-2.12132i", "3", "0", "0", "1+i" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Suppose that $f(x)$ is periodic with period $[-\\pi,\\,\\pi)$ and has the following complex Fourier coefficients: $\\ldots \\quad c_0=-5,\\quad c_1=4-3i,\\quad c_2=-1+4i,\\quad c_3=-2-3i,\\quad \\ldots$ (A) Compute the following complex Fourier coefficients. $c_{-3}=$ [ANS], $c_{-2}=$ [ANS], $c_{-1}=$ [ANS]\n(B) Compute the real Fourier coefficients. (Remember that $e^{i\\,kx}=\\cos(k\\,x)+i\\,\\sin(k\\,x)$.) $a_0=$ [ANS] $,\\ a_1=$ [ANS] $,\\ a_2=$ [ANS] $,\\ a_3=$ [ANS] $,\\ \\ldots$ $\\phantom{b_0=X,}\\qquad\\qquad b_1=$ [ANS] $,\\ b_2=$ [ANS] $,\\ b_3=$ [ANS] $,\\ \\ldots$ (C) Compute the complex Fourier coefficients of the following. (i) The derivative $f'(x)$. $c_{0}=$ [ANS], $c_{1}=$ [ANS], $c_{2}=$ [ANS], $c_{3}=$ [ANS]\n(ii) The shifted function $ f\\bigl(x+\\frac{\\pi}{3}\\bigr)$ $c_{0}=$ [ANS], $c_{1}=$ [ANS], $c_{2}=$ [ANS], $c_{3}=$ [ANS]\n(iii) The function $f(3x)$. $c_{0}=$ [ANS], $c_{1}=$ [ANS], $c_{2}=$ [ANS], $c_{3}=$ [ANS]", "answer_v2": [ "-2+3i", "-1-4i", "4+3i", "-5", "8", "-2", "-4", "6", "-8", "6", "0", "3+4i", "-8-2i", "9-6i", "-5", "4.59808+1.9641i", "-2.9641-2.86603i", "2+3i", "-5", "0", "0", "4-3i" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Suppose that $f(x)$ is periodic with period $[-\\pi,\\,\\pi)$ and has the following complex Fourier coefficients: $\\ldots \\quad c_0=-2,\\quad c_1=1-2i,\\quad c_2=-3i,\\quad c_3=-1+3i,\\quad \\ldots$ (A) Compute the following complex Fourier coefficients. $c_{-3}=$ [ANS], $c_{-2}=$ [ANS], $c_{-1}=$ [ANS]\n(B) Compute the real Fourier coefficients. (Remember that $e^{i\\,kx}=\\cos(k\\,x)+i\\,\\sin(k\\,x)$.) $a_0=$ [ANS] $,\\ a_1=$ [ANS] $,\\ a_2=$ [ANS] $,\\ a_3=$ [ANS] $,\\ \\ldots$ $\\phantom{b_0=X,}\\qquad\\qquad b_1=$ [ANS] $,\\ b_2=$ [ANS] $,\\ b_3=$ [ANS] $,\\ \\ldots$ (C) Compute the complex Fourier coefficients of the following. (i) The derivative $f'(x)$. $c_{0}=$ [ANS], $c_{1}=$ [ANS], $c_{2}=$ [ANS], $c_{3}=$ [ANS]\n(ii) The shifted function $ f\\bigl(x+\\frac{\\pi}{6}\\bigr)$ $c_{0}=$ [ANS], $c_{1}=$ [ANS], $c_{2}=$ [ANS], $c_{3}=$ [ANS]\n(iii) The function $f(3x)$. $c_{0}=$ [ANS], $c_{1}=$ [ANS], $c_{2}=$ [ANS], $c_{3}=$ [ANS]", "answer_v3": [ "-1-3i", "3i", "1+2i", "-2", "2", "0", "-2", "4", "6", "-6", "0", "2+i", "6", "-9-3i", "-2", "1.86603-1.23205i", "2.59808-1.5i", "-3-i", "-2", "0", "0", "1-2i" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "EX", "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0954", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "2", "keywords": [ "integral", "fourier", "series" ], "problem_v1": "(A) Compute the discrete inverse Fourier transform of $ \\vec{\\mathbf{c}}=\\Bigl(\\frac{7}{4}, \\frac{2+i}{4}, \\frac{1}{4}, \\frac{2-i}{4}\\Bigr)$. $\\mathcal{F}^{\\mathrm{-}1}\\Bigl\\lbrace \\vec{\\mathbf{c}} \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$\n(B) Compute the discrete inverse Fourier transform of $ \\vec{\\mathbf{d}}=\\Bigl(\\frac{-3}{3}, \\frac{-3+3\\sqrt{3}i}{6}, \\frac{-3-3\\sqrt{3}i}{6}\\Bigr)$. $\\mathcal{F}^{\\mathrm{-}1}\\Bigl\\lbrace \\vec{\\mathbf{d}} \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$", "answer_v1": [ "3", "1", "1", "2", "-2", "-2", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "(A) Compute the discrete inverse Fourier transform of $ \\vec{\\mathbf{c}}=\\Bigl(\\frac{-6}{4}, \\frac{-1-7i}{4}, \\frac{-12}{4}, \\frac{-1+7i}{4}\\Bigr)$. $\\mathcal{F}^{\\mathrm{-}1}\\Bigl\\lbrace \\vec{\\mathbf{c}} \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$\n(B) Compute the discrete inverse Fourier transform of $ \\vec{\\mathbf{d}}=\\Bigl(\\frac{0}{3}, \\frac{15-1\\sqrt{3}i}{6}, \\frac{15+1\\sqrt{3}i}{6}\\Bigr)$. $\\mathcal{F}^{\\mathrm{-}1}\\Bigl\\lbrace \\vec{\\mathbf{d}} \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$", "answer_v2": [ "-5", "5", "-4", "-2", "5", "-2", "-3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "(A) Compute the discrete inverse Fourier transform of $ \\vec{\\mathbf{c}}=\\Bigl(\\frac{-2}{4}, \\frac{0}{4}, \\frac{-6}{4}, \\frac{0}{4}\\Bigr)$. $\\mathcal{F}^{\\mathrm{-}1}\\Bigl\\lbrace \\vec{\\mathbf{c}} \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$\n(B) Compute the discrete inverse Fourier transform of $ \\vec{\\mathbf{d}}=\\Bigl(\\frac{-2}{3}, \\frac{-7+5\\sqrt{3}i}{6}, \\frac{-7-5\\sqrt{3}i}{6}\\Bigr)$. $\\mathcal{F}^{\\mathrm{-}1}\\Bigl\\lbrace \\vec{\\mathbf{d}} \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$", "answer_v3": [ "-2", "1", "-2", "1", "-3", "-2", "3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0955", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "2", "keywords": [ "integral", "fourier", "series" ], "problem_v1": "(A) Compute the (infinite) discrete convolution ${\\}\\quad \\bigl(3, 1, 1, 2\\bigr)\\ast \\bigl(-2,-2, 1, 1\\bigr)$. $f\\ast g=$ [ANS]\n(B) Compute the cyclic discrete convolution ${\\}\\quad \\bigl(3, 1, 1, 2\\bigr)\\circledast \\bigl(-2,-2, 1, 1\\bigr)$. $f\\circledast g=$ [ANS]\nPlease enter you answer in the form \"(x, x, x, x)\"--with commas separating entries, surrounded by parenthesis. Please enter you answer in the form \"(x, x, x, x)\"--with commas separating entries, surrounded by parenthesis.", "answer_v1": [ "(-6,-8,-1,-2,-2,3,2)", "(-8,-5,1,-2)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "(A) Compute the (infinite) discrete convolution ${\\}\\quad \\bigl(-5, 5,-4,-2\\bigr)\\ast \\bigl(5,-2,-3,-2\\bigr)$. $f\\ast g=$ [ANS]\n(B) Compute the cyclic discrete convolution ${\\}\\quad \\bigl(-5, 5,-4,-2\\bigr)\\circledast \\bigl(5,-2,-3,-2\\bigr)$. $f\\circledast g=$ [ANS]\nPlease enter you answer in the form \"(x, x, x, x)\"--with commas separating entries, surrounded by parenthesis. Please enter you answer in the form \"(x, x, x, x)\"--with commas separating entries, surrounded by parenthesis.", "answer_v2": [ "(-25,35,-15,-7,6,14,4)", "(-19,49,-11,-7)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "(A) Compute the (infinite) discrete convolution ${\\}\\quad \\bigl(-2, 1,-2, 1\\bigr)\\ast \\bigl(-3,-2, 3, 5\\bigr)$. $f\\ast g=$ [ANS]\n(B) Compute the cyclic discrete convolution ${\\}\\quad \\bigl(-2, 1,-2, 1\\bigr)\\circledast \\bigl(-3,-2, 3, 5\\bigr)$. $f\\circledast g=$ [ANS]\nPlease enter you answer in the form \"(x, x, x, x)\"--with commas separating entries, surrounded by parenthesis. Please enter you answer in the form \"(x, x, x, x)\"--with commas separating entries, surrounded by parenthesis.", "answer_v3": [ "(6,1,-2,-6,-3,-7,5)", "(3,-6,3,-6)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0956", "subject": "Calculus_-_single_variable", "topic": "Infinite sequences and series", "subtopic": "Fourier series", "level": "3", "keywords": [ "integral", "fourier", "series" ], "problem_v1": "In this problem you will use the fast fourier transform on $ {\\}\\quad \\vec{\\mathbf{f}}=\\bigl(2, 1, 1, 2,-1,-1, 1,-1 \\bigr)$\n(A) Split $\\vec{\\mathbf{f}}$ into its even and odd components: $\\vec{\\mathbf{f}}_{\\mathrm{even}}=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$ $\\vec{\\mathbf{f}}_{\\mathrm{odd}}\\;=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$\n(B) Compute the Fourier transforms of the even and odd components: $\\mathcal{F}\\Bigl\\lbrace \\vec{\\mathbf{f}}_{\\mathrm{even}} \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$ $\\mathcal{F}\\Bigl\\lbrace \\vec{\\mathbf{f}}_{\\mathrm{odd}}\\; \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$\n(C) Combine the Fourier transforms of the even and odd components to get the transform of $\\vec{\\mathbf{f}}$ $\\mathcal{F}_0\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\phantom{\\omega^2}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_1\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\overline{\\omega}\\phantom{^1}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_2\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\overline{\\omega}^2$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_3\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\overline{\\omega}^3$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_4\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\phantom{\\omega^2}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_5\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\overline{\\omega}\\phantom{^1}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_6\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\overline{\\omega}^2$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_7\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\overline{\\omega}^3$ [ANS] $\\Bigr)\\=\\ $ [ANS]", "answer_v1": [ "2", "1", "-1", "1", "1", "2", "-1", "-1", "0.75", "0.75", "-0.25", "0.75", "0.25", "0.5-0.75i", "-0.25", "0.5+0.75i", "0.75", "0.25", "0.5", "0.75", "0.5-0.75i", "0.286612-0.441942i", "-0.25", "-0.25", "-0.125+0.125i", "0.75", "0.5+0.75i", "0.463388-0.441942i", "0.75", "0.25", "0.25", "0.75", "0.5-0.75i", "0.463388+0.441942i", "-0.25", "-0.25", "-0.125-0.125i", "0.75", "0.5+0.75i", "0.286612+0.441942i" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "EX", "NV", "NV", "NV", "NV", "EX", "EX", "NV", "NV", "EX", "NV", "EX", "EX", "NV", "NV", "NV", "NV", "EX", "EX", "NV", "NV", "EX", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "In this problem you will use the fast fourier transform on $ {\\}\\quad \\vec{\\mathbf{f}}=\\bigl(-3, 3,-2,-1, 3,-1,-2,-1 \\bigr)$\n(A) Split $\\vec{\\mathbf{f}}$ into its even and odd components: $\\vec{\\mathbf{f}}_{\\mathrm{even}}=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$ $\\vec{\\mathbf{f}}_{\\mathrm{odd}}\\;=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$\n(B) Compute the Fourier transforms of the even and odd components: $\\mathcal{F}\\Bigl\\lbrace \\vec{\\mathbf{f}}_{\\mathrm{even}} \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$ $\\mathcal{F}\\Bigl\\lbrace \\vec{\\mathbf{f}}_{\\mathrm{odd}}\\; \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$\n(C) Combine the Fourier transforms of the even and odd components to get the transform of $\\vec{\\mathbf{f}}$ $\\mathcal{F}_0\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\phantom{\\omega^2}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_1\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\overline{\\omega}\\phantom{^1}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_2\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\overline{\\omega}^2$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_3\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\overline{\\omega}^3$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_4\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\phantom{\\omega^2}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_5\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\overline{\\omega}\\phantom{^1}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_6\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\overline{\\omega}^2$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_7\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\overline{\\omega}^3$ [ANS] $\\Bigr)\\=\\ $ [ANS]", "answer_v2": [ "-3", "-2", "3", "-2", "3", "-1", "-1", "-1", "-1", "-1.5", "1", "-1.5", "0", "1", "1", "1", "-1", "0", "-0.5", "-1.5", "1", "-0.396447-0.353553i", "1", "1", "0.5-0.5i", "-1.5", "1", "-1.10355-0.353553i", "-1", "0", "-0.5", "-1.5", "1", "-1.10355+0.353553i", "1", "1", "0.5+0.5i", "-1.5", "1", "-0.396447+0.353553i" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "In this problem you will use the fast fourier transform on $ {\\}\\quad \\vec{\\mathbf{f}}=\\bigl(-1, 1,-2,-2,-1, 2, 3, 3 \\bigr)$\n(A) Split $\\vec{\\mathbf{f}}$ into its even and odd components: $\\vec{\\mathbf{f}}_{\\mathrm{even}}=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$ $\\vec{\\mathbf{f}}_{\\mathrm{odd}}\\;=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$\n(B) Compute the Fourier transforms of the even and odd components: $\\mathcal{F}\\Bigl\\lbrace \\vec{\\mathbf{f}}_{\\mathrm{even}} \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$ $\\mathcal{F}\\Bigl\\lbrace \\vec{\\mathbf{f}}_{\\mathrm{odd}}\\; \\Bigr\\rbrace=\\Bigl($ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $,\\ $ [ANS] $\\Bigr)$\n(C) Combine the Fourier transforms of the even and odd components to get the transform of $\\vec{\\mathbf{f}}$ $\\mathcal{F}_0\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\phantom{\\omega^2}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_1\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\overline{\\omega}\\phantom{^1}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_2\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\overline{\\omega}^2$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_3\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $+\\ \\ \\overline{\\omega}^3$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_4\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\phantom{\\omega^2}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_5\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\overline{\\omega}\\phantom{^1}$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_6\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\overline{\\omega}^2$ [ANS] $\\Bigr)\\=\\ $ [ANS]\n$\\mathcal{F}_7\\Bigl\\lbrace \\vec{\\mathbf{f}} \\Bigr\\rbrace=\\frac{1}{2} \\Bigl($ [ANS] $-\\ \\ \\overline{\\omega}^3$ [ANS] $\\Bigr)\\=\\ $ [ANS]", "answer_v3": [ "-1", "-2", "-1", "3", "1", "-2", "2", "3", "-0.25", "1.25i", "-0.75", "-1.25i", "1", "-0.25+1.25i", "0.5", "-0.25-1.25i", "-0.25", "1", "0.375", "1.25i", "-0.25+1.25i", "0.353553+1.15533i", "-0.75", "0.5", "-0.375-0.25i", "-1.25i", "-0.25-1.25i", "-0.353553-0.0946699i", "-0.25", "1", "-0.625", "1.25i", "-0.25+1.25i", "-0.353553+0.0946699i", "-0.75", "0.5", "-0.375+0.25i", "-1.25i", "-0.25-1.25i", "0.353553-1.15533i" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "EX", "NV", "EX", "NV", "EX", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0957", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Curves", "level": "2", "keywords": [ "calculus", "parametric", "parametric equations" ], "problem_v1": "Which is a parametric equation for the curve $y=5-6x$? [ANS] A. $c(t)=\\left(t,6+t\\right)$ B. $c(t)=\\left(5t,6t\\right)$ C. $c(t)=\\left(t,5-6t\\right)$ D. $c(t)=\\left(t,5+t\\right)$", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Which is a parametric equation for the curve $y=-\\left(9+10x\\right)$? [ANS] A. $c(t)=\\left(t,10+t\\right)$ B. $c(t)=\\left(-9t,10t\\right)$ C. $c(t)=\\left(t,t-9\\right)$ D. $c(t)=\\left(t,-\\left(9+10t\\right)\\right)$", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Which is a parametric equation for the curve $y=-\\left(4+7x\\right)$? [ANS] A. $c(t)=\\left(t,7+t\\right)$ B. $c(t)=\\left(t,-\\left(4+7t\\right)\\right)$ C. $c(t)=\\left(t,t-4\\right)$ D. $c(t)=\\left(-4t,7t\\right)$", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_single_variable_0958", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Curves", "level": "1", "keywords": [ "calculus", "parametric", "parametric equations" ], "problem_v1": "Find the coordinates at times t=0, 3, 6 of a particle following the path $x=5+2t^{4}$, $y=5-4t^{2}$. t=0: [ANS]\nt=3: [ANS]\nt=6: [ANS]", "answer_v1": [ "(5,5)", "(167,-31)", "(2597,-139)" ], "answer_type_v1": [ "OL", "OL", "OL" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the coordinates at times t=0, 1, 3 of a particle following the path $x=9t-9$, $y=9t^{2}-3$. t=0: [ANS]\nt=1: [ANS]\nt=3: [ANS]", "answer_v2": [ "(-9,-3)", "(0,6)", "(18,78)" ], "answer_type_v2": [ "OL", "OL", "OL" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the coordinates at times t=0, 5, 10 of a particle following the path $x=2t^{2}-4$, $y=1-6t^{2}$. t=0: [ANS]\nt=5: [ANS]\nt=10: [ANS]", "answer_v3": [ "(-4,1)", "(46,-149)", "(196,-599)" ], "answer_type_v3": [ "OL", "OL", "OL" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0959", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Curves", "level": "2", "keywords": [ "parametric equation", "parametric" ], "problem_v1": "Assume time t runs from zero to $2\\pi$ and that the unit circle has been labled as a clock. Match each of the pairs of parametric equations with the best description of the curve from the following list. Enter the appropriate letter (A, B, C, D, E, F) in each blank.\nA. Starts at 12 o'clock and moves clockwise one time around. B. Starts at 6 o'clock and moves clockwise one time around. C. Starts at 3 o'clock and moves clockwise one time around. D. Starts at 9 o'clock and moves counterclockwise one time around. E. Starts at 3 o'clock and moves counterclockwise two times around. F. Starts at 3 o'clock and moves counterclockwise to 9 o'clock. [ANS] 1. $ x=\\cos(2t);$ $\\ y=\\sin(2t)$ [ANS] 2. $ x=\\cos(t);$ $\\ y=-\\sin(t)$ [ANS] 3. $ x=-\\cos(t);$ $\\ y=-\\sin(t)$ [ANS] 4. $ x=\\cos{\\frac{t}{2}};$ $\\ y=\\sin{\\frac{t}{2}}$ [ANS] 5. $ x=\\sin(t);$ $\\ y=\\cos(t)$", "answer_v1": [ "E", "C", "D", "F", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Assume time t runs from zero to $2\\pi$ and that the unit circle has been labled as a clock. Match each of the pairs of parametric equations with the best description of the curve from the following list. Enter the appropriate letter (A, B, C, D, E, F) in each blank.\nA. Starts at 12 o'clock and moves clockwise one time around. B. Starts at 6 o'clock and moves clockwise one time around. C. Starts at 3 o'clock and moves clockwise one time around. D. Starts at 9 o'clock and moves counterclockwise one time around. E. Starts at 3 o'clock and moves counterclockwise two times around. F. Starts at 3 o'clock and moves counterclockwise to 9 o'clock. [ANS] 1. $ x=\\sin(t);$ $\\ y=\\cos(t)$ [ANS] 2. $ x=\\cos{\\frac{t}{2}};$ $\\ y=\\sin{\\frac{t}{2}}$ [ANS] 3. $ x=-\\sin(t);$ $\\ y=-\\cos(t)$ [ANS] 4. $ x=-\\cos(t);$ $\\ y=-\\sin(t)$ [ANS] 5. $ x=\\cos(2t);$ $\\ y=\\sin(2t)$", "answer_v2": [ "A", "F", "B", "D", "E" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Assume time t runs from zero to $2\\pi$ and that the unit circle has been labled as a clock. Match each of the pairs of parametric equations with the best description of the curve from the following list. Enter the appropriate letter (A, B, C, D, E, F) in each blank.\nA. Starts at 12 o'clock and moves clockwise one time around. B. Starts at 6 o'clock and moves clockwise one time around. C. Starts at 3 o'clock and moves clockwise one time around. D. Starts at 9 o'clock and moves counterclockwise one time around. E. Starts at 3 o'clock and moves counterclockwise two times around. F. Starts at 3 o'clock and moves counterclockwise to 9 o'clock. [ANS] 1. $ x=-\\sin(t);$ $\\ y=-\\cos(t)$ [ANS] 2. $ x=\\cos(2t);$ $\\ y=\\sin(2t)$ [ANS] 3. $ x=\\cos(t);$ $\\ y=-\\sin(t)$ [ANS] 4. $ x=-\\cos(t);$ $\\ y=-\\sin(t)$ [ANS] 5. $ x=\\sin(t);$ $\\ y=\\cos(t)$", "answer_v3": [ "B", "E", "C", "D", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Calculus_-_single_variable_0960", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Curves", "level": "2", "keywords": [ "parametric equation", "parametric", "ellipse" ], "problem_v1": "The ellipse \\frac{x^2}{4^2}+\\frac{y^2}{7^2}=1 can be drawn counterclockwise with parametric equations. If x=a \\cos(t) with $a$ positive, then $a=$ [ANS]\nand $y=$ [ANS] (enter a function of $t$)", "answer_v1": [ "4", "7 * sin (t)" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "The ellipse \\frac{x^2}{1^2}+\\frac{y^2}{5^2}=1 can be drawn counterclockwise with parametric equations. If x=a \\cos(t) with $a$ positive, then $a=$ [ANS]\nand $y=$ [ANS] (enter a function of $t$)", "answer_v2": [ "1", "5 * sin (t)" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "The ellipse \\frac{x^2}{2^2}+\\frac{y^2}{5^2}=1 can be drawn counterclockwise with parametric equations. If x=a \\cos(t) with $a$ positive, then $a=$ [ANS]\nand $y=$ [ANS] (enter a function of $t$)", "answer_v3": [ "2", "5 * sin (t)" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0961", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Curves", "level": "2", "keywords": [ "parametric equation", "parametric" ], "problem_v1": "Suppose parametric equations for the line segment between $(7, 2)$ and $(5, 5)$ have the form: \\begin{array}{r@{\\,}c@{\\,}l} x &=& a+bt \\cr y &=& c+dt \\end{array} If the parametric curve starts at $(7, 2)$ when $t=0$ and ends at $(5, 5)$ at $t=1$, then find $a$, $b$, $c$, and $d$. $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS].", "answer_v1": [ "7", "-2", "2", "3" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose parametric equations for the line segment between $(-2, 9)$ and $(-1,-3)$ have the form: \\begin{array}{r@{\\,}c@{\\,}l} x &=& a+bt \\cr y &=& c+dt \\end{array} If the parametric curve starts at $(-2, 9)$ when $t=0$ and ends at $(-1,-3)$ at $t=1$, then find $a$, $b$, $c$, and $d$. $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS].", "answer_v2": [ "-2", "1", "9", "-12" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose parametric equations for the line segment between $(1, 2)$ and $(1, 1)$ have the form: \\begin{array}{r@{\\,}c@{\\,}l} x &=& a+bt \\cr y &=& c+dt \\end{array} If the parametric curve starts at $(1, 2)$ when $t=0$ and ends at $(1, 1)$ at $t=1$, then find $a$, $b$, $c$, and $d$. $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS].", "answer_v3": [ "1", "0", "2", "-1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0962", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Curves", "level": "3", "keywords": [ "parametric equation", "parametric", "trochoid" ], "problem_v1": "A bicycle wheel has radius R. Let P be a point on the spoke of a wheel at a distance d from the center of the wheel. The wheel begins to roll to the right along the x-axis. The curve traced out by P is given by the following parametric equations: $x=19 \\theta-13 \\sin(\\theta)$ $y=19-13 \\cos(\\theta)$ What must we have for R and d? R=[ANS]\nd=[ANS]", "answer_v1": [ "19", "13" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A bicycle wheel has radius R. Let P be a point on the spoke of a wheel at a distance d from the center of the wheel. The wheel begins to roll to the right along the x-axis. The curve traced out by P is given by the following parametric equations: $x=15 \\theta-5 \\sin(\\theta)$ $y=15-5 \\cos(\\theta)$ What must we have for R and d? R=[ANS]\nd=[ANS]", "answer_v2": [ "15", "5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A bicycle wheel has radius R. Let P be a point on the spoke of a wheel at a distance d from the center of the wheel. The wheel begins to roll to the right along the x-axis. The curve traced out by P is given by the following parametric equations: $x=15 \\theta-8 \\sin(\\theta)$ $y=15-8 \\cos(\\theta)$ What must we have for R and d? R=[ANS]\nd=[ANS]", "answer_v3": [ "15", "8" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0963", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Curves", "level": "3", "keywords": [ "parametric equations" ], "problem_v1": "Consider the parameterization of the unit circle given by $x=\\cos(\\ln\\!\\left(4t\\right))$, $y=\\sin(\\ln\\!\\left(4t\\right))$ for $t$ in $(0,\\infty)$. Describe in words and sketch how the circle is traced out, and use this to answer the following questions.\n(a) When is the parameterization tracing the circle out in a clockwise direction? [ANS]\n(Give your answer as a comma-separated list of intervals, for example, (0,1), (3,Inf)). Enter the word None if there are no such intervals. (b) When is the parameterization tracing the circle out in a counter-clockwise direction? [ANS]\n(Give your answer as a comma-separated list of intervals, for example, (0,1), (3,Inf)). Enter the word None if there are no such intervals. (c) Does the entire unit circle get traced by this parameterization? [ANS] A. yes B. no\n(d) Give a time $t$ at which the point being traced out on the circle is at $(1,0)$: $t=$ [ANS]", "answer_v1": [ "None", "(0,infinity)", "A", "0.25" ], "answer_type_v1": [ "OE", "INT", "MCS", "NV" ], "options_v1": [ [], [], [ "A", "B" ], [] ], "problem_v2": "Consider the parameterization of the unit circle given by $x=\\cos(t^{2}-t)$, $y=\\sin(t^{2}-t)$ for $t$ in $(-\\infty,\\infty)$. Describe in words and sketch how the circle is traced out, and use this to answer the following questions.\n(a) When is the parameterization tracing the circle out in a clockwise direction? [ANS]\n(Give your answer as a comma-separated list of intervals, for example, (0,1), (3,Inf)). Enter the word None if there are no such intervals. (b) When is the parameterization tracing the circle out in a counter-clockwise direction? [ANS]\n(Give your answer as a comma-separated list of intervals, for example, (0,1), (3,Inf)). Enter the word None if there are no such intervals. (c) Does the entire unit circle get traced by this parameterization? [ANS] A. yes B. no\n(d) Give a time $t$ at which the point being traced out on the circle is at $(1,0)$: $t=$ [ANS]", "answer_v2": [ "(-infinity,0.5)", "(0.5,infinity)", "A", "0" ], "answer_type_v2": [ "INT", "INT", "MCS", "NV" ], "options_v2": [ [], [], [ "A", "B" ], [] ], "problem_v3": "Consider the parameterization of the unit circle given by $x=\\cos(2t^{2}-t)$, $y=\\sin(2t^{2}-t)$ for $t$ in $(-\\infty,\\infty)$. Describe in words and sketch how the circle is traced out, and use this to answer the following questions.\n(a) When is the parameterization tracing the circle out in a clockwise direction? [ANS]\n(Give your answer as a comma-separated list of intervals, for example, (0,1), (3,Inf)). Enter the word None if there are no such intervals. (b) When is the parameterization tracing the circle out in a counter-clockwise direction? [ANS]\n(Give your answer as a comma-separated list of intervals, for example, (0,1), (3,Inf)). Enter the word None if there are no such intervals. (c) Does the entire unit circle get traced by this parameterization? [ANS] A. yes B. no\n(d) Give a time $t$ at which the point being traced out on the circle is at $(1,0)$: $t=$ [ANS]", "answer_v3": [ "(-infinity,0.25)", "(0.25,infinity)", "A", "0" ], "answer_type_v3": [ "INT", "INT", "MCS", "NV" ], "options_v3": [ [], [], [ "A", "B" ], [] ] }, { "id": "Calculus_-_single_variable_0964", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Curves", "level": "3", "keywords": [ "parametric equations" ], "problem_v1": "A line is parameterized by $x=7+6t$ and $y=6+7t$.\n(a) Which of the following points are on the section of the line obtained by restricting $t$ to nonnegative numbers (for each, enter Y if the point is on the section, and N if not)? $\\left(-17,-22\\right)$: [ANS]\n$\\left(1,-1\\right)$: [ANS]\n$\\left(13,13\\right)$: [ANS]\nThen, give one more point that is on the section of the line obtained by this restriction: [ANS]\n(b) What are the endpoints of the line segment obtained by restricting $t$ to $-3\\le t\\le-1$? left endpoint: [ANS]\nright endpoint: [ANS]\n(c) How should $t$ be restricted to give the part of the line above the $x$-axis (give your answer as an interval for $t$, for example, (3,8) or [-2,Inf))? $t$ must be in: [ANS]", "answer_v1": [ "N", "N", "Y", "(7,6)", "(-11,-15)", "(1,-1)", "(-0.857143,infinity)" ], "answer_type_v1": [ "TF", "TF", "TF", "INT", "INT", "INT", "INT" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "A line is parameterized by $x=2+8t$ and $y=3+4t$.\n(a) Which of the following points are on the section of the line obtained by restricting $t$ to nonnegative numbers (for each, enter Y if the point is on the section, and N if not)? $\\left(-6,-1\\right)$: [ANS]\n$\\left(18,11\\right)$: [ANS]\n$\\left(26,15\\right)$: [ANS]\nThen, give one more point that is on the section of the line obtained by this restriction: [ANS]\n(b) What are the endpoints of the line segment obtained by restricting $t$ to $-4\\le t\\le-2$? left endpoint: [ANS]\nright endpoint: [ANS]\n(c) How should $t$ be restricted to give the part of the line above the $x$-axis (give your answer as an interval for $t$, for example, (3,8) or [-2,Inf))? $t$ must be in: [ANS]", "answer_v2": [ "N", "Y", "Y", "(2,3)", "(-30,-13)", "(-14,-5)", "(-0.75,infinity)" ], "answer_type_v2": [ "TF", "TF", "TF", "INT", "INT", "INT", "INT" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "A line is parameterized by $x=4+6t$ and $y=3+5t$.\n(a) Which of the following points are on the section of the line obtained by restricting $t$ to nonnegative numbers (for each, enter Y if the point is on the section, and N if not)? $\\left(-20,-17\\right)$: [ANS]\n$\\left(-2,-2\\right)$: [ANS]\n$\\left(16,13\\right)$: [ANS]\nThen, give one more point that is on the section of the line obtained by this restriction: [ANS]\n(b) What are the endpoints of the line segment obtained by restricting $t$ to $-1\\le t\\le 2$? left endpoint: [ANS]\nright endpoint: [ANS]\n(c) How should $t$ be restricted to give the part of the line below the $x$-axis (give your answer as an interval for $t$, for example, (3,8) or [-2,Inf))? $t$ must be in: [ANS]", "answer_v3": [ "N", "N", "Y", "(4,3)", "(-2,-2)", "(16,13)", "(-infinity,-0.6)" ], "answer_type_v3": [ "TF", "TF", "TF", "INT", "INT", "INT", "INT" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0965", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Eliminating the parameter", "level": "2", "keywords": [ "calculus", "parametric", "parametric equations" ], "problem_v1": "Express $x=e^{-2t}$, $y=4e^{4t}$ in the form $y=f(x)$ by eliminating the parameter. y=[ANS]", "answer_v1": [ "4*e^{-4*[ln(x)]/2}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Express $x=e^{-5t}$, $y=6e^{2t}$ in the form $y=f(x)$ by eliminating the parameter. y=[ANS]", "answer_v2": [ "6*e^{-2*[ln(x)]/5}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Express $x=e^{-4t}$, $y=5e^{3t}$ in the form $y=f(x)$ by eliminating the parameter. y=[ANS]", "answer_v3": [ "5*e^{-3*[ln(x)]/4}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0966", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Eliminating the parameter", "level": "2", "keywords": [ "parametric equation", "parametric" ], "problem_v1": "Eliminate the parameter $t$ to find a Cartesian equation for \\begin{array}{r@{\\,}c@{\\,}l} x &=& 10-t \\cr y &=& 3-3 t \\end{array} The Cartesian equation has the form y=mx+b where $m=$ [ANS] and $b=$ [ANS].", "answer_v1": [ "3", "-27" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Eliminate the parameter $t$ to find a Cartesian equation for \\begin{array}{r@{\\,}c@{\\,}l} x &=&-17-t \\cr y &=& 18-2 t \\end{array} The Cartesian equation has the form y=mx+b where $m=$ [ANS] and $b=$ [ANS].", "answer_v2": [ "2", "52" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Eliminate the parameter $t$ to find a Cartesian equation for \\begin{array}{r@{\\,}c@{\\,}l} x &=&-8-t \\cr y &=& 4-2 t \\end{array} The Cartesian equation has the form y=mx+b where $m=$ [ANS] and $b=$ [ANS].", "answer_v3": [ "2", "20" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0967", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Eliminating the parameter", "level": "2", "keywords": [ "calculus", "parametric equation", "cartesian" ], "problem_v1": "Find a Cartesian equation relating $x$ and $y$ corresponding to the parametric equations x=e^{5 t} \\quad y=e^{-9 t} Write your answer in the form y=f(x) Answer: $y=$ [ANS]", "answer_v1": [ "x^{-1.8}" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a Cartesian equation relating $x$ and $y$ corresponding to the parametric equations x=e^{2 t} \\quad y=e^{-9 t} Write your answer in the form y=f(x) Answer: $y=$ [ANS]", "answer_v2": [ "x^{-1.8}" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a Cartesian equation relating $x$ and $y$ corresponding to the parametric equations x=e^{3 t} \\quad y=e^{-6 t} Write your answer in the form y=f(x) Answer: $y=$ [ANS]", "answer_v3": [ "x^{-2}" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0968", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Eliminating the parameter", "level": "3", "keywords": [ "parametric curve" ], "problem_v1": "Eliminate the parameter in $x=t^2+8 t$ and $y=t-3$ and then identify the parametric curve and sketch its image in the $xy$-plane on a piece of paper.\nEquation: [ANS]\nImage in the $xy$-plane: [ANS]", "answer_v1": [ "x = 1*(y+3)^2+8*(y+3)", "Parabola opening right" ], "answer_type_v1": [ "EQ", "MCS" ], "options_v1": [ [], [ "Circle", "Semicircle opening up", "Semicircle opening down", "Semicircle opening right", "Semicircle opening left", "Ellipse", "Ellipse opening up", "Ellipse opening down", "Ellipse opening right", "Ellipse opening left", "Hyperbola opening up", "Hyperbola opening down", "Hyperbola opening right", "Hyperbola opening left", "Parabola opening up", "Parabola opening down", "Parabola opening right", "Parabola opening left" ] ], "problem_v2": "Eliminate the parameter in $x=-t^2+2 t$ and $y=t-4$ and then identify the parametric curve and sketch its image in the $xy$-plane on a piece of paper.\nEquation: [ANS]\nImage in the $xy$-plane: [ANS]", "answer_v2": [ "x = -1*(y+4)^2+2*(y+4)", "Parabola opening left" ], "answer_type_v2": [ "EQ", "MCS" ], "options_v2": [ [], [ "Circle", "Semicircle opening up", "Semicircle opening down", "Semicircle opening right", "Semicircle opening left", "Ellipse", "Ellipse opening up", "Ellipse opening down", "Ellipse opening right", "Ellipse opening left", "Hyperbola opening up", "Hyperbola opening down", "Hyperbola opening right", "Hyperbola opening left", "Parabola opening up", "Parabola opening down", "Parabola opening right", "Parabola opening left" ] ], "problem_v3": "Eliminate the parameter in $x=-t^2+4 t$ and $y=t-3$ and then identify the parametric curve and sketch its image in the $xy$-plane on a piece of paper.\nEquation: [ANS]\nImage in the $xy$-plane: [ANS]", "answer_v3": [ "x = -1*(y+3)^2+4*(y+3)", "Parabola opening left" ], "answer_type_v3": [ "EQ", "MCS" ], "options_v3": [ [], [ "Circle", "Semicircle opening up", "Semicircle opening down", "Semicircle opening right", "Semicircle opening left", "Ellipse", "Ellipse opening up", "Ellipse opening down", "Ellipse opening right", "Ellipse opening left", "Hyperbola opening up", "Hyperbola opening down", "Hyperbola opening right", "Hyperbola opening left", "Parabola opening up", "Parabola opening down", "Parabola opening right", "Parabola opening left" ] ] }, { "id": "Calculus_-_single_variable_0969", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Eliminating the parameter", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Make the change of variables $x=u \\cos\\theta-v\\sin\\theta$ $y=u \\sin\\theta+v\\cos\\theta$ where the angle $0\\le \\theta < \\pi/2$ is chosen in order to eliminate the cross product term in $208x^2+288xy+292 y^2+2880x+2840 y=29100$ Then find the standard form of equation in the $(u, v)$ variables. (Enter a function of $(u,v)$.) [ANS] $=1.$", "answer_v1": [ "(u+ 5 )^2 / 10^2 +( v -3 )^2 / 20^2 " ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Make the change of variables $x=u \\cos\\theta-v\\sin\\theta$ $y=u \\sin\\theta+v\\cos\\theta$ where the angle $0\\le \\theta < \\pi/2$ is chosen in order to eliminate the cross product term in $145x^2+120xy+180 y^2+700x+600 y=21500$ Then find the standard form of equation in the $(u, v)$ variables. (Enter a function of $(u,v)$.) [ANS] $=1.$", "answer_v2": [ "(u+ 2 )^2 / 10^2 +( v -1 )^2 / 15^2 " ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Make the change of variables $x=u \\cos\\theta-v\\sin\\theta$ $y=u \\sin\\theta+v\\cos\\theta$ where the angle $0\\le \\theta < \\pi/2$ is chosen in order to eliminate the cross product term in $208x^2+288xy+292 y^2+1120x+1160 y=38300$ Then find the standard form of equation in the $(u, v)$ variables. (Enter a function of $(u,v)$.) [ANS] $=1.$", "answer_v3": [ "(u+ 2 )^2 / 10^2 +( v -1 )^2 / 20^2 " ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0970", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Eliminating the parameter", "level": "2", "keywords": [ "polar coordinates" ], "problem_v1": "Eliminate the parameter to find the cartesian equation of the curve: x=8 \\sin \\theta, \\; y=\\cos ^2 \\theta, \\;-\\frac{\\pi}{2} \\le \\theta \\le \\frac{\\pi}{2} The equation of the curve is: y=[ANS]\nfrom x=[ANS] to x=[ANS]", "answer_v1": [ "1-x^2/(8^2)", "-8", "8" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Eliminate the parameter to find the cartesian equation of the curve: x=2 \\sin \\theta, \\; y=\\cos ^2 \\theta, \\;-\\frac{\\pi}{2} \\le \\theta \\le \\frac{\\pi}{2} The equation of the curve is: y=[ANS]\nfrom x=[ANS] to x=[ANS]", "answer_v2": [ "1-x^2/(2^2)", "-2", "2" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Eliminate the parameter to find the cartesian equation of the curve: x=4 \\sin \\theta, \\; y=\\cos ^2 \\theta, \\;-\\frac{\\pi}{2} \\le \\theta \\le \\frac{\\pi}{2} The equation of the curve is: y=[ANS]\nfrom x=[ANS] to x=[ANS]", "answer_v3": [ "1-x^2/(4^2)", "-4", "4" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0971", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Eliminating the parameter", "level": "2", "keywords": [ "parametric equation", "Cartesian equation" ], "problem_v1": "Write the parametric equations\nx=\\sqrt{t}, \\quad y=7-t in Cartesian form.\n$y=$ [ANS] with $x\\ge 0$", "answer_v1": [ "7-x^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write the parametric equations\nx=\\sqrt{t}, \\quad y=1-t in Cartesian form.\n$y=$ [ANS] with $x\\ge 0$", "answer_v2": [ "1-x^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write the parametric equations\nx=\\sqrt{t}, \\quad y=3-t in Cartesian form.\n$y=$ [ANS] with $x\\ge 0$", "answer_v3": [ "3-x^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0972", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Eliminating the parameter", "level": "2", "keywords": [ "parametric equation", "Cartesian equation" ], "problem_v1": "Write the parametric equations\nx=5 \\sin \\theta, \\quad y=6 \\cos \\theta, \\quad 0 \\le \\theta \\le \\pi in the given Cartesian form.\n$\\frac{y^2}{36}=$ [ANS] with $x\\ge 0$.", "answer_v1": [ "1-x^2/5^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write the parametric equations\nx=2 \\sin \\theta, \\quad y=9 \\cos \\theta, \\quad 0 \\le \\theta \\le \\pi in the given Cartesian form.\n$\\frac{y^2}{81}=$ [ANS] with $x\\ge 0$.", "answer_v2": [ "1-x^2/2^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write the parametric equations\nx=3 \\sin \\theta, \\quad y=6 \\cos \\theta, \\quad 0 \\le \\theta \\le \\pi in the given Cartesian form.\n$\\frac{y^2}{36}=$ [ANS] with $x\\ge 0$.", "answer_v3": [ "1-x^2/3^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0973", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Tangents, velocity, and speed", "level": "2", "keywords": [ "calculus", "parametric", "polar", "conic" ], "problem_v1": "Determine the speed $s(t)$ of a particle with a given trajectory at a time $t_0$ (in units of meters and seconds). c(t)=(5 \\sin 6 t, 8 \\cos 6 t), \\, t_0=\\frac{\\pi}{4} [ANS]", "answer_v1": [ "48" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Determine the speed $s(t)$ of a particle with a given trajectory at a time $t_0$ (in units of meters and seconds). c(t)=(2 \\sin 3 t, 9 \\cos 3 t), \\, t_0=\\frac{\\pi}{4} [ANS]", "answer_v2": [ "19.5576" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Determine the speed $s(t)$ of a particle with a given trajectory at a time $t_0$ (in units of meters and seconds). c(t)=(3 \\sin 4 t, 8 \\cos 4 t), \\, t_0=\\frac{\\pi}{4} [ANS]", "answer_v3": [ "12" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0974", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Tangents, velocity, and speed", "level": "2", "keywords": [ "calculus", "parametric", "polar", "conic" ], "problem_v1": "Find the speed of the cycloid $c(t)=(8 t-8 \\sin t, \\, 8-8 \\cos t)$ at points where the tangent line is horizontal. [ANS]", "answer_v1": [ "16" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the speed of the cycloid $c(t)=(2 t-2 \\sin t, \\, 2-2 \\cos t)$ at points where the tangent line is horizontal. [ANS]", "answer_v2": [ "4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the speed of the cycloid $c(t)=(4 t-4 \\sin t, \\, 4-4 \\cos t)$ at points where the tangent line is horizontal. [ANS]", "answer_v3": [ "8" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0975", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Tangents, velocity, and speed", "level": "2", "keywords": [ "calculus", "parametric", "polar", "conic" ], "problem_v1": "Find the minimum speed of a particle with trajectory $c(t)=(t^3-8 t, \\, t^2+1)$ for $t \\ge 0$. Hint: it is easier to find the minimum of the square of the speed. [ANS]", "answer_v1": [ "sqrt(92/9)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the minimum speed of a particle with trajectory $c(t)=(t^3-2 t, \\, t^2+1)$ for $t \\ge 0$. Hint: it is easier to find the minimum of the square of the speed. [ANS]", "answer_v2": [ "sqrt(20/9)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the minimum speed of a particle with trajectory $c(t)=(t^3-4 t, \\, t^2+1)$ for $t \\ge 0$. Hint: it is easier to find the minimum of the square of the speed. [ANS]", "answer_v3": [ "sqrt(44/9)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0976", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Tangents, velocity, and speed", "level": "3", "keywords": [ "Parametric", "Line", "Tangent" ], "problem_v1": "The parametric form for the tangent line to the graph of $y=5x^{2}+x+1$ at $x=2$ is [ANS].", "answer_v1": [ "(2+t, 23+21t)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "The parametric form for the tangent line to the graph of $y=2x^{2}+5x-4$ at $x=-1$ is [ANS].", "answer_v2": [ "(-1+t, -7+t)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "The parametric form for the tangent line to the graph of $y=3x^{2}+x-2$ at $x=-2$ is [ANS].", "answer_v3": [ "(-2+t, 8-11t)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0977", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Tangents, velocity, and speed", "level": "3", "keywords": [ "parametric equation", "parametric" ], "problem_v1": "Suppose a curve is traced by the parametric equations x=5 \\big(\\sin(t)+\\cos(t)\\big)\ny=52-20\\cos^2(t)-40 \\sin(t) as $t$ runs from $0$ to $\\pi$. At what point $(x,y)$ on this curve is the tangent line horizontal? $x=$ [ANS]\n$y=$ [ANS]", "answer_v1": [ "5", "12" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose a curve is traced by the parametric equations x=2 \\big(\\sin(t)+\\cos(t)\\big)\ny=27-4\\cos^2(t)-8 \\sin(t) as $t$ runs from $0$ to $\\pi$. At what point $(x,y)$ on this curve is the tangent line horizontal? $x=$ [ANS]\n$y=$ [ANS]", "answer_v2": [ "2", "19" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose a curve is traced by the parametric equations x=3 \\big(\\sin(t)+\\cos(t)\\big)\ny=31-9\\cos^2(t)-18 \\sin(t) as $t$ runs from $0$ to $\\pi$. At what point $(x,y)$ on this curve is the tangent line horizontal? $x=$ [ANS]\n$y=$ [ANS]", "answer_v3": [ "3", "13" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0978", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Tangents, velocity, and speed", "level": "3", "keywords": [ "parametric equations" ], "problem_v1": "(a) On a separate sheet of paper, sketch the parameterized curve $x=t \\cos t, \\ y=t \\sin t$ for $0 \\leq t \\leq 4\\pi$. Use your graph to complete the following statement: At $t=5$, a particle moving along the curve in the direction of increasing $t$ is moving [ANS] and [ANS] (b) By calculating the position at $t=5$ and $t=5.01$, estimate the speed at $t=5$. speed $\\approx$ [ANS]\n(c) Use derivatives to calculate the speed at $t=5$ and compare your answer to part (b). speed=[ANS]", "answer_v1": [ "up", "to the right", "sqrt([\\frac{(1.46912-1.41831)}{0.01}]^2+[\\frac{(-4.78976--4.79462)}{0.01}]^2)", "sqrt(5.07828^2+0.459387^2)" ], "answer_type_v1": [ "MCS", "MCS", "NV", "NV" ], "options_v1": [ [ "up", "down", "neither up nor down" ], [ "to the left", "to the right", "neither to the left nor to the right" ], [], [] ], "problem_v2": "(a) On a separate sheet of paper, sketch the parameterized curve $x=t \\cos t, \\ y=t \\sin t$ for $0 \\leq t \\leq 4\\pi$. Use your graph to complete the following statement: At $t=2$, a particle moving along the curve in the direction of increasing $t$ is moving [ANS] and [ANS] (b) By calculating the position at $t=2$ and $t=2.01$, estimate the speed at $t=2$. speed $\\approx$ [ANS]\n(c) Use derivatives to calculate the speed at $t=2$ and compare your answer to part (b). speed=[ANS]", "answer_v2": [ "up", "to the left", "sqrt([\\frac{(-0.85469--0.832294)}{0.01}]^2+[\\frac{(1.81923-1.81859)}{0.01}]^2)", "sqrt((-2.23474)^2+0.0770038^2)" ], "answer_type_v2": [ "MCS", "MCS", "NV", "NV" ], "options_v2": [ [ "up", "down", "neither up nor down" ], [ "to the left", "to the right", "neither to the left nor to the right" ], [], [] ], "problem_v3": "(a) On a separate sheet of paper, sketch the parameterized curve $x=t \\cos t, \\ y=t \\sin t$ for $0 \\leq t \\leq 4\\pi$. Use your graph to complete the following statement: At $t=3$, a particle moving along the curve in the direction of increasing $t$ is moving [ANS] and [ANS] (b) By calculating the position at $t=3$ and $t=3.01$, estimate the speed at $t=3$. speed $\\approx$ [ANS]\n(c) Use derivatives to calculate the speed at $t=3$ and compare your answer to part (b). speed=[ANS]", "answer_v3": [ "down", "to the left", "sqrt([\\frac{(-2.98398--2.96998)}{0.01}]^2+[\\frac{(0.394952-0.42336)}{0.01}]^2)", "sqrt((-1.41335)^2+(-2.82886)^2)" ], "answer_type_v3": [ "MCS", "MCS", "NV", "NV" ], "options_v3": [ [ "up", "down", "neither up nor down" ], [ "to the left", "to the right", "neither to the left nor to the right" ], [], [] ] }, { "id": "Calculus_-_single_variable_0979", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Tangents, velocity, and speed", "level": "3", "keywords": [ "parametric equations" ], "problem_v1": "Two particles move in the $xy$-plane. At time $t$, the position of particle $A$ is given by $x(t)=5 t-5$ and $y(t)=3 t-k$, and the position of particle $B$ is given by $x(t)=4 t$ and $y(t)=t^2-2t-1$.\n(a) If $k=2$, do the particles ever collide? [ANS] A. yes B. no C. it is not possible to determine for certain\n(Be sure that you are able to explain your answer!) (Be sure that you are able to explain your answer!) (b) Find $k$ so that the two particles are certain to collide. $k=$ [ANS]\n(c) At the time the particle collide in (b), which is moving faster? [ANS] A. particle A B. particle B C. neither particle (they are moving at the same speed)", "answer_v1": [ "B", "1", "B" ], "answer_type_v1": [ "MCS", "NV", "MCS" ], "options_v1": [ [ "A", "B", "C" ], [], [ "A", "B", "C" ] ], "problem_v2": "Two particles move in the $xy$-plane. At time $t$, the position of particle $A$ is given by $x(t)=3 t-3$ and $y(t)=4 t-k$, and the position of particle $B$ is given by $x(t)=2 t$ and $y(t)=t^2-2t-1$.\n(a) If $k=8$, do the particles ever collide? [ANS] A. yes B. no C. it is not possible to determine for certain\n(Be sure that you are able to explain your answer!) (Be sure that you are able to explain your answer!) (b) Find $k$ so that the two particles are certain to collide. $k=$ [ANS]\n(c) At the time the particle collide in (b), which is moving faster? [ANS] A. particle A B. particle B C. neither particle (they are moving at the same speed)", "answer_v2": [ "B", "10", "A" ], "answer_type_v2": [ "MCS", "NV", "MCS" ], "options_v2": [ [ "A", "B", "C" ], [], [ "A", "B", "C" ] ], "problem_v3": "Two particles move in the $xy$-plane. At time $t$, the position of particle $A$ is given by $x(t)=3 t-3$ and $y(t)=3 t-k$, and the position of particle $B$ is given by $x(t)=2 t$ and $y(t)=t^2-2t-1$.\n(a) If $k=5$, do the particles ever collide? [ANS] A. yes B. no C. it is not possible to determine for certain\n(Be sure that you are able to explain your answer!) (Be sure that you are able to explain your answer!) (b) Find $k$ so that the two particles are certain to collide. $k=$ [ANS]\n(c) At the time the particle collide in (b), which is moving faster? [ANS] A. particle A B. particle B C. neither particle (they are moving at the same speed)", "answer_v3": [ "B", "7", "B" ], "answer_type_v3": [ "MCS", "NV", "MCS" ], "options_v3": [ [ "A", "B", "C" ], [], [ "A", "B", "C" ] ] }, { "id": "Calculus_-_single_variable_0980", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Tangents, velocity, and speed", "level": "4", "keywords": [ "calculus", "parametric equation", "tangential velocity", "minimum", "maximum" ], "problem_v1": "Consider an object moving in the plane according to the parametric equations x=40 t e^{\\frac{t}{40}}-1600 e^{\\frac{t}{40}}, \\quad y=240 e^{\\frac{t}{40}} where $t$ denotes time. Then the tangential velocity of the object at time $t$ is given by $v=$ [ANS]\nThe velocity has a relative maximum at time $t=$ [ANS]\nand a relative minimum at time $t=$ [ANS]", "answer_v1": [ "sqrt(t^2*e^{t/20}+6^2*e^{t/20})", "-39.0787840283389", "-0.921215971661088" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider an object moving in the plane according to the parametric equations x=40 t e^{\\frac{t}{40}}-1600 e^{\\frac{t}{40}}, \\quad y=120 e^{\\frac{t}{40}} where $t$ denotes time. Then the tangential velocity of the object at time $t$ is given by $v=$ [ANS]\nThe velocity has a relative maximum at time $t=$ [ANS]\nand a relative minimum at time $t=$ [ANS]", "answer_v2": [ "sqrt(t^2*e^{t/20}+3^2*e^{t/20})", "-39.7737199332852", "-0.226280066714811" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider an object moving in the plane according to the parametric equations x=26 t e^{\\frac{t}{26}}-676 e^{\\frac{t}{26}}, \\quad y=104 e^{\\frac{t}{26}} where $t$ denotes time. Then the tangential velocity of the object at time $t$ is given by $v=$ [ANS]\nThe velocity has a relative maximum at time $t=$ [ANS]\nand a relative minimum at time $t=$ [ANS]", "answer_v3": [ "sqrt(t^2*e^{t/13}+4^2*e^{t/13})", "-25.369316876853", "-0.630683123147019" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0981", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Tangents, velocity, and speed", "level": "2", "keywords": [ "parametric equation", "tangent line" ], "problem_v1": "Consider the curve given by the parametric equations\nx=t (t^2-147), \\quad y=6 (t^2-147) a.) Determine the point on the curve where the tangent is horizontal. $t=$ [ANS]\nb.) Determine the points $t_1$, $t_2$ where the tangent is vertical and $t_1 < t_2$.\n$t_1=$ [ANS]\n$t_2=$ [ANS]", "answer_v1": [ "0", "-7", "7" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the curve given by the parametric equations\nx=t (t^2-3), \\quad y=9 (t^2-3) a.) Determine the point on the curve where the tangent is horizontal. $t=$ [ANS]\nb.) Determine the points $t_1$, $t_2$ where the tangent is vertical and $t_1 < t_2$.\n$t_1=$ [ANS]\n$t_2=$ [ANS]", "answer_v2": [ "0", "-1", "1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the curve given by the parametric equations\nx=t (t^2-27), \\quad y=6 (t^2-27) a.) Determine the point on the curve where the tangent is horizontal. $t=$ [ANS]\nb.) Determine the points $t_1$, $t_2$ where the tangent is vertical and $t_1 < t_2$.\n$t_1=$ [ANS]\n$t_2=$ [ANS]", "answer_v3": [ "0", "-3", "3" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_single_variable_0982", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Tangents, velocity, and speed", "level": "2", "keywords": [ "parametric equation", "tangent line" ], "problem_v1": "If $x=e^t$ and $y=(t-7)^2$, find an equation $y=mx+b$ of the tangent to the curve at $(1,49)$.\n$m=$ [ANS]\n$b=$ [ANS]", "answer_v1": [ "-14", "2*7+49" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If $x=e^t$ and $y=(t-1)^2$, find an equation $y=mx+b$ of the tangent to the curve at $(1,1)$.\n$m=$ [ANS]\n$b=$ [ANS]", "answer_v2": [ "-2", "2*1+1" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If $x=e^t$ and $y=(t-3)^2$, find an equation $y=mx+b$ of the tangent to the curve at $(1,9)$.\n$m=$ [ANS]\n$b=$ [ANS]", "answer_v3": [ "-6", "2*3+9" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0983", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Tangents, velocity, and speed", "level": "2", "keywords": [ "parametric equation", "derivative" ], "problem_v1": "Given the parametric equations\nx=5 t-t^3, \\quad y=6-4 t Compute the derivative $dy/dx$ as a function of $t$.\nAnswer: [ANS]", "answer_v1": [ "-4/(5-3*t^2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Given the parametric equations\nx=2 t-t^3, \\quad y=9-2 t Compute the derivative $dy/dx$ as a function of $t$.\nAnswer: [ANS]", "answer_v2": [ "-2/(2-3*t^2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Given the parametric equations\nx=3 t-t^3, \\quad y=6-3 t Compute the derivative $dy/dx$ as a function of $t$.\nAnswer: [ANS]", "answer_v3": [ "-3/(3-3*t^2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0984", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Higher order derivatives", "level": "2", "keywords": [ "parametric equation", "first derivative", "second derivative" ], "problem_v1": "Given $x=e^{-t}$ and $y=t e^{8 t}$, find the following derivatives as functions of $t$.\n$dy/dx=$ [ANS]\n$d^2y/dx^2=$ [ANS]", "answer_v1": [ "-e^{9*t}-8*t*e^{9*t}", "(2*8+1)*e^{10*t}+8*(8+1)*t*e^{10*t}" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Given $x=e^{-t}$ and $y=t e^{2 t}$, find the following derivatives as functions of $t$.\n$dy/dx=$ [ANS]\n$d^2y/dx^2=$ [ANS]", "answer_v2": [ "-e^{3*t}-2*t*e^{3*t}", "(2*2+1)*e^{4*t}+2*(2+1)*t*e^{4*t}" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Given $x=e^{-t}$ and $y=t e^{4 t}$, find the following derivatives as functions of $t$.\n$dy/dx=$ [ANS]\n$d^2y/dx^2=$ [ANS]", "answer_v3": [ "-e^{5*t}-4*t*e^{5*t}", "(2*4+1)*e^{6*t}+4*(4+1)*t*e^{6*t}" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0985", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Arc length", "level": "3", "keywords": [ "calculus", "parametric", "polar", "conic" ], "problem_v1": "Use the Midpoint Rule with N=10, 20, 30, and 50 to approximate the given curve's length. c(t)=(\\cos t, e^{\\sin t}) \\, \\, for \\, 0 \\le t \\le 5 \\pi N=10: [ANS]\nN=20: [ANS]\nN=30: [ANS]\nN=50: [ANS]", "answer_v1": [ "20.0233", "18.1829", "18.0863", "18.0606" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Use the Midpoint Rule with N=10, 20, 30, and 50 to approximate the given curve's length. c(t)=(\\cos t, e^{\\sin t}) \\, \\, for \\, 0 \\le t \\le 2 \\pi N=10: [ANS]\nN=20: [ANS]\nN=30: [ANS]\nN=50: [ANS]", "answer_v2": [ "6.90373", "6.91504", "6.91495", "6.91495" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Use the Midpoint Rule with N=10, 20, 30, and 50 to approximate the given curve's length. c(t)=(\\cos t, e^{\\sin t}) \\, \\, for \\, 0 \\le t \\le 3 \\pi N=10: [ANS]\nN=20: [ANS]\nN=30: [ANS]\nN=50: [ANS]", "answer_v3": [ "11.1516", "11.1543", "11.1455", "11.14" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_single_variable_0986", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Arc length", "level": "3", "keywords": [ "calculus", "parametric", "polar", "conic" ], "problem_v1": "Calculate the length of the path over the given interval. c(t)=(5 t^2, 10 t^3), \\, 2 \\le t \\le 4 [ANS]", "answer_v1": [ "2*10/(3*18)*[145^{1.5}-37^{1.5}]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the length of the path over the given interval. c(t)=(3 t^2, 4 t^3), \\, 1 \\le t \\le 5 [ANS]", "answer_v2": [ "2*6/(3*8)*[101^{1.5}}-5^{1.5}]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the length of the path over the given interval. c(t)=(3 t^2, 4 t^3), \\, 1 \\le t \\le 4 [ANS]", "answer_v3": [ "2*6/(3*8)*[65^{1.5}-5^{1.5}]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0987", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Arc length", "level": "3", "keywords": [ "calculus", "parametric", "polar", "conic" ], "problem_v1": "Calculate the length of the path over the given interval. (\\sin \\theta-\\theta \\cos \\theta, \\cos \\theta+\\theta \\sin \\theta), \\, 0 \\le \\theta \\le 8 [ANS]", "answer_v1": [ "32" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the length of the path over the given interval. (\\sin \\theta-\\theta \\cos \\theta, \\cos \\theta+\\theta \\sin \\theta), \\, 0 \\le \\theta \\le 2 [ANS]", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the length of the path over the given interval. (\\sin \\theta-\\theta \\cos \\theta, \\cos \\theta+\\theta \\sin \\theta), \\, 0 \\le \\theta \\le 4 [ANS]", "answer_v3": [ "8" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0988", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Arc length", "level": "3", "keywords": [ "calculus", "parametric", "polar", "conic" ], "problem_v1": "Find the length of the spiral $c(t)=(t \\cos t, t \\sin t)$ for $0 \\le t \\le 5 \\pi$ to three decimal places. Hint: use the formula \\int \\sqrt{1+t^2} \\, dt=\\frac{1}{2} t \\sqrt{1+t^2}+\\frac{1}{2} \\ln \\left(t+\\sqrt{1+t^2} \\right)+C [ANS]", "answer_v1": [ "5*pi/2*(1+25*pi^2)^0.5+1/2*ln(5*pi+(1+25*pi^2)^0.5)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the length of the spiral $c(t)=(t \\cos t, t \\sin t)$ for $0 \\le t \\le 2 \\pi$ to three decimal places. Hint: use the formula \\int \\sqrt{1+t^2} \\, dt=\\frac{1}{2} t \\sqrt{1+t^2}+\\frac{1}{2} \\ln \\left(t+\\sqrt{1+t^2} \\right)+C [ANS]", "answer_v2": [ "2*pi/2*(1+4*pi^2)^0.5+1/2*ln(2*pi+(1+4*pi^2)^0.5)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the length of the spiral $c(t)=(t \\cos t, t \\sin t)$ for $0 \\le t \\le 3 \\pi$ to three decimal places. Hint: use the formula \\int \\sqrt{1+t^2} \\, dt=\\frac{1}{2} t \\sqrt{1+t^2}+\\frac{1}{2} \\ln \\left(t+\\sqrt{1+t^2} \\right)+C [ANS]", "answer_v3": [ "3*pi/2*(1+9*pi^2)^0.5+1/2*ln(3*pi+(1+9*pi^2)^0.5)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0989", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Arc length", "level": "3", "keywords": [ "calculus", "integration", "parametric equations", "arc length", "integral' 'distance' 'length", "Arc length", "parametric" ], "problem_v1": "You and your best friend Janine decide to play a game. You are in a land of make believe where you are a function, $f(t)$, and she is a function, $g(t)$. The two of you move together throughout this land with you (that is, $f(t)$) controlling your East/West movement and Janine (that is, $g(t)$) controlling your North/South movement. If your identity, $f(t)$, is given by f(t)=\\frac{(t^2+46)^{\\frac{3}{2}}}{3} and Janine's identity, $g(t)$, is given by g(t)=23 t then how many units of distance do the two of you cover between the Most Holy Point o' Beginnings, $t=0$, and The Buck Stops Here, $t=27$? We travel [ANS] units of distance.", "answer_v1": [ "7182" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You and your best friend Janine decide to play a game. You are in a land of make believe where you are a function, $f(t)$, and she is a function, $g(t)$. The two of you move together throughout this land with you (that is, $f(t)$) controlling your East/West movement and Janine (that is, $g(t)$) controlling your North/South movement. If your identity, $f(t)$, is given by f(t)=\\frac{(t^2+8)^{\\frac{3}{2}}}{3} and Janine's identity, $g(t)$, is given by g(t)=4 t then how many units of distance do the two of you cover between the Most Holy Point o' Beginnings, $t=0$, and The Buck Stops Here, $t=42$? We travel [ANS] units of distance.", "answer_v2": [ "24864" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You and your best friend Janine decide to play a game. You are in a land of make believe where you are a function, $f(t)$, and she is a function, $g(t)$. The two of you move together throughout this land with you (that is, $f(t)$) controlling your East/West movement and Janine (that is, $g(t)$) controlling your North/South movement. If your identity, $f(t)$, is given by f(t)=\\frac{(t^2+22)^{\\frac{3}{2}}}{3} and Janine's identity, $g(t)$, is given by g(t)=11 t then how many units of distance do the two of you cover between the Most Holy Point o' Beginnings, $t=0$, and The Buck Stops Here, $t=28$? We travel [ANS] units of distance.", "answer_v3": [ "7625.33" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0990", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Arc length", "level": "3", "keywords": [ "integral' 'distance' 'length", "Length", "Hypocycloid" ], "problem_v1": "Let L be the circle in the x-y plane with center the origin and radius 76. Let S be a moveable circle with radius 44. S is rolled along the inside of L without slipping while L remains fixed. A point P is marked on S before S is rolled and the path of P is studied. The initial position of P is (76,0). The initial position of the center of S is (32,0). After S has moved counterclockwise about the origin through an angle t the position of P is\nx=32 \\cos t+44 \\cos \\left(\\frac{8}{11} t \\right) y=32 \\sin t-44 \\sin \\left(\\frac{8}{11} t \\right) How far does P move before it returns to its initial position? Hint: You may use the formulas for cos(u+v) and sin(w/2). S makes several complete revolutions about the origin before P returns to (76,0). [ANS]", "answer_v1": [ "2816" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let L be the circle in the x-y plane with center the origin and radius 19. Let S be a moveable circle with radius 17. S is rolled along the inside of L without slipping while L remains fixed. A point P is marked on S before S is rolled and the path of P is studied. The initial position of P is (19,0). The initial position of the center of S is (2,0). After S has moved counterclockwise about the origin through an angle t the position of P is\nx=2 \\cos t+17 \\cos \\left(\\frac{2}{17} t \\right) y=2 \\sin t-17 \\sin \\left(\\frac{2}{17} t \\right) How far does P move before it returns to its initial position? Hint: You may use the formulas for cos(u+v) and sin(w/2). S makes several complete revolutions about the origin before P returns to (19,0). [ANS]", "answer_v2": [ "272" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let L be the circle in the x-y plane with center the origin and radius 38. Let S be a moveable circle with radius 24. S is rolled along the inside of L without slipping while L remains fixed. A point P is marked on S before S is rolled and the path of P is studied. The initial position of P is (38,0). The initial position of the center of S is (14,0). After S has moved counterclockwise about the origin through an angle t the position of P is\nx=14 \\cos t+24 \\cos \\left(\\frac{7}{12} t \\right) y=14 \\sin t-24 \\sin \\left(\\frac{7}{12} t \\right) How far does P move before it returns to its initial position? Hint: You may use the formulas for cos(u+v) and sin(w/2). S makes several complete revolutions about the origin before P returns to (38,0). [ANS]", "answer_v3": [ "1344" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0991", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Arc length", "level": "3", "keywords": [ "Parametric curve", "Length", "Integration", "integral' 'distance' 'length" ], "problem_v1": "Find the length of parametrized curve given by\nx(t)=6 t^2+12 t, \\quad y(t)=-t^3-3 t^2+9 t where $t$ goes from $0$ to $1$.\nAnswer: [ANS]", "answer_v1": [ "1 + 3 + 15" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the length of parametrized curve given by\nx(t)=-18 t^2+36 t, \\quad y(t)=-4 t^3+12 t^2+15 t where $t$ goes from $0$ to $1$.\nAnswer: [ANS]", "answer_v2": [ "4 + -12 + 39" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the length of parametrized curve given by\nx(t)=-3 t^2+6 t, \\quad y(t)=-t^3+3 t^2 where $t$ goes from $0$ to $1$.\nAnswer: [ANS]", "answer_v3": [ "1 + -3 + 6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0992", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Arc length", "level": "3", "keywords": [ "Integration", "Arc length", "parametric curve", "integral' 'distance' 'length", "parametric" ], "problem_v1": "If $x=16 \\cos^3 \\theta$ and $y=16 \\sin^3 \\theta$, find the total length of the curve swept out by the point $(x,y)$ as $\\theta$ ranges from $0$ to $2\\pi.$\nAnswer: [ANS]", "answer_v1": [ "16*6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $x=2 \\cos^3 \\theta$ and $y=2 \\sin^3 \\theta$, find the total length of the curve swept out by the point $(x,y)$ as $\\theta$ ranges from $0$ to $2\\pi.$\nAnswer: [ANS]", "answer_v2": [ "2*6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $x=8 \\cos^3 \\theta$ and $y=8 \\sin^3 \\theta$, find the total length of the curve swept out by the point $(x,y)$ as $\\theta$ ranges from $0$ to $2\\pi.$\nAnswer: [ANS]", "answer_v3": [ "8*6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0993", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Arc length", "level": "3", "keywords": [ "Integration", "Arc length", "parametric", "integral' 'distance' 'length" ], "problem_v1": "Consider the parametric curve given by the equations x(t)=t^2+29 t+8 y(t)=t^2+29 t+12 How many units of distance are covered by the point $P(t)=(x(t),y(t))$ between $t=0$ and $t=8$?\nAnswer: [ANS]", "answer_v1": [ "sqrt(2)*(8^2+29*8)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider the parametric curve given by the equations x(t)=t^2+4 t+42 y(t)=t^2+4 t-34 How many units of distance are covered by the point $P(t)=(x(t),y(t))$ between $t=0$ and $t=5$?\nAnswer: [ANS]", "answer_v2": [ "sqrt(2)*(5^2+4*5)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider the parametric curve given by the equations x(t)=t^2+13 t+10 y(t)=t^2+13 t-21 How many units of distance are covered by the point $P(t)=(x(t),y(t))$ between $t=0$ and $t=6$?\nAnswer: [ANS]", "answer_v3": [ "sqrt(2)*(6^2+13*6)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0994", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Area", "level": "3", "keywords": [ "parametric equation", "parametric", "area" ], "problem_v1": "The following parametric equations trace out a loop. \\begin{array}{r@{\\,}c@{\\,}l} x &=& 8-\\frac{5}{2} t^2 \\cr y &=&-\\frac{5}{6}t^3+5 t+2 \\end{array} Find the $t$ values at which the curve intersects itself: $t=\\pm$ [ANS]. What is the total area inside the loop? Area=[ANS].", "answer_v1": [ "2.44948974278318", "97.9795897113271" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The following parametric equations trace out a loop. \\begin{array}{r@{\\,}c@{\\,}l} x &=& 7-\\frac{2}{2} t^2 \\cr y &=&-\\frac{2}{6}t^3+2 t+1 \\end{array} Find the $t$ values at which the curve intersects itself: $t=\\pm$ [ANS]. What is the total area inside the loop? Area=[ANS].", "answer_v2": [ "2.44948974278318", "15.6767343538123" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The following parametric equations trace out a loop. \\begin{array}{r@{\\,}c@{\\,}l} x &=& 7-\\frac{3}{2} t^2 \\cr y &=&-\\frac{3}{6}t^3+3 t+1 \\end{array} Find the $t$ values at which the curve intersects itself: $t=\\pm$ [ANS]. What is the total area inside the loop? Area=[ANS].", "answer_v3": [ "2.44948974278318", "35.2726522960778" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_single_variable_0995", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Area", "level": "2", "keywords": [ "parametric", "area", "integration" ], "problem_v1": "Find the area of the region enclosed by the parametric equation\nx=t^3-7 t y=6 t^2 [ANS]", "answer_v1": [ "414.853805574928" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of the region enclosed by the parametric equation\nx=t^3-2 t y=9 t^2 [ANS]", "answer_v2": [ "27.1529003975634" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of the region enclosed by the parametric equation\nx=t^3-6 t y=4 t^2 [ANS]", "answer_v3": [ "188.120812245748" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0996", "subject": "Calculus_-_single_variable", "topic": "Parametric", "subtopic": "Area", "level": "3", "keywords": [ "parametric", "area", "integration", "integrals' 'area' 'parametric" ], "problem_v1": "Use the parametric equations of an ellipse $x=10 \\cos \\theta$ $y=15 \\sin \\theta$ $0 \\leq \\theta \\leq 2 \\pi$ to find the area that it encloses. [ANS]", "answer_v1": [ "471.238898038469" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the parametric equations of an ellipse $x=2 \\cos \\theta$ $y=23 \\sin \\theta$ $0 \\leq \\theta \\leq 2 \\pi$ to find the area that it encloses. [ANS]", "answer_v2": [ "144.51326206513" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the parametric equations of an ellipse $x=5 \\cos \\theta$ $y=15 \\sin \\theta$ $0 \\leq \\theta \\leq 2 \\pi$ to find the area that it encloses. [ANS]", "answer_v3": [ "235.619449019234" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0997", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Tangents", "level": "2", "keywords": [ "Polar", "Trigonometric", "Sin", "Cos", "Tangent", "Slope" ], "problem_v1": "Find the slope of the tangent line to the polar curve $\\small{r=\\mathrm{cos}(7 \\theta)}$ at $\\small{\\theta=} \\large{\\frac{\\pi}{4}}$. Enter as an integer or fraction in lowest terms. Slope=[ANS]", "answer_v1": [ "4/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the slope of the tangent line to the polar curve $\\small{r=\\mathrm{cos}(\\theta)}$ at $\\small{\\theta=} \\large{\\frac{\\pi}{4}}$. Enter as an integer or fraction in lowest terms. Slope=[ANS]", "answer_v2": [ "0" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the slope of the tangent line to the polar curve $\\small{r=\\mathrm{cos}(3 \\theta)}$ at $\\small{\\theta=} \\large{\\frac{\\pi}{4}}$. Enter as an integer or fraction in lowest terms. Slope=[ANS]", "answer_v3": [ "2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0998", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Tangents", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find the slope of the tangent to the curve $r=5+2\\cos\\theta$ at the value $\\theta=\\pi/2$ [ANS]", "answer_v1": [ "0.4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the slope of the tangent to the curve $r=-9+9\\cos\\theta$ at the value $\\theta=\\pi/2$ [ANS]", "answer_v2": [ "-1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the slope of the tangent to the curve $r=-4+2\\cos\\theta$ at the value $\\theta=\\pi/2$ [ANS]", "answer_v3": [ "-0.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_0999", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Area", "level": "4", "keywords": [ "calculus" ], "problem_v1": "Find the area of the part of the circle $r=8 \\sin\\theta+\\cos\\theta$ in the fourth quadrant. Answer: [ANS]", "answer_v1": [ "0.0207687" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of the part of the circle $r=2 \\sin\\theta+\\cos\\theta$ in the fourth quadrant. Answer: [ANS]", "answer_v2": [ "0.0795595" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of the part of the circle $r=4 \\sin\\theta+\\cos\\theta$ in the fourth quadrant. Answer: [ANS]", "answer_v3": [ "0.0411593" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_1000", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Area", "level": "4", "keywords": [ "polar' 'curve' 'area", "polar", "areas" ], "problem_v1": "Find the area of the region which is inside the polar curve\nr=7 \\cos (\\theta) and outside the curve\nr=5-3 \\cos (\\theta) The area is [ANS]", "answer_v1": [ "29.4050283953946" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of the region which is inside the polar curve\nr=7 \\cos (\\theta) and outside the curve\nr=4-1 \\cos (\\theta) The area is [ANS]", "answer_v2": [ "25.6980884852616" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of the region which is inside the polar curve\nr=5 \\cos (\\theta) and outside the curve\nr=3-1 \\cos (\\theta) The area is [ANS]", "answer_v3": [ "13.5338974990031" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_1001", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Area", "level": "4", "keywords": [ "polar' 'curve' 'area", "polar", "area", "polar coordinates" ], "problem_v1": "Find the area inside one leaf of the rose:\nr=5 \\sin (5 \\theta) The area is [ANS]", "answer_v1": [ "3.92699081698724" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area inside one leaf of the rose:\nr=2 \\sin (6 \\theta) The area is [ANS]", "answer_v2": [ "0.523598775598299" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area inside one leaf of the rose:\nr=3 \\sin (5 \\theta) The area is [ANS]", "answer_v3": [ "1.41371669411541" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_1002", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Area", "level": "4", "keywords": [ "Polar Coordinates", "Area", "Integration", "integral' 'polar' 'area" ], "problem_v1": "Find the area enclosed by the closed curve obtained by joining the ends of the spiral r=7 \\theta, 0 \\leq \\theta \\leq 3.5 by a straight line segment. [ANS]", "answer_v1": [ "350.145833333333" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area enclosed by the closed curve obtained by joining the ends of the spiral r=1 \\theta, 0 \\leq \\theta \\leq 5.6 by a straight line segment. [ANS]", "answer_v2": [ "29.2693333333333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area enclosed by the closed curve obtained by joining the ends of the spiral r=3 \\theta, 0 \\leq \\theta \\leq 3.7 by a straight line segment. [ANS]", "answer_v3": [ "75.9795" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_1003", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Area", "level": "4", "keywords": [ "Polar", "area", "limacon", "Integral" ], "problem_v1": "Find the area between the loops of the limacon $\\small{r=7(1+2 \\cos\\theta)}$. Area=[ANS]", "answer_v1": [ "49*[pi+3*sqrt(3)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area between the loops of the limacon $\\small{r=(1+2 \\cos\\theta)}$. Area=[ANS]", "answer_v2": [ "1*[pi+3*sqrt(3)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area between the loops of the limacon $\\small{r=3(1+2 \\cos\\theta)}$. Area=[ANS]", "answer_v3": [ "9*[pi+3*sqrt(3)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_1004", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Area", "level": "4", "keywords": [ "calculus", "integral", "polar coordinates" ], "problem_v1": "Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles $x^2+y^2=256$ and $x^2-16x+y^2=0$. [ANS]", "answer_v1": [ "100.530964914873" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles $x^2+y^2=4$ and $x^2-2x+y^2=0$. [ANS]", "answer_v2": [ "1.5707963267949" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles $x^2+y^2=64$ and $x^2-8x+y^2=0$. [ANS]", "answer_v3": [ "25.1327412287183" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_1005", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Arc length", "level": "4", "keywords": [ "calculus" ], "problem_v1": "Sketch the segment $r=\\sec\\theta$ for $0\\le \\theta\\le \\frac{\\pi}{8}$. Then compute its length in two ways: as an integral in polar coordinates and using trigonometry. $L=$ [ANS]", "answer_v1": [ "0.414213" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sketch the segment $r=\\sec\\theta$ for $0\\le \\theta\\le \\frac{\\pi}{3}$. Then compute its length in two ways: as an integral in polar coordinates and using trigonometry. $L=$ [ANS]", "answer_v2": [ "1.73205" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sketch the segment $r=\\sec\\theta$ for $0\\le \\theta\\le \\frac{\\pi}{5}$. Then compute its length in two ways: as an integral in polar coordinates and using trigonometry. $L=$ [ANS]", "answer_v3": [ "0.726542" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_1006", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Arc length", "level": "4", "keywords": [ "calculus" ], "problem_v1": "Find the length of the spiral $r=\\theta$ for $0\\le \\theta\\le 8$. Answer: [ANS]", "answer_v1": [ "33.6373" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the length of the spiral $r=\\theta$ for $0\\le \\theta\\le 2$. Answer: [ANS]", "answer_v2": [ "2.95789" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the length of the spiral $r=\\theta$ for $0\\le \\theta\\le 4$. Answer: [ANS]", "answer_v3": [ "9.29357" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_1007", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Arc length", "level": "4", "keywords": [ "calculus", "area", "\u2018arc length\u2019" ], "problem_v1": "On the interval $[0,7]$ the polar curve $r=10\\theta^{2}$ has arc length [ANS] units.", "answer_v1": [ "10/3*[(7^2+4)^{1.5}-8]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "On the interval $[0,12]$ the polar curve $r=\\theta^{2}$ has arc length [ANS] units.", "answer_v2": [ "1/3*[(12^2+4)^{1.5}-8]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "On the interval $[0,8]$ the polar curve $r=4\\theta^{2}$ has arc length [ANS] units.", "answer_v3": [ "4/3*[(8^2+4)^{1.5}-8]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_1008", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Arc length", "level": "2", "keywords": [ "Polar", "Trigonometric", "Exponential", "Arc Length", "Integral" ], "problem_v1": "Find the arc length of the polar curve $\\small{r=e^{8\\theta}}$ from $\\small{\\theta=0}$ to $\\small{\\theta=4}$. Keep all radicals in your answer, and enter $\\small{e}$ if appropriate. Arc Length=[ANS]", "answer_v1": [ "[sqrt(1+8^2)]/8*[e^32-1]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the arc length of the polar curve $\\small{r=e^{2\\theta}}$ from $\\small{\\theta=0}$ to $\\small{\\theta=5}$. Keep all radicals in your answer, and enter $\\small{e}$ if appropriate. Arc Length=[ANS]", "answer_v2": [ "[sqrt(1+2^2)]/2*[e^10-1]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the arc length of the polar curve $\\small{r=e^{4\\theta}}$ from $\\small{\\theta=0}$ to $\\small{\\theta=4}$. Keep all radicals in your answer, and enter $\\small{e}$ if appropriate. Arc Length=[ANS]", "answer_v3": [ "[sqrt(1+4^2)]/4*[e^16-1]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_single_variable_1009", "subject": "Calculus_-_single_variable", "topic": "Polar", "subtopic": "Arc length", "level": "4", "keywords": [ "Polar Coordinates", "Length", "Circles", "Integration", "integral' 'polar' 'area' 'length" ], "problem_v1": "Find the length of the entire perimeter of the region inside $r=17 \\sin \\theta$ but outside $r=3$.\nAnswer: [ANS]", "answer_v1": [ "(17+3)*2*(( pi/2 )-arcsin(3/17))" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the length of the entire perimeter of the region inside $r=6 \\sin \\theta$ but outside $r=4$.\nAnswer: [ANS]", "answer_v2": [ "(6+4)*2*(( pi/2 )-arcsin(4/6))" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the length of the entire perimeter of the region inside $r=10 \\sin \\theta$ but outside $r=3$.\nAnswer: [ANS]", "answer_v3": [ "(10+3)*2*(( pi/2 )-arcsin(3/10))" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] } ]