[ { "id": "Calculus_-_multivariable_0000", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "2", "keywords": [ "calculus" ], "problem_v1": "The function ${\\bf r} \\left(t \\right)$ traces a circle. Determine the radius, center, and plane containing the circle ${\\bf r} \\left(t \\right)=5 {\\bf i} \\,+$ $\\left(9 \\cos(t) \\right) {\\bf j} \\,+$ $\\left(9 \\sin(t) \\right) {\\bf k}$ Plane: $x=$ [ANS]\nCircle's Center: ([ANS], [ANS], [ANS]) Radius: [ANS]", "answer_v1": [ "5", "5", "0", "0", "9" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "The function ${\\bf r} \\left(t \\right)$ traces a circle. Determine the radius, center, and plane containing the circle ${\\bf r} \\left(t \\right)=-8 {\\bf i} \\,+$ $\\left(13 \\cos(t) \\right) {\\bf j} \\,+$ $\\left(13 \\sin(t) \\right) {\\bf k}$ Plane: $x=$ [ANS]\nCircle's Center: ([ANS], [ANS], [ANS]) Radius: [ANS]", "answer_v2": [ "-8", "-8", "0", "0", "13" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "The function ${\\bf r} \\left(t \\right)$ traces a circle. Determine the radius, center, and plane containing the circle ${\\bf r} \\left(t \\right)=-4 {\\bf i} \\,+$ $\\left(9 \\cos(t) \\right) {\\bf j} \\,+$ $\\left(9 \\sin(t) \\right) {\\bf k}$ Plane: $x=$ [ANS]\nCircle's Center: ([ANS], [ANS], [ANS]) Radius: [ANS]", "answer_v3": [ "-4", "-4", "0", "0", "9" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0001", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Use $\\cos(t)$ and $\\sin(t)$, with positive coefficients, to parametrize the intersection of the surfaces $x^2+y^2=64$ and $z=6x^{3}$. $\\bf r\\it (t)=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left. \\right>$", "answer_v1": [ "8*cos(t)", "8*sin(t)", "6*[8*cos(t)]^3" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Use $\\cos(t)$ and $\\sin(t)$, with positive coefficients, to parametrize the intersection of the surfaces $x^2+y^2=4$ and $z=8x^{2}$. $\\bf r\\it (t)=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left. \\right>$", "answer_v2": [ "2*cos(t)", "2*sin(t)", "8*[2*cos(t)]^2" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Use $\\cos(t)$ and $\\sin(t)$, with positive coefficients, to parametrize the intersection of the surfaces $x^2+y^2=16$ and $z=6x^{2}$. $\\bf r\\it (t)=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left. \\right>$", "answer_v3": [ "4*cos(t)", "4*sin(t)", "6*[4*cos(t)]^2" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0002", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find a parametrization, using $\\cos(t)$ and $\\sin(t)$, of the following curve: The intersection of the plane $y=5$ with the sphere $x^2+y^2+z^2=106$ $\\bf r\\it (t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v1": [ "9*cos(t)", "5", "9*sin(t)" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Find a parametrization, using $\\cos(t)$ and $\\sin(t)$, of the following curve: The intersection of the plane $y=7$ with the sphere $x^2+y^2+z^2=74$ $\\bf r\\it (t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v2": [ "5*cos(t)", "7", "5*sin(t)" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Find a parametrization, using $\\cos(t)$ and $\\sin(t)$, of the following curve: The intersection of the plane $y=5$ with the sphere $x^2+y^2+z^2=61$ $\\bf r\\it (t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v3": [ "6*cos(t)", "5", "6*sin(t)" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0003", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find a path that traces the circle in the plane $y=1$ with radius $r=2$ and center $(1,1,2)$ with constant speed 14. ${\\bf r\\it_1(s)}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left. \\right>$", "answer_v1": [ "1+2*cos(7*s)", "1", "2+2*sin(7*s)" ], "answer_type_v1": [ "EX", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Find a path that traces the circle in the plane $y=-4$ with radius $r=5$ and center $(5,-4,-2)$ with constant speed 10. ${\\bf r\\it_1(s)}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left. \\right>$", "answer_v2": [ "5+5*cos(2*s)", "-4", "5*sin(2*s)-2" ], "answer_type_v2": [ "EX", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Find a path that traces the circle in the plane $y=-2$ with radius $r=2$ and center $(1,-2,1)$ with constant speed 8. ${\\bf r\\it_1(s)}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left. \\right>$", "answer_v3": [ "1+2*cos(4*s)", "-2", "1+2*sin(4*s)" ], "answer_type_v3": [ "EX", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0004", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "3", "keywords": [ "calculus", "parametric", "vector", "3D", "three dimensions" ], "problem_v1": "$\\mathbf{r}_1(t)=\\left(8,-1,0\\right)+t \\left<2,-2,-2\\right>$ $\\mathbf{r}_2(t)=\\left(3,0,5\\right)+t \\left<1,1,-1\\right>$ Find the point of intersection, $P$, of the lines $\\mathbf{r}_1$ and $\\mathbf{r}_2$. $P$=[ANS]", "answer_v1": [ "(5,2,3)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "$\\mathbf{r}_1(t)=\\left(-5.5,-5,0\\right)+t \\left<-1,4,-2\\right>$ $\\mathbf{r}_2(t)=\\left(-12,7,-6\\right)+t \\left<-3,-2,1\\right>$ Find the point of intersection, $P$, of the lines $\\mathbf{r}_1$ and $\\mathbf{r}_2$. $P$=[ANS]", "answer_v2": [ "(-9,9,-7)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "$\\mathbf{r}_1(t)=\\left(-4,9.5,-2.5\\right)+t \\left<0,-3,-1\\right>$ $\\mathbf{r}_2(t)=\\left(-10,-6,-11\\right)+t \\left<3,4,3\\right>$ Find the point of intersection, $P$, of the lines $\\mathbf{r}_1$ and $\\mathbf{r}_2$. $P$=[ANS]", "answer_v3": [ "(-4,2,-5)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0005", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "2", "keywords": [ "calculus", "parametric", "vector", "3D", "three dimensions" ], "problem_v1": "Find the vector parameterization for the line which passes through the origin and is perpendicular to the xz-plane. [ANS] A. $\\left<0,t,0\\right>$ B. $\\left$ C. $\\left<0,0,t\\right>$ D. $\\left$", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D" ] ], "problem_v2": "Find the vector parameterization for the line which passes through the origin and is perpendicular to the xy-plane. [ANS] A. $\\left$ B. $\\left<0,t,0\\right>$ C. $\\left$ D. $\\left<0,0,t\\right>$", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D" ] ], "problem_v3": "Find the vector parameterization for the line which passes through the origin and is perpendicular to the xy-plane. [ANS] A. $\\left$ B. $\\left<0,0,t\\right>$ C. $\\left<0,t,0\\right>$ D. $\\left$", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_multivariable_0006", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "4", "keywords": [ "Vector", "Distance", "Point", "Line", "Dot Product", "Projection" ], "problem_v1": "Consider the line $L(t)=\\left<2+t,t-2,-\\left(2+t\\right)\\right>$ and the point $P=\\left(3,1,1\\right)$. How far is $P$ from the line $L$? [ANS].", "answer_v1": [ "4.32049" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider the line $L(t)=\\left<-\\left(2+3t\\right),5-2t,t-2\\right>$ and the point $P=\\left(-5,5,-4\\right)$. How far is $P$ from the line $L$? [ANS].", "answer_v2": [ "3.08221" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider the line $L(t)=\\left<1+3t,5t-3,4t-2\\right>$ and the point $P=\\left(-2,1,-2\\right)$. How far is $P$ from the line $L$? [ANS].", "answer_v3": [ "4.75184" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0007", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "3", "keywords": [ "Vector", "Parametric", "Geometry" ], "problem_v1": "Find the simplest vector parametric expression $\\vec{r}(t)$ for the line that passes through the points $P=\\left(3,1,1\\right)$ at time t=3 and $Q=\\left(5,-1,-1\\right)$ at time t=8.\n$\\vec{r}(t)=$ [ANS]", "answer_v1": [ "(0.4t+1.8,-0.4t+2.2,-0.4t+2.2)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the simplest vector parametric expression $\\vec{r}(t)$ for the line that passes through the points $P=\\left(-5,5,-4\\right)$ at time t=1 and $Q=\\left(-7,10,-6\\right)$ at time t=4.\n$\\vec{r}(t)=$ [ANS]", "answer_v2": [ "(-2/3*t-13/3, 5/3*t+10/3, -2/3*t-10/3)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the simplest vector parametric expression $\\vec{r}(t)$ for the line that passes through the points $P=\\left(-2,1,-2\\right)$ at time t=4 and $Q=\\left(-1,-2,-4\\right)$ at time t=12.\n$\\vec{r}(t)=$ [ANS]", "answer_v3": [ "(0.125t-7, -0.375t+2.5, -0.25t-1)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0008", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "3", "keywords": [ "Line", "Parametric", "Parallel" ], "problem_v1": "Give a vector parametric equation for the line that passes through the point $\\left(1,1,-1\\right)$, parallel to the line parametrized by $\\left<3+2t,1-2t,1-2t\\right>$:\n$L(t)$=[ANS].", "answer_v1": [ "(2t+1, -2t+1, -2t-1)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Give a vector parametric equation for the line that passes through the point $\\left(-3,-2,1\\right)$, parallel to the line parametrized by $\\left<-5-2t,5+5t,-4-2t\\right>$:\n$L(t)$=[ANS].", "answer_v2": [ "(-2t-3, 5t-2, -2t+1)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Give a vector parametric equation for the line that passes through the point $\\left(3,5,4\\right)$, parallel to the line parametrized by $\\left$:\n$L(t)$=[ANS].", "answer_v3": [ "(t+3, -3t+5, -2t+4)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0009", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "1", "keywords": [ "parametric equations" ], "problem_v1": "Write a parameterization for the curve in the $xy$-plane that is a vertical line through the point $(-2,-4)$. $x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v1": [ "-2", "t" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Write a parameterization for the curve in the $xy$-plane that is a vertical line through the point $(-8,-1)$. $x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v2": [ "-8", "t" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Write a parameterization for the curve in the $xy$-plane that is a vertical line through the point $(-6,-4)$. $x(t)=$ [ANS]\n$y(t)=$ [ANS]", "answer_v3": [ "-6", "t" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0010", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "4", "keywords": [ "parametric curve", "multivariable", "calculus" ], "problem_v1": "(a) Find a parametric equation for the line through the point $\\left(5,4,4\\right)$ and in the direction of $a\\vec i+b\\vec j+c\\vec k$. $\\vec r(t)=$ [ANS]\n(b) Find conditions on $a,b,c$ so that the line you found in part (a) goes through the origin. (Be sure you can give a reason for your answer.) Then use your work to give two distinct triples $a,b,c$ that result in the line passing through the origin. Choice 1: $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] ; Choice 2: $a=$ [ANS], $b=$ [ANS], $c=$ [ANS].", "answer_v1": [ "(1.404t+5, 2.71828t+4, 1.1771t+4)", "5", "4", "4", "-5", "-4", "-4" ], "answer_type_v1": [ "OL", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "(a) Find a parametric equation for the line through the point $\\left(1,6,1\\right)$ and in the direction of $a\\vec i+b\\vec j+c\\vec k$. $\\vec r(t)=$ [ANS]\n(b) Find conditions on $a,b,c$ so that the line you found in part (a) goes through the origin. (Be sure you can give a reason for your answer.) Then use your work to give two distinct triples $a,b,c$ that result in the line passing through the origin. Choice 1: $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] ; Choice 2: $a=$ [ANS], $b=$ [ANS], $c=$ [ANS].", "answer_v2": [ "(1.404t+1, 2,71828t+6,1.1771t+1)", "1", "6", "1", "-1", "-6", "-1" ], "answer_type_v2": [ "OL", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "(a) Find a parametric equation for the line through the point $\\left(2,4,2\\right)$ and in the direction of $a\\vec i+b\\vec j+c\\vec k$. $\\vec r(t)=$ [ANS]\n(b) Find conditions on $a,b,c$ so that the line you found in part (a) goes through the origin. (Be sure you can give a reason for your answer.) Then use your work to give two distinct triples $a,b,c$ that result in the line passing through the origin. Choice 1: $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] ; Choice 2: $a=$ [ANS], $b=$ [ANS], $c=$ [ANS].", "answer_v3": [ "(1.404t+2, 2.71828t+4, 1.1771t+2)", "2", "4", "2", "-2", "-4", "-2" ], "answer_type_v3": [ "OL", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0011", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "2", "keywords": [ "parametric curve", "multivariable", "calculus" ], "problem_v1": "Find a parameterization for the circle of radius 7 in the $xy$-plane, centered at the origin, clockwise. $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v1": [ "7*sin(t)", "7*cos(t)", "0" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find a parameterization for the circle of radius 2 in the $xy$-plane, centered at the origin, clockwise. $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v2": [ "2*sin(t)", "2*cos(t)", "0" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find a parameterization for the circle of radius 4 in the $xy$-plane, centered at the origin, clockwise. $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v3": [ "4*sin(t)", "4*cos(t)", "0" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0012", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "3", "keywords": [ "parametric curve", "multivariable", "calculus" ], "problem_v1": "Parameterize the line through $P=\\left(5,3\\right)$ and $Q=\\left(10,9\\right)$ so that the points $P$ and $Q$ correspond to the parameter values $t=8$ and $9$. $\\vec r(t)=$ [ANS]", "answer_v1": [ "(5+5*(t-8), 3+6*(t-8))" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Parameterize the line through $P=\\left(-4,8\\right)$ and $Q=\\left(-2,11\\right)$ so that the points $P$ and $Q$ correspond to the parameter values $t=15$ and $16$. $\\vec r(t)=$ [ANS]", "answer_v2": [ "(-4+2*(t-15), 8+3*(t-15))" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Parameterize the line through $P=\\left(-1,3\\right)$ and $Q=\\left(2,8\\right)$ so that the points $P$ and $Q$ correspond to the parameter values $t=7$ and $9$. $\\vec r(t)=$ [ANS]", "answer_v3": [ "(-1+3*(t-7)/2, 3+5*(t-7)/2)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0013", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "1", "keywords": [ "parametric curve", "multivariable", "calculus" ], "problem_v1": "Find the vector parametric equations for the line. in the direction of the vector $2\\,\\mathit{\\vec i}-2\\,\\mathit{\\vec j}-2\\,\\mathit{\\vec k}$ and through the point $\\left(3,1,1\\right)$. $\\vec r(t)=$ [ANS]", "answer_v1": [ "(2t+3, -2t+1, -2t+1)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the vector parametric equations for the line. in the direction of the vector $-2\\,\\mathit{\\vec i}+5\\,\\mathit{\\vec j}-2\\,\\mathit{\\vec k}$ and through the point $\\left(-5,5,-4\\right)$. $\\vec r(t)=$ [ANS]", "answer_v2": [ "(-2t-5, 5t+5, -2t-4)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the vector parametric equations for the line. in the direction of the vector $\\,\\mathit{\\vec i}-3\\,\\mathit{\\vec j}-2\\,\\mathit{\\vec k}$ and through the point $\\left(-2,1,-2\\right)$. $\\vec r(t)=$ [ANS]", "answer_v3": [ "(t-2, -3t+1, -2t_2)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0014", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "3", "keywords": [ "parametric curve", "multivariable", "calculus" ], "problem_v1": "Find a parameterization for the curve $y=7+1x^{4}$ that passes through the point $(0, 7, 2)$ when $t=-1$ and is parallel to the $xy$-plane. $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v1": [ "t+1", "7+1*(t+1)^4", "2" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find a parameterization for the curve $y=1+5x^{3}$ that passes through the point $(0, 1,-2)$ when $t=3$ and is parallel to the $xy$-plane. $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v2": [ "t-3", "1+5*(t-3)^3", "-2" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find a parameterization for the curve $y=3+1x^{3}$ that passes through the point $(0, 3, 1)$ when $t=-2$ and is parallel to the $xy$-plane. $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v3": [ "t+2", "3+1*(t+2)^3", "1" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0015", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "3", "keywords": [ "parametric curve", "multivariable", "calculus" ], "problem_v1": "Find parametric equations for the arc of a circle of radius 8 from $P=(0,0)$ to $Q=(16,0)$. $x(t)=$ [ANS], $y(t)=$ [ANS], with [ANS] $\\le t\\le$ [ANS]", "answer_v1": [ "8-8*cos(t)", "8*sin(t)", "0", "pi" ], "answer_type_v1": [ "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find parametric equations for the arc of a circle of radius 2 from $P=(0,0)$ to $Q=(4,0)$. $x(t)=$ [ANS], $y(t)=$ [ANS], with [ANS] $\\le t\\le$ [ANS]", "answer_v2": [ "2-2*cos(t)", "2*sin(t)", "0", "pi" ], "answer_type_v2": [ "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find parametric equations for the arc of a circle of radius 4 from $P=(0,0)$ to $Q=(8,0)$. $x(t)=$ [ANS], $y(t)=$ [ANS], with [ANS] $\\le t\\le$ [ANS]", "answer_v3": [ "4-4*cos(t)", "4*sin(t)", "0", "pi" ], "answer_type_v3": [ "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0016", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "5", "keywords": [ "parametric curve", "multivariable", "calculus" ], "problem_v1": "A plane directly above Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at 725 km/hour along a line at 8250 meters above the line joining Denver and Bismark. Bismark is about 850 km in the direction $60^{\\circ}$ north of east from Denver. Find parametric equations describing the plane's motion. Assume the origin is at sea level beneath Denver, that the $x$-axis points east and the $y$-axis points north, and that the earth is flat. Measure distances in kilometers and time in hours. $\\vec r(t)=$ [ANS]", "answer_v1": [ "(362.5t, 627.764t, -0.938235t+9.9)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "A plane directly above Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at 600 km/hour along a line at 9000 meters above the line joining Denver and Bismark. Bismark is about 850 km in the direction $60^{\\circ}$ north of east from Denver. Find parametric equations describing the plane's motion. Assume the origin is at sea level beneath Denver, that the $x$-axis points east and the $y$-axis points north, and that the earth is flat. Measure distances in kilometers and time in hours. $\\vec r(t)=$ [ANS]", "answer_v2": [ "(300t, 519.529t, -0.77647t+10.65)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "A plane directly above Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at 650 km/hour along a line at 8250 meters above the line joining Denver and Bismark. Bismark is about 850 km in the direction $60^{\\circ}$ north of east from Denver. Find parametric equations describing the plane's motion. Assume the origin is at sea level beneath Denver, that the $x$-axis points east and the $y$-axis points north, and that the earth is flat. Measure distances in kilometers and time in hours. $\\vec r(t)=$ [ANS]", "answer_v3": [ "(325t, 562.823t, -0.841176t+9.9)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0017", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "1", "keywords": [ "calculus", "vector", "parametric", "vector" ], "problem_v1": "Given a the vector equation $\\mathbf{r} (t)=(3+2 t) \\mathbf{i}+(1-2 t) \\mathbf{j}+(1-2 t) \\mathbf{k}$, rewrite this in terms of the parametric equations for the line.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS]\n$z(t)=$ [ANS]", "answer_v1": [ "3 + 2 * t", "1 + -2 * t", "1 + -2 * t" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Given a the vector equation $\\mathbf{r} (t)=(-5-2 t) \\mathbf{i}+(5+5 t) \\mathbf{j}+(-4-2 t) \\mathbf{k}$, rewrite this in terms of the parametric equations for the line.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS]\n$z(t)=$ [ANS]", "answer_v2": [ "-5 + -2 * t", "5 + 5 * t", "-4 + -2 * t" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Given a the vector equation $\\mathbf{r} (t)=(-2+1 t) \\mathbf{i}+(1-3 t) \\mathbf{j}+(-2-2 t) \\mathbf{k}$, rewrite this in terms of the parametric equations for the line.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS]\n$z(t)=$ [ANS]", "answer_v3": [ "-2 + 1 * t", "1 + -3 * t", "-2 + -2 * t" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0018", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "3", "keywords": [ "parametric curve" ], "problem_v1": "(a) Find a vector-parametric equation $\\vec{r}_1(t)=\\langle x(t), y(t), z(t) \\rangle$ for the shadow of the circular cylinder $x^2+z^2=4$ in the $xz$-plane.\nShadow: $\\vec{r}_1(t)=$ [ANS] for $0 \\leq t \\leq 2\\pi$.\n(b) Find a vector-parametric equation for intersection of the circular cylinder $x^2+z^2=4$ and the plane $8x+6 y+7 z=1$.\nIntersection: $\\vec{r}_2(t)=$ [ANS] for $0 \\leq t \\leq 2\\pi$.", "answer_v1": [ "(2*cos(t),0,2*sin(t))", "(2*cos(t), [1-16*cos(t)-14*sin(t)]/6, 2*sin(t))" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "(a) Find a vector-parametric equation $\\vec{r}_1(t)=\\langle x(t), y(t), z(t) \\rangle$ for the shadow of the circular cylinder $x^2+z^2=4$ in the $xz$-plane.\nShadow: $\\vec{r}_1(t)=$ [ANS] for $0 \\leq t \\leq 2\\pi$.\n(b) Find a vector-parametric equation for intersection of the circular cylinder $x^2+z^2=4$ and the plane $2x+9 y+3 z=1$.\nIntersection: $\\vec{r}_2(t)=$ [ANS] for $0 \\leq t \\leq 2\\pi$.", "answer_v2": [ "(2*cos(t), 0, 2*sin(t))", "(2*cos(t), [1-4*cos(t)-6*sin(t)]/9, 2*sin(t))" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "(a) Find a vector-parametric equation $\\vec{r}_1(t)=\\langle x(t), y(t), z(t) \\rangle$ for the shadow of the circular cylinder $x^2+z^2=8$ in the $xz$-plane.\nShadow: $\\vec{r}_1(t)=$ [ANS] for $0 \\leq t \\leq 2\\pi$.\n(b) Find a vector-parametric equation for intersection of the circular cylinder $x^2+z^2=8$ and the plane $4x+6 y+3 z=1$.\nIntersection: $\\vec{r}_2(t)=$ [ANS] for $0 \\leq t \\leq 2\\pi$.", "answer_v3": [ "(sqrt(8)*cos(t),0,sqrt(8)*sin(t))", "(sqrt(8)*cos(t),[1-4*sqrt(8)*cos(t)-3*sqrt(8)*sin(t)]/6,sqrt(8)*sin(t))" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0019", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "3", "keywords": [ "parametric curve" ], "problem_v1": "Find a vector-parametric equation for the intersection of the paraboloids $y=8x^2+6 z^2$ and $y=14-6x^2-8 z^2$.\nIntersection: $\\vec{r}(t)=$ [ANS] for $0 \\leq t \\leq 2\\pi$.", "answer_v1": [ "(cos(t),8*[cos(t)]^2+6*[sin(t)]^2,sin(t))" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find a vector-parametric equation for the intersection of the paraboloids $y=2x^2+9 z^2$ and $y=11-9x^2-2 z^2$.\nIntersection: $\\vec{r}(t)=$ [ANS] for $0 \\leq t \\leq 2\\pi$.", "answer_v2": [ "(cos(t),2*[cos(t)]^2+9*[sin(t)]^2,sin(t))" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find a vector-parametric equation for the intersection of the paraboloids $y=4x^2+6 z^2$ and $y=10-6x^2-4 z^2$.\nIntersection: $\\vec{r}(t)=$ [ANS] for $0 \\leq t \\leq 2\\pi$.", "answer_v3": [ "(cos(t),4*[cos(t)]^2+6*[sin(t)]^2,sin(t))" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0020", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "4", "keywords": [ "curve' 'arc length' 'intersection" ], "problem_v1": "Consider the paraboloid $z=x^2+y^2$. The plane $8x-5 y+z-5=0$ cuts the paraboloid, its intersection being a curve. Find \"the natural\" parametrization of this curve. Hint: The curve which is cut lies above a circle in the xy-plane which you should parametrize as a function of the variable t so that the circle is traversed counterclockwise exactly once as t goes from 0 to 2*pi, and the paramterization starts at the point on the circle with largest x coordinate. Using that as your starting point, give the parametrization of the curve on the surface.\n$\\mathbf{c}(t)=(x(t), y(t), z(t))$, where $x(t)=$ [ANS]\n$y(t)=$ [ANS]\n$z(t)=$ [ANS]", "answer_v1": [ "- 8/2 + 5.22015325445528*cos(t)", "- -5/2 + 5.22015325445528*sin(t)", "(- 8/2 + 5.22015325445528*cos(t))^2 + (- -5/2 + 5.22015325445528*sin(t))^2" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the paraboloid $z=x^2+y^2$. The plane $2x-2 y+z-9=0$ cuts the paraboloid, its intersection being a curve. Find \"the natural\" parametrization of this curve. Hint: The curve which is cut lies above a circle in the xy-plane which you should parametrize as a function of the variable t so that the circle is traversed counterclockwise exactly once as t goes from 0 to 2*pi, and the paramterization starts at the point on the circle with largest x coordinate. Using that as your starting point, give the parametrization of the curve on the surface.\n$\\mathbf{c}(t)=(x(t), y(t), z(t))$, where $x(t)=$ [ANS]\n$y(t)=$ [ANS]\n$z(t)=$ [ANS]", "answer_v2": [ "- 2/2 + 3.3166247903554*cos(t)", "- -2/2 + 3.3166247903554*sin(t)", "(- 2/2 + 3.3166247903554*cos(t))^2 + (- -2/2 + 3.3166247903554*sin(t))^2" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the paraboloid $z=x^2+y^2$. The plane $4x-5 y+z-8=0$ cuts the paraboloid, its intersection being a curve. Find \"the natural\" parametrization of this curve. Hint: The curve which is cut lies above a circle in the xy-plane which you should parametrize as a function of the variable t so that the circle is traversed counterclockwise exactly once as t goes from 0 to 2*pi, and the paramterization starts at the point on the circle with largest x coordinate. Using that as your starting point, give the parametrization of the curve on the surface.\n$\\mathbf{c}(t)=(x(t), y(t), z(t))$, where $x(t)=$ [ANS]\n$y(t)=$ [ANS]\n$z(t)=$ [ANS]", "answer_v3": [ "- 4/2 + 4.27200187265877*cos(t)", "- -5/2 + 4.27200187265877*sin(t)", "(- 4/2 + 4.27200187265877*cos(t))^2 + (- -5/2 + 4.27200187265877*sin(t))^2" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0021", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "3", "keywords": [ "Vector", "Parametric", "Geometry" ], "problem_v1": "Find parametric equations for the quarter-ellipse from $(5,0,8)$ to $(0,-3,8)$ centered at $(0,0,8)$ in the plane $z=8$. Use the interval $0 \\leq t \\leq \\pi/2$.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS] $z(t)=$ [ANS]", "answer_v1": [ "5*cos(t)", "-3*sin(t)", "8" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find parametric equations for the quarter-ellipse from $(3,0,9)$ to $(0,-2,9)$ centered at $(0,0,9)$ in the plane $z=9$. Use the interval $0 \\leq t \\leq \\pi/2$.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS] $z(t)=$ [ANS]", "answer_v2": [ "3*cos(t)", "-2*sin(t)", "9" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find parametric equations for the quarter-ellipse from $(4,0,6)$ to $(0,-3,6)$ centered at $(0,0,6)$ in the plane $z=6$. Use the interval $0 \\leq t \\leq \\pi/2$.\n$x(t)=$ [ANS]\n$y(t)=$ [ANS] $z(t)=$ [ANS]", "answer_v3": [ "4*cos(t)", "-3*sin(t)", "6" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0022", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "4", "keywords": [ "Vector", "Parametric", "Geometry" ], "problem_v1": "Find a vector parametrization of the ellipse centered at the origin in the xy-plane that has major diameter $14$ along the x-axis, minor diameter $10$ along the y-axis, and is oriented counter-clockwise. Your parametrization should make the point $(7,0)$ correspond to $t=0$. Use $t$ as the parameter in your answer.\n$\\vec{r}(t)=$ [ANS]", "answer_v1": [ "(7*cos(t),5*sin(t))" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find a vector parametrization of the ellipse centered at the origin in the xy-plane that has major diameter $8$ along the x-axis, minor diameter $4$ along the y-axis, and is oriented counter-clockwise. Your parametrization should make the point $(4,0)$ correspond to $t=0$. Use $t$ as the parameter in your answer.\n$\\vec{r}(t)=$ [ANS]", "answer_v2": [ "(4*cos(t),2*sin(t))" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find a vector parametrization of the ellipse centered at the origin in the xy-plane that has major diameter $10$ along the x-axis, minor diameter $6$ along the y-axis, and is oriented counter-clockwise. Your parametrization should make the point $(5,0)$ correspond to $t=0$. Use $t$ as the parameter in your answer.\n$\\vec{r}(t)=$ [ANS]", "answer_v3": [ "(5*cos(t),3*sin(t))" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0023", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "2", "keywords": [ "Vector", "Parametric", "Geometry" ], "problem_v1": "Find a vector parametrization of the curve $x=-2 z^2$ in the xz-plane. Use $t$ as the parameter in your answer.\n$\\vec{r}(t)=$ [ANS]", "answer_v1": [ "(-2*t^2,0,t)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find a vector parametrization of the curve $x=-5 z^2$ in the xz-plane. Use $t$ as the parameter in your answer.\n$\\vec{r}(t)=$ [ANS]", "answer_v2": [ "(-5*t^2,0,t)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find a vector parametrization of the curve $x=-4 z^2$ in the xz-plane. Use $t$ as the parameter in your answer.\n$\\vec{r}(t)=$ [ANS]", "answer_v3": [ "(-4*t^2,0,t)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0024", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "4", "keywords": [ "Vector", "Parametric", "Geometry" ], "problem_v1": "Consider a circle of radius $7$ centered at the origin in the $xy$-plane. Find a parametrization for the arc of this circle that lies in the fourth quadrant and has clockwise orientation. The point $(7,0)$ should correspond to $t=0$ and the point $(0,-7)$ should correspond to the right endpoint of the interval of $t$ values.\n$\\vec{r}(t)=$ [ANS] for $0 \\leq t \\leq$ [ANS]\nNote: Use $t$ as the parameter for all of your answers and write $\\vec{r}(t)$ in the form $< x(t), y(t) >$ without using $\\vec{i}$ and $\\vec{j}$. In order to have your answer checked for correctness, you must have answers in both answer blanks.", "answer_v1": [ "(7*cos(t),(-7)*sin(t))", "1.5708" ], "answer_type_v1": [ "OL", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider a circle of radius $3$ centered at the origin in the $xy$-plane. Find a parametrization for the arc of this circle that lies in the fourth quadrant and has clockwise orientation. The point $(3,0)$ should correspond to $t=0$ and the point $(0,-3)$ should correspond to the right endpoint of the interval of $t$ values.\n$\\vec{r}(t)=$ [ANS] for $0 \\leq t \\leq$ [ANS]\nNote: Use $t$ as the parameter for all of your answers and write $\\vec{r}(t)$ in the form $< x(t), y(t) >$ without using $\\vec{i}$ and $\\vec{j}$. In order to have your answer checked for correctness, you must have answers in both answer blanks.", "answer_v2": [ "(3*cos(t),(-3)*sin(t))", "1.5708" ], "answer_type_v2": [ "OL", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider a circle of radius $4$ centered at the origin in the $xy$-plane. Find a parametrization for the arc of this circle that lies in the fourth quadrant and has clockwise orientation. The point $(4,0)$ should correspond to $t=0$ and the point $(0,-4)$ should correspond to the right endpoint of the interval of $t$ values.\n$\\vec{r}(t)=$ [ANS] for $0 \\leq t \\leq$ [ANS]\nNote: Use $t$ as the parameter for all of your answers and write $\\vec{r}(t)$ in the form $< x(t), y(t) >$ without using $\\vec{i}$ and $\\vec{j}$. In order to have your answer checked for correctness, you must have answers in both answer blanks.", "answer_v3": [ "(4*cos(t),(-4)*sin(t))", "1.5708" ], "answer_type_v3": [ "OL", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0025", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "3", "keywords": [ "Vector", "Parametric", "Geometry" ], "problem_v1": "A particle passes through the point $P=\\left(3,1,1\\right)$ at time $t=5$, moving with constant velocity $\\vec{v}=\\left<2,-2,-2\\right>$. Find a parametric equation for the position of the particle in terms of the parameter $t$.\n$\\vec{r}(t)=$ [ANS]", "answer_v1": [ "(2t-7, -2t+11, -2t+11)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "A particle passes through the point $P=\\left(-5,5,-4\\right)$ at time $t=3$, moving with constant velocity $\\vec{v}=\\left<-2,5,-2\\right>$. Find a parametric equation for the position of the particle in terms of the parameter $t$.\n$\\vec{r}(t)=$ [ANS]", "answer_v2": [ "(-2t+1, 5t-10, -2t+2)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "A particle passes through the point $P=\\left(-2,1,-2\\right)$ at time $t=7$, moving with constant velocity $\\vec{v}=\\left<1,-3,-2\\right>$. Find a parametric equation for the position of the particle in terms of the parameter $t$.\n$\\vec{r}(t)=$ [ANS]", "answer_v3": [ "(t-9, -3t+22, -2t+12)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0026", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Parameterized curves", "level": "3", "keywords": [ "vector-valued function", "points", "intersection" ], "problem_v1": "Consider an object following the path of the two-dimensional vector-valued function $\\vec p(t)=\\left({4t^{2}+3t-2},{3t^{2}-3t-3} \\right)$. When does it pass through the point $({-1},{3})$? If it passes through that point, give the $t$ value. If it does not pass through that point, enter NONE as the answer. Answer: [ANS]\nWhen does it pass through the point $({20},{0})$? If it passes through that point, give the $t$ value. If it does not pass through that point, enter NONE as the answer. Answer: [ANS]\nWhen does it pass through the point $({160},{87})$? If it passes through that point, give the $t$ value. If it does not pass through that point, enter NONE as the answer. Answer: [ANS]\nWhen is it at rest? If it is at rest at some point, give the $t$ value. If it is never at rest, enter NONE as the answer. Answer: [ANS]", "answer_v1": [ "-1", "NONE", "6", "NONE" ], "answer_type_v1": [ "OE", "OE", "OE", "OE" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider an object following the path of the two-dimensional vector-valued function $\\vec p(t)=\\left({2t^{2}-2t-4},{-\\left(2t^{2}+3t+3\\right)} \\right)$. When does it pass through the point $({-4},{-2})$? If it passes through that point, give the $t$ value. If it does not pass through that point, enter NONE as the answer. Answer: [ANS]\nWhen does it pass through the point $({-4},{-8})$? If it passes through that point, give the $t$ value. If it does not pass through that point, enter NONE as the answer. Answer: [ANS]\nWhen does it pass through the point $({36},{-68})$? If it passes through that point, give the $t$ value. If it does not pass through that point, enter NONE as the answer. Answer: [ANS]\nWhen is it at rest? If it is at rest at some point, give the $t$ value. If it is never at rest, enter NONE as the answer. Answer: [ANS]", "answer_v2": [ "NONE", "1", "5", "NONE" ], "answer_type_v2": [ "OE", "OE", "OE", "OE" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider an object following the path of the two-dimensional vector-valued function $\\vec p(t)=\\left({2t^{2}+2t-2},{4t^{2}-4t-2} \\right)$. When does it pass through the point $({10},{46})$? If it passes through that point, give the $t$ value. If it does not pass through that point, enter NONE as the answer. Answer: [ANS]\nWhen does it pass through the point $({22},{22})$? If it passes through that point, give the $t$ value. If it does not pass through that point, enter NONE as the answer. Answer: [ANS]\nWhen does it pass through the point $({142},{223})$? If it passes through that point, give the $t$ value. If it does not pass through that point, enter NONE as the answer. Answer: [ANS]\nWhen is it at rest? If it is at rest at some point, give the $t$ value. If it is never at rest, enter NONE as the answer. Answer: [ANS]", "answer_v3": [ "-3", "3", "NONE", "NONE" ], "answer_type_v3": [ "OE", "OE", "OE", "OE" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0027", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Limits and continuity", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Evaluate the limit: $\\lim_{t \\to 5} \\left=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v1": [ "25", "10", "0.2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Evaluate the limit: $\\lim_{t \\to-8} \\left=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v2": [ "64", "-64", "-0.125" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Evaluate the limit: $\\lim_{t \\to-4} \\left=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v3": [ "16", "-8", "-0.25" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0028", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Limits and continuity", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Evaluate the limit: ${ \\lim_{h \\to 0} \\frac{{\\bf r}(t+h)-{\\bf r}(t)}{h} }$ for ${\\bf r}(t)=\\left$ ${\\bf r}^{\\prime}(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v1": [ "-3*t^{-4}", "cos(t)", "0" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Evaluate the limit: ${ \\lim_{h \\to 0} \\frac{{\\bf r}(t+h)-{\\bf r}(t)}{h} }$ for ${\\bf r}(t)=\\left$ ${\\bf r}^{\\prime}(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v2": [ "-9*t^(-10)", "cos(t)", "0" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Evaluate the limit: ${ \\lim_{h \\to 0} \\frac{{\\bf r}(t+h)-{\\bf r}(t)}{h} }$ for ${\\bf r}(t)=\\left$ ${\\bf r}^{\\prime}(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v3": [ "-7*t^{-8}", "cos(t)", "0" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0029", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Let ${\\bf r}(t)=\\left< t^2,1-t,4 t \\right>$. Calculate the derivative of ${\\bf r}(t)\\cdot {\\bf a}(t)$ at $t=6$, assuming that ${\\bf a}(6)=\\left< 2, 4,-4 \\right>$ and ${\\bf a}'(6)=\\left<-3, 1, 1 \\right>$ ${ \\frac{\\,d}{\\,dt} {\\bf r}(t)\\cdot {\\bf a}(t)|_{t=6}=}$ [ANS]", "answer_v1": [ "-85" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let ${\\bf r}(t)=\\left< t^2,1-t,4 t \\right>$. Calculate the derivative of ${\\bf r}(t)\\cdot {\\bf a}(t)$ at $t=9$, assuming that ${\\bf a}(9)=\\left<-7,-3, 8 \\right>$ and ${\\bf a}'(9)=\\left<-3,-6,-3 \\right>$ ${ \\frac{\\,d}{\\,dt} {\\bf r}(t)\\cdot {\\bf a}(t)|_{t=9}=}$ [ANS]", "answer_v2": [ "-394" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let ${\\bf r}(t)=\\left< t^2,1-t,4 t \\right>$. Calculate the derivative of ${\\bf r}(t)\\cdot {\\bf a}(t)$ at $t=6$, assuming that ${\\bf a}(6)=\\left<-4, 1,-6 \\right>$ and ${\\bf a}'(6)=\\left<-3, 6, 8 \\right>$ ${ \\frac{\\,d}{\\,dt} {\\bf r}(t)\\cdot {\\bf a}(t)|_{t=6}=}$ [ANS]", "answer_v3": [ "-19" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0030", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Compute the derivative of ${\\bf r}(t)=\\left$ ${ \\frac{\\,d{\\bf r}}{\\,dt} =\\,} \\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v1": [ "1", "16*t^15", "13*t^12" ], "answer_type_v1": [ "NV", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Compute the derivative of ${\\bf r}(t)=\\left$ ${ \\frac{\\,d{\\bf r}}{\\,dt} =\\,} \\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v2": [ "1", "4*t^3", "19*t^18" ], "answer_type_v2": [ "NV", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Compute the derivative of ${\\bf r}(t)=\\left$ ${ \\frac{\\,d{\\bf r}}{\\,dt} =\\,} \\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v3": [ "1", "8*t^7", "13*t^12" ], "answer_type_v3": [ "NV", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0031", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Use the appropriate Product Rule to evaluate the derivative, where ${\\bf r}_{1}(t)=\\left<5 t,2,-t^{7}\\right>, {\\bf r}_{2}(t)=\\left<5,e^{t},-8\\right>$ $ \\frac{\\,d}{\\,dt} \\left({\\bf r}_1(t)\\cdot {\\bf r}_2(t)\\right)=$ [ANS]", "answer_v1": [ "2*e^t+56*t^6+25" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the appropriate Product Rule to evaluate the derivative, where ${\\bf r}_{1}(t)=\\left<-9 t,9,-t^{3}\\right>, {\\bf r}_{2}(t)=\\left<-3,e^{t},-2\\right>$ $ \\frac{\\,d}{\\,dt} \\left({\\bf r}_1(t)\\cdot {\\bf r}_2(t)\\right)=$ [ANS]", "answer_v2": [ "9*e^t+6*t^2+27" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the appropriate Product Rule to evaluate the derivative, where ${\\bf r}_{1}(t)=\\left<-4 t,2,-t^{4}\\right>, {\\bf r}_{2}(t)=\\left<1,e^{t},-9\\right>$ $ \\frac{\\,d}{\\,dt} \\left({\\bf r}_1(t)\\cdot {\\bf r}_2(t)\\right)=$ [ANS]", "answer_v3": [ "2*e^t+36*t^3-4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0032", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Evaluate $ \\frac{\\,d}{\\,dt} {\\bf r}(g(t))$ using the Chain Rule: ${\\bf r}(t)=\\left, \\ \\ g(t)=6 t+4$ $ \\frac{\\,d}{\\,dt} {\\bf r}(g(t))=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v1": [ "6*e^(6*t+4)", "48*e^[8*(6*t+4)]", "0" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Evaluate $ \\frac{\\,d}{\\,dt} {\\bf r}(g(t))$ using the Chain Rule: ${\\bf r}(t)=\\left, \\ \\ g(t)=3 t-3$ $ \\frac{\\,d}{\\,dt} {\\bf r}(g(t))=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v2": [ "3*e^(3*t-3)", "6*e^[2*(3*t-3)]", "0" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Evaluate $ \\frac{\\,d}{\\,dt} {\\bf r}(g(t))$ using the Chain Rule: ${\\bf r}(t)=\\left, \\ \\ g(t)=4 t+1$ $ \\frac{\\,d}{\\,dt} {\\bf r}(g(t))=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v3": [ "4*e^(4*t+1)", "16*e^[4*(4*t+1)]", "0" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0033", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "2", "keywords": [ "Derivative", "Parametric", "Path" ], "problem_v1": "The derivative of $f(t)= \\left<3+\\tan\\!\\left(t\\right),t\\sin\\!\\left(t\\right), \\frac{1}{4-t} \\right>$ is $Df(t)$=[ANS].", "answer_v1": [ "([sec(t)]^2,sin(t)+t*cos(t),1/[(4-t)^2])" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "The derivative of $f(t)= \\left<\\tan\\!\\left(t\\right)-5,5t\\sin\\!\\left(t\\right), \\frac{1}{1-t} \\right>$ is $Df(t)$=[ANS].", "answer_v2": [ "([sec(t)]^2,5*sin(t)+5*t*cos(t),1/[(1-t)^2])" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "The derivative of $f(t)= \\left<\\tan\\!\\left(t\\right)-2,t\\sin\\!\\left(t\\right), \\frac{1}{2-t} \\right>$ is $Df(t)$=[ANS].", "answer_v3": [ "([sec(t)]^2,sin(t)+t*cos(t),1/[(2-t)^2])" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0034", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "4", "keywords": [ "Derivative", "Parametric", "Path" ], "problem_v1": "The parametric form of the tangent line to the image of $f(t)=\\left<5t^{2}, \\frac{4}{t} ,t+1\\right>$ at $t=2$ is\n$L(t)$=[ANS].", "answer_v1": [ "(20t+20, -t+2, t+3)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "The parametric form of the tangent line to the image of $f(t)=\\left<2t^{2}, \\frac{5}{t} ,t-4\\right>$ at $t=-2$ is\n$L(t)$=[ANS].", "answer_v2": [ "(-8t+8, -1.25t-2.5, t-6)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "The parametric form of the tangent line to the image of $f(t)=\\left<3t^{2}, \\frac{4}{t} ,t-2\\right>$ at $t=1$ is\n$L(t)$=[ANS].", "answer_v3": [ "(6t+3,-4t+4,t-1)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0035", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "2", "keywords": [ "tangent", "differentiation", "vector" ], "problem_v1": "Let f(x, y)=x^2+\\sin(xy), \\quad x(t)=8 t+6, \\quad \\text{and}\\quad y(t)=6 t^2+7 t. The parametric curve $\\mathbf{c}(t)=(x(t), y(t), f(x(t), y(t)))$ lies on the surface $z=f(x,y)$. Find the tangent vector to this curve at $t=1$.\nTangent vector $=($ [ANS], [ANS], [ANS] $)$", "answer_v1": [ "8", "19", "585.687319041071" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let f(x, y)=x^2+\\sin(xy), \\quad x(t)=2 t+9, \\quad \\text{and}\\quad y(t)=3 t^2+4 t. The parametric curve $\\mathbf{c}(t)=(x(t), y(t), f(x(t), y(t)))$ lies on the surface $z=f(x,y)$. Find the tangent vector to this curve at $t=1$.\nTangent vector $=($ [ANS], [ANS], [ANS] $)$", "answer_v2": [ "2", "10", "40.1590960653292" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let f(x, y)=x^2+\\sin(xy), \\quad x(t)=4 t+6, \\quad \\text{and}\\quad y(t)=4 t^2+6 t. The parametric curve $\\mathbf{c}(t)=(x(t), y(t), f(x(t), y(t)))$ lies on the surface $z=f(x,y)$. Find the tangent vector to this curve at $t=1$.\nTangent vector $=($ [ANS], [ANS], [ANS] $)$", "answer_v3": [ "4", "14", "235.217397011783" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0036", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "5", "keywords": [ "Vector", "Parametric", "Geometry" ], "problem_v1": "Suppose $\\vec{r}(t)=\\cos\\!\\left(\\pi t\\right)\\boldsymbol{i}+\\sin\\!\\left(\\pi t\\right)\\boldsymbol{j}+5t\\boldsymbol{k}$ represents the position of a particle on a helix, where $z$ is the height of the particle.\n(a) What is $t$ when the particle has height $20$? $t=$ [ANS]\n(b) What is the velocity of the particle when its height is $20$? $\\vec{v}=$ [ANS]\n(c) When the particle has height $20$, it leaves the helix and moves along the tangent line at the constant velocity found in part (b). Find a vector parametric equation for the position of the particle (in terms of the original parameter $t$) as it moves along this tangent line. $L(t)=$ [ANS]", "answer_v1": [ "4", "(0,3,14159,5)", "(1,(t-4)*3.14159, 25)" ], "answer_type_v1": [ "NV", "OL", "OL" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $\\vec{r}(t)=\\cos\\!\\left(\\pi t\\right)\\boldsymbol{i}+\\sin\\!\\left(\\pi t\\right)\\boldsymbol{j}+2t\\boldsymbol{k}$ represents the position of a particle on a helix, where $z$ is the height of the particle.\n(a) What is $t$ when the particle has height $10$? $t=$ [ANS]\n(b) What is the velocity of the particle when its height is $10$? $\\vec{v}=$ [ANS]\n(c) When the particle has height $10$, it leaves the helix and moves along the tangent line at the constant velocity found in part (b). Find a vector parametric equation for the position of the particle (in terms of the original parameter $t$) as it moves along this tangent line. $L(t)=$ [ANS]", "answer_v2": [ "5", "(0,-3.14159,2)", "(-1,-(t-5)*3.14159,12)" ], "answer_type_v2": [ "NV", "OL", "OL" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $\\vec{r}(t)=\\cos\\!\\left(\\pi t\\right)\\boldsymbol{i}+\\sin\\!\\left(\\pi t\\right)\\boldsymbol{j}+3t\\boldsymbol{k}$ represents the position of a particle on a helix, where $z$ is the height of the particle.\n(a) What is $t$ when the particle has height $12$? $t=$ [ANS]\n(b) What is the velocity of the particle when its height is $12$? $\\vec{v}=$ [ANS]\n(c) When the particle has height $12$, it leaves the helix and moves along the tangent line at the constant velocity found in part (b). Find a vector parametric equation for the position of the particle (in terms of the original parameter $t$) as it moves along this tangent line. $L(t)=$ [ANS]", "answer_v3": [ "4", "(0,3.14159,3)", "(1,(t-4)*3.14159,15)" ], "answer_type_v3": [ "NV", "OL", "OL" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0037", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "2", "keywords": [ "calculus", "Jacobian", "derivative' 'matrix' 'jacobians" ], "problem_v1": "Consider the function $f: \\mathbb{R}^2 \\to \\mathbb{R}^2$ given by $f(x,y)=(e^{7x},\\ \\sin(6xy))$. The derivative (Jacobian matrix) of $f$ is $f'(x,y)=Df(x,y)=\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}$ where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS].", "answer_v1": [ "7 * exp(7 * x)", "0", "6 * y * cos(6 * x * y)", "6 * x * cos(6 * x * y)" ], "answer_type_v1": [ "EX", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the function $f: \\mathbb{R}^2 \\to \\mathbb{R}^2$ given by $f(x,y)=(e^{x},\\ \\sin(9xy))$. The derivative (Jacobian matrix) of $f$ is $f'(x,y)=Df(x,y)=\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}$ where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS].", "answer_v2": [ "1 * exp(1 * x)", "0", "9 * y * cos(9 * x * y)", "9 * x * cos(9 * x * y)" ], "answer_type_v2": [ "EX", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the function $f: \\mathbb{R}^2 \\to \\mathbb{R}^2$ given by $f(x,y)=(e^{3x},\\ \\sin(6xy))$. The derivative (Jacobian matrix) of $f$ is $f'(x,y)=Df(x,y)=\\begin{bmatrix}a&b\\\\c&d\\end{bmatrix}$ where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS].", "answer_v3": [ "3 * exp(3 * x)", "0", "6 * y * cos(6 * x * y)", "6 * x * cos(6 * x * y)" ], "answer_type_v3": [ "EX", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0038", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "2", "keywords": [ "calculus", "Jacobian", "derivative", "linear approximation" ], "problem_v1": "Consider a function $f: \\mathbb{R}^2 \\to \\mathbb{R}^2$ for which $f(1, 1)=(7, 6)$ and $f'(1, 1)=\\begin{bmatrix}1&2\\\\-2&-2\\end{bmatrix}$. The local linearization of $f$ at $(1, 1)$ is $L(x,y)=($ [ANS], [ANS] $)$.", "answer_v1": [ "7 + 1 * (x - 1) + 2 * (y - 1)", "6 + -2 * (x - 1) + -2 * (y - 1)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Consider a function $f: \\mathbb{R}^2 \\to \\mathbb{R}^2$ for which $f(-3,-2)=(1, 9)$ and $f'(-3,-2)=\\begin{bmatrix}-4&-2\\\\5&-2\\end{bmatrix}$. The local linearization of $f$ at $(-3,-2)$ is $L(x,y)=($ [ANS], [ANS] $)$.", "answer_v2": [ "1 + -4 * (x - -3) + -2 * (y - -2)", "9 + 5 * (x - -3) + -2 * (y - -2)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Consider a function $f: \\mathbb{R}^2 \\to \\mathbb{R}^2$ for which $f(3, 5)=(3, 6)$ and $f'(3, 5)=\\begin{bmatrix}-2&1\\\\-3&-2\\end{bmatrix}$. The local linearization of $f$ at $(3, 5)$ is $L(x,y)=($ [ANS], [ANS] $)$.", "answer_v3": [ "3 + -2 * (x - 3) + 1 * (y - 5)", "6 + -3 * (x - 3) + -2 * (y - 5)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0039", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "3", "keywords": [ "vector-valued function", "ballistics", "gravity", "derivative" ], "problem_v1": "Consider a cannonball launched along the path\n$\\vec p(t)=({700t+60},{600t-150},{38t-16t^{2}}),$ with length units in feet and time units in seconds. When does it return to the height from which it was launched? (The coordinate system is right-handed, with $z$ the vertical axis.) Answer: $t=$ [ANS]\nWhat is its velocity then? Answer: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Momentum is mass times speed, usually measured in $ \\frac{\\text{kg m}}{\\text{s} }$, but here we can measure it in $ \\frac{\\text{kg ft}}{\\text{s} }$. If the cannonball has mass 25 kg, what is its momentum upon impact with the ground? Answer: [ANS] $ \\frac{\\text{kg ft}}{\\text{s} }$", "answer_v1": [ "2.375", "700", "600", "-38", "23068.4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Consider a cannonball launched along the path\n$\\vec p(t)=({100t+90},{200t-350},{45t-16t^{2}}),$ with length units in feet and time units in seconds. When does it return to the height from which it was launched? (The coordinate system is right-handed, with $z$ the vertical axis.) Answer: $t=$ [ANS]\nWhat is its velocity then? Answer: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Momentum is mass times speed, usually measured in $ \\frac{\\text{kg m}}{\\text{s} }$, but here we can measure it in $ \\frac{\\text{kg ft}}{\\text{s} }$. If the cannonball has mass 25 kg, what is its momentum upon impact with the ground? Answer: [ANS] $ \\frac{\\text{kg ft}}{\\text{s} }$", "answer_v2": [ "2.8125", "100", "200", "-45", "5702.25" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Consider a cannonball launched along the path\n$\\vec p(t)=({300t+60},{300t-250},{37t-16t^{2}}),$ with length units in feet and time units in seconds. When does it return to the height from which it was launched? (The coordinate system is right-handed, with $z$ the vertical axis.) Answer: $t=$ [ANS]\nWhat is its velocity then? Answer: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Momentum is mass times speed, usually measured in $ \\frac{\\text{kg m}}{\\text{s} }$, but here we can measure it in $ \\frac{\\text{kg ft}}{\\text{s} }$. If the cannonball has mass 25 kg, what is its momentum upon impact with the ground? Answer: [ANS] $ \\frac{\\text{kg ft}}{\\text{s} }$", "answer_v3": [ "2.3125", "300", "300", "-37", "10646.9" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0040", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "3", "keywords": [ "vector-valued function", "function application", "derivatives" ], "problem_v1": "Compute the starting and ending positions (at times $t=0$ and $t=1$, respectively) for the path of motion described by the following vector-valued function. Function: $\\vec f(t)=\\left({5t},{4t^{2}},{4t^{2}-4}\\right)$ Starting point: ([ANS], [ANS], [ANS]) Ending point: ([ANS], [ANS], [ANS]) Now compute the derivative of that same vector-valued function. Answer: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Now compute the starting and ending velocities for that same vector-valued function. Starting velocity: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Ending velocity: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v1": [ "0", "0", "-4", "5", "4", "0", "5", "4*2*t", "4*2*t", "5", "0", "0", "5", "8", "8" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Compute the starting and ending positions (at times $t=0$ and $t=1$, respectively) for the path of motion described by the following vector-valued function. Function: $\\vec f(t)=\\left({2t},{5t^{4}},{2t^{2}-3}\\right)$ Starting point: ([ANS], [ANS], [ANS]) Ending point: ([ANS], [ANS], [ANS]) Now compute the derivative of that same vector-valued function. Answer: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Now compute the starting and ending velocities for that same vector-valued function. Starting velocity: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Ending velocity: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v2": [ "0", "0", "-3", "2", "5", "-1", "2", "5*4*t^3", "2*2*t", "2", "0", "0", "2", "20", "4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Compute the starting and ending positions (at times $t=0$ and $t=1$, respectively) for the path of motion described by the following vector-valued function. Function: $\\vec f(t)=\\left({3t},{4t^{2}},{3t^{3}-4}\\right)$ Starting point: ([ANS], [ANS], [ANS]) Ending point: ([ANS], [ANS], [ANS]) Now compute the derivative of that same vector-valued function. Answer: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Now compute the starting and ending velocities for that same vector-valued function. Starting velocity: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$ Ending velocity: $\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v3": [ "0", "0", "-4", "3", "4", "-1", "3", "4*2*t", "3*3*t^2", "3", "0", "0", "3", "8", "9" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0041", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Derivatives", "level": "3", "keywords": [ "vector-valued function", "acceleration", "velocity", "derivatives" ], "problem_v1": "An animation has two scenes, each of them being one second long. In the first scene, an object moves along the path\n$\\vec p_1(t)=\\left(8t^2+7t,-4t^2+7,-5t^2-7t-6 \\right)$ for $0 \\leq t \\leq 1$. At the end of the scene, the object\u2019s acceleration immediately becomes zero, and remains zero throughout the second scene. The vector-valued function $\\vec p_2(t)$ for the object\u2019s path in the second scene also uses $0 \\leq t \\leq 1$, so that $t=0$ in the second scene is the same moment as $t=1$ in the first scene. What is $\\vec p_2(t)$? Answer: $\\vec p_2(t)=($ [ANS], [ANS], [ANS] $)$", "answer_v1": [ "8+7+(8*2+7)*t", "-4+7+(-4)*2*t", "-5+(-7)+(-6)+[-5*2+(-7)]*t" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "An animation has two scenes, each of them being one second long. In the first scene, an object moves along the path\n$\\vec p_1(t)=\\left(2t^2-3t,-10t^2-3,-7t^2-7t-9 \\right)$ for $0 \\leq t \\leq 1$. At the end of the scene, the object\u2019s acceleration immediately becomes zero, and remains zero throughout the second scene. The vector-valued function $\\vec p_2(t)$ for the object\u2019s path in the second scene also uses $0 \\leq t \\leq 1$, so that $t=0$ in the second scene is the same moment as $t=1$ in the first scene. What is $\\vec p_2(t)$? Answer: $\\vec p_2(t)=($ [ANS], [ANS], [ANS] $)$", "answer_v2": [ "2+(-3)+[2*2+(-3)]*t", "-10+(-3)+(-10)*2*t", "-7+(-7)+(-9)+[-7*2+(-7)]*t" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "An animation has two scenes, each of them being one second long. In the first scene, an object moves along the path\n$\\vec p_1(t)=\\left(4t^2+4t,-3t^2+9,-9t^2-4t+2 \\right)$ for $0 \\leq t \\leq 1$. At the end of the scene, the object\u2019s acceleration immediately becomes zero, and remains zero throughout the second scene. The vector-valued function $\\vec p_2(t)$ for the object\u2019s path in the second scene also uses $0 \\leq t \\leq 1$, so that $t=0$ in the second scene is the same moment as $t=1$ in the first scene. What is $\\vec p_2(t)$? Answer: $\\vec p_2(t)=($ [ANS], [ANS], [ANS] $)$", "answer_v3": [ "4+4+(4*2+4)*t", "-3+9+(-3)*2*t", "-9+(-4)+2+[-9*2+(-4)]*t" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0042", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Integrals", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Evaluate the integral:\n\\int_{0}^{t} (9 s {\\bf i}+18 s^2 {\\bf j}+13 {\\bf k}) \\,ds Answer: [ANS] ${\\bf i} \\,+$ [ANS] ${\\bf j} \\,+$ [ANS] ${\\bf k}$", "answer_v1": [ "4.5*t^2", "6*t^3", "13*t" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Evaluate the integral:\n\\int_{0}^{t} (3 s {\\bf i}+30 s^2 {\\bf j}+4 {\\bf k}) \\,ds Answer: [ANS] ${\\bf i} \\,+$ [ANS] ${\\bf j} \\,+$ [ANS] ${\\bf k}$", "answer_v2": [ "1.5*t^2", "10*t^3", "4*t" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Evaluate the integral:\n\\int_{0}^{t} (5 s {\\bf i}+21 s^2 {\\bf j}+7 {\\bf k}) \\,ds Answer: [ANS] ${\\bf i} \\,+$ [ANS] ${\\bf j} \\,+$ [ANS] ${\\bf k}$", "answer_v3": [ "2.5*t^2", "7*t^3", "7*t" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0043", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Integrals", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find the solution ${\\bf r\\it(t)}$ of the differential equation with the given initial condition:\n{\\bf r\\it'(t)}=\\left<\\sin 8 t,\\sin 6 t,7 t\\right>, {\\bf r\\it(0)}=\\left<7,4,4\\right> ${\\bf r\\it(t)}=\\langle$ [ANS] $,$ [ANS] $,$ [ANS] $\\rangle$", "answer_v1": [ "7.125-[cos(8*t)]/8", "4.16667-[cos(6*t)]/6", "4+3.5*t^2" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the solution ${\\bf r\\it(t)}$ of the differential equation with the given initial condition:\n{\\bf r\\it'(t)}=\\left<\\sin 2 t,\\sin 9 t,3 t\\right>, {\\bf r\\it(0)}=\\left<4,9,4\\right> ${\\bf r\\it(t)}=\\langle$ [ANS] $,$ [ANS] $,$ [ANS] $\\rangle$", "answer_v2": [ "4.5-[cos(2*t)]/2", "9.11111-[cos(9*t)]/9", "4+1.5*t^2" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the solution ${\\bf r\\it(t)}$ of the differential equation with the given initial condition:\n{\\bf r\\it'(t)}=\\left<\\sin 4 t,\\sin 6 t,5 t\\right>, {\\bf r\\it(0)}=\\left<6,3,4\\right> ${\\bf r\\it(t)}=\\langle$ [ANS] $,$ [ANS] $,$ [ANS] $\\rangle$", "answer_v3": [ "6.25-[cos(4*t)]/4", "3.16667-[cos(6*t)]/6", "4+2.5*t^2" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0044", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Integrals", "level": "2", "keywords": [ "calculus" ], "problem_v1": "A bee with velocity vector $\\bf r\\it'(t)$ starts out at $(1, 1, 2)$ at $t=0$ and flies around for $8$ seconds. Where is the bee located at time $8$ if $\\int^{8}_{0}{\\bf r\\it'(u)du}=0$? $P=($ [ANS], [ANS], [ANS] $)$", "answer_v1": [ "1", "1", "2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A bee with velocity vector $\\bf r\\it'(t)$ starts out at $(5,-4,-2)$ at $t=0$ and flies around for $1$ seconds. Where is the bee located at time $1$ if $\\int^{1}_{0}{\\bf r\\it'(u)du}=0$? $P=($ [ANS], [ANS], [ANS] $)$", "answer_v2": [ "5", "-4", "-2" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A bee with velocity vector $\\bf r\\it'(t)$ starts out at $(1,-2, 1)$ at $t=0$ and flies around for $4$ seconds. Where is the bee located at time $4$ if $\\int^{4}_{0}{\\bf r\\it'(u)du}=0$? $P=($ [ANS], [ANS], [ANS] $)$", "answer_v3": [ "1", "-2", "1" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0045", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Integrals", "level": "2", "keywords": [ "vector' 'integral", "integral", "vector function", "integral' 'vector function" ], "problem_v1": "Evaluate $\\int_{0}^{8}(t\\mathbf{i}+t^2\\mathbf{j}+t^3\\mathbf{k})dt$=[ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$.", "answer_v1": [ "32", "170.666666666667", "1024" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Evaluate $\\int_{0}^{1}(t\\mathbf{i}+t^2\\mathbf{j}+t^3\\mathbf{k})dt$=[ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$.", "answer_v2": [ "0.5", "0.333333333333333", "0.25" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Evaluate $\\int_{0}^{4}(t\\mathbf{i}+t^2\\mathbf{j}+t^3\\mathbf{k})dt$=[ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$.", "answer_v3": [ "8", "21.3333333333333", "64" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0046", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Integrals", "level": "2", "keywords": [ "calculus", "vector", "derivative", "integral", "vector' 'integral' 'derivative", "vector function", "parametric" ], "problem_v1": "If $\\mathbf{r}(t)=\\cos(3t)\\mathbf{i}+\\sin(3t)\\mathbf{j}+2t \\mathbf{k}$ compute $\\mathbf{r}'(t)$=[ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] ${\\mathbf{k}}$ and $\\int{\\mathbf{r}}(t)\\, dt$=[ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}+\\mathbf{C}$ with $\\mathbf{C}$ a constant vector.", "answer_v1": [ "-3*sin(3*t)", "3*cos(3*t)", "2*1", "1/3*sin(3*t)", "-1/3*cos(3*t)", "1*t^2" ], "answer_type_v1": [ "EX", "EX", "NV", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "If $\\mathbf{r}(t)=\\cos(-5t)\\mathbf{i}+\\sin(-5t)\\mathbf{j}+10t \\mathbf{k}$ compute $\\mathbf{r}'(t)$=[ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] ${\\mathbf{k}}$ and $\\int{\\mathbf{r}}(t)\\, dt$=[ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}+\\mathbf{C}$ with $\\mathbf{C}$ a constant vector.", "answer_v2": [ "-(-5)*sin(-5*t)", "-5*cos(-5*t)", "2*5", "1/-5*sin(-5*t)", "-1/-5*cos(-5*t)", "5*t^2" ], "answer_type_v2": [ "EX", "EX", "NV", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "If $\\mathbf{r}(t)=\\cos(-2t)\\mathbf{i}+\\sin(-2t)\\mathbf{j}+2t \\mathbf{k}$ compute $\\mathbf{r}'(t)$=[ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] ${\\mathbf{k}}$ and $\\int{\\mathbf{r}}(t)\\, dt$=[ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}+\\mathbf{C}$ with $\\mathbf{C}$ a constant vector.", "answer_v3": [ "-(-2)*sin(-2*t)", "-2*cos(-2*t)", "2*1", "1/-2*sin(-2*t)", "-1/-2*cos(-2*t)", "1*t^2" ], "answer_type_v3": [ "EX", "EX", "NV", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0047", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "2", "keywords": [ "calculus" ], "problem_v1": "For a plane curve $\\mathbf{r}(t)=\\langle x(t),y(t)\\rangle$, \\kappa(t)= \\frac{|x'(t)y''(t)-x''(t)y'(t)|}{(x'(t)^2+y'(t)^2)^{3/2} }. Use this equation to compute the curvature at the given point. \\mathbf{r}(t)=\\langle 3t^{4},t^{4}\\rangle, t=2. $\\kappa(2)=$ [ANS]", "answer_v1": [ "|96*48-144*32|/[(96^2+32^2)^(3/2)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "For a plane curve $\\mathbf{r}(t)=\\langle x(t),y(t)\\rangle$, \\kappa(t)= \\frac{|x'(t)y''(t)-x''(t)y'(t)|}{(x'(t)^2+y'(t)^2)^{3/2} }. Use this equation to compute the curvature at the given point. \\mathbf{r}(t)=\\langle-5t^{2},5t^{3}\\rangle, t=5. $\\kappa(5)=$ [ANS]", "answer_v2": [ "|-50*150--10*375|/([(-50)^2+375^2]^(3/2))" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "For a plane curve $\\mathbf{r}(t)=\\langle x(t),y(t)\\rangle$, \\kappa(t)= \\frac{|x'(t)y''(t)-x''(t)y'(t)|}{(x'(t)^2+y'(t)^2)^{3/2} }. Use this equation to compute the curvature at the given point. \\mathbf{r}(t)=\\langle-2t^{3},t^{4}\\rangle, t=2. $\\kappa(2)=$ [ANS]", "answer_v3": [ "|-24*48--24*32|/([(-24)^2+32^2]^(3/2))" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0048", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find the curvature of the plane curve y=2t^{3} at the point $t=2$. $\\kappa(2)=$ [ANS]", "answer_v1": [ "0.0017316" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the curvature of the plane curve y=-4t^{4} at the point $t=1$. $\\kappa(1)=$ [ANS]", "answer_v2": [ "0.0116504" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the curvature of the plane curve y=-2t^{3} at the point $t=1$. $\\kappa(1)=$ [ANS]", "answer_v3": [ "0.0533186" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0049", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find the speed: $\\bf r\\it(t)=\\left<\\cosh\\!\\left(t\\right),\\cosh\\!\\left(t\\right),2t\\right>$ at $t=4$. $v(4)=$ [ANS]", "answer_v1": [ "38.6456" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the speed: $\\bf r\\it(t)=\\left<\\cosh\\!\\left(t\\right),\\cosh\\!\\left(t\\right),-2t\\right>$ at $t=0$. $v(0)=$ [ANS]", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the speed: $\\bf r\\it(t)=\\left<\\cosh\\!\\left(t\\right),\\cosh\\!\\left(t\\right),t\\right>$ at $t=1$. $v(1)=$ [ANS]", "answer_v3": [ "1.93964" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0050", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find an arc length parametrization $\\bf r\\it_1(s)$ of ${\\bf r}({\\it t})=\\left$. Assume $t(s)=0$ when $s=0$, and $t'(0)>0$. ${\\bf r\\it_1(s)}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left. \\right>$", "answer_v1": [ "(1+s/[sqrt(66)])*sin(ln(1+s/[sqrt(66)]))", "(1+s/[sqrt(66)])*cos(ln(1+s/[sqrt(66)]))", "8*(1+s/[sqrt(66)])" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Find an arc length parametrization $\\bf r\\it_1(s)$ of ${\\bf r}({\\it t})=\\left$. Assume $t(s)=0$ when $s=0$, and $t'(0)>0$. ${\\bf r\\it_1(s)}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left. \\right>$", "answer_v2": [ "(1+s/[sqrt(3)])*sin(ln(1+s/[sqrt(3)]))", "(1+s/[sqrt(3)])*cos(ln(1+s/[sqrt(3)]))", "1*(1+s/[sqrt(3)])" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Find an arc length parametrization $\\bf r\\it_1(s)$ of ${\\bf r}({\\it t})=\\left$. Assume $t(s)=0$ when $s=0$, and $t'(0)>0$. ${\\bf r\\it_1(s)}=\\left< \\right.$ [ANS], [ANS], [ANS] $\\left. \\right>$", "answer_v3": [ "(1+s/[sqrt(18)])*sin(ln(1+s/[sqrt(18)]))", "(1+s/[sqrt(18)])*cos(ln(1+s/[sqrt(18)]))", "4*(1+s/[sqrt(18)])" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0051", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Compute the length of the curve ${\\bf r\\it(t)}=\\left<-5 t,6 t-4,-4 t-3\\right>$ over the interval $0 \\le t \\le 8$ $L=$ [ANS]", "answer_v1": [ "70.1997" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the length of the curve ${\\bf r\\it(t)}=\\left<-2 t,3 t+8,-7 t-3\\right>$ over the interval $0 \\le t \\le 3$ $L=$ [ANS]", "answer_v2": [ "23.622" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the length of the curve ${\\bf r\\it(t)}=\\left<-5 t,4 t-6,-5 t-3\\right>$ over the interval $0 \\le t \\le 5$ $L=$ [ANS]", "answer_v3": [ "40.6202" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0052", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find the speed at the given value of t: ${\\bf r\\it(t)}=\\left, t=8$ $v(8)=$ [ANS]", "answer_v1": [ "1.00778" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the speed at the given value of t: ${\\bf r\\it(t)}=\\left, t=2$ $v(2)=$ [ANS]", "answer_v2": [ "1.11803" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the speed at the given value of t: ${\\bf r\\it(t)}=\\left, t=4$ $v(4)=$ [ANS]", "answer_v3": [ "1.03078" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0053", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Evaluate $s(t)=\\int^{t}_{-\\infty}||\\bf r\\it'(u)||du$ for the Bernoulli spiral $\\bf r\\it(t)=\\left$. It is convenient to take $-\\infty$ as the lower limit since $s(-\\infty)=0$. Then use $s$ to obtain an arc length parametrization of $\\bf r\\it(t)$. $\\bf r\\it_1(s)=\\left< \\right.$ [ANS], [ANS] $\\left. \\right>$", "answer_v1": [ "s/8.06226*cos(8*ln(s/8.06226))", "s/8.06226*sin(8*ln(s/8.06226))" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Evaluate $s(t)=\\int^{t}_{-\\infty}||\\bf r\\it'(u)||du$ for the Bernoulli spiral $\\bf r\\it(t)=\\left$. It is convenient to take $-\\infty$ as the lower limit since $s(-\\infty)=0$. Then use $s$ to obtain an arc length parametrization of $\\bf r\\it(t)$. $\\bf r\\it_1(s)=\\left< \\right.$ [ANS], [ANS] $\\left. \\right>$", "answer_v2": [ "s/2.23607*cos(2*ln(s/2.23607))", "s/2.23607*sin(2*ln(s/2.23607))" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Evaluate $s(t)=\\int^{t}_{-\\infty}||\\bf r\\it'(u)||du$ for the Bernoulli spiral $\\bf r\\it(t)=\\left$. It is convenient to take $-\\infty$ as the lower limit since $s(-\\infty)=0$. Then use $s$ to obtain an arc length parametrization of $\\bf r\\it(t)$. $\\bf r\\it_1(s)=\\left< \\right.$ [ANS], [ANS] $\\left. \\right>$", "answer_v3": [ "s/4.12311*cos(4*ln(s/4.12311))", "s/4.12311*sin(4*ln(s/4.12311))" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0054", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "2", "keywords": [ "vector' 'multivariable' 'curvature", "Vector", "Curvature", "calculus" ], "problem_v1": "Find the curvature $\\kappa (t)$ of the curve $\\mathbf{r} (t)=\\left(3 \\sin t \\right) \\mathbf{i}+\\left(3 \\sin t \\right) \\mathbf{j}+\\left(1 \\cos t \\right) \\mathbf{k}$ [ANS]", "answer_v1": [ "(2*( 3*1 )^2)**.5/(( 1*sin(t))^2 + 2*( 3*cos(t))^2 )^{1.5} " ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the curvature $\\kappa (t)$ of the curve $\\mathbf{r} (t)=\\left(-5 \\sin t \\right) \\mathbf{i}+\\left(-5 \\sin t \\right) \\mathbf{j}+\\left(5 \\cos t \\right) \\mathbf{k}$ [ANS]", "answer_v2": [ "(2*( -5*5 )^2)**.5/(( 5*sin(t))^2 + 2*( -5*cos(t))^2 )^{1.5} " ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the curvature $\\kappa (t)$ of the curve $\\mathbf{r} (t)=\\left(-2 \\sin t \\right) \\mathbf{i}+\\left(-2 \\sin t \\right) \\mathbf{j}+\\left(1 \\cos t \\right) \\mathbf{k}$ [ANS]", "answer_v3": [ "(2*( -2*1 )^2)**.5/(( 1*sin(t))^2 + 2*( -2*cos(t))^2 )^{1.5} " ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0055", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "2", "keywords": [ "vector", "multivariable", "curvature", "Vector", "Curvature" ], "problem_v1": "Find the curvature of $y=\\sin \\left(3x \\right)$ at $x= \\frac{\\pi}{4} $. [ANS]", "answer_v1": [ "0.493382200218159" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the curvature of $y=\\sin \\left(-5x \\right)$ at $x= \\frac{\\pi}{4} $. [ANS]", "answer_v2": [ "0.356389055055324" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the curvature of $y=\\sin \\left(-2x \\right)$ at $x= \\frac{\\pi}{4} $. [ANS]", "answer_v3": [ "4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0056", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "2", "keywords": [ "vector' 'parametric' 'multivariable' 'length", "Vector", "Parametric", "Length", "curve' 'length" ], "problem_v1": "Find the length of the given curve: \\mathbf{r} \\left(t \\right)=\\left(-2 t,-2 \\sin t,-2 \\cos t \\right) where $-3 \\leq t \\leq 3$. [ANS]", "answer_v1": [ "16.9705627484771" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the length of the given curve: \\mathbf{r} \\left(t \\right)=\\left(-5 t, 5 \\sin t, 5 \\cos t \\right) where $-5 \\leq t \\leq 2$. [ANS]", "answer_v2": [ "49.4974746830583" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the length of the given curve: \\mathbf{r} \\left(t \\right)=\\left(-3 t,-2 \\sin t,-2 \\cos t \\right) where $-1 \\leq t \\leq 5$. [ANS]", "answer_v3": [ "21.6333076527839" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0057", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "3", "keywords": [ "vector' 'parametric' 'multivariable' 'length", "Vector", "Parametric", "Reparametrize" ], "problem_v1": "Starting from the point $\\left(3, 1, 1 \\right)$ reparametrize the curve \\mathbf{r} \\left(t \\right)=\\left(3+2 t \\right) \\mathbf{i}+\\left(1-1 t \\right) \\mathbf{j}+\\left(1-1 t \\right) \\mathbf{k} in terms of arclength.\n$\\mathbf{r} \\left(t \\left(s \\right) \\right)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$", "answer_v1": [ "3 + s*2 / 2.44948974278318", "1 + s*-1 / 2.44948974278318", "1 + s*-1 / 2.44948974278318" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Starting from the point $\\left(-5, 5,-4 \\right)$ reparametrize the curve \\mathbf{r} \\left(t \\right)=\\left(-5-1 t \\right) \\mathbf{i}+\\left(5+3 t \\right) \\mathbf{j}+\\left(-4-1 t \\right) \\mathbf{k} in terms of arclength.\n$\\mathbf{r} \\left(t \\left(s \\right) \\right)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$", "answer_v2": [ "-5 + s*-1 / 3.3166247903554", "5 + s*3 / 3.3166247903554", "-4 + s*-1 / 3.3166247903554" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Starting from the point $\\left(-2, 1,-2 \\right)$ reparametrize the curve \\mathbf{r} \\left(t \\right)=\\left(-2-2 t \\right) \\mathbf{i}+\\left(1-1 t \\right) \\mathbf{j}+\\left(-2+2 t \\right) \\mathbf{k} in terms of arclength.\n$\\mathbf{r} \\left(t \\left(s \\right) \\right)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$", "answer_v3": [ "-2 + s*-2 / 3", "1 + s*-1 / 3", "-2 + s*2 / 3" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0058", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "5", "keywords": [ "vector' 'acceleration' 'multivariable' 'velocity' 'curvature", "calculus", "vector", "curvature" ], "problem_v1": "A factory has a machine which bends wire at a rate of 6 unit(s) of curvature per second. How long does it take to bend a straight wire into a circle of radius 8? [ANS] seconds", "answer_v1": [ "0.0208333333333333" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A factory has a machine which bends wire at a rate of 10 unit(s) of curvature per second. How long does it take to bend a straight wire into a circle of radius 1? [ANS] seconds", "answer_v2": [ "0.1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A factory has a machine which bends wire at a rate of 7 unit(s) of curvature per second. How long does it take to bend a straight wire into a circle of radius 4? [ANS] seconds", "answer_v3": [ "0.0357142857142857" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0059", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "5", "keywords": [ "Vector", "Projectile", "vector' 'acceleration' 'multivariable' 'velocity' 'projectile" ], "problem_v1": "A gun has a muzzle speed of 90 meters per second. What angle of elevation should be used to hit an object 180 meters away? Neglect air resistance and use $g=9.8\\, \\textrm{m}/\\textrm{sec}^{2}$ as the acceleration of gravity.\nAnswer: [ANS] radians", "answer_v1": [ "0.109768509523908" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A gun has a muzzle speed of 50 meters per second. What angle of elevation should be used to hit an object 200 meters away? Neglect air resistance and use $g=9.8\\, \\textrm{m}/\\textrm{sec}^{2}$ as the acceleration of gravity.\nAnswer: [ANS] radians", "answer_v2": [ "0.450541778920686" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A gun has a muzzle speed of 60 meters per second. What angle of elevation should be used to hit an object 180 meters away? Neglect air resistance and use $g=9.8\\, \\textrm{m}/\\textrm{sec}^{2}$ as the acceleration of gravity.\nAnswer: [ANS] radians", "answer_v3": [ "0.256044876467074" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0060", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "2", "keywords": [ "acceleration", "velocity", "vector function", "arc length " ], "problem_v1": "Find the length of the curve x=3+t,\\quad y=1+2t, z=-\\left(2+2t\\right), for $3 \\le t \\le 5$. length=[ANS]\n(Think of second way that you could calculate this length, too, and see that you get the same result.) (Think of second way that you could calculate this length, too, and see that you get the same result.)", "answer_v1": [ "sqrt(1*1+2*2+-2*-2)*(5-3)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the length of the curve x=5t-5,\\quad y=-\\left(4+2t\\right), z=5-2t, for $1 \\le t \\le 2$. length=[ANS]\n(Think of second way that you could calculate this length, too, and see that you get the same result.) (Think of second way that you could calculate this length, too, and see that you get the same result.)", "answer_v2": [ "sqrt(5*5+-2*-2+-2*-2)*(2-1)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the length of the curve x=t-2,\\quad y=t-2, z=-\\left(3+2t\\right), for $4 \\le t \\le 6$. length=[ANS]\n(Think of second way that you could calculate this length, too, and see that you get the same result.) (Think of second way that you could calculate this length, too, and see that you get the same result.)", "answer_v3": [ "sqrt(1*1+1*1+-2*-2)*(6-4)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0061", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "3", "keywords": [ "line integral", "arc length" ], "problem_v1": "Find the length of the curve $\\boldsymbol{\\vec{r}}(t)=\\left$ for $0 \\leq t \\leq 3$.\nArc length=[ANS]", "answer_v1": [ "sqrt(3)*[e^(3/9)-1]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the length of the curve $\\boldsymbol{\\vec{r}}(t)=\\left$ for $0 \\leq t \\leq 5$.\nArc length=[ANS]", "answer_v2": [ "sqrt(3)*[e^(5/6)-1]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the length of the curve $\\boldsymbol{\\vec{r}}(t)=\\left$ for $0 \\leq t \\leq 4$.\nArc length=[ANS]", "answer_v3": [ "sqrt(3)*[e^(4/7)-1]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0062", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "3", "keywords": [ "line integral", "arc length" ], "problem_v1": "Find the length of the curve $\\boldsymbol{\\vec{r}}(t)=\\left<7t, \\frac{t^{3}}{3} , \\frac{\\sqrt{14}t^{2}}{2} \\right>$ for $-2 \\leq t \\leq 4$.\nArc length=[ANS]", "answer_v1": [ "7*(4--2)+[4^3-(-2)^3]/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the length of the curve $\\boldsymbol{\\vec{r}}(t)=\\left<3t, \\frac{t^{3}}{3} , \\frac{\\sqrt{6}t^{2}}{2} \\right>$ for $-1 \\leq t \\leq 1$.\nArc length=[ANS]", "answer_v2": [ "3*(1--1)+[1^3-(-1)^3]/3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the length of the curve $\\boldsymbol{\\vec{r}}(t)=\\left<4t, \\frac{t^{3}}{3} , \\frac{\\sqrt{8}t^{2}}{2} \\right>$ for $-2 \\leq t \\leq 2$.\nArc length=[ANS]", "answer_v3": [ "4*(2--2)+[2^3-(-2)^3]/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0063", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "3", "keywords": [ "line integral", "arc length" ], "problem_v1": "In 1932, Giuseppe Momo was commissioned to build the famous Vatican Museum double spiral staircase. Suppose that it takes you one hour to stroll at a constant speed up one spiral of this staircase, which has a radius of 26 feet and a height of 54 feet and makes 6 revolutions.\n(a) Assuming the spiral staircase is centered about the $z$-axis, find a vector parametric equation for the helical path you take from the point $(26,0,0)$ to the point $(26,0,54)$ that makes $6$ revolutions during the time interval $0 \\leq t \\leq 1$.\n$\\boldsymbol{\\vec{r}}(t)=$ [ANS]\n(b) How far did you walk? [ANS] feet", "answer_v1": [ "(26*cos(12*pi*t),26*sin(12*pi*t),54*t)", "sqrt((26*12*pi)^2+54^2)" ], "answer_type_v1": [ "OL", "NV" ], "options_v1": [ [], [] ], "problem_v2": "In 1932, Giuseppe Momo was commissioned to build the famous Vatican Museum double spiral staircase. Suppose that it takes you one hour to stroll at a constant speed up one spiral of this staircase, which has a radius of 16 feet and a height of 56 feet and makes 7 revolutions.\n(a) Assuming the spiral staircase is centered about the $z$-axis, find a vector parametric equation for the helical path you take from the point $(16,0,0)$ to the point $(16,0,56)$ that makes $7$ revolutions during the time interval $0 \\leq t \\leq 1$.\n$\\boldsymbol{\\vec{r}}(t)=$ [ANS]\n(b) How far did you walk? [ANS] feet", "answer_v2": [ "(16*cos(14*pi*t),16*sin(14*pi*t),56*t)", "sqrt((16*14*pi)^2+56^2)" ], "answer_type_v2": [ "OL", "NV" ], "options_v2": [ [], [] ], "problem_v3": "In 1932, Giuseppe Momo was commissioned to build the famous Vatican Museum double spiral staircase. Suppose that it takes you one hour to stroll at a constant speed up one spiral of this staircase, which has a radius of 19 feet and a height of 48 feet and makes 6 revolutions.\n(a) Assuming the spiral staircase is centered about the $z$-axis, find a vector parametric equation for the helical path you take from the point $(19,0,0)$ to the point $(19,0,48)$ that makes $6$ revolutions during the time interval $0 \\leq t \\leq 1$.\n$\\boldsymbol{\\vec{r}}(t)=$ [ANS]\n(b) How far did you walk? [ANS] feet", "answer_v3": [ "(19*cos(12*pi*t),19*sin(12*pi*t),48*t)", "sqrt((19*12*pi)^2+48^2)" ], "answer_type_v3": [ "OL", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0064", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "2", "keywords": [ "curve", "arc length" ], "problem_v1": "Consider the curve $ \\mathbf{r}=(e^{3t} \\cos(t), e^{3t} \\sin(t), e^{3t})$. Compute the arclength function $s(t)$: (with initial point $t=0$). [ANS]", "answer_v1": [ "sqrt(2*3*3 + 1*1) * (exp(3*t) - 1) / 3" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Consider the curve $ \\mathbf{r}=(e^{-5t} \\cos(6t), e^{-5t} \\sin(6t), e^{-5t})$. Compute the arclength function $s(t)$: (with initial point $t=0$). [ANS]", "answer_v2": [ "sqrt(2*-5*-5 + 6*6) * (exp(-5*t) - 1) / -5" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Consider the curve $ \\mathbf{r}=(e^{-2t} \\cos(2t), e^{-2t} \\sin(2t), e^{-2t})$. Compute the arclength function $s(t)$: (with initial point $t=0$). [ANS]", "answer_v3": [ "sqrt(2*-2*-2 + 2*2) * (exp(-2*t) - 1) / -2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0065", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "3", "keywords": [ "calculus", "arc length" ], "problem_v1": "Find the arclength of the curve $\\mathbf r(t)=\\langle 8 \\sqrt2 t, e^{8 t}, e^{-8 t}\\rangle$, $0 \\le t \\le 1$ [ANS]", "answer_v1": [ "2980.9576515791" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the arclength of the curve $\\mathbf r(t)=\\langle 2 \\sqrt2 t, e^{2 t}, e^{-2 t}\\rangle$, $0 \\le t \\le 1$ [ANS]", "answer_v2": [ "7.25372081569404" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the arclength of the curve $\\mathbf r(t)=\\langle 4 \\sqrt2 t, e^{4 t}, e^{-4 t}\\rangle$, $0 \\le t \\le 1$ [ANS]", "answer_v3": [ "54.5798343942555" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0066", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Arc length and curvature", "level": "3", "keywords": [ "calculus", "arc length", "curve' 'arc length" ], "problem_v1": "Consider the path $\\mathbf{r}(t)=(16 t, 8 t^2, 8\\ln t)$ defined for $t > 0$. Find the length of the curve between the points $(16, 8, 0)$ and $(64, 128, 8 \\ln(4))$. [ANS]", "answer_v1": [ "120+8*ln(4)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider the path $\\mathbf{r}(t)=(4 t, 2 t^2, 2\\ln t)$ defined for $t > 0$. Find the length of the curve between the points $(4, 2, 0)$ and $(20, 50, 2 \\ln(5))$. [ANS]", "answer_v2": [ "48+2*ln(5)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider the path $\\mathbf{r}(t)=(8 t, 4 t^2, 4\\ln t)$ defined for $t > 0$. Find the length of the curve between the points $(8, 4, 0)$ and $(32, 64, 4 \\ln(4))$. [ANS]", "answer_v3": [ "60+4*ln(4)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0067", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Fictional planet Hlipod has a day that lasts $5$ Earth days, and a mass $M\\approx 6.067\\times 10^{23}\\ \\text{kg}$. Hlipod has a satellite that is in \"geosynchronous orbit,\" meaning its orbit is circular and the orbital period is the length of a day on Hlipod (so $5$ Earth days). Use Kepler's Third Law to determine the semimajor axis $a$ for the satellite, and thus its orbital radius. Then determine the orbital velocity, using the formula v=\\sqrt{ \\frac{GM}{R} }. $a=$ [ANS]\n$v=$ [ANS]", "answer_v1": [ "[432000^2*(6.673E-11)*(6.067E+23)/(4*pi^2)]^(1/3)", "[(6.673E-11)*(6.067E+23)/(5.76281E+7)]^(1/2)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Fictional planet Hlipod has a day that lasts $2$ Earth days, and a mass $M\\approx 6.014\\times 10^{30}\\ \\text{kg}$. Hlipod has a satellite that is in \"geosynchronous orbit,\" meaning its orbit is circular and the orbital period is the length of a day on Hlipod (so $2$ Earth days). Use Kepler's Third Law to determine the semimajor axis $a$ for the satellite, and thus its orbital radius. Then determine the orbital velocity, using the formula v=\\sqrt{ \\frac{GM}{R} }. $a=$ [ANS]\n$v=$ [ANS]", "answer_v2": [ "[172800^2*(6.673E-11)*(6.014E+30)/(4*pi^2)]^(1/3)", "[(6.673E-11)*(6.014E+30)/(6.72054E+9)]^(1/2)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Fictional planet Hlipod has a day that lasts $3$ Earth days, and a mass $M\\approx 6.025\\times 10^{22}\\ \\text{kg}$. Hlipod has a satellite that is in \"geosynchronous orbit,\" meaning its orbit is circular and the orbital period is the length of a day on Hlipod (so $3$ Earth days). Use Kepler's Third Law to determine the semimajor axis $a$ for the satellite, and thus its orbital radius. Then determine the orbital velocity, using the formula v=\\sqrt{ \\frac{GM}{R} }. $a=$ [ANS]\n$v=$ [ANS]", "answer_v3": [ "[259200^2*(6.673E-11)*(6.025E+22)/(4*pi^2)]^(1/3)", "[(6.673E-11)*(6.025E+22)/(1.89844E+7)]^(1/2)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0068", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "calculus" ], "problem_v1": "It can be shown that if an object orbiting a star of mass $M$ in a circular orbit of radius $R$ has speed $v$, then M= \\frac{R v^2}{G} . Suppose a star orbits the center of the galaxy it is contained in with an orbit that is nearly circular with radius $R=3.5\\times 10^{18}$ and velocity $v=260\\ \\text{km/s}$. Use the result above to estimate the mass of the portion of the galaxy inside the star's orbit (place all of this mass at the center of the orbit). Mass=[ANS]", "answer_v1": [ "(3.5E+18)*260^2/(6.673E-11)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "It can be shown that if an object orbiting a star of mass $M$ in a circular orbit of radius $R$ has speed $v$, then M= \\frac{R v^2}{G} . Suppose a star orbits the center of the galaxy it is contained in with an orbit that is nearly circular with radius $R=2.1\\times 10^{20}$ and velocity $v=210\\ \\text{km/s}$. Use the result above to estimate the mass of the portion of the galaxy inside the star's orbit (place all of this mass at the center of the orbit). Mass=[ANS]", "answer_v2": [ "(2.1E+20)*210^2/(6.673E-11)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "It can be shown that if an object orbiting a star of mass $M$ in a circular orbit of radius $R$ has speed $v$, then M= \\frac{R v^2}{G} . Suppose a star orbits the center of the galaxy it is contained in with an orbit that is nearly circular with radius $R=2.6\\times 10^{18}$ and velocity $v=230\\ \\text{km/s}$. Use the result above to estimate the mass of the portion of the galaxy inside the star's orbit (place all of this mass at the center of the orbit). Mass=[ANS]", "answer_v3": [ "(2.6E+18)*230^2/(6.673E-11)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0069", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "calculus" ], "problem_v1": "The fictional planet Snazbort has a fictional moon Pingdol. Pingdol has an orbital period of $11.58$ days and a semimajor axis of $62.84\\times 10^{9}$. Use Kepler's Third Law to estimate the mass of Snazbort. Mass=[ANS]", "answer_v1": [ "4*pi^2/(6.673E-11)*(6.284E+10)^3/[(1.00051E+6)^2]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The fictional planet Snazbort has a fictional moon Pingdol. Pingdol has an orbital period of $6.93$ days and a semimajor axis of $16.65\\times 10^{8}$. Use Kepler's Third Law to estimate the mass of Snazbort. Mass=[ANS]", "answer_v2": [ "4*pi^2/(6.673E-11)*(1.665E+9)^3/(598752^2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The fictional planet Snazbort has a fictional moon Pingdol. Pingdol has an orbital period of $8.6$ days and a semimajor axis of $29.3\\times 10^{9}$. Use Kepler's Third Law to estimate the mass of Snazbort. Mass=[ANS]", "answer_v3": [ "4*pi^2/(6.673E-11)*(2.93E+10)^3/(743040^2)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0070", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Kepler's Third Law states that $T^2/a^3$ has the same value for each planetary orbit. One of the following orbit periods for the fictional planets below was measured incorrectly. Identify the planet with incorrect data, and determine what its orbital period should be. \\begin{array}{lcccc} \\text{Planet} & \\text{Snizbort} & \\text{Woolat} & \\text{Bingrel} & \\text{Pevneq} \\\\ a\\ (10^{10}\\text{m}) & 5.613 & 12.344 & 15.007 & 48.442 \\\\ T\\ (\\text{Earth years}) & 0.23 & 0.75 & 2.6 & 5.83 \\end{array} The planet whose data was wrongly measured was [ANS]. It should have $T=$ [ANS].", "answer_v1": [ "BINGREL", "(0.000299*15.0074^3)^(1/2)" ], "answer_type_v1": [ "MCS", "NV" ], "options_v1": [ [ "Snizbort", "Woolat", "Bingrel", "Pevneq" ], [] ], "problem_v2": "Kepler's Third Law states that $T^2/a^3$ has the same value for each planetary orbit. One of the following orbit periods for the fictional planets below was measured incorrectly. Identify the planet with incorrect data, and determine what its orbital period should be. \\begin{array}{lcccc} \\text{Planet} & \\text{Snizbort} & \\text{Woolat} & \\text{Bingrel} & \\text{Pevneq} \\\\ a\\ (10^{10}\\text{m}) & 3.556 & 8.937 & 19.111 & 36.544 \\\\ T\\ (\\text{Earth years}) & 0.3 & 0.71 & 2.22 & 5.87 \\end{array} The planet whose data was wrongly measured was [ANS]. It should have $T=$ [ANS].", "answer_v2": [ "SNIZBORT", "(0.000706*3.55689^3)^(1/2)" ], "answer_type_v2": [ "MCS", "NV" ], "options_v2": [ [ "Snizbort", "Woolat", "Bingrel", "Pevneq" ], [] ], "problem_v3": "Kepler's Third Law states that $T^2/a^3$ has the same value for each planetary orbit. One of the following orbit periods for the fictional planets below was measured incorrectly. Identify the planet with incorrect data, and determine what its orbital period should be. \\begin{array}{lcccc} \\text{Planet} & \\text{Snizbort} & \\text{Woolat} & \\text{Bingrel} & \\text{Pevneq} \\\\ a\\ (10^{10}\\text{m}) & 6.247 & 12.234 & 26.208 & 40.858 \\\\ T\\ (\\text{Earth years}) & 0.27 & 0.74 & 2.32 & 5.84 \\end{array} The planet whose data was wrongly measured was [ANS]. It should have $T=$ [ANS].", "answer_v3": [ "PEVNEQ", "(0.000299*40.8588^3)^(1/2)" ], "answer_type_v3": [ "MCS", "NV" ], "options_v3": [ [ "Snizbort", "Woolat", "Bingrel", "Pevneq" ], [] ] }, { "id": "Calculus_-_multivariable_0071", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find $\\mathbf{v}(t)$ given the acceleration \\mathbf{a}(t)=6\\boldsymbol{j} and initial condition \\mathbf{v}(0)=\\boldsymbol{i}. $\\mathbf{v}(t)=$ [ANS]\nUsage: To enter a vector, for example $\\langle x,y,z\\rangle$, type \" $<$ x, y, z $>$ \"", "answer_v1": [ "t*6*j+i" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find $\\mathbf{v}(t)$ given the acceleration \\mathbf{a}(t)=2\\boldsymbol{k} and initial condition \\mathbf{v}(0)=\\boldsymbol{i}. $\\mathbf{v}(t)=$ [ANS]\nUsage: To enter a vector, for example $\\langle x,y,z\\rangle$, type \" $<$ x, y, z $>$ \"", "answer_v2": [ "t*2*k+i" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find $\\mathbf{v}(t)$ given the acceleration \\mathbf{a}(t)=3\\boldsymbol{j} and initial condition \\mathbf{v}(0)=\\boldsymbol{k}. $\\mathbf{v}(t)=$ [ANS]\nUsage: To enter a vector, for example $\\langle x,y,z\\rangle$, type \" $<$ x, y, z $>$ \"", "answer_v3": [ "t*3*j+k" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0072", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find $\\mathbf{r}(t)$ and $\\mathbf{v}(t)$ given acceleration \\mathbf{a}(t)=t^{4}\\boldsymbol{k}, initial velocity \\mathbf{v}(0)=\\boldsymbol{i}, and initial position \\mathbf{r}(0)=2\\boldsymbol{j}. $\\mathbf{v}(t)=$ [ANS]\n$\\mathbf{r}(t)=$ [ANS]\nUsage: To enter a vector, for example $\\langle x,y,z\\rangle$, type \" $<$ x, y, z $>$ \"", "answer_v1": [ "1/5*t^5*k+i", "1/30*t^6*k+t*i+2*j" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Find $\\mathbf{r}(t)$ and $\\mathbf{v}(t)$ given acceleration \\mathbf{a}(t)=t^{1}\\boldsymbol{j}, initial velocity \\mathbf{v}(0)=6\\boldsymbol{i}, and initial position \\mathbf{r}(0)=-5\\boldsymbol{k}. $\\mathbf{v}(t)=$ [ANS]\n$\\mathbf{r}(t)=$ [ANS]\nUsage: To enter a vector, for example $\\langle x,y,z\\rangle$, type \" $<$ x, y, z $>$ \"", "answer_v2": [ "1/2*t^2*j+6*i", "1/6*t^3*j+t*6*i+-5*k" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Find $\\mathbf{r}(t)$ and $\\mathbf{v}(t)$ given acceleration \\mathbf{a}(t)=t^{2}\\boldsymbol{j}, initial velocity \\mathbf{v}(0)=\\boldsymbol{k}, and initial position \\mathbf{r}(0)=-3\\boldsymbol{i}. $\\mathbf{v}(t)=$ [ANS]\n$\\mathbf{r}(t)=$ [ANS]\nUsage: To enter a vector, for example $\\langle x,y,z\\rangle$, type \" $<$ x, y, z $>$ \"", "answer_v3": [ "1/3*t^3*j+k", "1/12*t^4*j+t*k+-3*i" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0073", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Calculate the velocity and acceleration vectors, and speed for \\mathbf{r}(t)=\\langle \\cos\\!\\left(4t\\right),\\cos\\!\\left(t\\right),\\sin\\!\\left(4t\\right) \\rangle when $t= \\frac{7 \\pi}{4} $. Velocity: [ANS]\nAcceleration: [ANS]\nSpeed: [ANS]\nUsage: To enter a vector, for example $\\langle x,y,z\\rangle$, type \" $<$ x, y, z $>$ \"", "answer_v1": [ "(0,0.707107,-4)", "(16,-0.707107,0)", "4.06202" ], "answer_type_v1": [ "OL", "OL", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Calculate the velocity and acceleration vectors, and speed for \\mathbf{r}(t)=\\langle \\sin\\!\\left(t\\right),\\cos\\!\\left(3t\\right),\\sin\\!\\left(3t\\right) \\rangle when $t= \\frac{\\pi}{3} $. Velocity: [ANS]\nAcceleration: [ANS]\nSpeed: [ANS]\nUsage: To enter a vector, for example $\\langle x,y,z\\rangle$, type \" $<$ x, y, z $>$ \"", "answer_v2": [ "(0.5,0,-3)", "(-0.866025,9,0)", "3.04138" ], "answer_type_v2": [ "OL", "OL", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Calculate the velocity and acceleration vectors, and speed for \\mathbf{r}(t)=\\langle \\cos\\!\\left(t\\right),\\sin\\!\\left(4t\\right),\\cos\\!\\left(4t\\right) \\rangle when $t= \\frac{3 \\pi}{4} $. Velocity: [ANS]\nAcceleration: [ANS]\nSpeed: [ANS]\nUsage: To enter a vector, for example $\\langle x,y,z\\rangle$, type \" $<$ x, y, z $>$ \"", "answer_v3": [ "(-0.707107,-4,0)", "(0.707107,0,16)", "4.06202" ], "answer_type_v3": [ "OL", "OL", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0074", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Calculate $\\mathbf{r}'(t)$, $\\mathbf{T}(t)$, and $\\mathbf{T}(2)$ where \\mathbf{r}(t)=\\langle 4t^{3}, 3t^{3}\\rangle. $\\mathbf{r}'(t)=\\langle$ [ANS] $,$ [ANS] $\\rangle$. $\\mathbf{T}(t)=\\langle$ [ANS] $,$ [ANS] $\\rangle$. $\\mathbf{T}(2)=\\langle$ [ANS] $,$ [ANS] $\\rangle$.", "answer_v1": [ "4*3*t^2", "3*3*t^2", "12*t^2/[sqrt((12*t^2)^2+(9*t^2)^2)]", "9*t^2/[sqrt((12*t^2)^2+(9*t^2)^2)]", "0.8", "0.6" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Calculate $\\mathbf{r}'(t)$, $\\mathbf{T}(t)$, and $\\mathbf{T}(4)$ where \\mathbf{r}(t)=\\langle t^{2}, 4t^{2}\\rangle. $\\mathbf{r}'(t)=\\langle$ [ANS] $,$ [ANS] $\\rangle$. $\\mathbf{T}(t)=\\langle$ [ANS] $,$ [ANS] $\\rangle$. $\\mathbf{T}(4)=\\langle$ [ANS] $,$ [ANS] $\\rangle$.", "answer_v2": [ "2*t", "4*2*t", "2*t/[sqrt((2*t)^2+(8*t)^2)]", "8*t/[sqrt((2*t)^2+(8*t)^2)]", "0.242536", "0.970143" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Calculate $\\mathbf{r}'(t)$, $\\mathbf{T}(t)$, and $\\mathbf{T}(1)$ where \\mathbf{r}(t)=\\langle 2t^{2}, 3t^{3}\\rangle. $\\mathbf{r}'(t)=\\langle$ [ANS] $,$ [ANS] $\\rangle$. $\\mathbf{T}(t)=\\langle$ [ANS] $,$ [ANS] $\\rangle$. $\\mathbf{T}(1)=\\langle$ [ANS] $,$ [ANS] $\\rangle$.", "answer_v3": [ "2*2*t", "3*3*t^2", "4*t/[sqrt((4*t)^2+(9*t^2)^2)]", "9*t^2/[sqrt((4*t)^2+(9*t^2)^2)]", "0.406138", "0.913812" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0075", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "Derivative", "Parametric", "Path", "Velocity", "Acceleration" ], "problem_v1": "Consider the parametric curve $f(t)=\\left<\\left(t+3\\right)^{6},t\\!\\left(1-t\\right)\\right>$. The velocity vector is $\\overline{v}(t)$=[ANS]\nand the acceleration vector is $\\overline{a}(t)$=[ANS].", "answer_v1": [ "(6*(t+3)^5,1-t-t)", "(6*5*(t+3)^4,-2)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Consider the parametric curve $f(t)=\\left<\\left(t-5\\right)^{4},t\\!\\left(1-5t\\right)\\right>$. The velocity vector is $\\overline{v}(t)$=[ANS]\nand the acceleration vector is $\\overline{a}(t)$=[ANS].", "answer_v2": [ "(4*(t-5)^3,1-5*t-5*t)", "(4*3*(t-5)^2,-10)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Consider the parametric curve $f(t)=\\left<\\left(t-2\\right)^{5},t\\!\\left(1-t\\right)\\right>$. The velocity vector is $\\overline{v}(t)$=[ANS]\nand the acceleration vector is $\\overline{a}(t)$=[ANS].", "answer_v3": [ "(5*(t-2)^4,1-t-t)", "(5*4*(t-2)^3,-2)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0076", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "3", "keywords": [ "Derivative", "Parametric", "Path" ], "problem_v1": "Consider a particle moving along a curve so that it is at position $f(t)=\\left<3t^{3},t^{2}\\right>$ at time $t$. Then the particle is moving parallel to $\\left<1,2\\right>$ at time $t$=[ANS] at the point [ANS], when the direction of motion is [ANS].", "answer_v1": [ "0.111111", "(0.00411523,0.0123457)", "(0.111111,0.222222)" ], "answer_type_v1": [ "NV", "OL", "OL" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider a particle moving along a curve so that it is at position $f(t)=\\left<-5t^{3},5t^{2}\\right>$ at time $t$. Then the particle is moving parallel to $\\left<1,-2\\right>$ at time $t$=[ANS] at the point [ANS], when the direction of motion is [ANS].", "answer_v2": [ "0.333333", "(-0.185185,0.555556)", "(-1.66667,3.33333)" ], "answer_type_v2": [ "NV", "OL", "OL" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider a particle moving along a curve so that it is at position $f(t)=\\left<-2t^{3},t^{2}\\right>$ at time $t$. Then the particle is moving parallel to $\\left<1,-2\\right>$ at time $t$=[ANS] at the point [ANS], when the direction of motion is [ANS].", "answer_v3": [ "0.166667", "(-0.00925926,0.0277778)", "(-0.166667,0.333333)" ], "answer_type_v3": [ "NV", "OL", "OL" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0077", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "3", "keywords": [ "Vector", "Helix", "Tangent", "Normal", "Binormal", "Curvature" ], "problem_v1": "Consider the helix $\\mathbf{r} (t)=\\left(\\cos(2 t), \\sin(2 t), 1 t \\right)$. Compute, at $t= \\frac{\\pi}{6} $: A. The unit tangent vector $\\mathbf{T}=$ ([ANS], [ANS], [ANS]) B. The unit normal vector $\\mathbf{N}=$ ([ANS], [ANS], [ANS]) C. The unit binormal vector $\\mathbf{B}=$ ([ANS], [ANS], [ANS]) D. The curvature $\\kappa=$ [ANS]", "answer_v1": [ "-0.774596669241483", "0.447213595499958", "0.447213595499958", "-0.5", "-0.866025403784439", "0", "0.387298334620742", "-0.223606797749979", "0.894427190999916", "0.8" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the helix $\\mathbf{r} (t)=\\left(\\cos(-4 t), \\sin(-4 t), 4 t \\right)$. Compute, at $t= \\frac{\\pi}{6} $: A. The unit tangent vector $\\mathbf{T}=$ ([ANS], [ANS], [ANS]) B. The unit normal vector $\\mathbf{N}=$ ([ANS], [ANS], [ANS]) C. The unit binormal vector $\\mathbf{B}=$ ([ANS], [ANS], [ANS]) D. The curvature $\\kappa=$ [ANS]", "answer_v2": [ "-0.612372435695794", "0.353553390593274", "0.707106781186547", "0.5", "0.866025403784439", "0", "-0.612372435695794", "0.353553390593274", "-0.707106781186547", "0.5" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the helix $\\mathbf{r} (t)=\\left(\\cos(-2 t), \\sin(-2 t), 1 t \\right)$. Compute, at $t= \\frac{\\pi}{6} $: A. The unit tangent vector $\\mathbf{T}=$ ([ANS], [ANS], [ANS]) B. The unit normal vector $\\mathbf{N}=$ ([ANS], [ANS], [ANS]) C. The unit binormal vector $\\mathbf{B}=$ ([ANS], [ANS], [ANS]) D. The curvature $\\kappa=$ [ANS]", "answer_v3": [ "-0.774596669241483", "-0.447213595499958", "0.447213595499958", "-0.5", "0.866025403784439", "0", "-0.387298334620742", "-0.223606797749979", "-0.894427190999916", "0.8" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0078", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "vector' 'acceleration' 'multivariable' 'velocity", "Vector", "Velocity", "Acceleration" ], "problem_v1": "If $\\mathbf{r} (t)=\\cos (5 t) \\mathbf{i}+\\sin (5 t) \\mathbf{j}+2 t \\mathbf{k}$, compute: A. The velocity vector $\\mathbf{v} (t)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$ B. The acceleration vector $\\mathbf{a} (t)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$\nNote: the coefficients in your answers must be entered in the form of expressions in the variable $t$ ; e.g. \"5 cos(2t)\"", "answer_v1": [ "- 5 * sin( 5 * t )", "5 * cos( 5 * t )", "2", "- (5)^2 * cos( 5 * t )", "- (5)^2 * sin( 5 * t )", "0" ], "answer_type_v1": [ "EX", "EX", "NV", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "If $\\mathbf{r} (t)=\\cos (-9 t) \\mathbf{i}+\\sin (-9 t) \\mathbf{j}+9 t \\mathbf{k}$, compute: A. The velocity vector $\\mathbf{v} (t)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$ B. The acceleration vector $\\mathbf{a} (t)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$\nNote: the coefficients in your answers must be entered in the form of expressions in the variable $t$ ; e.g. \"5 cos(2t)\"", "answer_v2": [ "- -9 * sin( -9 * t )", "-9 * cos( -9 * t )", "9", "- (-9)^2 * cos( -9 * t )", "- (-9)^2 * sin( -9 * t )", "0" ], "answer_type_v2": [ "EX", "EX", "NV", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "If $\\mathbf{r} (t)=\\cos (-4 t) \\mathbf{i}+\\sin (-4 t) \\mathbf{j}+2 t \\mathbf{k}$, compute: A. The velocity vector $\\mathbf{v} (t)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$ B. The acceleration vector $\\mathbf{a} (t)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$\nNote: the coefficients in your answers must be entered in the form of expressions in the variable $t$ ; e.g. \"5 cos(2t)\"", "answer_v3": [ "- -4 * sin( -4 * t )", "-4 * cos( -4 * t )", "2", "- (-4)^2 * cos( -4 * t )", "- (-4)^2 * sin( -4 * t )", "0" ], "answer_type_v3": [ "EX", "EX", "NV", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0079", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "4", "keywords": [ "parametric curve", "multivariable", "calculus" ], "problem_v1": "Two particles are traveling through space. At time $t$ the first particle is at the point (1-t, 1-t, 2+t) and the second particle is at (3+t, 0-2 t, 0-t) Describe the two paths in words. Then determine if the two particles collide, cross, or neither cross nor collide to fill in the following. We see that [ANS]. If they collide or their paths cross, indicate where (if neither is true, enter none). A collision or crossing occurs at $(x,y,z)=$ [ANS].", "answer_v1": [ "the paths cross, and the particles collide there", "(2,2,1)" ], "answer_type_v1": [ "MCS", "OL" ], "options_v1": [ [ "the paths cross", "and the particles collide there", "and the particles collide there", "the paths of the particles cross", "but with no collision", "but with no collision", "neither of these is true" ], [] ], "problem_v2": "Two particles are traveling through space. At time $t$ the first particle is at the point (5+3 t,-4-t,-2-2 t) and the second particle is at (-7-3 t, 0+t, 6+2 t) Describe the two paths in words. Then determine if the two particles collide, cross, or neither cross nor collide to fill in the following. We see that [ANS]. If they collide or their paths cross, indicate where (if neither is true, enter none). A collision or crossing occurs at $(x,y,z)=$ [ANS].", "answer_v2": [ "the paths cross, and the particles collide there", "(-1,-2,2)" ], "answer_type_v2": [ "MCS", "OL" ], "options_v2": [ [ "the paths cross", "and the particles collide there", "and the particles collide there", "the paths of the particles cross", "but with no collision", "but with no collision", "neither of these is true" ], [] ], "problem_v3": "Two particles are traveling through space. At time $t$ the first particle is at the point (1-2 t,-2-t, 1+2 t) and the second particle is at (-24+3 t, 3-2 t, 16-t) Describe the two paths in words. Then determine if the two particles collide, cross, or neither cross nor collide to fill in the following. We see that [ANS]. If they collide or their paths cross, indicate where (if neither is true, enter none). A collision or crossing occurs at $(x,y,z)=$ [ANS].", "answer_v3": [ "the paths cross, and the particles collide there", "(-9,-7,11)" ], "answer_type_v3": [ "MCS", "OL" ], "options_v3": [ [ "the paths cross", "and the particles collide there", "and the particles collide there", "the paths of the particles cross", "but with no collision", "but with no collision", "neither of these is true" ], [] ] }, { "id": "Calculus_-_multivariable_0080", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "5", "keywords": [ "acceleration", "velocity", "vector function", "arc length " ], "problem_v1": "At time $t=0$ an object is moving with velocity vector $\\vec v=k \\vec i+\\vec j$ and acceleration vector $\\vec a=4 \\vec i+3 \\vec j$. For what value of $k$ is it possible that the object could be in uniform circular motion about some point in the plane? $k=$ [ANS]", "answer_v1": [ "-3*1/4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "At time $t=0$ an object is moving with velocity vector $\\vec v=k \\vec i-2 \\vec j$ and acceleration vector $\\vec a=\\vec i+4 \\vec j$. For what value of $k$ is it possible that the object could be in uniform circular motion about some point in the plane? $k=$ [ANS]", "answer_v2": [ "-4*-2/1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "At time $t=0$ an object is moving with velocity vector $\\vec v=k \\vec i-2 \\vec j$ and acceleration vector $\\vec a=2 \\vec i+3 \\vec j$. For what value of $k$ is it possible that the object could be in uniform circular motion about some point in the plane? $k=$ [ANS]", "answer_v3": [ "-3*-2/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0081", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "5", "keywords": [ "acceleration", "velocity", "vector function", "arc length " ], "problem_v1": "Suppose $\\vec{r}(t)=\\cos t\\,\\vec i+\\sin t\\, \\vec j+4 t\\,\\vec k$ represents the position of a particle on a helix, where $z$ is the height of the particle above the ground.\n(a) Is the particle ever moving downward? [ANS] If the particle is moving downward, when is this? When $t$ is in [ANS]\n(Enter none if it is never moving downward; otherwise, enter an interval or comma-separated list of intervals, e.g., (0,3], [4,5].) (b) When does the particle reach a point 14 units above the ground? When $t=$ [ANS]\n(c) What is the velocity of the particle when it is 14 units above the ground? $\\vec v=$ [ANS]\n(d) When it is 14 units above the ground, the particle leaves the helix and moves along the tangent. Find parametric equations for this tangent line (pick $t$ so that it is continuous through the time when the particle leaves the helix). $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v1": [ "no", "none", "14/4", "0.350783i-0.936457j+4k", "cos(14/4)-sin(14/4)*(t-14/4)", "sin(14/4)+cos(14/4)*(t-14/4)", "4*14/4+4*(t-14/4)" ], "answer_type_v1": [ "MCS", "OE", "NV", "EX", "EX", "EX", "EX" ], "options_v1": [ [ "yes", "no" ], [], [], [], [], [], [] ], "problem_v2": "Suppose $\\vec{r}(t)=\\cos t\\,\\vec i+\\sin t\\, \\vec j+2 t\\,\\vec k$ represents the position of a particle on a helix, where $z$ is the height of the particle above the ground.\n(a) Is the particle ever moving downward? [ANS] If the particle is moving downward, when is this? When $t$ is in [ANS]\n(Enter none if it is never moving downward; otherwise, enter an interval or comma-separated list of intervals, e.g., (0,3], [4,5].) (b) When does the particle reach a point 20 units above the ground? When $t=$ [ANS]\n(c) What is the velocity of the particle when it is 20 units above the ground? $\\vec v=$ [ANS]\n(d) When it is 20 units above the ground, the particle leaves the helix and moves along the tangent. Find parametric equations for this tangent line (pick $t$ so that it is continuous through the time when the particle leaves the helix). $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v2": [ "no", "none", "20/2", "0.544021i-0.839072j+2k", "cos(20/2)-sin(20/2)*(t-20/2)", "sin(20/2)+cos(20/2)*(t-20/2)", "2*20/2+2*(t-20/2)" ], "answer_type_v2": [ "MCS", "OE", "NV", "EX", "EX", "EX", "EX" ], "options_v2": [ [ "yes", "no" ], [], [], [], [], [], [] ], "problem_v3": "Suppose $\\vec{r}(t)=\\cos t\\,\\vec i+\\sin t\\, \\vec j+2 t\\,\\vec k$ represents the position of a particle on a helix, where $z$ is the height of the particle above the ground.\n(a) Is the particle ever moving downward? [ANS] If the particle is moving downward, when is this? When $t$ is in [ANS]\n(Enter none if it is never moving downward; otherwise, enter an interval or comma-separated list of intervals, e.g., (0,3], [4,5].) (b) When does the particle reach a point 14 units above the ground? When $t=$ [ANS]\n(c) What is the velocity of the particle when it is 14 units above the ground? $\\vec v=$ [ANS]\n(d) When it is 14 units above the ground, the particle leaves the helix and moves along the tangent. Find parametric equations for this tangent line (pick $t$ so that it is continuous through the time when the particle leaves the helix). $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v3": [ "no", "none", "14/2", "-0.656987i+0.753902j+2k", "cos(14/2)-sin(14/2)*(t-14/2)", "sin(14/2)+cos(14/2)*(t-14/2)", "2*14/2+2*(t-14/2)" ], "answer_type_v3": [ "MCS", "OE", "NV", "EX", "EX", "EX", "EX" ], "options_v3": [ [ "yes", "no" ], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0082", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "5", "keywords": [ "acceleration", "velocity", "vector function", "arc length " ], "problem_v1": "A child wanders slowly down a circular staircase from the top of a tower. With $x,y,z$ in feet and the origin at the base of the tower, her position $t$ minutes from the start is given by x=25\\cos t,\\quad y=25 \\sin t, \\quad z=100-5 t.\n(a) How tall is the tower? height=[ANS] ft (b) When does the child reach the bottom? time=[ANS] minutes (c) What is her speed at time $t$? speed=[ANS] ft/min (d) What is her acceleration at time $t$? acceleration=[ANS] ft/min ${}^2$", "answer_v1": [ "100", "100/5", "25.4951", "-25*cos(t)*i-25*sin(t)*j+0*k" ], "answer_type_v1": [ "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A child wanders slowly down a circular staircase from the top of a tower. With $x,y,z$ in feet and the origin at the base of the tower, her position $t$ minutes from the start is given by x=10\\cos t,\\quad y=10 \\sin t, \\quad z=120-5 t.\n(a) How tall is the tower? height=[ANS] ft (b) When does the child reach the bottom? time=[ANS] minutes (c) What is her speed at time $t$? speed=[ANS] ft/min (d) What is her acceleration at time $t$? acceleration=[ANS] ft/min ${}^2$", "answer_v2": [ "120", "120/5", "11.1803", "-10*cos(t)*i-10*sin(t)*j+0*k" ], "answer_type_v2": [ "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A child wanders slowly down a circular staircase from the top of a tower. With $x,y,z$ in feet and the origin at the base of the tower, her position $t$ minutes from the start is given by x=15\\cos t,\\quad y=15 \\sin t, \\quad z=100-5 t.\n(a) How tall is the tower? height=[ANS] ft (b) When does the child reach the bottom? time=[ANS] minutes (c) What is her speed at time $t$? speed=[ANS] ft/min (d) What is her acceleration at time $t$? acceleration=[ANS] ft/min ${}^2$", "answer_v3": [ "100", "100/5", "15.8114", "-15*cos(t)*i-15*sin(t)*j+0*k" ], "answer_type_v3": [ "NV", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0083", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "3", "keywords": [ "acceleration", "velocity", "vector function", "arc length " ], "problem_v1": "Find the velocity $\\vec{v}(t)$ and the speed $\\|\\vec{v}(t)\\|$ of a particle with position x=3t^{2}-24t+48,\\quad y=7,\\quad z=8t^{3}-48t^{2}. $\\vec v(t)=$ [ANS]\nspeed $\\|\\vec v(t)\\|=$ [ANS]\nFind any times at which the particle stops. $t=$ [ANS]\n(Give your answer as a time or list of times, comma separated.) (Give your answer as a time or list of times, comma separated.)", "answer_v1": [ "6*(t-4)i+24*t*(t-4)k", "sqrt([6*(t-4)]^2+0^2+[24*t*(t-4)]^2)", "4" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the velocity $\\vec{v}(t)$ and the speed $\\|\\vec{v}(t)\\|$ of a particle with position x=2t^{3}-3t^{2},\\quad y=5t^{2}-10t+5,\\quad z=4. $\\vec v(t)=$ [ANS]\nspeed $\\|\\vec v(t)\\|=$ [ANS]\nFind any times at which the particle stops. $t=$ [ANS]\n(Give your answer as a time or list of times, comma separated.) (Give your answer as a time or list of times, comma separated.)", "answer_v2": [ "6*t*(t-1)i+10*(t-1)j", "sqrt([6*t*(t-1)]^2+[10*(t-1)]^2+0^2)", "1" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the velocity $\\vec{v}(t)$ and the speed $\\|\\vec{v}(t)\\|$ of a particle with position x=4t^{2}-16t+16,\\quad y=5,\\quad z=4t^{3}-12t^{2}. $\\vec v(t)=$ [ANS]\nspeed $\\|\\vec v(t)\\|=$ [ANS]\nFind any times at which the particle stops. $t=$ [ANS]\n(Give your answer as a time or list of times, comma separated.) (Give your answer as a time or list of times, comma separated.)", "answer_v3": [ "8*(t-2)i+12*t*(t-2)k", "sqrt([8*(t-2)]^2+0^2+[12*t*(t-2)]^2)", "2" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0084", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "acceleration", "velocity", "vector function", "arc length " ], "problem_v1": "A particle starts at the point $P=(4,1,2)$ and moves along a straight line toward $Q=(6,0,1)$ at a speed of 5 cm/sec. Let $x$, $y$, $z$ be measured in centimeters and $t$ be measured in seconds.\n(a) Find the particle's velocity vector. $\\vec v(t)=$ [ANS]\n(b) Find parametric equations for the particle's motion. $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v1": [ "(4.08248,-2.04124,-2.04124)", "4+4.08248*t", "1+-2.04124*t", "2+-2.04124*t" ], "answer_type_v1": [ "OL", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A particle starts at the point $P=(-6,6,-5)$ and moves along a straight line toward $Q=(-7,9,-6)$ at a speed of 3 cm/sec. Let $x$, $y$, $z$ be measured in centimeters and $t$ be measured in seconds.\n(a) Find the particle's velocity vector. $\\vec v(t)=$ [ANS]\n(b) Find parametric equations for the particle's motion. $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v2": [ "(-0.904534,2.7136,-0.904534)", "-6+-0.904534*t", "6+2.7136*t", "-5+-0.904534*t" ], "answer_type_v2": [ "OL", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A particle starts at the point $P=(-3,2,-3)$ and moves along a straight line toward $Q=(-5,1,-1)$ at a speed of 7 cm/sec. Let $x$, $y$, $z$ be measured in centimeters and $t$ be measured in seconds.\n(a) Find the particle's velocity vector. $\\vec v(t)=$ [ANS]\n(b) Find parametric equations for the particle's motion. $x(t)=$ [ANS], $y(t)=$ [ANS], $z(t)=$ [ANS]", "answer_v3": [ "(-4.66667,-2.33333,4.66667)", "-3+-4.66667*t", "2+-2.33333*t", "-3+4.66667*t" ], "answer_type_v3": [ "OL", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0085", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "acceleration", "velocity", "vector function", "arc length " ], "problem_v1": "Find the velocity and acceleration vectors for the motion described by x=7+t^{2}, \\quad y=6+2t^{2}, \\quad z=3-2t^{2}. $\\vec v(t)=$ [ANS]\n$\\vec a(t)=$ [ANS]", "answer_v1": [ "(2*t, 4*t,4*t)", "(2,4,-4)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Find the velocity and acceleration vectors for the motion described by x=1+5t^{2}, \\quad y=2-2t^{2}, \\quad z=9-2t^{2}. $\\vec v(t)=$ [ANS]\n$\\vec a(t)=$ [ANS]", "answer_v2": [ "(10*t,-4*t,-4*t)", "(10,-4,-4)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Find the velocity and acceleration vectors for the motion described by x=3+t^{2}, \\quad y=3+t^{2}, \\quad z=2-2t^{2}. $\\vec v(t)=$ [ANS]\n$\\vec a(t)=$ [ANS]", "answer_v3": [ "(2*t,2*t,-4*t)", "(2,2,-4)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0086", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "5", "keywords": [ "acceleration", "velocity", "vector function", "arc length " ], "problem_v1": "You bicycle along a straight flat road with a safety light attached to one foot. Your bike moves at a speed of $25$ km/hr and your foot moves in a circle of radius $24$ cm centered $34$ cm above the ground, making one revolution per second.\n(a) Find parametric equations for $x$ and $y$ which describe the path traced out by the light, where $y$ is distance (in cm) above the ground and $x$ the horizontal distance (in cm) starting position of the center of the circle around which your foot moves. Assuming the light starts 34 cm above the ground, at the front of its rotation. $x(t)=$ [ANS], $y(t)=$ [ANS]. On a separate sheet of paper, sketch the path that your equations describe. (b) How fast (in revolutions/sec) would your foot have to be rotating if an observer standing at the side of the road sees the light moving backward? Rotate at [ANS] revolutions/second.", "answer_v1": [ "25*1000*100*t/3600+24*cos(2*pi*t)", "34-24*sin(2*pi*t)", "25*1000*100/(3600*24*2*pi)" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "You bicycle along a straight flat road with a safety light attached to one foot. Your bike moves at a speed of $10$ km/hr and your foot moves in a circle of radius $28$ cm centered $38$ cm above the ground, making one revolution per second.\n(a) Find parametric equations for $x$ and $y$ which describe the path traced out by the light, where $y$ is distance (in cm) above the ground and $x$ the horizontal distance (in cm) starting position of the center of the circle around which your foot moves. Assuming the light starts 38 cm above the ground, at the front of its rotation. $x(t)=$ [ANS], $y(t)=$ [ANS]. On a separate sheet of paper, sketch the path that your equations describe. (b) How fast (in revolutions/sec) would your foot have to be rotating if an observer standing at the side of the road sees the light moving backward? Rotate at [ANS] revolutions/second.", "answer_v2": [ "10*1000*100*t/3600+28*cos(2*pi*t)", "38-28*sin(2*pi*t)", "10*1000*100/(3600*28*2*pi)" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "You bicycle along a straight flat road with a safety light attached to one foot. Your bike moves at a speed of $15$ km/hr and your foot moves in a circle of radius $24$ cm centered $34$ cm above the ground, making one revolution per second.\n(a) Find parametric equations for $x$ and $y$ which describe the path traced out by the light, where $y$ is distance (in cm) above the ground and $x$ the horizontal distance (in cm) starting position of the center of the circle around which your foot moves. Assuming the light starts 34 cm above the ground, at the front of its rotation. $x(t)=$ [ANS], $y(t)=$ [ANS]. On a separate sheet of paper, sketch the path that your equations describe. (b) How fast (in revolutions/sec) would your foot have to be rotating if an observer standing at the side of the road sees the light moving backward? Rotate at [ANS] revolutions/second.", "answer_v3": [ "15*1000*100*t/3600+24*cos(2*pi*t)", "34-24*sin(2*pi*t)", "15*1000*100/(3600*24*2*pi)" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0087", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "3", "keywords": [ "acceleration", "velocity", "vector function", "arc length " ], "problem_v1": "A particle moves on a circle of radius 8 cm, centered at the origin, in the $xy$-plane ($x$ and $y$ measured in centimeters). At time $t=0$ it starts at the point $(0,8)$ and moves counterclockwise as $t$ increases, going once around the circle in 9 seconds at constant speed.\n(a) Write a parameterization for the particle's motion. $x(t)=$ [ANS], $y(t)=$ [ANS]\n(b) What is the particle's speed? speed=[ANS] cm/s", "answer_v1": [ "-8*sin(2*pi*t/9)", "8*cos(2*pi*t/9)", "5.58505" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A particle moves on a circle of radius 3 cm, centered at the origin, in the $xy$-plane ($x$ and $y$ measured in centimeters). At time $t=0$ it starts at the point $(0,3)$ and moves counterclockwise as $t$ increases, going once around the circle in 12 seconds at constant speed.\n(a) Write a parameterization for the particle's motion. $x(t)=$ [ANS], $y(t)=$ [ANS]\n(b) What is the particle's speed? speed=[ANS] cm/s", "answer_v2": [ "-3*sin(2*pi*t/12)", "3*cos(2*pi*t/12)", "1.5708" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A particle moves on a circle of radius 5 cm, centered at the origin, in the $xy$-plane ($x$ and $y$ measured in centimeters). At time $t=0$ it starts at the point $(0,5)$ and moves counterclockwise as $t$ increases, going once around the circle in 9 seconds at constant speed.\n(a) Write a parameterization for the particle's motion. $x(t)=$ [ANS], $y(t)=$ [ANS]\n(b) What is the particle's speed? speed=[ANS] cm/s", "answer_v3": [ "-5*sin(2*pi*t/9)", "5*cos(2*pi*t/9)", "3.49066" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0088", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "5", "keywords": [ "acceleration", "velocity", "vector function", "arc length " ], "problem_v1": "The table below gives $x$ and $y$ coordinates of a particle in the plane at time $t$.\n$\\begin{array}{cccccccccc}\\hline t & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\\\ \\hline x & 5 & 8 & 10 & 11 & 10 & 7 & 6 & 7 & 9 \\\\ \\hline y & 4 & 3 & 4 & 6 & 9 & 11 & 12 & 11 & 10 \\\\ \\hline \\end{array}$\nAssuming that the particle moves smoothly and that the points given show all the major features of the motion, estimate the following quantities:\n(a) The velocity vector and speed at time $t=3.5$. $\\vec v(t)=$ [ANS]\nspeed=[ANS]\n(b) Any times when the particle is moving parallel to the $y$-axis. times=[ANS]\n(Enter the times as a comma separated list, or enter none if there are none.) Any times when the particle has come to a stop. times=[ANS]\n(Enter the times as a comma separated list, or enter none if there are none.)", "answer_v1": [ "(3,-2)", "sqrt((9-6)^2+(10-12)^2)", "(1.5, 3)", "3" ], "answer_type_v1": [ "OL", "NV", "UOL", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The table below gives $x$ and $y$ coordinates of a particle in the plane at time $t$.\n$\\begin{array}{cccccccccc}\\hline t & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\\\ \\hline x & 2 & 1 & 2 & 4 & 7 & 9 & 10 & 9 & 8 \\\\ \\hline y & 5 & 8 & 10 & 11 & 10 & 7 & 6 & 7 & 9 \\\\ \\hline \\end{array}$\nAssuming that the particle moves smoothly and that the points given show all the major features of the motion, estimate the following quantities:\n(a) The velocity vector and speed at time $t=1$. $\\vec v(t)=$ [ANS]\nspeed=[ANS]\n(b) Any times when the particle is moving parallel to the $y$-axis. times=[ANS]\n(Enter the times as a comma separated list, or enter none if there are none.) Any times when the particle has come to a stop. times=[ANS]\n(Enter the times as a comma separated list, or enter none if there are none.)", "answer_v2": [ "(3,3)", "sqrt((4-1)^2+(11-8)^2)", "(0.5, 3)", "3" ], "answer_type_v2": [ "OL", "NV", "UOL", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The table below gives $x$ and $y$ coordinates of a particle in the plane at time $t$.\n$\\begin{array}{cccccccccc}\\hline t & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 & 3.5 & 4 \\\\ \\hline x & 3 & 6 & 8 & 9 & 8 & 5 & 4 & 5 & 7 \\\\ \\hline y & 4 & 3 & 4 & 6 & 9 & 11 & 12 & 11 & 10 \\\\ \\hline \\end{array}$\nAssuming that the particle moves smoothly and that the points given show all the major features of the motion, estimate the following quantities:\n(a) The velocity vector and speed at time $t=1$. $\\vec v(t)=$ [ANS]\nspeed=[ANS]\n(b) Any times when the particle is moving parallel to the $y$-axis. times=[ANS]\n(Enter the times as a comma separated list, or enter none if there are none.) Any times when the particle has come to a stop. times=[ANS]\n(Enter the times as a comma separated list, or enter none if there are none.)", "answer_v3": [ "(3,3)", "sqrt((9-6)^2+(6-3)^2)", "(1.5, 3)", "3" ], "answer_type_v3": [ "OL", "NV", "UOL", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0089", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "3", "keywords": [ "calculus", "vector", "curvature", "trajectory", "acceleration", "vector' 'acceleration' 'trajectory' 'velocity", "trajectory' 'vector" ], "problem_v1": "A particle in space undergoes a constant nonzero acceleration. Depending on the circumstances, the particle's trajectory can be held by the following curves. Enter T or F depending on whether the statement is true or false. (You must enter T or F--True and False will not work.) [ANS] 1. a parabola [ANS] 2. a hyperbola [ANS] 3. a straight line [ANS] 4. an ellipse [ANS] 5. a circle", "answer_v1": [ "T", "F", "T", "F", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "A particle in space undergoes a constant nonzero acceleration. Depending on the circumstances, the particle's trajectory can be held by the following curves. Enter T or F depending on whether the statement is true or false. (You must enter T or F--True and False will not work.) [ANS] 1. an ellipse [ANS] 2. a hyperbola [ANS] 3. a circle [ANS] 4. a parabola [ANS] 5. a straight line", "answer_v2": [ "F", "F", "F", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "A particle in space undergoes a constant nonzero acceleration. Depending on the circumstances, the particle's trajectory can be held by the following curves. Enter T or F depending on whether the statement is true or false. (You must enter T or F--True and False will not work.) [ANS] 1. a straight line [ANS] 2. a hyperbola [ANS] 3. a circle [ANS] 4. an ellipse [ANS] 5. a parabola", "answer_v3": [ "T", "F", "F", "F", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0090", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "5", "keywords": [ "Vector", "Parametric", "Geometry" ], "problem_v1": "A stone is thrown from a rooftop at time $t=0$ seconds. Its position at time $t$ is given by $\\vec{r}(t)=8t\\boldsymbol{i}-3t\\boldsymbol{j}+\\left(24.5-4.9t^{2}\\right)\\boldsymbol{k}.$ The origin is at the base of the building, which is standing on flat ground. Distance is measured in meters. The vector $\\boldsymbol{i}$ points east, $\\boldsymbol{j}$ points north, and $\\boldsymbol{k}$ points up.\n(a) How high is the rooftop? [ANS] meters.\n(b) When does the stone hit the ground? [ANS] seconds.\n(c) Where does the stone hit the ground? [ANS] (in meters)\n(d) How fast is the stone moving when it hits the ground? [ANS] (in meters per second)", "answer_v1": [ "24.5", "sqrt(5)", "17.8885i-6.7082j", "23.5202" ], "answer_type_v1": [ "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "A stone is thrown from a rooftop at time $t=0$ seconds. Its position at time $t$ is given by $\\vec{r}(t)=12t\\boldsymbol{i}-5t\\boldsymbol{j}+\\left(9.8-4.9t^{2}\\right)\\boldsymbol{k}.$ The origin is at the base of the building, which is standing on flat ground. Distance is measured in meters. The vector $\\boldsymbol{i}$ points east, $\\boldsymbol{j}$ points north, and $\\boldsymbol{k}$ points up.\n(a) How high is the rooftop? [ANS] meters.\n(b) When does the stone hit the ground? [ANS] seconds.\n(c) Where does the stone hit the ground? [ANS] (in meters)\n(d) How fast is the stone moving when it hits the ground? [ANS] (in meters per second)", "answer_v2": [ "9.8", "sqrt(2)", "16.9706i-7.07107j", "19.0021" ], "answer_type_v2": [ "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "A stone is thrown from a rooftop at time $t=0$ seconds. Its position at time $t$ is given by $\\vec{r}(t)=8t\\boldsymbol{i}-4t\\boldsymbol{j}+\\left(14.7-4.9t^{2}\\right)\\boldsymbol{k}.$ The origin is at the base of the building, which is standing on flat ground. Distance is measured in meters. The vector $\\boldsymbol{i}$ points east, $\\boldsymbol{j}$ points north, and $\\boldsymbol{k}$ points up.\n(a) How high is the rooftop? [ANS] meters.\n(b) When does the stone hit the ground? [ANS] seconds.\n(c) Where does the stone hit the ground? [ANS] (in meters)\n(d) How fast is the stone moving when it hits the ground? [ANS] (in meters per second)", "answer_v3": [ "14.7", "sqrt(3)", "13.8564i-6.9282j", "19.1865" ], "answer_type_v3": [ "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0091", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "3", "keywords": [ "Vector", "Parametric", "Geometry" ], "problem_v1": "Suppose the displacement of a particle in motion at time $t$ is given by the parametric equations $x(t)=\\left(3t-1\\right)^{3}, \\quad y(t)=6, \\quad z(t)=243t^{4}-108t^{3}.$\n(a) Find the speed of the particle when $t=1$. Speed=[ANS]\n(b) Find $t$ when the particle is stationary. $t$=[ANS]", "answer_v1": [ "648.999", "0.333333" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose the displacement of a particle in motion at time $t$ is given by the parametric equations $x(t)=\\left(2t-1\\right)^{2}, \\quad y(t)=9, \\quad z(t)=16t^{3}-12t^{2}.$\n(a) Find the speed of the particle when $t=3$. Speed=[ANS]\n(b) Find $t$ when the particle is stationary. $t$=[ANS]", "answer_v2": [ "360.555", "0.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose the displacement of a particle in motion at time $t$ is given by the parametric equations $x(t)=\\left(2t-1\\right)^{2}, \\quad y(t)=6, \\quad z(t)=48t^{4}-32t^{3}.$\n(a) Find the speed of the particle when $t=1$. Speed=[ANS]\n(b) Find $t$ when the particle is stationary. $t$=[ANS]", "answer_v3": [ "96.0833", "0.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0092", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "Vector", "Parametric", "Geometry" ], "problem_v1": "Suppose the position of a particle in motion at time $t$ is given by the vector parametric equation $\\vec{r}(t)=\\langle 8t, 6t^{3}-6t^{2} \\rangle$.\n(a) Find the velocity of the particle at time $t$. $\\vec{v}(t)$=[ANS]\n(b) Find the acceleration of the particle at time $t$. $\\vec{a}(t)$=[ANS]", "answer_v1": [ "(8,18*t^2-12*t)", "(0,36*t-12)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Suppose the position of a particle in motion at time $t$ is given by the vector parametric equation $\\vec{r}(t)=\\langle 2t, 9t^{3}-3t^{2} \\rangle$.\n(a) Find the velocity of the particle at time $t$. $\\vec{v}(t)$=[ANS]\n(b) Find the acceleration of the particle at time $t$. $\\vec{a}(t)$=[ANS]", "answer_v2": [ "(2,27*t^2-6*t)", "(0,54*t-6)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Suppose the position of a particle in motion at time $t$ is given by the vector parametric equation $\\vec{r}(t)=\\langle 4t, 6t^{3}-4t^{2} \\rangle$.\n(a) Find the velocity of the particle at time $t$. $\\vec{v}(t)$=[ANS]\n(b) Find the acceleration of the particle at time $t$. $\\vec{a}(t)$=[ANS]", "answer_v3": [ "(4,18*t^2-8*t)", "(0,36*t-8)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0093", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "2", "keywords": [ "calculus", "kepler" ], "problem_v1": "A satellite in a circular orbit 800 miles above the surface of the Earth. What is the period of the orbit?\nYou may use the following constants:\nRadius of the Earth: 4000 miles Gravitational Constant: $6.67 \\times 10^{-11} \\; \\mathrm{m}^3/(\\mathrm{kg} \\cdot \\mathrm{s}^2)$ Mass of earth: $5.98 \\times 10^{24}\\; \\mathrm{kg}$ Number of Meters in a mile: 1609\nPeriod=[ANS] seconds", "answer_v1": [ "6752.46406443037" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A satellite in a circular orbit 200 miles above the surface of the Earth. What is the period of the orbit?\nYou may use the following constants:\nRadius of the Earth: 4000 miles Gravitational Constant: $6.67 \\times 10^{-11} \\; \\mathrm{m}^3/(\\mathrm{kg} \\cdot \\mathrm{s}^2)$ Mass of earth: $5.98 \\times 10^{24}\\; \\mathrm{kg}$ Number of Meters in a mile: 1609\nPeriod=[ANS] seconds", "answer_v2": [ "5526.80779122533" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A satellite in a circular orbit 400 miles above the surface of the Earth. What is the period of the orbit?\nYou may use the following constants:\nRadius of the Earth: 4000 miles Gravitational Constant: $6.67 \\times 10^{-11} \\; \\mathrm{m}^3/(\\mathrm{kg} \\cdot \\mathrm{s}^2)$ Mass of earth: $5.98 \\times 10^{24}\\; \\mathrm{kg}$ Number of Meters in a mile: 1609\nPeriod=[ANS] seconds", "answer_v3": [ "5926.24279448317" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0094", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "4", "keywords": [ "calculus", "motion", "centripetal" ], "problem_v1": "A body of mass 8 kg moves in a (counterclockwise) circular path of radius 7 meters, making one revolution every 11 seconds. You may assume the circle is in the xy-plane, and so you may ignore the third component. A. Compute the centripetal force acting on the body. $($ [ANS], [ANS] $)$ B. Compute the magnitude of that force. [ANS]", "answer_v1": [ "- 8 * 3.99839065002337^2 / 7 * cos(3.99839065002337 * t / 7)", "- 8 * 3.99839065002337^2 / 7 * sin(3.99839065002337 * t / 7)", "18.2710031887935" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A body of mass 2 kg moves in a (counterclockwise) circular path of radius 10 meters, making one revolution every 6 seconds. You may assume the circle is in the xy-plane, and so you may ignore the third component. A. Compute the centripetal force acting on the body. $($ [ANS], [ANS] $)$ B. Compute the magnitude of that force. [ANS]", "answer_v2": [ "- 2 * 10.471975511966^2 / 10 * cos(10.471975511966 * t / 10)", "- 2 * 10.471975511966^2 / 10 * sin(10.471975511966 * t / 10)", "21.932454224643" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A body of mass 4 kg moves in a (counterclockwise) circular path of radius 7 meters, making one revolution every 8 seconds. You may assume the circle is in the xy-plane, and so you may ignore the third component. A. Compute the centripetal force acting on the body. $($ [ANS], [ANS] $)$ B. Compute the magnitude of that force. [ANS]", "answer_v3": [ "- 4 * 5.49778714378214^2 / 7 * cos(5.49778714378214 * t / 7)", "- 4 * 5.49778714378214^2 / 7 * sin(5.49778714378214 * t / 7)", "17.2718077019064" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0095", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "3", "keywords": [ "calculus", "vector", "derivative" ], "problem_v1": "Consider the vector function $\\mathbf r(t)=\\langle t, t^{8}, t^{6}\\rangle$ Compute $\\mathbf r'(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$ $\\mathbf T(1)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$ $\\mathbf r''(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$ $\\mathbf r'(t)\\times \\mathbf r''(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v1": [ "1", "8* t**(8 - 1)", "6 * t**(6 - 1)", "0.0995037190209989", "0.796029752167991", "0.597022314125993", "0", "8*(8-1)*t**(8-2)", "6*(6-1)*t**(6-2)", "8*6*(6-8)*t**(8+6 -3)", "-6*(6-1)*t**(6-2)", "8*(8-1)*t**(8-2)" ], "answer_type_v1": [ "NV", "EX", "EX", "NV", "NV", "NV", "NV", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the vector function $\\mathbf r(t)=\\langle t, t^{2}, t^{9}\\rangle$ Compute $\\mathbf r'(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$ $\\mathbf T(1)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$ $\\mathbf r''(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$ $\\mathbf r'(t)\\times \\mathbf r''(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v2": [ "1", "2* t**(2 - 1)", "9 * t**(9 - 1)", "0.107832773203438", "0.215665546406877", "0.970494958830946", "0", "2*(2-1)*t**(2-2)", "9*(9-1)*t**(9-2)", "2*9*(9-2)*t**(2+9 -3)", "-9*(9-1)*t**(9-2)", "2*(2-1)*t**(2-2)" ], "answer_type_v2": [ "NV", "EX", "EX", "NV", "NV", "NV", "NV", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the vector function $\\mathbf r(t)=\\langle t, t^{4}, t^{6}\\rangle$ Compute $\\mathbf r'(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$ $\\mathbf T(1)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$ $\\mathbf r''(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$ $\\mathbf r'(t)\\times \\mathbf r''(t)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$", "answer_v3": [ "1", "4* t**(4 - 1)", "6 * t**(6 - 1)", "0.137360563948689", "0.549442255794756", "0.824163383692134", "0", "4*(4-1)*t**(4-2)", "6*(6-1)*t**(6-2)", "4*6*(6-4)*t**(4+6 -3)", "-6*(6-1)*t**(6-2)", "4*(4-1)*t**(4-2)" ], "answer_type_v3": [ "NV", "EX", "EX", "NV", "NV", "NV", "NV", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0096", "subject": "Calculus_-_multivariable", "topic": "Calculus of vector valued functions", "subtopic": "Frames, motions, and other applications", "level": "3", "keywords": [ "vectors", "acceleration" ], "problem_v1": "If $\\mathbf{r} (t)=4 t \\mathbf{i}+5 t^2 \\mathbf{j}+2 t \\mathbf{k}$, compute the tangential and normal components of the acceleration vector. Tangential component $\\; a_T (t)=$ [ANS]\nNormal component $\\; a_N (t)=$ [ANS]", "answer_v1": [ "4*5^2*t/sqrt((4)^2+4*(5*t)^2+(2)^2)", "2*5*sqrt((4)^2+(2)^2)/sqrt((4)^2+4*(5*t)^2+(2)^2)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "If $\\mathbf{r} (t)=-7 t \\mathbf{i}+8 t^2 \\mathbf{j}-6 t \\mathbf{k}$, compute the tangential and normal components of the acceleration vector. Tangential component $\\; a_T (t)=$ [ANS]\nNormal component $\\; a_N (t)=$ [ANS]", "answer_v2": [ "4*8^2*t/sqrt((-7)^2+4*(8*t)^2+(-6)^2)", "2*8*sqrt((-7)^2+(-6)^2)/sqrt((-7)^2+4*(8*t)^2+(-6)^2)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "If $\\mathbf{r} (t)=-3 t \\mathbf{i}+5 t^2 \\mathbf{j}-4 t \\mathbf{k}$, compute the tangential and normal components of the acceleration vector. Tangential component $\\; a_T (t)=$ [ANS]\nNormal component $\\; a_N (t)=$ [ANS]", "answer_v3": [ "4*5^2*t/sqrt((-3)^2+4*(5*t)^2+(-4)^2)", "2*5*sqrt((-3)^2+(-4)^2)/sqrt((-3)^2+4*(5*t)^2+(-4)^2)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0097", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Evaluate the function at the specified points. $h(x,y,z)=xy^{-4}z, \\left(3,4,-4\\right), \\left(-2,4,-4\\right)$ At $\\left(3,4,-4\\right)$: [ANS]\nAt $\\left(-2,4,-4\\right)$: [ANS]", "answer_v1": [ "-0.046875", "0.03125" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Evaluate the function at the specified points. $h(x,y,z)=x^{-2}yz, \\left(3,3,-3\\right), \\left(5,2,-3\\right)$ At $\\left(3,3,-3\\right)$: [ANS]\nAt $\\left(5,2,-3\\right)$: [ANS]", "answer_v2": [ "-1", "-0.24" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Evaluate the function at the specified points. $h(x,y,z)=xy^{-2}z, \\left(-4,3,6\\right), \\left(3,3,-4\\right)$ At $\\left(-4,3,6\\right)$: [ANS]\nAt $\\left(3,3,-4\\right)$: [ANS]", "answer_v3": [ "-2.66667", "-1.33333" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0098", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Evaluate the function at the specified points. $f(x,y)=y+xy^{5}, \\left(1,2\\right), \\left(-2,-2\\right), \\left(1,1\\right)$ At $\\left(1,2\\right)$: [ANS]\nAt $\\left(-2,-2\\right)$: [ANS]\nAt $\\left(1,1\\right)$: [ANS]", "answer_v1": [ "34", "62", "2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Evaluate the function at the specified points. $f(x,y)=y+xy^{2}, \\left(-4,-2\\right), \\left(5,-2\\right), \\left(-3,-2\\right)$ At $\\left(-4,-2\\right)$: [ANS]\nAt $\\left(5,-2\\right)$: [ANS]\nAt $\\left(-3,-2\\right)$: [ANS]", "answer_v2": [ "-18", "18", "-14" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Evaluate the function at the specified points. $f(x,y)=y+xy^{3}, \\left(-2,1\\right), \\left(-3,-2\\right), \\left(3,5\\right)$ At $\\left(-2,1\\right)$: [ANS]\nAt $\\left(-3,-2\\right)$: [ANS]\nAt $\\left(3,5\\right)$: [ANS]", "answer_v3": [ "-1", "22", "380" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0099", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "1", "keywords": [ "Function", "Multi-Variable", "Multivariate", "Multivariable" ], "problem_v1": "The formula $y=x^2-2x+3$ defines a function $y=f(x)$. Similarly, the equation $ x= \\frac{s\\!\\left(1+t\\right)}{1-u^{2} }$ defines a function [ANS]=$f($ [ANS] $)$.", "answer_v1": [ "x", "(s, t, u)" ], "answer_type_v1": [ "EX", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "The formula $y=x^2-2x+3$ defines a function $y=f(x)$. Similarly, the equation $ z=\\cos\\!\\left(xy\\right)$ defines a function [ANS]=$f($ [ANS] $)$.", "answer_v2": [ "z", "(x, y)" ], "answer_type_v2": [ "EX", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "The formula $y=x^2-2x+3$ defines a function $y=f(x)$. Similarly, the equation $ s=3u\\!\\left(t^{2}-v\\right)$ defines a function [ANS]=$f($ [ANS] $)$.", "answer_v3": [ "s", "(t, u, v)" ], "answer_type_v3": [ "EX", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0100", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "1", "keywords": [ "Function", "Multi-Variable", "Multivariate", "Multivariable", "Coordinate Function" ], "problem_v1": "The function $f(s,t)=\\left(\\sin\\!\\left(t\\right)\\cos\\!\\left(s\\right),2\\tan\\!\\left(t\\right)\\right)$ has how many coordinate functions? [ANS]\nEach coordinate function is of the form $f_i\\colon {\\bf R}^n\\to{\\bf R}^m$ where $n$=[ANS] and $m$=[ANS]. The second coordinate function is: $f_{2}($ [ANS] $)$=[ANS].", "answer_v1": [ "2", "2", "1", "(s, t)", "2*tan(t)" ], "answer_type_v1": [ "NV", "NV", "NV", "UOL", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "The function $f(u,v)=\\left(u^{2}+5,u-v^{3}\\right)$ has how many coordinate functions? [ANS]\nEach coordinate function is of the form $f_i\\colon {\\bf R}^n\\to{\\bf R}^m$ where $n$=[ANS] and $m$=[ANS]. The second coordinate function is: $f_{2}($ [ANS] $)$=[ANS].", "answer_v2": [ "2", "2", "1", "(u, v)", "u-v^3" ], "answer_type_v2": [ "NV", "NV", "NV", "UOL", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "The function $f(t)=\\left(\\cos\\!\\left(t\\right),\\sin\\!\\left(t\\right)\\right)$ has how many coordinate functions? [ANS]\nEach coordinate function is of the form $f_i\\colon {\\bf R}^n\\to{\\bf R}^m$ where $n$=[ANS] and $m$=[ANS]. The second coordinate function is: $f_{2}($ [ANS] $)$=[ANS].", "answer_v3": [ "2", "1", "1", "t", "sin(t)" ], "answer_type_v3": [ "NV", "NV", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0101", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "1", "keywords": [ "Function", "Multi-Variable", "Multivariate", "Multivariable" ], "problem_v1": "Suppose $f(t)=\\left(3t-t^{2},1-t\\right)$. Compute the following values:\n$\\begin{array}{ccc}\\hline f(2) &=& [ANS] \\\\ \\hline f(3) &=& [ANS] \\\\ \\hline f(1-s) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "(2,-1)", "(0,-2)", "(3*(1-s)-(1-s)^2,1-(1-s))" ], "answer_type_v1": [ "OL", "OL", "OL" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $f(t)=\\left(-\\left(5t+t^{2}\\right),5-t\\right)$. Compute the following values:\n$\\begin{array}{ccc}\\hline f(-5) &=& [ANS] \\\\ \\hline f(-2) &=& [ANS] \\\\ \\hline f(1-s) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "(0,10)", "(6,7)", "(-[5*(1-s)+(1-s)^2],5-(1-s))" ], "answer_type_v2": [ "OL", "OL", "OL" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $f(t)=\\left(-\\left(2t+t^{2}\\right),1-t\\right)$. Compute the following values:\n$\\begin{array}{ccc}\\hline f(-3) &=& [ANS] \\\\ \\hline f(1) &=& [ANS] \\\\ \\hline f(1-s) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "(-3,4)", "(-3,0)", "(-[2*(1-s)+(1-s)^2],1-(1-s))" ], "answer_type_v3": [ "OL", "OL", "OL" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0102", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "1", "keywords": [ "Function", "Multi-Variable", "Multivariate", "Multivariable" ], "problem_v1": "Suppose $f(x,y)=xy+4$. Compute the following values:\n$\\begin{array}{ccc}\\hline f(x+4y,x-4y) &=& [ANS] \\\\ \\hline f(xy,5x^2y^3) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "(x+4*y)*(x-4*y)+4", "x*y*5*x^2*y^3+4" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Suppose $f(x,y)=xy-7$. Compute the following values:\n$\\begin{array}{ccc}\\hline f(x+6y,x-6y) &=& [ANS] \\\\ \\hline f(xy,2x^2y^3) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "(x+6*y)*(x-6*y)-7", "x*y*2*x^2*y^3-7" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Suppose $f(x,y)=xy-3$. Compute the following values:\n$\\begin{array}{ccc}\\hline f(x+5y,x-5y) &=& [ANS] \\\\ \\hline f(xy,3x^2y^3) &=& [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "(x+5*y)*(x-5*y)-3", "x*y*3*x^2*y^3-3" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0103", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "1", "keywords": [ "Function", "Multi-Variable", "Multivariate", "Multivariable" ], "problem_v1": "The function f(s,t)=\\left(\\cos\\!\\left(s\\right),\\sin\\!\\left(s\\right),\\cos\\!\\left(t\\right),\\sin\\!\\left(t\\right)\\right) has [ANS] inputs and [ANS] outputs. Thus it is of the form f\\colon {\\bf R}^n \\to {\\bf R}^m where $n$=[ANS] and $m$=[ANS].", "answer_v1": [ "2", "4", "2", "4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The function f(x,y)=x^{2}-y^{3} has [ANS] inputs and [ANS] outputs. Thus it is of the form f\\colon {\\bf R}^n \\to {\\bf R}^m where $n$=[ANS] and $m$=[ANS].", "answer_v2": [ "2", "1", "2", "1" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The function f(u,v)=\\left(u^{2}-v^{2},2uv\\right) has [ANS] inputs and [ANS] outputs. Thus it is of the form f\\colon {\\bf R}^n \\to {\\bf R}^m where $n$=[ANS] and $m$=[ANS].", "answer_v3": [ "2", "2", "2", "2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0104", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "1", "keywords": [ "Function", "Multi-Variable", "Multivariate", "Multivariable" ], "problem_v1": "The function f(s,t,u)=\\left(s+u,4t,st+\\ln\\!\\left(u\\right),t-u^{2}\\right) has [ANS] input variables and a [ANS] dimensional output. Thus it is of the form f \\colon \\mathcal{U}\\subseteq\\mathbb{R}^n \\to \\mathbb{R}^m where $\\mathcal{U}$ is a nonempty open subset on which $f$ is defined, and $n$=[ANS] and $m$=[ANS].", "answer_v1": [ "3", "4", "3", "4" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The function f(u,v)=\\left(u^{2}-v^{2},2uv\\right) has [ANS] input variables and a [ANS] dimensional output. Thus it is of the form f \\colon \\mathcal{U}\\subseteq\\mathbb{R}^n \\to \\mathbb{R}^m where $\\mathcal{U}$ is a nonempty open subset on which $f$ is defined, and $n$=[ANS] and $m$=[ANS].", "answer_v2": [ "2", "2", "2", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The function f(x,y,z)=x^{2}+y^{2}+z^{2} has [ANS] input variables and a [ANS] dimensional output. Thus it is of the form f \\colon \\mathcal{U}\\subseteq\\mathbb{R}^n \\to \\mathbb{R}^m where $\\mathcal{U}$ is a nonempty open subset on which $f$ is defined, and $n$=[ANS] and $m$=[ANS].", "answer_v3": [ "3", "1", "3", "1" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0105", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "1", "keywords": [ "Function", "Multi-Variable", "Multivariate", "Multivariable" ], "problem_v1": "Consider the system of equations below: \\begin{array}{rcl}u\\kern-6pt &=& \\kern-6pt\\sin\\!\\left(t\\right)\\cos\\!\\left(s\\right)\\\\v\\kern-6pt &=& \\kern-6pt2\\tan\\!\\left(t\\right).\\end{array} These equations represent a function f\\colon {\\bf R}^n\\to{\\bf R}^m where $n$=[ANS] and $m$=[ANS].\n$\\begin{array}{cc}\\hline The input variables are & [ANS]. \\\\ \\hline The output variables are & [ANS]. \\\\ \\hline \\end{array}$\n(List the variable names separated by commas.) (List the variable names separated by commas.)", "answer_v1": [ "2", "2", "(s, t)", "(u, v)" ], "answer_type_v1": [ "NV", "NV", "UOL", "UOL" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the system of equations below: \\begin{array}{rcl}x\\kern-6pt &=& \\kern-6ptu^{2}+5\\\\y\\kern-6pt &=& \\kern-6ptu-v^{3}.\\end{array} These equations represent a function f\\colon {\\bf R}^n\\to{\\bf R}^m where $n$=[ANS] and $m$=[ANS].\n$\\begin{array}{cc}\\hline The input variables are & [ANS]. \\\\ \\hline The output variables are & [ANS]. \\\\ \\hline \\end{array}$\n(List the variable names separated by commas.) (List the variable names separated by commas.)", "answer_v2": [ "2", "2", "(u, v)", "(x, y)" ], "answer_type_v2": [ "NV", "NV", "UOL", "UOL" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the system of equations below: \\begin{array}{rcl}x\\kern-6pt &=& \\kern-6pt\\cos\\!\\left(t\\right)\\\\y\\kern-6pt &=& \\kern-6pt\\sin\\!\\left(t\\right).\\end{array} These equations represent a function f\\colon {\\bf R}^n\\to{\\bf R}^m where $n$=[ANS] and $m$=[ANS].\n$\\begin{array}{cc}\\hline The input variables are & [ANS]. \\\\ \\hline The output variables are & [ANS]. \\\\ \\hline \\end{array}$\n(List the variable names separated by commas.) (List the variable names separated by commas.)", "answer_v3": [ "1", "2", "t", "(x, y)" ], "answer_type_v3": [ "NV", "NV", "EX", "UOL" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0106", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "2", "keywords": [ "multivariable", "functions" ], "problem_v1": "The temperature adjusted for wind-chill, $w$, is a temperature which tells you how cold it feels, as a result of the combination of wind and temperature [see]. See the table below, which gives temperature adjusted for wind-chill, $w$, as a function of temperature $T$ and wind speed $s$.\n$\\begin{array}{ccccccccc}\\hline & T=35 & T=30 & T=25 & T=20 & T=15 & T=10 & T=5 & T=0 \\\\ \\hline s=5 & 31 & 25 & 19 & 13 & 7 & 1 &-5 &-11 \\\\ \\hline s=10 & 27 & 21 & 15 & 9 & 3 &-4 &-10 &-16 \\\\ \\hline s=15 & 25 & 19 & 13 & 6 & 0 &-7 &-13 &-19 \\\\ \\hline s=20 & 24 & 17 & 11 & 4 &-2 &-9 &-15 &-22 \\\\ \\hline s=25 & 23 & 16 & 9 & 3 &-4 &-11 &-17 &-24 \\\\ \\hline \\end{array}$\nUse this table to make tables of the temperature adjusted for wind-chill ($w$) as a function of temperature for wind speeds 15 and 25 mph: $s=15$:\n$\\begin{array}{ccccccccc}\\hline independent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline dependent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n$s=25$:\n$\\begin{array}{ccccccccc}\\hline independent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline dependent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "T", "35", "30", "25", "20", "15", "10", "5", "0", "w", "25", "19", "13", "6", "0", "-7", "-13", "-19", "T", "35", "30", "25", "20", "15", "10", "5", "0", "w", "23", "16", "9", "3", "-4", "-11", "-17", "-24" ], "answer_type_v1": [ "TF", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "TF", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "The temperature adjusted for wind-chill, $w$, is a temperature which tells you how cold it feels, as a result of the combination of wind and temperature [see]. See the table below, which gives temperature adjusted for wind-chill, $w$, as a function of temperature $T$ and wind speed $s$.\n$\\begin{array}{ccccccccc}\\hline & T=35 & T=30 & T=25 & T=20 & T=15 & T=10 & T=5 & T=0 \\\\ \\hline s=5 & 31 & 25 & 19 & 13 & 7 & 1 &-5 &-11 \\\\ \\hline s=10 & 27 & 21 & 15 & 9 & 3 &-4 &-10 &-16 \\\\ \\hline s=15 & 25 & 19 & 13 & 6 & 0 &-7 &-13 &-19 \\\\ \\hline s=20 & 24 & 17 & 11 & 4 &-2 &-9 &-15 &-22 \\\\ \\hline s=25 & 23 & 16 & 9 & 3 &-4 &-11 &-17 &-24 \\\\ \\hline \\end{array}$\nUse this table to make tables of the temperature adjusted for wind-chill ($w$) as a function of temperature for wind speeds 5 and 25 mph: $s=5$:\n$\\begin{array}{ccccccccc}\\hline independent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline dependent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n$s=25$:\n$\\begin{array}{ccccccccc}\\hline independent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline dependent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "T", "35", "30", "25", "20", "15", "10", "5", "0", "w", "31", "25", "19", "13", "7", "1", "-5", "-11", "T", "35", "30", "25", "20", "15", "10", "5", "0", "w", "23", "16", "9", "3", "-4", "-11", "-17", "-24" ], "answer_type_v2": [ "TF", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "TF", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "The temperature adjusted for wind-chill, $w$, is a temperature which tells you how cold it feels, as a result of the combination of wind and temperature [see]. See the table below, which gives temperature adjusted for wind-chill, $w$, as a function of temperature $T$ and wind speed $s$.\n$\\begin{array}{ccccccccc}\\hline & T=35 & T=30 & T=25 & T=20 & T=15 & T=10 & T=5 & T=0 \\\\ \\hline s=5 & 31 & 25 & 19 & 13 & 7 & 1 &-5 &-11 \\\\ \\hline s=10 & 27 & 21 & 15 & 9 & 3 &-4 &-10 &-16 \\\\ \\hline s=15 & 25 & 19 & 13 & 6 & 0 &-7 &-13 &-19 \\\\ \\hline s=20 & 24 & 17 & 11 & 4 &-2 &-9 &-15 &-22 \\\\ \\hline s=25 & 23 & 16 & 9 & 3 &-4 &-11 &-17 &-24 \\\\ \\hline \\end{array}$\nUse this table to make tables of the temperature adjusted for wind-chill ($w$) as a function of temperature for wind speeds 5 and 20 mph: $s=5$:\n$\\begin{array}{ccccccccc}\\hline independent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline dependent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$\n$s=20$:\n$\\begin{array}{ccccccccc}\\hline independent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline dependent variable=[ANS]=& [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "T", "35", "30", "25", "20", "15", "10", "5", "0", "w", "31", "25", "19", "13", "7", "1", "-5", "-11", "T", "35", "30", "25", "20", "15", "10", "5", "0", "w", "24", "17", "11", "4", "-2", "-9", "-15", "-22" ], "answer_type_v3": [ "TF", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "TF", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0107", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "2", "keywords": [ "linear", "functions", "multivariable", "plane" ], "problem_v1": "The table below gives a partial table of values for a linear function. Fill in the blanks.\n$\\begin{array}{cccc}\\hline x\\backslash y &-1.0 & 0.0 & 1.0 \\\\ \\hline 2.0 & 10 & [ANS] & [ANS] \\\\ \\hline 3.0 & [ANS] & 17 & 20 \\\\ \\hline \\end{array}$", "answer_v1": [ "13", "16", "14" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The table below gives a partial table of values for a linear function. Fill in the blanks.\n$\\begin{array}{cccc}\\hline x\\backslash y &-1.0 & 0.0 & 1.0 \\\\ \\hline 2.0 & 0 & [ANS] & [ANS] \\\\ \\hline 3.0 & [ANS] & 5 & 9 \\\\ \\hline \\end{array}$", "answer_v2": [ "4", "8", "1" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The table below gives a partial table of values for a linear function. Fill in the blanks.\n$\\begin{array}{cccc}\\hline x\\backslash y &-1.0 & 0.0 & 1.0 \\\\ \\hline 2.0 & 6 & [ANS] & [ANS] \\\\ \\hline 3.0 & [ANS] & 9 & 10 \\\\ \\hline \\end{array}$", "answer_v3": [ "7", "8", "8" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0108", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "2", "keywords": [ "linear", "functions", "multivariable", "plane" ], "problem_v1": "The table below gives the number of calories burned per minute for someone roller-blading, as a function of the person's weight in pounds and speed in miles per hour [from the August 28,1994, issue of Parade Magazine].\ncalories burned per minute\n$\\begin{array}{ccccc}\\hline weight \\backslashspeed & 8 & 9 & 10 & 11 \\\\ \\hline 120 & 4.2 & 5.8 & 7.4 & 8.9 \\\\ \\hline 140 & 5.1 & 6.7 & 8.3 & 9.9 \\\\ \\hline 160 & 6.1 & 7.7 & 9.2 & 10.8 \\\\ \\hline 180 & 7 & 8.6 & 10.2 & 11.7 \\\\ \\hline 200 & 7.9 & 9.5 & 11.1 & 12.6 \\\\ \\hline \\end{array}$\n(a) Suppose that a 160 lb person and a 200 person both go 12 miles, the first at 11 mph and the second at 10 mph. How many calories does the 160 lb person burn? [ANS]\nHow many calories does the 200 lb person burn? [ANS]\n(b) We might also be interested in the number of calories each person burns per pound of their weight. How many calories per pound does the 160 lb person burn? [ANS]\nHow many calories per pound does the 200 lb person burn? [ANS]", "answer_v1": [ "720*10.8/11", "720*11.1/10", "706.909/160", "799.2/200" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The table below gives the number of calories burned per minute for someone roller-blading, as a function of the person's weight in pounds and speed in miles per hour [from the August 28,1994, issue of Parade Magazine].\ncalories burned per minute\n$\\begin{array}{ccccc}\\hline weight \\backslashspeed & 8 & 9 & 10 & 11 \\\\ \\hline 120 & 4.2 & 5.8 & 7.4 & 8.9 \\\\ \\hline 140 & 5.1 & 6.7 & 8.3 & 9.9 \\\\ \\hline 160 & 6.1 & 7.7 & 9.2 & 10.8 \\\\ \\hline 180 & 7 & 8.6 & 10.2 & 11.7 \\\\ \\hline 200 & 7.9 & 9.5 & 11.1 & 12.6 \\\\ \\hline \\end{array}$\n(a) Suppose that a 180 lb person and a 200 person both go 6 miles, the first at 11 mph and the second at 9 mph. How many calories does the 180 lb person burn? [ANS]\nHow many calories does the 200 lb person burn? [ANS]\n(b) We might also be interested in the number of calories each person burns per pound of their weight. How many calories per pound does the 180 lb person burn? [ANS]\nHow many calories per pound does the 200 lb person burn? [ANS]", "answer_v2": [ "360*11.7/11", "360*9.5/9", "382.909/180", "380/200" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The table below gives the number of calories burned per minute for someone roller-blading, as a function of the person's weight in pounds and speed in miles per hour [from the August 28,1994, issue of Parade Magazine].\ncalories burned per minute\n$\\begin{array}{ccccc}\\hline weight \\backslashspeed & 8 & 9 & 10 & 11 \\\\ \\hline 120 & 4.2 & 5.8 & 7.4 & 8.9 \\\\ \\hline 140 & 5.1 & 6.7 & 8.3 & 9.9 \\\\ \\hline 160 & 6.1 & 7.7 & 9.2 & 10.8 \\\\ \\hline 180 & 7 & 8.6 & 10.2 & 11.7 \\\\ \\hline 200 & 7.9 & 9.5 & 11.1 & 12.6 \\\\ \\hline \\end{array}$\n(a) Suppose that a 160 lb person and a 180 person both go 8 miles, the first at 10 mph and the second at 9 mph. How many calories does the 160 lb person burn? [ANS]\nHow many calories does the 180 lb person burn? [ANS]\n(b) We might also be interested in the number of calories each person burns per pound of their weight. How many calories per pound does the 160 lb person burn? [ANS]\nHow many calories per pound does the 180 lb person burn? [ANS]", "answer_v3": [ "480*9.2/10", "480*8.6/9", "441.6/160", "458.667/180" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0109", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "2", "keywords": [ "linear", "functions", "multivariable", "plane" ], "problem_v1": "Find the equation for a linear function $f(x,y)$ with the values given in the table below:\n$\\begin{array}{ccccc}\\hline x\\backslash y &-1 & 0 & 1 & 2 \\\\ \\hline 0 & 0.5 & 1 & 1.5 & 2 \\\\ \\hline 1 & 2 & 2.5 & 3 & 3.5 \\\\ \\hline 2 & 3.5 & 4 & 4.5 & 5 \\\\ \\hline 3 & 5 & 5.5 & 6 & 6.5 \\\\ \\hline \\end{array}$\n$f(x,y)=$ [ANS]", "answer_v1": [ "1.5*x+0.5*y+1" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the equation for a linear function $f(x,y)$ with the values given in the table below:\n$\\begin{array}{ccccc}\\hline x\\backslash y &-1 & 0 & 1 & 2 \\\\ \\hline 0 &-6.5 &-4 &-1.5 & 1 \\\\ \\hline 1 &-9 &-6.5 &-4 &-1.5 \\\\ \\hline 2 &-11.5 &-9 &-6.5 &-4 \\\\ \\hline 3 &-14 &-11.5 &-9 &-6.5 \\\\ \\hline \\end{array}$\n$f(x,y)=$ [ANS]", "answer_v2": [ "-2.5*x+2.5*y+-4" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the equation for a linear function $f(x,y)$ with the values given in the table below:\n$\\begin{array}{ccccc}\\hline x\\backslash y &-1 & 0 & 1 & 2 \\\\ \\hline 0 &-2.5 &-2 &-1.5 &-1 \\\\ \\hline 1 &-3.5 &-3 &-2.5 &-2 \\\\ \\hline 2 &-4.5 &-4 &-3.5 &-3 \\\\ \\hline 3 &-5.5 &-5 &-4.5 &-4 \\\\ \\hline \\end{array}$\n$f(x,y)=$ [ANS]", "answer_v3": [ "-1*x+0.5*y+-2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0110", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "3", "keywords": [ "contour", "graphs", "multivariable", "functions" ], "problem_v1": "Values of $f(x,y)= \\frac{1}{2} \\!\\left(x+y-4\\right)\\!\\left(x+y-1\\right)+y$ are in the table below.\n(a) Find a pattern in the table. Make a conjecture and use it to complete the table without computation. Check by using the formula for $f$. (b) Using the formula, check that the pattern holds for all $x \\ge 1$ and $y\\ge 1$.\n$\\begin{array}{ccccccc}\\hline x\\backslash y & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline 1 & 0 & 1 & 3 & 6 & 10 & 15 \\\\ \\hline 2 & 0 & 2 & 5 & 9 & 14 & [ANS] \\\\ \\hline 3 & 1 & 4 & 8 & 13 & [ANS] & [ANS] \\\\ \\hline 4 & 3 & 7 & 12 & [ANS] & [ANS] & [ANS] \\\\ \\hline 5 & 6 & 11 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 6 & 10 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v1": [ "20", "19", "26", "18", "25", "33", "17", "24", "32", "41", "16", "23", "31", "40", "50" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Values of $f(x,y)= \\frac{1}{2} \\!\\left(x+y-5\\right)\\!\\left(x+y-4\\right)+y$ are in the table below.\n(a) Find a pattern in the table. Make a conjecture and use it to complete the table without computation. Check by using the formula for $f$. (b) Using the formula, check that the pattern holds for all $x \\ge 1$ and $y\\ge 1$.\n$\\begin{array}{ccccccc}\\hline x\\backslash y & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline 1 & 4 & 3 & 3 & 4 & 6 & 9 \\\\ \\hline 2 & 2 & 2 & 3 & 5 & 8 & [ANS] \\\\ \\hline 3 & 1 & 2 & 4 & 7 & [ANS] & [ANS] \\\\ \\hline 4 & 1 & 3 & 6 & [ANS] & [ANS] & [ANS] \\\\ \\hline 5 & 2 & 5 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 6 & 4 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v2": [ "12", "11", "16", "10", "15", "21", "9", "14", "20", "27", "8", "13", "19", "26", "34" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Values of $f(x,y)= \\frac{1}{2} \\!\\left(x+y-4\\right)\\!\\left(x+y-3\\right)+y$ are in the table below.\n(a) Find a pattern in the table. Make a conjecture and use it to complete the table without computation. Check by using the formula for $f$. (b) Using the formula, check that the pattern holds for all $x \\ge 1$ and $y\\ge 1$.\n$\\begin{array}{ccccccc}\\hline x\\backslash y & 1 & 2 & 3 & 4 & 5 & 6 \\\\ \\hline 1 & 2 & 2 & 3 & 5 & 8 & 12 \\\\ \\hline 2 & 1 & 2 & 4 & 7 & 11 & [ANS] \\\\ \\hline 3 & 1 & 3 & 6 & 10 & [ANS] & [ANS] \\\\ \\hline 4 & 2 & 5 & 9 & [ANS] & [ANS] & [ANS] \\\\ \\hline 5 & 4 & 8 & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline 6 & 7 & [ANS] & [ANS] & [ANS] & [ANS] & [ANS] \\\\ \\hline \\end{array}$", "answer_v3": [ "16", "15", "21", "14", "20", "27", "13", "19", "26", "34", "12", "18", "25", "33", "42" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0111", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "3", "keywords": [ "graph", "image" ], "problem_v1": "Let $f: \\mathbb{R} \\to \\mathbb{R}^2$ be defined by $f(t)=\\left(25t^{2}, 5t^{3}\\right)$.\n(a) The graph of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS]. The image of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS].\n(b) The graph of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, t \\in$ [ANS] $\\Big\\rbrace.$\n(c) The image of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, t \\in$ [ANS] $\\Big\\rbrace.$", "answer_v1": [ "3", "2", "(t,25*t^2,5*t^3)", "(-infinity,infinity)", "(25*t^2,5*t^3)", "(-infinity,infinity)" ], "answer_type_v1": [ "NV", "NV", "OL", "INT", "INT", "INT" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let $f: \\mathbb{R} \\to \\mathbb{R}^2$ be defined by $f(t)=\\left(4t^{2}, 2t^{3}\\right)$.\n(a) The graph of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS]. The image of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS].\n(b) The graph of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, t \\in$ [ANS] $\\Big\\rbrace.$\n(c) The image of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, t \\in$ [ANS] $\\Big\\rbrace.$", "answer_v2": [ "3", "2", "(t,4*t^2,2*t^3)", "(-infinity,infinity)", "(4*t^2,2*t^3)", "(-infinity,infinity)" ], "answer_type_v2": [ "NV", "NV", "OL", "INT", "INT", "INT" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let $f: \\mathbb{R} \\to \\mathbb{R}^2$ be defined by $f(t)=\\left(9t^{2}, 3t^{3}\\right)$.\n(a) The graph of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS]. The image of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS].\n(b) The graph of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, t \\in$ [ANS] $\\Big\\rbrace.$\n(c) The image of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, t \\in$ [ANS] $\\Big\\rbrace.$", "answer_v3": [ "3", "2", "(t,9*t^2,3*t^3)", "(-infinity,infinity)", "(9*t^2,3*t^3)", "(-infinity,infinity)" ], "answer_type_v3": [ "NV", "NV", "OL", "INT", "INT", "INT" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0112", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "3", "keywords": [ "graph", "image" ], "problem_v1": "Let $f: \\lbrack-8,8 \\rbrack \\to \\mathbb{R}$ be defined by $f(x)=\\sqrt{64-x^{2}}$ and set $y=f(x)$.\n(a) The graph of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS]. The image of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS].\n(b) The graph of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, x \\in$ [ANS] $\\Big\\rbrace.$\n(c) The image of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, y \\in$ [ANS] $\\Big\\rbrace.$", "answer_v1": [ "2", "1", "(x,1*sqrt(64-x^2))", "[-8,8]", "y", "[0,8]" ], "answer_type_v1": [ "NV", "NV", "OL", "INT", "EX", "INT" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Let $f: \\lbrack-2,2 \\rbrack \\to \\mathbb{R}$ be defined by $f(x)=\\sqrt{4-x^{2}}$ and set $y=f(x)$.\n(a) The graph of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS]. The image of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS].\n(b) The graph of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, x \\in$ [ANS] $\\Big\\rbrace.$\n(c) The image of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, y \\in$ [ANS] $\\Big\\rbrace.$", "answer_v2": [ "2", "1", "(x,1*sqrt(4-x^2))", "[-2,2]", "y", "[0,2]" ], "answer_type_v2": [ "NV", "NV", "OL", "INT", "EX", "INT" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Let $f: \\lbrack-4,4 \\rbrack \\to \\mathbb{R}$ be defined by $f(x)=\\sqrt{16-x^{2}}$ and set $y=f(x)$.\n(a) The graph of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS]. The image of $f$ is a subset of $\\mathbb{R}^k$ for $k=$ [ANS].\n(b) The graph of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, x \\in$ [ANS] $\\Big\\rbrace.$\n(c) The image of $f$ is the set $\\Big\\lbrace$ [ANS] $\\,\\Big|\\, y \\in$ [ANS] $\\Big\\rbrace.$", "answer_v3": [ "2", "1", "(x,1*sqrt(16-x^2))", "[-4,4]", "y", "[0,4]" ], "answer_type_v3": [ "NV", "NV", "OL", "INT", "EX", "INT" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0113", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "2", "keywords": [ "vector", "function", "boundary" ], "problem_v1": "Each of the following functions has a maximal domain on which it is continuous and that domain has a boundary. Match the verbal description of this boundary with the function by putting the letter corresponding to the boundary to the left of the letter labelling the function. [ANS] 1. $ f(x, y, z)= \\frac{1}{x^2+y^2+z^2} $ [ANS] 2. $ f(x, y, z)= \\frac{z}{1-x^2-y^2} $ [ANS] 3. $f(x, y)=x \\ln{y}$ [ANS] 4. $ f(x, y)= \\frac{1}{4-x^2-y^2} $ [ANS] 5. $ f(x, y, z)= \\frac{xyz}{x^2+y^2-z} $\nA. a circular cylinder B. a straight line C. one point D. a circle E. a circular parabaloid", "answer_v1": [ "C", "A", "B", "D", "E" ], "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": "Each of the following functions has a maximal domain on which it is continuous and that domain has a boundary. Match the verbal description of this boundary with the function by putting the letter corresponding to the boundary to the left of the letter labelling the function. [ANS] 1. $ f(x, y)= \\frac{1}{4-x^2-y^2} $ [ANS] 2. $ f(x, y, z)= \\frac{1}{x^2+y^2+z^2} $ [ANS] 3. $f(x, y)=e^{1/(x-y)}$ [ANS] 4. $ f(x, y, z)= \\frac{z}{1-x^2-y^2} $ [ANS] 5. $ f(x, y, z)= \\frac{xyz}{x^2+y^2-z} $\nA. one point B. a straight line C. a circular cylinder D. a circular parabaloid E. a circle", "answer_v2": [ "E", "A", "B", "C", "D" ], "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": "Each of the following functions has a maximal domain on which it is continuous and that domain has a boundary. Match the verbal description of this boundary with the function by putting the letter corresponding to the boundary to the left of the letter labelling the function. [ANS] 1. $ f(x, y)= \\frac{1}{4-x^2-y^2} $ [ANS] 2. $ f(x, y, z)= \\frac{xyz}{x^2+y^2-z} $ [ANS] 3. $ f(x, y, z)= \\frac{z}{1-x^2-y^2} $ [ANS] 4. $f(x, y)=e^{1/(x-y)}$ [ANS] 5. $ f(x, y, z)= \\frac{1}{x^2+y^2+z^2} $\nA. a circle B. a circular parabaloid C. a circular cylinder D. a straight line E. one point", "answer_v3": [ "A", "B", "C", "D", "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_-_multivariable_0114", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "1", "keywords": [ "Multivariable", "Graph" ], "problem_v1": "A car rental company charges a one-time application fee of 35 dollars, 50 dollars per day, and 13 cents per mile for its cars.\n(a) Write a formula for the cost, $C$, of renting a car as a function of the number of days, $d$, and the number of miles driven, $m$. $C=$ [ANS]\n(b) If $C=f(d, m)$, then $f(5, 650)=$ [ANS]", "answer_v1": [ "35+50*d+13*m/100", "369.5" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A car rental company charges a one-time application fee of 20 dollars, 60 dollars per day, and 10 cents per mile for its cars.\n(a) Write a formula for the cost, $C$, of renting a car as a function of the number of days, $d$, and the number of miles driven, $m$. $C=$ [ANS]\n(b) If $C=f(d, m)$, then $f(4, 980)=$ [ANS]", "answer_v2": [ "20+60*d+10*m/100", "358" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A car rental company charges a one-time application fee of 25 dollars, 55 dollars per day, and 11 cents per mile for its cars.\n(a) Write a formula for the cost, $C$, of renting a car as a function of the number of days, $d$, and the number of miles driven, $m$. $C=$ [ANS]\n(b) If $C=f(d, m)$, then $f(4, 600)=$ [ANS]", "answer_v3": [ "25+55*d+11*m/100", "311" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0115", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Notation, domain, and range", "level": "2", "keywords": [ "Multivariable", "Graph" ], "problem_v1": "Consider the concentration, C, (in mg/liter) of a drug in the blood as a function of the amount of drug given, x, and the time since injection, t. For $0 \\leq x \\leq 6$ mg and $t \\geq 0$ hours, we have C=f(x,t)=28te^{-\\left(6-x\\right)t}\n$f(2,4)=$ [ANS]\nGive a practical interpretation of your answer: $f(2, 4)$ is [ANS] A. the concentration of a 4 mg dose in the blood 2 hours after injection. B. the amount of a 2 mg dose in the blood 4 hours after injection. C. the change in concentration of a 2 mg dose in the blood 4 hours after injection. D. the change in concentration of a 4 mg dose in the blood 2 hours after injection. E. the amount of a 4 mg dose in the blood 2 hours after injection. F. the concentration of a 2 mg dose in the blood 4 hours after injection.", "answer_v1": [ "1.26039E-05", "F" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Consider the concentration, C, (in mg/liter) of a drug in the blood as a function of the amount of drug given, x, and the time since injection, t. For $0 \\leq x \\leq 4$ mg and $t \\geq 0$ hours, we have C=f(x,t)=24te^{-\\left(4-x\\right)t}\n$f(3,2)=$ [ANS]\nGive a practical interpretation of your answer: $f(3, 2)$ is [ANS] A. the change in concentration of a 2 mg dose in the blood 3 hours after injection. B. the amount of a 3 mg dose in the blood 2 hours after injection. C. the change in concentration of a 3 mg dose in the blood 2 hours after injection. D. the concentration of a 2 mg dose in the blood 3 hours after injection. E. the concentration of a 3 mg dose in the blood 2 hours after injection. F. the amount of a 2 mg dose in the blood 3 hours after injection.", "answer_v2": [ "6.49609", "E" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Consider the concentration, C, (in mg/liter) of a drug in the blood as a function of the amount of drug given, x, and the time since injection, t. For $0 \\leq x \\leq 4$ mg and $t \\geq 0$ hours, we have C=f(x,t)=26te^{-\\left(4-x\\right)t}\n$f(2,3)=$ [ANS]\nGive a practical interpretation of your answer: $f(2, 3)$ is [ANS] A. the change in concentration of a 3 mg dose in the blood 2 hours after injection. B. the concentration of a 3 mg dose in the blood 2 hours after injection. C. the amount of a 2 mg dose in the blood 3 hours after injection. D. the amount of a 3 mg dose in the blood 2 hours after injection. E. the concentration of a 2 mg dose in the blood 3 hours after injection. F. the change in concentration of a 2 mg dose in the blood 3 hours after injection.", "answer_v3": [ "0.193343", "E" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Calculus_-_multivariable_0116", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Surfaces", "level": "2", "keywords": [ "Multivariable", "Geometry", "Surface", "Ellipsoid", "Paraboloid", "Hyperboloid", "Plane", "Cylinder" ], "problem_v1": "Match the surfaces with the appropriate descriptions. [ANS] 1. $z=4$ [ANS] 2. $z=2x+3y$ [ANS] 3. $x^{2}+y^{2}=5$ [ANS] 4. $z=x^{2}$ [ANS] 5. $z=y^{2}-2x^{2}$ [ANS] 6. $z=2x^{2}+3y^{2}$ [ANS] 7. $x^{2}+2y^{2}+3z^{2}=1$\nA. ellipsoid B. circular cylinder C. parabolic cylinder D. nonhorizontal plane E. horizontal plane F. hyperbolic paraboloid G. elliptic paraboloid", "answer_v1": [ "E", "D", "B", "C", "F", "G", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v2": "Match the surfaces with the appropriate descriptions. [ANS] 1. $z=y^{2}-2x^{2}$ [ANS] 2. $z=2x+3y$ [ANS] 3. $z=2x^{2}+3y^{2}$ [ANS] 4. $z=4$ [ANS] 5. $x^{2}+y^{2}=5$ [ANS] 6. $x^{2}+2y^{2}+3z^{2}=1$ [ANS] 7. $z=x^{2}$\nA. circular cylinder B. elliptic paraboloid C. ellipsoid D. parabolic cylinder E. hyperbolic paraboloid F. nonhorizontal plane G. horizontal plane", "answer_v2": [ "E", "F", "B", "G", "A", "C", "D" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ], "problem_v3": "Match the surfaces with the appropriate descriptions. [ANS] 1. $x^{2}+y^{2}=5$ [ANS] 2. $z=2x^{2}+3y^{2}$ [ANS] 3. $z=4$ [ANS] 4. $z=x^{2}$ [ANS] 5. $z=y^{2}-2x^{2}$ [ANS] 6. $z=2x+3y$ [ANS] 7. $x^{2}+2y^{2}+3z^{2}=1$\nA. horizontal plane B. elliptic paraboloid C. nonhorizontal plane D. hyperbolic paraboloid E. circular cylinder F. ellipsoid G. parabolic cylinder", "answer_v3": [ "E", "B", "A", "G", "D", "C", "F" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ], [ "A", "B", "C", "D", "E", "F", "G" ] ] }, { "id": "Calculus_-_multivariable_0117", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Surfaces", "level": "2", "keywords": [ "graphing", "multivariable", "functions" ], "problem_v1": "For each of the following functions, decide whether its graph could be a bowl, a plate, or neither. Consider a plate to be any fairly flat surface and a bowl to be anything that could hold water, assuming the positive $z$-axis is up.\n(a) $z= \\frac{xy}{|xy|} $: [ANS] (b) $z=-e^{-x^2-y^2}$: [ANS] (c) $z=3$: [ANS] (d) $z=4\\cos(3x)-y$: [ANS]", "answer_v1": [ "plate", "bowl", "plate", "neither" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "bowl", "plate", "neither" ], [ "bowl", "plate", "neither" ], [ "bowl", "plate", "neither" ], [ "bowl", "plate", "neither" ] ], "problem_v2": "For each of the following functions, decide whether its graph could be a bowl, a plate, or neither. Consider a plate to be any fairly flat surface and a bowl to be anything that could hold water, assuming the positive $z$-axis is up.\n(a) $z=x^2+y^2$: [ANS] (b) $z=- \\frac{\\sin(x)y}{x} $: [ANS] (c) $x+y+z=1$: [ANS] (d) $z=3$: [ANS]", "answer_v2": [ "bowl", "neither", "plate", "plate" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "bowl", "plate", "neither" ], [ "bowl", "plate", "neither" ], [ "bowl", "plate", "neither" ], [ "bowl", "plate", "neither" ] ], "problem_v3": "For each of the following functions, decide whether its graph could be a bowl, a plate, or neither. Consider a plate to be any fairly flat surface and a bowl to be anything that could hold water, assuming the positive $z$-axis is up.\n(a) $z=-\\sqrt{5-x^2-y^2}$: [ANS] (b) $z=x-0.2\\sin(y)$: [ANS] (c) $x+y+z=1$: [ANS] (d) $z=-e^{-x^2-y^2}$: [ANS]", "answer_v3": [ "bowl", "plate", "plate", "bowl" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "bowl", "plate", "neither" ], [ "bowl", "plate", "neither" ], [ "bowl", "plate", "neither" ], [ "bowl", "plate", "neither" ] ] }, { "id": "Calculus_-_multivariable_0118", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Surfaces", "level": "2", "keywords": [ "Multivariable", "Graph" ], "problem_v1": "(a) Describe the set of points whose distance from the z-axis equals the distance from the xy-plane. [ANS] A. A cone opening along the y-axis B. A cylinder opening along the z-axis C. A cone opening along the z-axis D. A cylinder opening along the x-axis E. A cone opening along the x-axis F. A cylinder opening along the y-axis\n(b) Find the equation for the set of points whose distance from the z-axis equals the distance from the xy-plane. [ANS] A. $x^2=y^2+z^2$ B. $x^2+z^2=r^2$ C. $y^2=x^2+z^2$ D. $x^2+y^2=r^2$ E. $z^2=x^2+y^2$ F. $y^2+z^2=r^2$", "answer_v1": [ "C", "E" ], "answer_type_v1": [ "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "(a) Describe the set of points whose distance from the x-axis equals the distance from the yz-plane. [ANS] A. A cone opening along the x-axis B. A cylinder opening along the z-axis C. A cylinder opening along the y-axis D. A cone opening along the y-axis E. A cone opening along the z-axis F. A cylinder opening along the x-axis\n(b) Find the equation for the set of points whose distance from the x-axis equals the distance from the yz-plane. [ANS] A. $y^2+z^2=r^2$ B. $x^2+y^2=r^2$ C. $y^2=x^2+z^2$ D. $x^2+z^2=r^2$ E. $x^2=y^2+z^2$ F. $z^2=x^2+y^2$", "answer_v2": [ "A", "E" ], "answer_type_v2": [ "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "(a) Describe the set of points whose distance from the x-axis equals the distance from the yz-plane. [ANS] A. A cylinder opening along the y-axis B. A cylinder opening along the z-axis C. A cone opening along the y-axis D. A cone opening along the x-axis E. A cone opening along the z-axis F. A cylinder opening along the x-axis\n(b) Find the equation for the set of points whose distance from the x-axis equals the distance from the yz-plane. [ANS] A. $x^2+y^2=r^2$ B. $y^2+z^2=r^2$ C. $x^2=y^2+z^2$ D. $x^2+z^2=r^2$ E. $y^2=x^2+z^2$ F. $z^2=x^2+y^2$", "answer_v3": [ "D", "C" ], "answer_type_v3": [ "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Calculus_-_multivariable_0119", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Surfaces", "level": "2", "keywords": [ "Multivariable", "Graph" ], "problem_v1": "Sketch a graph of each surface. Then, match each surface to its brief description in words. $\\begin{array}{cccc}\\hline & [ANS]GHHII 1. z=x^2+25 y^2+4 [ANS]GHHII 2. x^2+y^2=9 [ANS]GHHII 3. z^2=x^2+25 y^2 [ANS]GHHII 4. x^2+y^2+z^2=16 [ANS]GHHII 5. z=16x^2-y^2 & & A.Hyperbolic paraboloid (saddle) B.Elliptic paraboloid C.Sphere D.Elliptical cylinder E.Circular cylinder F.Skew line G.Cone H.Circle I.Sinusoidal cylinder \\\\ \\hline \\end{array}$", "answer_v1": [ "B", "E", "G", "C", "A" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ], "problem_v2": "Sketch a graph of each surface. Then, match each surface to its brief description in words. $\\begin{array}{cccc}\\hline & [ANS]GHHII 1. x^2+y^2+z^2=25 [ANS]GHHII 2. z=9x^2-y^2 [ANS]GHHII 3. x^2+y^2=9 [ANS]GHHII 4. z=x^2+4 y^2+6 [ANS]GHHII 5. z^2=x^2+16 y^2 & & A.Elliptical cylinder B.Cone C.Circular cylinder D.Elliptic paraboloid E.Skew line F.Circle G.Sinusoidal cylinder H.Sphere I.Hyperbolic paraboloid (saddle) \\\\ \\hline \\end{array}$", "answer_v2": [ "H", "I", "C", "D", "B" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ], "problem_v3": "Sketch a graph of each surface. Then, match each surface to its brief description in words. $\\begin{array}{cccc}\\hline & [ANS]GHHII 1. x^2+y^2=16 [ANS]GHHII 2. z^2=x^2+9 y^2 [ANS]GHHII 3. z=x^2+9 y^2+5 [ANS]GHHII 4. x^2+y^2+z^2=49 [ANS]GHHII 5. z=49x^2-y^2 & & A.Circular cylinder B.Cone C.Sphere D.Sinusoidal cylinder E.Hyperbolic paraboloid (saddle) F.Elliptical cylinder G.Skew line H.Elliptic paraboloid I.Circle \\\\ \\hline \\end{array}$", "answer_v3": [ "A", "B", "H", "C", "E" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ] }, { "id": "Calculus_-_multivariable_0120", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find the equation of the hyperboloid of one sheet passing through the points $(\\pm 7,0,0),(0,\\pm 6,0)$ and $(\\pm 14,0,6),(0,\\pm 12,6)$ [ANS] $=1$", "answer_v1": [ "(x/7)^2+(y/6)^2-(z/3.4641)^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the equation of the hyperboloid of one sheet passing through the points $(\\pm 2,0,0),(0,\\pm 8,0)$ and $(\\pm 4,0,3),(0,\\pm 16,3)$ [ANS] $=1$", "answer_v2": [ "(x/2)^2+(y/8)^2-(z/1.73205)^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the equation of the hyperboloid of one sheet passing through the points $(\\pm 4,0,0),(0,\\pm 6,0)$ and $(\\pm 8,0,3),(0,\\pm 12,3)$ [ANS] $=1$", "answer_v3": [ "(x/4)^2+(y/6)^2-(z/1.73205)^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0121", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "2", "keywords": [ "calculus" ], "problem_v1": "State whether the equation $z^{}=\\left( \\frac{x}{7} \\right)^2+\\left( \\frac{y}{6} \\right)^2$ defines: [ANS]", "answer_v1": [ "AN ELLIPTIC PARABOLOID" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A hyperbolic paraboloid", "An elliptic paraboloid", "An elliptic cone", "None of these" ] ], "problem_v2": "State whether the equation $z^{}=\\left( \\frac{x}{2} \\right)^2-\\left( \\frac{y}{8} \\right)^2$ defines: [ANS]", "answer_v2": [ "A HYPERBOLIC PARABOLOID" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A hyperbolic paraboloid", "An elliptic paraboloid", "An elliptic cone", "None of these" ] ], "problem_v3": "State whether the equation $z^{2}=\\left( \\frac{x}{4} \\right)^2-\\left( \\frac{y}{6} \\right)^2$ defines: [ANS]", "answer_v3": [ "AN ELLIPTIC CONE" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A hyperbolic paraboloid", "An elliptic paraboloid", "An elliptic cone", "None of these" ] ] }, { "id": "Calculus_-_multivariable_0122", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find the equation of the ellipsoid passing through the points $(\\pm 7,0,0),(0,\\pm 5,0)$ and $(0,0,\\pm 5)$ [ANS] $=1$", "answer_v1": [ "(x/7)^2+(y/5)^2+(z/5)^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the equation of the ellipsoid passing through the points $(\\pm 2,0,0),(0,\\pm 8,0)$ and $(0,0,\\pm 2)$ [ANS] $=1$", "answer_v2": [ "(x/2)^2+(y/8)^2+(z/2)^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the equation of the ellipsoid passing through the points $(\\pm 4,0,0),(0,\\pm 5,0)$ and $(0,0,\\pm 3)$ [ANS] $=1$", "answer_v3": [ "(x/4)^2+(y/5)^2+(z/3)^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0123", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "3", "keywords": [ "Sphere", "Center", "Radius", "equation" ], "problem_v1": "Find the center and radius of the sphere $x^2-10x+y^2-4y+z^2-6z=26$ Center: ([ANS], [ANS], [ANS]) Radius: [ANS]", "answer_v1": [ "5", "2", "3", "8" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the center and radius of the sphere $x^2+18x+y^2-18y+z^2+14z=-195$ Center: ([ANS], [ANS], [ANS]) Radius: [ANS]", "answer_v2": [ "-9", "9", "-7", "4" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the center and radius of the sphere $x^2+8x+y^2-4y+z^2+10z=-9$ Center: ([ANS], [ANS], [ANS]) Radius: [ANS]", "answer_v3": [ "-4", "2", "-5", "6" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0124", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "2", "keywords": [ "Sphere", "Center", "Radius", "equation" ], "problem_v1": "Find an equation of the sphere with center (3, 1, 1) and radius 4. [ANS]=0 Note that you must move everything to the left hand side of the equation and that we desire the coefficients of the quadratic terms to be 1.", "answer_v1": [ "x^2 + y^2 + z^2 + (-6*x + -2*y + -2*z) + ((3)^2 + (1)^2 + (1)^2) - (4)^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an equation of the sphere with center (-5, 5,-4) and radius 2. [ANS]=0 Note that you must move everything to the left hand side of the equation and that we desire the coefficients of the quadratic terms to be 1.", "answer_v2": [ "x^2 + y^2 + z^2 + (10*x + -10*y + 8*z) + ((-5)^2 + (5)^2 + (-4)^2) - (2)^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an equation of the sphere with center (-2, 1,-2) and radius 3. [ANS]=0 Note that you must move everything to the left hand side of the equation and that we desire the coefficients of the quadratic terms to be 1.", "answer_v3": [ "x^2 + y^2 + z^2 + (4*x + -2*y + 4*z) + ((-2)^2 + (1)^2 + (-2)^2) - (3)^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0125", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "2", "keywords": [ "Sphere", "Diameter", "equation" ], "problem_v1": "Find the equation of a sphere if one of its diameters has endpoints: (-3,-6,-5) and (13, 10, 11). [ANS]=0 Note that you must move everything to the left hand side of the equation and that we desire the coefficients of the quadratic terms to be 1.", "answer_v1": [ "x^2 + y^2 + z^2 - 2*(5*x + 2*y + 3*z) - 3*((8)^2) + ((5)^2 + (2)^2 + (3)^2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the equation of a sphere if one of its diameters has endpoints: (-13, 5,-11) and (-5, 13,-3). [ANS]=0 Note that you must move everything to the left hand side of the equation and that we desire the coefficients of the quadratic terms to be 1.", "answer_v2": [ "x^2 + y^2 + z^2 - 2*(-9*x + 9*y + -7*z) - 3*((4)^2) + ((-9)^2 + (9)^2 + (-7)^2)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the equation of a sphere if one of its diameters has endpoints: (-10,-4,-11) and (2, 8, 1). [ANS]=0 Note that you must move everything to the left hand side of the equation and that we desire the coefficients of the quadratic terms to be 1.", "answer_v3": [ "x^2 + y^2 + z^2 - 2*(-4*x + 2*y + -5*z) - 3*((6)^2) + ((-4)^2 + (2)^2 + (-5)^2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0126", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "2", "keywords": [ "multivariable", "functions" ], "problem_v1": "Find the equation of the sphere of radius 7 centered at $(5, 5, 6)$. This sphere is given by: [ANS]", "answer_v1": [ "(x-5)^2+(y-5)^2+(z-6)^2 = 49" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Find the equation of the sphere of radius 2 centered at $(8, 2, 3)$. This sphere is given by: [ANS]", "answer_v2": [ "(x-8)^2+(y-2)^2+(z-3)^2 = 4" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Find the equation of the sphere of radius 4 centered at $(5, 3, 5)$. This sphere is given by: [ANS]", "answer_v3": [ "(x-5)^2+(y-3)^2+(z-5)^2 = 16" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0127", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "2", "keywords": [ "multivariable", "functions" ], "problem_v1": "Find the equations of planes that just touch the sphere $(x-4)^2+(y-3)^2+(z-4)^2=16$ and are parallel to\n(a) The $xy$-plane: [ANS] and [ANS]\n(b) The $yz$-plane: [ANS] and [ANS]\n(c) The $xz$-plane: [ANS] and [ANS]", "answer_v1": [ "z = 8", "z = 0", "x = 8", "x = 0", "y = 7", "y = -1" ], "answer_type_v1": [ "EQ", "EQ", "EQ", "EQ", "EQ", "EQ" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find the equations of planes that just touch the sphere $(x-1)^2+(y-5)^2+(z-1)^2=9$ and are parallel to\n(a) The $xy$-plane: [ANS] and [ANS]\n(b) The $yz$-plane: [ANS] and [ANS]\n(c) The $xz$-plane: [ANS] and [ANS]", "answer_v2": [ "z = 4", "z = -2", "x = 4", "x = -2", "y = 8", "y = 2" ], "answer_type_v2": [ "EQ", "EQ", "EQ", "EQ", "EQ", "EQ" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find the equations of planes that just touch the sphere $(x-2)^2+(y-4)^2+(z-2)^2=16$ and are parallel to\n(a) The $xy$-plane: [ANS] and [ANS]\n(b) The $yz$-plane: [ANS] and [ANS]\n(c) The $xz$-plane: [ANS] and [ANS]", "answer_v3": [ "z = 6", "z = -2", "x = 6", "x = -2", "y = 8", "y = 0" ], "answer_type_v3": [ "EQ", "EQ", "EQ", "EQ", "EQ", "EQ" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0128", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "2", "keywords": [ "multivariable", "functions", "level surface" ], "problem_v1": "Use the catalog of surfaces in your textbook to identify the surfaces given below.\n(a) $z- \\frac{x^{2}}{7} + \\frac{y^{2}}{5} =0$: [ANS] (b) $ \\frac{x^{2}}{7} + \\frac{y^{2}}{5} -z=0$: [ANS] (c) $z^{2}-\\left(7x^{2}+7y^{2}\\right)=1$: [ANS]", "answer_v1": [ "Hyperbolic paraboloid", "Elliptical paraboloid", "Hyperboloid of two sheets" ], "answer_type_v1": [ "MCS", "MCS", "MCS" ], "options_v1": [ [ "Elliptical paraboloid", "Hyperbolic paraboloid", "Ellipsoid", "Hyperboloid of one sheet", "Hyperboloid of two sheets", "Cone", "Plane", "Cylindrical surface", "Parabolic cylinder" ], [ "Elliptical paraboloid", "Hyperbolic paraboloid", "Ellipsoid", "Hyperboloid of one sheet", "Hyperboloid of two sheets", "Cone", "Plane", "Cylindrical surface", "Parabolic cylinder" ], [ "Elliptical paraboloid", "Hyperbolic paraboloid", "Ellipsoid", "Hyperboloid of one sheet", "Hyperboloid of two sheets", "Cone", "Plane", "Cylindrical surface", "Parabolic cylinder" ] ], "problem_v2": "Use the catalog of surfaces in your textbook to identify the surfaces given below.\n(a) $z^{2}-\\left(x^{2}+y^{2}\\right)=1$: [ANS] (b) $x^{2}+ \\frac{y^{2}}{8} -z=0$: [ANS] (c) $x^{2}+y^{2}+z^{2}=8$: [ANS]", "answer_v2": [ "Hyperboloid of two sheets", "Elliptical paraboloid", "Ellipsoid" ], "answer_type_v2": [ "MCS", "MCS", "MCS" ], "options_v2": [ [ "Elliptical paraboloid", "Hyperbolic paraboloid", "Ellipsoid", "Hyperboloid of one sheet", "Hyperboloid of two sheets", "Cone", "Plane", "Cylindrical surface", "Parabolic cylinder" ], [ "Elliptical paraboloid", "Hyperbolic paraboloid", "Ellipsoid", "Hyperboloid of one sheet", "Hyperboloid of two sheets", "Cone", "Plane", "Cylindrical surface", "Parabolic cylinder" ], [ "Elliptical paraboloid", "Hyperbolic paraboloid", "Ellipsoid", "Hyperboloid of one sheet", "Hyperboloid of two sheets", "Cone", "Plane", "Cylindrical surface", "Parabolic cylinder" ] ], "problem_v3": "Use the catalog of surfaces in your textbook to identify the surfaces given below.\n(a) $z^{2}-\\left(3x^{2}+3y^{2}\\right)=0$: [ANS] (b) $z- \\frac{x^{2}}{3} + \\frac{y^{2}}{5} =0$: [ANS] (c) $z^{2}-\\left(3x^{2}+3y^{2}\\right)=1$: [ANS]", "answer_v3": [ "Cone", "Hyperbolic paraboloid", "Hyperboloid of two sheets" ], "answer_type_v3": [ "MCS", "MCS", "MCS" ], "options_v3": [ [ "Elliptical paraboloid", "Hyperbolic paraboloid", "Ellipsoid", "Hyperboloid of one sheet", "Hyperboloid of two sheets", "Cone", "Plane", "Cylindrical surface", "Parabolic cylinder" ], [ "Elliptical paraboloid", "Hyperbolic paraboloid", "Ellipsoid", "Hyperboloid of one sheet", "Hyperboloid of two sheets", "Cone", "Plane", "Cylindrical surface", "Parabolic cylinder" ], [ "Elliptical paraboloid", "Hyperbolic paraboloid", "Ellipsoid", "Hyperboloid of one sheet", "Hyperboloid of two sheets", "Cone", "Plane", "Cylindrical surface", "Parabolic cylinder" ] ] }, { "id": "Calculus_-_multivariable_0129", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "3", "keywords": [ "quadric", "quadrics", "quadric surface", "quadric surfaces", "cylinder", "cylinders" ], "problem_v1": "By completing the square, put the equation\n8x^{2}+y^{2}+7z^{2}=14-16x-12y+14z into the standard form $a (x-x_0)^2+b(y-y_0)^2+c(z-z_0)^2=d$. Then, use your answer to sketch a graph of this quadric surface on paper. [ANS] $\\big($ [ANS] $\\big)^2+$ [ANS] $\\big($ [ANS] $\\big)^2+$ [ANS] $\\big($ [ANS] $\\big)^2=$ [ANS]", "answer_v1": [ "8", "x+1", "1", "y+6", "7", "z-1", "65" ], "answer_type_v1": [ "NV", "EX", "NV", "EX", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "By completing the square, put the equation\n2x^{2}+y^{2}+3z^{2}=15-4x-18y+6z into the standard form $a (x-x_0)^2+b(y-y_0)^2+c(z-z_0)^2=d$. Then, use your answer to sketch a graph of this quadric surface on paper. [ANS] $\\big($ [ANS] $\\big)^2+$ [ANS] $\\big($ [ANS] $\\big)^2+$ [ANS] $\\big($ [ANS] $\\big)^2=$ [ANS]", "answer_v2": [ "2", "x+1", "1", "y+9", "3", "z-1", "101" ], "answer_type_v2": [ "NV", "EX", "NV", "EX", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "By completing the square, put the equation\n4x^{2}+y^{2}+3z^{2}=15-8x-12y+6z into the standard form $a (x-x_0)^2+b(y-y_0)^2+c(z-z_0)^2=d$. Then, use your answer to sketch a graph of this quadric surface on paper. [ANS] $\\big($ [ANS] $\\big)^2+$ [ANS] $\\big($ [ANS] $\\big)^2+$ [ANS] $\\big($ [ANS] $\\big)^2=$ [ANS]", "answer_v3": [ "4", "x+1", "1", "y+6", "3", "z-1", "58" ], "answer_type_v3": [ "NV", "EX", "NV", "EX", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0130", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "3", "keywords": [ "quadric", "quadrics", "quadric surface", "quadric surfaces", "cylinder", "cylinders" ], "problem_v1": "Identify the type of quadric surface defined by the equation\n6x^{2}+6y^{2}-z^{2}=-36, and find all x-, y-, and z-intercepts of the resulting graph. Sketch the graph of this quadric surface on paper.\nThe quadric surface is a/an [ANS] with x-intercepts when $x=$ [ANS], y-intercepts when $y=$ [ANS], and z-intercepts when $z=$ [ANS].\nEnter your answers as comma separated lists, or enter NONE if there are no intercepts of a particular type.", "answer_v1": [ "hyperboloid of two sheets", "NONE", "NONE", "(6, -6)" ], "answer_type_v1": [ "MCS", "OE", "OE", "UOL" ], "options_v1": [ [ "elliptic cylinder", "hyperbolic cylinder", "parabolic cylinder", "elliptic paraboloid", "hyperbolic paraboloid", "ellipsoid", "hyperboloid of one sheet", "hyperboloid of two sheets", "elliptic cone" ], [], [], [] ], "problem_v2": "Identify the type of quadric surface defined by the equation\n2x^{2}+2y^{2}-z^{2}=-4, and find all x-, y-, and z-intercepts of the resulting graph. Sketch the graph of this quadric surface on paper.\nThe quadric surface is a/an [ANS] with x-intercepts when $x=$ [ANS], y-intercepts when $y=$ [ANS], and z-intercepts when $z=$ [ANS].\nEnter your answers as comma separated lists, or enter NONE if there are no intercepts of a particular type.", "answer_v2": [ "hyperboloid of two sheets", "NONE", "NONE", "(2, -2)" ], "answer_type_v2": [ "MCS", "OE", "OE", "UOL" ], "options_v2": [ [ "elliptic cylinder", "hyperbolic cylinder", "parabolic cylinder", "elliptic paraboloid", "hyperbolic paraboloid", "ellipsoid", "hyperboloid of one sheet", "hyperboloid of two sheets", "elliptic cone" ], [], [], [] ], "problem_v3": "Identify the type of quadric surface defined by the equation\n3x^{2}+3y^{2}-z^{2}=-9, and find all x-, y-, and z-intercepts of the resulting graph. Sketch the graph of this quadric surface on paper.\nThe quadric surface is a/an [ANS] with x-intercepts when $x=$ [ANS], y-intercepts when $y=$ [ANS], and z-intercepts when $z=$ [ANS].\nEnter your answers as comma separated lists, or enter NONE if there are no intercepts of a particular type.", "answer_v3": [ "hyperboloid of two sheets", "NONE", "NONE", "(3, -3)" ], "answer_type_v3": [ "MCS", "OE", "OE", "UOL" ], "options_v3": [ [ "elliptic cylinder", "hyperbolic cylinder", "parabolic cylinder", "elliptic paraboloid", "hyperbolic paraboloid", "ellipsoid", "hyperboloid of one sheet", "hyperboloid of two sheets", "elliptic cone" ], [], [], [] ] }, { "id": "Calculus_-_multivariable_0131", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "2", "keywords": [ "Graph", "Multivariable", "Level Curve" ], "problem_v1": "Identify the following surfaces as an elliptical paraboloid, hyperbolic paraboloid, a hyperboloid of one sheet, a hyperboloid of two sheets, a cone, a circular cylinder, an elliptical cylinder, or a parabolic cylinder, and identify the axis of symmetry as the x-axis, the y-axis, or the z-axis.\na. [ANS] [ANS] $-6x^2+7 y^2-z^2=0$\nb. [ANS] [ANS] $-8x^2-6 y^2+z^2=1$\nc. [ANS] [ANS] $4x^2+4 y^2-z=0$\nd. [ANS] [ANS] $6 y^2+6 z^2=1$\ne. [ANS] [ANS] $7 y^2-4x^2-8 z^2+1=0$", "answer_v1": [ "cone", "along the y-axis", "hyperboloid of two sheets", "along the z-axis", "elliptical paraboloid", "along the z-axis", "circular cylinder", "along the x-axis", "hyperboloid of one sheet", "along the y-axis" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ] ], "problem_v2": "Identify the following surfaces as an elliptical paraboloid, hyperbolic paraboloid, a hyperboloid of one sheet, a hyperboloid of two sheets, a cone, a circular cylinder, an elliptical cylinder, or a parabolic cylinder, and identify the axis of symmetry as the x-axis, the y-axis, or the z-axis.\na. [ANS] [ANS] $-2x^2-9 y^2+z^2=1$\nb. [ANS] [ANS] $9x^2+4 y^2-z=0$\nc. [ANS] [ANS] $4 y^2-9x^2-2 z^2+1=0$\nd. [ANS] [ANS] $3 y^2+3 z^2=1$\ne. [ANS] [ANS] $-3x^2+4 y^2-z^2=0$", "answer_v2": [ "hyperboloid of two sheets", "along the z-axis", "elliptical paraboloid", "along the z-axis", "hyperboloid of one sheet", "along the y-axis", "circular cylinder", "along the x-axis", "cone", "along the y-axis" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ] ], "problem_v3": "Identify the following surfaces as an elliptical paraboloid, hyperbolic paraboloid, a hyperboloid of one sheet, a hyperboloid of two sheets, a cone, a circular cylinder, an elliptical cylinder, or a parabolic cylinder, and identify the axis of symmetry as the x-axis, the y-axis, or the z-axis.\na. [ANS] [ANS] $8 y^2+8 z^2=1$\nb. [ANS] [ANS] $3x^2+4 y^2-z=0$\nc. [ANS] [ANS] $6 y^2-3x^2-4 z^2+1=0$\nd. [ANS] [ANS] $-4x^2-6 y^2+z^2=1$\ne. [ANS] [ANS] $-4x^2+6 y^2-z^2=0$", "answer_v3": [ "circular cylinder", "along the x-axis", "elliptical paraboloid", "along the z-axis", "hyperboloid of one sheet", "along the y-axis", "hyperboloid of two sheets", "along the z-axis", "cone", "along the y-axis" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ], [ "elliptical paraboloid", "hyperbolic paraboloid", "hyperboloid of one sheet", "hyperboloid of two sheets", "cone", "circular cylinder", "elliptical cylinder", "parabolic cylinder" ], [ "along the x-axis", "along the y-axis", "along the z-axis" ] ] }, { "id": "Calculus_-_multivariable_0132", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Quadratic surfaces", "level": "2", "keywords": [ "Implicit", "Sphere" ], "problem_v1": "Find an equation for the sphere centered at $(3,-2, 2)$ with radius $4$. [ANS]", "answer_v1": [ "(x-3)^2+[y-(-2)]^2+(z-2)^2= 16" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Find an equation for the sphere centered at $(-5, 5,-2)$ with radius $6$. [ANS]", "answer_v2": [ "[x-(-5)]^2+(y-5)^2+[z-(-2)]^2= 36" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Find an equation for the sphere centered at $(-4, 4, 1)$ with radius $3$. [ANS]", "answer_v3": [ "[x-(-4)]^2+(y-4)^2+(z-1)^2= 9" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0133", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Surfaces in other coordinate systems", "level": "3", "keywords": [ "parametrize' 'surface' 'level" ], "problem_v1": "Express the surface x=7\\cos\\theta \\sin\\phi, y=8\\sin\\theta \\sin\\phi, z=6\\cos\\phi as a level surface $f(x,y,z)=112896$, $f(x,y,z)=$ [ANS]\nThis surface is a [ANS]\nA. paraboloid B. hyperboloid C. ellipsoid D. sphere E. None of these", "answer_v1": [ "6*6*8*8*x*x + 6*6*7*7*y*y + 7*7*8*8*z*z", "C" ], "answer_type_v1": [ "EX", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Express the surface x=2\\cos\\theta \\sin\\phi, y=3\\sin\\theta \\sin\\phi, z=8\\cos\\phi as a level surface $f(x,y,z)=2304$, $f(x,y,z)=$ [ANS]\nThis surface is a [ANS]\nA. ellipsoid B. paraboloid C. hyperboloid D. sphere E. None of these", "answer_v2": [ "8*8*3*3*x*x + 8*8*2*2*y*y + 2*2*3*3*z*z", "A" ], "answer_type_v2": [ "EX", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Express the surface x=4\\cos\\theta \\sin\\phi, y=5\\sin\\theta \\sin\\phi, z=6\\cos\\phi as a level surface $f(x,y,z)=14400$, $f(x,y,z)=$ [ANS]\nThis surface is a [ANS]\nA. sphere B. paraboloid C. ellipsoid D. hyperboloid E. None of these", "answer_v3": [ "6*6*5*5*x*x + 6*6*4*4*y*y + 4*4*5*5*z*z", "C" ], "answer_type_v3": [ "EX", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Calculus_-_multivariable_0134", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Parameterized surfaces", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Compute the normal vector at $(8,6)$ and use it to estimate the area of the small patch of the surface $\\Phi(u,v)=(u^2-v^2,u+v,u-v)$ defined by 8 \\le u \\le 8.6,\\qquad 6 \\le v \\le 6.7 [ANS]", "answer_v1": [ "11.9091" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the normal vector at $(2,9)$ and use it to estimate the area of the small patch of the surface $\\Phi(u,v)=(u^2-v^2,u+v,u-v)$ defined by 2 \\le u \\le 2.2,\\qquad 9 \\le v \\le 9.4 [ANS]", "answer_v2": [ "2.09227" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the normal vector at $(4,6)$ and use it to estimate the area of the small patch of the surface $\\Phi(u,v)=(u^2-v^2,u+v,u-v)$ defined by 4 \\le u \\le 4.3,\\qquad 6 \\le v \\le 6.5 [ANS]", "answer_v3": [ "3.07409" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0135", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Parameterized surfaces", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Calculate ${\\mathbf{T}}_u$, ${\\mathbf{T}}_v$, and $\\mathbf{n}(u,v)$ for the parametrized surface at the given point. Then find the equation of the tangent plane to the surface at that point. $\\Phi(u,v)=(2u+v,u-4v,8 u)$ ; $\\qquad u=6\\text{,}\\quad v=6$ ${\\mathbf{T}}_u=$ [ANS], ${\\mathbf{T}}_v=$ [ANS], $\\mathbf{n}(u,v)=$ [ANS]\nThe tangent plane: [ANS] $=9z$", "answer_v1": [ "(2,1,8)", "(1,-4,0)", "(32,8,-9)", "32*x+8*y" ], "answer_type_v1": [ "OL", "OL", "OL", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Calculate ${\\mathbf{T}}_u$, ${\\mathbf{T}}_v$, and $\\mathbf{n}(u,v)$ for the parametrized surface at the given point. Then find the equation of the tangent plane to the surface at that point. $\\Phi(u,v)=(2u+v,u-4v,1 u)$ ; $\\qquad u=9\\text{,}\\quad v=3$ ${\\mathbf{T}}_u=$ [ANS], ${\\mathbf{T}}_v=$ [ANS], $\\mathbf{n}(u,v)=$ [ANS]\nThe tangent plane: [ANS] $=9z$", "answer_v2": [ "(2,1,1)", "(1,-4,0)", "(4,1,-9)", "4*x+y" ], "answer_type_v2": [ "OL", "OL", "OL", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Calculate ${\\mathbf{T}}_u$, ${\\mathbf{T}}_v$, and $\\mathbf{n}(u,v)$ for the parametrized surface at the given point. Then find the equation of the tangent plane to the surface at that point. $\\Phi(u,v)=(2u+v,u-4v,4 u)$ ; $\\qquad u=6\\text{,}\\quad v=4$ ${\\mathbf{T}}_u=$ [ANS], ${\\mathbf{T}}_v=$ [ANS], $\\mathbf{n}(u,v)=$ [ANS]\nThe tangent plane: [ANS] $=9z$", "answer_v3": [ "(2,1,4)", "(1,-4,0)", "(16,4,-9)", "16*x+4*y" ], "answer_type_v3": [ "OL", "OL", "OL", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0136", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Parameterized surfaces", "level": "3", "keywords": [ "vector' 'parametric' 'multivariable", "Vector", "Parametric", "Geometry" ], "problem_v1": "Match the parametric equations with the verbal descriptions of the surfaces by putting the letter of the verbal description to the left of the letter of the parametric equation. [ANS] 1. $\\mathbf{r} \\left(u, v \\right)=u \\mathbf{i}+v \\mathbf{j}+\\left(2u-3v \\right) \\mathbf{k}$ [ANS] 2. $\\mathbf{r} \\left(u, v \\right)=u \\mathbf{i}+u \\cos v \\mathbf{j}+u \\sin v \\mathbf{k}$ [ANS] 3. $\\mathbf{r} \\left(u, v \\right)=u \\cos v \\mathbf{i}+u \\sin v \\mathbf{j}+u^{2} \\mathbf{k}$ [ANS] 4. $\\mathbf{r} \\left(u, v \\right)=u \\mathbf{i}+\\cos v \\mathbf{j}+\\sin v \\mathbf{k}$\nA. plane B. circular paraboloid C. cone D. circular cylinder", "answer_v1": [ "A", "C", "B", "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": "Match the parametric equations with the verbal descriptions of the surfaces by putting the letter of the verbal description to the left of the letter of the parametric equation. [ANS] 1. $\\mathbf{r} \\left(u, v \\right)=u \\mathbf{i}+u \\cos v \\mathbf{j}+u \\sin v \\mathbf{k}$ [ANS] 2. $\\mathbf{r} \\left(u, v \\right)=u \\cos v \\mathbf{i}+u \\sin v \\mathbf{j}+u^{2} \\mathbf{k}$ [ANS] 3. $\\mathbf{r} \\left(u, v \\right)=u \\mathbf{i}+v \\mathbf{j}+\\left(2u-3v \\right) \\mathbf{k}$ [ANS] 4. $\\mathbf{r} \\left(u, v \\right)=u \\mathbf{i}+\\cos v \\mathbf{j}+\\sin v \\mathbf{k}$\nA. plane B. circular cylinder C. circular paraboloid D. cone", "answer_v2": [ "D", "C", "A", "B" ], "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 parametric equations with the verbal descriptions of the surfaces by putting the letter of the verbal description to the left of the letter of the parametric equation. [ANS] 1. $\\mathbf{r} \\left(u, v \\right)=u \\mathbf{i}+u \\cos v \\mathbf{j}+u \\sin v \\mathbf{k}$ [ANS] 2. $\\mathbf{r} \\left(u, v \\right)=u \\mathbf{i}+\\cos v \\mathbf{j}+\\sin v \\mathbf{k}$ [ANS] 3. $\\mathbf{r} \\left(u, v \\right)=u \\mathbf{i}+v \\mathbf{j}+\\left(2u-3v \\right) \\mathbf{k}$ [ANS] 4. $\\mathbf{r} \\left(u, v \\right)=u \\cos v \\mathbf{i}+u \\sin v \\mathbf{j}+u^{2} \\mathbf{k}$\nA. cone B. circular cylinder C. circular paraboloid D. plane", "answer_v3": [ "A", "B", "D", "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_-_multivariable_0137", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Parameterized surfaces", "level": "3", "keywords": [ "parametric surfaces", "parametric equations", "multivariable", "functions" ], "problem_v1": "(a) Does the plane $\\vec r(s,t)=\\left(3-2s\\right)\\,\\mathit{\\vec i}+\\left(1+2t\\right)\\,\\mathit{\\vec j}+\\left(s-3t\\right)\\,\\mathit{\\vec k}$ contain the point $\\left(5,-5,9\\right)$? [ANS] (b) Find the $z$-component of the point $(1,-3, z_0)$ so that it lies on the plane. $z_0=$ [ANS]\nFor what values of $s$ and $t$ is this the case? $s=$ [ANS]\n$t=$ [ANS]", "answer_v1": [ "no", "7", "1", "-2" ], "answer_type_v1": [ "TF", "NV", "NV", "NV" ], "options_v1": [ [ "yes", "no" ], [], [], [] ], "problem_v2": "(a) Does the plane $\\vec r(s,t)=\\left(t-5\\right)\\,\\mathit{\\vec i}-2s\\,\\mathit{\\vec j}+\\left(2t-\\left(4+3s\\right)\\right)\\,\\mathit{\\vec k}$ contain the point $\\left(-7,4,-2\\right)$? [ANS] (b) Find the $z$-component of the point $(-7, 2, z_0)$ so that it lies on the plane. $z_0=$ [ANS]\nFor what values of $s$ and $t$ is this the case? $s=$ [ANS]\n$t=$ [ANS]", "answer_v2": [ "yes", "-5", "-1", "-2" ], "answer_type_v2": [ "TF", "NV", "NV", "NV" ], "options_v2": [ [ "yes", "no" ], [], [], [] ], "problem_v3": "(a) Does the plane $\\vec r(s,t)=-\\left(\\left(2+3s\\right)\\,\\mathit{\\vec i}+\\left(2s+3t\\right)\\,\\mathit{\\vec j}+\\left(2+2t\\right)\\,\\mathit{\\vec k}\\right)$ contain the point $\\left(-5,-13,-8\\right)$? [ANS] (b) Find the $z$-component of the point $(-8,-10, z_0)$ so that it lies on the plane. $z_0=$ [ANS]\nFor what values of $s$ and $t$ is this the case? $s=$ [ANS]\n$t=$ [ANS]", "answer_v3": [ "no", "-6", "2", "2" ], "answer_type_v3": [ "TF", "NV", "NV", "NV" ], "options_v3": [ [ "yes", "no" ], [], [], [] ] }, { "id": "Calculus_-_multivariable_0138", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Parameterized surfaces", "level": "4", "keywords": [ "parametric surfaces", "parametric equations", "multivariable", "functions" ], "problem_v1": "Suppose you are standing at a point on the equator of a sphere, parameterized by spherical coordinates $\\theta$, and $\\phi$. Let \"north\" point to the top of the sphere and \"west\" and \"east\" similarly be chosen as if the sphere were a globe. What can you say about your initial $\\phi$ coordinate? $\\phi_0=$ [ANS]\nLet your initial $\\theta$ coordinate be $\\theta=a$. If you go east two thirds of the way around the equator and halfway up toward the south pole along a longitude, what are your new $\\theta$ and $\\phi$ coordinates? $\\theta=$ [ANS]\n$\\phi=$ [ANS]", "answer_v1": [ "pi/2", "a+4*pi/3", "3*pi/4" ], "answer_type_v1": [ "NV", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose you are standing at a point on the equator of a sphere, parameterized by spherical coordinates $\\theta$, and $\\phi$. Let \"north\" point to the top of the sphere and \"west\" and \"east\" similarly be chosen as if the sphere were a globe. What can you say about your initial $\\phi$ coordinate? $\\phi_0=$ [ANS]\nLet your initial $\\theta$ coordinate be $\\theta=a$. If you go east one quarter of the way around the equator and halfway up toward the south pole along a longitude, what are your new $\\theta$ and $\\phi$ coordinates? $\\theta=$ [ANS]\n$\\phi=$ [ANS]", "answer_v2": [ "pi/2", "a+pi/2", "3*pi/4" ], "answer_type_v2": [ "NV", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose you are standing at a point on the equator of a sphere, parameterized by spherical coordinates $\\theta$, and $\\phi$. Let \"north\" point to the top of the sphere and \"west\" and \"east\" similarly be chosen as if the sphere were a globe. What can you say about your initial $\\phi$ coordinate? $\\phi_0=$ [ANS]\nLet your initial $\\theta$ coordinate be $\\theta=a$. If you go east one third of the way around the equator and halfway up toward the south pole along a longitude, what are your new $\\theta$ and $\\phi$ coordinates? $\\theta=$ [ANS]\n$\\phi=$ [ANS]", "answer_v3": [ "pi/2", "a+2*pi/3", "3*pi/4" ], "answer_type_v3": [ "NV", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0139", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Parameterized surfaces", "level": "3", "keywords": [ "parametric surfaces", "parametric equations", "multivariable", "functions" ], "problem_v1": "Find parametric equations for the sphere centered at the origin and with radius 8. Use the parameters $s$ and $t$ in your answer. $x(s,t)=$ [ANS], $y(s,t)=$ [ANS], and $z(s,t)=$ [ANS], where [ANS] $\\le s\\le$ [ANS] and [ANS] $\\le t\\le$ [ANS].", "answer_v1": [ "8*cos(t)*sin(s)", "8*sin(t)*sin(s)", "8*cos(s)", "0", "2*pi", "0", "pi" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Find parametric equations for the sphere centered at the origin and with radius 3. Use the parameters $s$ and $t$ in your answer. $x(s,t)=$ [ANS], $y(s,t)=$ [ANS], and $z(s,t)=$ [ANS], where [ANS] $\\le s\\le$ [ANS] and [ANS] $\\le t\\le$ [ANS].", "answer_v2": [ "3*cos(t)*sin(s)", "3*sin(t)*sin(s)", "3*cos(s)", "0", "2*pi", "0", "pi" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Find parametric equations for the sphere centered at the origin and with radius 5. Use the parameters $s$ and $t$ in your answer. $x(s,t)=$ [ANS], $y(s,t)=$ [ANS], and $z(s,t)=$ [ANS], where [ANS] $\\le s\\le$ [ANS] and [ANS] $\\le t\\le$ [ANS].", "answer_v3": [ "5*cos(t)*sin(s)", "5*sin(t)*sin(s)", "5*cos(s)", "0", "2*pi", "0", "pi" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0140", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Parameterized surfaces", "level": "3", "keywords": [ "parametric surfaces", "parametric equations", "multivariable", "functions" ], "problem_v1": "Parameterize the plane that contains the three points $(3, 1, 1)$, $(6,-6,-4)$, and $(30, 30, 20)$. $\\vec r(s,t)=$ [ANS]\n(Use $s$ and $t$ for the parameters in your parameterization, and enter your vector as a single vector, with angle brackets: e.g., as <\u00a01+s+t, s-t, 3-t\u00a0>.)", "answer_v1": [ "(3+3*s+27*t,1-7*s+29*t,1-5*s+19*t)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Parameterize the plane that contains the three points $(-5, 5,-4)$, $(-4, 12,-4)$, and $(10, 15, 30)$. $\\vec r(s,t)=$ [ANS]\n(Use $s$ and $t$ for the parameters in your parameterization, and enter your vector as a single vector, with angle brackets: e.g., as <\u00a01+s+t, s-t, 3-t\u00a0>.)", "answer_v2": [ "(s-5+15*t,5+7*s+10*t,(-4)+34*t)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Parameterize the plane that contains the three points $(-2, 1,-2)$, $(2,-8,-4)$, and $(40, 50, 45)$. $\\vec r(s,t)=$ [ANS]\n(Use $s$ and $t$ for the parameters in your parameterization, and enter your vector as a single vector, with angle brackets: e.g., as <\u00a01+s+t, s-t, 3-t\u00a0>.)", "answer_v3": [ "(4*s-2+42*t,1-9*s+49*t,47*t-(2+2*s))" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0141", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Parameterized surfaces", "level": "3", "keywords": [ "parametric surfaces", "parametric equations", "multivariable", "functions" ], "problem_v1": "Parameterize a vase formed by rotating the curve $z=8 \\sqrt{x-3},\\,3\\leq x\\leq 6$, around the $z$-axis. Use $s$ and $t$ for your parameters. $x(s,t)=$ [ANS], $y(s,t)=$ [ANS], and $z(s,t)=$ [ANS], with [ANS] $\\le s\\le$ [ANS] and [ANS] $\\le t\\le$ [ANS]", "answer_v1": [ "s*cos(t)", "s*sin(t)", "8*sqrt(s-3)", "3", "6", "0", "2*pi" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Parameterize a vase formed by rotating the curve $z=2 \\sqrt{x-1},\\,1\\leq x\\leq 6$, around the $z$-axis. Use $s$ and $t$ for your parameters. $x(s,t)=$ [ANS], $y(s,t)=$ [ANS], and $z(s,t)=$ [ANS], with [ANS] $\\le s\\le$ [ANS] and [ANS] $\\le t\\le$ [ANS]", "answer_v2": [ "s*cos(t)", "s*sin(t)", "2*sqrt(s-1)", "1", "6", "0", "2*pi" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Parameterize a vase formed by rotating the curve $z=4 \\sqrt{x-1},\\,1\\leq x\\leq 5$, around the $z$-axis. Use $s$ and $t$ for your parameters. $x(s,t)=$ [ANS], $y(s,t)=$ [ANS], and $z(s,t)=$ [ANS], with [ANS] $\\le s\\le$ [ANS] and [ANS] $\\le t\\le$ [ANS]", "answer_v3": [ "s*cos(t)", "s*sin(t)", "4*sqrt(s-1)", "1", "5", "0", "2*pi" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0142", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Parameterized surfaces", "level": "5", "keywords": [ "parametric surfaces", "parametric equations", "multivariable", "functions" ], "problem_v1": "A city is described parametrically by the equation \\vec r=(a \\vec i+b \\vec j+c \\vec k)+s\\vec{v_1}+t\\vec{v_2} where \\vec{v}_1=4\\,\\mathit{\\vec i}-\\,\\mathit{\\vec j}-\\,\\mathit{\\vec k} and \\vec{v}_2=-\\,\\mathit{\\vec i}+3\\,\\mathit{\\vec j}-7\\,\\mathit{\\vec k}. A city block is a rectangle determined by $\\vec{v}_1$ and $\\vec{v}_2$. East is in the direction of $\\vec{v}_1$ and north is in the direction of $\\vec{v}_2$. Starting at the point $(a,b,c)$, you walk $3$ blocks east, $5$ blocks north, $2$ blocks west and $3$ blocks south.\n(a) What are the parameters of the point where you end up? $s=$ [ANS]\n$t=$ [ANS]\n(b) What are the coordinates at that point? $x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v1": [ "1", "2", "a+2", "b+5", "c+-15" ], "answer_type_v1": [ "NV", "NV", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "A city is described parametrically by the equation \\vec r=(a \\vec i+b \\vec j+c \\vec k)+s\\vec{v_1}+t\\vec{v_2} where \\vec{v}_1=3\\,\\mathit{\\vec i}-\\,\\mathit{\\vec j}-\\,\\mathit{\\vec k} and \\vec{v}_2=-\\,\\mathit{\\vec i}+5\\,\\mathit{\\vec j}-8\\,\\mathit{\\vec k}. A city block is a rectangle determined by $\\vec{v}_1$ and $\\vec{v}_2$. East is in the direction of $\\vec{v}_1$ and north is in the direction of $\\vec{v}_2$. Starting at the point $(a,b,c)$, you walk $1$ block east, $2$ blocks north, $3$ blocks west and $1$ block south.\n(a) What are the parameters of the point where you end up? $s=$ [ANS]\n$t=$ [ANS]\n(b) What are the coordinates at that point? $x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v2": [ "-2", "1", "a+-7", "b+7", "c+-6" ], "answer_type_v2": [ "NV", "NV", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "A city is described parametrically by the equation \\vec r=(a \\vec i+b \\vec j+c \\vec k)+s\\vec{v_1}+t\\vec{v_2} where \\vec{v}_1=4\\,\\mathit{\\vec i}-\\,\\mathit{\\vec j}-\\,\\mathit{\\vec k} and \\vec{v}_2=-\\,\\mathit{\\vec i}+2\\,\\mathit{\\vec j}-6\\,\\mathit{\\vec k}. A city block is a rectangle determined by $\\vec{v}_1$ and $\\vec{v}_2$. East is in the direction of $\\vec{v}_1$ and north is in the direction of $\\vec{v}_2$. Starting at the point $(a,b,c)$, you walk $5$ blocks east, $5$ blocks north, $7$ blocks west and $2$ blocks south.\n(a) What are the parameters of the point where you end up? $s=$ [ANS]\n$t=$ [ANS]\n(b) What are the coordinates at that point? $x=$ [ANS]\n$y=$ [ANS]\n$z=$ [ANS]", "answer_v3": [ "-2", "3", "a+-11", "b+8", "c+-16" ], "answer_type_v3": [ "NV", "NV", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0143", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Parameterized surfaces", "level": "3", "keywords": [ "calculus", "tangent plane", "partial derivative" ], "problem_v1": "Find an equation of the tangent plane (in the variables x, y and z) to the parametric surface $\\mathbf{r}(u,v)=\\langle 3u, u^2+v, 2v^2 \\rangle$ at the point $(-3, 0, 2)$. [ANS]", "answer_v1": [ "4 * 1* 2 * -1 * -1 *(x - 3 * -1) - 2 * 3 * 2 * -1 * (y - (1 * (-1*-1) + 1 * -1)) + 3 * 1 * (z - 2 * (-1*-1))=0" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Find an equation of the tangent plane (in the variables x, y and z) to the parametric surface $\\mathbf{r}(u,v)=\\langle-5u, 5u^2-4v,-2v^2 \\rangle$ at the point $(-15, 49,-2)$. [ANS]", "answer_v2": [ "4 * 5* -2 * 3 * -1 *(x - -5 * 3) - 2 * -5 * -2 * -1 * (y - (5 * (3*3) + -4 * -1)) + -5 * -4 * (z - -2 * (-1*-1))=0" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Find an equation of the tangent plane (in the variables x, y and z) to the parametric surface $\\mathbf{r}(u,v)=\\langle-2u, u^2-2v, v^2 \\rangle$ at the point $(4, 6, 1)$. [ANS]", "answer_v3": [ "4 * 1* 1 * -2 * -1 *(x - -2 * -2) - 2 * -2 * 1 * -1 * (y - (1 * (-2*-2) + -2 * -1)) + -2 * -2 * (z - 1 * (-1*-1))=0" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0144", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "function", "level", "surface", "curve", "Multivariable", "Graph", "Contour", "vector", "function" ], "problem_v1": "Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface. [ANS] 1. $z= \\frac{1}{x-1} $ [ANS] 2. $z=xy$ [ANS] 3. $z=2x+3y$ [ANS] 4. $z=2x^2+3y^2$ [ANS] 5. $z=\\sqrt{(x^2+y^2)}$ [ANS] 6. $z=\\sqrt{(25-x^2-y^2)}$ [ANS] 7. $z=x^2+y^2$\nA. a collection of equally spaced parallel lines B. a collection of concentric ellipses C. two straight lines and a collection of hyperbolas D. a collection of unequally spaced parallel lines E. a collection of equally spaced concentric circles F. a collection of unequally spaced concentric circles", "answer_v1": [ "D", "C", "A", "B", "E", "F", "F" ], "answer_type_v1": [ "MCS", "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" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface. [ANS] 1. $z=\\sqrt{(x^2+y^2)}$ [ANS] 2. $z=xy$ [ANS] 3. $z=\\sqrt{(25-x^2-y^2)}$ [ANS] 4. $z= \\frac{1}{x-1} $ [ANS] 5. $z=2x+3y$ [ANS] 6. $z=x^2+y^2$ [ANS] 7. $z=2x^2+3y^2$\nA. a collection of equally spaced parallel lines B. a collection of unequally spaced concentric circles C. a collection of concentric ellipses D. a collection of equally spaced concentric circles E. two straight lines and a collection of hyperbolas F. a collection of unequally spaced parallel lines", "answer_v2": [ "D", "E", "B", "F", "A", "B", "C" ], "answer_type_v2": [ "MCS", "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" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface. [ANS] 1. $z=2x+3y$ [ANS] 2. $z=\\sqrt{(25-x^2-y^2)}$ [ANS] 3. $z= \\frac{1}{x-1} $ [ANS] 4. $z=2x^2+3y^2$ [ANS] 5. $z=\\sqrt{(x^2+y^2)}$ [ANS] 6. $z=xy$ [ANS] 7. $z=x^2+y^2$\nA. a collection of unequally spaced parallel lines B. two straight lines and a collection of hyperbolas C. a collection of equally spaced concentric circles D. a collection of equally spaced parallel lines E. a collection of unequally spaced concentric circles F. a collection of concentric ellipses", "answer_v3": [ "D", "E", "A", "F", "C", "B", "E" ], "answer_type_v3": [ "MCS", "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" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Calculus_-_multivariable_0145", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "Contour", "Topography", "Gradient" ], "problem_v1": "Frodo and Sam are studying a topographic map of Mordor. Place the letter describing contour lines on a map to the left of the number describing a possible goal. [ANS] 1. If Frodo and Sam want to go directly uphill, they should go: [ANS] 2. If Frodo and Sam want to find Mount Doom, they should look for: [ANS] 3. If Frodo and Sam want to find the River Anduin, they should look for: [ANS] 4. If Frodo and Sam want to find a level route, they should look at:\nA. Perpendicular to the contour lines B. Parallel contour lines C. Concentric contour lines D. Single contour lines", "answer_v1": [ "A", "C", "B", "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": "Frodo and Sam are studying a topographic map of Mordor. Place the letter describing contour lines on a map to the left of the number describing a possible goal. [ANS] 1. If Frodo and Sam want to find Mount Doom, they should look for: [ANS] 2. If Frodo and Sam want to find the River Anduin, they should look for: [ANS] 3. If Frodo and Sam want to go directly uphill, they should go: [ANS] 4. If Frodo and Sam want to find a level route, they should look at:\nA. Perpendicular to the contour lines B. Single contour lines C. Parallel contour lines D. Concentric contour lines", "answer_v2": [ "D", "C", "A", "B" ], "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": "Frodo and Sam are studying a topographic map of Mordor. Place the letter describing contour lines on a map to the left of the number describing a possible goal. [ANS] 1. If Frodo and Sam want to find Mount Doom, they should look for: [ANS] 2. If Frodo and Sam want to find a level route, they should look at: [ANS] 3. If Frodo and Sam want to go directly uphill, they should go: [ANS] 4. If Frodo and Sam want to find the River Anduin, they should look for:\nA. Concentric contour lines B. Single contour lines C. Parallel contour lines D. Perpendicular to the contour lines", "answer_v3": [ "A", "B", "D", "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_-_multivariable_0146", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "multivariable", "functions", "level surface" ], "problem_v1": "Find a function $f(x,y,z)$ whose level surface $f=5$ is the graph of the function $g(x,y)=7x+5y$. $f(x,y,z)=$ [ANS]", "answer_v1": [ "7*x+5*y-z+5" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a function $f(x,y,z)$ whose level surface $f=2$ is the graph of the function $g(x,y)=x+8y$. $f(x,y,z)=$ [ANS]", "answer_v2": [ "x+8*y-z+2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a function $f(x,y,z)$ whose level surface $f=3$ is the graph of the function $g(x,y)=3x+5y$. $f(x,y,z)=$ [ANS]", "answer_v3": [ "3*x+5*y-z+3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0147", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "multivariable", "functions", "level surface" ], "problem_v1": "The surface $S$ is the graph of $f(x,y)=\\sqrt{16-y^{2}}$. Be sure you can explain why $S$ is the upper half of a circular cylinder of radius 4, centered along the $x$-axis. Write three or four sentences on a sheet of paper that clearly demonstrate this. Find a level surface $g(x,y,z)=c$ representing $S$. $g(x,y,z)=$ [ANS], with $c=$ [ANS]\n(Note that because your answers for $g$ and $c$ are interdependent, you cannot get partial credit for this problem.)", "answer_v1": [ "sqrt(16-y^2)-z", "0" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The surface $S$ is the graph of $f(x,y)=\\sqrt{1-y^{2}}$. Be sure you can explain why $S$ is the upper half of a circular cylinder of radius 1, centered along the $x$-axis. Write three or four sentences on a sheet of paper that clearly demonstrate this. Find a level surface $g(x,y,z)=c$ representing $S$. $g(x,y,z)=$ [ANS], with $c=$ [ANS]\n(Note that because your answers for $g$ and $c$ are interdependent, you cannot get partial credit for this problem.)", "answer_v2": [ "sqrt(1-y^2)-z", "0" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The surface $S$ is the graph of $f(x,y)=\\sqrt{4-y^{2}}$. Be sure you can explain why $S$ is the upper half of a circular cylinder of radius 2, centered along the $x$-axis. Write three or four sentences on a sheet of paper that clearly demonstrate this. Find a level surface $g(x,y,z)=c$ representing $S$. $g(x,y,z)=$ [ANS], with $c=$ [ANS]\n(Note that because your answers for $g$ and $c$ are interdependent, you cannot get partial credit for this problem.)", "answer_v3": [ "sqrt(4-y^2)-z", "0" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0148", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "multivariable", "functions", "level surface" ], "problem_v1": "Describe in your own words the level surfaces of the function $g(x,y,z)=4x+3y+4z$. Think what the intersections of these surfaces with the three coordinate planes look like.\n(a) Find a formula for intersections of the level surfaces with the $xz$-plane: $z=$ [ANS]\n(Your answer may involve an arbitrary constant.) (Your answer may involve an arbitrary constant.) (b) Find a formula for intersections of the level surfaces with the $yz$-plane: $z=$ [ANS]\n(Your answer may involve an arbitrary constant.) (Your answer may involve an arbitrary constant.) (c) Find a formula for intersections of the level surfaces with the $xy$-plane: $y=$ [ANS]\n(Your answer may involve an arbitrary constant.) (Your answer may involve an arbitrary constant.)", "answer_v1": [ "(c-4*x)/4", "(c-3*y)/4", "(c-4*x)/3" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Describe in your own words the level surfaces of the function $g(x,y,z)=x+5y+z$. Think what the intersections of these surfaces with the three coordinate planes look like.\n(a) Find a formula for intersections of the level surfaces with the $xz$-plane: $z=$ [ANS]\n(Your answer may involve an arbitrary constant.) (Your answer may involve an arbitrary constant.) (b) Find a formula for intersections of the level surfaces with the $yz$-plane: $z=$ [ANS]\n(Your answer may involve an arbitrary constant.) (Your answer may involve an arbitrary constant.) (c) Find a formula for intersections of the level surfaces with the $xy$-plane: $y=$ [ANS]\n(Your answer may involve an arbitrary constant.) (Your answer may involve an arbitrary constant.)", "answer_v2": [ "c-x", "c-5*y", "(c-x)/5" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Describe in your own words the level surfaces of the function $g(x,y,z)=2x+4y+2z$. Think what the intersections of these surfaces with the three coordinate planes look like.\n(a) Find a formula for intersections of the level surfaces with the $xz$-plane: $z=$ [ANS]\n(Your answer may involve an arbitrary constant.) (Your answer may involve an arbitrary constant.) (b) Find a formula for intersections of the level surfaces with the $yz$-plane: $z=$ [ANS]\n(Your answer may involve an arbitrary constant.) (Your answer may involve an arbitrary constant.) (c) Find a formula for intersections of the level surfaces with the $xy$-plane: $y=$ [ANS]\n(Your answer may involve an arbitrary constant.) (Your answer may involve an arbitrary constant.)", "answer_v3": [ "(c-2*x)/2", "(c-4*y)/2", "(c-2*x)/4" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0149", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "multivariable", "functions", "level surface" ], "problem_v1": "Find a formula for a function $g(x,y,z)$ whose level surfaces are planes parallel to the plane $z=7x+6y-6$. $g(x,y,z)=$ [ANS]", "answer_v1": [ "7*x+6*y-z" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find a formula for a function $g(x,y,z)$ whose level surfaces are planes parallel to the plane $z=2x+8y-3$. $g(x,y,z)=$ [ANS]", "answer_v2": [ "2*x+8*y-z" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find a formula for a function $g(x,y,z)$ whose level surfaces are planes parallel to the plane $z=4x+6y-3$. $g(x,y,z)=$ [ANS]", "answer_v3": [ "4*x+6*y-z" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0150", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "multivariable", "functions", "level surface" ], "problem_v1": "Write the level surface $5= \\frac{4x^{2}+3y}{z} $ as the graph of a function $f(x,y).$ $f(x,y)=$ [ANS]", "answer_v1": [ "(4*x^2+3*y)/5" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write the level surface $2=x+5y^{2}+\\sqrt{z}$ as the graph of a function $f(x,y).$ $f(x,y)=$ [ANS]", "answer_v2": [ "(2-x-5*y^2)^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Write the level surface $3= \\frac{2x^{2}+4y}{z} $ as the graph of a function $f(x,y).$ $f(x,y)=$ [ANS]", "answer_v3": [ "(2*x^2+4*y)/3" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0151", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "linear", "functions", "multivariable", "plane" ], "problem_v1": "Let $f$ be the linear function $f(x,y)=c+m x+n y$, where $c, m, n$ are constants and $n \\neq 0$.\nNote: For all parts of this problem, your answers must be in expanded form--that is, may not contain any parentheses--and may involve the constants $c, m, n$ and $k$.\n(a) Find and expand the equation of the contour that goes through the value $f(x,y)=k$. $y=$ [ANS]\n(b) Note that your equation is a line. What is the slope of the line? slope=[ANS]\n(c) Now suppose that we look at the point $(x,y)=(3,6)$. Find and simplify $f(3,6)$: $f(3, 6)=$ [ANS]\nFind and simplify $f(3+n,6-m)$: $f(3+n, 6-m)=$ [ANS]\n(d) What do you notice about your result in (c)? Confirm that this works in general by calculating $f(x+m,y-n)$ for any $x$ and $y$ (simplify your expression by expanding out terms): $f(x+n, y-m)=$ [ANS]\nFinally, be sure that you can explain how this last result is related the equation of the line and slope that you found in (a) and (b).", "answer_v1": [ "k/n-c/n-m*x/n", "-m/n", "c+m*3+n*6", "c+m*3+n*6", "c+m*x+n*y" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Let $f$ be the linear function $f(x,y)=c+m x+n y$, where $c, m, n$ are constants and $n \\neq 0$.\nNote: For all parts of this problem, your answers must be in expanded form--that is, may not contain any parentheses--and may involve the constants $c, m, n$ and $k$.\n(a) Find and expand the equation of the contour that goes through the value $f(x,y)=k$. $y=$ [ANS]\n(b) Note that your equation is a line. What is the slope of the line? slope=[ANS]\n(c) Now suppose that we look at the point $(x,y)=(1,6)$. Find and simplify $f(1,6)$: $f(1, 6)=$ [ANS]\nFind and simplify $f(1+n,6-m)$: $f(1+n, 6-m)=$ [ANS]\n(d) What do you notice about your result in (c)? Confirm that this works in general by calculating $f(x+m,y-n)$ for any $x$ and $y$ (simplify your expression by expanding out terms): $f(x+n, y-m)=$ [ANS]\nFinally, be sure that you can explain how this last result is related the equation of the line and slope that you found in (a) and (b).", "answer_v2": [ "k/n-c/n-m*x/n", "-m/n", "c+m*1+n*6", "c+m*1+n*6", "c+m*x+n*y" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Let $f$ be the linear function $f(x,y)=c+m x+n y$, where $c, m, n$ are constants and $n \\neq 0$.\nNote: For all parts of this problem, your answers must be in expanded form--that is, may not contain any parentheses--and may involve the constants $c, m, n$ and $k$.\n(a) Find and expand the equation of the contour that goes through the value $f(x,y)=k$. $y=$ [ANS]\n(b) Note that your equation is a line. What is the slope of the line? slope=[ANS]\n(c) Now suppose that we look at the point $(x,y)=(1,5)$. Find and simplify $f(1,5)$: $f(1, 5)=$ [ANS]\nFind and simplify $f(1+n,5-m)$: $f(1+n, 5-m)=$ [ANS]\n(d) What do you notice about your result in (c)? Confirm that this works in general by calculating $f(x+m,y-n)$ for any $x$ and $y$ (simplify your expression by expanding out terms): $f(x+n, y-m)=$ [ANS]\nFinally, be sure that you can explain how this last result is related the equation of the line and slope that you found in (a) and (b).", "answer_v3": [ "k/n-c/n-m*x/n", "-m/n", "c+m*1+n*5", "c+m*1+n*5", "c+m*x+n*y" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0152", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "graphing", "multivariable", "functions" ], "problem_v1": "By setting one variable constant, find a plane that intersects the graph of $z=9x^{2}-3y^{2}+4$ in a:\n(a) Parabola opening upward: the plane [ANS]=[ANS]\n(Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (b) Parabola opening downward: the plane [ANS]=[ANS]\n(Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (c) Pair of intersecting straight lines: the plane [ANS]=[ANS]\n(Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (Give your answer by specifying the variable in the first answer blank and a value for it in the second.)", "answer_v1": [ "y", "0", "x", "0", "z", "4" ], "answer_type_v1": [ "EX", "NV", "EX", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "By setting one variable constant, find a plane that intersects the graph of $z=2y^{2}-6x^{2}+2$ in a:\n(a) Parabola opening upward: the plane [ANS]=[ANS]\n(Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (b) Parabola opening downward: the plane [ANS]=[ANS]\n(Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (c) Pair of intersecting straight lines: the plane [ANS]=[ANS]\n(Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (Give your answer by specifying the variable in the first answer blank and a value for it in the second.)", "answer_v2": [ "x", "0", "y", "0", "z", "2" ], "answer_type_v2": [ "EX", "NV", "EX", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "By setting one variable constant, find a plane that intersects the graph of $z=2y^{2}-6x^{2}+3$ in a:\n(a) Parabola opening upward: the plane [ANS]=[ANS]\n(Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (b) Parabola opening downward: the plane [ANS]=[ANS]\n(Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (c) Pair of intersecting straight lines: the plane [ANS]=[ANS]\n(Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (Give your answer by specifying the variable in the first answer blank and a value for it in the second.)", "answer_v3": [ "x", "0", "y", "0", "z", "3" ], "answer_type_v3": [ "EX", "NV", "EX", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0155", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "3", "keywords": [ "graph", "image" ], "problem_v1": "Let $f: \\mathbb{R} \\to \\mathbb{R}$ be defined by $f(x)=4x^{2}+5$ and set $y=f(x)$.\n(a) The level sets of $f$ are subsets of $\\mathbb{R}^k$ for $k=$ [ANS]\n(b) Find the level set of $f$ at level $y=8$. If there is more than one answer, enter your answers as a comma separated list. [ANS]", "answer_v1": [ "1", "(0.866025, -0.866025)" ], "answer_type_v1": [ "NV", "UOL" ], "options_v1": [ [], [] ], "problem_v2": "Let $f: \\mathbb{R} \\to \\mathbb{R}$ be defined by $f(x)=5x^{2}+2$ and set $y=f(x)$.\n(a) The level sets of $f$ are subsets of $\\mathbb{R}^k$ for $k=$ [ANS]\n(b) Find the level set of $f$ at level $y=6$. If there is more than one answer, enter your answers as a comma separated list. [ANS]", "answer_v2": [ "1", "(0.894427, -0.894427)" ], "answer_type_v2": [ "NV", "UOL" ], "options_v2": [ [], [] ], "problem_v3": "Let $f: \\mathbb{R} \\to \\mathbb{R}$ be defined by $f(x)=4x^{2}+3$ and set $y=f(x)$.\n(a) The level sets of $f$ are subsets of $\\mathbb{R}^k$ for $k=$ [ANS]\n(b) Find the level set of $f$ at level $y=7$. If there is more than one answer, enter your answers as a comma separated list. [ANS]", "answer_v3": [ "1", "(1, -1)" ], "answer_type_v3": [ "NV", "UOL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0156", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "vector' 'function' 'level' 'surface' 'curve", "function", "level", "surface", "curve", "vector", "function" ], "problem_v1": "For each function below, consider its level surfaces, where $w=\\cdots-2,-1,0,1,2\\cdots$ if negative $w$ 's are allowed, or $w=1,2,3,\\cdots$ if negative $w$ 's are not allowed. Match each function with the verbal description of its level surfaces by placing the letter of the verbal description to the left of the number of the function. [ANS] 1. $w=x^2+2y^2+3z^2$ [ANS] 2. $w=\\sqrt{(x^2+2y^2+3z^2)}$ [ANS] 3. $w=x+2y+3z$ [ANS] 4. $w=\\sqrt{(x^2+y^2+z^2)}$ [ANS] 5. $w=x^2+y^2+z^2$ [ANS] 6. $w=\\sqrt{(x+2y+3z)}$ [ANS] 7. $w=x^2-y^2-z^2$\nA. two cones and two collections of hyperboloids B. a collection of equally spaced parallel planes C. a collection of equally spaced concentric spheres D. a collection of concentric ellipsoids E. a collection of unequally spaced concentric spheres F. a collection of unequally spaced parallel planes", "answer_v1": [ "D", "D", "B", "C", "E", "F", "A" ], "answer_type_v1": [ "MCS", "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" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v2": "For each function below, consider its level surfaces, where $w=\\cdots-2,-1,0,1,2\\cdots$ if negative $w$ 's are allowed, or $w=1,2,3,\\cdots$ if negative $w$ 's are not allowed. Match each function with the verbal description of its level surfaces by placing the letter of the verbal description to the left of the number of the function. [ANS] 1. $w=x^2+y^2+z^2$ [ANS] 2. $w=\\sqrt{(x^2+2y^2+3z^2)}$ [ANS] 3. $w=\\sqrt{(x+2y+3z)}$ [ANS] 4. $w=x^2+2y^2+3z^2$ [ANS] 5. $w=x+2y+3z$ [ANS] 6. $w=x^2-y^2-z^2$ [ANS] 7. $w=\\sqrt{(x^2+y^2+z^2)}$\nA. a collection of equally spaced parallel planes B. a collection of unequally spaced parallel planes C. two cones and two collections of hyperboloids D. a collection of equally spaced concentric spheres E. a collection of unequally spaced concentric spheres F. a collection of concentric ellipsoids", "answer_v2": [ "E", "F", "B", "F", "A", "C", "D" ], "answer_type_v2": [ "MCS", "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" ], [ "A", "B", "C", "D", "E", "F" ] ], "problem_v3": "For each function below, consider its level surfaces, where $w=\\cdots-2,-1,0,1,2\\cdots$ if negative $w$ 's are allowed, or $w=1,2,3,\\cdots$ if negative $w$ 's are not allowed. Match each function with the verbal description of its level surfaces by placing the letter of the verbal description to the left of the number of the function. [ANS] 1. $w=x+2y+3z$ [ANS] 2. $w=\\sqrt{(x+2y+3z)}$ [ANS] 3. $w=x^2+2y^2+3z^2$ [ANS] 4. $w=\\sqrt{(x^2+y^2+z^2)}$ [ANS] 5. $w=x^2+y^2+z^2$ [ANS] 6. $w=\\sqrt{(x^2+2y^2+3z^2)}$ [ANS] 7. $w=x^2-y^2-z^2$\nA. a collection of unequally spaced parallel planes B. a collection of concentric ellipsoids C. a collection of unequally spaced concentric spheres D. a collection of equally spaced parallel planes E. two cones and two collections of hyperboloids F. a collection of equally spaced concentric spheres", "answer_v3": [ "D", "A", "B", "F", "C", "B", "E" ], "answer_type_v3": [ "MCS", "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" ], [ "A", "B", "C", "D", "E", "F" ] ] }, { "id": "Calculus_-_multivariable_0157", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "Graph", "Multivariable", "Level Curve" ], "problem_v1": "Sketch a contour diagram of each function. Then, decide whether its contours are predominantly lines, parabolas, ellipses, or hyperbolas.\n[ANS] 1. $z=x^2-4 y^2$ [ANS] 2. $z=-5x^2$ [ANS] 3. $z=x^2+4 y^2$ [ANS] 4. $z=y-4x^2$", "answer_v1": [ "HYPERBOLAS", "LINES", "ELLIPSES", "Parabolas" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ], [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ], [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ], [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ] ], "problem_v2": "Sketch a contour diagram of each function. Then, decide whether its contours are predominantly lines, parabolas, ellipses, or hyperbolas.\n[ANS] 1. $z=x^2+2 y^2$ [ANS] 2. $z=-2x^2$ [ANS] 3. $z=x^2-3 y^2$ [ANS] 4. $z=y-5x^2$", "answer_v2": [ "ELLIPSES", "LINES", "HYPERBOLAS", "Parabolas" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ], [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ], [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ], [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ] ], "problem_v3": "Sketch a contour diagram of each function. Then, decide whether its contours are predominantly lines, parabolas, ellipses, or hyperbolas.\n[ANS] 1. $z=x^2-4 y^2$ [ANS] 2. $z=x^2+3 y^2$ [ANS] 3. $z=y-4x^2$ [ANS] 4. $z=-3x^2$", "answer_v3": [ "HYPERBOLAS", "ELLIPSES", "PARABOLAS", "Lines" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ], [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ], [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ], [ "Lines", "Parabolas", "Ellipses", "Hyperbolas" ] ] }, { "id": "Calculus_-_multivariable_0158", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "Graph", "Multivariable", "Level Curve" ], "problem_v1": "A manufacturer sells aardvark masks at a price of \\$220 per mask and butterfly masks at a price of \\$530 per mask. A quantity of a aardvark masks and b butterfly masks is sold at a total cost of \\$550 to the manufacturer.\n(a) Express the manufacturer's profit, P, as a function of a and b. $P(a,b)=$ [ANS] dollars.\n(b) The curves of constant profit in the ab-plane are [ANS] A. parabolas B. lines C. hyperbolas D. circles E. ellipses", "answer_v1": [ "220*a+530*b-550", "B" ], "answer_type_v1": [ "EX", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D", "E" ] ], "problem_v2": "A manufacturer sells aardvark masks at a price of \\$290 per mask and butterfly masks at a price of \\$430 per mask. A quantity of a aardvark masks and b butterfly masks is sold at a total cost of \\$300 to the manufacturer.\n(a) Express the manufacturer's profit, P, as a function of a and b. $P(a,b)=$ [ANS] dollars.\n(b) The curves of constant profit in the ab-plane are [ANS] A. hyperbolas B. parabolas C. lines D. ellipses E. circles", "answer_v2": [ "290*a+430*b-300", "C" ], "answer_type_v2": [ "EX", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D", "E" ] ], "problem_v3": "A manufacturer sells aardvark masks at a price of \\$220 per mask and butterfly masks at a price of \\$450 per mask. A quantity of a aardvark masks and b butterfly masks is sold at a total cost of \\$400 to the manufacturer.\n(a) Express the manufacturer's profit, P, as a function of a and b. $P(a,b)=$ [ANS] dollars.\n(b) The curves of constant profit in the ab-plane are [ANS] A. parabolas B. hyperbolas C. ellipses D. lines E. circles", "answer_v3": [ "220*a+450*b-400", "D" ], "answer_type_v3": [ "EX", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D", "E" ] ] }, { "id": "Calculus_-_multivariable_0159", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "Graph", "Multivariable", "Level Curve" ], "problem_v1": "Decide whether the level surfaces of each function are concentric circular cylinders, concentric spheres, cones, elliptical paraboloids, hyperbolic paraboloids, hyperboloids of one sheet, hyperboloids of two sheets, parabolic cylinders, or parallel planes.\n[ANS] 1. $f(x,y,z)=\\cos(8x+y+z)$ [ANS] 2. $f(x,y,z)=4y^2-4x^2-z$ [ANS] 3. $f(x,y,z)=e^{-(x^2+y^2+z^2)}$ [ANS] 4. $f(x,y,z)=6x^2-y$ [ANS] 5. $f(x,y,z)=7x^2+y^2-z$ [ANS] 6. $f(x,y,z)=\\sin(\\sqrt{6(x^2+y^2+z^2)})$ [ANS] 7. $f(x,y,z)=x+y-6 z$ [ANS] 8. $f(x,y,z)=\\ln(\\sqrt{y^2+z^2})$", "answer_v1": [ "PARALLEL PLANES", "HYPERBOLIC PARABOLOIDS", "CONCENTRIC SPHERES", "Parabolic cylinders", "Elliptical paraboloids", "Concentric spheres", "Parallel planes", "Concentric circular cylinders" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ] ], "problem_v2": "Decide whether the level surfaces of each function are concentric circular cylinders, concentric spheres, cones, elliptical paraboloids, hyperbolic paraboloids, hyperboloids of one sheet, hyperboloids of two sheets, parabolic cylinders, or parallel planes.\n[ANS] 1. $f(x,y,z)=\\sin(\\sqrt{3(x^2+y^2+z^2)})$ [ANS] 2. $f(x,y,z)=9y^2-4x^2-z$ [ANS] 3. $f(x,y,z)=4x^2+y^2-z$ [ANS] 4. $f(x,y,z)=\\cos(2x+y+z)$ [ANS] 5. $f(x,y,z)=e^{-(x^2+y^2+z^2)}$ [ANS] 6. $f(x,y,z)=x+y-9 z$ [ANS] 7. $f(x,y,z)=3x^2-y$ [ANS] 8. $f(x,y,z)=\\ln(\\sqrt{y^2+z^2})$", "answer_v2": [ "CONCENTRIC SPHERES", "HYPERBOLIC PARABOLOIDS", "ELLIPTICAL PARABOLOIDS", "Parallel planes", "Concentric spheres", "Parallel planes", "Parabolic cylinders", "Concentric circular cylinders" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ] ], "problem_v3": "Decide whether the level surfaces of each function are concentric circular cylinders, concentric spheres, cones, elliptical paraboloids, hyperbolic paraboloids, hyperboloids of one sheet, hyperboloids of two sheets, parabolic cylinders, or parallel planes.\n[ANS] 1. $f(x,y,z)=4x^2-y$ [ANS] 2. $f(x,y,z)=6x^2+y^2-z$ [ANS] 3. $f(x,y,z)=\\ln(\\sqrt{y^2+z^2})$ [ANS] 4. $f(x,y,z)=e^{-(x^2+y^2+z^2)}$ [ANS] 5. $f(x,y,z)=3y^2-4x^2-z$ [ANS] 6. $f(x,y,z)=x+y-6 z$ [ANS] 7. $f(x,y,z)=\\sin(\\sqrt{8(x^2+y^2+z^2)})$ [ANS] 8. $f(x,y,z)=\\cos(4x+y+z)$", "answer_v3": [ "PARABOLIC CYLINDERS", "ELLIPTICAL PARABOLOIDS", "CONCENTRIC CIRCULAR CYLINDERS", "Concentric spheres", "Hyperbolic paraboloids", "Parallel planes", "Concentric spheres", "Parallel planes" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ], [ "Concentric circular cylinders", "Concentric spheres", "Cones", "Elliptical paraboloids", "Hyperbolic paraboloids", "Hyperboloids of one sheet", "Hyperboloids of two sheets", "Parabolic cylinders", "Parallel planes" ] ] }, { "id": "Calculus_-_multivariable_0160", "subject": "Calculus_-_multivariable", "topic": "Concepts for multivariable functions", "subtopic": "Traces, contours, and level sets", "level": "2", "keywords": [ "Multivariable", "Graph" ], "problem_v1": "Consider the sphere $(x-5)^2+(y-4)^2+(z-3)^2=25$\n(a) Does the sphere intersect each of the following planes at zero points, at one point, at two points, in a line, or in a circle?\nThe sphere intersects the yz-plane [ANS] The sphere intersects the xz-plane [ANS] The sphere intersects the xy-plane [ANS]\n(b) Does the sphere intersect each of the following coordinate axes at zero points, at one point, at two points, or in a line?\nThe sphere intersects the x-axis [ANS] The sphere intersects the y-axis [ANS] The sphere intersects the z-axis [ANS]", "answer_v1": [ "at one point", "in a circle", "in a circle", "at one point", "at zero points", "at zero points" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "at zero points", "at one point", "at two points", "in a line", "in a circle" ], [ "at zero points", "at one point", "at two points", "in a line", "in a circle" ], [ "at zero points", "at one point", "at two points", "in a line", "in a circle" ], [ "at zero points", "at one point", "at two points", "in a line" ], [ "at zero points", "at one point", "at two points", "in a line" ], [ "at zero points", "at one point", "at two points", "in a line" ] ], "problem_v2": "Consider the sphere $(x-3)^2+(y-5)^2+(z-4)^2=25$\n(a) Does the sphere intersect each of the following planes at zero points, at one point, at two points, in a line, or in a circle?\nThe sphere intersects the xz-plane [ANS] The sphere intersects the xy-plane [ANS] The sphere intersects the yz-plane [ANS]\n(b) Does the sphere intersect each of the following coordinate axes at zero points, at one point, at two points, or in a line?\nThe sphere intersects the z-axis [ANS] The sphere intersects the x-axis [ANS] The sphere intersects the y-axis [ANS]", "answer_v2": [ "at one point", "in a circle", "in a circle", "at zero points", "at zero points", "at one point" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "at zero points", "at one point", "at two points", "in a line", "in a circle" ], [ "at zero points", "at one point", "at two points", "in a line", "in a circle" ], [ "at zero points", "at one point", "at two points", "in a line", "in a circle" ], [ "at zero points", "at one point", "at two points", "in a line" ], [ "at zero points", "at one point", "at two points", "in a line" ], [ "at zero points", "at one point", "at two points", "in a line" ] ], "problem_v3": "Consider the sphere $(x-3)^2+(y-5)^2+(z-4)^2=25$\n(a) Does the sphere intersect each of the following planes at zero points, at one point, at two points, in a line, or in a circle?\nThe sphere intersects the yz-plane [ANS] The sphere intersects the xy-plane [ANS] The sphere intersects the xz-plane [ANS]\n(b) Does the sphere intersect each of the following coordinate axes at zero points, at one point, at two points, or in a line?\nThe sphere intersects the x-axis [ANS] The sphere intersects the y-axis [ANS] The sphere intersects the z-axis [ANS]", "answer_v3": [ "in a circle", "in a circle", "at one point", "at zero points", "at one point", "at zero points" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "at zero points", "at one point", "at two points", "in a line", "in a circle" ], [ "at zero points", "at one point", "at two points", "in a line", "in a circle" ], [ "at zero points", "at one point", "at two points", "in a line", "in a circle" ], [ "at zero points", "at one point", "at two points", "in a line" ], [ "at zero points", "at one point", "at two points", "in a line" ], [ "at zero points", "at one point", "at two points", "in a line" ] ] }, { "id": "Calculus_-_multivariable_0161", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Limits and continuity", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Use continuity to evaluate the limit. $ \\lim_{(x,y) \\to (3,3)} \\frac{e^{x^{3}}-e^{-y^{3}}}{x+y} =$ [ANS]", "answer_v1": [ "8.86747E+10" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use continuity to evaluate the limit. $ \\lim_{(x,y) \\to (-5,-5)} \\frac{e^{x^{4}}-e^{-y^{4}}}{x+y} =$ [ANS]", "answer_v2": [ "-2.71676E+270" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use continuity to evaluate the limit. $ \\lim_{(x,y) \\to (-2,-2)} \\frac{e^{x^{3}}-e^{-y^{3}}}{x+y} =$ [ANS]", "answer_v3": [ "745.239" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0162", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Limits and continuity", "level": "2", "keywords": [ "continuity", "limits", "multivariable", "functions" ], "problem_v1": "In this problem we show that the function f(x,y)= \\frac{6x^{2}-y^{2}}{x^{2} +y^{2}} does not have a limit as $(x,y)\\to (0,0)$.\n(a) Suppose that we consider $(x,y)\\to (0,0)$ along the curve $y=2x$. Find the limit in this case: $\\lim\\limits_{(x,2x)\\to(0,0)} \\frac{6x^{2}-y^{2}}{x^{2} +y^{2}}=$ [ANS]\n(b) Now consider $(x,y)\\to (0,0)$ along the curve $y=3x$. Find the limit in this case: $\\lim\\limits_{(x,3x)\\to(0,0)} \\frac{6x^{2}-y^{2}}{x^{2} +y^{2}}=$ [ANS]\n(c) Note that the results from\n(a) and (b) indicate that $f$ has no limit as $(x,y)\\to (0,0)$ (be sure you can explain why!) (be sure you can explain why!). To show this more generally, consider $(x,y)\\to (0,0)$ along the curve $y=m x$, for arbitrary $m$. Find the limit in this case: $\\lim\\limits_{(x,m x)\\to(0,0)} \\frac{6x^{2}-y^{2}}{x^{2} +y^{2}}=$ [ANS]\n(Be sure that you can explain how this result also indicates that $f$ has no limit as $(x,y)\\to(0,0)$.", "answer_v1": [ "(6-2^2)/(1+2^2)", "(6-3^2)/(1+3^2)", "(6-m^2)/(1+m^2)" ], "answer_type_v1": [ "NV", "NV", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "In this problem we show that the function f(x,y)= \\frac{3x^{2}-y}{x^{2} +y} does not have a limit as $(x,y)\\to (0,0)$.\n(a) Suppose that we consider $(x,y)\\to (0,0)$ along the curve $y=4x^2$. Find the limit in this case: $\\lim\\limits_{(x,4x^2)\\to(0,0)} \\frac{3x^{2}-y}{x^{2} +y}=$ [ANS]\n(b) Now consider $(x,y)\\to (0,0)$ along the curve $y=5x^2$. Find the limit in this case: $\\lim\\limits_{(x,5x^2)\\to(0,0)} \\frac{3x^{2}-y}{x^{2} +y}=$ [ANS]\n(c) Note that the results from\n(a) and (b) indicate that $f$ has no limit as $(x,y)\\to (0,0)$ (be sure you can explain why!) (be sure you can explain why!). To show this more generally, consider $(x,y)\\to (0,0)$ along the curve $y=m x^2$, for arbitrary $m$. Find the limit in this case: $\\lim\\limits_{(x,m x^2)\\to(0,0)} \\frac{3x^{2}-y}{x^{2} +y}=$ [ANS]\n(Be sure that you can explain how this result also indicates that $f$ has no limit as $(x,y)\\to(0,0)$.", "answer_v2": [ "(3-4^1)/(1+4^1)", "(3-5^1)/(1+5^1)", "(3-m)/(1+m)" ], "answer_type_v2": [ "NV", "NV", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "In this problem we show that the function f(x,y)= \\frac{5x^{2}-y}{x^{2} +y} does not have a limit as $(x,y)\\to (0,0)$.\n(a) Suppose that we consider $(x,y)\\to (0,0)$ along the curve $y=2x^2$. Find the limit in this case: $\\lim\\limits_{(x,2x^2)\\to(0,0)} \\frac{5x^{2}-y}{x^{2} +y}=$ [ANS]\n(b) Now consider $(x,y)\\to (0,0)$ along the curve $y=3x^2$. Find the limit in this case: $\\lim\\limits_{(x,3x^2)\\to(0,0)} \\frac{5x^{2}-y}{x^{2} +y}=$ [ANS]\n(c) Note that the results from\n(a) and (b) indicate that $f$ has no limit as $(x,y)\\to (0,0)$ (be sure you can explain why!) (be sure you can explain why!). To show this more generally, consider $(x,y)\\to (0,0)$ along the curve $y=m x^2$, for arbitrary $m$. Find the limit in this case: $\\lim\\limits_{(x,m x^2)\\to(0,0)} \\frac{5x^{2}-y}{x^{2} +y}=$ [ANS]\n(Be sure that you can explain how this result also indicates that $f$ has no limit as $(x,y)\\to(0,0)$.", "answer_v3": [ "(5-2^1)/(1+2^1)", "(5-3^1)/(1+3^1)", "(5-m)/(1+m)" ], "answer_type_v3": [ "NV", "NV", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0163", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Limits and continuity", "level": "2", "keywords": [ "continuity", "limits", "multivariable", "functions" ], "problem_v1": "What value of $c$ makes the following function continuous at $(0,0)$? f(x,y)=\\cases{x^{3}+y^{3}+4, & (x,y)\\ne (0,0)\\cr c, & (x,y)=(0,0)\\cr} $c=$ [ANS]", "answer_v1": [ "4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What value of $c$ makes the following function continuous at $(0,0)$? f(x,y)=\\cases{x+y^{4}+1, & (x,y)\\ne (0,0)\\cr c, & (x,y)=(0,0)\\cr} $c=$ [ANS]", "answer_v2": [ "1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What value of $c$ makes the following function continuous at $(0,0)$? f(x,y)=\\cases{x+y^{3}+2, & (x,y)\\ne (0,0)\\cr c, & (x,y)=(0,0)\\cr} $c=$ [ANS]", "answer_v3": [ "2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0164", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Limits and continuity", "level": "3", "keywords": [ "Multivariable", "limit" ], "problem_v1": "Examine the behavior of $ f(x,y)= \\frac{x^{4}y^{4}}{\\left(8x^{2} +6y^{4}\\right)^{3}}$ as $(x,y)$ approaches $(0,0)$.\nUse \"infinity\" for \" $\\infty$ \" and \"-infinity\" for \" $-\\infty$ \". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\".\n(a) Taking a linear approach to the origin along the $y$-axis, we find $ \\lim_{(x,y) \\to (0,0),\\ x=0} \\Bigg( \\frac{x^{4}y^{4}}{\\left(8x^{2} +6y^{4}\\right)^{3}} \\Bigg)=$ [ANS].\n(b) Taking a linear approach to the origin along the $x$-axis, we find $ \\lim_{(x,y) \\to (0,0),\\ y=0} \\Bigg( \\frac{x^{4}y^{4}}{\\left(8x^{2} +6y^{4}\\right)^{3}} \\Bigg)=$ [ANS].\n(c) Taking a linear approach to the origin along a line $y=m x$ with $m \\ne 0$ we find $ \\lim_{(x,y) \\to (0,0),\\ y=mx} \\Bigg( \\frac{x^{4}y^{4}}{\\left(8x^{2} +6y^{4}\\right)^{3}} \\Bigg)=\\lim_{x \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].\nThought question: Do your answers to parts\n(a)-(c) allow you to conclude that the limit exists?\n(d) Taking an approach to the origin along the parabola $x=y^2$ we find $ \\lim_{(x,y) \\to (0,0),\\ x=y^2} \\Bigg( \\frac{x^{4}y^{4}}{\\left(8x^{2} +6y^{4}\\right)^{3}} \\Bigg)=\\lim_{y \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].\n(e) Find the value of the limit. Be sure you can explain your answer. $ \\lim_{(x,y) \\to (0,0)} \\Bigg( \\frac{x^{4}y^{4}}{\\left(8x^{2} +6y^{4}\\right)^{3}} \\Bigg)=$ [ANS].", "answer_v1": [ "0", "0", "m^4*x^2/[(8+6*m^4*x^2)^3]", "0", "1/(14^3)", "1/(14^3)", "DNE" ], "answer_type_v1": [ "NV", "NV", "EX", "NV", "NV", "NV", "OE" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Examine the behavior of $ f(x,y)= \\frac{x^{4}y^{4}}{\\left(2x^{2} +9y^{4}\\right)^{3}}$ as $(x,y)$ approaches $(0,0)$.\nUse \"infinity\" for \" $\\infty$ \" and \"-infinity\" for \" $-\\infty$ \". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\".\n(a) Taking a linear approach to the origin along the $y$-axis, we find $ \\lim_{(x,y) \\to (0,0),\\ x=0} \\Bigg( \\frac{x^{4}y^{4}}{\\left(2x^{2} +9y^{4}\\right)^{3}} \\Bigg)=$ [ANS].\n(b) Taking a linear approach to the origin along the $x$-axis, we find $ \\lim_{(x,y) \\to (0,0),\\ y=0} \\Bigg( \\frac{x^{4}y^{4}}{\\left(2x^{2} +9y^{4}\\right)^{3}} \\Bigg)=$ [ANS].\n(c) Taking a linear approach to the origin along a line $y=m x$ with $m \\ne 0$ we find $ \\lim_{(x,y) \\to (0,0),\\ y=mx} \\Bigg( \\frac{x^{4}y^{4}}{\\left(2x^{2} +9y^{4}\\right)^{3}} \\Bigg)=\\lim_{x \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].\nThought question: Do your answers to parts\n(a)-(c) allow you to conclude that the limit exists?\n(d) Taking an approach to the origin along the parabola $x=y^2$ we find $ \\lim_{(x,y) \\to (0,0),\\ x=y^2} \\Bigg( \\frac{x^{4}y^{4}}{\\left(2x^{2} +9y^{4}\\right)^{3}} \\Bigg)=\\lim_{y \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].\n(e) Find the value of the limit. Be sure you can explain your answer. $ \\lim_{(x,y) \\to (0,0)} \\Bigg( \\frac{x^{4}y^{4}}{\\left(2x^{2} +9y^{4}\\right)^{3}} \\Bigg)=$ [ANS].", "answer_v2": [ "0", "0", "m^4*x^2/[(2+9*m^4*x^2)^3]", "0", "1/(11^3)", "1/(11^3)", "DNE" ], "answer_type_v2": [ "NV", "NV", "EX", "NV", "NV", "NV", "OE" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Examine the behavior of $ f(x,y)= \\frac{x^{4}y^{4}}{\\left(4x^{2} +6y^{4}\\right)^{3}}$ as $(x,y)$ approaches $(0,0)$.\nUse \"infinity\" for \" $\\infty$ \" and \"-infinity\" for \" $-\\infty$ \". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\".\n(a) Taking a linear approach to the origin along the $y$-axis, we find $ \\lim_{(x,y) \\to (0,0),\\ x=0} \\Bigg( \\frac{x^{4}y^{4}}{\\left(4x^{2} +6y^{4}\\right)^{3}} \\Bigg)=$ [ANS].\n(b) Taking a linear approach to the origin along the $x$-axis, we find $ \\lim_{(x,y) \\to (0,0),\\ y=0} \\Bigg( \\frac{x^{4}y^{4}}{\\left(4x^{2} +6y^{4}\\right)^{3}} \\Bigg)=$ [ANS].\n(c) Taking a linear approach to the origin along a line $y=m x$ with $m \\ne 0$ we find $ \\lim_{(x,y) \\to (0,0),\\ y=mx} \\Bigg( \\frac{x^{4}y^{4}}{\\left(4x^{2} +6y^{4}\\right)^{3}} \\Bigg)=\\lim_{x \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].\nThought question: Do your answers to parts\n(a)-(c) allow you to conclude that the limit exists?\n(d) Taking an approach to the origin along the parabola $x=y^2$ we find $ \\lim_{(x,y) \\to (0,0),\\ x=y^2} \\Bigg( \\frac{x^{4}y^{4}}{\\left(4x^{2} +6y^{4}\\right)^{3}} \\Bigg)=\\lim_{y \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].\n(e) Find the value of the limit. Be sure you can explain your answer. $ \\lim_{(x,y) \\to (0,0)} \\Bigg( \\frac{x^{4}y^{4}}{\\left(4x^{2} +6y^{4}\\right)^{3}} \\Bigg)=$ [ANS].", "answer_v3": [ "0", "0", "m^4*x^2/[(4+6*m^4*x^2)^3]", "0", "1/(10^3)", "1/(10^3)", "DNE" ], "answer_type_v3": [ "NV", "NV", "EX", "NV", "NV", "NV", "OE" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0165", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Limits and continuity", "level": "4", "keywords": [ "Multivariable", "limit" ], "problem_v1": "Examine the behavior of $ f(x,y)= \\frac{6x^{2.5}}{x^{2} +y^{2}}$ as $(x,y)$ approaches $(0,0)$.\n(a) Changing to polar coordinates, we find\n$ \\lim_{(x,y) \\to (0,0)} \\Big( \\frac{6x^{2.5}}{x^{2} +y^{2}} \\Big)=\\lim_{r \\to 0^+,\\ \\theta=anything} \\Big($ [ANS] $\\Big)=$ [ANS].\nUse \"theta\" for $\\theta$.\nUse \"infinity\" for \" $\\infty$ \" and \"-infinity\" for \" $-\\infty$ \". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\".\n(b) Since $f(0,0)$ is undefined, $f$ has a discontinuity at $(x,y)=(0,0)$. Is it possible to define a function $g: \\mathbb{R}^2 \\to \\mathbb{R}$ such that $g(x,y)=f(x,y)$ for all $(x,y) \\ne (0,0)$ and $g$ is continuous everywhere? If so, what would the value of $g(0,0)$ be? If there is no continuous function $g$, enter DNE.\n$g(0,0)=$ [ANS]", "answer_v1": [ "6*r^0.5*[cos(theta)]^2.5", "0", "0" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Examine the behavior of $ f(x,y)= \\frac{9x^{2.1}}{x^{2} +y^{2}}$ as $(x,y)$ approaches $(0,0)$.\n(a) Changing to polar coordinates, we find\n$ \\lim_{(x,y) \\to (0,0)} \\Big( \\frac{9x^{2.1}}{x^{2} +y^{2}} \\Big)=\\lim_{r \\to 0^+,\\ \\theta=anything} \\Big($ [ANS] $\\Big)=$ [ANS].\nUse \"theta\" for $\\theta$.\nUse \"infinity\" for \" $\\infty$ \" and \"-infinity\" for \" $-\\infty$ \". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\".\n(b) Since $f(0,0)$ is undefined, $f$ has a discontinuity at $(x,y)=(0,0)$. Is it possible to define a function $g: \\mathbb{R}^2 \\to \\mathbb{R}$ such that $g(x,y)=f(x,y)$ for all $(x,y) \\ne (0,0)$ and $g$ is continuous everywhere? If so, what would the value of $g(0,0)$ be? If there is no continuous function $g$, enter DNE.\n$g(0,0)=$ [ANS]", "answer_v2": [ "9*r^0.1*[cos(theta)]^2.1", "0", "0" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Examine the behavior of $ f(x,y)= \\frac{6x^{2.1}}{x^{2} +y^{2}}$ as $(x,y)$ approaches $(0,0)$.\n(a) Changing to polar coordinates, we find\n$ \\lim_{(x,y) \\to (0,0)} \\Big( \\frac{6x^{2.1}}{x^{2} +y^{2}} \\Big)=\\lim_{r \\to 0^+,\\ \\theta=anything} \\Big($ [ANS] $\\Big)=$ [ANS].\nUse \"theta\" for $\\theta$.\nUse \"infinity\" for \" $\\infty$ \" and \"-infinity\" for \" $-\\infty$ \". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\". Use \"DNE\" for \"Does not exist\".\n(b) Since $f(0,0)$ is undefined, $f$ has a discontinuity at $(x,y)=(0,0)$. Is it possible to define a function $g: \\mathbb{R}^2 \\to \\mathbb{R}$ such that $g(x,y)=f(x,y)$ for all $(x,y) \\ne (0,0)$ and $g$ is continuous everywhere? If so, what would the value of $g(0,0)$ be? If there is no continuous function $g$, enter DNE.\n$g(0,0)=$ [ANS]", "answer_v3": [ "6*r^0.1*[cos(theta)]^2.1", "0", "0" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0166", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Limits and continuity", "level": "3", "keywords": [ "Multivariable", "limit" ], "problem_v1": "(a) $ \\lim_{(u,v,w) \\to (2\\pi,7,3)} \\langle e^{u^2-v^2}, 4w-6 \\rangle$ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].\n(b) $ \\lim_{(x,y) \\to (\\pi,2e)} \\frac{\\sin(xy)}{xy} $ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].\n(c) $ \\lim_{t \\to 5} 3\\cos(t) \\boldsymbol{\\vec{i}}+2\\sin(t) \\boldsymbol{\\vec{j}}+4t \\boldsymbol{\\vec{k}}$ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].\n(d) $ \\lim_{(x,y,z) \\to (4,1,2)} \\frac{x^2+3z}{9y} $ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].", "answer_v1": [ "3", "2", "2", "1", "1", "3", "3", "1" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "(a) $ \\lim_{(x,y,z) \\to (4,1,2)} \\frac{x^2+3z}{9y} $ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].\n(b) $ \\lim_{(u,v,w) \\to (2\\pi,7,3)} \\langle e^{u^2-v^2}, 4w-6 \\rangle$ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].\n(c) $ \\lim_{(x,y) \\to (\\pi,2e)} \\frac{\\sin(xy)}{xy} $ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].\n(d) $ \\lim_{t \\to 5} 3\\cos(t) \\boldsymbol{\\vec{i}}+2\\sin(t) \\boldsymbol{\\vec{j}}+4t \\boldsymbol{\\vec{k}}$ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].", "answer_v2": [ "3", "1", "3", "2", "2", "1", "1", "3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "(a) $ \\lim_{(x,y) \\to (\\pi,2e)} \\frac{\\sin(xy)}{xy} $ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].\n(b) $ \\lim_{t \\to 5} 3\\cos(t) \\boldsymbol{\\vec{i}}+2\\sin(t) \\boldsymbol{\\vec{j}}+4t \\boldsymbol{\\vec{k}}$ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].\n(c) $ \\lim_{(x,y,z) \\to (4,1,2)} \\frac{x^2+3z}{9y} $ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].\n(d) $ \\lim_{(u,v,w) \\to (2\\pi,7,3)} \\langle e^{u^2-v^2}, 4w-6 \\rangle$ is the limit of a function $f: \\mathbb{R}^m \\to \\mathbb{R}^n$ with $m=$ [ANS] and $n=$ [ANS].", "answer_v3": [ "2", "1", "1", "3", "3", "1", "3", "2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0167", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Limits and continuity", "level": "2", "keywords": [ "vector' 'limit" ], "problem_v1": "Find the limit, if it exists, or type DNE if it does not exist. A. $ \\lim_{(x, y) \\rightarrow (3, 1)} e^{\\sqrt{(4x^2+4y^2)}}=$ [ANS]\nB. $ \\lim_{(x, y) \\rightarrow (0, 0)} \\frac{2x^2}{2x^2+3 y^2} =$ [ANS]", "answer_v1": [ "558.109578497093", "DNE" ], "answer_type_v1": [ "NV", "OE" ], "options_v1": [ [], [] ], "problem_v2": "Find the limit, if it exists, or type DNE if it does not exist. A. $ \\lim_{(x, y) \\rightarrow (-5, 5)} e^{\\sqrt{(1x^2+2y^2)}}=$ [ANS]\nB. $ \\lim_{(x, y) \\rightarrow (0, 0)} \\frac{5x^2}{2x^2+1 y^2} =$ [ANS]", "answer_v2": [ "5769.00002069627", "DNE" ], "answer_type_v2": [ "NV", "OE" ], "options_v2": [ [], [] ], "problem_v3": "Find the limit, if it exists, or type DNE if it does not exist. A. $ \\lim_{(x, y) \\rightarrow (-2, 1)} e^{\\sqrt{(2x^2+3y^2)}}=$ [ANS]\nB. $ \\lim_{(x, y) \\rightarrow (0, 0)} \\frac{2x^2}{2x^2+5 y^2} =$ [ANS]", "answer_v3": [ "27.5671484533409", "DNE" ], "answer_type_v3": [ "NV", "OE" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0168", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Limits and continuity", "level": "2", "keywords": [ "calculus", "limits", "vector' 'limit", "Multivariable", "limit", "vector", "limit" ], "problem_v1": "Find the limit, if it exists, or type N if it does not exist. $ \\lim_{(x, y, z) \\rightarrow (4, 3, 4)} \\frac{4 z e^{x^2+y^2}}{4x^2+3 y^2+4 z^2} =$ [ANS]", "answer_v1": [ "7432763802.56886" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the limit, if it exists, or type N if it does not exist. $ \\lim_{(x, y, z) \\rightarrow (1, 5, 1)} \\frac{3 z e^{x^2+y^2}}{x^2+5 y^2+z^2} =$ [ANS]", "answer_v2": [ "4623534080.99619" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the limit, if it exists, or type N if it does not exist. $ \\lim_{(x, y, z) \\rightarrow (2, 4, 2)} \\frac{4 z e^{x^2+y^2}}{2x^2+4 y^2+2 z^2} =$ [ANS]", "answer_v3": [ "48516519.540979" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0169", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Given $u(x,t)=t^{2}e^{- \\frac{x^{2}}{2t} }$, compute: $u_{xx}=$ [ANS]", "answer_v1": [ "x^2*e^(-[x^2/(2*t)])-t*e^(-[x^2/(2*t)])" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Given $u(x,t)=t^{-3}e^{- \\frac{x^{2}}{4t} }$, compute: $u_{xx}=$ [ANS]", "answer_v2": [ "0.25*t^{-5}*x^2*e^(-[x^2/(4*t)])-0.5*t^{-4}*e^(-[x^2/(4*t)])" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Given $u(x,t)= \\frac{1}{t} e^{- \\frac{x^{2}}{2t} }$, compute: $u_{xx}=$ [ANS]", "answer_v3": [ "t^{-3}*x^2*e^(-[x^2/(2*t)])-t^{-3}*e^(-[x^2/(2*t)])" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0170", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Compute the partial derivatives: $z= \\frac{7x}{\\sqrt{x^{2} +y^{2}}}$ $ \\frac{\\partial{z}}{\\partial{x} }=$ [ANS]\n$ \\frac{\\partial{z}}{\\partial{y} }=$ [ANS]", "answer_v1": [ "7*y^2/[(x^2+y^2)^1.5]", "-(7*x*y/[(x^2+y^2)^1.5])" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Compute the partial derivatives: $z= \\frac{x}{\\sqrt{x^{2} +y^{2}}}$ $ \\frac{\\partial{z}}{\\partial{x} }=$ [ANS]\n$ \\frac{\\partial{z}}{\\partial{y} }=$ [ANS]", "answer_v2": [ "y^2/[(x^2+y^2)^1.5]", "-(x*y/[(x^2+y^2)^1.5])" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Compute the partial derivatives: $z= \\frac{3x}{\\sqrt{x^{2} +y^{2}}}$ $ \\frac{\\partial{z}}{\\partial{x} }=$ [ANS]\n$ \\frac{\\partial{z}}{\\partial{y} }=$ [ANS]", "answer_v3": [ "3*y^2/[(x^2+y^2)^1.5]", "-(3*x*y/[(x^2+y^2)^1.5])" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0171", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "3", "keywords": [ "calculus" ], "problem_v1": "The plane $y=1$ intersects the surface $z=x^{5}+7xy-y^{6}$ in a certain curve. Find the slope of the tangent line of this curve at the point $P=(1,1,7)$. $m=$ [ANS]", "answer_v1": [ "12" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The plane $y=1$ intersects the surface $z=x^{7}+2xy-y^{3}$ in a certain curve. Find the slope of the tangent line of this curve at the point $P=(1,1,2)$. $m=$ [ANS]", "answer_v2": [ "9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The plane $y=1$ intersects the surface $z=x^{6}+4xy-y^{4}$ in a certain curve. Find the slope of the tangent line of this curve at the point $P=(1,1,4)$. $m=$ [ANS]", "answer_v3": [ "10" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0172", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Compute the partial derivatives: $z=\\ln\\!\\left(x^{6}+y^{5}\\right)$ $ \\frac{\\partial{z}}{\\partial{x} }=$ [ANS]\n$ \\frac{\\partial{z}}{\\partial{y} }=$ [ANS]", "answer_v1": [ "6*x^5/(x^6+y^5)", "5*y^4/(x^6+y^5)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Compute the partial derivatives: $z=\\ln\\!\\left(x^{2}+y^{7}\\right)$ $ \\frac{\\partial{z}}{\\partial{x} }=$ [ANS]\n$ \\frac{\\partial{z}}{\\partial{y} }=$ [ANS]", "answer_v2": [ "2*x/(x^2+y^7)", "7*y^6/(x^2+y^7)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Compute the partial derivatives: $z=\\ln\\!\\left(x^{3}+y^{5}\\right)$ $ \\frac{\\partial{z}}{\\partial{x} }=$ [ANS]\n$ \\frac{\\partial{z}}{\\partial{y} }=$ [ANS]", "answer_v3": [ "3*x^2/(x^3+y^5)", "5*y^4/(x^3+y^5)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0173", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Compute the partial derivative: $f(x,y)=\\cos\\!\\left(x^{6}-5y\\right)$ $f_{y}(0,\\pi)=$ [ANS]", "answer_v1": [ "0" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the partial derivative: $f(x,y)=\\sin\\!\\left(x^{2}-7y\\right)$ $f_{y}(0,\\pi)=$ [ANS]", "answer_v2": [ "7" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the partial derivative: $f(x,y)=\\sin\\!\\left(x^{3}-5y\\right)$ $f_{y}(0,\\pi)=$ [ANS]", "answer_v3": [ "5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0174", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Use the Quotient Rule to compute: $ \\frac{\\partial}{\\partial{y} }\\frac {y}{y+3x}=$ [ANS]", "answer_v1": [ "3*x/[(y+3*x)^2]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Use the Quotient Rule to compute: $ \\frac{\\partial}{\\partial{y} }\\frac {y}{y-5x}=$ [ANS]", "answer_v2": [ "-(5*x/[(y-5*x)^2])" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Use the Quotient Rule to compute: $ \\frac{\\partial}{\\partial{y} }\\frac {y}{y-2x}=$ [ANS]", "answer_v3": [ "-(2*x/[(y-2*x)^2])" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0175", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Given $f(x,y)=5x^{5}y+2xy^{5}$. Compute: $ \\frac{\\partial^2f}{\\partial{x} ^2}=$ [ANS]\n$ \\frac{\\partial^2f}{\\partial{y} ^2}=$ [ANS]", "answer_v1": [ "100*x^3*y", "40*x*y^3" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Given $f(x,y)=8xy^{3}-8x^{2}y$. Compute: $ \\frac{\\partial^2f}{\\partial{x} ^2}=$ [ANS]\n$ \\frac{\\partial^2f}{\\partial{y} ^2}=$ [ANS]", "answer_v2": [ "(-16)*y", "48*x*y" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Given $f(x,y)=2xy^{4}-4x^{3}y$. Compute: $ \\frac{\\partial^2f}{\\partial{x} ^2}=$ [ANS]\n$ \\frac{\\partial^2f}{\\partial{y} ^2}=$ [ANS]", "answer_v3": [ "-24*x*y", "24*x*y^2" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0176", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "Derivative", "Partial" ], "problem_v1": "Let $z=5e^{x^{4}y^{3}}$. Then:\n$\\begin{array}{ccc}\\hline \\frac{\\partial z}{\\partial x} &=& [ANS] \\\\ \\hline \\\\ \\hline \\frac{\\partial z}{\\partial y} &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v1": [ "20*x^3*y^3*e^(x^4*y^3)", "15*x^4*y^2*e^(x^4*y^3)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let $z=2e^{x^{5}y^{2}}$. Then:\n$\\begin{array}{ccc}\\hline \\frac{\\partial z}{\\partial x} &=& [ANS] \\\\ \\hline \\\\ \\hline \\frac{\\partial z}{\\partial y} &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v2": [ "10*x^4*y^2*e^(x^5*y^2)", "4*x^5*y*e^(x^5*y^2)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let $z=3e^{x^{4}y^{2}}$. Then:\n$\\begin{array}{ccc}\\hline \\frac{\\partial z}{\\partial x} &=& [ANS] \\\\ \\hline \\\\ \\hline \\frac{\\partial z}{\\partial y} &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v3": [ "12*x^3*y^2*e^(x^4*y^2)", "6*x^4*y*e^(x^4*y^2)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0177", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "Derivative", "Partial" ], "problem_v1": "Let $z=\\cos\\!\\left(3y^{2}+x\\right)$. Then: The rate of change in $z$ at $(1,2)$ as we change $x$ but hold $y$ fixed is [ANS], and The rate of change in $z$ at $(1,2)$ as we change $y$ but hold $x$ fixed is [ANS].", "answer_v1": [ "-0.420167", "-5.042" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $z=\\cos\\!\\left(5x-5y^{2}\\right)$. Then: The rate of change in $z$ at $(-4,-2)$ as we change $x$ but hold $y$ fixed is [ANS], and The rate of change in $z$ at $(-4,-2)$ as we change $y$ but hold $x$ fixed is [ANS].", "answer_v2": [ "3.72557", "14.9023" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $z=\\cos\\!\\left(x-2y^{2}\\right)$. Then: The rate of change in $z$ at $(-2,1)$ as we change $x$ but hold $y$ fixed is [ANS], and The rate of change in $z$ at $(-2,1)$ as we change $y$ but hold $x$ fixed is [ANS].", "answer_v3": [ "-0.756802", "3.02721" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0178", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "Derivative", "Partial" ], "problem_v1": "Let $f(x,y,z)= \\frac{x^{2}-5y^{2}}{y^{2} +4z^{2}}$. Then\n$\\begin{array}{ccc}\\hline f_x(x,y,z) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(x,y,z) &=& [ANS] \\\\ \\hline \\\\ \\hline f_z(x,y,z) &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v1": [ "2*x/(y^2+4*z^2)", "-2*y*(x^2+20*z^2)/[(y^2+4*z^2)^2]", "(-8)*z*(x^2-5*y^2)/[(y^2+4*z^2)^2]" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $f(x,y,z)= \\frac{x^{2}-2y^{2}}{y^{2} +6z^{2}}$. Then\n$\\begin{array}{ccc}\\hline f_x(x,y,z) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(x,y,z) &=& [ANS] \\\\ \\hline \\\\ \\hline f_z(x,y,z) &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v2": [ "2*x/(y^2+6*z^2)", "-2*y*(x^2+12*z^2)/[(y^2+6*z^2)^2]", "(-12)*z*(x^2-2*y^2)/[(y^2+6*z^2)^2]" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $f(x,y,z)= \\frac{x^{2}-3y^{2}}{y^{2} +5z^{2}}$. Then\n$\\begin{array}{ccc}\\hline f_x(x,y,z) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(x,y,z) &=& [ANS] \\\\ \\hline \\\\ \\hline f_z(x,y,z) &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v3": [ "2*x/(y^2+5*z^2)", "-2*y*(x^2+15*z^2)/[(y^2+5*z^2)^2]", "(-10)*z*(x^2-3*y^2)/[(y^2+5*z^2)^2]" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0179", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "Derivative", "Partial" ], "problem_v1": "Let $f(x,y)=5x^{4}y^{3}$. Then\n$\\begin{array}{ccc}\\hline f_x(x,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_x(2,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_x(x,-2) &=& [ANS] \\\\ \\hline \\\\ \\hline f_x(2,-2) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(x,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(2,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(x,-2) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(2,-2) &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v1": [ "5*4*x^3*y^3", "160*y^3", "-8*5*4*x^3", "-1280", "5*x^4*3*y^2", "80*3*y^2", "12*5*x^4", "960" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let $f(x,y)=2x^{5}y^{2}$. Then\n$\\begin{array}{ccc}\\hline f_x(x,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_x(-2,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_x(x,5) &=& [ANS] \\\\ \\hline \\\\ \\hline f_x(-2,5) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(x,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(-2,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(x,5) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(-2,5) &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v2": [ "2*5*x^4*y^2", "160*y^2", "25*2*5*x^4", "4000", "2*x^5*2*y", "-64*2*y", "10*2*x^5", "-640" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let $f(x,y)=3x^{4}y^{2}$. Then\n$\\begin{array}{ccc}\\hline f_x(x,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_x(1,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_x(x,-3) &=& [ANS] \\\\ \\hline \\\\ \\hline f_x(1,-3) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(x,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(1,y) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(x,-3) &=& [ANS] \\\\ \\hline \\\\ \\hline f_y(1,-3) &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v3": [ "3*4*x^3*y^2", "12*y^2", "9*3*4*x^3", "108", "3*x^4*2*y", "3*2*y", "-6*3*x^4", "-18" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0180", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "Derivative", "Partial", "Higher" ], "problem_v1": "Let $f(x,y)=\\left(3x-y\\right)^{7}$. Then\n$\\begin{array}{ccc}\\hline \\frac{\\partial^2\\!f}{\\partial x\\partial y} &=& [ANS] \\\\ \\hline \\\\ \\hline \\frac{\\partial^3\\!f}{\\partial x\\partial y\\partial x} &=& [ANS] \\\\ \\hline \\\\ \\hline \\frac{\\partial^3\\!f}{\\partial x^2\\partial y} &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v1": [ "-126*(3*x-y)^5", "-1890*(3*x-y)^4", "-1890*(3*x-y)^4" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $f(x,y)=\\left(-\\left(5x+y\\right)\\right)^{9}$. Then\n$\\begin{array}{ccc}\\hline \\frac{\\partial^2\\!f}{\\partial x\\partial y} &=& [ANS] \\\\ \\hline \\\\ \\hline \\frac{\\partial^3\\!f}{\\partial x\\partial y\\partial x} &=& [ANS] \\\\ \\hline \\\\ \\hline \\frac{\\partial^3\\!f}{\\partial x^2\\partial y} &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v2": [ "360*[-(5*x+y)]^7", "-12600*[-(5*x+y)]^6", "-12600*[-(5*x+y)]^6" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $f(x,y)=\\left(-\\left(2x+y\\right)\\right)^{8}$. Then\n$\\begin{array}{ccc}\\hline \\frac{\\partial^2\\!f}{\\partial x\\partial y} &=& [ANS] \\\\ \\hline \\\\ \\hline \\frac{\\partial^3\\!f}{\\partial x\\partial y\\partial x} &=& [ANS] \\\\ \\hline \\\\ \\hline \\frac{\\partial^3\\!f}{\\partial x^2\\partial y} &=& [ANS] \\\\ \\hline \\\\ \\hline \\end{array}$", "answer_v3": [ "112*[-(2*x+y)]^6", "-1344*[-(2*x+y)]^5", "-1344*[-(2*x+y)]^5" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0182", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "partial", "derivative", "multivariable", "functions" ], "problem_v1": "Use difference quotients with $\\Delta x=0.1$ and $\\Delta y=0.1$ to estimate $f_x(4,3)$ and $f_y(4,3)$ where f(x,y)=e^{-x}\\sin\\!\\left(y\\right). $f_x(4,3) \\approx$ [ANS]\n$f_y(4,3) \\approx$ [ANS]\nThen give better estimates by using $\\Delta x=0.01$ and $\\Delta y=0.01$. $f_x(4,3) \\approx$ [ANS]\n$f_y(4,3) \\approx$ [ANS]", "answer_v1": [ "-0.00245967", "-0.0182313", "-0.00257182", "-0.018145" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Use difference quotients with $\\Delta x=0.1$ and $\\Delta y=0.1$ to estimate $f_x(1,4)$ and $f_y(1,4)$ where f(x,y)=e^{-x}\\sin\\!\\left(y\\right). $f_x(1,4) \\approx$ [ANS]\n$f_y(1,4) \\approx$ [ANS]\nThen give better estimates by using $\\Delta x=0.01$ and $\\Delta y=0.01$. $f_x(1,4) \\approx$ [ANS]\n$f_y(1,4) \\approx$ [ANS]", "answer_v2": [ "0.264944", "-0.226152", "0.277025", "-0.239066" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Use difference quotients with $\\Delta x=0.1$ and $\\Delta y=0.1$ to estimate $f_x(2,3)$ and $f_y(2,3)$ where f(x,y)=e^{-x}\\sin\\!\\left(y\\right). $f_x(2,3) \\approx$ [ANS]\n$f_y(2,3) \\approx$ [ANS]\nThen give better estimates by using $\\Delta x=0.01$ and $\\Delta y=0.01$. $f_x(2,3) \\approx$ [ANS]\n$f_y(2,3) \\approx$ [ANS]", "answer_v3": [ "-0.0181746", "-0.134712", "-0.0190033", "-0.134074" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0183", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "5", "keywords": [ "partial", "derivative", "multivariable", "functions" ], "problem_v1": "An experiment to measure the toxicity of formaldehyde yielded the data in the table below. The values show the percent, $P=f(t,c)$, of rats surviving an exposure to formaldehyde at a concentration of $c$ (in parts per million, ppm) after $t$ months.\n$\\begin{array}{ccccccc}\\hline & t=14 & t=16 & t=18 & t=20 & t=22 & t=24 \\\\ \\hline c=0 & 100 & 100 & 100 & 99 & 97 & 95 \\\\ \\hline c=2 & 100 & 99 & 98 & 97 & 95 & 92 \\\\ \\hline c=6 & 96 & 95 & 93 & 90 & 86 & 80 \\\\ \\hline c=15 & 96 & 93 & 82 & 70 & 58 & 36 \\\\ \\hline \\end{array}$\n(a) Estimate $f_t(22,6)$: $f_t(22, 6)\\approx$ [ANS]\n(b) Estimate $f_c(22,6)$: $f_c(22, 6)\\approx$ [ANS]\n(Be sure that you can give the practical meaning of these two values in terms of formaldehyde toxicity.) (Be sure that you can give the practical meaning of these two values in terms of formaldehyde toxicity.)", "answer_v1": [ "(80-90)/(24-20)", "(58-95)/(15-2)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "An experiment to measure the toxicity of formaldehyde yielded the data in the table below. The values show the percent, $P=f(t,c)$, of rats surviving an exposure to formaldehyde at a concentration of $c$ (in parts per million, ppm) after $t$ months.\n$\\begin{array}{ccccccc}\\hline & t=14 & t=16 & t=18 & t=20 & t=22 & t=24 \\\\ \\hline c=0 & 100 & 100 & 100 & 99 & 97 & 95 \\\\ \\hline c=2 & 100 & 99 & 98 & 97 & 95 & 92 \\\\ \\hline c=6 & 96 & 95 & 93 & 90 & 86 & 80 \\\\ \\hline c=15 & 96 & 93 & 82 & 70 & 58 & 36 \\\\ \\hline \\end{array}$\n(a) Estimate $f_t(16,6)$: $f_t(16, 6)\\approx$ [ANS]\n(b) Estimate $f_c(16,6)$: $f_c(16, 6)\\approx$ [ANS]\n(Be sure that you can give the practical meaning of these two values in terms of formaldehyde toxicity.) (Be sure that you can give the practical meaning of these two values in terms of formaldehyde toxicity.)", "answer_v2": [ "(93-96)/(18-14)", "(93-99)/(15-2)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "An experiment to measure the toxicity of formaldehyde yielded the data in the table below. The values show the percent, $P=f(t,c)$, of rats surviving an exposure to formaldehyde at a concentration of $c$ (in parts per million, ppm) after $t$ months.\n$\\begin{array}{ccccccc}\\hline & t=14 & t=16 & t=18 & t=20 & t=22 & t=24 \\\\ \\hline c=0 & 100 & 100 & 100 & 99 & 97 & 95 \\\\ \\hline c=2 & 100 & 99 & 98 & 97 & 95 & 92 \\\\ \\hline c=6 & 96 & 95 & 93 & 90 & 86 & 80 \\\\ \\hline c=15 & 96 & 93 & 82 & 70 & 58 & 36 \\\\ \\hline \\end{array}$\n(a) Estimate $f_t(18,6)$: $f_t(18, 6)\\approx$ [ANS]\n(b) Estimate $f_c(18,6)$: $f_c(18, 6)\\approx$ [ANS]\n(Be sure that you can give the practical meaning of these two values in terms of formaldehyde toxicity.) (Be sure that you can give the practical meaning of these two values in terms of formaldehyde toxicity.)", "answer_v3": [ "(90-95)/(20-16)", "(82-98)/(15-2)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0184", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "5", "keywords": [ "derivatives", "partial", "functions", "multivariable" ], "problem_v1": "A one-meter long bar is heated unevenly, with temperature in ${}^\\circ$ C at a distance $x$ meters from one end at time $t$ given by H(x,t)=140e^{-0.15t}\\sin\\!\\left(\\pi x\\right) $0\\le x\\le 1$. On a sheet of paper, sketch a graph of $H$ against $x$ for $t=0$ and $t=1$. Think about why your graphs make sense.\n(a) Calculate each of: $H_x(0.2,t)=$ [ANS]\n$H_x(0.8,t)=$ [ANS]. (Be sure that you can say in words what the practical interpretation (in terms of temperature) of these two partial derivatives is, and why each has the sign that it does.) (Be sure that you can say in words what the practical interpretation (in terms of temperature) of these two partial derivatives is, and why each has the sign that it does.) (b) Calculate: $H_t(x,t)=$ [ANS]. (Again, be sure that you understand why it has the sign that it does, and what its interpretation in terms of temperature is.) (Again, be sure that you understand why it has the sign that it does, and what its interpretation in terms of temperature is.)", "answer_v1": [ "140*e^(-0.15*t)*pi*cos(pi*0.2)", "140*e^(-0.15*t)*pi*cos(pi*0.8)", "-140*0.15*e^(-0.15*t)*ln(e)*sin(pi*x)" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "A one-meter long bar is heated unevenly, with temperature in ${}^\\circ$ C at a distance $x$ meters from one end at time $t$ given by H(x,t)=80e^{-0.2t}\\sin\\!\\left(\\pi x\\right) $0\\le x\\le 1$. On a sheet of paper, sketch a graph of $H$ against $x$ for $t=0$ and $t=1$. Think about why your graphs make sense.\n(a) Calculate each of: $H_x(0.1,t)=$ [ANS]\n$H_x(0.9,t)=$ [ANS]. (Be sure that you can say in words what the practical interpretation (in terms of temperature) of these two partial derivatives is, and why each has the sign that it does.) (Be sure that you can say in words what the practical interpretation (in terms of temperature) of these two partial derivatives is, and why each has the sign that it does.) (b) Calculate: $H_t(x,t)=$ [ANS]. (Again, be sure that you understand why it has the sign that it does, and what its interpretation in terms of temperature is.) (Again, be sure that you understand why it has the sign that it does, and what its interpretation in terms of temperature is.)", "answer_v2": [ "80*e^(-0.2*t)*pi*cos(pi*0.1)", "80*e^(-0.2*t)*pi*cos(pi*0.9)", "-80*0.2*e^(-0.2*t)*ln(e)*sin(pi*x)" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "A one-meter long bar is heated unevenly, with temperature in ${}^\\circ$ C at a distance $x$ meters from one end at time $t$ given by H(x,t)=100e^{-0.15t}\\sin\\!\\left(\\pi x\\right) $0\\le x\\le 1$. On a sheet of paper, sketch a graph of $H$ against $x$ for $t=0$ and $t=1$. Think about why your graphs make sense.\n(a) Calculate each of: $H_x(0.1,t)=$ [ANS]\n$H_x(0.9,t)=$ [ANS]. (Be sure that you can say in words what the practical interpretation (in terms of temperature) of these two partial derivatives is, and why each has the sign that it does.) (Be sure that you can say in words what the practical interpretation (in terms of temperature) of these two partial derivatives is, and why each has the sign that it does.) (b) Calculate: $H_t(x,t)=$ [ANS]. (Again, be sure that you understand why it has the sign that it does, and what its interpretation in terms of temperature is.) (Again, be sure that you understand why it has the sign that it does, and what its interpretation in terms of temperature is.)", "answer_v3": [ "100*e^(-0.15*t)*pi*cos(pi*0.1)", "100*e^(-0.15*t)*pi*cos(pi*0.9)", "-100*0.15*e^(-0.15*t)*ln(e)*sin(pi*x)" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0185", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "derivatives", "partial", "functions", "multivariable" ], "problem_v1": "Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. $ \\frac{\\partial}{\\partial M} \\left( \\frac{4\\pi r^{5/2}}{\\sqrt{GM} }\\right)=$ [ANS]", "answer_v1": [ "-(4*pi*r^2.5*1/[2*sqrt(G*M)]*G/([sqrt(G*M)]^2))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. $ \\frac{\\partial}{\\partial M} \\left( \\frac{2\\pi r^{7/2}}{\\sqrt{GM} }\\right)=$ [ANS]", "answer_v2": [ "-(2*pi*r^3.5*1/[2*sqrt(G*M)]*G/([sqrt(G*M)]^2))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. $ \\frac{\\partial}{\\partial M} \\left( \\frac{2\\pi r^{5/2}}{\\sqrt{GM} }\\right)=$ [ANS]", "answer_v3": [ "-(2*pi*r^2.5*1/[2*sqrt(G*M)]*G/([sqrt(G*M)]^2))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0186", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "derivatives", "partial", "functions", "multivariable" ], "problem_v1": "Find the partial derivatives indicated. Assume the variables are restricted to a domain on which the function is defined. f(x,y)=x^{5}+6x^{4}y. $f_x(3,2)=$ [ANS]\n$f_y(3,2)=$ [ANS]", "answer_v1": [ "1701", "486" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the partial derivatives indicated. Assume the variables are restricted to a domain on which the function is defined. f(x,y)=x^{2}+3x^{5}y. $f_x(2,4)=$ [ANS]\n$f_y(2,4)=$ [ANS]", "answer_v2": [ "964", "96" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the partial derivatives indicated. Assume the variables are restricted to a domain on which the function is defined. f(x,y)=x^{3}+4x^{4}y. $f_x(2,2)=$ [ANS]\n$f_y(2,2)=$ [ANS]", "answer_v3": [ "268", "64" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0187", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "3", "keywords": [ "derivatives", "partial", "functions", "multivariable" ], "problem_v1": "Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. $ \\frac{\\partial}{\\partial x} (x^{2}e^{\\sqrt{5xy}})=$ [ANS]", "answer_v1": [ "2*x*e^[sqrt(5*x*y)]+x^2*e^[sqrt(5*x*y)]*1/[2*sqrt(5*x*y)]*5*y*ln(e)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. $ \\frac{\\partial}{\\partial x} (x^{3}e^{\\sqrt{2xy}})=$ [ANS]", "answer_v2": [ "3*x^2*e^[sqrt(2*x*y)]+x^3*e^[sqrt(2*x*y)]*1/[2*sqrt(2*x*y)]*2*y*ln(e)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. $ \\frac{\\partial}{\\partial x} (x^{2}e^{\\sqrt{3xy}})=$ [ANS]", "answer_v3": [ "2*x*e^[sqrt(3*x*y)]+x^2*e^[sqrt(3*x*y)]*1/[2*sqrt(3*x*y)]*3*y*ln(e)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0188", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "derivatives", "partial", "functions", "multivariable" ], "problem_v1": "Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. z=\\sin\\!\\left(6x^{4}y-6xy^{4}\\right). $z_x=$ [ANS]", "answer_v1": [ "(6*4*x^3*y-6*y^4)*cos(6*x^4*y-6*x*y^4)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. z=\\sin\\!\\left(3x^{5}y-3xy^{3}\\right). $z_x=$ [ANS]", "answer_v2": [ "(3*5*x^4*y-3*y^3)*cos(3*x^5*y-3*x*y^3)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the partial derivative indicated. Assume the variables are restricted to a domain on which the function is defined. z=\\sin\\!\\left(4x^{4}y-4xy^{3}\\right). $z_x=$ [ANS]", "answer_v3": [ "(4*4*x^3*y-4*y^3)*cos(4*x^4*y-4*x*y^3)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0189", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "derivatives", "partial", "functions", "multivariable" ], "problem_v1": "Let f(w,z)=e^{w\\ln\\!\\left(z\\right)}.\n(a) Use difference quotients with $h=0.02$ to approximate each of the partial derivatives: $f_w(5, 4) \\approx$ [ANS]\n$f_z(5, 4) \\approx$ [ANS]. (b) Now evaluate the partial derivatives exactly: $f_w(5, 4)=$ [ANS]\n$f_z(5, 4)=$ [ANS]", "answer_v1": [ "28.79/0.02", "25.86/0.02", "1419.57", "1280" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let f(w,z)=e^{w\\ln\\!\\left(z\\right)}.\n(a) Use difference quotients with $h=0.01$ to approximate each of the partial derivatives: $f_w(2, 5) \\approx$ [ANS]\n$f_z(2, 5) \\approx$ [ANS]. (b) Now evaluate the partial derivatives exactly: $f_w(2, 5)=$ [ANS]\n$f_z(2, 5)=$ [ANS]", "answer_v2": [ "0.4056/0.01", "0.1001/0.01", "40.2359", "10" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let f(w,z)=e^{w\\ln\\!\\left(z\\right)}.\n(a) Use difference quotients with $h=0.01$ to approximate each of the partial derivatives: $f_w(3, 4) \\approx$ [ANS]\n$f_z(3, 4) \\approx$ [ANS]. (b) Now evaluate the partial derivatives exactly: $f_w(3, 4)=$ [ANS]\n$f_z(3, 4)=$ [ANS]", "answer_v3": [ "0.8934/0.01", "0.4812/0.01", "88.7228", "48" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0190", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "3", "keywords": [ "derivatives", "partial", "functions", "multivariable" ], "problem_v1": "Consider the partial derivatives f_{x}(x,y)=5x^{4}y^{6}-16x^{3}y, f_{y}(x,y)=6x^{5}y^{5}-4x^{4}. Is there a function $f$ which has these partial derivatives? [ANS] If so, what is it? $f=$ [ANS]\n(Enter none if there is no such function.) Are there any others? [ANS]", "answer_v1": [ "Yes", "x^5*y^6-4*x^4*y", "Yes" ], "answer_type_v1": [ "TF", "EX", "TF" ], "options_v1": [ [ "Yes", "No" ], [], [ "Yes", "No" ] ], "problem_v2": "Consider the partial derivatives f_{x}(x,y)=3x^{2}y^{6}-6xy, f_{y}(x,y)=6x^{3}y^{5}-3x^{2}. Is there a function $f$ which has these partial derivatives? [ANS] If so, what is it? $f=$ [ANS]\n(Enter none if there is no such function.) Are there any others? [ANS]", "answer_v2": [ "Yes", "x^3*y^6-3*x^2*y", "Yes" ], "answer_type_v2": [ "TF", "EX", "TF" ], "options_v2": [ [ "Yes", "No" ], [], [ "Yes", "No" ] ], "problem_v3": "Consider the partial derivatives f_{x}(x,y)=3x^{2}y^{5}-12x^{2}y, f_{y}(x,y)=5x^{3}y^{4}-4x^{3}. Is there a function $f$ which has these partial derivatives? [ANS] If so, what is it? $f=$ [ANS]\n(Enter none if there is no such function.) Are there any others? [ANS]", "answer_v3": [ "Yes", "x^3*y^5-4*x^3*y", "Yes" ], "answer_type_v3": [ "TF", "EX", "TF" ], "options_v3": [ [ "Yes", "No" ], [], [ "Yes", "No" ] ] }, { "id": "Calculus_-_multivariable_0191", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "chain rule", "multivariable", "functions", "calculus", "derivatives", "second derivative" ], "problem_v1": "Calculate all four second-order partial derivatives and check that $f_{xy}=f_{yx}$. Assume the variables are restricted to a domain on which the function is defined. f(x,y)=5\\sin\\!\\left(5x\\right)\\cos\\!\\left(5y\\right) $f_{xx}=$ [ANS]\n$f_{yy}=$ [ANS]\n$f_{xy}=$ [ANS]\n$f_{yx}=$ [ANS]", "answer_v1": [ "-125*sin(5*x)*cos(5*y)", "-125*sin(5*x)*cos(5*y)", "-125*cos(5*x)*sin(5*y)", "-125*cos(5*x)*sin(5*y)" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Calculate all four second-order partial derivatives and check that $f_{xy}=f_{yx}$. Assume the variables are restricted to a domain on which the function is defined. f(x,y)=2\\sin\\!\\left(7x\\right)\\cos\\!\\left(2y\\right) $f_{xx}=$ [ANS]\n$f_{yy}=$ [ANS]\n$f_{xy}=$ [ANS]\n$f_{yx}=$ [ANS]", "answer_v2": [ "-98*sin(7*x)*cos(2*y)", "-8*sin(7*x)*cos(2*y)", "-28*cos(7*x)*sin(2*y)", "-28*cos(7*x)*sin(2*y)" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Calculate all four second-order partial derivatives and check that $f_{xy}=f_{yx}$. Assume the variables are restricted to a domain on which the function is defined. f(x,y)=3\\sin\\!\\left(5x\\right)\\cos\\!\\left(3y\\right) $f_{xx}=$ [ANS]\n$f_{yy}=$ [ANS]\n$f_{xy}=$ [ANS]\n$f_{yx}=$ [ANS]", "answer_v3": [ "-75*sin(5*x)*cos(3*y)", "-27*sin(5*x)*cos(3*y)", "-45*cos(5*x)*sin(3*y)", "-45*cos(5*x)*sin(3*y)" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0192", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "chain rule", "multivariable", "functions", "calculus", "derivatives", "second derivative" ], "problem_v1": "Find the quadratic Taylor polynomial $Q(x,y)$ approximating f(x,y)=e^{y}\\cos\\!\\left(3x\\right) about $(0,0)$. $Q(x,y)=$ [ANS]", "answer_v1": [ "1+y-4.5*x^2+0.5*y^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the quadratic Taylor polynomial $Q(x,y)$ approximating f(x,y)=e^{y}\\sin\\!\\left(5x\\right) about $(0,0)$. $Q(x,y)=$ [ANS]", "answer_v2": [ "5*x+5*x*y" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the quadratic Taylor polynomial $Q(x,y)$ approximating f(x,y)=e^{x}\\cos\\!\\left(4y\\right) about $(0,0)$. $Q(x,y)=$ [ANS]", "answer_v3": [ "1+x+0.5*x^2-8*y^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0193", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "chain rule", "multivariable", "functions", "calculus", "derivatives", "second derivative" ], "problem_v1": "Assume $z$ is a smooth function of $x$ and $y$. If $z_{xy}=3y$, what can you say about each of the following?\n(a) $z_{yx}$=[ANS]\n(b) $z_{xyx}$=[ANS]\n(c) $z_{xyy}$=[ANS]", "answer_v1": [ "3*y", "0", "3" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Assume $z$ is a smooth function of $x$ and $y$. If $z_{xy}=-5y$, what can you say about each of the following?\n(a) $z_{yx}$=[ANS]\n(b) $z_{xyx}$=[ANS]\n(c) $z_{xyy}$=[ANS]", "answer_v2": [ "-5*y", "0", "-5" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Assume $z$ is a smooth function of $x$ and $y$. If $z_{xy}=-2y$, what can you say about each of the following?\n(a) $z_{yx}$=[ANS]\n(b) $z_{xyx}$=[ANS]\n(c) $z_{xyy}$=[ANS]", "answer_v3": [ "-2*y", "0", "-2" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0194", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "3", "keywords": [ "partial derivative" ], "problem_v1": "Find the indicated partial derivatives of $f(x,y)=8x^{2}-6y^{2}$ using the limit definition. The limits need to be reduced as much as possible before they are evaluated.\n$ f_x(x,y)=\\lim_{h \\to 0} \\frac{f(x+h,y)-f(x,y)}{h} =\\lim_{h \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].\n$ f_y(x,y)=\\lim_{h \\to 0} \\frac{f(x,y+h)-f(x,y)}{h} =\\lim_{h \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].", "answer_v1": [ "16*x+8*h", "8*2*x", "(-12)*y-6*h", "-(6*2*y)" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the indicated partial derivatives of $f(x,y)=2x^{2}-9y^{2}$ using the limit definition. The limits need to be reduced as much as possible before they are evaluated.\n$ f_x(x,y)=\\lim_{h \\to 0} \\frac{f(x+h,y)-f(x,y)}{h} =\\lim_{h \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].\n$ f_y(x,y)=\\lim_{h \\to 0} \\frac{f(x,y+h)-f(x,y)}{h} =\\lim_{h \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].", "answer_v2": [ "4*x+2*h", "2*2*x", "(-18)*y-9*h", "-(9*2*y)" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the indicated partial derivatives of $f(x,y)=4x^{2}-6y^{2}$ using the limit definition. The limits need to be reduced as much as possible before they are evaluated.\n$ f_x(x,y)=\\lim_{h \\to 0} \\frac{f(x+h,y)-f(x,y)}{h} =\\lim_{h \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].\n$ f_y(x,y)=\\lim_{h \\to 0} \\frac{f(x,y+h)-f(x,y)}{h} =\\lim_{h \\to 0} \\Bigg($ [ANS] $\\Bigg)=$ [ANS].", "answer_v3": [ "8*x+4*h", "4*2*x", "(-12)*y-6*h", "-(6*2*y)" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0195", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "partial' 'derivative", "Multivariable", "derivative' 'partial", "vector", "partial", "derivative" ], "problem_v1": "Find the first partial derivatives of $ f(x,y)= \\frac{4x-3y}{4x+3y} $ at the point $(x,y)=(3, 3)$. $ \\frac{\\partial f}{\\partial x} (3, 3)=$ [ANS]\n$ \\frac{\\partial f}{\\partial y} (3, 3)=$ [ANS]", "answer_v1": [ "0.163265306122449", "-0.163265306122449" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the first partial derivatives of $ f(x,y)= \\frac{x-4y}{x+4y} $ at the point $(x,y)=(1, 2)$. $ \\frac{\\partial f}{\\partial x} (1, 2)=$ [ANS]\n$ \\frac{\\partial f}{\\partial y} (1, 2)=$ [ANS]", "answer_v2": [ "0.197530864197531", "-0.0987654320987654" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the first partial derivatives of $ f(x,y)= \\frac{2x-3y}{2x+3y} $ at the point $(x,y)=(2, 3)$. $ \\frac{\\partial f}{\\partial x} (2, 3)=$ [ANS]\n$ \\frac{\\partial f}{\\partial y} (2, 3)=$ [ANS]", "answer_v3": [ "0.21301775147929", "-0.142011834319527" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0196", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "partial' 'derivative" ], "problem_v1": "Find the first partial derivatives of $ f(x,y,z)=z \\ \\arctan\\left( \\frac{y}{2x} \\right)$.\nCaution: ay/bx and ay/(bx) are very different, the former equals axy/b. Parentheses are your friends! A. $ \\frac{\\partial f}{\\partial x} =$ [ANS]\nB. $ \\frac{\\partial f}{\\partial y} =$ [ANS]\nC. $ \\frac{\\partial f}{\\partial z} =$ [ANS]", "answer_v1": [ "-(2 * 1 * y * z)/((2 * x)^2 + (1 * y)^2)", "(2 * 1 * x * z)/((2 * x)^2 + (1 * y)^2)", "arctan((1*y)/(2*x))" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the first partial derivatives of $ f(x,y,z)=z \\ \\arctan\\left( \\frac{-y}{x} \\right)$.\nCaution: ay/bx and ay/(bx) are very different, the former equals axy/b. Parentheses are your friends! A. $ \\frac{\\partial f}{\\partial x} =$ [ANS]\nB. $ \\frac{\\partial f}{\\partial y} =$ [ANS]\nC. $ \\frac{\\partial f}{\\partial z} =$ [ANS]", "answer_v2": [ "-(-4 * 4 * y * z)/((-4 * x)^2 + (4 * y)^2)", "(-4 * 4 * x * z)/((-4 * x)^2 + (4 * y)^2)", "arctan((4*y)/(-4*x))" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the first partial derivatives of $ f(x,y,z)=z \\ \\arctan\\left( \\frac{-y}{2x} \\right)$.\nCaution: ay/bx and ay/(bx) are very different, the former equals axy/b. Parentheses are your friends! A. $ \\frac{\\partial f}{\\partial x} =$ [ANS]\nB. $ \\frac{\\partial f}{\\partial y} =$ [ANS]\nC. $ \\frac{\\partial f}{\\partial z} =$ [ANS]", "answer_v3": [ "-(-2 * 1 * y * z)/((-2 * x)^2 + (1 * y)^2)", "(-2 * 1 * x * z)/((-2 * x)^2 + (1 * y)^2)", "arctan((1*y)/(-2*x))" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0198", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "4", "keywords": [ "Second-order partial derivatives" ], "problem_v1": "Find the second-order Taylor polynomial for $ f(x,y)=5x^{2}y+4xy^{3}$ at the point $(1,2)$.\n$Q(x,y)=$ [ANS]", "answer_v1": [ "42+52*(x-1)+53*(y-2)+20*(x-1)^2/2+48*(y-2)^2/2+58*(x-1)*(y-2)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the second-order Taylor polynomial for $ f(x,y)=2x^{2}y+5xy^{3}$ at the point $(-2,-1)$.\n$Q(x,y)=$ [ANS]", "answer_v2": [ "2+3*(x+2)-22*(y+1)-4*(x+2)^2/2+60*(y+1)^2/2+7*(x+2)*(y+1)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the second-order Taylor polynomial for $ f(x,y)=3x^{2}y+4xy^{3}$ at the point $(-2,-2)$.\n$Q(x,y)=$ [ANS]", "answer_v3": [ "40-8*(x+2)-84*(y+2)-12*(x+2)^2/2+96*(y+2)^2/2+36*(x+2)*(y+2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0199", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "Second-order partial derivatives" ], "problem_v1": "Calculate all four second-order partial derivatives of $ f(x,y)=\\left(5x+4y\\right)e^{y}$.\n$f_{xx} \\, (x,y)=$ [ANS]\n$f_{xy} \\, (x,y)=$ [ANS]\n$f_{yx} \\, (x,y)=$ [ANS]\n$f_{yy} \\, (x,y)=$ [ANS]", "answer_v1": [ "0", "5*e^y*ln(e)", "5*e^y*ln(e)", "4*e^y*ln(e)+4*e^y*ln(e)+(5*x+4*y)*e^y*ln(e)*ln(e)" ], "answer_type_v1": [ "NV", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Calculate all four second-order partial derivatives of $ f(x,y)=\\left(2x+5y\\right)e^{y}$.\n$f_{xx} \\, (x,y)=$ [ANS]\n$f_{xy} \\, (x,y)=$ [ANS]\n$f_{yx} \\, (x,y)=$ [ANS]\n$f_{yy} \\, (x,y)=$ [ANS]", "answer_v2": [ "0", "2*e^y*ln(e)", "2*e^y*ln(e)", "5*e^y*ln(e)+5*e^y*ln(e)+(2*x+5*y)*e^y*ln(e)*ln(e)" ], "answer_type_v2": [ "NV", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Calculate all four second-order partial derivatives of $ f(x,y)=\\left(3x+4y\\right)e^{y}$.\n$f_{xx} \\, (x,y)=$ [ANS]\n$f_{xy} \\, (x,y)=$ [ANS]\n$f_{yx} \\, (x,y)=$ [ANS]\n$f_{yy} \\, (x,y)=$ [ANS]", "answer_v3": [ "0", "3*e^y*ln(e)", "3*e^y*ln(e)", "4*e^y*ln(e)+4*e^y*ln(e)+(3*x+4*y)*e^y*ln(e)*ln(e)" ], "answer_type_v3": [ "NV", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0200", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Partial derivatives", "level": "2", "keywords": [ "Second-order partial derivatives" ], "problem_v1": "If $z_{xy}=8 y$ and all of the second order partial derivatives of $z$ are continuous, then\n(a) $z_{yx}=$ [ANS]\n(b) $z_{xyx}=$ [ANS]\n(c) $z_{xyy}=$ [ANS]", "answer_v1": [ "8*y", "0", "8" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "If $z_{xy}=3 y$ and all of the second order partial derivatives of $z$ are continuous, then\n(a) $z_{yx}=$ [ANS]\n(b) $z_{xyx}=$ [ANS]\n(c) $z_{xyy}=$ [ANS]", "answer_v2": [ "3*y", "0", "3" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "If $z_{xy}=5 y$ and all of the second order partial derivatives of $z$ are continuous, then\n(a) $z_{yx}=$ [ANS]\n(b) $z_{xyx}=$ [ANS]\n(c) $z_{xyy}=$ [ANS]", "answer_v3": [ "5*y", "0", "5" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0201", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Calculate the derivative using implicit differentiation: \\frac{\\partial w}{\\partial z} , \\quad x^{6}w+w^{8}+wz^2+6 yz=0 $ \\frac{\\partial w}{\\partial z} =$ [ANS]", "answer_v1": [ "-[(2*w*z+6*y)/(x^6+8*w^7+z^2)]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Calculate the derivative using implicit differentiation: \\frac{\\partial w}{\\partial z} , \\quad x^{9}w+w^{2}+wz^2+3 yz=0 $ \\frac{\\partial w}{\\partial z} =$ [ANS]", "answer_v2": [ "-[(2*w*z+3*y)/(x^9+2*w+z^2)]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Calculate the derivative using implicit differentiation: \\frac{\\partial w}{\\partial z} , \\quad x^{6}w+w^{4}+wz^2+4 yz=0 $ \\frac{\\partial w}{\\partial z} =$ [ANS]", "answer_v3": [ "-[(2*w*z+4*y)/(x^6+4*w^3+z^2)]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0202", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Use the Chain Rule to calculate the partial derivatives $g(x,y)=\\cos\\!\\left(x^{2}+y^{2}\\right), \\quad x=u+v, y=2u-2v$ $ \\frac{\\partial{g}}{\\partial{u} }=$ [ANS]\n$ \\frac{\\partial{g}}{\\partial{v} }=$ [ANS]\nExpress your answer in terms of the independent variables $u,v$", "answer_v1": [ "-(10*u-6*v)*sin(5*u^2+5*v^2-6*u*v)", "-(10*v-6*u)*sin(5*u^2+5*v^2-6*u*v)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Use the Chain Rule to calculate the partial derivatives $g(x,y)=\\sin\\!\\left(x^{2}+y^{2}\\right), \\quad x=4u-3v, y=4v-u$ $ \\frac{\\partial{g}}{\\partial{u} }=$ [ANS]\n$ \\frac{\\partial{g}}{\\partial{v} }=$ [ANS]\nExpress your answer in terms of the independent variables $u,v$", "answer_v2": [ "(34*u-32*v)*cos(17*u^2+25*v^2-32*u*v)", "(50*v-32*u)*cos(17*u^2+25*v^2-32*u*v)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Use the Chain Rule to calculate the partial derivatives $g(x,y)=\\sin\\!\\left(x^{2}+y^{2}\\right), \\quad x=u-2v, y=-\\left(3u+v\\right)$ $ \\frac{\\partial{g}}{\\partial{u} }=$ [ANS]\n$ \\frac{\\partial{g}}{\\partial{v} }=$ [ANS]\nExpress your answer in terms of the independent variables $u,v$", "answer_v3": [ "(20*u+2*v)*cos(10*u^2+5*v^2+2*u*v)", "(10*v+2*u)*cos(10*u^2+5*v^2+2*u*v)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0203", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Let $f(x,y,z)=x^{4}y^{3}+z^{3}$ and $x=s^{3}t, y=st^{2}$, and $z=s^{2}t^{2}$.\n(a) Calculate the primary derivatives $ \\frac{\\partial{f}}{\\partial{x} }=$ [ANS]\n$ \\frac{\\partial{f}}{\\partial{y} }=$ [ANS]\n$ \\frac{\\partial{f}}{\\partial{z} }=$ [ANS]\n(b) Calculate $ \\frac{\\partial{x}}{\\partial{s} }=$ [ANS]\n$ \\frac{\\partial{y}}{\\partial{s} }=$ [ANS]\n$ \\frac{\\partial{z}}{\\partial{s} }=$ [ANS]\n(c) Use the Chain Rule to compute $ \\frac{\\partial{f}}{\\partial{s} }=$ [ANS]\nIn (c) express your answer in terms of the independent variables $t,s$", "answer_v1": [ "4*x^3*y^3", "3*x^4*y^2", "3*z^2", "3*s^2*t", "t^2", "2*s*t^2", "15*s^14*t^10+6*s^5*t^6" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Let $f(x,y,z)=xy^{4}+z$ and $x=s^{2}t^{3}, y=st$, and $z=st^{2}$.\n(a) Calculate the primary derivatives $ \\frac{\\partial{f}}{\\partial{x} }=$ [ANS]\n$ \\frac{\\partial{f}}{\\partial{y} }=$ [ANS]\n$ \\frac{\\partial{f}}{\\partial{z} }=$ [ANS]\n(b) Calculate $ \\frac{\\partial{x}}{\\partial{s} }=$ [ANS]\n$ \\frac{\\partial{y}}{\\partial{s} }=$ [ANS]\n$ \\frac{\\partial{z}}{\\partial{s} }=$ [ANS]\n(c) Use the Chain Rule to compute $ \\frac{\\partial{f}}{\\partial{s} }=$ [ANS]\nIn (c) express your answer in terms of the independent variables $t,s$", "answer_v2": [ "y^4", "4*x*y^3", "1", "2*s*t^3", "t", "t^2", "6*s^5*t^7+t^2" ], "answer_type_v2": [ "EX", "EX", "NV", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Let $f(x,y,z)=x^{2}y^{3}+z^{2}$ and $x=s^{2}t, y=s^{2}t^{3}$, and $z=s^{3}t^{3}$.\n(a) Calculate the primary derivatives $ \\frac{\\partial{f}}{\\partial{x} }=$ [ANS]\n$ \\frac{\\partial{f}}{\\partial{y} }=$ [ANS]\n$ \\frac{\\partial{f}}{\\partial{z} }=$ [ANS]\n(b) Calculate $ \\frac{\\partial{x}}{\\partial{s} }=$ [ANS]\n$ \\frac{\\partial{y}}{\\partial{s} }=$ [ANS]\n$ \\frac{\\partial{z}}{\\partial{s} }=$ [ANS]\n(c) Use the Chain Rule to compute $ \\frac{\\partial{f}}{\\partial{s} }=$ [ANS]\nIn (c) express your answer in terms of the independent variables $t,s$", "answer_v3": [ "2*x*y^3", "3*x^2*y^2", "2*z", "2*s*t", "2*s*t^3", "3*s^2*t^3", "10*s^9*t^11+6*s^5*t^6" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0204", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Use the Chain Rule to evaluate the partial derivative $ \\frac{\\partial g}{\\partial \\theta} $ at the point $(r,\\theta)=(2\\sqrt 2, \\frac{\\pi}4)$ where $g(x,y)=\\frac{1}{8x+6 y^2} $, $x=r\\sin\\theta$, $y=r\\cos\\theta$. $ \\frac{\\partial g}{\\partial \\theta} \\bigg|_{(r,\\theta)=\\left(2\\sqrt{2}, \\frac{\\pi}{4} \\right)}=$ [ANS]", "answer_v1": [ "0.02" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the Chain Rule to evaluate the partial derivative $ \\frac{\\partial g}{\\partial \\theta} $ at the point $(r,\\theta)=(2\\sqrt 2, \\frac{\\pi}4)$ where $g(x,y)=\\frac{1}{2x+9 y^2} $, $x=r\\sin\\theta$, $y=r\\cos\\theta$. $ \\frac{\\partial g}{\\partial \\theta} \\bigg|_{(r,\\theta)=\\left(2\\sqrt{2}, \\frac{\\pi}{4} \\right)}=$ [ANS]", "answer_v2": [ "0.0425" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the Chain Rule to evaluate the partial derivative $ \\frac{\\partial g}{\\partial \\theta} $ at the point $(r,\\theta)=(2\\sqrt 2, \\frac{\\pi}4)$ where $g(x,y)=\\frac{1}{4x+6 y^2} $, $x=r\\sin\\theta$, $y=r\\cos\\theta$. $ \\frac{\\partial g}{\\partial \\theta} \\bigg|_{(r,\\theta)=\\left(2\\sqrt{2}, \\frac{\\pi}{4} \\right)}=$ [ANS]", "answer_v3": [ "0.0390625" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0205", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Calculate the derivatives using implicit differentiation: $ \\frac{\\partial U}{\\partial T} $ and $ \\frac{\\partial T}{\\partial U} $ of $(TU-V)^2\\ln(W-UV)=1$ at $(T,U,V,W)=(6,1,8,16)$ $ \\frac{\\partial U}{\\partial T} \\bigg|_{(6,1,8,16)}=$ [ANS]\n$ \\frac{\\partial T}{\\partial U} \\bigg|_{(6,1,8,16)}=$ [ANS]", "answer_v1": [ "-0.1543", "-6.4809" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Calculate the derivatives using implicit differentiation: $ \\frac{\\partial U}{\\partial T} $ and $ \\frac{\\partial T}{\\partial U} $ of $(TU-V)^2\\ln(W-UV)=1$ at $(T,U,V,W)=(9,1,2,4)$ $ \\frac{\\partial U}{\\partial T} \\bigg|_{(9,1,2,4)}=$ [ANS]\n$ \\frac{\\partial T}{\\partial U} \\bigg|_{(9,1,2,4)}=$ [ANS]", "answer_v2": [ "-0.253128", "-3.95057" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Calculate the derivatives using implicit differentiation: $ \\frac{\\partial U}{\\partial T} $ and $ \\frac{\\partial T}{\\partial U} $ of $(TU-V)^2\\ln(W-UV)=1$ at $(T,U,V,W)=(6,1,4,8)$ $ \\frac{\\partial U}{\\partial T} \\bigg|_{(6,1,4,8)}=$ [ANS]\n$ \\frac{\\partial T}{\\partial U} \\bigg|_{(6,1,4,8)}=$ [ANS]", "answer_v3": [ "-0.189442", "-5.27865" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0206", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "2", "keywords": [ "Multivariable", "differential", "Differential" ], "problem_v1": "Find the differential of the function $w=x \\sin \\left(4 y z^{3} \\right)$. dw=[ANS] dx+[ANS] dy+[ANS] dz Note: Your answers should be expressions of x, y and z; e.g. \"3xy+4z\"", "answer_v1": [ "sin( 4 * y * z^3 )", "4 * x * z^3 * cos( 4 * y * z^3 )", "4 * 3 * x * y * z**(3-1) * cos( 4 * y * z^3 )" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the differential of the function $w=x \\sin \\left(1 y z^{4} \\right)$. dw=[ANS] dx+[ANS] dy+[ANS] dz Note: Your answers should be expressions of x, y and z; e.g. \"3xy+4z\"", "answer_v2": [ "sin( 1 * y * z^4 )", "1 * x * z^4 * cos( 1 * y * z^4 )", "1 * 4 * x * y * z**(4-1) * cos( 1 * y * z^4 )" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the differential of the function $w=x \\sin \\left(2 y z^{3} \\right)$. dw=[ANS] dx+[ANS] dy+[ANS] dz Note: Your answers should be expressions of x, y and z; e.g. \"3xy+4z\"", "answer_v3": [ "sin( 2 * y * z^3 )", "2 * x * z^3 * cos( 2 * y * z^3 )", "2 * 3 * x * y * z**(3-1) * cos( 2 * y * z^3 )" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0207", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "3", "keywords": [ "Multivariable", "Derivative", "related rates" ], "problem_v1": "In a simple electric circuit, Ohm's law states that $V=IR$, where V is the voltage in volts, I is the current in amperes, and R is the resistance in ohms. Assume that, as the battery wears out, the voltage decreases at 0.04 volts per second and, as the resistor heats up, the resistance is increasing at 0.03 ohms per second. When the resistance is 300 ohms and the current is 0.03 amperes, at what rate is the current changing? [ANS] amperes per second", "answer_v1": [ "-0.000136333333333333" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "In a simple electric circuit, Ohm's law states that $V=IR$, where V is the voltage in volts, I is the current in amperes, and R is the resistance in ohms. Assume that, as the battery wears out, the voltage decreases at 0.01 volts per second and, as the resistor heats up, the resistance is increasing at 0.04 ohms per second. When the resistance is 100 ohms and the current is 0.02 amperes, at what rate is the current changing? [ANS] amperes per second", "answer_v2": [ "-0.000108" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "In a simple electric circuit, Ohm's law states that $V=IR$, where V is the voltage in volts, I is the current in amperes, and R is the resistance in ohms. Assume that, as the battery wears out, the voltage decreases at 0.02 volts per second and, as the resistor heats up, the resistance is increasing at 0.03 ohms per second. When the resistance is 200 ohms and the current is 0.03 amperes, at what rate is the current changing? [ANS] amperes per second", "answer_v3": [ "-0.0001045" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0208", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "2", "keywords": [ "Multivariable", "Chain Rule", "Derivative", "Partial" ], "problem_v1": "Suppose $z=x^{2} \\sin y$, $x=3 s^{2}+2 t^{2}$, $y=2 s t$. A. Use the chain rule to find $ \\frac{\\partial z}{\\partial s} $ and $ \\frac{\\partial z}{\\partial t} $ as functions of x, y, s and t. $ \\frac{\\partial z}{\\partial s} =$ [ANS]\n$ \\frac{\\partial z}{\\partial t} =$ [ANS]\nB. Find the numerical values of $ \\frac{\\partial z}{\\partial s} $ and $ \\frac{\\partial z}{\\partial t} $ when $\\left(s, t \\right)=\\left(2,-1 \\right)$. $ \\frac{\\partial z}{\\partial s} \\left(2,-1 \\right)=$ [ANS]\n$ \\frac{\\partial z}{\\partial t} \\left(2,-1 \\right)=$ [ANS]", "answer_v1": [ "( 4 * 3 * s * x * sin(y) ) + ( 2 * 1 * t * x^2 * cos(y) )", "( 4 * 2 * t * x * sin(y) ) + ( 2 * 1 * s * x^2 * cos(y) )", "510.513937802", "-597.21847823156" ], "answer_type_v1": [ "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose $z=x^{2} \\sin y$, $x=1 s^{2}+3 t^{2}$, $y=-4 s t$. A. Use the chain rule to find $ \\frac{\\partial z}{\\partial s} $ and $ \\frac{\\partial z}{\\partial t} $ as functions of x, y, s and t. $ \\frac{\\partial z}{\\partial s} =$ [ANS]\n$ \\frac{\\partial z}{\\partial t} =$ [ANS]\nB. Find the numerical values of $ \\frac{\\partial z}{\\partial s} $ and $ \\frac{\\partial z}{\\partial t} $ when $\\left(s, t \\right)=\\left(-1, 3 \\right)$. $ \\frac{\\partial z}{\\partial s} \\left(-1, 3 \\right)=$ [ANS]\n$ \\frac{\\partial z}{\\partial t} \\left(-1, 3 \\right)=$ [ANS]", "answer_v2": [ "( 4 * 1 * s * x * sin(y) ) + ( 2 * -2 * t * x^2 * cos(y) )", "( 4 * 3 * t * x * sin(y) ) + ( 2 * -2 * s * x^2 * cos(y) )", "-7878.88187693924", "2105.46051324066" ], "answer_type_v2": [ "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose $z=x^{2} \\sin y$, $x=1 s^{2}+2 t^{2}$, $y=-4 s t$. A. Use the chain rule to find $ \\frac{\\partial z}{\\partial s} $ and $ \\frac{\\partial z}{\\partial t} $ as functions of x, y, s and t. $ \\frac{\\partial z}{\\partial s} =$ [ANS]\n$ \\frac{\\partial z}{\\partial t} =$ [ANS]\nB. Find the numerical values of $ \\frac{\\partial z}{\\partial s} $ and $ \\frac{\\partial z}{\\partial t} $ when $\\left(s, t \\right)=\\left(-2,-1 \\right)$. $ \\frac{\\partial z}{\\partial s} \\left(-2,-1 \\right)=$ [ANS]\n$ \\frac{\\partial z}{\\partial t} \\left(-2,-1 \\right)=$ [ANS]", "answer_v3": [ "( 4 * 1 * s * x * sin(y) ) + ( 2 * -2 * t * x^2 * cos(y) )", "( 4 * 2 * t * x * sin(y) ) + ( 2 * -2 * s * x^2 * cos(y) )", "26.537190969482", "5.58518610104163" ], "answer_type_v3": [ "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0209", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "2", "keywords": [ "chain rule", "multivariable", "functions", "calculus", "derivatives " ], "problem_v1": "If z=\\left(x+y\\right)e^{y},\\qquad x=5t,\\qquad y=4-t^{2}, find $dz/dt$ using the chain rule. Assume the variables are restricted to domains on which the functions are defined. $ \\frac{dz}{dt} =$ [ANS]", "answer_v1": [ "5*e^(4-t^2)-(1+5*t+4-t^2)*e^(4-t^2)*2*t" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If z=\\left(x+y\\right)e^{x},\\qquad x=2t,\\qquad y=5-t^{2}, find $dz/dt$ using the chain rule. Assume the variables are restricted to domains on which the functions are defined. $ \\frac{dz}{dt} =$ [ANS]", "answer_v2": [ "2*(1+2*t+5-t^2)*e^(2*t)-2*t*e^(2*t)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If z=\\left(x+y\\right)e^{x},\\qquad x=3t,\\qquad y=4-t^{2}, find $dz/dt$ using the chain rule. Assume the variables are restricted to domains on which the functions are defined. $ \\frac{dz}{dt} =$ [ANS]", "answer_v3": [ "3*(1+3*t+4-t^2)*e^(3*t)-2*t*e^(3*t)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0210", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "3", "keywords": [ "chain rule", "multivariable", "functions", "calculus", "derivatives " ], "problem_v1": "Air pressure decreases at a rate of 2.25 pascals per kilometer in the eastward direction. In addition, the air pressure is decreasing at a constant rate with respect to time everywhere. A ship sailing eastward at 12 km/hour past an island takes barometer readings and records a pressure drop of 60 pascals in 2 hours. Estimate the time rate of change of air pressure on the island. (A pascal is a unit of air pressure.) Time rate of change of air pressure=[ANS] Pa/hr", "answer_v1": [ "-3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Air pressure decreases at a rate of 1 pascals per kilometer in the eastward direction. In addition, the air pressure is increasing at a constant rate with respect to time everywhere. A ship sailing eastward at 16 km/hour past an island takes barometer readings and records a pressure drop of 30 pascals in 2 hours. Estimate the time rate of change of air pressure on the island. (A pascal is a unit of air pressure.) Time rate of change of air pressure=[ANS] Pa/hr", "answer_v2": [ "1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Air pressure decreases at a rate of 1.5 pascals per kilometer in the eastward direction. In addition, the air pressure is decreasing at a constant rate with respect to time everywhere. A ship sailing eastward at 12 km/hour past an island takes barometer readings and records a pressure drop of 40 pascals in 2 hours. Estimate the time rate of change of air pressure on the island. (A pascal is a unit of air pressure.) Time rate of change of air pressure=[ANS] Pa/hr", "answer_v3": [ "-2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0211", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "2", "keywords": [ "chain rule", "multivariable", "functions", "calculus", "derivatives " ], "problem_v1": "Let $F(u,v)$ be a function of two variables. Let $F_u(u,v)=G(u,v)$, and $F_v(u,v)=H(u,v)$. Find $f'(x)$ for each of the following cases (your answers should be written in terms of $G$ and $H$).\n(a) $f(x)=F(x,3)$: then $f'(x)=$ [ANS]\n(b) $f(x)=F(4,x)$: then $f'(x)=$ [ANS]\n(c) $f(x)=F(x,x)$: then $f'(x)=$ [ANS]\n(d) $f(x)=F(5x,x^{4})$: then $f'(x)=$ [ANS]", "answer_v1": [ "G(x,3)", "H(4,x)", "G(x,x)+H(x,x)", "5*G(5*x,x^4)+4*x^3*H(5*x,x^4)" ], "answer_type_v1": [ "OE", "OE", "OE", "OE" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $F(u,v)$ be a function of two variables. Let $F_u(u,v)=G(u,v)$, and $F_v(u,v)=H(u,v)$. Find $f'(x)$ for each of the following cases (your answers should be written in terms of $G$ and $H$).\n(a) $f(x)=F(x,5)$: then $f'(x)=$ [ANS]\n(b) $f(x)=F(1,x)$: then $f'(x)=$ [ANS]\n(c) $f(x)=F(x,x)$: then $f'(x)=$ [ANS]\n(d) $f(x)=F(2x,x^{3})$: then $f'(x)=$ [ANS]", "answer_v2": [ "G(x,5)", "H(1,x)", "G(x,x)+H(x,x)", "2*G(2*x,x^3)+3*x^2*H(2*x,x^3)" ], "answer_type_v2": [ "OE", "OE", "OE", "OE" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $F(u,v)$ be a function of two variables. Let $F_u(u,v)=G(u,v)$, and $F_v(u,v)=H(u,v)$. Find $f'(x)$ for each of the following cases (your answers should be written in terms of $G$ and $H$).\n(a) $f(x)=F(x,4)$: then $f'(x)=$ [ANS]\n(b) $f(x)=F(2,x)$: then $f'(x)=$ [ANS]\n(c) $f(x)=F(x,x)$: then $f'(x)=$ [ANS]\n(d) $f(x)=F(3x,x^{4})$: then $f'(x)=$ [ANS]", "answer_v3": [ "G(x,4)", "H(2,x)", "G(x,x)+H(x,x)", "3*G(3*x,x^4)+4*x^3*H(3*x,x^4)" ], "answer_type_v3": [ "OE", "OE", "OE", "OE" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0212", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "2", "keywords": [ "chain rule", "multivariable", "functions", "calculus", "derivatives " ], "problem_v1": "Let $z=g(u,v,w,x,y)$ and $u(r,s,t),v(r,s,t),w(r,s,t),x(r,s,t),y(r,s,t)$. How many terms are there in the expression for $\\partial z/\\partial r$? [ANS] terms", "answer_v1": [ "5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $z=g(u,v)$ and $u(r,s,t),v(r,s,t)$. How many terms are there in the expression for $\\partial z/\\partial r$? [ANS] terms", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $z=g(u,v,w)$ and $u(r,s,t),v(r,s,t),w(r,s,t)$. How many terms are there in the expression for $\\partial z/\\partial r$? [ANS] terms", "answer_v3": [ "3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0213", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "2", "keywords": [ "chain rule", "multivariable", "functions", "calculus", "derivatives " ], "problem_v1": "If z=\\left(6x+y\\right)e^{x},\\qquad x=\\ln\\!\\left(u\\right),\\qquad y=v, find $\\partial z/\\partial u$ and $\\partial z/\\partial v$. The variables are restricted to domains on which the functions are defined. $\\partial z/\\partial u=$ [ANS]\n$\\partial z/\\partial v=$ [ANS]", "answer_v1": [ "(6*e^{ln(u)}+[6*ln(u)+v]*e^{ln(u)}*ln(e))*1/u", "e^[ln(u)]*1" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "If z=\\left(x+y\\right)e^{x+y},\\qquad x=u,\\qquad y=\\ln\\!\\left(v\\right), find $\\partial z/\\partial u$ and $\\partial z/\\partial v$. The variables are restricted to domains on which the functions are defined. $\\partial z/\\partial u=$ [ANS]\n$\\partial z/\\partial v=$ [ANS]", "answer_v2": [ "(e^{u+ln(v)}+[u+ln(v)]*e^{u+ln(v)}*ln(e))*1", "(e^{u+ln(v)}+[u+ln(v)]*e^{u+ln(v)}*ln(e))*1/v" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "If z=\\left(3x+y\\right)e^{x},\\qquad x=\\ln\\!\\left(u\\right),\\qquad y=v, find $\\partial z/\\partial u$ and $\\partial z/\\partial v$. The variables are restricted to domains on which the functions are defined. $\\partial z/\\partial u=$ [ANS]\n$\\partial z/\\partial v=$ [ANS]", "answer_v3": [ "(3*e^{ln(u)}+[3*ln(u)+v]*e^{ln(u)}*ln(e))*1/u", "e^{ln(u)}*1" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0214", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "2", "keywords": [ "chain rule", "multivariable", "functions", "calculus", "derivatives " ], "problem_v1": "Given $z=f(x,y),\\quad x=x(u,v),\\quad y=y(u,v)$, with $x(1,1)=2$ and $y(1,1)=3$, calculate $z_u(1,1)$ in terms of some of the values given in the table below.\n$\\begin{array}{cccc}\\hline f_x(1,1)=p & f_y(1,1)=a & x_u(1,1)=q & y_u(1,1)=c \\\\ \\hline f_x(2,3)=-3 & f_y(2,3)=d & x_v(1,1)=b & y_v(1,1)=r \\\\ \\hline \\end{array}$\n$z_u(1,1)=$ [ANS]", "answer_v1": [ "-3*q+d*c" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Given $z=f(x,y),\\quad x=x(u,v),\\quad y=y(u,v)$, with $x(1,3)=3$ and $y(1,3)=5$, calculate $z_u(1,3)$ in terms of some of the values given in the table below.\n$\\begin{array}{cccc}\\hline f_x(1,3)=a & f_y(1,3)=s & x_u(1,3)=c & y_u(1,3)=1 \\\\ \\hline f_x(3,5)=-5 & f_y(3,5)=5 & x_v(1,3)=p & y_v(1,3)=q \\\\ \\hline \\end{array}$\n$z_u(1,3)=$ [ANS]", "answer_v2": [ "-5*c+5*1" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Given $z=f(x,y),\\quad x=x(u,v),\\quad y=y(u,v)$, with $x(4,2)=2$ and $y(4,2)=1$, calculate $z_v(4,2)$ in terms of some of the values given in the table below.\n$\\begin{array}{cccc}\\hline f_x(4,2)=d & f_y(4,2)=-4 & x_u(4,2)=s & y_u(4,2)=4 \\\\ \\hline f_x(2,1)=b & f_y(2,1)=r & x_v(4,2)=q & y_v(4,2)=a \\\\ \\hline \\end{array}$\n$z_v(4,2)=$ [ANS]", "answer_v3": [ "b*q+r*a" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0215", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "3", "keywords": [ "Chain Rule" ], "problem_v1": "Suppose $xy^{3}z^{2}-1=3xy-z$. Compute $ \\frac{\\partial z}{\\partial x} $ and $ \\frac{\\partial z}{\\partial y} $ at the point $(1,2,-1)$.\n$ \\left. \\frac{\\partial z}{\\partial x} \\right|_{(1,2,-1)}=$ [ANS]\n$ \\left. \\frac{\\partial z}{\\partial y} \\right|_{(1,2,-1)}=$ [ANS]", "answer_v1": [ "[3*2-2^3*(-1)^2]/[1+1*2^3*2*(-1)^1]", "[3*1-1*3*2^2*(-1)^2]/[1+1*2^3*2*(-1)^1]" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose $xy^{2}z^{3}+59=4xy-z$. Compute $ \\frac{\\partial z}{\\partial x} $ and $ \\frac{\\partial z}{\\partial y} $ at the point $(-2,-1,3)$.\n$ \\left. \\frac{\\partial z}{\\partial x} \\right|_{(-2,-1,3)}=$ [ANS]\n$ \\left. \\frac{\\partial z}{\\partial y} \\right|_{(-2,-1,3)}=$ [ANS]", "answer_v2": [ "[4*-1-(-1)^2*3^3]/[1+-2*(-1)^2*3*3^2]", "[4*-2--2*2*(-1)^1*3^3]/[1+-2*(-1)^2*3*3^2]" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose $xy^{2}z^{3}+20=3xy-z$. Compute $ \\frac{\\partial z}{\\partial x} $ and $ \\frac{\\partial z}{\\partial y} $ at the point $(-2,-1,2)$.\n$ \\left. \\frac{\\partial z}{\\partial x} \\right|_{(-2,-1,2)}=$ [ANS]\n$ \\left. \\frac{\\partial z}{\\partial y} \\right|_{(-2,-1,2)}=$ [ANS]", "answer_v3": [ "[3*-1-(-1)^2*2^3]/[1+-2*(-1)^2*3*2^2]", "[3*-2--2*2*(-1)^1*2^3]/[1+-2*(-1)^2*3*2^2]" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0216", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "4", "keywords": [ "derivative", "differentiation", "matrix" ], "problem_v1": "Let $z=f(u,v)=\\sin u \\cos v$, $u=3x^2+y$, $v=x+2y$, and put $g(x,y)=\\left(u(x,y), v(x,y)\\right)$.\nThe derivative matrix $\\mathbf{D}(f\\circ g)(x,y)=$ $($ [ANS], [ANS] $)$ (Leaving your answer in terms of $u, v, x, y$)", "answer_v1": [ "2 * 3 * x * cos(u) * cos(v) - 1 * sin(u) * sin(v)", "1 * cos(u) * cos(v) - 2 * sin(u) * sin(v)" ], "answer_type_v1": [ "EX", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Let $z=f(u,v)=\\sin u \\cos v$, $u=-5x^2+5y$, $v=-4x-2y$, and put $g(x,y)=\\left(u(x,y), v(x,y)\\right)$.\nThe derivative matrix $\\mathbf{D}(f\\circ g)(x,y)=$ $($ [ANS], [ANS] $)$ (Leaving your answer in terms of $u, v, x, y$)", "answer_v2": [ "2 * -5 * x * cos(u) * cos(v) - -4 * sin(u) * sin(v)", "5 * cos(u) * cos(v) - -2 * sin(u) * sin(v)" ], "answer_type_v2": [ "EX", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Let $z=f(u,v)=\\sin u \\cos v$, $u=-2x^2+y$, $v=-2x+y$, and put $g(x,y)=\\left(u(x,y), v(x,y)\\right)$.\nThe derivative matrix $\\mathbf{D}(f\\circ g)(x,y)=$ $($ [ANS], [ANS] $)$ (Leaving your answer in terms of $u, v, x, y$)", "answer_v3": [ "2 * -2 * x * cos(u) * cos(v) - -2 * sin(u) * sin(v)", "1 * cos(u) * cos(v) - 1 * sin(u) * sin(v)" ], "answer_type_v3": [ "EX", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0217", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "2", "keywords": [ "chain rule", "differentiation", "Multivariable", "Derivative", "Chain Rule" ], "problem_v1": "Suppose $ w= \\frac{x}{y} + \\frac{y}{z} $, where $x=e^{4t},\\ y=2+\\sin \\left(3t \\right)$, and $z=2+\\cos \\left(6t \\right)$. A) Use the chain rule to find $ \\frac{dw}{dt} $ as a function of x, y, z, and t. Do not rewrite x, y, and z in terms of t, and do not rewrite $e^{4t}$ as x. $ \\frac{dw}{dt} $=[ANS]\nNote: You may want to use exp() for the exponential function. Your answer should be an expression in x, y, z, and t; e.g. \"3x-4y\" B) Use part A to evaluate $ \\frac{dw}{dt} $ when $t=0$. [ANS]", "answer_v1": [ "( 4 * exp(4 * t) / y ) + ( ( -x / y^2 + 1/z ) * 3 * cos(
3 * t ) ) + ( -y / z^2 ) * ( - 6 * sin( 6 * t ) )", "2.25" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose $ w= \\frac{x}{y} + \\frac{y}{z} $, where $x=e^{t},\\ y=2+\\sin \\left(5t \\right)$, and $z=2+\\cos \\left(3t \\right)$. A) Use the chain rule to find $ \\frac{dw}{dt} $ as a function of x, y, z, and t. Do not rewrite x, y, and z in terms of t, and do not rewrite $e^{t}$ as x. $ \\frac{dw}{dt} $=[ANS]\nNote: You may want to use exp() for the exponential function. Your answer should be an expression in x, y, z, and t; e.g. \"3x-4y\" B) Use part A to evaluate $ \\frac{dw}{dt} $ when $t=0$. [ANS]", "answer_v2": [ "( 1 * exp(1 * t) / y ) + ( ( -x / y^2 + 1/z ) * 5 * cos(
5 * t ) ) + ( -y / z^2 ) * ( - 3 * sin( 3 * t ) )", "0.916666666666667" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose $ w= \\frac{x}{y} + \\frac{y}{z} $, where $x=e^{2t},\\ y=2+\\sin \\left(4t \\right)$, and $z=2+\\cos \\left(4t \\right)$. A) Use the chain rule to find $ \\frac{dw}{dt} $ as a function of x, y, z, and t. Do not rewrite x, y, and z in terms of t, and do not rewrite $e^{2t}$ as x. $ \\frac{dw}{dt} $=[ANS]\nNote: You may want to use exp() for the exponential function. Your answer should be an expression in x, y, z, and t; e.g. \"3x-4y\" B) Use part A to evaluate $ \\frac{dw}{dt} $ when $t=0$. [ANS]", "answer_v3": [ "( 2 * exp(2 * t) / y ) + ( ( -x / y^2 + 1/z ) * 4 * cos(
4 * t ) ) + ( -y / z^2 ) * ( - 4 * sin( 4 * t ) )", "1.33333333333333" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0218", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "2", "keywords": [ "tangent' 'slope" ], "problem_v1": "Find the slope of the tangent line to the curve $ \\sqrt{3x+3y}+\\sqrt{2xy}=\\sqrt{36}+\\sqrt{70}$ at the point $(7,5)$. The slope is [ANS].", "answer_v1": [ "-0.780017930143874" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the slope of the tangent line to the curve $ \\sqrt{x+2y}+\\sqrt{4xy}=\\sqrt{17}+\\sqrt{32}$ at the point $(1,8)$. The slope is [ANS].", "answer_v2": [ "-4.94841350859117" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the slope of the tangent line to the curve $ \\sqrt{2x+3y}+\\sqrt{xy}=\\sqrt{21}+\\sqrt{15}$ at the point $(3,5)$. The slope is [ANS].", "answer_v3": [ "-1.20862677511686" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0219", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Chain rule", "level": "2", "keywords": [ "Chain Rule" ], "problem_v1": "If $z=\\left(x+y\\right)e^{y}$ and $x=6t$ and $y=1-t^{2}$, find the following derivative using the chain rule. Enter your answer as a function of $t$.\n$ \\frac{dz}{dt} =$ [ANS]", "answer_v1": [ "(6-2*t)*e^(1-t^2)-(6*t+1-t^2)*e^(1-t^2)*2*t*ln(e)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If $z=\\left(x+y\\right)e^{y}$ and $x=2t$ and $y=1-t^{2}$, find the following derivative using the chain rule. Enter your answer as a function of $t$.\n$ \\frac{dz}{dt} =$ [ANS]", "answer_v2": [ "(2-2*t)*e^(1-t^2)-(2*t+1-t^2)*e^(1-t^2)*2*t*ln(e)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If $z=\\left(x+y\\right)e^{y}$ and $x=3t$ and $y=1-t^{2}$, find the following derivative using the chain rule. Enter your answer as a function of $t$.\n$ \\frac{dz}{dt} =$ [ANS]", "answer_v3": [ "(3-2*t)*e^(1-t^2)-(3*t+1-t^2)*e^(1-t^2)*2*t*ln(e)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0220", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "Multivariable", "Geometry", "Vector" ], "problem_v1": "Find the equation of the plane through the point $(3,5,0)$ that is tangent to the surface xy^2+3ye^z=z+90 at $(3,5,0)$. Write it in the form indicated below.\nEquation: $25(x-3)+$ [ANS] $(y-$ [ANS]) $+$ [ANS] $(z-$ [ANS]) $=0$", "answer_v1": [ "33", "5", "14", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Find the equation of the plane through the point $(3,-8,0)$ that is tangent to the surface xy^2+3ye^z=z+168 at $(3,-8,0)$. Write it in the form indicated below.\nEquation: $64(x-3)+$ [ANS] $(y-$ [ANS]) $+$ [ANS] $(z-$ [ANS]) $=0$", "answer_v2": [ "-45", "-8", "-25", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Find the equation of the plane through the point $(3,-4,0)$ that is tangent to the surface xy^2+3ye^z=z+36 at $(3,-4,0)$. Write it in the form indicated below.\nEquation: $16(x-3)+$ [ANS] $(y-$ [ANS]) $+$ [ANS] $(z-$ [ANS]) $=0$", "answer_v3": [ "-21", "-4", "-13", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0221", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Use the linear approximation to estimate $(1.99)^{3}(0.99)^{3}\\approx$ [ANS]\nCompare with the value given by a calculator and compute the percentage error: Error=[ANS] $\\%$", "answer_v1": [ "7.64", "0.085494" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Use the linear approximation to estimate $(-2.97)^{2}(2.99)^{2}\\approx$ [ANS]\nCompare with the value given by a calculator and compute the percentage error: Error=[ANS] $\\%$", "answer_v2": [ "78.84", "0.0250167" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Use the linear approximation to estimate $(-1.02)^{2}(0.99)^{3}\\approx$ [ANS]\nCompare with the value given by a calculator and compute the percentage error: Error=[ANS] $\\%$", "answer_v3": [ "1.01", "0.0496207" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0222", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find the point on the graph of $z=2x^{2}+y^{2}$ at which vector $\\mathbf{n}=\\left<4,4,-1 \\right>$ is normal to the tangent plane. $P=$ [ANS]", "answer_v1": [ "(1,2,6)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the point on the graph of $z=4y^{2}-4x^{2}$ at which vector $\\mathbf{n}=\\left<-96,48,3 \\right>$ is normal to the tangent plane. $P=$ [ANS]", "answer_v2": [ "(-4,-2,-48)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the point on the graph of $z=y^{2}-2x^{2}$ at which vector $\\mathbf{n}=\\left<16,4,-2 \\right>$ is normal to the tangent plane. $P=$ [ANS]", "answer_v3": [ "(-2,1,-7)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0223", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "4", "keywords": [ "calculus" ], "problem_v1": "The volume of a cylinder of radius $r$ and height $h$ is $V=\\pi r^2h$. Calculate the percentage increase in $V$ if $r$ is increased by $3.1 \\%$ and $h$ is increased by $3.3 \\%$. Hint: Use the linear approximation to show that $ \\frac{\\Delta V}{V} \\approx \\frac{2\\Delta r}{r} + \\frac{\\Delta h}{h} $ $ \\frac{\\Delta V}{V} \\times 100\\%=$ [ANS] $\\%$ The volume of a certain cylinder $V$ is determined by measuring $r$ and $h$. Which will lead to a greater error in $V$: [ANS] A. a $1\\%$ error in $r$ is equivalent to a $1\\%$ error in $h$ B. a $1\\%$ error in $h$ C. a $1\\%$ error in $r$", "answer_v1": [ "9.5", "C" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "A", "B", "C" ] ], "problem_v2": "The volume of a cylinder of radius $r$ and height $h$ is $V=\\pi r^2h$. Calculate the percentage increase in $V$ if $r$ is increased by $4.7 \\%$ and $h$ is increased by $1.1 \\%$. Hint: Use the linear approximation to show that $ \\frac{\\Delta V}{V} \\approx \\frac{2\\Delta r}{r} + \\frac{\\Delta h}{h} $ $ \\frac{\\Delta V}{V} \\times 100\\%=$ [ANS] $\\%$ The volume of a certain cylinder $V$ is determined by measuring $r$ and $h$. Which will lead to a greater error in $V$: [ANS] A. a $1\\%$ error in $r$ B. a $1\\%$ error in $h$ C. a $1\\%$ error in $r$ is equivalent to a $1\\%$ error in $h$", "answer_v2": [ "10.5", "A" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "A", "B", "C" ] ], "problem_v3": "The volume of a cylinder of radius $r$ and height $h$ is $V=\\pi r^2h$. Calculate the percentage increase in $V$ if $r$ is increased by $3.2 \\%$ and $h$ is increased by $1.7 \\%$. Hint: Use the linear approximation to show that $ \\frac{\\Delta V}{V} \\approx \\frac{2\\Delta r}{r} + \\frac{\\Delta h}{h} $ $ \\frac{\\Delta V}{V} \\times 100\\%=$ [ANS] $\\%$ The volume of a certain cylinder $V$ is determined by measuring $r$ and $h$. Which will lead to a greater error in $V$: [ANS] A. a $1\\%$ error in $r$ is equivalent to a $1\\%$ error in $h$ B. a $1\\%$ error in $r$ C. a $1\\%$ error in $h$", "answer_v3": [ "8.1", "B" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "A", "B", "C" ] ] }, { "id": "Calculus_-_multivariable_0224", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Use the linear approximation of $f(x,y)=e^{3x^{2}+2y}$ at $(0,0)$ to estimate $f(0.01,0.02)\\approx$ [ANS]", "answer_v1": [ "1.04" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the linear approximation of $f(x,y)=e^{x^{2}+3y}$ at $(0,0)$ to estimate $f(-0.02,-0.01)\\approx$ [ANS]", "answer_v2": [ "0.97" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the linear approximation of $f(x,y)=e^{x^{2}+2y}$ at $(0,0)$ to estimate $f(-0.02,-0.02)\\approx$ [ANS]", "answer_v3": [ "0.96" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0225", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find an equation of the tangent plane at the given point: $F(r,s)=r^{4}s^{-0.5}+s^{-4}, \\qquad (1,1)$ $z=$ [ANS]", "answer_v1": [ "4*r-4.5*s+2.5" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an equation of the tangent plane at the given point: $F(r,s)=3r^{3}s^{-0.5}-2s^{-2}, \\qquad (-2,1)$ $z=$ [ANS]", "answer_v2": [ "36*r+16*s+30" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an equation of the tangent plane at the given point: $F(r,s)=r^{3}s^{-0.5}-2s^{-4}, \\qquad (-1,1)$ $z=$ [ANS]", "answer_v3": [ "3*r+8.5*s-8.5" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0226", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "4", "keywords": [ "calculus" ], "problem_v1": "The volume $V$ of a cylinder is computed using the values $8 \\text{m}$ for the diameter and $6.7 \\text{m}$ for the height. Use the linear approximation to estimate the maximum error in $V$ if each of these values has a possible error of at most $7 \\%$. Percentage error in $V$: [ANS] $\\%$", "answer_v1": [ "21" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The volume $V$ of a cylinder is computed using the values $2.6 \\text{m}$ for the diameter and $9.5 \\text{m}$ for the height. Use the linear approximation to estimate the maximum error in $V$ if each of these values has a possible error of at most $3 \\%$. Percentage error in $V$: [ANS] $\\%$", "answer_v2": [ "9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The volume $V$ of a cylinder is computed using the values $4.5 \\text{m}$ for the diameter and $6.9 \\text{m}$ for the height. Use the linear approximation to estimate the maximum error in $V$ if each of these values has a possible error of at most $4 \\%$. Percentage error in $V$: [ANS] $\\%$", "answer_v3": [ "12" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0227", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Estimate $f(2.01,1.02)$ given that $f(2,1)=-2, f_x(2,1)=-2$ and $f_y(2,1)=1$. $f(2.01,1.02) \\approx$ [ANS]", "answer_v1": [ "-2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Estimate $f(-3.02,2.99)$ given that $f(-3,3)=5, f_x(-3,3)=-2$ and $f_y(-3,3)=-3$. $f(-3.02,2.99) \\approx$ [ANS]", "answer_v2": [ "5.07" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Estimate $f(-1.02,0.98)$ given that $f(-1,1)=-2, f_x(-1,1)=3$ and $f_y(-1,1)=5$. $f(-1.02,0.98) \\approx$ [ANS]", "answer_v3": [ "-2.16" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0228", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "4", "keywords": [ "calculus" ], "problem_v1": "The monthly payment for a home loan is given by a function $f(P,r,N)$ where $P$ is the principal (the initial size of the loan), $r$ the interest rate, and $N$ the length of the loan in months. Interest rates are expressed as a decimal: A \\% interest rate is denoted by $r=0.075$. If $P=200000, r=0.075$, and $N=288$ (a 24-year loan), then the monthly payment is $f(200000,0.075,288)=1241$. Furthermore, with these values we have \\frac{\\partial{f}}{\\partial{P} }=0.0072, \\quad \\frac{\\partial{f}}{\\partial{r} }=7871, \\quad \\frac{\\partial{f}}{\\partial{N} }=-1.6145 Estimate:\n(a) The change in monthly payment per $2000$ increase in loan principal: $\\Delta f \\approx$ [ANS] dollars (b) The change in monthly payment if the interest rate changes from $r=0.075$ to $r=0.08$: $\\Delta f \\approx$ [ANS] dollars (c) The change in monthly payment if the length of the loan changes from $24$ to $26$ years: $\\Delta f \\approx$ [ANS] dollars", "answer_v1": [ "14.4", "39.355", "-38.748" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The monthly payment for a home loan is given by a function $f(P,r,N)$ where $P$ is the principal (the initial size of the loan), $r$ the interest rate, and $N$ the length of the loan in months. Interest rates are expressed as a decimal: A \\% interest rate is denoted by $r=0.035$. If $P=500000, r=0.035$, and $N=204$ (a 17-year loan), then the monthly payment is $f(500000,0.035,204)=602$. Furthermore, with these values we have \\frac{\\partial{f}}{\\partial{P} }=0.0057, \\quad \\frac{\\partial{f}}{\\partial{r} }=6641, \\quad \\frac{\\partial{f}}{\\partial{N} }=-1.5179 Estimate:\n(a) The change in monthly payment per $2000$ increase in loan principal: $\\Delta f \\approx$ [ANS] dollars (b) The change in monthly payment if the interest rate changes from $r=0.035$ to $r=0.05$: $\\Delta f \\approx$ [ANS] dollars (c) The change in monthly payment if the length of the loan changes from $17$ to $16$ years: $\\Delta f \\approx$ [ANS] dollars", "answer_v2": [ "11.4", "99.615", "18.2148" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The monthly payment for a home loan is given by a function $f(P,r,N)$ where $P$ is the principal (the initial size of the loan), $r$ the interest rate, and $N$ the length of the loan in months. Interest rates are expressed as a decimal: A \\% interest rate is denoted by $r=0.05$. If $P=200000, r=0.05$, and $N=228$ (a 19-year loan), then the monthly payment is $f(200000,0.05,228)=935$. Furthermore, with these values we have \\frac{\\partial{f}}{\\partial{P} }=0.0086, \\quad \\frac{\\partial{f}}{\\partial{r} }=9365, \\quad \\frac{\\partial{f}}{\\partial{N} }=-1.6999 Estimate:\n(a) The change in monthly payment per $4500$ increase in loan principal: $\\Delta f \\approx$ [ANS] dollars (b) The change in monthly payment if the interest rate changes from $r=0.05$ to $r=0.055$: $\\Delta f \\approx$ [ANS] dollars (c) The change in monthly payment if the length of the loan changes from $19$ to $16$ years: $\\Delta f \\approx$ [ANS] dollars", "answer_v3": [ "38.7", "46.825", "61.1964" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0229", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find the linear approximation to $f(x,y,z)= \\frac{xy}{z} $ at the point $(2,1,1)$: $f(x,y,z)\\approx$ [ANS]", "answer_v1": [ "x+2*y-2*z" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the linear approximation to $f(x,y,z)= \\frac{xy}{z} $ at the point $(-3,3,-2)$: $f(x,y,z)\\approx$ [ANS]", "answer_v2": [ "1.5*y-1.5*x+2.25*z" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the linear approximation to $f(x,y,z)= \\frac{xy}{z} $ at the point $(-1,1,-1)$: $f(x,y,z)\\approx$ [ANS]", "answer_v3": [ "y-x+z" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0230", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "Tangent", "Plane", "Multivariable", "Implicit" ], "problem_v1": "Let $f(x,y)=3x^{2}+xy+y^{2}$. Then an implicit equation for the tangent plane to the graph of $f$ at the point $(2,-1)$ is [ANS].", "answer_v1": [ "z-11*x = -11" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Let $f(x,y)=5xy-5x^{2}-4y^{2}$. Then an implicit equation for the tangent plane to the graph of $f$ at the point $(-1,3)$ is [ANS].", "answer_v2": [ "29*y-25*x+z = 56" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Let $f(x,y)=xy-2x^{2}-2y^{2}$. Then an implicit equation for the tangent plane to the graph of $f$ at the point $(0,-2)$ is [ANS].", "answer_v3": [ "2*x-8*y+z = 8" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0231", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "Tangent", "Plane", "Multivariable", "Implicit", "Parametric" ], "problem_v1": "Consider the ellipsoid $x^{2}+y^{2}+5z^{2}=10$. The implicit form of the tangent plane to this ellipsoid at $\\left(-1,-2,-1\\right)$ is [ANS]. The parametric form of the line through this point that is perpendicular to that tangent plane is $L(t)$=[ANS].", "answer_v1": [ "x+2*y+5*z = -10", "(-t-1,-2t-2,-5t-1)" ], "answer_type_v1": [ "EQ", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Consider the ellipsoid $4x^{2}+y^{2}+2z^{2}=10$. The implicit form of the tangent plane to this ellipsoid at $\\left(-1,-2,-1\\right)$ is [ANS]. The parametric form of the line through this point that is perpendicular to that tangent plane is $L(t)$=[ANS].", "answer_v2": [ "4*x+2*y+2*z = -10", "(-4t-1,-2t-2,-2t-1)" ], "answer_type_v2": [ "EQ", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Consider the ellipsoid $4x^{2}+y^{2}+z^{2}=17$. The implicit form of the tangent plane to this ellipsoid at $\\left(1,2,3\\right)$ is [ANS]. The parametric form of the line through this point that is perpendicular to that tangent plane is $L(t)$=[ANS].", "answer_v3": [ "4*x+2*y+3*z = 17", "(4t+1,2t+2,3t+3)" ], "answer_type_v3": [ "EQ", "OL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0232", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "Multivariable", "Derivative", "Directional' 'gradient", "Parametric", "Tangent", "Normal" ], "problem_v1": "Consider the surface 16x^{2}+9 y^{2}+16 z^{2}=41 and the point $P=\\left(1, 1, 1 \\right)$ on this surface. A. Starting with the equation $x=1+32 t$, find equations for y and z which combine with this equation to give parametric equations for the normal line through P. $y=$ [ANS]\n$z=$ [ANS]\nNote: Your answers should be expressions of t; B. Find an equation for the tangent plane through P. $z=$ [ANS]\nNote: Your answers should be expressions of x and y; e.g. \"3xy+2y\"", "answer_v1": [ "1 + 2 * 3^2 * t", "1 + 2 * 4^2 * t", "-(4/4)^2 * x - (3/4)^2 * y + (4^2 + 3^2 + 4^2) / 4^2" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the surface 1x^{2}+25 y^{2}+1 z^{2}=27 and the point $P=\\left(1, 1, 1 \\right)$ on this surface. A. Starting with the equation $x=1+2 t$, find equations for y and z which combine with this equation to give parametric equations for the normal line through P. $y=$ [ANS]\n$z=$ [ANS]\nNote: Your answers should be expressions of t; B. Find an equation for the tangent plane through P. $z=$ [ANS]\nNote: Your answers should be expressions of x and y; e.g. \"3xy+2y\"", "answer_v2": [ "1 + 2 * 5^2 * t", "1 + 2 * 1^2 * t", "-(1/1)^2 * x - (5/1)^2 * y + (1^2 + 5^2 + 1^2) / 1^2" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the surface 4x^{2}+16 y^{2}+4 z^{2}=24 and the point $P=\\left(1, 1, 1 \\right)$ on this surface. A. Starting with the equation $x=1+8 t$, find equations for y and z which combine with this equation to give parametric equations for the normal line through P. $y=$ [ANS]\n$z=$ [ANS]\nNote: Your answers should be expressions of t; B. Find an equation for the tangent plane through P. $z=$ [ANS]\nNote: Your answers should be expressions of x and y; e.g. \"3xy+2y\"", "answer_v3": [ "1 + 2 * 4^2 * t", "1 + 2 * 2^2 * t", "-(2/2)^2 * x - (4/2)^2 * y + (2^2 + 4^2 + 2^2) / 2^2" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0233", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "Multivariable", "Tangent" ], "problem_v1": "Find the equation of the tangent plane to the surface $z=4 y^{2}-1x^{2}$ at the point $\\left(1, 2, 15 \\right)$. z=[ANS]\nNote: Your answer should be an expression of x and y; e.g. \"3x-4y+6\"", "answer_v1": [ "( -2 * 1 * 1 * x ) + (2 * 4 * 2 * y ) + ( 1* 1 - 4 * 4 )" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the equation of the tangent plane to the surface $z=16 y^{2}-16x^{2}$ at the point $\\left(-3,-1,-128 \\right)$. z=[ANS]\nNote: Your answer should be an expression of x and y; e.g. \"3x-4y+6\"", "answer_v2": [ "( -2 * 16 * -3 * x ) + (2 * 16 * -1 * y ) + ( 16* 9 - 16 * 1 )" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the equation of the tangent plane to the surface $z=4 y^{2}-1x^{2}$ at the point $\\left(-2, 0,-4 \\right)$. z=[ANS]\nNote: Your answer should be an expression of x and y; e.g. \"3x-4y+6\"", "answer_v3": [ "( -2 * 1 * -2 * x ) + (2 * 4 * 0 * y ) + ( 1* 4 - 4 * 0 )" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0234", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "Multivariable", "Parametric", "Tangent" ], "problem_v1": "Find an equation of the tangent plane to the parametric surface $x=3 r \\cos \\theta$, $y=1 r \\sin \\theta$, $z=r$ at the point $\\left(3 \\sqrt{2}, 1 \\sqrt{2}, 2 \\right)$ when $r=2$, $\\theta=\\pi/4$. z=[ANS]\nNote: Your answer should be an expression of x and y; e.g. \"3x-4y\"", "answer_v1": [ "2 + 0.235702260395516 * (x- ((3)*sqrt(2))) + 0.707106781186547 * (y- ((1)*sqrt(2)))" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an equation of the tangent plane to the parametric surface $x=-5 r \\cos \\theta$, $y=5 r \\sin \\theta$, $z=r$ at the point $\\left(-5 \\sqrt{2}, 5 \\sqrt{2}, 2 \\right)$ when $r=2$, $\\theta=\\pi/4$. z=[ANS]\nNote: Your answer should be an expression of x and y; e.g. \"3x-4y\"", "answer_v2": [ "2 + -0.14142135623731 * (x- ((-5)*sqrt(2))) + 0.14142135623731 * (y- ((5)*sqrt(2)))" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an equation of the tangent plane to the parametric surface $x=-2 r \\cos \\theta$, $y=1 r \\sin \\theta$, $z=r$ at the point $\\left(-2 \\sqrt{2}, 1 \\sqrt{2}, 2 \\right)$ when $r=2$, $\\theta=\\pi/4$. z=[ANS]\nNote: Your answer should be an expression of x and y; e.g. \"3x-4y\"", "answer_v3": [ "2 + -0.353553390593274 * (x- ((-2)*sqrt(2))) + 0.707106781186547 * (y- ((1)*sqrt(2)))" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0235", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "Multivariable", "Linearization", "Linearization" ], "problem_v1": "Find the linearization $L \\left(x, y \\right)$ of the function $f\\left(x, y \\right)=\\sqrt{109-16x^{2}-9 y^{2}}$ at $\\left(2,-2 \\right)$. $L \\left(x, y \\right)=$ [ANS]\nNote: Your answer should be an expression in x and y; e.g. \"3x-5y+9\"", "answer_v1": [ "-10.6666666666667 * ( x - 2 ) + 6 * ( y - -2 ) + 3" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the linearization $L \\left(x, y \\right)$ of the function $f\\left(x, y \\right)=\\sqrt{258-1x^{2}-16 y^{2}}$ at $\\left(-1, 4 \\right)$. $L \\left(x, y \\right)=$ [ANS]\nNote: Your answer should be an expression in x and y; e.g. \"3x-5y+9\"", "answer_v2": [ "1 * ( x - -1 ) + -64 * ( y - 4 ) + 1" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the linearization $L \\left(x, y \\right)$ of the function $f\\left(x, y \\right)=\\sqrt{85-4x^{2}-9 y^{2}}$ at $\\left(0,-3 \\right)$. $L \\left(x, y \\right)=$ [ANS]\nNote: Your answer should be an expression in x and y; e.g. \"3x-5y+9\"", "answer_v3": [ "0 * ( x - 0 ) + 13.5 * ( y - -3 ) + 2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0236", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "differentiable function", "multivariable", "functions", "derivatives", "calculus" ], "problem_v1": "List the points in the $xy$-plane, if any, at which the function $z=7+\\sqrt{\\left(x-1\\right)^{2}+\\left(y-2\\right)^{2}}$ is not differentiable. points=[ANS]\n(If there is more than one point, give them as a comma-separated list, e.g., (1,2),(3,4) ; if the function is differentiable everywhere, enter the word none.)", "answer_v1": [ "(1,2)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "List the points in the $xy$-plane, if any, at which the function $z=1+\\sqrt{\\left(x-7\\right)^{2}+\\left(y+6\\right)^{2}}$ is not differentiable. points=[ANS]\n(If there is more than one point, give them as a comma-separated list, e.g., (1,2),(3,4) ; if the function is differentiable everywhere, enter the word none.)", "answer_v2": [ "(7,-6)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "List the points in the $xy$-plane, if any, at which the function $z=3+\\sqrt{\\left(x-2\\right)^{2}+\\left(y+4\\right)^{2}}$ is not differentiable. points=[ANS]\n(If there is more than one point, give them as a comma-separated list, e.g., (1,2),(3,4) ; if the function is differentiable everywhere, enter the word none.)", "answer_v3": [ "(2,-4)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0237", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "linearization", "differentials", "multivariable", "functions" ], "problem_v1": "Find the differential of the function $f(x,y)=xe^{y}$ at $(1,0)$. $df=$ [ANS]", "answer_v1": [ "1*dx+1*dy" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the differential of the function $f(x,y)=ye^{-x}$ at $(0,3)$. $df=$ [ANS]", "answer_v2": [ "-3*dx+1*dy" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the differential of the function $f(x,y)=ye^{-x}$ at $(0,1)$. $df=$ [ANS]", "answer_v3": [ "-1*dx+1*dy" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0238", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "linearization", "differentials", "multivariable", "functions" ], "problem_v1": "A fluid moves through a tube of length 1 meter and radius $r=0.007 \\pm 0.0002$ meters under a pressure $p=4\\cdot 10^5 \\pm 1750$ pascals, at a rate $v=0.375 \\cdot 10^{-9}$ $\\hbox{m}^3$ per unit time. Use differentials to estimate the maximum error in the viscosity $\\eta$ given by \\eta= \\frac{\\pi}{8} \\frac{p r^4}{v} . maximum error $\\approx$ [ANS]", "answer_v1": [ "pi/8*(0.007^4*1750/0.375+4*4*10^5*0.007^3*0.0002/0.375)*10^9" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A fluid moves through a tube of length 1 meter and radius $r=0.002 \\pm 0.00025$ meters under a pressure $p=1\\cdot 10^5 \\pm 1250$ pascals, at a rate $v=0.875 \\cdot 10^{-9}$ $\\hbox{m}^3$ per unit time. Use differentials to estimate the maximum error in the viscosity $\\eta$ given by \\eta= \\frac{\\pi}{8} \\frac{p r^4}{v} . maximum error $\\approx$ [ANS]", "answer_v2": [ "pi/8*(0.002^4*1250/0.875+4*1*10^5*0.002^3*0.00025/0.875)*10^9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A fluid moves through a tube of length 1 meter and radius $r=0.004 \\pm 0.0002$ meters under a pressure $p=2\\cdot 10^5 \\pm 1500$ pascals, at a rate $v=0.25 \\cdot 10^{-9}$ $\\hbox{m}^3$ per unit time. Use differentials to estimate the maximum error in the viscosity $\\eta$ given by \\eta= \\frac{\\pi}{8} \\frac{p r^4}{v} . maximum error $\\approx$ [ANS]", "answer_v3": [ "pi/8*(0.004^4*1500/0.25+4*2*10^5*0.004^3*0.0002/0.25)*10^9" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0239", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "1", "keywords": [ "linearization", "differentials", "multivariable", "functions" ], "problem_v1": "Find the equation of the tangent plane to the surface determined by x^4 y^3+z-45=0 at $x=4$, $y=3$. $z=$ [ANS]", "answer_v1": [ "-[6867+6912*(x-4)+6912*(y-3)]" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the equation of the tangent plane to the surface determined by x^2 y^4+z-25=0 at $x=3$, $y=5$. $z=$ [ANS]", "answer_v2": [ "-[5600+3750*(x-3)+4500*(y-5)]" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the equation of the tangent plane to the surface determined by x^2 y^3+z-30=0 at $x=3$, $y=2$. $z=$ [ANS]", "answer_v3": [ "-[42+48*(x-3)+108*(y-2)]" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0240", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "linearization", "differentials", "multivariable", "functions" ], "problem_v1": "Find the equation of the tangent plane to z=e^{y}+x+x^{5}+6 at the point $(4, 0, 1035)$. $z=$ [ANS]", "answer_v1": [ "1035+1281*(x-4)+y" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the equation of the tangent plane to z=e^{x}+y+y^{2}+10 at the point $(0, 1, 13)$. $z=$ [ANS]", "answer_v2": [ "13+x+3*(y-1)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the equation of the tangent plane to z=e^{y}+x+x^{3}+7 at the point $(2, 0, 18)$. $z=$ [ANS]", "answer_v3": [ "18+13*(x-2)+y" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0241", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "linearization", "differentials", "multivariable", "functions" ], "problem_v1": "The gas equation for one mole of oxygen relates its pressure, $P$ (in atmospheres), its temperature, $T$ (in K), and its volume, $V$ (in cubic decimeters, dm ${}^3$): T=16.574\\cdot{1 \\over V}-0.52754\\cdot{1 \\over V^2}-0.3879\\,P+12.187\\,V\\,P.\n(a) Find the temperature $T$ and differential $dT$ if the volume is 36 dm ${}^3$ and the pressure is 1.5 atmosphere. $T=$ [ANS]\n$dT=$ [ANS]\n(b) Use your answer to part (a) to estimate how much the pressure would have to change if the volume increased by 2.5 dm ${}^3$ and the temperature remained constant. change in pressure=[ANS]", "answer_v1": [ "657.976", "18.2677*dV+438.344*dP", "-1*18.2677*2.5/438.344" ], "answer_type_v1": [ "NV", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The gas equation for one mole of oxygen relates its pressure, $P$ (in atmospheres), its temperature, $T$ (in K), and its volume, $V$ (in cubic decimeters, dm ${}^3$): T=16.574\\cdot{1 \\over V}-0.52754\\cdot{1 \\over V^2}-0.3879\\,P+12.187\\,V\\,P.\n(a) Find the temperature $T$ and differential $dT$ if the volume is 20 dm ${}^3$ and the pressure is 2 atmosphere. $T=$ [ANS]\n$dT=$ [ANS]\n(b) Use your answer to part (a) to estimate how much the volume would have to change if the pressure increased by 0.1 atmosphere and the temperature remained constant. change in volume=[ANS]", "answer_v2": [ "487.532", "24.3327*dV+243.352*dP", "-1*243.352*0.1/24.3327" ], "answer_type_v2": [ "NV", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The gas equation for one mole of oxygen relates its pressure, $P$ (in atmospheres), its temperature, $T$ (in K), and its volume, $V$ (in cubic decimeters, dm ${}^3$): T=16.574\\cdot{1 \\over V}-0.52754\\cdot{1 \\over V^2}-0.3879\\,P+12.187\\,V\\,P.\n(a) Find the temperature $T$ and differential $dT$ if the volume is 26 dm ${}^3$ and the pressure is 1.5 atmosphere. $T=$ [ANS]\n$dT=$ [ANS]\n(b) Use your answer to part (a) to estimate how much the volume would have to change if the pressure increased by 0.15 atmosphere and the temperature remained constant. change in volume=[ANS]", "answer_v3": [ "475.348", "18.256*dV+316.474*dP", "-1*316.474*0.15/18.256" ], "answer_type_v3": [ "NV", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0242", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "linearization", "differentials", "multivariable", "functions" ], "problem_v1": "Find the differential of $f(x,y)=\\sqrt{x^{3}+y^{2}}$ at the point $(2,3)$. $df=$ [ANS]\nThen use the differential to estimate $f(2.04,2.96)$. $f(2.04,2.96) \\approx$ [ANS]", "answer_v1": [ "3*2^2/[2*sqrt(2^3+3^2)]*dx+2*3^1/[2*sqrt(2^3+3^2)]*dy", "4.15221" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the differential of $f(x,y)=\\sqrt{x^{2}+y^{3}}$ at the point $(3,2)$. $df=$ [ANS]\nThen use the differential to estimate $f(2.96,2.1)$. $f(2.96,2.1) \\approx$ [ANS]", "answer_v2": [ "2*3^1/[2*sqrt(3^2+2^3)]*dx+3*2^2/[2*sqrt(3^2+2^3)]*dy", "4.23952" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the differential of $f(x,y)=\\sqrt{x^{2}+y^{3}}$ at the point $(2,1)$. $df=$ [ANS]\nThen use the differential to estimate $f(2.02,0.94)$. $f(2.02,0.94) \\approx$ [ANS]", "answer_v3": [ "2*2^1/[2*sqrt(2^2+1^3)]*dx+3*1^2/[2*sqrt(2^2+1^3)]*dy", "2.21371" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0243", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "linear", "functions", "multivariable", "plane" ], "problem_v1": "Suppose that $z$ is a linear function of $x$ and $y$ with slope 3 in the $x$ direction and slope 1 in the $y$ direction.\n(a) A change of $0.1$ in $x$ and $0.2$ in $y$ produces what change in $z$? change in $z=$ [ANS]\n(b) If $z=5$ when $x=4$ and $y=4$, what is the value of $z$ when $x=4$ and $y=3.9$? $z=$ [ANS]", "answer_v1": [ "3*0.1+1*0.2", "5+3*(4-4)+1*(3.9-4)" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that $z$ is a linear function of $x$ and $y$ with slope-5 in the $x$ direction and slope 5 in the $y$ direction.\n(a) A change of $-0.4$ in $x$ and $-0.2$ in $y$ produces what change in $z$? change in $z=$ [ANS]\n(b) If $z=3$ when $x=8$ and $y=4$, what is the value of $z$ when $x=7.8$ and $y=4$? $z=$ [ANS]", "answer_v2": [ "-5*-0.4+5*-0.2", "3+-5*(7.8-8)+5*(4-4)" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that $z$ is a linear function of $x$ and $y$ with slope-2 in the $x$ direction and slope 1 in the $y$ direction.\n(a) A change of $-0.2$ in $x$ and $0.1$ in $y$ produces what change in $z$? change in $z=$ [ANS]\n(b) If $z=7$ when $x=3$ and $y=4$, what is the value of $z$ when $x=3.2$ and $y=4.2$? $z=$ [ANS]", "answer_v3": [ "-2*-0.2+1*0.1", "7+-2*(3.2-3)+1*(4.2-4)" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0244", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "gradient", "directional derivative", "derivatives" ], "problem_v1": "(a) For $g(x,y)=\\sqrt{x^{2}+4y+21}$, $\\nabla g(4,3)=$ [ANS]\n(b) Find the best linear approximation of $g(x,y)$ for $(x,y)$ near $(4,3)$. Linear approximation=[ANS]\n(c) Use the approximation in part (b) to estimate $g(4.02, 2.99)$. $g(4.02,2.99) \\approx$ [ANS]", "answer_v1": [ "(0.571429,0.285714)", "7+4/7*(x-4)+4/(2*7)*(y-3)", "7.00857" ], "answer_type_v1": [ "OL", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "(a) For $g(x,y)=\\sqrt{x^{2}+y+10}$, $\\nabla g(1,5)=$ [ANS]\n(b) Find the best linear approximation of $g(x,y)$ for $(x,y)$ near $(1,5)$. Linear approximation=[ANS]\n(c) Use the approximation in part (b) to estimate $g(0.99, 5.03)$. $g(0.99,5.03) \\approx$ [ANS]", "answer_v2": [ "(0.25,0.125)", "4+1/4*(x-1)+1/(2*4)*(y-5)", "4.00125" ], "answer_type_v2": [ "OL", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "(a) For $g(x,y)=\\sqrt{x^{2}+2y+13}$, $\\nabla g(2,4)=$ [ANS]\n(b) Find the best linear approximation of $g(x,y)$ for $(x,y)$ near $(2,4)$. Linear approximation=[ANS]\n(c) Use the approximation in part (b) to estimate $g(1.98, 3.99)$. $g(1.98,3.99) \\approx$ [ANS]", "answer_v3": [ "(0.4,0.2)", "5+2/5*(x-2)+2/(2*5)*(y-4)", "4.99" ], "answer_type_v3": [ "OL", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0245", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "chain rule", "multivariable", "functions", "calculus", "derivatives", "second derivative" ], "problem_v1": "0.1/0.1 Find the linear, $L(x,y)$, and quadratic, $Q(x,y)$, Taylor polynomials valid near $(4,0)$ for f(x,y)=\\sin\\!\\left(x-4\\right)\\cos\\!\\left(y\\right). $L(x,y)=$ [ANS]\n$Q(x,y)=$ [ANS]\nFind the approximations for $f(4.1,0.1)$ generated by these, and compare with the exact value of the function: At $(4.1,0.1)$, $L(4.1,0.1)=$ [ANS], $Q(4.1,0.1)=$ [ANS], $f(4.1,0.1)=$ [ANS].", "answer_v1": [ "x-4", "x-4", "0.1", "0.1", "0.0993347" ], "answer_type_v1": [ "EX", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "0.3/-0.3 Find the linear, $L(x,y)$, and quadratic, $Q(x,y)$, Taylor polynomials valid near $(0,1)$ for f(x,y)=\\cos\\!\\left(x\\right)\\sin\\!\\left(y-1\\right). $L(x,y)=$ [ANS]\n$Q(x,y)=$ [ANS]\nFind the approximations for $f(0.3,0.7)$ generated by these, and compare with the exact value of the function: At $(0.3,0.7)$, $L(0.3,0.7)=$ [ANS], $Q(0.3,0.7)=$ [ANS], $f(0.3,0.7)=$ [ANS].", "answer_v2": [ "y-1", "y-1", "-0.3", "-0.3", "-0.282321" ], "answer_type_v2": [ "EX", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "0.1/-0.2 Find the linear, $L(x,y)$, and quadratic, $Q(x,y)$, Taylor polynomials valid near $(2,0)$ for f(x,y)=\\sin\\!\\left(x-2\\right)\\cos\\!\\left(y\\right). $L(x,y)=$ [ANS]\n$Q(x,y)=$ [ANS]\nFind the approximations for $f(2.1,-0.2)$ generated by these, and compare with the exact value of the function: At $(2.1,-0.2)$, $L(2.1,-0.2)=$ [ANS], $Q(2.1,-0.2)=$ [ANS], $f(2.1,-0.2)=$ [ANS].", "answer_v3": [ "x-2", "x-2", "0.1", "0.1", "0.0978434" ], "answer_type_v3": [ "EX", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0246", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "chain rule", "multivariable", "functions", "calculus", "derivatives", "second derivative" ], "problem_v1": "Find the best quadratic approximation, $Q(x,y)$ of $\\sqrt{9+x+2y}$ for $(x,y)$ near $(0,0)$, $Q(x,y)=$ [ANS]", "answer_v1": [ "3+0.166667*x+0.333333*y-0.00925926*x^2/2-0.037037*y^2/2-0.0185185*x*y" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the best quadratic approximation, $Q(x,y)$ of $\\sqrt{1+6x-5y}$ for $(x,y)$ near $(0,0)$, $Q(x,y)=$ [ANS]", "answer_v2": [ "1+3*x-2.5*y-9*x^2/2-6.25*y^2/2+7.5*x*y" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find the best quadratic approximation, $Q(x,y)$ of $\\sqrt{1+2x-3y}$ for $(x,y)$ near $(0,0)$, $Q(x,y)=$ [ANS]", "answer_v3": [ "1+x-1.5*y-x^2/2-2.25*y^2/2+1.5*x*y" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0247", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "gradient", "directional derivative", "multivariable", "functions" ], "problem_v1": "Check that the point $(1,-1,1)$ lies on the given surface. Then, viewing the surface as a level surface for a function $f(x,y,z)$, find a vector normal to the surface and an equation for the tangent plane to the surface at $(1,-1,1)$. 4x^{2}-3y^{2}+3z^{2}=4 vector normal=[ANS]\ntangent plane: $z=$ [ANS]", "answer_v1": [ "(8,6,6)", "1-[8*(x-1)+6*(y+1)]/6" ], "answer_type_v1": [ "OL", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Check that the point $(-1,1,1)$ lies on the given surface. Then, viewing the surface as a level surface for a function $f(x,y,z)$, find a vector normal to the surface and an equation for the tangent plane to the surface at $(-1,1,1)$. x^{2}-4y^{2}+z^{2}=-2 vector normal=[ANS]\ntangent plane: $z=$ [ANS]", "answer_v2": [ "(-2,-8,+2)", "[2*(x+1)+8*(y-1)]/2+1" ], "answer_type_v2": [ "OL", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Check that the point $(1,-1,2)$ lies on the given surface. Then, viewing the surface as a level surface for a function $f(x,y,z)$, find a vector normal to the surface and an equation for the tangent plane to the surface at $(1,-1,2)$. 2x^{2}-3y^{2}+2z^{2}=7 vector normal=[ANS]\ntangent plane: $z=$ [ANS]", "answer_v3": [ "(4,6,8)", "2-[4*(x-1)+6*(y+1)]/8" ], "answer_type_v3": [ "OL", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0248", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "gradient", "directional derivative", "multivariable", "functions" ], "problem_v1": "A differentiable function $f(x,y)$ has the property that $f(4,3)=5$ and $f_x(4,3)=3$ and $f_y(4,3)=-3$. Find the equation of the tangent plane at the point on the surface $z=f(x,y)$ where $x=4$, $y=3$. $z=$ [ANS]", "answer_v1": [ "5+3*(x-4)+-3*(y-3)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "A differentiable function $f(x,y)$ has the property that $f(1,5)=2$ and $f_x(1,5)=-2$ and $f_y(1,5)=7$. Find the equation of the tangent plane at the point on the surface $z=f(x,y)$ where $x=1$, $y=5$. $z=$ [ANS]", "answer_v2": [ "2+-2*(x-1)+7*(y-5)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "A differentiable function $f(x,y)$ has the property that $f(2,4)=2$ and $f_x(2,4)=1$ and $f_y(2,4)=-4$. Find the equation of the tangent plane at the point on the surface $z=f(x,y)$ where $x=2$, $y=4$. $z=$ [ANS]", "answer_v3": [ "2+1*(x-2)+-4*(y-4)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0249", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "differentiation' 'function" ], "problem_v1": "Consider the surface $F(x,y,z)=x^{8}z^{6}+\\sin(y^{6}z^{6})+5=0.$ Describe the set of points on the surface for which it is not possible to define the surface as the graph of a differentiable function $z=f(x,y)$. Your answer should be in the form $g(x,y,z)=0$ [ANS] $=0.$", "answer_v1": [ "x**8 * 6 * z**(6-1) + cos(y**6 * z**6) * 6 * y**6 * z**(6-1)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Consider the surface $F(x,y,z)=x^{2}z^{3}+\\sin(y^{9}z^{3})-3=0.$ Describe the set of points on the surface for which it is not possible to define the surface as the graph of a differentiable function $z=f(x,y)$. Your answer should be in the form $g(x,y,z)=0$ [ANS] $=0.$", "answer_v2": [ "x^2 * 3 * z**(3-1) + cos(y**9 * z^3) * 3 * y**9 * z**(3-1)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Consider the surface $F(x,y,z)=x^{4}z^{4}+\\sin(y^{6}z^{4})+1=0.$ Describe the set of points on the surface for which it is not possible to define the surface as the graph of a differentiable function $z=f(x,y)$. Your answer should be in the form $g(x,y,z)=0$ [ANS] $=0.$", "answer_v3": [ "x^4 * 4 * z**(4-1) + cos(y**6 * z^4) * 4 * y**6 * z**(4-1)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0250", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "tangent' 'plane" ], "problem_v1": "Find an equation of the tangent plane to the surface $xy \\sin(\\pi z/3)=21$ at the point $(\\sqrt{3}, 14, 1)$. The equation should have the form $g(x,y,z)=0$, and the coefficient of x should be $7 \\sqrt{3}$. [ANS] $=0.$", "answer_v1": [ "sqrt(3)* 7 * (x - sqrt(3)) + 3 * (y - 14)/2 + 7 * 3.14159265358979 * (z - 1)/sqrt(3)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find an equation of the tangent plane to the surface $xy \\sin(\\pi z/3)=3$ at the point $(\\sqrt{3}, 2, 1)$. The equation should have the form $g(x,y,z)=0$, and the coefficient of x should be $1 \\sqrt{3}$. [ANS] $=0.$", "answer_v2": [ "sqrt(3)* 1 * (x - sqrt(3)) + 3 * (y - 2)/2 + 1 * 3.14159265358979 * (z - 1)/sqrt(3)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find an equation of the tangent plane to the surface $xy \\sin(\\pi z/3)=9$ at the point $(\\sqrt{3}, 6, 1)$. The equation should have the form $g(x,y,z)=0$, and the coefficient of x should be $3 \\sqrt{3}$. [ANS] $=0.$", "answer_v3": [ "sqrt(3)* 3 * (x - sqrt(3)) + 3 * (y - 6)/2 + 3 * 3.14159265358979 * (z - 1)/sqrt(3)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0251", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "tangent plane' 'normal vector" ], "problem_v1": "Consider the surface $xyz=64$. A. Find the unit normal vector to the surface at the point $(4, 4, 4)$ with positive first coordinate. $($ [ANS], [ANS], [ANS] $)$ B. Find the equation of the tangent plane to the surface at the given point. Express your answer in the form $ax+by+cz+d=0$, normalized so that $a=16$. [ANS]=0.", "answer_v1": [ "0.577350269189626", "0.577350269189626", "0.577350269189626", "4*4*(x - 4) + 4*4*(y - 4) + 4*4*(z - 4)" ], "answer_type_v1": [ "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the surface $xyz=10$. A. Find the unit normal vector to the surface at the point $(1, 5, 2)$ with positive first coordinate. $($ [ANS], [ANS], [ANS] $)$ B. Find the equation of the tangent plane to the surface at the given point. Express your answer in the form $ax+by+cz+d=0$, normalized so that $a=10$. [ANS]=0.", "answer_v2": [ "0.880450906325624", "0.176090181265125", "0.440225453162812", "5*2*(x - 1) + 1*2*(y - 5) + 1*5*(z - 2)" ], "answer_type_v2": [ "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the surface $xyz=24$. A. Find the unit normal vector to the surface at the point $(2, 4, 3)$ with positive first coordinate. $($ [ANS], [ANS], [ANS] $)$ B. Find the equation of the tangent plane to the surface at the given point. Express your answer in the form $ax+by+cz+d=0$, normalized so that $a=12$. [ANS]=0.", "answer_v3": [ "0.768221279597376", "0.384110639798688", "0.512147519731584", "4*3*(x - 2) + 2*3*(y - 4) + 2*4*(z - 3)" ], "answer_type_v3": [ "NV", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0252", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "3", "keywords": [ "parametrize' 'surface' 'tangent plane" ], "problem_v1": "Consider $x=h(y,z)$ as a parametrized surface $\\Phi(y,z)$ in the natural way. Write the equation of the tangent plane to the surface at the point $(3, 1, 1)$ [with the coefficient of x being 1] given that $ \\frac{\\partial h}{\\partial y} (1,1)=2$ and $ \\frac{\\partial h}{\\partial z} (1,1)=-2$. [ANS] $=0$.", "answer_v1": [ "x - 3 - 2*(y - 1) - -2 * (z - 1)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Consider $x=h(y,z)$ as a parametrized surface $\\Phi(y,z)$ in the natural way. Write the equation of the tangent plane to the surface at the point $(-5, 5,-4)$ [with the coefficient of x being 1] given that $ \\frac{\\partial h}{\\partial y} (5,-4)=-2$ and $ \\frac{\\partial h}{\\partial z} (5,-4)=5$. [ANS] $=0$.", "answer_v2": [ "x - -5 - -2*(y - 5) - 5 * (z - -4)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Consider $x=h(y,z)$ as a parametrized surface $\\Phi(y,z)$ in the natural way. Write the equation of the tangent plane to the surface at the point $(-2, 1,-2)$ [with the coefficient of x being 1] given that $ \\frac{\\partial h}{\\partial y} (1,-2)=1$ and $ \\frac{\\partial h}{\\partial z} (1,-2)=-3$. [ANS] $=0$.", "answer_v3": [ "x - -2 - 1*(y - 1) - -3 * (z - -2)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0253", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "Linear function" ], "problem_v1": "Find the implicit equation of the tangent plane to $z=\\ln\\!\\left(x^{2}+1\\right)+y^{2}$ at the point $\\left(0,6,36\\right)$. [ANS]", "answer_v1": [ "12*y-z = 36" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Find the implicit equation of the tangent plane to $z=\\ln\\!\\left(x^{2}+1\\right)+y^{2}$ at the point $\\left(0,2,4\\right)$. [ANS]", "answer_v2": [ "4*y-z = 4" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Find the implicit equation of the tangent plane to $z=\\ln\\!\\left(x^{2}+1\\right)+y^{2}$ at the point $\\left(0,3,9\\right)$. [ANS]", "answer_v3": [ "6*y-z = 9" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0254", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "Derivative", "Differential", "Tangent Plane" ], "problem_v1": "A student was asked to find the equation of the tangent plane to the surface $z=x^{4}-y^{5}$ at the point $(x,y)=(3,4)$. The student's answer was $z=-943+4x^{3} (x-3)-\\left(5y^{4}\\right) (y-4).$\n(a) At a glance, how do you know this is wrong. What mistakes did the student make? Select all that apply. [ANS] A. The partial derivatives were not evaluated a the point. B. The-943 should not be in the answer. C. The answer is not a linear function. D. The (x-3) and (y-4) should be x and y. E. All of the above\n(b) Find the correct equation for the tangent plane. $z=$ [ANS]", "answer_v1": [ "AC", "108*(x-3)-943-1280*(y-4)" ], "answer_type_v1": [ "MCM", "EX" ], "options_v1": [ [ "A", "B", "C", "D", "E" ], [] ], "problem_v2": "A student was asked to find the equation of the tangent plane to the surface $z=x^{2}-y^{3}$ at the point $(x,y)=(5,1)$. The student's answer was $z=24+2x (x-5)-\\left(3y^{2}\\right) (y-1).$\n(a) At a glance, how do you know this is wrong. What mistakes did the student make? Select all that apply. [ANS] A. The 24 should not be in the answer. B. The (x-5) and (y-1) should be x and y. C. The partial derivatives were not evaluated a the point. D. The answer is not a linear function. E. All of the above\n(b) Find the correct equation for the tangent plane. $z=$ [ANS]", "answer_v2": [ "CD", "24+10*(x-5)-3*(y-1)" ], "answer_type_v2": [ "MCM", "EX" ], "options_v2": [ [ "A", "B", "C", "D", "E" ], [] ], "problem_v3": "A student was asked to find the equation of the tangent plane to the surface $z=x^{2}-y^{3}$ at the point $(x,y)=(4,2)$. The student's answer was $z=8+2x (x-4)-\\left(3y^{2}\\right) (y-2).$\n(a) At a glance, how do you know this is wrong. What mistakes did the student make? Select all that apply. [ANS] A. The answer is not a linear function. B. The partial derivatives were not evaluated a the point. C. The (x-4) and (y-2) should be x and y. D. The 8 should not be in the answer. E. All of the above\n(b) Find the correct equation for the tangent plane. $z=$ [ANS]", "answer_v3": [ "AB", "8+8*(x-4)-12*(y-2)" ], "answer_type_v3": [ "MCM", "EX" ], "options_v3": [ [ "A", "B", "C", "D", "E" ], [] ] }, { "id": "Calculus_-_multivariable_0255", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "Derivative", "Differential", "Tangent Plane" ], "problem_v1": "One mole of ammonia gas is contained in a vessel which is capable of changing its volume (a compartment sealed by a piston, for example). The total energy $U$ (in Joules) of the ammonia is a function of the volume $V$ (in cubic meters) of the container, and the temperature $T$ (in degrees Kelvin) of the gas. The differential $dU$ is given by $dU=840 dV+27.32 dT$.\n(a) How does the energy change if the volume is held constant and the temperature is increased slightly? [ANS] A. it increases slightly B. it does not change C. it decreases slightly\n(b) How does the energy change if the temperature is held constant and the volume is decreased slightly? [ANS] A. it decreases slightly B. it does not change C. it increases slightly\n(c) Find the approximate change in energy if the gas is compressed by 400 cubic centimeters and heated by 4 degrees Kelvin. Change in energy=[ANS] J. Please include in your answer.", "answer_v1": [ "A", "A", "108.944" ], "answer_type_v1": [ "MCS", "MCS", "NV" ], "options_v1": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [] ], "problem_v2": "One mole of ammonia gas is contained in a vessel which is capable of changing its volume (a compartment sealed by a piston, for example). The total energy $U$ (in Joules) of the ammonia is a function of the volume $V$ (in cubic meters) of the container, and the temperature $T$ (in degrees Kelvin) of the gas. The differential $dU$ is given by $dU=840 dV+27.32 dT$.\n(a) How does the energy change if the volume is held constant and the temperature is decreased slightly? [ANS] A. it decreases slightly B. it increases slightly C. it does not change\n(b) How does the energy change if the temperature is held constant and the volume is increased slightly? [ANS] A. it decreases slightly B. it increases slightly C. it does not change\n(c) Find the approximate change in energy if the gas is compressed by 250 cubic centimeters and heated by 6 degrees Kelvin. Change in energy=[ANS] J. Please include in your answer.", "answer_v2": [ "A", "B", "163.71" ], "answer_type_v2": [ "MCS", "MCS", "NV" ], "options_v2": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [] ], "problem_v3": "One mole of ammonia gas is contained in a vessel which is capable of changing its volume (a compartment sealed by a piston, for example). The total energy $U$ (in Joules) of the ammonia is a function of the volume $V$ (in cubic meters) of the container, and the temperature $T$ (in degrees Kelvin) of the gas. The differential $dU$ is given by $dU=840 dV+27.32 dT$.\n(a) How does the energy change if the volume is held constant and the temperature is decreased slightly? [ANS] A. it does not change B. it decreases slightly C. it increases slightly\n(b) How does the energy change if the temperature is held constant and the volume is increased slightly? [ANS] A. it increases slightly B. it decreases slightly C. it does not change\n(c) Find the approximate change in energy if the gas is compressed by 300 cubic centimeters and heated by 3 degrees Kelvin. Change in energy=[ANS] J. Please include in your answer.", "answer_v3": [ "B", "A", "81.708" ], "answer_type_v3": [ "MCS", "MCS", "NV" ], "options_v3": [ [ "A", "B", "C" ], [ "A", "B", "C" ], [] ] }, { "id": "Calculus_-_multivariable_0256", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "Derivative", "Differential", "Tangent Plane" ], "problem_v1": "Suppose $ f(x,t)=x^{4}\\sin\\!\\left(5t\\right)$.\n(a) At any point $(x,t)$, the differential is $df=$ [ANS]\n(b) At the point $(1,\\pi/4)$, the differential is $df=$ [ANS]\n(c) At the point $(1,\\pi/4)$ with $dx=0.2$ and $dt=-0.4$, the differential is $df=$ [ANS]", "answer_v1": [ "4*x^3*sin(5*t)*dx+5*x^4*cos(5*t)*dt", "-(2.82843*dx+3.53553*dt)", "0.848526" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $ f(x,t)=x^{4}\\sin\\!\\left(2t\\right)$.\n(a) At any point $(x,t)$, the differential is $df=$ [ANS]\n(b) At the point $(-2,\\pi/3)$, the differential is $df=$ [ANS]\n(c) At the point $(-2,\\pi/3)$ with $dx=-0.5$ and $dt=0.4$, the differential is $df=$ [ANS]", "answer_v2": [ "4*x^3*sin(2*t)*dx+2*x^4*cos(2*t)*dt", "-(27.7128*dx+16*dt)", "7.4564" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $ f(x,t)=x^{4}\\sin\\!\\left(3t\\right)$.\n(a) At any point $(x,t)$, the differential is $df=$ [ANS]\n(b) At the point $(-2,\\pi/4)$, the differential is $df=$ [ANS]\n(c) At the point $(-2,\\pi/4)$ with $dx=0.2$ and $dt=-0.4$, the differential is $df=$ [ANS]", "answer_v3": [ "4*x^3*sin(3*t)*dx+3*x^4*cos(3*t)*dt", "-(22.6274*dx+33.9411*dt)", "9.05096" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0257", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "Derivative", "Differential", "Tangent Plane" ], "problem_v1": "Suppose $ F(m,r)= \\frac{Gm}{r^{2} }$.\n(a) At any point $(m,r)$, the differential is $dF=$ [ANS]\n(b) At the point $(250,8)$, the differential is $dF=$ [ANS]\n(c) At the point $(250,8)$ with $dm=0.4$ and $dr=-0.4$, the differential is $dF=$ [ANS]", "answer_v1": [ "G*r^2/[(r^2)^2]*dm+[-(G*m*2*r/[(r^2)^2])]*dr", "0.153125*dm-9.57031*dr", "3.88937" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $ F(m,r)= \\frac{Gm}{r^{2} }$.\n(a) At any point $(m,r)$, the differential is $dF=$ [ANS]\n(b) At the point $(110,10)$, the differential is $dF=$ [ANS]\n(c) At the point $(110,10)$ with $dm=-0.2$ and $dr=0.1$, the differential is $dF=$ [ANS]", "answer_v2": [ "G*r^2/[(r^2)^2]*dm+[-(G*m*2*r/[(r^2)^2])]*dr", "0.098*dm-2.156*dr", "-0.2352" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $ F(m,r)= \\frac{Gm}{r^{2} }$.\n(a) At any point $(m,r)$, the differential is $dF=$ [ANS]\n(b) At the point $(160,8)$, the differential is $dF=$ [ANS]\n(c) At the point $(160,8)$ with $dm=-0.3$ and $dr=0.4$, the differential is $dF=$ [ANS]", "answer_v3": [ "G*r^2/[(r^2)^2]*dm+[-(G*m*2*r/[(r^2)^2])]*dr", "0.153125*dm-6.125*dr", "-2.49594" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0258", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "Linear function" ], "problem_v1": "Find the implicit equation of the tangent plane to $z=e^{y}+x+x^{2}+10$ at the point $\\left(6,0,53\\right)$. [ANS]", "answer_v1": [ "13*x+y-z = 25" ], "answer_type_v1": [ "EQ" ], "options_v1": [ [] ], "problem_v2": "Find the implicit equation of the tangent plane to $z=e^{y}+x+x^{2}+12$ at the point $\\left(2,0,19\\right)$. [ANS]", "answer_v2": [ "5*x+y-z = -9" ], "answer_type_v2": [ "EQ" ], "options_v2": [ [] ], "problem_v3": "Find the implicit equation of the tangent plane to $z=e^{y}+x+x^{2}+10$ at the point $\\left(3,0,23\\right)$. [ANS]", "answer_v3": [ "7*x+y-z = -2" ], "answer_type_v3": [ "EQ" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0259", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "2", "keywords": [ "Derivative", "Differential", "Tangent Plane" ], "problem_v1": "An unevenly heated metal plate has temperature $T(x,y)$ in degrees Celsius at a point $(x,y)$. If $T(2,1)=131$, $T_x \\, (2,1)=17$, and $T_y \\, (2,1)=-13$, estimate the temperature at the point $(2.05,0.96)$.\n$T(2.05,0.96) \\approx$ [ANS] degC. Please include in your answer.", "answer_v1": [ "132.37" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "An unevenly heated metal plate has temperature $T(x,y)$ in degrees Celsius at a point $(x,y)$. If $T(2,1)=107$, $T_x \\, (2,1)=12$, and $T_y \\, (2,1)=-8$, estimate the temperature at the point $(2.02,0.95)$.\n$T(2.02,0.95) \\approx$ [ANS] degC. Please include in your answer.", "answer_v2": [ "107.64" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "An unevenly heated metal plate has temperature $T(x,y)$ in degrees Celsius at a point $(x,y)$. If $T(2,1)=114$, $T_x \\, (2,1)=15$, and $T_y \\, (2,1)=-14$, estimate the temperature at the point $(2.03,0.96)$.\n$T(2.03,0.96) \\approx$ [ANS] degC. Please include in your answer.", "answer_v3": [ "115.01" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0260", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Differentiability, linearization and tangent planes", "level": "4", "keywords": [ "Gradient", "Directional Derivative" ], "problem_v1": "Check that the point $\\left(-2,2,4\\right)$ lies on the surface $\\cos\\!\\left(x+y\\right)=e^{xz+8}$.\n(a) View this surface as a level surface for a function $f(x,y,z)$. Find a vector normal to the surface at the point $\\left(-2,2,4\\right)$. [ANS]\n(b) Find an implicit equation for the tangent plane to the surface at $\\left(-2,2,4\\right)$. [ANS]", "answer_v1": [ "(-4,0,2)", "2*z-4*x = 16" ], "answer_type_v1": [ "OL", "EQ" ], "options_v1": [ [], [] ], "problem_v2": "Check that the point $\\left(-5,5,5\\right)$ lies on the surface $\\cos\\!\\left(x+y\\right)=e^{xz+25}$.\n(a) View this surface as a level surface for a function $f(x,y,z)$. Find a vector normal to the surface at the point $\\left(-5,5,5\\right)$. [ANS]\n(b) Find an implicit equation for the tangent plane to the surface at $\\left(-5,5,5\\right)$. [ANS]", "answer_v2": [ "(-5,0,5)", "5*z-5*x = 50" ], "answer_type_v2": [ "OL", "EQ" ], "options_v2": [ [], [] ], "problem_v3": "Check that the point $\\left(-4,4,4\\right)$ lies on the surface $\\cos\\!\\left(x+y\\right)=e^{xz+16}$.\n(a) View this surface as a level surface for a function $f(x,y,z)$. Find a vector normal to the surface at the point $\\left(-4,4,4\\right)$. [ANS]\n(b) Find an implicit equation for the tangent plane to the surface at $\\left(-4,4,4\\right)$. [ANS]", "answer_v3": [ "(-4,0,4)", "4*z-4*x = 32" ], "answer_type_v3": [ "OL", "EQ" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0261", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Let $f(x,y)=xe^{x^2-y}$ and $P=(7,49)$.\n(a) Calculate $\\|\\nabla f_P\\|$. (b) Find the rate of change of $f$ in the direction $\\nabla f_P$. (c) Find the rate of change of $f$ in the direction of a vector making an angle of $45^\\circ$ with $\\nabla f_P$. Answers:\n(a) [ANS]\n(b) [ANS]\n(c) [ANS]", "answer_v1": [ "99.2472", "99.2472", "70.1783" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $f(x,y)=xe^{x^2-y}$ and $P=(1,1)$.\n(a) Calculate $\\|\\nabla f_P\\|$. (b) Find the rate of change of $f$ in the direction $\\nabla f_P$. (c) Find the rate of change of $f$ in the direction of a vector making an angle of $45^\\circ$ with $\\nabla f_P$. Answers:\n(a) [ANS]\n(b) [ANS]\n(c) [ANS]", "answer_v2": [ "3.16228", "3.16228", "2.23607" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $f(x,y)=xe^{x^2-y}$ and $P=(3,9)$.\n(a) Calculate $\\|\\nabla f_P\\|$. (b) Find the rate of change of $f$ in the direction $\\nabla f_P$. (c) Find the rate of change of $f$ in the direction of a vector making an angle of $45^\\circ$ with $\\nabla f_P$. Answers:\n(a) [ANS]\n(b) [ANS]\n(c) [ANS]", "answer_v3": [ "19.2354", "19.2354", "13.6015" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0262", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Calculate the directional derivative of $f(x,y)=x^{3}y$ in the direction of $\\mathbf{v}=-\\mathbf{i}+\\mathbf{j}$ at the point $P=(1,1)$. Remember to normalize the direction vector. $D_uf(1,1)=$ [ANS]", "answer_v1": [ "-1.41421" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the directional derivative of $f(x,y)=xy^{2}$ in the direction of $\\mathbf{v}=3 \\mathbf{i}-\\mathbf{j}$ at the point $P=(-2,2)$. Remember to normalize the direction vector. $D_uf(-2,2)=$ [ANS]", "answer_v2": [ "6.32456" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the directional derivative of $f(x,y)=xy^{2}$ in the direction of $\\mathbf{v}=-2 \\mathbf{i}-\\mathbf{j}$ at the point $P=(-1,1)$. Remember to normalize the direction vector. $D_uf(-1,1)=$ [ANS]", "answer_v3": [ "0" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0263", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Calculate the gradient of $h(x,y,z)=x^{2}yz$ $\\nabla h=$ [ANS]", "answer_v1": [ "(2*x*y*z,x^2*z,x^2*y)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Calculate the gradient of $h(x,y,z)=x^{-3}y^{3}z^{-2}$ $\\nabla h=$ [ANS]", "answer_v2": [ "(-3*x^{-4}*y^3*z^{-3},3*x^{-3}*y^2*z^{-3},-2*x^{-3}*y^3*z^{-3})" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Calculate the gradient of $h(x,y,z)= \\frac{1}{x} yz^{-2}$ $\\nabla h=$ [ANS]", "answer_v3": [ "(-x^{-3}*y*z^{-3},1/x*z^{-3},-2*1/x*y*z^{-3})" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0264", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Let $f(x,y)=x^{3}y^{2}$ and $c(t)=\\left(2t^{2},2t^{3}\\right)$\n(a) Calculate: $\\nabla f \\cdot c'(t)=$ [ANS]\n(b) Use the Chain Rule for Paths to evaluate $ \\frac{d}{dt} f(c(t))$ at $t=-1$. $ \\frac{d}{dt} f(c(-1))=$ [ANS]", "answer_v1": [ "12*x^2*y^2*t+12*x^3*y*t^2", "-384" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x,y)=xy^{3}$ and $c(t)=\\left(t^{2},t^{3}\\right)$\n(a) Calculate: $\\nabla f \\cdot c'(t)=$ [ANS]\n(b) Use the Chain Rule for Paths to evaluate $ \\frac{d}{dt} f(c(t))$ at $t=1$. $ \\frac{d}{dt} f(c(1))=$ [ANS]", "answer_v2": [ "2*y^3*t+9*x*y^2*t^2", "11" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x,y)=xy^{2}$ and $c(t)=\\left(t^{2},2t^{3}\\right)$\n(a) Calculate: $\\nabla f \\cdot c'(t)=$ [ANS]\n(b) Use the Chain Rule for Paths to evaluate $ \\frac{d}{dt} f(c(t))$ at $t=-1$. $ \\frac{d}{dt} f(c(-1))=$ [ANS]", "answer_v3": [ "2*y^2*t+12*x*y*t^2", "-32" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0265", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Calculate the directional derivative of $f(x,y)=\\cot^{-1}\\!\\left(xy\\right)$ in the direction of $\\mathbf{v}=2 \\mathbf{i}-2\\mathbf{j}$ at the point $P=(0.5,0.5)$. Remember to normalize the direction vector. $D_uf(0.5,0.5)=$ [ANS]", "answer_v1": [ "0" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the directional derivative of $f(x,y)=\\sin^{-1}\\!\\left(xy\\right)$ in the direction of $\\mathbf{v}=-2 \\mathbf{i}+5\\mathbf{j}$ at the point $P=(0.5,0.5)$. Remember to normalize the direction vector. $D_uf(0.5,0.5)=$ [ANS]", "answer_v2": [ "0.287678" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the directional derivative of $f(x,y)=\\cos^{-1}\\!\\left(xy\\right)$ in the direction of $\\mathbf{v}=\\mathbf{i}-3\\mathbf{j}$ at the point $P=(0.5,0.5)$. Remember to normalize the direction vector. $D_uf(0.5,0.5)=$ [ANS]", "answer_v3": [ "0.326599" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0266", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Calculate the gradient of $f(x,y)=\\sin\\!\\left(4x^{2}+y\\right)$ $\\nabla f(x,y)=$ [ANS]", "answer_v1": [ "(8*x*cos(4*x^2+y),cos(4*x^2+y))" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Calculate the gradient of $f(x,y)=\\cos\\!\\left(x^{2}+5y\\right)$ $\\nabla f(x,y)=$ [ANS]", "answer_v2": [ "(-2*x*sin(x^2+5*y),-5*sin(x^2+5*y))" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Calculate the gradient of $f(x,y)=\\cos\\!\\left(2x^{2}+y\\right)$ $\\nabla f(x,y)=$ [ANS]", "answer_v3": [ "(-4*x*sin(2*x^2+y),-[sin(2*x^2+y)])" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0267", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "Gradient", "Directional", "Derivative" ], "problem_v1": "Let $f(x,y)=xe^{4y}-ye^{3x}$. Then $\\nabla \\!f$=[ANS], and $D_{\\boldsymbol{u}} f(1,-1)$ in the direction of the vector $\\left<1,2\\right>$ is [ANS].", "answer_v1": [ "(e^(4*y)-3*y*e^(3*x),4*x*e^(4*y)-e^(3*x))", "9.05624" ], "answer_type_v1": [ "OL", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x,y)=xe^{y}-ye^{5x}$. Then $\\nabla \\!f$=[ANS], and $D_{\\boldsymbol{u}} f(1,-1)$ in the direction of the vector $\\left<-2,-1\\right>$ is [ANS].", "answer_v2": [ "(e^y-5*y*e^(5*x),x*e^y-e^(5*x))", "-597.845" ], "answer_type_v2": [ "OL", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x,y)=xe^{2y}-ye^{4x}$. Then $\\nabla \\!f$=[ANS], and $D_{\\boldsymbol{u}} f(1,-1)$ in the direction of the vector $\\left<-2,0\\right>$ is [ANS].", "answer_v3": [ "(e^(2*y)-4*y*e^(4*x),2*x*e^(2*y)-e^(4*x))", "-218.528" ], "answer_type_v3": [ "OL", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0268", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "Gradient" ], "problem_v1": "Let $f(x,y)=3x^{3}+xy^{2}$. Then the direction in which $f$ is increasing the fastest at the point $(1,2)$ is [ANS], and the rate of increase in that direction is [ANS].", "answer_v1": [ "(13,4)", "13.6015" ], "answer_type_v1": [ "OL", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $f(x,y)=5xy^{2}-5x^{3}$. Then the direction in which $f$ is increasing the fastest at the point $(-2,-1)$ is [ANS], and the rate of increase in that direction is [ANS].", "answer_v2": [ "(-55,20)", "58.5235" ], "answer_type_v2": [ "OL", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $f(x,y)=xy^{2}-2x^{3}$. Then the direction in which $f$ is increasing the fastest at the point $(-2,-2)$ is [ANS], and the rate of increase in that direction is [ANS].", "answer_v3": [ "(-20,8)", "21.5407" ], "answer_type_v3": [ "OL", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0269", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "vector' 'gradient' 'multivariable", "gradient", "vector", "field", "Vector Fields", "Gradient" ], "problem_v1": "Compute the gradient vector fields of the following functions: A. $f(x, y)=8x^2+6 y^2$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j$ B. $f(x, y)=x^{7} y^{8},$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j$ C. $f(x, y)=8x+6 y$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j$ D. $f(x, y, z)=8x+6 y+7 z$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j+$ [ANS] $\\bf k$ E. $f(x, y, z)=8x^2+6 y^2+7 z^2$ $\\nabla f(x, y, z)=$ [ANS] $\\bf i+$ [ANS] $\\bf j+$ [ANS] $\\bf k$", "answer_v1": [ "2*8*x", "2*6*y", "(7*x^(7 -1))*y^(8)", "(8*y^(8 -1))*x^(7)", "8", "6", "8", "6", "7", "2*8*x", "2*6*y", "2*7*z" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Compute the gradient vector fields of the following functions: A. $f(x, y)=1x^2+10 y^2$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j$ B. $f(x, y)=x^{2} y^{4},$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j$ C. $f(x, y)=1x+10 y$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j$ D. $f(x, y, z)=1x+10 y+2 z$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j+$ [ANS] $\\bf k$ E. $f(x, y, z)=1x^2+10 y^2+2 z^2$ $\\nabla f(x, y, z)=$ [ANS] $\\bf i+$ [ANS] $\\bf j+$ [ANS] $\\bf k$", "answer_v2": [ "2*1*x", "2*10*y", "(2*x^(2 -1))*y^(4)", "(4*y^(4 -1))*x^(2)", "1", "10", "1", "10", "2", "2*1*x", "2*10*y", "2*2*z" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Compute the gradient vector fields of the following functions: A. $f(x, y)=4x^2+7 y^2$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j$ B. $f(x, y)=x^{3} y^{6},$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j$ C. $f(x, y)=4x+7 y$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j$ D. $f(x, y, z)=4x+7 y+3 z$ $\\nabla f(x, y)=$ [ANS] $\\bf i+$ [ANS] $\\bf j+$ [ANS] $\\bf k$ E. $f(x, y, z)=4x^2+7 y^2+3 z^2$ $\\nabla f(x, y, z)=$ [ANS] $\\bf i+$ [ANS] $\\bf j+$ [ANS] $\\bf k$", "answer_v3": [ "2*4*x", "2*7*y", "(3*x^(3 -1))*y^(6)", "(6*y^(6 -1))*x^(3)", "4", "7", "4", "7", "3", "2*4*x", "2*7*y", "2*3*z" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0270", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "4", "keywords": [ "Multivariable", "Derivative", "Directional' 'gradient", "Gradient", "Directional Derivative" ], "problem_v1": "Suppose $f \\left(x, y \\right)= \\frac{x}{y} $, $P=\\left(2, 1 \\right)$ and $\\mathbf{v}=1 \\mathbf{i}+2 \\mathbf{j}$. A. Find the gradient of $f$. $(\\nabla f)(x,y)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}$ Note: Your answers should be expressions of x and y; e.g. \"3x-4y\" B. Find the gradient of $f$ at the point $P$. $\\left(\\nabla f \\right) \\left(P \\right)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}$ Note: Your answers should be numbers C. Find the directional derivative of $f$ at $P$ in the direction of $\\mathbf{v}$. $(D_{\\mathbf{u}} f)(P)=$ [ANS]\nNote: Your answer should be a number D. Find the maximum rate of change of $f$ at $P$. [ANS]\nNote: Your answer should be a number E. Find the (unit) direction vector $\\mathbf{w}$ in which the maximum rate of change occurs at $P$. $\\mathbf{w}=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}$ Note: Your answers should be numbers", "answer_v1": [ "1/y", "-x/(y^2)", "1", "-2", "-1.34164", "2.23607", "0.447214", "-0.894427" ], "answer_type_v1": [ "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Suppose $f \\left(x, y \\right)= \\frac{x}{y} $, $P=\\left(-4, 4 \\right)$ and $\\mathbf{v}=-3 \\mathbf{i}-1 \\mathbf{j}$. A. Find the gradient of $f$. $(\\nabla f)(x,y)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}$ Note: Your answers should be expressions of x and y; e.g. \"3x-4y\" B. Find the gradient of $f$ at the point $P$. $\\left(\\nabla f \\right) \\left(P \\right)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}$ Note: Your answers should be numbers C. Find the directional derivative of $f$ at $P$ in the direction of $\\mathbf{v}$. $(D_{\\mathbf{u}} f)(P)=$ [ANS]\nNote: Your answer should be a number D. Find the maximum rate of change of $f$ at $P$. [ANS]\nNote: Your answer should be a number E. Find the (unit) direction vector $\\mathbf{w}$ in which the maximum rate of change occurs at $P$. $\\mathbf{w}=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}$ Note: Your answers should be numbers", "answer_v2": [ "1/y", "-x/(y^2)", "0.25", "0.25", "-0.316228", "0.353553", "0.707107", "0.707107" ], "answer_type_v2": [ "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Suppose $f \\left(x, y \\right)= \\frac{x}{y} $, $P=\\left(-2, 1 \\right)$ and $\\mathbf{v}=-2 \\mathbf{i}-3 \\mathbf{j}$. A. Find the gradient of $f$. $(\\nabla f)(x,y)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}$ Note: Your answers should be expressions of x and y; e.g. \"3x-4y\" B. Find the gradient of $f$ at the point $P$. $\\left(\\nabla f \\right) \\left(P \\right)=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}$ Note: Your answers should be numbers C. Find the directional derivative of $f$ at $P$ in the direction of $\\mathbf{v}$. $(D_{\\mathbf{u}} f)(P)=$ [ANS]\nNote: Your answer should be a number D. Find the maximum rate of change of $f$ at $P$. [ANS]\nNote: Your answer should be a number E. Find the (unit) direction vector $\\mathbf{w}$ in which the maximum rate of change occurs at $P$. $\\mathbf{w}=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}$ Note: Your answers should be numbers", "answer_v3": [ "1/y", "-x/(y^2)", "1", "2", "-2.2188", "2.23607", "0.447214", "0.894427" ], "answer_type_v3": [ "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0271", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "Multivariable", "Derivative", "Directional" ], "problem_v1": "If $f \\left(x, y \\right)=2x^{2}+1 y^{2}$, find the value of the directional derivative at the point $\\left(1, 2 \\right)$ in the direction given by the angle $\\theta= \\frac{2 \\pi}{2} $. [ANS]", "answer_v1": [ "-4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $f \\left(x, y \\right)=-4x^{2}+4 y^{2}$, find the value of the directional derivative at the point $\\left(-3,-1 \\right)$ in the direction given by the angle $\\theta= \\frac{2 \\pi}{6} $. [ANS]", "answer_v2": [ "5.07179676972449" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $f \\left(x, y \\right)=-2x^{2}+1 y^{2}$, find the value of the directional derivative at the point $\\left(-2,-3 \\right)$ in the direction given by the angle $\\theta= \\frac{2 \\pi}{3} $. [ANS]", "answer_v3": [ "-9.19615242270663" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0272", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "2", "keywords": [ "Vector Fields", "Gradient" ], "problem_v1": "A. $f(x, y)=5x^{4}+2y^{3}$ $\\nabla f(x, y)=$ [ANS]\nB. $f(x, y)=5x^{4}y^{3}$ $\\nabla f(x, y)=$ [ANS]\nC. $f(x,y,z)=5x^{4}+2y^{3}+3z^{3}$ $\\nabla f(x,y,z)=$ [ANS]\nD. $f(x,y,z)=5x^{4}y^{3}z^{3}$ $\\nabla f(x,y,z)=$ [ANS]\nE. $f(x, y)=\\cos^{4}\\!\\left(5x\\right)+\\sin^{3}\\!\\left(2y\\right)$ $\\nabla f(x, y)=$ [ANS]", "answer_v1": [ "(20*x^3,6*y^2)", "(20*x^3*y^3,3*5*x^4*y^2)", "(20*x^3,6*y^2,9*z^2)", "(20*x^3*y^3*z^3,3*5*x^4*y^2*z^3,3*5*x^4*y^3*z^2)", "(-5*4*[cos(5*x)]^3*sin(5*x),2*3*[sin(2*y)]^2*cos(2*y))" ], "answer_type_v1": [ "OL", "OL", "OL", "OL", "OL" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "A. $f(x, y)=9y^{5}-9x^{3}$ $\\nabla f(x, y)=$ [ANS]\nB. $f(x, y)=-9x^{3}y^{5}$ $\\nabla f(x, y)=$ [ANS]\nC. $f(x,y,z)=9y^{5}-9x^{3}-7z^{3}$ $\\nabla f(x,y,z)=$ [ANS]\nD. $f(x,y,z)=-9x^{3}y^{5}z^{3}$ $\\nabla f(x,y,z)=$ [ANS]\nE. $f(x, y)=\\cos^{3}\\!\\left(-9x\\right)+\\sin^{5}\\!\\left(9y\\right)$ $\\nabla f(x, y)=$ [ANS]", "answer_v2": [ "(-(27*x^2),45*y^4)", "(-(27*x^2*y^5),-(5*9*x^3*y^4))", "(-(27*x^2),45*y^4,-(21*z^2))", "(-(27*x^2*y^5*z^3),-(5*9*x^3*y^4*z^3),-(3*9*x^3*y^5*z^2))", "(9*3*[cos(-9*x)]^2*sin(-9*x),9*5*[sin(9*y)]^4*cos(9*y))" ], "answer_type_v2": [ "OL", "OL", "OL", "OL", "OL" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "A. $f(x, y)=2y^{2}-4x^{4}$ $\\nabla f(x, y)=$ [ANS]\nB. $f(x, y)=-4x^{4}y^{2}$ $\\nabla f(x, y)=$ [ANS]\nC. $f(x,y,z)=2y^{2}-4x^{4}-5z^{3}$ $\\nabla f(x,y,z)=$ [ANS]\nD. $f(x,y,z)=-4x^{4}y^{2}z^{3}$ $\\nabla f(x,y,z)=$ [ANS]\nE. $f(x, y)=\\cos^{4}\\!\\left(-4x\\right)+\\sin^{2}\\!\\left(2y\\right)$ $\\nabla f(x, y)=$ [ANS]", "answer_v3": [ "(-(16*x^3),4*y)", "(-(16*x^3*y^2),-(2*4*x^4*y))", "(-(16*x^3),4*y,-(15*z^2))", "(-(16*x^3*y^2*z^3),-(2*4*x^4*y*z^3),-(3*4*x^4*y^2*z^2))", "(4*4*[cos(-4*x)]^3*sin(-4*x),2*2*sin(2*y)*cos(2*y))" ], "answer_type_v3": [ "OL", "OL", "OL", "OL", "OL" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0273", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "parametric curve", "multivariable", "calculus" ], "problem_v1": "Find a vector parametric equation for the line perpendicular to the surface $z=4x^2+3 y^2$ at the point $(1,2,16)$. $\\vec r(t)=$ [ANS]", "answer_v1": [ "(8t+1,12t+2,-t+16)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find a vector parametric equation for the line perpendicular to the surface $z=x^2+5 y^2$ at the point $(-2,-1,9)$. $\\vec r(t)=$ [ANS]", "answer_v2": [ "(-4t-2,-10t-1,-t+9)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find a vector parametric equation for the line perpendicular to the surface $z=2x^2+4 y^2$ at the point $(-2,-2,24)$. $\\vec r(t)=$ [ANS]", "answer_v3": [ "(-8t-2,-16t-2,-t+24)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0274", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "derivatives" ], "problem_v1": "Find the gradient of the given function. Assume the variables are restricted to a domain on which the function is defined. f(a,b)= \\frac{5a+4b}{5a-4b} . $\\nabla f=$ [ANS]", "answer_v1": [ "(-40*b/[(5*a-4*b)^2],40*a/[(5*a-4*b)^2])" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the gradient of the given function. Assume the variables are restricted to a domain on which the function is defined. f(a,b)= \\frac{2a+5b}{2a-5b} . $\\nabla f=$ [ANS]", "answer_v2": [ "(-20*b/[(2*a-5*b)^2],20*a/[(2*a-5*b)^2])" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the gradient of the given function. Assume the variables are restricted to a domain on which the function is defined. f(a,b)= \\frac{3a+4b}{3a-4b} . $\\nabla f=$ [ANS]", "answer_v3": [ "(-24*b/[(3*a-4*b)^2],24*a/[(3*a-4*b)^2])" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0275", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "4", "keywords": [ "gradient", "directional derivative", "derivatives" ], "problem_v1": "You are climbing a mountain by the steepest route at a slope of $25^\\circ$ when you come upon a trail branching off at a $30^\\circ$ angle from yours. What is the angle of ascent of the branch trail? angle=[ANS] (in degrees)", "answer_v1": [ "180*atan(tan(25*pi/180)*cos(30*pi/180))/pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "You are climbing a mountain by the steepest route at a slope of $10^\\circ$ when you come upon a trail branching off at a $45^\\circ$ angle from yours. What is the angle of ascent of the branch trail? angle=[ANS] (in degrees)", "answer_v2": [ "180*atan(tan(10*pi/180)*cos(45*pi/180))/pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "You are climbing a mountain by the steepest route at a slope of $15^\\circ$ when you come upon a trail branching off at a $30^\\circ$ angle from yours. What is the angle of ascent of the branch trail? angle=[ANS] (in degrees)", "answer_v3": [ "180*atan(tan(15*pi/180)*cos(30*pi/180))/pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0276", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "derivatives" ], "problem_v1": "Find the gradient $\\nabla f$ of the function $f$ given the differential. df=\\left(4x^{3}+1\\right)ye^{x}dx+x^{4}e^{x}dy $\\nabla f=$ [ANS]", "answer_v1": [ "((4*x^3+1)*y*e^x,x^4*e^x)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the gradient $\\nabla f$ of the function $f$ given the differential. df=2ye^{x}dx+xe^{x}dy $\\nabla f=$ [ANS]", "answer_v2": [ "(2*y*e^x,x*e^x)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the gradient $\\nabla f$ of the function $f$ given the differential. df=\\left(2x+1\\right)ye^{x}dx+x^{2}e^{x}dy $\\nabla f=$ [ANS]", "answer_v3": [ "((2*x+1)*y*e^x,x^2*e^x)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0277", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "derivatives" ], "problem_v1": "Find the gradient of the given function. Assume the variables are restricted to a domain on which the function is defined. f(r,t)=r^{2}\\sin\\!\\left(3t\\right). $\\nabla f=$ [ANS]", "answer_v1": [ "(2*r*sin(3*t),3*r^2*cos(3*t))" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the gradient of the given function. Assume the variables are restricted to a domain on which the function is defined. f(r,t)=r^{1}\\cos\\!\\left(5t\\right). $\\nabla f=$ [ANS]", "answer_v2": [ "(cos(5*t),-5*r*sin(5*t))" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the gradient of the given function. Assume the variables are restricted to a domain on which the function is defined. f(r,t)=r^{1}\\cos\\!\\left(4t\\right). $\\nabla f=$ [ANS]", "answer_v3": [ "(cos(4*t),-4*r*sin(4*t))" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0278", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "derivatives" ], "problem_v1": "For each of the following pairs of functions $f$ and $g$, determine if the level curves of the functions cross at right angles, and find their gradients at the indicated point.\n(a) $f(x,y)=4x+3y$, $g(x,y)=4x-3y$. Do the level curves of $f$ and $g$ cross at right angles? [ANS] $\\nabla f(5, 4)=$ [ANS]\n$\\nabla g(5, 4)=$ [ANS]\n(b) $f(x,y)=x^{2}-y$, $g(x,y)=2y+\\ln\\!\\left(\\left|x\\right|\\right)$. Do the level curves of $f$ and $g$ cross at right angles? [ANS] $\\nabla f(5, 4)=$ [ANS]\n$\\nabla g(5, 4)=$ [ANS]", "answer_v1": [ "no", "(4,3)", "(4,-3)", "yes", "(10,-1)", "(1/5,2)" ], "answer_type_v1": [ "TF", "OL", "OL", "TF", "OL", "OL" ], "options_v1": [ [ "yes", "no" ], [], [], [ "yes", "no" ], [], [] ], "problem_v2": "For each of the following pairs of functions $f$ and $g$, determine if the level curves of the functions cross at right angles, and find their gradients at the indicated point.\n(a) $f(x,y)=3x+5y$, $g(x,y)=5x-3y$. Do the level curves of $f$ and $g$ cross at right angles? [ANS] $\\nabla f(1, 6)=$ [ANS]\n$\\nabla g(1, 6)=$ [ANS]\n(b) $f(x,y)=x^{3}-y$, $g(x,y)=3y+\\ln\\!\\left(\\left|x\\right|\\right)$. Do the level curves of $f$ and $g$ cross at right angles? [ANS] $\\nabla f(1, 6)=$ [ANS]\n$\\nabla g(1, 6)=$ [ANS]", "answer_v2": [ "yes", "(3,5)", "(5,-3)", "no", "(3,-1)", "(1,3)" ], "answer_type_v2": [ "TF", "OL", "OL", "TF", "OL", "OL" ], "options_v2": [ [ "yes", "no" ], [], [], [ "yes", "no" ], [], [] ], "problem_v3": "For each of the following pairs of functions $f$ and $g$, determine if the level curves of the functions cross at right angles, and find their gradients at the indicated point.\n(a) $f(x,y)=4x+2y$, $g(x,y)=2x-4y$. Do the level curves of $f$ and $g$ cross at right angles? [ANS] $\\nabla f(2, 4)=$ [ANS]\n$\\nabla g(2, 4)=$ [ANS]\n(b) $f(x,y)=x^{4}-y$, $g(x,y)=4y+\\ln\\!\\left(\\left|x\\right|\\right)$. Do the level curves of $f$ and $g$ cross at right angles? [ANS] $\\nabla f(2, 4)=$ [ANS]\n$\\nabla g(2, 4)=$ [ANS]", "answer_v3": [ "yes", "(4,2)", "(2,-4)", "no", "(4*2^3,-1)", "(1/2,4)" ], "answer_type_v3": [ "TF", "OL", "OL", "TF", "OL", "OL" ], "options_v3": [ [ "yes", "no" ], [], [], [ "yes", "no" ], [], [] ] }, { "id": "Calculus_-_multivariable_0279", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "4", "keywords": [ "gradient", "directional derivative", "derivatives" ], "problem_v1": "(a) What is the rate of change of $f(x,y)=5xy+y^{2}$ at the point $(3,3)$ in the direction $\\vec v=2\\boldsymbol{i}-\\boldsymbol{j}$? $f_{\\vec v}=$ [ANS]\n(b) What is the direction of maximum rate of change of $f$ at $(3,3)$? direction=[ANS]\n(Give your answer as a vector.) (Give your answer as a vector.) (c) What is the maximum rate of change? maximum rate of change=[ANS]", "answer_v1": [ "4.02492", "(15,21)", "25.807" ], "answer_type_v1": [ "NV", "OL", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "(a) What is the rate of change of $f(x,y)=2xy+y^{2}$ at the point $(4,1)$ in the direction $\\vec v=-\\boldsymbol{i}+3\\boldsymbol{j}$? $f_{\\vec v}=$ [ANS]\n(b) What is the direction of maximum rate of change of $f$ at $(4,1)$? direction=[ANS]\n(Give your answer as a vector.) (Give your answer as a vector.) (c) What is the maximum rate of change? maximum rate of change=[ANS]", "answer_v2": [ "8.85438", "(2,10)", "10.198" ], "answer_type_v2": [ "NV", "OL", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "(a) What is the rate of change of $f(x,y)=3xy+y^{2}$ at the point $(3,2)$ in the direction $\\vec v=-2\\boldsymbol{i}-\\boldsymbol{j}$? $f_{\\vec v}=$ [ANS]\n(b) What is the direction of maximum rate of change of $f$ at $(3,2)$? direction=[ANS]\n(Give your answer as a vector.) (Give your answer as a vector.) (c) What is the maximum rate of change? maximum rate of change=[ANS]", "answer_v3": [ "-11.1803", "(6,13)", "14.3178" ], "answer_type_v3": [ "NV", "OL", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0280", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "derivatives" ], "problem_v1": "Find the gradient of the function $f(r,h)=2\\pi rh+\\pi r^{2}$ at the point $(4,4)$. $\\nabla f(4,4)=$ [ANS]", "answer_v1": [ "(4*2*pi+4*2*pi,4*2*pi)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the gradient of the function $f(r,h)=2\\pi rh+\\pi r^{2}$ at the point $(1,5)$. $\\nabla f(1,5)=$ [ANS]", "answer_v2": [ "(5*2*pi+2*pi,2*pi)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the gradient of the function $f(r,h)=2\\pi rh+\\pi r^{2}$ at the point $(2,4)$. $\\nabla f(2,4)=$ [ANS]", "answer_v3": [ "(4*2*pi+2*2*pi,2*2*pi)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0281", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "derivatives" ], "problem_v1": "A car is driving northwest at $v$ mph across a sloping plain whose height, in feet above sea level, at a point $N$ miles north and $E$ miles east of a city is given by h(N,E)=4000+125 N+75 E.\n(a) At what rate is the height above sea level changing with respect to distance in the direction the car is driving? rate=[ANS]\n(b) Express the rate of change of the height of the car with respect to time in terms of $v$. rate=[ANS]", "answer_v1": [ "(125-75)/[sqrt(2)]", "(125-75)*v/[sqrt(2)]" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "A car is driving northwest at $v$ mph across a sloping plain whose height, in feet above sea level, at a point $N$ miles north and $E$ miles east of a city is given by h(N,E)=1250+175 N+25 E.\n(a) At what rate is the height above sea level changing with respect to distance in the direction the car is driving? rate=[ANS]\n(b) Express the rate of change of the height of the car with respect to time in terms of $v$. rate=[ANS]", "answer_v2": [ "(175-25)/[sqrt(2)]", "(175-25)*v/[sqrt(2)]" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "A car is driving northwest at $v$ mph across a sloping plain whose height, in feet above sea level, at a point $N$ miles north and $E$ miles east of a city is given by h(N,E)=2250+125 N+50 E.\n(a) At what rate is the height above sea level changing with respect to distance in the direction the car is driving? rate=[ANS]\n(b) Express the rate of change of the height of the car with respect to time in terms of $v$. rate=[ANS]", "answer_v3": [ "(125-50)/[sqrt(2)]", "(125-50)*v/[sqrt(2)]" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0282", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "derivatives" ], "problem_v1": "Find the gradient of the given function. Assume the variables are restricted to a domain on which the function is defined. z=\\left(4x+3y\\right)e^{4y} $\\nabla z=$ [ANS]", "answer_v1": [ "(4*e^(4*y),[3+4*(4*x+3*y)]*e^(4*y))" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the gradient of the given function. Assume the variables are restricted to a domain on which the function is defined. z=\\left(x+5y\\right)e^{y} $\\nabla z=$ [ANS]", "answer_v2": [ "(e^y,(5+x+5*y)*e^y)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the gradient of the given function. Assume the variables are restricted to a domain on which the function is defined. z=\\left(2x+4y\\right)e^{2y} $\\nabla z=$ [ANS]", "answer_v3": [ "(2*e^(2*y),[4+2*(2*x+4*y)]*e^(2*y))" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0283", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "4", "keywords": [ "gradient", "directional derivative", "derivatives" ], "problem_v1": "(a) Find the point $Q$ that is a distance 0.1 from the point $P=(5, 6)$ in the direction of $\\mathbf{v}=\\left<2,1\\right>$. Give five decimal places in your answer. $Q=$ ([ANS], [ANS]) (b) Use $P$ and $Q$ to approximate the directional derivative of $f(x,y)=\\sqrt{x+y}$ at $P$, in the direction of $\\mathbf{v}$. $f_{v} \\approx$ [ANS]\n(c) Give the exact value for the directional derivative you estimated in part (b). $f_{v}=$ [ANS]", "answer_v1": [ "5+0.1*2/2.23607", "6+0.1*1/2.23607", "0.2017/0.1", "1/[2*sqrt(5+1*6)]*2/2.23607+1/[2*sqrt(5+1*6)]*1/2.23607" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "(a) Find the point $Q$ that is a distance 0.1 from the point $P=(2, 4)$ in the direction of $\\mathbf{v}=\\left<-3,3\\right>$. Give five decimal places in your answer. $Q=$ ([ANS], [ANS]) (b) Use $P$ and $Q$ to approximate the directional derivative of $f(x,y)=\\sqrt{x+3y}$ at $P$, in the direction of $\\mathbf{v}$. $f_{v} \\approx$ [ANS]\n(c) Give the exact value for the directional derivative you estimated in part (b). $f_{v}=$ [ANS]", "answer_v2": [ "2+0.1*-3/4.24264", "4+0.1*3/4.24264", "0.1885/0.1", "1/[2*sqrt(2+3*4)]*-3/4.24264+3/[2*sqrt(2+3*4)]*3/4.24264" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "(a) Find the point $Q$ that is a distance 0.1 from the point $P=(3, 5)$ in the direction of $\\mathbf{v}=\\left<-1,1\\right>$. Give five decimal places in your answer. $Q=$ ([ANS], [ANS]) (b) Use $P$ and $Q$ to approximate the directional derivative of $f(x,y)=\\sqrt{x+y}$ at $P$, in the direction of $\\mathbf{v}$. $f_{v} \\approx$ [ANS]\n(c) Give the exact value for the directional derivative you estimated in part (b). $f_{v}=$ [ANS]", "answer_v3": [ "3+0.1*-1/1.41421", "5+0.1*1/1.41421", "0/0.1", "1/[2*sqrt(3+1*5)]*-1/1.41421+1/[2*sqrt(3+1*5)]*1/1.41421" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0284", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "multivariable", "functions" ], "problem_v1": "Find the gradient of the function $f(x,y,z)=zy^{4}$, at the point $(0,-1,-1)$ $\\nabla f(0,-1,-1)=$ [ANS]", "answer_v1": [ "(0,4,1)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find the gradient of the function $f(x,y,z)=xz^{3}$, at the point $(1,0,-1)$ $\\nabla f(1,0,-1)=$ [ANS]", "answer_v2": [ "(-1,0,3)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the gradient of the function $f(x,y,z)=xz^{3}$, at the point $(-1,0,1)$ $\\nabla f(-1,0,1)=$ [ANS]", "answer_v3": [ "(1,0,-3)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0285", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "multivariable", "functions" ], "problem_v1": "If the gradient of $f$ is $\\nabla f=2z\\,\\mathit{\\vec i}+x^{2}\\,\\mathit{\\vec j}+yx\\,\\mathit{\\vec k}$ and the point $P=(1, 2,-3)$ lies on the level surface $f(x, y, z)=0$, find an equation for the tangent plane to the surface at the point $P$. $z=$ [ANS]", "answer_v1": [ "-([y-2-6*(x-1)]/2+3)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If the gradient of $f$ is $\\nabla f=z^{3}\\,\\mathit{\\vec j}-3y\\,\\mathit{\\vec i}-3xz\\,\\mathit{\\vec k}$ and the point $P=(-7,-4, 1)$ lies on the level surface $f(x, y, z)=0$, find an equation for the tangent plane to the surface at the point $P$. $z=$ [ANS]", "answer_v2": [ "1-[12*(x+7)+y+4]/21" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "If the gradient of $f$ is $\\nabla f=x^{2}\\,\\mathit{\\vec j}-y\\,\\mathit{\\vec i}-2zx\\,\\mathit{\\vec k}$ and the point $P=(6, 9, 8)$ lies on the level surface $f(x, y, z)=0$, find an equation for the tangent plane to the surface at the point $P$. $z=$ [ANS]", "answer_v3": [ "[36*(y-9)-9*(x-6)]/96+8" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0286", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "multivariable", "functions" ], "problem_v1": "The concentration of salt in a fluid at $(x,y,z)$ is given by $F(x,y,z)=4x^{2}+3y^{4}+3x^{2}z^{2}$ mg/cm ${}^3$. You are at the point $(1,-1,-1)$.\n(a) In which direction should you move if you want the concentration to increase the fastest? direction: [ANS]\n(Give your answer as a vector.) (Give your answer as a vector.) (b) You start to move in the direction you found in part (a) at a speed of $5$ cm/sec. How fast is the concentration changing? rate of change=[ANS]", "answer_v1": [ "(14,-12,-6)", "5*sqrt(14*14+-12*-12+-6*-6)" ], "answer_type_v1": [ "OL", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The concentration of salt in a fluid at $(x,y,z)$ is given by $F(x,y,z)=x^{2}+4y^{4}+x^{2}z^{2}$ mg/cm ${}^3$. You are at the point $(-1,1,-1)$.\n(a) In which direction should you move if you want the concentration to increase the fastest? direction: [ANS]\n(Give your answer as a vector.) (Give your answer as a vector.) (b) You start to move in the direction you found in part (a) at a speed of $3$ cm/sec. How fast is the concentration changing? rate of change=[ANS]", "answer_v2": [ "(-4,16,-2)", "3*sqrt(-4*-4+16*16+-2*-2)" ], "answer_type_v2": [ "OL", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The concentration of salt in a fluid at $(x,y,z)$ is given by $F(x,y,z)=2x^{2}+3y^{4}+2x^{2}z^{2}$ mg/cm ${}^3$. You are at the point $(1,-1,-1)$.\n(a) In which direction should you move if you want the concentration to increase the fastest? direction: [ANS]\n(Give your answer as a vector.) (Give your answer as a vector.) (b) You start to move in the direction you found in part (a) at a speed of $7$ cm/sec. How fast is the concentration changing? rate of change=[ANS]", "answer_v3": [ "(8,-12,-4)", "7*sqrt(8*8+-12*-12+-4*-4)" ], "answer_type_v3": [ "OL", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0287", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "multivariable", "functions" ], "problem_v1": "Find the directional derivative of $f(x,y,z)=zy+x^{4}$, at $(1,2,3)$ in the direction of $\\vec v=\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j}+\\,\\mathit{\\vec k}$. $f_{\\vec u}=$ [ANS]", "answer_v1": [ "(4+3+2)/[sqrt(3)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the directional derivative of $f(x,y,z)=xz+y^{3}$, at $(3,1,2)$ in the direction of $\\vec v=\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j}+\\,\\mathit{\\vec k}$. $f_{\\vec u}=$ [ANS]", "answer_v2": [ "(2+3+3)/[sqrt(3)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the directional derivative of $f(x,y,z)=xz+y^{3}$, at $(1,2,3)$ in the direction of $\\vec v=\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j}+\\,\\mathit{\\vec k}$. $f_{\\vec u}=$ [ANS]", "answer_v3": [ "(3+12+1)/[sqrt(3)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0288", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "multivariable", "functions" ], "problem_v1": "Find the gradient of the function $f(p,q,r)=\\ln\\!\\left(q\\right)+e^{ \\frac{r}{4} }+e^{p}$. $\\mbox{grad} f=$ [ANS]", "answer_v1": [ "e^p*ln(e)i+1/qj+0.25*e^(r/4)*ln(e)k" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find the gradient of the function $f(p,q,r)=e^{r}+e^{p}+\\ln\\!\\left(q\\right)$. $\\mbox{grad} f=$ [ANS]", "answer_v2": [ "(e^p*ln(e),1/q,e^r*ln(e))" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find the gradient of the function $f(p,q,r)=\\ln\\!\\left(q\\right)+e^{p}+e^{ \\frac{r}{2} }$. $\\mbox{grad} f=$ [ANS]", "answer_v3": [ "(e^p*ln(e),1/q,0.5*e^(r/2)*ln(e))" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0289", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "4", "keywords": [ "gradient", "directional derivative", "multivariable", "functions" ], "problem_v1": "Consider the function $f(x,y)=\\left(e^{x}-3x\\right)\\sin\\!\\left(y\\right)$. Suppose $S$ is the surface $z=f(x,y).$\n(a) Find a vector which is perpendicular to the level curve of $f$ through the point $(4,3)$ in the direction in which $f$ decreases most rapidly. vector=[ANS]\n(b) Suppose $\\vec v=3\\vec i+5 \\vec j+a \\vec k$ is a vector in 3-space which is tangent to the surface $S$ at the point $P$ lying on the surface above $(4,3)$. What is $a$? $a=$ [ANS]", "answer_v1": [ "(-7.28153,42.1718)", "7.28153*3+-42.1718*5" ], "answer_type_v1": [ "OL", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider the function $f(x,y)=\\left(e^{x}-5x\\right)\\cos\\!\\left(y\\right)$. Suppose $S$ is the surface $z=f(x,y).$\n(a) Find a vector which is perpendicular to the level curve of $f$ through the point $(2,6)$ in the direction in which $f$ decreases most rapidly. vector=[ANS]\n(b) Suppose $\\vec v=3\\vec i+2 \\vec j+a \\vec k$ is a vector in 3-space which is tangent to the surface $S$ at the point $P$ lying on the surface above $(2,6)$. What is $a$? $a=$ [ANS]", "answer_v2": [ "(-2.2939,0.729538)", "2.2939*3+-0.729538*2" ], "answer_type_v2": [ "OL", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider the function $f(x,y)=\\left(e^{x}-4x\\right)\\cos\\!\\left(y\\right)$. Suppose $S$ is the surface $z=f(x,y).$\n(a) Find a vector which is perpendicular to the level curve of $f$ through the point $(3,3)$ in the direction in which $f$ decreases most rapidly. vector=[ANS]\n(b) Suppose $\\vec v=3\\vec i+7 \\vec j+a \\vec k$ is a vector in 3-space which is tangent to the surface $S$ at the point $P$ lying on the surface above $(3,3)$. What is $a$? $a=$ [ANS]", "answer_v3": [ "(15.9246,1.14103)", "-15.9246*3+-1.14103*7" ], "answer_type_v3": [ "OL", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0290", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "4", "keywords": [ "gradient", "directional derivative", "multivariable", "functions" ], "problem_v1": "At an altitude of $h$ feet above sea level, the air pressure, $P$, in inches of mercury (in Hg), is given by P=30 e^{-3.23 \\times 10^{-5}h}. An unpressurized seaplane takes off at an angle of $40^{\\circ}$ to the horizontal and a speed of $110$ mph. What is the rate of change of pressure in the plane with respect to time at take-off, in inches of mercury per second? rate of change=[ANS]", "answer_v1": [ "5280*110/3600*30*3.23*10^{-5}*cos((90+40)*pi/180)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "At an altitude of $h$ feet above sea level, the air pressure, $P$, in inches of mercury (in Hg), is given by P=30 e^{-3.23 \\times 10^{-5}h}. An unpressurized seaplane takes off at an angle of $15^{\\circ}$ to the horizontal and a speed of $130$ mph. What is the rate of change of pressure in the plane with respect to time at take-off, in inches of mercury per second? rate of change=[ANS]", "answer_v2": [ "5280*130/3600*30*3.23*10^{-5}*cos((90+15)*pi/180)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "At an altitude of $h$ feet above sea level, the air pressure, $P$, in inches of mercury (in Hg), is given by P=30 e^{-3.23 \\times 10^{-5}h}. An unpressurized seaplane takes off at an angle of $25^{\\circ}$ to the horizontal and a speed of $110$ mph. What is the rate of change of pressure in the plane with respect to time at take-off, in inches of mercury per second? rate of change=[ANS]", "answer_v3": [ "5280*110/3600*30*3.23*10^{-5}*cos((90+25)*pi/180)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0291", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient", "directional derivative", "multivariable", "functions" ], "problem_v1": "You are standing above the point $(4,3)$ on the surface $z=15-\\left(2x^{2}+3y^{2}\\right)$.\n(a) In which direction should you walk to descend fastest? (Give your answer as a unit 2-vector.) direction=[ANS]\n(b) If you start to move in this direction, what is the slope of your path? slope=[ANS]", "answer_v1": [ "(0.664364,0.747409)", "-1*sqrt(4*(2*2*4*4+3*3*3*3))" ], "answer_type_v1": [ "OL", "NV" ], "options_v1": [ [], [] ], "problem_v2": "You are standing above the point $(1,5)$ on the surface $z=30-\\left(x^{2}+2y^{2}\\right)$.\n(a) In which direction should you walk to descend fastest? (Give your answer as a unit 2-vector.) direction=[ANS]\n(b) If you start to move in this direction, what is the slope of your path? slope=[ANS]", "answer_v2": [ "(0.0995037,0.995037)", "-1*sqrt(4*(1*1*1*1+2*2*5*5))" ], "answer_type_v2": [ "OL", "NV" ], "options_v2": [ [], [] ], "problem_v3": "You are standing above the point $(2,4)$ on the surface $z=15-\\left(x^{2}+2y^{2}\\right)$.\n(a) In which direction should you walk to descend fastest? (Give your answer as a unit 2-vector.) direction=[ANS]\n(b) If you start to move in this direction, what is the slope of your path? slope=[ANS]", "answer_v3": [ "(0.242536,0.970143)", "-1*sqrt(4*(1*1*2*2+2*2*4*4))" ], "answer_type_v3": [ "OL", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0292", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "4", "keywords": [ "Jacobian", "Jacobian matrix" ], "problem_v1": "Consider the function\nz=f(x,y)=x^{3}+5xy^{2}.\n(a) Find $f(-2,1)$. $z=f(-2,1)=$ [ANS]\n(b) Find a function $g(x,y,z)$ whose level zero set is equal to the graph of $z=f(x,y)$ and such that the coefficient of $z$ in $g(x,y,z)$ is $1$. The level set $g(x,y,z)=$ [ANS] $=0$ is the same as the graph of $z=f(x,y)$.\n(c) Find the gradient of $g$. Write your answer as a row vector of the general form $(a, b, c)$. $\\nabla g(x,y,z)=$ [ANS]\n(d) Use $\\nabla g$ to find a vector $\\vec{n}$ perpendicular (or normal) to the graph of $z=f(x,y)$ at the point $\\left(-2,1,-18\\right)$. Write your answer as a row vector of the general form $(a, b, c)$. $\\vec{n}=$ [ANS]\n(e) Find an equation for the tangent plane to $z=f(x,y)$ at the point $\\left(-2,1,-18\\right)$. Enter your answer as an equation. [ANS]", "answer_v1": [ "-18", "z-(x^3+5*x*y^2)", "(-(3*x^2+5*y^2),-(5*x*2*y),1)", "(-17,20,1)", "20*y-17*x+z = 36" ], "answer_type_v1": [ "NV", "EX", "OL", "OL", "EQ" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Consider the function\nz=f(x,y)=x^{2}+6xy^{3}.\n(a) Find $f(-2,-3)$. $z=f(-2,-3)=$ [ANS]\n(b) Find a function $g(x,y,z)$ whose level zero set is equal to the graph of $z=f(x,y)$ and such that the coefficient of $z$ in $g(x,y,z)$ is $1$. The level set $g(x,y,z)=$ [ANS] $=0$ is the same as the graph of $z=f(x,y)$.\n(c) Find the gradient of $g$. Write your answer as a row vector of the general form $(a, b, c)$. $\\nabla g(x,y,z)=$ [ANS]\n(d) Use $\\nabla g$ to find a vector $\\vec{n}$ perpendicular (or normal) to the graph of $z=f(x,y)$ at the point $\\left(-2,-3,328\\right)$. Write your answer as a row vector of the general form $(a, b, c)$. $\\vec{n}=$ [ANS]\n(e) Find an equation for the tangent plane to $z=f(x,y)$ at the point $\\left(-2,-3,328\\right)$. Enter your answer as an equation. [ANS]", "answer_v2": [ "328", "z-(x^2+6*x*y^3)", "(-(2*x+6*y^3),-(6*x*3*y^2),1)", "(166,324,1)", "166*x+324*y+z = -976" ], "answer_type_v2": [ "NV", "EX", "OL", "OL", "EQ" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Consider the function\nz=f(x,y)=x^{2}+5xy^{3}.\n(a) Find $f(-3,-2)$. $z=f(-3,-2)=$ [ANS]\n(b) Find a function $g(x,y,z)$ whose level zero set is equal to the graph of $z=f(x,y)$ and such that the coefficient of $z$ in $g(x,y,z)$ is $1$. The level set $g(x,y,z)=$ [ANS] $=0$ is the same as the graph of $z=f(x,y)$.\n(c) Find the gradient of $g$. Write your answer as a row vector of the general form $(a, b, c)$. $\\nabla g(x,y,z)=$ [ANS]\n(d) Use $\\nabla g$ to find a vector $\\vec{n}$ perpendicular (or normal) to the graph of $z=f(x,y)$ at the point $\\left(-3,-2,129\\right)$. Write your answer as a row vector of the general form $(a, b, c)$. $\\vec{n}=$ [ANS]\n(e) Find an equation for the tangent plane to $z=f(x,y)$ at the point $\\left(-3,-2,129\\right)$. Enter your answer as an equation. [ANS]", "answer_v3": [ "129", "z-(x^2+5*x*y^3)", "(-(2*x+5*y^3),-(5*x*3*y^2),1)", "(46,180,1)", "46*x+180*y+z = -369" ], "answer_type_v3": [ "NV", "EX", "OL", "OL", "EQ" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0293", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "4", "keywords": [ "gradient' 'derivative", "Multivariable' 'gradient" ], "problem_v1": "Suppose that distances are measured in lightyears and that the temperature T of a gaseous nebula is inversely proportional to the distance from a fixed point, which we take to be the origin. Suppose that the temperature 1 lightyear from the origin is 800 degrees celsius. Find the gradient of T at $\\left(x, y, z \\right)$. $\\nabla T=($ [ANS], [ANS], [ANS] $)$ Note: Your answers should be expressions of x, y and z; e.g. \"3x-4yz\"", "answer_v1": [ "(-100 * 8 * x) / ( x^2 + y^2 + z^2 )**(3/2)", "(-100 * 8 * y) / ( x^2 + y^2 + z^2 )**(3/2)", "(-100 * 8 * z) / ( x^2 + y^2 + z^2 )**(3/2)" ], "answer_type_v1": [ "EX", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose that distances are measured in lightyears and that the temperature T of a gaseous nebula is inversely proportional to the distance from a fixed point, which we take to be the origin. Suppose that the temperature 1 lightyear from the origin is 100 degrees celsius. Find the gradient of T at $\\left(x, y, z \\right)$. $\\nabla T=($ [ANS], [ANS], [ANS] $)$ Note: Your answers should be expressions of x, y and z; e.g. \"3x-4yz\"", "answer_v2": [ "(-100 * 1 * x) / ( x^2 + y^2 + z^2 )**(3/2)", "(-100 * 1 * y) / ( x^2 + y^2 + z^2 )**(3/2)", "(-100 * 1 * z) / ( x^2 + y^2 + z^2 )**(3/2)" ], "answer_type_v2": [ "EX", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose that distances are measured in lightyears and that the temperature T of a gaseous nebula is inversely proportional to the distance from a fixed point, which we take to be the origin. Suppose that the temperature 1 lightyear from the origin is 400 degrees celsius. Find the gradient of T at $\\left(x, y, z \\right)$. $\\nabla T=($ [ANS], [ANS], [ANS] $)$ Note: Your answers should be expressions of x, y and z; e.g. \"3x-4yz\"", "answer_v3": [ "(-100 * 4 * x) / ( x^2 + y^2 + z^2 )**(3/2)", "(-100 * 4 * y) / ( x^2 + y^2 + z^2 )**(3/2)", "(-100 * 4 * z) / ( x^2 + y^2 + z^2 )**(3/2)" ], "answer_type_v3": [ "EX", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0294", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "gradient' 'derivative" ], "problem_v1": "Consider a function $f(x,y)$ at the point $(5, 4)$. At that point the function has directional derivatives: $ \\frac{5}{\\sqrt{52} }$ in the direction (parallel to) $(6, 4)$, and $ \\frac{5}{\\sqrt{50} }$ in the direction (parallel to) $(5, 5)$.\nThe gradient of $f$ at the point $(5, 4)$ is $($ [ANS], [ANS] $)$.", "answer_v1": [ "0.5", "0.5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider a function $f(x,y)$ at the point $(1, 6)$. At that point the function has directional derivatives: $ \\frac{2}{\\sqrt{40} }$ in the direction (parallel to) $(2, 6)$, and $ \\frac{3}{\\sqrt{50} }$ in the direction (parallel to) $(1, 7)$.\nThe gradient of $f$ at the point $(1, 6)$ is $($ [ANS], [ANS] $)$.", "answer_v2": [ "-0.5", "0.5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider a function $f(x,y)$ at the point $(2, 4)$. At that point the function has directional derivatives: $ \\frac{3}{\\sqrt{25} }$ in the direction (parallel to) $(3, 4)$, and $ \\frac{4}{\\sqrt{29} }$ in the direction (parallel to) $(2, 5)$.\nThe gradient of $f$ at the point $(2, 4)$ is $($ [ANS], [ANS] $)$.", "answer_v3": [ "-0.142857142857143", "0.857142857142857" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0295", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "4", "keywords": [ "Vector", "Parametric", "Geometry" ], "problem_v1": "Consider the line perpendicular to the surface $z=x^{2}+y^{2}$ at the point where $x=3$ and $y=1.$ Find a vector parametric equation for this line in terms of the parameter $t.$ $L(t)$=[ANS].", "answer_v1": [ "(6t+3,2t+1,-t+10)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Consider the line perpendicular to the surface $z=x^{2}+y^{2}$ at the point where $x=-5$ and $y=5.$ Find a vector parametric equation for this line in terms of the parameter $t.$ $L(t)$=[ANS].", "answer_v2": [ "(-10t-5,10t+5,-t+50)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Consider the line perpendicular to the surface $z=x^{2}+y^{2}$ at the point where $x=-2$ and $y=1.$ Find a vector parametric equation for this line in terms of the parameter $t.$ $L(t)$=[ANS].", "answer_v3": [ "(-4t-2,2t+1,-t+5)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0296", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "Gradient", "Directional Derivative" ], "problem_v1": "For each function $f(x,y,z)$, determine whether its gradient points radially outward from the origin, radially inward toward the origin, radially outward from the z-axis, or radially inward toward the z-axis.\n[ANS] 1. $f(x,y,z)=x^2+y^2$ [ANS] 2. $ f(x,y,z)= \\frac{1}{x^2+y^2+z^2} $ [ANS] 3. $ f(x,y,z)= \\frac{1}{x^2+y^2} $ [ANS] 4. $f(x,y,z)=x^2+y^2+z^2$", "answer_v1": [ "POINTS RADIALLY OUTWARD FROM THE Z-AXIS", "POINTS RADIALLY INWARD TOWARD THE ORIGIN", "POINTS RADIALLY INWARD TOWARD THE Z-AXIS", "Points radially outward from the origin" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ], [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ], [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ], [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ] ], "problem_v2": "For each function $f(x,y,z)$, determine whether its gradient points radially outward from the origin, radially inward toward the origin, radially outward from the z-axis, or radially inward toward the z-axis.\n[ANS] 1. $ f(x,y,z)= \\frac{1}{x^2+y^2+z^2} $ [ANS] 2. $ f(x,y,z)= \\frac{1}{x^2+y^2} $ [ANS] 3. $f(x,y,z)=x^2+y^2$ [ANS] 4. $f(x,y,z)=x^2+y^2+z^2$", "answer_v2": [ "POINTS RADIALLY INWARD TOWARD THE ORIGIN", "POINTS RADIALLY INWARD TOWARD THE Z-AXIS", "POINTS RADIALLY OUTWARD FROM THE Z-AXIS", "Points radially outward from the origin" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ], [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ], [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ], [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ] ], "problem_v3": "For each function $f(x,y,z)$, determine whether its gradient points radially outward from the origin, radially inward toward the origin, radially outward from the z-axis, or radially inward toward the z-axis.\n[ANS] 1. $ f(x,y,z)= \\frac{1}{x^2+y^2+z^2} $ [ANS] 2. $f(x,y,z)=x^2+y^2+z^2$ [ANS] 3. $f(x,y,z)=x^2+y^2$ [ANS] 4. $ f(x,y,z)= \\frac{1}{x^2+y^2} $", "answer_v3": [ "POINTS RADIALLY INWARD TOWARD THE ORIGIN", "POINTS RADIALLY OUTWARD FROM THE ORIGIN", "POINTS RADIALLY OUTWARD FROM THE Z-AXIS", "Points radially inward toward the z-axis" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ], [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ], [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ], [ "Points radially outward from the origin", "Points radially inward toward the origin", "Points radially outward from the z-axis", "Points radially inward toward the z-axis" ] ] }, { "id": "Calculus_-_multivariable_0297", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "Gradient", "Directional Derivative" ], "problem_v1": "Find the directional derivative of $ f(x,y,z)=4xy+z^{2}$ at the point $(1,1,2)$ in the direction of the maximum rate of change of $f$.\n$f_{\\boldsymbol{u}} \\, (1,1,2)=D_{\\boldsymbol{u}} \\, f(1,1,2)=$ [ANS]", "answer_v1": [ "6.9282" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the directional derivative of $ f(x,y,z)=2xy+z^{2}$ at the point $(5,-4,-2)$ in the direction of the maximum rate of change of $f$.\n$f_{\\boldsymbol{u}} \\, (5,-4,-2)=D_{\\boldsymbol{u}} \\, f(5,-4,-2)=$ [ANS]", "answer_v2": [ "13.4164" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the directional derivative of $ f(x,y,z)=2xy+z^{2}$ at the point $(1,-2,1)$ in the direction of the maximum rate of change of $f$.\n$f_{\\boldsymbol{u}} \\, (1,-2,1)=D_{\\boldsymbol{u}} \\, f(1,-2,1)=$ [ANS]", "answer_v3": [ "4.89898" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0298", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "Gradient", "Directional Derivative" ], "problem_v1": "Are the following statements true or false?\n[ANS] 1. If $\\vec{u}$ is a unit vector, then $f_{\\vec{u}} (a,b)$ is a vector. [ANS] 2. The gradient vector $\\nabla f(a,b)$ is tangent to the contour of $f$ at $(a,b)$. [ANS] 3. Suppose $f_x(a,b)$ and $f_y(a,b)$ both exist. Then there is always a direction in which the rate of change of $f$ at $(a,b)$ is zero. [ANS] 4. If $\\vec{u}$ is perpendicular to $\\nabla f(a,b)$, then $f_{\\vec{u}} \\, (a,b)=\\langle 0, 0 \\rangle$. [ANS] 5. $\\nabla f(a,b)$ is a vector in 3-dimensional space. [ANS] 6. If $f(x,y)$ has $f_x(a,b)=0$ and $f_y(a,b)=0$ at the point $(a,b)$, then $f$ is constant everywhere. [ANS] 7. $f_{\\vec{u}} \\, (a,b)=|| \\nabla f(a,b) ||$. [ANS] 8. $f_{\\vec{u}} \\, (a,b)$ is parallel to $\\vec{u}$.", "answer_v1": [ "FALSE", "FALSE", "TRUE", "False", "False", "False", "False", "False" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v2": "Are the following statements true or false?\n[ANS] 1. $\\nabla f(a,b)$ is a vector in 3-dimensional space. [ANS] 2. The gradient vector $\\nabla f(a,b)$ is tangent to the contour of $f$ at $(a,b)$. [ANS] 3. $f_{\\vec{u}} \\, (a,b)=|| \\nabla f(a,b) ||$. [ANS] 4. $f_{\\vec{u}} \\, (a,b)$ is parallel to $\\vec{u}$. [ANS] 5. If $\\vec{u}$ is a unit vector, then $f_{\\vec{u}} (a,b)$ is a vector. [ANS] 6. If $\\vec{u}$ is perpendicular to $\\nabla f(a,b)$, then $f_{\\vec{u}} \\, (a,b)=\\langle 0, 0 \\rangle$. [ANS] 7. If $f(x,y)$ has $f_x(a,b)=0$ and $f_y(a,b)=0$ at the point $(a,b)$, then $f$ is constant everywhere. [ANS] 8. Suppose $f_x(a,b)$ and $f_y(a,b)$ both exist. Then there is always a direction in which the rate of change of $f$ at $(a,b)$ is zero.", "answer_v2": [ "FALSE", "FALSE", "FALSE", "False", "False", "False", "False", "True" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v3": "Are the following statements true or false?\n[ANS] 1. If $f(x,y)$ has $f_x(a,b)=0$ and $f_y(a,b)=0$ at the point $(a,b)$, then $f$ is constant everywhere. [ANS] 2. The gradient vector $\\nabla f(a,b)$ is tangent to the contour of $f$ at $(a,b)$. [ANS] 3. If $\\vec{u}$ is perpendicular to $\\nabla f(a,b)$, then $f_{\\vec{u}} \\, (a,b)=\\langle 0, 0 \\rangle$. [ANS] 4. Suppose $f_x(a,b)$ and $f_y(a,b)$ both exist. Then there is always a direction in which the rate of change of $f$ at $(a,b)$ is zero. [ANS] 5. $f_{\\vec{u}} \\, (a,b)=|| \\nabla f(a,b) ||$. [ANS] 6. If $\\vec{u}$ is a unit vector, then $f_{\\vec{u}} (a,b)$ is a vector. [ANS] 7. $\\nabla f(a,b)$ is a vector in 3-dimensional space. [ANS] 8. $f_{\\vec{u}} \\, (a,b)$ is parallel to $\\vec{u}$.", "answer_v3": [ "FALSE", "FALSE", "FALSE", "True", "False", "False", "False", "False" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ] }, { "id": "Calculus_-_multivariable_0299", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "Gradient", "Directional Derivative" ], "problem_v1": "If $ f(x,y,z)=xe^{4y}\\sin\\!\\left(7z\\right)$, then the gradient is\n$\\nabla f (x,y,z)=$ [ANS]", "answer_v1": [ "(e^(4*y)*sin(7*z),4*x*e^(4*y)*ln(e)*sin(7*z),7*x*e^(4*y)*cos(7*z))" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "If $ f(x,y,z)=xe^{2y}\\sin\\!\\left(9z\\right)$, then the gradient is\n$\\nabla f (x,y,z)=$ [ANS]", "answer_v2": [ "(e^(2*y)*sin(9*z),2*x*e^(2*y)*ln(e)*sin(9*z),9*x*e^(2*y)*cos(9*z))" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "If $ f(x,y,z)=xe^{2y}\\sin\\!\\left(7z\\right)$, then the gradient is\n$\\nabla f (x,y,z)=$ [ANS]", "answer_v3": [ "(e^(2*y)*sin(7*z),2*x*e^(2*y)*ln(e)*sin(7*z),7*x*e^(2*y)*cos(7*z))" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0300", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "Gradient", "Directional Derivative" ], "problem_v1": "Suppose $ f(x,y)=\\sin\\!\\left( \\frac{2y}{x} \\right)$ and $\\boldsymbol{u}$ is the unit vector in the direction of $\\left<1,1\\right>$. Then,\n(a) $\\nabla f (x,y)=$ [ANS]\n(b) $\\nabla f (6,\\pi)=$ [ANS]\n(c) $f_{\\boldsymbol{u}} \\, (6,\\pi)=D_{\\boldsymbol{u}} \\, f(6,\\pi)=$ [ANS]", "answer_v1": [ "(-2*y/(x^2)*cos(2*y/x),2*x/(x^2)*cos(2*y/x))", "(-0.0872665,0.166667)", "0.0561444" ], "answer_type_v1": [ "OL", "OL", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $ f(x,y)=\\sin\\!\\left( \\frac{2y}{x} \\right)$ and $\\boldsymbol{u}$ is the unit vector in the direction of $\\left<3,-2\\right>$. Then,\n(a) $\\nabla f (x,y)=$ [ANS]\n(b) $\\nabla f (2,\\pi)=$ [ANS]\n(c) $f_{\\boldsymbol{u}} \\, (2,\\pi)=D_{\\boldsymbol{u}} \\, f(2,\\pi)=$ [ANS]", "answer_v2": [ "(-2*y/(x^2)*cos(2*y/x),2*x/(x^2)*cos(2*y/x))", "(1.5708,-1)", "1.86168" ], "answer_type_v2": [ "OL", "OL", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $ f(x,y)=\\sin\\!\\left( \\frac{2y}{x} \\right)$ and $\\boldsymbol{u}$ is the unit vector in the direction of $\\left<1,-2\\right>$. Then,\n(a) $\\nabla f (x,y)=$ [ANS]\n(b) $\\nabla f (2,\\pi)=$ [ANS]\n(c) $f_{\\boldsymbol{u}} \\, (2,\\pi)=D_{\\boldsymbol{u}} \\, f(2,\\pi)=$ [ANS]", "answer_v3": [ "(-2*y/(x^2)*cos(2*y/x),2*x/(x^2)*cos(2*y/x))", "(1.5708,-1)", "1.59691" ], "answer_type_v3": [ "OL", "OL", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0301", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "Gradient", "Directional Derivative" ], "problem_v1": "View the curve $(y-x)^2+2=xy-3$ as a contour of $f(x,y)$.\n(a) Use $\\nabla f (3,2)$ to find a vector normal to the curve at $(3,2)$. [ANS]\n(b) Use your answer to part (a) to find an implicit equation for the tangent line to the curve at $(3,2)$. [ANS]", "answer_v1": [ "(0,-5)", "y = 2" ], "answer_type_v1": [ "OL", "EQ" ], "options_v1": [ [], [] ], "problem_v2": "View the curve $(y-x)^2+2=xy-3$ as a contour of $f(x,y)$.\n(a) Use $\\nabla f (-3,-2)$ to find a vector normal to the curve at $(-3,-2)$. [ANS]\n(b) Use your answer to part (a) to find an implicit equation for the tangent line to the curve at $(-3,-2)$. [ANS]", "answer_v2": [ "(0,5)", "y = -2" ], "answer_type_v2": [ "OL", "EQ" ], "options_v2": [ [], [] ], "problem_v3": "View the curve $(y-x)^2+2=xy-3$ as a contour of $f(x,y)$.\n(a) Use $\\nabla f (-2,-3)$ to find a vector normal to the curve at $(-2,-3)$. [ANS]\n(b) Use your answer to part (a) to find an implicit equation for the tangent line to the curve at $(-2,-3)$. [ANS]", "answer_v3": [ "(5,0)", "x = -2" ], "answer_type_v3": [ "OL", "EQ" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0302", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "4", "keywords": [ "Gradient", "Directional Derivative" ], "problem_v1": "The temperature at any point in the plane is given by $ T(x,y)= \\frac{180}{x^{2} +y^{2}+3}$.\n(a) What shape are the level curves of $T$? [ANS] A. ellipses B. parabolas C. circles D. lines E. hyperbolas F. none of the above\n(b) At what point on the plane is it hottest? [ANS]\nWhat is the maximum temperature? [ANS]\n(c) Find the direction of the greatest increase in temperature at the point $(2,-1)$. [ANS]\nWhat is the value of this maximum rate of change, that is, the maximum value of the directional derivative at $(2,-1)$? [ANS]\n(d) Find the direction of the greatest decrease in temperature at the point $(2,-1)$. [ANS]\nWhat is the value of this most negative rate of change, that is, the minimum value of the directional derivative at $(2,-1)$? [ANS]", "answer_v1": [ "C", "(0,0)", "60", "(-11.25,5.625)", "12.5779", "(11.25,-5.625)", "-12.5779" ], "answer_type_v1": [ "MCS", "OL", "NV", "OL", "NV", "OL", "NV" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F" ], [], [], [], [], [], [] ], "problem_v2": "The temperature at any point in the plane is given by $ T(x,y)= \\frac{100}{x^{2} +y^{2}+4}$.\n(a) What shape are the level curves of $T$? [ANS] A. parabolas B. hyperbolas C. lines D. circles E. ellipses F. none of the above\n(b) At what point on the plane is it hottest? [ANS]\nWhat is the maximum temperature? [ANS]\n(c) Find the direction of the greatest increase in temperature at the point $(-1,3)$. [ANS]\nWhat is the value of this maximum rate of change, that is, the maximum value of the directional derivative at $(-1,3)$? [ANS]\n(d) Find the direction of the greatest decrease in temperature at the point $(-1,3)$. [ANS]\nWhat is the value of this most negative rate of change, that is, the minimum value of the directional derivative at $(-1,3)$? [ANS]", "answer_v2": [ "D", "(0,0)", "25", "(1.02041,-3.06122)", "3.22681", "(-1.02041,3.06122)", "-3.22681" ], "answer_type_v2": [ "MCS", "OL", "NV", "OL", "NV", "OL", "NV" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F" ], [], [], [], [], [], [] ], "problem_v3": "The temperature at any point in the plane is given by $ T(x,y)= \\frac{130}{x^{2} +y^{2}+3}$.\n(a) What shape are the level curves of $T$? [ANS] A. parabolas B. circles C. ellipses D. lines E. hyperbolas F. none of the above\n(b) At what point on the plane is it hottest? [ANS]\nWhat is the maximum temperature? [ANS]\n(c) Find the direction of the greatest increase in temperature at the point $(-2,-1)$. [ANS]\nWhat is the value of this maximum rate of change, that is, the maximum value of the directional derivative at $(-2,-1)$? [ANS]\n(d) Find the direction of the greatest decrease in temperature at the point $(-2,-1)$. [ANS]\nWhat is the value of this most negative rate of change, that is, the minimum value of the directional derivative at $(-2,-1)$? [ANS]", "answer_v3": [ "B", "(0,0)", "43.3333", "(8.125,4.0625)", "9.08403", "(-8.125,-4.0625)", "-9.08403" ], "answer_type_v3": [ "MCS", "OL", "NV", "OL", "NV", "OL", "NV" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F" ], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0303", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "Gradient", "Directional Derivative" ], "problem_v1": "At a certain point on a heated metal plate, the greatest rate of temperature increase, 5 degrees Celsius per meter, is toward the northeast. If an object at this point moves directly north, at what rate is the temperature increasing? [ANS] degrees Celsius per meter", "answer_v1": [ "5/[sqrt(2)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "At a certain point on a heated metal plate, the greatest rate of temperature increase, 2 degrees Celsius per meter, is toward the northeast. If an object at this point moves directly north, at what rate is the temperature increasing? [ANS] degrees Celsius per meter", "answer_v2": [ "2/[sqrt(2)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "At a certain point on a heated metal plate, the greatest rate of temperature increase, 3 degrees Celsius per meter, is toward the northeast. If an object at this point moves directly north, at what rate is the temperature increasing? [ANS] degrees Celsius per meter", "answer_v3": [ "3/[sqrt(2)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0304", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "Gradient", "Directional Derivative" ], "problem_v1": "(a) Find the directional derivative of $z=x^{2}y$ at $(4,3)$ in the direction of $\\pi/3$ with the positive x-axis. [ANS]\n(b) In which direction is the directional derivative the largest at the point $(4,3)$? Enter your answer as a vector whose length is the largest value of the directional derivative. [ANS]", "answer_v1": [ "25.8564", "(24,16)" ], "answer_type_v1": [ "NV", "OL" ], "options_v1": [ [], [] ], "problem_v2": "(a) Find the directional derivative of $z=x^{2}y$ at $(1,5)$ in the direction of $\\pi/2$ with the positive x-axis. [ANS]\n(b) In which direction is the directional derivative the largest at the point $(1,5)$? Enter your answer as a vector whose length is the largest value of the directional derivative. [ANS]", "answer_v2": [ "1", "(10,1)" ], "answer_type_v2": [ "NV", "OL" ], "options_v2": [ [], [] ], "problem_v3": "(a) Find the directional derivative of $z=x^{2}y$ at $(2,4)$ in the direction of $\\pi/2$ with the positive x-axis. [ANS]\n(b) In which direction is the directional derivative the largest at the point $(2,4)$? Enter your answer as a vector whose length is the largest value of the directional derivative. [ANS]", "answer_v3": [ "4", "(16,4)" ], "answer_type_v3": [ "NV", "OL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0305", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Directional derivatives and the gradient", "level": "3", "keywords": [ "calculus", "normal", "gradient", "parametrize", "surface", "sphere" ], "problem_v1": "A sphere of radius 5 is centered at the origin. It may be viewed as a parametrized surface: $\\mathbf{r}(\\theta,\\phi)=(5 \\cos\\theta \\sin\\phi, 5 \\sin\\theta \\sin\\phi, 5 \\cos\\phi)$, a level surface of the function $f(x,y,z)=x^2+y^2+z^2$, or as the graph of the function $g(x,y)=\\sqrt{25-x^2-y^2}$. Consider the sphere at the point $(2.50000, 2.50000, 3.53553)$ (corresponding to $(\\theta, \\phi)=(\\pi/4, \\pi/4)$). A) Find the normal vector $\\mathbf{r}_\\theta \\times \\mathbf{r}_\\phi$ at the given point: $($ [ANS], [ANS], [ANS] $)$ B) Find the gradient of $f$ at the indicated point: $($ [ANS], [ANS], [ANS] $)$ They should be parallel....", "answer_v1": [ "-8.83883476483184", "-8.83883476483184", "-12.4999861904424", "5", "5", "7.07106" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "A sphere of radius 2 is centered at the origin. It may be viewed as a parametrized surface: $\\mathbf{r}(\\theta,\\phi)=(2 \\cos\\theta \\sin\\phi, 2 \\sin\\theta \\sin\\phi, 2 \\cos\\phi)$, a level surface of the function $f(x,y,z)=x^2+y^2+z^2$, or as the graph of the function $g(x,y)=\\sqrt{4-x^2-y^2}$. Consider the sphere at the point $(1.00000, 1.00000, 1.41421)$ (corresponding to $(\\theta, \\phi)=(\\pi/4, \\pi/4)$). A) Find the normal vector $\\mathbf{r}_\\theta \\times \\mathbf{r}_\\phi$ at the given point: $($ [ANS], [ANS], [ANS] $)$ B) Find the gradient of $f$ at the indicated point: $($ [ANS], [ANS], [ANS] $)$ They should be parallel....", "answer_v2": [ "-1.41421356237309", "-1.41421356237309", "-1.99999496204365", "2", "2", "2.82842" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "A sphere of radius 3 is centered at the origin. It may be viewed as a parametrized surface: $\\mathbf{r}(\\theta,\\phi)=(3 \\cos\\theta \\sin\\phi, 3 \\sin\\theta \\sin\\phi, 3 \\cos\\phi)$, a level surface of the function $f(x,y,z)=x^2+y^2+z^2$, or as the graph of the function $g(x,y)=\\sqrt{9-x^2-y^2}$. Consider the sphere at the point $(1.50000, 1.50000, 2.12132)$ (corresponding to $(\\theta, \\phi)=(\\pi/4, \\pi/4)$). A) Find the normal vector $\\mathbf{r}_\\theta \\times \\mathbf{r}_\\phi$ at the given point: $($ [ANS], [ANS], [ANS] $)$ B) Find the gradient of $f$ at the indicated point: $($ [ANS], [ANS], [ANS] $)$ They should be parallel....", "answer_v3": [ "-3.18198051533946", "-3.18198051533946", "-4.49999927119994", "3", "3", "4.24264" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0306", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find the critical point of the function $f(x,y)=4e^{x}-3xe^{y}$. $c=$ [ANS]\nUse the Second Derivative Test to determine whether it is [ANS] A. test fails B. a local minimum C. a saddle point D. a local maximum", "answer_v1": [ "(0,0.287682)", "C" ], "answer_type_v1": [ "OL", "MCS" ], "options_v1": [ [], [ "A", "B", "C", "D" ] ], "problem_v2": "Find the critical point of the function $f(x,y)=e^{x}-5xe^{y}$. $c=$ [ANS]\nUse the Second Derivative Test to determine whether it is [ANS] A. test fails B. a local minimum C. a local maximum D. a saddle point", "answer_v2": [ "(0,-1.60944)", "D" ], "answer_type_v2": [ "OL", "MCS" ], "options_v2": [ [], [ "A", "B", "C", "D" ] ], "problem_v3": "Find the critical point of the function $f(x,y)=2e^{x}-4xe^{y}$. $c=$ [ANS]\nUse the Second Derivative Test to determine whether it is [ANS] A. test fails B. a saddle point C. a local maximum D. a local minimum", "answer_v3": [ "(0,-0.693147)", "B" ], "answer_type_v3": [ "OL", "MCS" ], "options_v3": [ [], [ "A", "B", "C", "D" ] ] }, { "id": "Calculus_-_multivariable_0307", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Determine the global extreme values of the function on the given set without using calculus $f(x,y)=7e^{-\\left(6x^{2}+6y^{2}\\right)}, \\quad 6x^{2}+6y^{2} \\le 3$ $f_{min}=$ [ANS]\n$f_{max}=$ [ANS]", "answer_v1": [ "0.348509", "7" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Determine the global extreme values of the function on the given set without using calculus $f(x,y)=e^{-\\left(9x^{2}+2y^{2}\\right)}, \\quad 9x^{2}+2y^{2} \\le 2$ $f_{min}=$ [ANS]\n$f_{max}=$ [ANS]", "answer_v2": [ "0.135335", "1" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Determine the global extreme values of the function on the given set without using calculus $f(x,y)=3e^{-\\left(6x^{2}+3y^{2}\\right)}, \\quad 6x^{2}+3y^{2} \\le 3$ $f_{min}=$ [ANS]\n$f_{max}=$ [ANS]", "answer_v3": [ "0.149361", "3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0310", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "4", "keywords": [ "Critical", "Point", "Saddle", "Maximum", "Minimum" ], "problem_v1": "Consider the function $f(x,y)=x^{2}y+y^{3}-75y$.\n$\\begin{array}{c}\\hline fhas [ANS]no critical pointno critical pointat \\left(0,-5\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(5\\sqrt{3},0\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(-5\\sqrt{3},0\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(0,0\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(0,5\\right). \\\\ \\hline \\end{array}$", "answer_v1": [ "a maximum", "a saddle", "a saddle", "no critical point", "a minimum" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ] ], "problem_v2": "Consider the function $f(x,y)=x^{2}y+y^{3}-12y$.\n$\\begin{array}{c}\\hline fhas [ANS]no critical pointno critical pointat \\left(-2\\sqrt{3},0\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(0,0\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(0,-2\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(2\\sqrt{3},0\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(0,2\\right). \\\\ \\hline \\end{array}$", "answer_v2": [ "a saddle", "no critical point", "a maximum", "a saddle", "a minimum" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ] ], "problem_v3": "Consider the function $f(x,y)=x^{2}y+y^{3}-27y$.\n$\\begin{array}{c}\\hline fhas [ANS]no critical pointno critical pointat \\left(3\\sqrt{3},0\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(0,3\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(0,-3\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(0,0\\right). \\\\ \\hline \\\\ \\hline fhas [ANS]no critical pointno critical pointat \\left(-3\\sqrt{3},0\\right). \\\\ \\hline \\end{array}$", "answer_v3": [ "a saddle", "a minimum", "a maximum", "no critical point", "a saddle" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ], [ "a maximum", "a minimum", "a saddle", "some other critical point", "no critical point" ] ] }, { "id": "Calculus_-_multivariable_0311", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "4", "keywords": [ "critical points", "minimum", "maximum", "multivariable", "calculus", "gradient", "maximum", "minimum", "level curve", "Extrema" ], "problem_v1": "For each of the following functions, find the maximum and minimum values of the function on the circular disk: $x^{2}+y^{2} \\leq 1$. Do this by looking at the level curves and gradients. (A) $f(x, y)=x+y+4$: maximum value=[ANS]\nminimum value=[ANS]\n(B) $f(x, y)=4x^{2}+5 y^{2}$: maximum value=[ANS]\nminimum value=[ANS]\n(C) $f(x, y)=4x^{2}-5 y^{2}$: maximum value=[ANS]\nminimum value=[ANS]", "answer_v1": [ "5.41421356237309", "2.58578643762691", "5", "0", "4", "-5" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "For each of the following functions, find the maximum and minimum values of the function on the circular disk: $x^{2}+y^{2} \\leq 1$. Do this by looking at the level curves and gradients. (A) $f(x, y)=x+y+1$: maximum value=[ANS]\nminimum value=[ANS]\n(B) $f(x, y)=1x^{2}+2 y^{2}$: maximum value=[ANS]\nminimum value=[ANS]\n(C) $f(x, y)=1x^{2}-2 y^{2}$: maximum value=[ANS]\nminimum value=[ANS]", "answer_v2": [ "2.41421356237309", "-0.414213562373095", "2", "0", "1", "-2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "For each of the following functions, find the maximum and minimum values of the function on the circular disk: $x^{2}+y^{2} \\leq 1$. Do this by looking at the level curves and gradients. (A) $f(x, y)=x+y+2$: maximum value=[ANS]\nminimum value=[ANS]\n(B) $f(x, y)=2x^{2}+3 y^{2}$: maximum value=[ANS]\nminimum value=[ANS]\n(C) $f(x, y)=2x^{2}-3 y^{2}$: maximum value=[ANS]\nminimum value=[ANS]", "answer_v3": [ "3.41421356237309", "0.585786437626905", "3", "0", "2", "-3" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0312", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "5", "keywords": [ "minimum' 'triangle' 'multivariable' 'surface area", "Volume" ], "problem_v1": "You are to manufacture a rectangular box with 3 dimensions x, y and z, and volume $v=4096$. Find the dimensions which minimize the surface area of this box. x=[ANS]\ny=[ANS]\nz=[ANS]", "answer_v1": [ "16", "16", "16" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "You are to manufacture a rectangular box with 3 dimensions x, y and z, and volume $v=27$. Find the dimensions which minimize the surface area of this box. x=[ANS]\ny=[ANS]\nz=[ANS]", "answer_v2": [ "3", "3", "3" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "You are to manufacture a rectangular box with 3 dimensions x, y and z, and volume $v=343$. Find the dimensions which minimize the surface area of this box. x=[ANS]\ny=[ANS]\nz=[ANS]", "answer_v3": [ "7", "7", "7" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0313", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "5", "keywords": [ "critical points' 'minimum' 'maximum' 'multivariable", "Multivariable", "Geometry" ], "problem_v1": "Find the coordinates of the point (x, y, z) on the plane z=4x+3 y+3 which is closest to the origin. x=[ANS]\ny=[ANS]\nz=[ANS]", "answer_v1": [ "-0.461538461538462", "-0.346153846153846", "0.115384615384615" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the coordinates of the point (x, y, z) on the plane z=1x+4 y+1 which is closest to the origin. x=[ANS]\ny=[ANS]\nz=[ANS]", "answer_v2": [ "-0.0555555555555556", "-0.222222222222222", "0.0555555555555556" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the coordinates of the point (x, y, z) on the plane z=2x+3 y+2 which is closest to the origin. x=[ANS]\ny=[ANS]\nz=[ANS]", "answer_v3": [ "-0.285714285714286", "-0.428571428571429", "0.142857142857143" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0314", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "4", "keywords": [ "critical points' 'minimum' 'maximum' 'multivariable", "Extrema" ], "problem_v1": "Suppose $f(x, y)=x^{2}+y^{2}-8x-6 y+4$ (A) How many critical points does $f$ have in $\\mathbf{R}^{2}$? [ANS]\n(B) If there is a local minimum, what is the value of the discriminant D at that point? If there is none, type N. [ANS]\n(C) If there is a local maximum, what is the value of the discriminant D at that point? If there is none, type N. [ANS]\n(D) If there is a saddle point, what is the value of the discriminant D at that point? If there is none, type N. [ANS]\n(E) What is the maximum value of $f$ on $\\mathbf{R}^{2}$? If there is none, type N. [ANS]\n(F) What is the minimum value of $f$ on $\\bf{R}^{2}$? If there is none, type N. [ANS]", "answer_v1": [ "1", "4", "N", "N", "N", "4 - 4^2 - 3^2" ], "answer_type_v1": [ "NV", "NV", "OE", "OE", "OE", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Suppose $f(x, y)=x^{2}+y^{2}-2x-10 y+1$ (A) How many critical points does $f$ have in $\\mathbf{R}^{2}$? [ANS]\n(B) If there is a local minimum, what is the value of the discriminant D at that point? If there is none, type N. [ANS]\n(C) If there is a local maximum, what is the value of the discriminant D at that point? If there is none, type N. [ANS]\n(D) If there is a saddle point, what is the value of the discriminant D at that point? If there is none, type N. [ANS]\n(E) What is the maximum value of $f$ on $\\mathbf{R}^{2}$? If there is none, type N. [ANS]\n(F) What is the minimum value of $f$ on $\\bf{R}^{2}$? If there is none, type N. [ANS]", "answer_v2": [ "1", "4", "N", "N", "N", "1 - 1^2 - 5^2" ], "answer_type_v2": [ "NV", "NV", "OE", "OE", "OE", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Suppose $f(x, y)=x^{2}+y^{2}-4x-8 y+2$ (A) How many critical points does $f$ have in $\\mathbf{R}^{2}$? [ANS]\n(B) If there is a local minimum, what is the value of the discriminant D at that point? If there is none, type N. [ANS]\n(C) If there is a local maximum, what is the value of the discriminant D at that point? If there is none, type N. [ANS]\n(D) If there is a saddle point, what is the value of the discriminant D at that point? If there is none, type N. [ANS]\n(E) What is the maximum value of $f$ on $\\mathbf{R}^{2}$? If there is none, type N. [ANS]\n(F) What is the minimum value of $f$ on $\\bf{R}^{2}$? If there is none, type N. [ANS]", "answer_v3": [ "1", "4", "N", "N", "N", "2 - 2^2 - 4^2" ], "answer_type_v3": [ "NV", "NV", "OE", "OE", "OE", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0315", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "5", "keywords": [ "critical points' 'minimum' 'maximum' 'multivariable" ], "problem_v1": "Find the dimensions of a rectangular box, open at the top, having volume 763, and requiring the least amount of material for its construction. length=[ANS]\nwidth $\\hspace{0.8pt}$=[ANS]\nheight=[ANS]", "answer_v1": [ "(2*763)^(1/3)", "(2*763)^(1/3)", "11.5129/2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Find the dimensions of a rectangular box, open at the top, having volume 119, and requiring the least amount of material for its construction. length=[ANS]\nwidth $\\hspace{0.8pt}$=[ANS]\nheight=[ANS]", "answer_v2": [ "(2*119)^(1/3)", "(2*119)^(1/3)", "6.19715/2" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Find the dimensions of a rectangular box, open at the top, having volume 341, and requiring the least amount of material for its construction. length=[ANS]\nwidth $\\hspace{0.8pt}$=[ANS]\nheight=[ANS]", "answer_v3": [ "(2*341)^(1/3)", "(2*341)^(1/3)", "8.80227/2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0316", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "4", "keywords": [ "Extrema", "critical points' 'minimum' 'maximum' 'multivariable", "calculus", "critical point" ], "problem_v1": "The function f(x, y)=xy(1-8x-6 y) has 4 critical points. List them and select the type of critical point.\nPoints should be entered as ordered pairs and listed in increasing lexicographic order. By that we mean that $(x, y)$ comes before $(z, w)$ if $x < z$ or if $x=z$ and $y < w$.\nFirst point [ANS] of type [ANS] Second point [ANS] of type [ANS] Third point [ANS] of type [ANS] Fourth point [ANS] of type [ANS]", "answer_v1": [ "(0,0)", "Saddle", "(0,1/6)", "Saddle", "(1/24,1/18)", "Maximum", "(1/8,0)", "Saddle" ], "answer_type_v1": [ "OL", "MCS", "OL", "MCS", "OL", "MCS", "OL", "MCS" ], "options_v1": [ [], [ "Maximum", "Minimum", "Saddle" ], [], [ "Maximum", "Minimum", "Saddle" ], [], [ "Maximum", "Minimum", "Saddle" ], [], [ "Maximum", "Minimum", "Saddle" ] ], "problem_v2": "The function f(x, y)=xy(1-1x-10 y) has 4 critical points. List them and select the type of critical point.\nPoints should be entered as ordered pairs and listed in increasing lexicographic order. By that we mean that $(x, y)$ comes before $(z, w)$ if $x < z$ or if $x=z$ and $y < w$.\nFirst point [ANS] of type [ANS] Second point [ANS] of type [ANS] Third point [ANS] of type [ANS] Fourth point [ANS] of type [ANS]", "answer_v2": [ "(0,0)", "Saddle", "(0,1/10)", "Saddle", "(1/3,1/30)", "Maximum", "(1/1,0)", "Saddle" ], "answer_type_v2": [ "OL", "MCS", "OL", "MCS", "OL", "MCS", "OL", "MCS" ], "options_v2": [ [], [ "Maximum", "Minimum", "Saddle" ], [], [ "Maximum", "Minimum", "Saddle" ], [], [ "Maximum", "Minimum", "Saddle" ], [], [ "Maximum", "Minimum", "Saddle" ] ], "problem_v3": "The function f(x, y)=xy(1-4x-7 y) has 4 critical points. List them and select the type of critical point.\nPoints should be entered as ordered pairs and listed in increasing lexicographic order. By that we mean that $(x, y)$ comes before $(z, w)$ if $x < z$ or if $x=z$ and $y < w$.\nFirst point [ANS] of type [ANS] Second point [ANS] of type [ANS] Third point [ANS] of type [ANS] Fourth point [ANS] of type [ANS]", "answer_v3": [ "(0,0)", "Saddle", "(0,1/7)", "Saddle", "(1/12,1/21)", "Maximum", "(1/4,0)", "Saddle" ], "answer_type_v3": [ "OL", "MCS", "OL", "MCS", "OL", "MCS", "OL", "MCS" ], "options_v3": [ [], [ "Maximum", "Minimum", "Saddle" ], [], [ "Maximum", "Minimum", "Saddle" ], [], [ "Maximum", "Minimum", "Saddle" ], [], [ "Maximum", "Minimum", "Saddle" ] ] }, { "id": "Calculus_-_multivariable_0317", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "4", "keywords": [ "maximum", "minimum", "absolute maximum", "absolute minimum", "local maximum", "local minimum", "extrema", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "Let $f(x,y)=4/x+5/y+6x+7 y$ in the region $R$ where $x, y > 0$. Explain why $f$ must have a global minimum at some point in $R$ (note that $R$ is unbounded---how does this influence your explanation?). Then find the global minimum. minimum=[ANS]", "answer_v1": [ "2*sqrt(6*4)+2*sqrt(7*5)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $f(x,y)=1/x+2/y+3x+4 y$ in the region $R$ where $x, y > 0$. Explain why $f$ must have a global minimum at some point in $R$ (note that $R$ is unbounded---how does this influence your explanation?). Then find the global minimum. minimum=[ANS]", "answer_v2": [ "2*sqrt(3*1)+2*sqrt(4*2)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $f(x,y)=2/x+3/y+4x+5 y$ in the region $R$ where $x, y > 0$. Explain why $f$ must have a global minimum at some point in $R$ (note that $R$ is unbounded---how does this influence your explanation?). Then find the global minimum. minimum=[ANS]", "answer_v3": [ "2*sqrt(4*2)+2*sqrt(5*3)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0318", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "5", "keywords": [ "maximum", "minimum", "absolute maximum", "absolute minimum", "local maximum", "local minimum", "extrema", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "A company operates two plants which manufacture the same item and whose total cost functions are C_1=9+0.03 q_1^2 \\quad\\mbox{and}\\quad C_2=4.4+0.05 q_2^2, where $q_1$ and $q_2$ are the quantities produced by each plant. The total quantity demanded, $q=q_1+q_2$, is related to the price, $p$, by p=40-0.05 q. How much should each plant produce in order to maximize the company's profit? $q_1=$ [ANS]\n$q_2=$ [ANS]\nAdapted from M. Rosser, Basic Mathematics for Economists, p. 318 (New York: Routledge, 1993). Adapted from M. Rosser, Basic Mathematics for Economists, p. 318 (New York: Routledge, 1993).", "answer_v1": [ "40/[2*(2*0.03+0.05)]", "40*0.03/[2*(2*0.03+0.05)*0.05]" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A company operates two plants which manufacture the same item and whose total cost functions are C_1=6+0.01 q_1^2 \\quad\\mbox{and}\\quad C_2=5.4+0.02 q_2^2, where $q_1$ and $q_2$ are the quantities produced by each plant. The total quantity demanded, $q=q_1+q_2$, is related to the price, $p$, by p=80-0.02 q. How much should each plant produce in order to maximize the company's profit? $q_1=$ [ANS]\n$q_2=$ [ANS]\nAdapted from M. Rosser, Basic Mathematics for Economists, p. 318 (New York: Routledge, 1993). Adapted from M. Rosser, Basic Mathematics for Economists, p. 318 (New York: Routledge, 1993).", "answer_v2": [ "80/[2*(2*0.01+0.02)]", "80*0.01/[2*(2*0.01+0.02)*0.02]" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A company operates two plants which manufacture the same item and whose total cost functions are C_1=7+0.02 q_1^2 \\quad\\mbox{and}\\quad C_2=4.4+0.04 q_2^2, where $q_1$ and $q_2$ are the quantities produced by each plant. The total quantity demanded, $q=q_1+q_2$, is related to the price, $p$, by p=40-0.04 q. How much should each plant produce in order to maximize the company's profit? $q_1=$ [ANS]\n$q_2=$ [ANS]\nAdapted from M. Rosser, Basic Mathematics for Economists, p. 318 (New York: Routledge, 1993). Adapted from M. Rosser, Basic Mathematics for Economists, p. 318 (New York: Routledge, 1993).", "answer_v3": [ "40/[2*(2*0.02+0.04)]", "40*0.02/[2*(2*0.02+0.04)*0.04]" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0319", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "5", "keywords": [ "maximum", "minimum", "absolute maximum", "absolute minimum", "local maximum", "local minimum", "extrema", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "What is the shortest distance from the surface $x y+12x+z^2=137$ to the origin? distance=[ANS]", "answer_v1": [ "sqrt(89)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "What is the shortest distance from the surface $x y+3x+z^2=24$ to the origin? distance=[ANS]", "answer_v2": [ "sqrt(21)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "What is the shortest distance from the surface $x y+6x+z^2=41$ to the origin? distance=[ANS]", "answer_v3": [ "sqrt(29)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0320", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "4", "keywords": [ "maximum", "minimum", "absolute maximum", "absolute minimum", "local maximum", "local minimum", "extrema", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "Find the parabola of the form $y=ax^2+b$ which best fits the points $(1,0)$, $(4,4)$, $(6,8)$ by minimizing the sum of squares, $S$, given by S=(a+b)^2+(16 a+b-4)^2+(36 a+b-8)^2. $y=$ [ANS] $x^2+$ [ANS]", "answer_v1": [ "-[(1-2*4^2+6^2)*4+(1+4^2-2*6^2)*8]/(2*[1+4^4-6^2+6^4-4^2*(1+6^2)])", "([1+6^4-4^2*(1+6^2)]*4+(1+4^4-6^2-4^2*6^2)*8)/(2*[1+4^4-6^2+6^4-4^2*(1+6^2)])" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the parabola of the form $y=ax^2+b$ which best fits the points $(-1,0)$, $(5,5)$, $(6,10)$ by minimizing the sum of squares, $S$, given by S=(a+b)^2+(25 a+b-5)^2+(36 a+b-10)^2. $y=$ [ANS] $x^2+$ [ANS]", "answer_v2": [ "-[(1-2*5^2+6^2)*5+(1+5^2-2*6^2)*10]/(2*[1+5^4-6^2+6^4-5^2*(1+6^2)])", "([1+6^4-5^2*(1+6^2)]*5+(1+5^4-6^2-5^2*6^2)*10)/(2*[1+5^4-6^2+6^4-5^2*(1+6^2)])" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the parabola of the form $y=ax^2+b$ which best fits the points $(-1,0)$, $(4,4)$, $(5,8)$ by minimizing the sum of squares, $S$, given by S=(a+b)^2+(16 a+b-4)^2+(25 a+b-8)^2. $y=$ [ANS] $x^2+$ [ANS]", "answer_v3": [ "-[(1-2*4^2+5^2)*4+(1+4^2-2*5^2)*8]/(2*[1+4^4-5^2+5^4-4^2*(1+5^2)])", "([1+5^4-4^2*(1+5^2)]*4+(1+4^4-5^2-4^2*5^2)*8)/(2*[1+4^4-5^2+5^4-4^2*(1+5^2)])" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0321", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "5", "keywords": [ "maximum", "minimum", "absolute maximum", "absolute minimum", "local maximum", "local minimum", "extrema", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "A closed rectangular box has volume $50\\,\\mbox{cm}^3$. What are the lengths of the edges giving the minimum surface area? lengths=[ANS]\n(Give the three lengths as a comma separated list.) (Give the three lengths as a comma separated list.)", "answer_v1": [ "(3.68403, 3.68403, 3.68403)" ], "answer_type_v1": [ "UOL" ], "options_v1": [ [] ], "problem_v2": "A closed rectangular box has volume $22\\,\\mbox{cm}^3$. What are the lengths of the edges giving the minimum surface area? lengths=[ANS]\n(Give the three lengths as a comma separated list.) (Give the three lengths as a comma separated list.)", "answer_v2": [ "(2.80204, 2.80204, 2.80204)" ], "answer_type_v2": [ "UOL" ], "options_v2": [ [] ], "problem_v3": "A closed rectangular box has volume $32\\,\\mbox{cm}^3$. What are the lengths of the edges giving the minimum surface area? lengths=[ANS]\n(Give the three lengths as a comma separated list.) (Give the three lengths as a comma separated list.)", "answer_v3": [ "(3.1748, 3.1748, 3.1748)" ], "answer_type_v3": [ "UOL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0323", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "4", "keywords": [ "extrema", "multivariable", "functions", "calculus", "derivatives" ], "problem_v1": "Find $A$ and $B$ so that $f(x,y)=x^2+Ax+y^2+B$ has a local minimum at the point $(5, 0)$, with $z$-coordinate 35. $A=$ [ANS]\n$B=$ [ANS]", "answer_v1": [ "-10", "35+5*5" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find $A$ and $B$ so that $f(x,y)=x^2+Ay+y^2+B$ has a local minimum at the point $(0, 8)$, with $z$-coordinate 10. $A=$ [ANS]\n$B=$ [ANS]", "answer_v2": [ "-16", "10+8*8" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find $A$ and $B$ so that $f(x,y)=x^2+Ay+y^2+B$ has a local minimum at the point $(0, 5)$, with $z$-coordinate 15. $A=$ [ANS]\n$B=$ [ANS]", "answer_v3": [ "-10", "15+5*5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0324", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "4", "keywords": [ "extrema", "multivariable", "functions", "calculus", "derivatives" ], "problem_v1": "The function k(x,y)=e^{-y^{2}}\\cos\\!\\left(4x\\right) has a critical point at $(0,0)$. What is the value of $D$ at this critical point? $D=$ [ANS]\nWhat type of critical point is it? [ANS]", "answer_v1": [ "2*16", "maximum" ], "answer_type_v1": [ "NV", "MCS" ], "options_v1": [ [], [ "maximum", "minimum", "saddle point", "point with unknown behavior" ] ], "problem_v2": "The function k(x,y)=e^{-y^{2}}\\cos\\!\\left(x\\right) has a critical point at $(0,0)$. What is the value of $D$ at this critical point? $D=$ [ANS]\nWhat type of critical point is it? [ANS]", "answer_v2": [ "2*1", "maximum" ], "answer_type_v2": [ "NV", "MCS" ], "options_v2": [ [], [ "maximum", "minimum", "saddle point", "point with unknown behavior" ] ], "problem_v3": "The function k(x,y)=e^{-y^{2}}\\cos\\!\\left(2x\\right) has a critical point at $(0,0)$. What is the value of $D$ at this critical point? $D=$ [ANS]\nWhat type of critical point is it? [ANS]", "answer_v3": [ "2*4", "maximum" ], "answer_type_v3": [ "NV", "MCS" ], "options_v3": [ [], [ "maximum", "minimum", "saddle point", "point with unknown behavior" ] ] }, { "id": "Calculus_-_multivariable_0325", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "4", "keywords": [ "extrema", "multivariable", "functions", "calculus", "derivatives" ], "problem_v1": "(a) Find the critical point of $f(x,y)=\\left(4x^{2}-y\\right)\\!\\left(x^{2}+3y\\right)$. $(x,y)=$ [ANS]\n(b) Find the discriminant, $D(x,y)$ for the function. $D(x,y)=$ [ANS]\nwhat is the value of the discriminant at the critical point? $D=$ [ANS]\nWhat does this tell us about the critical point? [ANS] (c) Sketch contours near the critical point to determine, or confirm, whether it is a local maximum, a local minimum, a saddle point, or none of these. [ANS]", "answer_v1": [ "(0,0)", "[48*x^2+2*(12-1)*y]*-6-[2*(12-1)*x]^2", "0", "The discriminant doesn", "The critical point is a saddle point" ], "answer_type_v1": [ "OL", "EX", "NV", "MCS", "MCS" ], "options_v1": [ [], [], [], [ "The critical point is a local maximum", "The critical point is a local minimum", "The critical point is a saddle point", "The discriminant doesn't tell us anything" ], [ "The critical point is a local maximum", "The critical point is a local minimum", "The critical point is a saddle point", "The discriminant doesn't tell us anything" ] ], "problem_v2": "(a) Find the critical point of $f(x,y)=\\left(x^{2}-y\\right)\\!\\left(x^{2}+4y\\right)$. $(x,y)=$ [ANS]\n(b) Find the discriminant, $D(x,y)$ for the function. $D(x,y)=$ [ANS]\nwhat is the value of the discriminant at the critical point? $D=$ [ANS]\nWhat does this tell us about the critical point? [ANS] (c) Sketch contours near the critical point to determine, or confirm, whether it is a local maximum, a local minimum, a saddle point, or none of these. [ANS]", "answer_v2": [ "(0,0)", "[12*x^2+2*(4-1)*y]*-8-[2*(4-1)*x]^2", "0", "The discriminant doesn", "The critical point is a saddle point" ], "answer_type_v2": [ "OL", "EX", "NV", "MCS", "MCS" ], "options_v2": [ [], [], [], [ "The critical point is a local maximum", "The critical point is a local minimum", "The critical point is a saddle point", "The discriminant doesn't tell us anything" ], [ "The critical point is a local maximum", "The critical point is a local minimum", "The critical point is a saddle point", "The discriminant doesn't tell us anything" ] ], "problem_v3": "(a) Find the critical point of $f(x,y)=\\left(2x^{2}-y\\right)\\!\\left(x^{2}+3y\\right)$. $(x,y)=$ [ANS]\n(b) Find the discriminant, $D(x,y)$ for the function. $D(x,y)=$ [ANS]\nwhat is the value of the discriminant at the critical point? $D=$ [ANS]\nWhat does this tell us about the critical point? [ANS] (c) Sketch contours near the critical point to determine, or confirm, whether it is a local maximum, a local minimum, a saddle point, or none of these. [ANS]", "answer_v3": [ "(0,0)", "[24*x^2+2*(6-1)*y]*-6-[2*(6-1)*x]^2", "0", "The discriminant doesn", "The critical point is a saddle point" ], "answer_type_v3": [ "OL", "EX", "NV", "MCS", "MCS" ], "options_v3": [ [], [], [], [ "The critical point is a local maximum", "The critical point is a local minimum", "The critical point is a saddle point", "The discriminant doesn't tell us anything" ], [ "The critical point is a local maximum", "The critical point is a local minimum", "The critical point is a saddle point", "The discriminant doesn't tell us anything" ] ] }, { "id": "Calculus_-_multivariable_0327", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Extreme values and optimization", "level": "3", "keywords": [ "lagrange multipliers", "maximum", "minimum", "absolute maximum", "absolute minimum", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "A company manufactures $x$ units of one item and $y$ units of another. The total cost in dollars, $C$, of producing these two items is approximated by the function C=6x^2+3xy+6 y^2+900.\n(a) If the production quota for the total number of items (both types combined) is 22, find the minimum production cost. cost=[ANS]\n(b) Estimate the additional production cost or savings if the production quota is raised to 23 or lowered to 21. production cost or savings=[ANS]", "answer_v1": [ "6*11*11+3*11*11+6*11*11+900", "2*6*11+3*11" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A company manufactures $x$ units of one item and $y$ units of another. The total cost in dollars, $C$, of producing these two items is approximated by the function C=3x^2+2xy+2 y^2+700.\n(a) If the production quota for the total number of items (both types combined) is 45, find the minimum production cost. cost=[ANS]\n(b) Estimate the additional production cost or savings if the production quota is raised to 46 or lowered to 44. production cost or savings=[ANS]", "answer_v2": [ "3*15*15+2*15*30+2*30*30+700", "2*3*15+2*30" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A company manufactures $x$ units of one item and $y$ units of another. The total cost in dollars, $C$, of producing these two items is approximated by the function C=4x^2+2xy+3 y^2+800.\n(a) If the production quota for the total number of items (both types combined) is 55, find the minimum production cost. cost=[ANS]\n(b) Estimate the additional production cost or savings if the production quota is raised to 56 or lowered to 54. production cost or savings=[ANS]", "answer_v3": [ "4*22*22+2*22*33+3*33*33+800", "2*4*22+2*33" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0328", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Determine the global extreme values of the function $f(x,y)=4x^{3}-6y, \\quad 0 \\le x,y \\le 1$ $f_{min}=$ [ANS]\n$f_{max}=$ [ANS]", "answer_v1": [ "-6", "4" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Determine the global extreme values of the function $f(x,y)=x^{3}-8y, \\quad 0 \\le x,y \\le 1$ $f_{min}=$ [ANS]\n$f_{max}=$ [ANS]", "answer_v2": [ "-8", "1" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Determine the global extreme values of the function $f(x,y)=2x^{3}-6y, \\quad 0 \\le x,y \\le 1$ $f_{min}=$ [ANS]\n$f_{max}=$ [ANS]", "answer_v3": [ "-6", "2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0329", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "4", "keywords": [ "calculus" ], "problem_v1": "The surface area of a right-circular cone of radius $r$ and height $h$ is $S=\\pi r\\sqrt{r^2+h^2}$, and its volume is $V=\\frac13\\pi r^2h$.\n(a) Determine $h$ and $r$ for the cone with given surface area $S=7$ and maximal volume $V$. $h=$ [ANS], $\\quad$ $r=$ [ANS]\n(b) What is the ratio $h/r$ for a cone with given volume $V=6$ and minimal surface area $S$? $ \\frac{h}{r} =$ [ANS]\n(c) Does a cone with given volume $V$ and maximal surface area exist? [ANS] A. no B. yes", "answer_v1": [ "1.60402", "1.13421", "1.41421", "A" ], "answer_type_v1": [ "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [ "A", "B" ] ], "problem_v2": "The surface area of a right-circular cone of radius $r$ and height $h$ is $S=\\pi r\\sqrt{r^2+h^2}$, and its volume is $V=\\frac13\\pi r^2h$.\n(a) Determine $h$ and $r$ for the cone with given surface area $S=1$ and maximal volume $V$. $h=$ [ANS], $\\quad$ $r=$ [ANS]\n(b) What is the ratio $h/r$ for a cone with given volume $V=9$ and minimal surface area $S$? $ \\frac{h}{r} =$ [ANS]\n(c) Does a cone with given volume $V$ and maximal surface area exist? [ANS] A. yes B. no", "answer_v2": [ "0.606261", "0.428691", "1.41421", "B" ], "answer_type_v2": [ "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [ "A", "B" ] ], "problem_v3": "The surface area of a right-circular cone of radius $r$ and height $h$ is $S=\\pi r\\sqrt{r^2+h^2}$, and its volume is $V=\\frac13\\pi r^2h$.\n(a) Determine $h$ and $r$ for the cone with given surface area $S=3$ and maximal volume $V$. $h=$ [ANS], $\\quad$ $r=$ [ANS]\n(b) What is the ratio $h/r$ for a cone with given volume $V=6$ and minimal surface area $S$? $ \\frac{h}{r} =$ [ANS]\n(c) Does a cone with given volume $V$ and maximal surface area exist? [ANS] A. yes B. no", "answer_v3": [ "1.05008", "0.742515", "1.41421", "B" ], "answer_type_v3": [ "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [ "A", "B" ] ] }, { "id": "Calculus_-_multivariable_0330", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find the minimum and maximum values of the function subject to the given constraint\n$f(x,y)=9x^{2}+4y^{2}$, $\\quad 3x+16y=2$\nEnter DNE if such a value does not exist. $f_{min}=$ [ANS]\n$f_{max}=$ [ANS]", "answer_v1": [ "0.0615385", "DNE" ], "answer_type_v1": [ "NV", "OE" ], "options_v1": [ [], [] ], "problem_v2": "Find the minimum and maximum values of the function subject to the given constraint\n$f(x,y)=5x^{2}+y^{2}$, $\\quad 5x+2y=5$\nEnter DNE if such a value does not exist. $f_{min}=$ [ANS]\n$f_{max}=$ [ANS]", "answer_v2": [ "2.77778", "DNE" ], "answer_type_v2": [ "NV", "OE" ], "options_v2": [ [], [] ], "problem_v3": "Find the minimum and maximum values of the function subject to the given constraint\n$f(x,y)=8x^{2}+2y^{2}$, $\\quad 4x+4y=2$\nEnter DNE if such a value does not exist. $f_{min}=$ [ANS]\n$f_{max}=$ [ANS]", "answer_v3": [ "0.4", "DNE" ], "answer_type_v3": [ "NV", "OE" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0331", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Use Lagrange multipliers to find the point $(a,b)$ on the graph of $y=e^{7x}$, where the value $ab$ is as small as possible. $P=$ [ANS]", "answer_v1": [ "(-0.142857,0.367879)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Use Lagrange multipliers to find the point $(a,b)$ on the graph of $y=e^{1x}$, where the value $ab$ is as small as possible. $P=$ [ANS]", "answer_v2": [ "(-1,0.367879)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Use Lagrange multipliers to find the point $(a,b)$ on the graph of $y=e^{3x}$, where the value $ab$ is as small as possible. $P=$ [ANS]", "answer_v3": [ "(-0.333333,0.367879)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0332", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find the maximum value of f(x,y,z)=xy+xz+yz-4xyz subject to the constraints $x+y+z=7$ and $x,y,z \\ge 0$.\n$f_{max}=$ [ANS]", "answer_v1": [ "12.25" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the maximum value of f(x,y,z)=xy+xz+yz-4xyz subject to the constraints $x+y+z=1$ and $x,y,z \\ge 0$.\n$f_{max}=$ [ANS]", "answer_v2": [ "0.25" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the maximum value of f(x,y,z)=xy+xz+yz-4xyz subject to the constraints $x+y+z=3$ and $x,y,z \\ge 0$.\n$f_{max}=$ [ANS]", "answer_v3": [ "2.25" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0333", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find the maximum value of $f(x,y)=x^{5}y^{6}$ for $x,y \\geq 0$ on the unit circle. $f_{max}=$ [ANS]", "answer_v1": [ "0.0226058" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the maximum value of $f(x,y)=x^{2}y^{9}$ for $x,y \\geq 0$ on the unit circle. $f_{max}=$ [ANS]", "answer_v2": [ "0.073699" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the maximum value of $f(x,y)=x^{3}y^{6}$ for $x,y \\geq 0$ on the unit circle. $f_{max}=$ [ANS]", "answer_v3": [ "0.0570222" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0334", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Find the minimum and maximum values of the function $f(x,y,z)=3x+2y+4z$ subject to the constraint $x^2+2y^2+6z^2=49$. $f_{max}=$ [ANS]\n$f_{min}=$ [ANS]", "answer_v1": [ "25.8779", "-25.8779" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the minimum and maximum values of the function $f(x,y,z)=3x+2y+4z$ subject to the constraint $x^2+2y^2+6z^2=1$. $f_{max}=$ [ANS]\n$f_{min}=$ [ANS]", "answer_v2": [ "3.69685", "-3.69685" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the minimum and maximum values of the function $f(x,y,z)=3x+2y+4z$ subject to the constraint $x^2+2y^2+6z^2=9$. $f_{max}=$ [ANS]\n$f_{min}=$ [ANS]", "answer_v3": [ "11.0905", "-11.0905" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0335", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "4", "keywords": [ "critical points' 'minimum' 'maximum' 'multivariable", "Extrema" ], "problem_v1": "Find the maximum and minimum values of $f(x, y)=16\\!x^{2}+17\\!y^{2}$ on the disk D: $x^{2}+y^{2} \\leq 1$. maximum value: [ANS]\nminimum value: [ANS]", "answer_v1": [ "17", "0" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the maximum and minimum values of $f(x, y)=2\\!x^{2}+3\\!y^{2}$ on the disk D: $x^{2}+y^{2} \\leq 1$. maximum value: [ANS]\nminimum value: [ANS]", "answer_v2": [ "3", "0" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the maximum and minimum values of $f(x, y)=7\\!x^{2}+8\\!y^{2}$ on the disk D: $x^{2}+y^{2} \\leq 1$. maximum value: [ANS]\nminimum value: [ANS]", "answer_v3": [ "8", "0" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0336", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "4", "keywords": [ "critical points' 'minimum' 'maximum' 'multivariable", "Extrema", "Multivariable", "Geometry" ], "problem_v1": "Find the maximum and minimum values of $f(x, y)=xy$ on the ellipse $8\\!x^{2}+y^{2}=6$. maximum value=[ANS]\nminimum value=[ANS]", "answer_v1": [ "1.06066017177982", "-1.06066017177982" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the maximum and minimum values of $f(x, y)=xy$ on the ellipse $2\\!x^{2}+y^{2}=9$. maximum value=[ANS]\nminimum value=[ANS]", "answer_v2": [ "3.18198051533946", "-3.18198051533946" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the maximum and minimum values of $f(x, y)=xy$ on the ellipse $4\\!x^{2}+y^{2}=6$. maximum value=[ANS]\nminimum value=[ANS]", "answer_v3": [ "1.5", "-1.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0337", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "5", "keywords": [ "maximum", "minimum", "absolute maximum", "absolute minimum", "local maximum", "local minimum", "extrema", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "A closed rectangular box with faces parallel to the coordinate planes has one bottom corner at the origin and the opposite top corner in the first octant on the plane $5x+4 y+z=1$. What is the maximum volume of such a box? volume=[ANS]", "answer_v1": [ "1/15*1/12*1/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A closed rectangular box with faces parallel to the coordinate planes has one bottom corner at the origin and the opposite top corner in the first octant on the plane $2x+6 y+z=1$. What is the maximum volume of such a box? volume=[ANS]", "answer_v2": [ "1/6*1/18*1/3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A closed rectangular box with faces parallel to the coordinate planes has one bottom corner at the origin and the opposite top corner in the first octant on the plane $3x+5 y+z=1$. What is the maximum volume of such a box? volume=[ANS]", "answer_v3": [ "1/9*1/15*1/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0338", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "5", "keywords": [ "maximum", "minimum", "absolute maximum", "absolute minimum", "local maximum", "local minimum", "extrema", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "Design a rectangular milk carton box of width $w$, length $l$, and height $h$ which holds $526 \\mbox{cm}^3$ of milk. The sides of the box cost $2 \\ \\mbox{cent}/\\mbox{cm}^2$ and the top and bottom cost $4\\ \\mbox{cent}/\\mbox{cm}^2$. Find the dimensions of the box that minimize the total cost of materials used. dimensions=[ANS]\n(Enter your answer as a comma separated list of lengths.) (Enter your answer as a comma separated list of lengths.)", "answer_v1": [ "(12.8139, 6.40695, 6.40698)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Design a rectangular milk carton box of width $w$, length $l$, and height $h$ which holds $458 \\mbox{cm}^3$ of milk. The sides of the box cost $3 \\ \\mbox{cent}/\\mbox{cm}^2$ and the top and bottom cost $4\\ \\mbox{cent}/\\mbox{cm}^2$. Find the dimensions of the box that minimize the total cost of materials used. dimensions=[ANS]\n(Enter your answer as a comma separated list of lengths.) (Enter your answer as a comma separated list of lengths.)", "answer_v2": [ "(9.33787, 7.0034, 7.0034)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Design a rectangular milk carton box of width $w$, length $l$, and height $h$ which holds $480 \\mbox{cm}^3$ of milk. The sides of the box cost $2 \\ \\mbox{cent}/\\mbox{cm}^2$ and the top and bottom cost $3\\ \\mbox{cent}/\\mbox{cm}^2$. Find the dimensions of the box that minimize the total cost of materials used. dimensions=[ANS]\n(Enter your answer as a comma separated list of lengths.) (Enter your answer as a comma separated list of lengths.)", "answer_v3": [ "(10.2599, 6.83993, 6.83985)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0339", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "3", "keywords": [ "lagrange multipliers", "maximum", "minimum", "absolute maximum", "absolute minimum", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "Use Lagrange multipliers to find the maximum and minimum values of $f(x,y)=3x+5 y$, subject to the constraint $x^2+y^2\\le 3$. maximum=[ANS]\nminimum=[ANS]\n(For either value, enter DNE if there is no such value.)", "answer_v1": [ "sqrt((3*3+5*5)*3)", "-[sqrt((3*3+5*5)*3)]" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Use Lagrange multipliers to find the maximum and minimum values of $f(x,y)=x+7 y$, subject to the constraint $x^2+y^2\\le 1$. maximum=[ANS]\nminimum=[ANS]\n(For either value, enter DNE if there is no such value.)", "answer_v2": [ "sqrt((1*1+7*7)*1)", "-[sqrt((1*1+7*7)*1)]" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Use Lagrange multipliers to find the maximum and minimum values of $f(x,y)=x+5 y$, subject to the constraint $x^2+y^2\\le 2$. maximum=[ANS]\nminimum=[ANS]\n(For either value, enter DNE if there is no such value.)", "answer_v3": [ "sqrt((1*1+5*5)*2)", "-[sqrt((1*1+5*5)*2)]" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0340", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "2", "keywords": [ "lagrange multipliers", "maximum", "minimum", "absolute maximum", "absolute minimum", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "For each value of $\\lambda$ the function $h(x,y)=x^2+y^2-\\lambda(2x+6 y-18)$ has a minimum value $m(\\lambda)$.\n(a) Find $m(\\lambda)$ $m(\\lambda)=$ [ANS]\n(Use the letter L for $\\lambda$ in your expression.) (b) For which value of $\\lambda$ is $m(\\lambda)$ the largest, and what is that maximum value? $\\lambda=$ [ANS]\nmaximum $m(\\lambda)=$ [ANS]\n(c) Find the minimum value of $f(x,y)=x^2+y^2$ subject to the constraint $2x+6 y=18$ using the method of Lagrange multipliers and evaluate $\\lambda$. minimum $f$=[ANS]\n$\\lambda=$ [ANS]\n(How are these results related to your result in part (b)?) (How are these results related to your result in part (b)?)", "answer_v1": [ "18*L-(2*2+6*6)*L^2/4", "2*18/(2*2+6*6)", "18*18/(2*2+6*6)", "18*18/(2*2+6*6)", "2*18/(2*2+6*6)" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "For each value of $\\lambda$ the function $h(x,y)=x^2+y^2-\\lambda(2x+8 y-15)$ has a minimum value $m(\\lambda)$.\n(a) Find $m(\\lambda)$ $m(\\lambda)=$ [ANS]\n(Use the letter L for $\\lambda$ in your expression.) (b) For which value of $\\lambda$ is $m(\\lambda)$ the largest, and what is that maximum value? $\\lambda=$ [ANS]\nmaximum $m(\\lambda)=$ [ANS]\n(c) Find the minimum value of $f(x,y)=x^2+y^2$ subject to the constraint $2x+8 y=15$ using the method of Lagrange multipliers and evaluate $\\lambda$. minimum $f$=[ANS]\n$\\lambda=$ [ANS]\n(How are these results related to your result in part (b)?) (How are these results related to your result in part (b)?)", "answer_v2": [ "15*L-(2*2+8*8)*L^2/4", "2*15/(2*2+8*8)", "15*15/(2*2+8*8)", "15*15/(2*2+8*8)", "2*15/(2*2+8*8)" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "For each value of $\\lambda$ the function $h(x,y)=x^2+y^2-\\lambda(2x+6 y-16)$ has a minimum value $m(\\lambda)$.\n(a) Find $m(\\lambda)$ $m(\\lambda)=$ [ANS]\n(Use the letter L for $\\lambda$ in your expression.) (b) For which value of $\\lambda$ is $m(\\lambda)$ the largest, and what is that maximum value? $\\lambda=$ [ANS]\nmaximum $m(\\lambda)=$ [ANS]\n(c) Find the minimum value of $f(x,y)=x^2+y^2$ subject to the constraint $2x+6 y=16$ using the method of Lagrange multipliers and evaluate $\\lambda$. minimum $f$=[ANS]\n$\\lambda=$ [ANS]\n(How are these results related to your result in part (b)?) (How are these results related to your result in part (b)?)", "answer_v3": [ "16*L-(2*2+6*6)*L^2/4", "2*16/(2*2+6*6)", "16*16/(2*2+6*6)", "16*16/(2*2+6*6)", "2*16/(2*2+6*6)" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0341", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "3", "keywords": [ "lagrange multipliers", "maximum", "minimum", "absolute maximum", "absolute minimum", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "(a) If $\\sum_{i=1}^3x_i=4$, find the values of $x_1, x_2, x_3$ making $\\sum_{i=1}^3 {x_i}^2$ minimum. $x_1, x_2, x_3=$ [ANS]\n(Give your values as a comma separated list.) (Give your values as a comma separated list.) (b) Generalize the result of part (a) to find the minimum value of $\\sum_{i=1}^n{x_i}^2$ subject to $\\sum_{i=1}^n x_i=4$. minimum value=[ANS]", "answer_v1": [ "(1.33333, 1.33333, 1.33333)", "4*4/n" ], "answer_type_v1": [ "OL", "EX" ], "options_v1": [ [], [] ], "problem_v2": "(a) If $\\sum_{i=1}^3x_i=1$, find the values of $x_1, x_2, x_3$ making $\\sum_{i=1}^3 {x_i}^2$ minimum. $x_1, x_2, x_3=$ [ANS]\n(Give your values as a comma separated list.) (Give your values as a comma separated list.) (b) Generalize the result of part (a) to find the minimum value of $\\sum_{i=1}^n{x_i}^2$ subject to $\\sum_{i=1}^n x_i=1$. minimum value=[ANS]", "answer_v2": [ "(0.333333, 0.333333, 0.333333)", "1*1/n" ], "answer_type_v2": [ "OL", "EX" ], "options_v2": [ [], [] ], "problem_v3": "(a) If $\\sum_{i=1}^3x_i=2$, find the values of $x_1, x_2, x_3$ making $\\sum_{i=1}^3 {x_i}^2$ minimum. $x_1, x_2, x_3=$ [ANS]\n(Give your values as a comma separated list.) (Give your values as a comma separated list.) (b) Generalize the result of part (a) to find the minimum value of $\\sum_{i=1}^n{x_i}^2$ subject to $\\sum_{i=1}^n x_i=2$. minimum value=[ANS]", "answer_v3": [ "(0.666667, 0.666667, 0.666667)", "2*2/n" ], "answer_type_v3": [ "OL", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0342", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "3", "keywords": [ "lagrange multipliers", "maximum", "minimum", "absolute maximum", "absolute minimum", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "The maximum value of $f(x, y)$ subject to the constraint $g(x, y)=260$ is $7300$. The method of Lagrange multipliers gives $\\lambda=25$. Find an approximate value for the maximum of $f(x, y)$ subject to the constraint $g(x, y)=262$.} $f_{max} \\approx$ [ANS]", "answer_v1": [ "7300+2*25" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The maximum value of $f(x, y)$ subject to the constraint $g(x, y)=210$ is $5200$. The method of Lagrange multipliers gives $\\lambda=35$. Find an approximate value for the maximum of $f(x, y)$ subject to the constraint $g(x, y)=208$.} $f_{max} \\approx$ [ANS]", "answer_v2": [ "5200+-2*35" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The maximum value of $f(x, y)$ subject to the constraint $g(x, y)=230$ is $5900$. The method of Lagrange multipliers gives $\\lambda=30$. Find an approximate value for the maximum of $f(x, y)$ subject to the constraint $g(x, y)=232$.} $f_{max} \\approx$ [ANS]", "answer_v3": [ "5900+2*30" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0343", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "3", "keywords": [ "lagrange multipliers", "maximum", "minimum", "absolute maximum", "absolute minimum", "calculus", "differentiation", "multivariable", "functions" ], "problem_v1": "Find the minimum distance from the point $(1,4,13)$ to the paraboloid given by the equation $z=x^2+y^2$. Minimum distance=[ANS]\nNote: If you need to find roots of a polynomial of degree $\\geq 3$, you may want to use a calculator of computer to do so numerically. Also be sure that you can give a geometric justification for your answer.", "answer_v1": [ "sqrt(0.262822)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the minimum distance from the point $(1,1,15)$ to the paraboloid given by the equation $z=x^2+y^2$. Minimum distance=[ANS]\nNote: If you need to find roots of a polynomial of degree $\\geq 3$, you may want to use a calculator of computer to do so numerically. Also be sure that you can give a geometric justification for your answer.", "answer_v2": [ "sqrt(5.94541)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the minimum distance from the point $(1,2,13)$ to the paraboloid given by the equation $z=x^2+y^2$. Minimum distance=[ANS]\nNote: If you need to find roots of a polynomial of degree $\\geq 3$, you may want to use a calculator of computer to do so numerically. Also be sure that you can give a geometric justification for your answer.", "answer_v3": [ "sqrt(1.83985)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0344", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "4", "keywords": [], "problem_v1": "Suppose that all you do in a day is work, play and sleep. Let $x_1$ be the number of hours per day you spend playing, $x_2$ is the number of hours you spend sleeping, and $x_3$ is the number of hours you spend working. Suppose that sleeping is free, but playing costs you \\$16 an hour. Furthermore, you spend all the money you earn working on playing. The utility you get from sleeping and playing is given by a Cobb-Douglas utility function: $U=x_1^{a_1}x_2^{a_2}$, where $a_1+a_2=1$ NOTE: By construction, $x_2$ can represent the hours you spend consuming anything that is free, and $x_1$ can be the number of hours consuming goods you have to pay for. This does not change the question, it is just interesting that this set-up can be applied to a more general setting. Let $a_1= \\frac{2}{3} $ If your hourly wage is $w$. Find the number of hours you should work a day $(x_3^*)$ in order to maximize your utility as a function of $w$. $x_3^*=$ [ANS]\nIf $w=16$: $x_3^*=$ [ANS]\nThe easiest way to solve this question is using Lagrange multipliers. In general: suppose you want to maximize $f(x_1,x_2,x_3)$, subject to $h(x_1,x_2,x_3)=0$, and $g(x_1,x_2,x_3)=0$. We define the Lagrange function to be: $ \\Lambda(x_1,x_2,x_3,\\lambda_1,\\lambda_2)=f(x_1,x_2,x_3)-\\lambda_1 g(x_1,x_2,x_3)-\\lambda_2 h(x_1,x_2,x_3)$ $f(x_1,x_2,x_3)$ is maximized when all of the partial derivatives of the Lagrange function are equal to $0$.\n(you will lose 50\\% of your points if you do)", "answer_v1": [ "256/(16+w)", "8" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose that all you do in a day is work, play and sleep. Let $x_1$ be the number of hours per day you spend playing, $x_2$ is the number of hours you spend sleeping, and $x_3$ is the number of hours you spend working. Suppose that sleeping is free, but playing costs you \\$20 an hour. Furthermore, you spend all the money you earn working on playing. The utility you get from sleeping and playing is given by a Cobb-Douglas utility function: $U=x_1^{a_1}x_2^{a_2}$, where $a_1+a_2=1$ NOTE: By construction, $x_2$ can represent the hours you spend consuming anything that is free, and $x_1$ can be the number of hours consuming goods you have to pay for. This does not change the question, it is just interesting that this set-up can be applied to a more general setting. Let $a_1= \\frac{3}{4} $ If your hourly wage is $w$. Find the number of hours you should work a day $(x_3^*)$ in order to maximize your utility as a function of $w$. $x_3^*=$ [ANS]\nIf $w=40$: $x_3^*=$ [ANS]\nThe easiest way to solve this question is using Lagrange multipliers. In general: suppose you want to maximize $f(x_1,x_2,x_3)$, subject to $h(x_1,x_2,x_3)=0$, and $g(x_1,x_2,x_3)=0$. We define the Lagrange function to be: $ \\Lambda(x_1,x_2,x_3,\\lambda_1,\\lambda_2)=f(x_1,x_2,x_3)-\\lambda_1 g(x_1,x_2,x_3)-\\lambda_2 h(x_1,x_2,x_3)$ $f(x_1,x_2,x_3)$ is maximized when all of the partial derivatives of the Lagrange function are equal to $0$.\n(you will lose 50\\% of your points if you do)", "answer_v2": [ "360/(20+w)", "6" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose that all you do in a day is work, play and sleep. Let $x_1$ be the number of hours per day you spend playing, $x_2$ is the number of hours you spend sleeping, and $x_3$ is the number of hours you spend working. Suppose that sleeping is free, but playing costs you \\$15 an hour. Furthermore, you spend all the money you earn working on playing. The utility you get from sleeping and playing is given by a Cobb-Douglas utility function: $U=x_1^{a_1}x_2^{a_2}$, where $a_1+a_2=1$ NOTE: By construction, $x_2$ can represent the hours you spend consuming anything that is free, and $x_1$ can be the number of hours consuming goods you have to pay for. This does not change the question, it is just interesting that this set-up can be applied to a more general setting. Let $a_1= \\frac{3}{4} $ If your hourly wage is $w$. Find the number of hours you should work a day $(x_3^*)$ in order to maximize your utility as a function of $w$. $x_3^*=$ [ANS]\nIf $w=30$: $x_3^*=$ [ANS]\nThe easiest way to solve this question is using Lagrange multipliers. In general: suppose you want to maximize $f(x_1,x_2,x_3)$, subject to $h(x_1,x_2,x_3)=0$, and $g(x_1,x_2,x_3)=0$. We define the Lagrange function to be: $ \\Lambda(x_1,x_2,x_3,\\lambda_1,\\lambda_2)=f(x_1,x_2,x_3)-\\lambda_1 g(x_1,x_2,x_3)-\\lambda_2 h(x_1,x_2,x_3)$ $f(x_1,x_2,x_3)$ is maximized when all of the partial derivatives of the Lagrange function are equal to $0$.\n(you will lose 50\\% of your points if you do)", "answer_v3": [ "270/(15+w)", "6" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0345", "subject": "Calculus_-_multivariable", "topic": "Differentiation of multivariable functions", "subtopic": "Lagrange multipliers and constrained optimization", "level": "4", "keywords": [], "problem_v1": "Suppose that you have five consumption choices: good $x_1 \\cdots x_5$. An indifference surface is the set of consumption choices with a CONSTANT utility. For example if $(x_1, \\cdots,x_5)=(2,1,1,1,1)$ gives the same utility as $(x_1, \\cdots, x_5)=(1,1,1,1,2)$ than these are both points on the same indifference surface. An indifference map is the set of all indifference surface for EVERY given utility. Consider the following utility map: $ U=\\sum_{i=1}^{5}{\\ln({x_i-a_i})}$ Where $(a_1, \\cdots, a_5)=(7,6,6,7,4)$ The budget constraint gives the set of possible consumption choices with a given income. If you have an income of \\$666 and the price of good $x_i$ is given by $p_i$. The equation for the budget line is given by: $ 666=\\sum_{i=1}^{5}{p_i x_i}$. A utility maximizing combination of goods $x_1 \\cdots x_5$ occurs when the surface given by the budget constraint is tangent to an indifference surface. Find $x_1$ as a function of $p_1 \\cdots p_5$ $x_1=$ [ANS]\n(Use p1 for $p_1$ and likewise for $p_2,p_3,p_4,p_5$. The easiest way to solve this question is using Lagrange multiplier. We define the Lagrange function to be: $ \\Lambda(x_1,\\cdots,x_5,\\lambda)=U(x_1,\\cdots,x_5)-\\lambda\\left(\\sum_{i=1}^{5}{p_i x_i}-666\\right)$ Utility is maximized when all of the partial derivatives of the Lagrange function are equal to $0$.", "answer_v1": [ "(666-6*p2-6*p3-7*p4-4*p5)/p1" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Suppose that you have five consumption choices: good $x_1 \\cdots x_5$. An indifference surface is the set of consumption choices with a CONSTANT utility. For example if $(x_1, \\cdots,x_5)=(2,1,1,1,1)$ gives the same utility as $(x_1, \\cdots, x_5)=(1,1,1,1,2)$ than these are both points on the same indifference surface. An indifference map is the set of all indifference surface for EVERY given utility. Consider the following utility map: $ U=\\sum_{i=1}^{5}{\\ln({x_i-a_i})}$ Where $(a_1, \\cdots, a_5)=(2,8,3,4,8)$ The budget constraint gives the set of possible consumption choices with a given income. If you have an income of \\$658 and the price of good $x_i$ is given by $p_i$. The equation for the budget line is given by: $ 658=\\sum_{i=1}^{5}{p_i x_i}$. A utility maximizing combination of goods $x_1 \\cdots x_5$ occurs when the surface given by the budget constraint is tangent to an indifference surface. Find $x_1$ as a function of $p_1 \\cdots p_5$ $x_1=$ [ANS]\n(Use p1 for $p_1$ and likewise for $p_2,p_3,p_4,p_5$. The easiest way to solve this question is using Lagrange multiplier. We define the Lagrange function to be: $ \\Lambda(x_1,\\cdots,x_5,\\lambda)=U(x_1,\\cdots,x_5)-\\lambda\\left(\\sum_{i=1}^{5}{p_i x_i}-658\\right)$ Utility is maximized when all of the partial derivatives of the Lagrange function are equal to $0$.", "answer_v2": [ "(658-8*p2-3*p3-4*p4-8*p5)/p1" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Suppose that you have five consumption choices: good $x_1 \\cdots x_5$. An indifference surface is the set of consumption choices with a CONSTANT utility. For example if $(x_1, \\cdots,x_5)=(2,1,1,1,1)$ gives the same utility as $(x_1, \\cdots, x_5)=(1,1,1,1,2)$ than these are both points on the same indifference surface. An indifference map is the set of all indifference surface for EVERY given utility. Consider the following utility map: $ U=\\sum_{i=1}^{5}{\\ln({x_i-a_i})}$ Where $(a_1, \\cdots, a_5)=(4,6,3,5,3)$ The budget constraint gives the set of possible consumption choices with a given income. If you have an income of \\$672 and the price of good $x_i$ is given by $p_i$. The equation for the budget line is given by: $ 672=\\sum_{i=1}^{5}{p_i x_i}$. A utility maximizing combination of goods $x_1 \\cdots x_5$ occurs when the surface given by the budget constraint is tangent to an indifference surface. Find $x_1$ as a function of $p_1 \\cdots p_5$ $x_1=$ [ANS]\n(Use p1 for $p_1$ and likewise for $p_2,p_3,p_4,p_5$. The easiest way to solve this question is using Lagrange multiplier. We define the Lagrange function to be: $ \\Lambda(x_1,\\cdots,x_5,\\lambda)=U(x_1,\\cdots,x_5)-\\lambda\\left(\\sum_{i=1}^{5}{p_i x_i}-672\\right)$ Utility is maximized when all of the partial derivatives of the Lagrange function are equal to $0$.", "answer_v3": [ "(672-6*p2-3*p3-5*p4-3*p5)/p1" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0346", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "3", "keywords": [ "double integral", "iterated integral" ], "problem_v1": "Consider the solid that lies above the rectangle (in the xy-plane) $R=[-2, 2] \\times [0, 2]$, and below the surface $z=x^2-7 y+14$. (A) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the largest possible Riemann sum. Riemann sum=[ANS]\n(B) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the smallest possible Riemann sum. Riemann sum=[ANS]\n(C) Using iterated integrals, compute the exact value of the volume. Volume=[ANS]", "answer_v1": [ "116", "28", "200/3" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Consider the solid that lies above the rectangle (in the xy-plane) $R=[-2, 2] \\times [0, 2]$, and below the surface $z=x^2-1 y+2$. (A) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the largest possible Riemann sum. Riemann sum=[ANS]\n(B) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the smallest possible Riemann sum. Riemann sum=[ANS]\n(C) Using iterated integrals, compute the exact value of the volume. Volume=[ANS]", "answer_v2": [ "44", "4", "56/3" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Consider the solid that lies above the rectangle (in the xy-plane) $R=[-2, 2] \\times [0, 2]$, and below the surface $z=x^2-3 y+6$. (A) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the largest possible Riemann sum. Riemann sum=[ANS]\n(B) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the smallest possible Riemann sum. Riemann sum=[ANS]\n(C) Using iterated integrals, compute the exact value of the volume. Volume=[ANS]", "answer_v3": [ "68", "12", "104/3" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0347", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Evaluate the double integral of the function over the rectangle:\n\\int\\!\\!\\int _{\\mathcal{R}} \\frac{y}{x+1} \\,dA,\\quad\\mathcal{R}=[0,23]\\times[0,18] Answer: [ANS]", "answer_v1": [ "514.845" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the double integral of the function over the rectangle:\n\\int\\!\\!\\int _{\\mathcal{R}} \\frac{y}{x+1} \\,dA,\\quad\\mathcal{R}=[0,3]\\times[0,28] Answer: [ANS]", "answer_v2": [ "543.427" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the double integral of the function over the rectangle:\n\\int\\!\\!\\int _{\\mathcal{R}} \\frac{y}{x+1} \\,dA,\\quad\\mathcal{R}=[0,10]\\times[0,20] Answer: [ANS]", "answer_v3": [ "479.579" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0348", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Evaluate the integral:\n\\int\\!\\!\\int _{\\mathcal{R}} e^{3x+4 y}\\,dA,\\quad\\mathcal{R}=[0,8]\\times[0,6] Answer: [ANS]", "answer_v1": [ "5.84728E+19" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the integral:\n\\int\\!\\!\\int _{\\mathcal{R}} e^{2x+3 y}\\,dA,\\quad\\mathcal{R}=[0,1]\\times[0,10] Answer: [ANS]", "answer_v2": [ "1.13794E+13" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the integral:\n\\int\\!\\!\\int _{\\mathcal{R}} e^{2x+3 y}\\,dA,\\quad\\mathcal{R}=[0,4]\\times[0,7] Answer: [ANS]", "answer_v3": [ "6.55003E+11" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0349", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Let $S_{N,N}$ be the Riemann sum for $\\int_0^1\\int_0^1 8 e^{x^3-y^3}\\,dy\\,dx$ using the regular partition and the lower left-hand vertex of each subrectangle as sample points. Use a computer algebra system to calculate $S_{N,N}$ for $N=40, 65, 115$. $S_{40,40}=$ [ANS]\n$S_{65,65}=$ [ANS]\n$S_{115,115}=$ [ANS]", "answer_v1": [ "8.6157", "8.63594", "8.65016" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $S_{N,N}$ be the Riemann sum for $\\int_0^1\\int_0^1 2 e^{x^3-y^3}\\,dy\\,dx$ using the regular partition and the lower left-hand vertex of each subrectangle as sample points. Use a computer algebra system to calculate $S_{N,N}$ for $N=49, 74, 124$. $S_{49,49}=$ [ANS]\n$S_{74,74}=$ [ANS]\n$S_{124,124}=$ [ANS]", "answer_v2": [ "2.15633", "2.15998", "2.16288" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $S_{N,N}$ be the Riemann sum for $\\int_0^1\\int_0^1 4 e^{x^3-y^3}\\,dy\\,dx$ using the regular partition and the lower left-hand vertex of each subrectangle as sample points. Use a computer algebra system to calculate $S_{N,N}$ for $N=40, 65, 115$. $S_{40,40}=$ [ANS]\n$S_{65,65}=$ [ANS]\n$S_{115,115}=$ [ANS]", "answer_v3": [ "4.30785", "4.31797", "4.32508" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0350", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Let $\\mathcal{R}=[0,1] \\times [0,1]$. Estimate $\\iint_{\\mathcal{R}}8 (x+y)\\,dA$ by computing two different Riemann sums, each with at least six rectangles. $\\iint_{\\mathcal{R}}8 (x+y)\\,dA\\approx$ [ANS]", "answer_v1": [ "8.66667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $\\mathcal{R}=[0,1] \\times [0,1]$. Estimate $\\iint_{\\mathcal{R}}2 (x+y)\\,dA$ by computing two different Riemann sums, each with at least six rectangles. $\\iint_{\\mathcal{R}}2 (x+y)\\,dA\\approx$ [ANS]", "answer_v2": [ "2.16667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $\\mathcal{R}=[0,1] \\times [0,1]$. Estimate $\\iint_{\\mathcal{R}}4 (x+y)\\,dA$ by computing two different Riemann sums, each with at least six rectangles. $\\iint_{\\mathcal{R}}4 (x+y)\\,dA\\approx$ [ANS]", "answer_v3": [ "4.33333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0351", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Compute the Riemann sum $S_{4,3}$ to estimate the double integral of $f(x,y)=8xy$ over $\\mathcal{R}=[1,3] \\times [1,2.5]$. Use the regular partition and upper-right vertices of the subrectangles as sample points. $S_{4,3}=$ [ANS]", "answer_v1": [ "108" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the Riemann sum $S_{4,3}$ to estimate the double integral of $f(x,y)=2xy$ over $\\mathcal{R}=[1,3] \\times [1,2.5]$. Use the regular partition and upper-right vertices of the subrectangles as sample points. $S_{4,3}=$ [ANS]", "answer_v2": [ "27" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the Riemann sum $S_{4,3}$ to estimate the double integral of $f(x,y)=4xy$ over $\\mathcal{R}=[1,3] \\times [1,2.5]$. Use the regular partition and upper-right vertices of the subrectangles as sample points. $S_{4,3}=$ [ANS]", "answer_v3": [ "54" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0352", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Evaluate the iterated integral:\n\\int_{0}^{7} \\int_{6}^{7} \\sqrt{x+4 y} \\,dx \\,dy Answer: [ANS]", "answer_v1": [ "31.0105" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the iterated integral:\n\\int_{0}^{1} \\int_{3}^{4} \\sqrt{x+4 y} \\,dx \\,dy Answer: [ANS]", "answer_v2": [ "2.33107" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the iterated integral:\n\\int_{0}^{3} \\int_{2}^{5} \\sqrt{x+4 y} \\,dx \\,dy Answer: [ANS]", "answer_v3": [ "27.1979" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0353", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "integral' 'double' 'multivariable", "double integral' 'iterated integral" ], "problem_v1": "If $ \\int_{3}^{6} f(x) \\: dx=-2$ and $ \\int_{1}^{4} g(x) \\: dx=-2$, what is the value of $ \\int\\!\\!\\int_{D} f(x)\\!g(y) \\: dA$ where $D$ is the rectangle: $3 \\leq x \\leq 6, \\ \\ 1 \\leq y \\leq 4$? [ANS]", "answer_v1": [ "4" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $ \\int_{-5}^{-1} f(x) \\: dx=5$ and $ \\int_{-4}^{-2} g(x) \\: dx=-2$, what is the value of $ \\int\\!\\!\\int_{D} f(x)\\!g(y) \\: dA$ where $D$ is the rectangle: $-5 \\leq x \\leq-1, \\ \\-4 \\leq y \\leq-2$? [ANS]", "answer_v2": [ "-10" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $ \\int_{-2}^{1} f(x) \\: dx=-3$ and $ \\int_{-2}^{1} g(x) \\: dx=-2$, what is the value of $ \\int\\!\\!\\int_{D} f(x)\\!g(y) \\: dA$ where $D$ is the rectangle: $-2 \\leq x \\leq 1, \\ \\-2 \\leq y \\leq 1$? [ANS]", "answer_v3": [ "6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0354", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "3", "keywords": [ "integral' 'iterated' 'multivariable" ], "problem_v1": "$\\int_{0}^{ \\frac{2\\pi}{2} }\\int_{0}^{ \\frac{3\\pi}{2} } \\left(x\\sin\\!\\left(y\\right)-y\\cos\\!\\left(x\\right)\\right) dxdy=$ [ANS]", "answer_v1": [ "pi^2/8*(3^2*[1-cos(2*pi/2)]-2^2*sin(3*pi/2))" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$\\int_{0}^{ \\frac{3\\pi}{2} }\\int_{0}^{ \\frac{1\\pi}{2} } \\left(x\\sin\\!\\left(y\\right)-y\\cos\\!\\left(x\\right)\\right) dxdy=$ [ANS]", "answer_v2": [ "pi^2/8*(1^2*[1-cos(3*pi/2)]-3^2*sin(1*pi/2))" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$\\int_{0}^{ \\frac{2\\pi}{2} }\\int_{0}^{ \\frac{1\\pi}{2} } \\left(x\\sin\\!\\left(y\\right)-y\\cos\\!\\left(x\\right)\\right) dxdy=$ [ANS]", "answer_v3": [ "pi^2/8*(1^2*[1-cos(2*pi/2)]-2^2*sin(1*pi/2))" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0355", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "3", "keywords": [ "integral' 'iterated' 'multivariable" ], "problem_v1": "To compute $\\int_{7}^{12}\\int_{8}^{12} \\left( \\frac{4xy}{\\sqrt{x^{2} +y^{2}+4}} \\right) dxdy$ you do the u substitution: u=[ANS]\ndu=[ANS]\n$\\int_{7}^{12}\\int_{8}^{12} \\left( \\frac{4xy}{\\sqrt{x^{2} +y^{2}+4}} \\right) dxdy=$ [ANS]", "answer_v1": [ "x^2+y^2+4", "2*x*dx", "1/3*4*(8^2+7^2+4)^(3/2)-1/3*4*(12^2+7^2+4)^(3/2)-1/3*4*(8^2+12^2+4)^(3/2)+1/3*4*(12^2+12^2+4)^(3/2)" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "To compute $\\int_{2}^{5}\\int_{1}^{7} \\left( \\frac{10xy}{\\sqrt{x^{2} +y^{2}+4}} \\right) dxdy$ you do the u substitution: u=[ANS]\ndu=[ANS]\n$\\int_{2}^{5}\\int_{1}^{7} \\left( \\frac{10xy}{\\sqrt{x^{2} +y^{2}+4}} \\right) dxdy=$ [ANS]", "answer_v2": [ "x^2+y^2+4", "2*x*dx", "1/3*10*(1^2+2^2+4)^(3/2)-1/3*10*(7^2+2^2+4)^(3/2)-1/3*10*(1^2+5^2+4)^(3/2)+1/3*10*(7^2+5^2+4)^(3/2)" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "To compute $\\int_{3}^{7}\\int_{4}^{8} \\left( \\frac{3xy}{\\sqrt{x^{2} +y^{2}+4}} \\right) dxdy$ you do the u substitution: u=[ANS]\ndu=[ANS]\n$\\int_{3}^{7}\\int_{4}^{8} \\left( \\frac{3xy}{\\sqrt{x^{2} +y^{2}+4}} \\right) dxdy=$ [ANS]", "answer_v3": [ "x^2+y^2+4", "2*x*dx", "1/3*3*(4^2+3^2+4)^(3/2)-1/3*3*(8^2+3^2+4)^(3/2)-1/3*3*(4^2+7^2+4)^(3/2)+1/3*3*(8^2+7^2+4)^(3/2)" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0356", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "3", "keywords": [ "integral' 'iterated' 'multivariable" ], "problem_v1": "$\\int_{4}^{8}\\int_{4}^{9} \\frac{x}{\\left(xy+4\\right)^{5} } dydx=$ [ANS]\nNow try the integral $\\int_{4}^{9}\\int_{4}^{8} \\frac{x}{\\left(xy+4\\right)^{5} } dxdy$ to see why the order of integration is important.", "answer_v1": [ "2.03405296504821E-06" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$\\int_{1}^{7}\\int_{1}^{4} \\frac{x}{\\left(xy+10\\right)^{5} } dydx=$ [ANS]\nNow try the integral $\\int_{1}^{4}\\int_{1}^{7} \\frac{x}{\\left(xy+10\\right)^{5} } dxdy$ to see why the order of integration is important.", "answer_v2": [ "3.84351135544719E-05" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$\\int_{2}^{6}\\int_{2}^{6} \\frac{x}{\\left(xy+3\\right)^{5} } dydx=$ [ANS]\nNow try the integral $\\int_{2}^{6}\\int_{2}^{6} \\frac{x}{\\left(xy+3\\right)^{5} } dxdy$ to see why the order of integration is important.", "answer_v3": [ "0.000105250395902357" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0357", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "3", "keywords": [ "integral' 'iterated' 'multivariable" ], "problem_v1": "$\\int_{0}^{\\ln (8)}\\int_{0}^{\\ln(7)} e^{3x+5y} dydx=$ [ANS]", "answer_v1": [ "(8^3-1)*(7^5-1)/(3*5)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$\\int_{0}^{\\ln (2)}\\int_{0}^{\\ln(10)} e^{-\\left(7x+3y\\right)} dydx=$ [ANS]", "answer_v2": [ "[2^{-7}-1]*[10^{-3}-1]/(-7*-3)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$\\int_{0}^{\\ln (4)}\\int_{0}^{\\ln(7)} e^{y-5x} dydx=$ [ANS]", "answer_v3": [ "[4^{-5}-1]*(7^1-1)/(-5*1)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0358", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "double integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Using Riemann sums with four subdivisions in each direction, find upper and lower bounds for the volume under the graph of $f(x,y)=5+3xy$ above the rectangle $R$ with $0\\le x\\le 4,\\quad 0\\le y \\le 5$. upper bound=[ANS]\nlower bound=[ANS]", "answer_v1": [ "1*1.25*455", "1*1.25*215" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Using Riemann sums with four subdivisions in each direction, find upper and lower bounds for the volume under the graph of $f(x,y)=2+2xy$ above the rectangle $R$ with $0\\le x\\le 1,\\quad 0\\le y \\le 6$. upper bound=[ANS]\nlower bound=[ANS]", "answer_v2": [ "0.25*1.5*107", "0.25*1.5*59" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Using Riemann sums with four subdivisions in each direction, find upper and lower bounds for the volume under the graph of $f(x,y)=3+2xy$ above the rectangle $R$ with $0\\le x\\le 2,\\quad 0\\le y \\le 5$. upper bound=[ANS]\nlower bound=[ANS]", "answer_v3": [ "0.5*1.25*173", "0.5*1.25*93" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0359", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "double integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Values of $f(x,y)$ are shown in the table below.\n$\\begin{array}{cccc}\\hline & x=3 & x=3.2 & x=3.4 \\\\ \\hline y=5 & 6 & 7 & 10 \\\\ \\hline y=5.4 & 7 & 8 & 13 \\\\ \\hline y=5.8 & 8 & 9 & 4 \\\\ \\hline \\end{array}$ Let $R$ be the rectangle $3 \\leq x \\leq 3.4$, $5 \\leq y \\leq 5.8$. Find the values of Riemann sums which are reasonable over-and under-estimates for $\\int_R f(x,y) \\,dA$ with $\\Delta x=0.2$ and $\\Delta y=0.4$. over-estimate: [ANS]\nunder-estimate: [ANS]", "answer_v1": [ "43*0.2*0.4", "24*0.2*0.4" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Values of $f(x,y)$ are shown in the table below.\n$\\begin{array}{cccc}\\hline & x=1 & x=1.1 & x=1.2 \\\\ \\hline y=3 & 10 & 11 & 13 \\\\ \\hline y=3.2 & 9 & 10 & 11 \\\\ \\hline y=3.4 & 8 & 9 & 17 \\\\ \\hline \\end{array}$ Let $R$ be the rectangle $1 \\leq x \\leq 1.2$, $3 \\leq y \\leq 3.4$. Find the values of Riemann sums which are reasonable over-and under-estimates for $\\int_R f(x,y) \\,dA$ with $\\Delta x=0.1$ and $\\Delta y=0.2$. over-estimate: [ANS]\nunder-estimate: [ANS]", "answer_v2": [ "51*0.1*0.2", "36*0.1*0.2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Values of $f(x,y)$ are shown in the table below.\n$\\begin{array}{cccc}\\hline & x=1 & x=1.1 & x=1.2 \\\\ \\hline y=3 & 6 & 7 & 10 \\\\ \\hline y=3.3 & 7 & 8 & 13 \\\\ \\hline y=3.6 & 8 & 9 & 4 \\\\ \\hline \\end{array}$ Let $R$ be the rectangle $1 \\leq x \\leq 1.2$, $3 \\leq y \\leq 3.6$. Find the values of Riemann sums which are reasonable over-and under-estimates for $\\int_R f(x,y) \\,dA$ with $\\Delta x=0.1$ and $\\Delta y=0.3$. over-estimate: [ANS]\nunder-estimate: [ANS]", "answer_v3": [ "43*0.1*0.3", "24*0.1*0.3" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0360", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "double integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Let $R$ be the rectangle with vertices $(0,0)$, $(8,0)$, $(8,8)$, and $(0,8)$ and let $f(x,y)=\\sqrt{2xy}$.\n(a) Find reasonable upper and lower bounds for $\\int_{R}f\\,dA$ without subdividing $R$. upper bound=[ANS]\nlower bound=[ANS]\n(b) Estimate $\\int_{R}f\\,dA$ three ways: by partitioning $R$ into four subrectangles and evaluating $f$ at its maximum and minimum values on each subrectangle, and then by considering the average of these (over and under) estimates. overestimate: $\\int_{R}f\\,dA \\approx$ [ANS]\nunderestimate: $\\int_{R}f\\,dA \\approx$ [ANS]\naverage: $\\int_{R}f\\,dA \\approx$ [ANS]", "answer_v1": [ "64*sqrt(2*8*8)", "0", "64/4*[sqrt(2*64/4)+2*sqrt(2*64/2)+sqrt(2*64)]", "64/4*sqrt(2*64/4)", "64/8*[2*sqrt(2*64/4)+2*sqrt(2*64/2)+sqrt(2*64)]" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Let $R$ be the rectangle with vertices $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$ and let $f(x,y)=\\sqrt{4xy}$.\n(a) Find reasonable upper and lower bounds for $\\int_{R}f\\,dA$ without subdividing $R$. upper bound=[ANS]\nlower bound=[ANS]\n(b) Estimate $\\int_{R}f\\,dA$ three ways: by partitioning $R$ into four subrectangles and evaluating $f$ at its maximum and minimum values on each subrectangle, and then by considering the average of these (over and under) estimates. overestimate: $\\int_{R}f\\,dA \\approx$ [ANS]\nunderestimate: $\\int_{R}f\\,dA \\approx$ [ANS]\naverage: $\\int_{R}f\\,dA \\approx$ [ANS]", "answer_v2": [ "4*sqrt(4*2*2)", "0", "4/4*[sqrt(4*4/4)+2*sqrt(4*4/2)+sqrt(4*4)]", "4/4*sqrt(4*4/4)", "4/8*[2*sqrt(4*4/4)+2*sqrt(4*4/2)+sqrt(4*4)]" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Let $R$ be the rectangle with vertices $(0,0)$, $(4,0)$, $(4,4)$, and $(0,4)$ and let $f(x,y)=\\sqrt{2xy}$.\n(a) Find reasonable upper and lower bounds for $\\int_{R}f\\,dA$ without subdividing $R$. upper bound=[ANS]\nlower bound=[ANS]\n(b) Estimate $\\int_{R}f\\,dA$ three ways: by partitioning $R$ into four subrectangles and evaluating $f$ at its maximum and minimum values on each subrectangle, and then by considering the average of these (over and under) estimates. overestimate: $\\int_{R}f\\,dA \\approx$ [ANS]\nunderestimate: $\\int_{R}f\\,dA \\approx$ [ANS]\naverage: $\\int_{R}f\\,dA \\approx$ [ANS]", "answer_v3": [ "16*sqrt(2*4*4)", "0", "16/4*[sqrt(2*16/4)+2*sqrt(2*16/2)+sqrt(2*16)]", "16/4*sqrt(2*16/4)", "16/8*[2*sqrt(2*16/4)+2*sqrt(2*16/2)+sqrt(2*16)]" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0361", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "4", "keywords": [ "double integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "The table below gives values of $f(x,y)$, the number of milligrams of mosquito larvae per square meter in a swamp.\n$\\begin{array}{cccc}\\hline & x=0 & x=5 & x=10 \\\\ \\hline y=0 & 1 & 3 & 6 \\\\ \\hline y=4 & 2 & 5 & 10 \\\\ \\hline y=8 & 4 & 9 & 15 \\\\ \\hline \\end{array}$\nIf $x$ and $y$ are in meters and $R$ is the rectangle $0 \\leq x \\leq 10$, $0 \\leq y \\leq 8$, estimate $\\int _R {f(x,y)dA}$. $\\int _R {f(x,y)dA} \\approx$ [ANS] mg", "answer_v1": [ "500" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The table below gives values of $f(x,y)$, the number of milligrams of mosquito larvae per square meter in a swamp.\n$\\begin{array}{cccc}\\hline & x=0 & x=2 & x=4 \\\\ \\hline y=0 & 1 & 2 & 3 \\\\ \\hline y=6 & 2 & 4 & 6 \\\\ \\hline y=12 & 5 & 8 & 12 \\\\ \\hline \\end{array}$\nIf $x$ and $y$ are in meters and $R$ is the rectangle $0 \\leq x \\leq 4$, $0 \\leq y \\leq 12$, estimate $\\int _R {f(x,y)dA}$. $\\int _R {f(x,y)dA} \\approx$ [ANS] mg", "answer_v2": [ "234" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The table below gives values of $f(x,y)$, the number of milligrams of mosquito larvae per square meter in a swamp.\n$\\begin{array}{cccc}\\hline & x=0 & x=3 & x=6 \\\\ \\hline y=0 & 1 & 2 & 4 \\\\ \\hline y=5 & 2 & 4 & 8 \\\\ \\hline y=10 & 4 & 8 & 14 \\\\ \\hline \\end{array}$\nIf $x$ and $y$ are in meters and $R$ is the rectangle $0 \\leq x \\leq 6$, $0 \\leq y \\leq 10$, estimate $\\int _R {f(x,y)dA}$. $\\int _R {f(x,y)dA} \\approx$ [ANS] mg", "answer_v3": [ "322.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0362", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "double integral' 'iterated integral" ], "problem_v1": "Find $\\iint_R f(x,y)\\, dA$ where $f(x,y)=x$ and $R=[4, 8] \\times [1, 6]$. $\\iint_R f(x,y)\\, dA=$ [ANS]", "answer_v1": [ "120" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $\\iint_R f(x,y)\\, dA$ where $f(x,y)=x$ and $R=[0, 6] \\times [-4,-1]$. $\\iint_R f(x,y)\\, dA=$ [ANS]", "answer_v2": [ "54" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $\\iint_R f(x,y)\\, dA$ where $f(x,y)=x$ and $R=[1, 5] \\times [-2, 2]$. $\\iint_R f(x,y)\\, dA=$ [ANS]", "answer_v3": [ "48" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0363", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "double integral' 'iterated integral", "Multiple Integral", "integral' 'double' 'multivariable", "Integral", "Double Integral", "calculus" ], "problem_v1": "Calculate the double integral $\\int \\int_{\\mathbf{R}} (8x+6y+48)\\: dA$ where $\\mathbf{R}$ is the region: $0 \\leq x \\leq 3, 0 \\leq y \\leq 4$. [ANS]", "answer_v1": [ "864" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the double integral $\\int \\int_{\\mathbf{R}} (2x+10y+20)\\: dA$ where $\\mathbf{R}$ is the region: $0 \\leq x \\leq 5, 0 \\leq y \\leq 1$. [ANS]", "answer_v2": [ "150" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the double integral $\\int \\int_{\\mathbf{R}} (4x+8y+32)\\: dA$ where $\\mathbf{R}$ is the region: $0 \\leq x \\leq 4, 0 \\leq y \\leq 2$. [ANS]", "answer_v3": [ "384" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0364", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "Double Integral" ], "problem_v1": "Evaluate the following integral.\n$ \\int_{1}^{5} \\!\\! \\int_{0}^{5} (5x^{2}+y^{2}) \\, dx \\, dy=$ [ANS]", "answer_v1": [ "1040" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the following integral.\n$ \\int_{1}^{2} \\!\\! \\int_{0}^{3} (2x^{2}+y^{2}) \\, dx \\, dy=$ [ANS]", "answer_v2": [ "25" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the following integral.\n$ \\int_{1}^{3} \\!\\! \\int_{0}^{4} (3x^{2}+y^{2}) \\, dx \\, dy=$ [ANS]", "answer_v3": [ "162.667" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0365", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over rectangles", "level": "2", "keywords": [ "Double Integral" ], "problem_v1": "Suppose $f(x,y)=16-x^{2}-y^{2}$ and $R$ is the rectangle with vertices (0,0), (6,0), (6,4), (0,4). In each part, estimate $ \\iint\\limits_R f(x,y) \\, dA$ using Riemann sums. For underestimates or overestimates, consistently use either the lower left-hand corner or the upper right-hand corner of each rectangle in a subdivision, as appropriate.\n(a) Without subdividing $R$, Underestimate=[ANS]\nOverestimate=[ANS]\n(b) By partitioning $R$ into four equal-sized rectangles. Underestimate=[ANS]\nOverestimate=[ANS]", "answer_v1": [ "-864", "384", "-396", "228" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Suppose $f(x,y)=1-x^{2}-y^{2}$ and $R$ is the rectangle with vertices (0,0), (6,0), (6,4), (0,4). In each part, estimate $ \\iint\\limits_R f(x,y) \\, dA$ using Riemann sums. For underestimates or overestimates, consistently use either the lower left-hand corner or the upper right-hand corner of each rectangle in a subdivision, as appropriate.\n(a) Without subdividing $R$, Underestimate=[ANS]\nOverestimate=[ANS]\n(b) By partitioning $R$ into four equal-sized rectangles. Underestimate=[ANS]\nOverestimate=[ANS]", "answer_v2": [ "-1224", "24", "-756", "-132" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Suppose $f(x,y)=4-x^{2}-y^{2}$ and $R$ is the rectangle with vertices (0,0), (6,0), (6,4), (0,4). In each part, estimate $ \\iint\\limits_R f(x,y) \\, dA$ using Riemann sums. For underestimates or overestimates, consistently use either the lower left-hand corner or the upper right-hand corner of each rectangle in a subdivision, as appropriate.\n(a) Without subdividing $R$, Underestimate=[ANS]\nOverestimate=[ANS]\n(b) By partitioning $R$ into four equal-sized rectangles. Underestimate=[ANS]\nOverestimate=[ANS]", "answer_v3": [ "-1152", "96", "-684", "-60" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0366", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Iterated integrals and Fubini's theorem", "level": "2", "keywords": [ "iterated integral", "double integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Evaluate the integral \\int_0^{2}\\int_y^{2} \\cos(x^2)\\ dx\\,dy by reversing the order of integration. With order reversed, \\int_a^b\\int_c^d \\cos(x^2)\\ dy\\,dx, where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], and $d=$ [ANS]. Evaluating the integral, $\\int_0^{2}\\int_y^{2} \\cos(x^2)\\ dx\\,dy=$ [ANS]", "answer_v1": [ "0", "2", "0", "x", "1/2*sin(2*2)" ], "answer_type_v1": [ "NV", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Evaluate the integral \\int_0^{3}\\int_y^{3} \\sin(x^2)\\ dx\\,dy by reversing the order of integration. With order reversed, \\int_a^b\\int_c^d \\sin(x^2)\\ dy\\,dx, where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], and $d=$ [ANS]. Evaluating the integral, $\\int_0^{3}\\int_y^{3} \\sin(x^2)\\ dx\\,dy=$ [ANS]", "answer_v2": [ "0", "3", "0", "x", "1/2*[1-cos(3*3)]" ], "answer_type_v2": [ "NV", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Evaluate the integral \\int_0^{2}\\int_y^{2} \\sin(x^2)\\ dx\\,dy by reversing the order of integration. With order reversed, \\int_a^b\\int_c^d \\sin(x^2)\\ dy\\,dx, where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], and $d=$ [ANS]. Evaluating the integral, $\\int_0^{2}\\int_y^{2} \\sin(x^2)\\ dx\\,dy=$ [ANS]", "answer_v3": [ "0", "2", "0", "x", "1/2*[1-cos(2*2)]" ], "answer_type_v3": [ "NV", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0367", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Iterated integrals and Fubini's theorem", "level": "3", "keywords": [ "calculus", "iterated integral" ], "problem_v1": "Consider the integral $ \\int_0^7 \\int_0^{\\sqrt{49-y}} f(x,y) dx dy$. If we change the order of integration we obtain the sum of two integrals: $ \\int_a^b \\int_{g_1(x)}^{g_2(x)} f(x,y) dy dx+ \\int_c^d \\int_{g_3(x)}^{g_4(x)} f(x,y) dy dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]\n$c=$ [ANS] $d=$ [ANS]\n$g_3(x)=$ [ANS] $g_4(x)=$ [ANS]", "answer_v1": [ "0", "6.48074069840786", "0", "7", "6.48074069840786", "7", "0", "49-x^2" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Consider the integral $ \\int_0^2 \\int_0^{\\sqrt{4-y}} f(x,y) dx dy$. If we change the order of integration we obtain the sum of two integrals: $ \\int_a^b \\int_{g_1(x)}^{g_2(x)} f(x,y) dy dx+ \\int_c^d \\int_{g_3(x)}^{g_4(x)} f(x,y) dy dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]\n$c=$ [ANS] $d=$ [ANS]\n$g_3(x)=$ [ANS] $g_4(x)=$ [ANS]", "answer_v2": [ "0", "1.4142135623731", "0", "2", "1.4142135623731", "2", "0", "4-x^2" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Consider the integral $ \\int_0^4 \\int_0^{\\sqrt{16-y}} f(x,y) dx dy$. If we change the order of integration we obtain the sum of two integrals: $ \\int_a^b \\int_{g_1(x)}^{g_2(x)} f(x,y) dy dx+ \\int_c^d \\int_{g_3(x)}^{g_4(x)} f(x,y) dy dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]\n$c=$ [ANS] $d=$ [ANS]\n$g_3(x)=$ [ANS] $g_4(x)=$ [ANS]", "answer_v3": [ "0", "3.46410161513775", "0", "4", "3.46410161513775", "4", "0", "16-x^2" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0368", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Iterated integrals and Fubini's theorem", "level": "3", "keywords": [ "calculus", "iterated integral" ], "problem_v1": "Consider the integral $ \\int_1^{13} \\int_0^{6 \\ln{x}} f(x,y) dy dx$. Sketch the region of integration and change the order of integration. $ \\int_a^b \\int_{g_1(y)}^{g_2(y)} f(x,y) dx dy$ $a=$ [ANS] $b=$ [ANS]\n$g_1(y)=$ [ANS] $g_2(y)=$ [ANS]", "answer_v1": [ "0", "15.3896961447692", "exp(y/6)", "13" ], "answer_type_v1": [ "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the integral $ \\int_1^{3} \\int_0^{8 \\ln{x}} f(x,y) dy dx$. Sketch the region of integration and change the order of integration. $ \\int_a^b \\int_{g_1(y)}^{g_2(y)} f(x,y) dx dy$ $a=$ [ANS] $b=$ [ANS]\n$g_1(y)=$ [ANS] $g_2(y)=$ [ANS]", "answer_v2": [ "0", "8.78889830934488", "exp(y/8)", "3" ], "answer_type_v2": [ "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the integral $ \\int_1^{6} \\int_0^{6 \\ln{x}} f(x,y) dy dx$. Sketch the region of integration and change the order of integration. $ \\int_a^b \\int_{g_1(y)}^{g_2(y)} f(x,y) dx dy$ $a=$ [ANS] $b=$ [ANS]\n$g_1(y)=$ [ANS] $g_2(y)=$ [ANS]", "answer_v3": [ "0", "10.7505568153683", "exp(y/6)", "6" ], "answer_type_v3": [ "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0369", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Iterated integrals and Fubini's theorem", "level": "3", "keywords": [ "calculus", "iterated integral" ], "problem_v1": "In evaluating a double integral over a region $D$, a sum of iterated integrals was obtained as follows:\n\\iint_D f(x,y)\\, dA=\\int_0^{7} \\int_0^{(3/7)y} f(x,y)\\, dx dy+\\int_{7}^{10} \\int_0^{10-y} f(x,y)\\, dx dy \\,. Sketch the region $D$ and express the double integral as an iterated integral with reversed order of integration.\n$ \\int_a^b \\int_{g_1(x)}^{g_2(x)} f(x,y)\\, dy dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]", "answer_v1": [ "0", "3", "7*x/3", "10-x" ], "answer_type_v1": [ "NV", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "In evaluating a double integral over a region $D$, a sum of iterated integrals was obtained as follows:\n\\iint_D f(x,y)\\, dA=\\int_0^{2} \\int_0^{5y} f(x,y)\\, dx dy+\\int_{2}^{12} \\int_0^{12-y} f(x,y)\\, dx dy \\,. Sketch the region $D$ and express the double integral as an iterated integral with reversed order of integration.\n$ \\int_a^b \\int_{g_1(x)}^{g_2(x)} f(x,y)\\, dy dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]", "answer_v2": [ "0", "10", "2*x/10", "12-x" ], "answer_type_v2": [ "NV", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "In evaluating a double integral over a region $D$, a sum of iterated integrals was obtained as follows:\n\\iint_D f(x,y)\\, dA=\\int_0^{4} \\int_0^{(5/4)y} f(x,y)\\, dx dy+\\int_{4}^{9} \\int_0^{9-y} f(x,y)\\, dx dy \\,. Sketch the region $D$ and express the double integral as an iterated integral with reversed order of integration.\n$ \\int_a^b \\int_{g_1(x)}^{g_2(x)} f(x,y)\\, dy dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]", "answer_v3": [ "0", "5", "4*x/5", "9-x" ], "answer_type_v3": [ "NV", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0370", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over general regions", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Calculate the integral of $f(x,y)=8x$ over the region $\\mathcal{D}$ bounded above by $y=x(2-x)$ and below by $x=y(2-y)$. Hint: Apply the quadratic formula to the lower boundary curve to solve for $y$ as a function of $x$. Answer: [ANS]", "answer_v1": [ "1.46667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the integral of $f(x,y)=2x$ over the region $\\mathcal{D}$ bounded above by $y=x(2-x)$ and below by $x=y(2-y)$. Hint: Apply the quadratic formula to the lower boundary curve to solve for $y$ as a function of $x$. Answer: [ANS]", "answer_v2": [ "0.366667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the integral of $f(x,y)=4x$ over the region $\\mathcal{D}$ bounded above by $y=x(2-x)$ and below by $x=y(2-y)$. Hint: Apply the quadratic formula to the lower boundary curve to solve for $y$ as a function of $x$. Answer: [ANS]", "answer_v3": [ "0.733333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0371", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over general regions", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Calculate the average height above the $x$-axis of a point in the region $0\\le x\\le a$, $0\\le y \\le x^2$ for $a=23$. $\\overline{H}=$ [ANS]", "answer_v1": [ "158.7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the average height above the $x$-axis of a point in the region $0\\le x\\le a$, $0\\le y \\le x^2$ for $a=4$. $\\overline{H}=$ [ANS]", "answer_v2": [ "4.8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the average height above the $x$-axis of a point in the region $0\\le x\\le a$, $0\\le y \\le x^2$ for $a=11$. $\\overline{H}=$ [ANS]", "answer_v3": [ "36.3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0372", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over general regions", "level": "2", "keywords": [ "integral' 'double' 'multivariable", "Double Integral", "Multiple Integral" ], "problem_v1": "Evaluate the double integral $ I=\\int\\!\\!\\int_{\\mathbf{D}} xy \\: d\\!A$ where $\\mathbf{D}$ is the triangular region with vertices $(0, 0), (5, 0), (0, 4)$. [ANS]", "answer_v1": [ "16.6666666666667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the double integral $ I=\\int\\!\\!\\int_{\\mathbf{D}} xy \\: d\\!A$ where $\\mathbf{D}$ is the triangular region with vertices $(0, 0), (1, 0), (0, 6)$. [ANS]", "answer_v2": [ "1.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the double integral $ I=\\int\\!\\!\\int_{\\mathbf{D}} xy \\: d\\!A$ where $\\mathbf{D}$ is the triangular region with vertices $(0, 0), (2, 0), (0, 4)$. [ANS]", "answer_v3": [ "2.66666666666667" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0373", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over general regions", "level": "2", "keywords": [ "double integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "For each of the integrals below, decide (without calculation) whether the integrals are positive, negative, or zero. Let $D$ be the region inside the unit circle centered on the origin, $L$ be the left half of $D$, $R$ be the right half of $D$.\n(a) $\\int_{L} (4 y^3+y^5)\\, dA$ is [ANS] (b) $\\int_{R} 4(x^2+y^2)\\, dA$ is [ANS] (c) $\\int_{R} (4 y^2+y^4)\\, dA$ is [ANS] (d) $\\int_{L} 4x\\, dA$ is [ANS]", "answer_v1": [ "zero", "positive", "positive", "negative" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "positive", "negative", "zero" ], [ "positive", "negative", "zero" ], [ "positive", "negative", "zero" ], [ "positive", "negative", "zero" ] ], "problem_v2": "For each of the integrals below, decide (without calculation) whether the integrals are positive, negative, or zero. Let $D$ be the region inside the unit circle centered on the origin, $L$ be the left half of $D$, $R$ be the right half of $D$.\n(a) $\\int_{L} (x^2+x^4)\\, dA$ is [ANS] (b) $\\int_{D} x\\, dA$ is [ANS] (c) $\\int_{L} x y\\, dA$ is [ANS] (d) $\\int_{R} (y^2+y^4)\\, dA$ is [ANS]", "answer_v2": [ "positive", "zero", "zero", "positive" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "positive", "negative", "zero" ], [ "positive", "negative", "zero" ], [ "positive", "negative", "zero" ], [ "positive", "negative", "zero" ] ], "problem_v3": "For each of the integrals below, decide (without calculation) whether the integrals are positive, negative, or zero. Let $D$ be the region inside the unit circle centered on the origin, $L$ be the left half of $D$, $T$ be the top half of $D$.\n(a) $\\int_{L} (2 y^3+y^5)\\, dA$ is [ANS] (b) $\\int_{T} 2 y\\, dA$ is [ANS] (c) $\\int_{T} 2(x^2+y^2)\\, dA$ is [ANS] (d) $\\int_{T} 2x\\, dA$ is [ANS]", "answer_v3": [ "zero", "positive", "positive", "zero" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "positive", "negative", "zero" ], [ "positive", "negative", "zero" ], [ "positive", "negative", "zero" ], [ "positive", "negative", "zero" ] ] }, { "id": "Calculus_-_multivariable_0374", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over general regions", "level": "2", "keywords": [ "iterated integral", "double integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Evaluate the integral $\\int_R 6xy\\,dA$, where $R$ is the triangle $7x+y \\le 7,\\ x \\ge 0,\\ y \\ge 0$. $\\int_R 6xy\\,dA=$ [ANS]", "answer_v1": [ "49*6/24" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the integral $\\int_R 8xy\\,dA$, where $R$ is the triangle $2x+y \\le 2,\\ x \\ge 0,\\ y \\ge 0$. $\\int_R 8xy\\,dA=$ [ANS]", "answer_v2": [ "4*8/24" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the integral $\\int_R 6xy\\,dA$, where $R$ is the triangle $4x+y \\le 4,\\ x \\ge 0,\\ y \\ge 0$. $\\int_R 6xy\\,dA=$ [ANS]", "answer_v3": [ "16*6/24" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0375", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over general regions", "level": "2", "keywords": [ "iterated integral", "double integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Let $f(x,y)=x^2e^{x^2}$ and let $R$ be the triangle bounded by the lines $x=5$, $x=y/3$, and $y=x$ in the $xy$-plane.\n(a) Express $\\int_R{f}\\,dA$ as a double integral in two different ways by filling in the values for the integrals below. (For one of these it will be necessary to write the double integral as a sum of two integrals, as indicated; for the other, it can be written as a single integral.) $\\int_R{f}\\,dA=\\int_a^b\\int_c^d f(x,y)\\, d$ [ANS] $d$ [ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], and $d=$ [ANS]. And $\\int_R{f}\\,dA=\\int_a^b\\int_c^d f(x,y)\\, d$ [ANS] $d$ [ANS] $+\\int_m^n\\int_p^q f(x,y)\\, d$ [ANS] $d$ [ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], $m=$ [ANS], $n=$ [ANS], $p=$ [ANS], and $q=$ [ANS]. (b) Evaluate one of your integrals to find the value of $\\int_R{f}\\,dA$. $\\int_R{f}\\,dA=$ [ANS]", "answer_v1": [ "y", "x", "0", "5", "x", "3*x", "x", "y", "x", "y", "0", "5", "y/3", "y", "5", "15", "y/3", "5", "[(5*5-1)*e^25/2+1/2]*(3-1)" ], "answer_type_v1": [ "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "NV", "EX", "EX", "NV", "NV", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $f(x,y)=x^2e^{x^2}$ and let $R$ be the triangle bounded by the lines $x=2$, $x=y/3$, and $y=x$ in the $xy$-plane.\n(a) Express $\\int_R{f}\\,dA$ as a double integral in two different ways by filling in the values for the integrals below. (For one of these it will be necessary to write the double integral as a sum of two integrals, as indicated; for the other, it can be written as a single integral.) $\\int_R{f}\\,dA=\\int_a^b\\int_c^d f(x,y)\\, d$ [ANS] $d$ [ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], and $d=$ [ANS]. And $\\int_R{f}\\,dA=\\int_a^b\\int_c^d f(x,y)\\, d$ [ANS] $d$ [ANS] $+\\int_m^n\\int_p^q f(x,y)\\, d$ [ANS] $d$ [ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], $m=$ [ANS], $n=$ [ANS], $p=$ [ANS], and $q=$ [ANS]. (b) Evaluate one of your integrals to find the value of $\\int_R{f}\\,dA$. $\\int_R{f}\\,dA=$ [ANS]", "answer_v2": [ "y", "x", "0", "2", "x", "3*x", "x", "y", "x", "y", "0", "2", "y/3", "y", "2", "6", "y/3", "2", "[(2*2-1)*e^4/2+1/2]*(3-1)" ], "answer_type_v2": [ "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "NV", "EX", "EX", "NV", "NV", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $f(x,y)=x^2e^{x^2}$ and let $R$ be the triangle bounded by the lines $x=3$, $x=y/3$, and $y=x$ in the $xy$-plane.\n(a) Express $\\int_R{f}\\,dA$ as a double integral in two different ways by filling in the values for the integrals below. (For one of these it will be necessary to write the double integral as a sum of two integrals, as indicated; for the other, it can be written as a single integral.) $\\int_R{f}\\,dA=\\int_a^b\\int_c^d f(x,y)\\, d$ [ANS] $d$ [ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], and $d=$ [ANS]. And $\\int_R{f}\\,dA=\\int_a^b\\int_c^d f(x,y)\\, d$ [ANS] $d$ [ANS] $+\\int_m^n\\int_p^q f(x,y)\\, d$ [ANS] $d$ [ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], $m=$ [ANS], $n=$ [ANS], $p=$ [ANS], and $q=$ [ANS]. (b) Evaluate one of your integrals to find the value of $\\int_R{f}\\,dA$. $\\int_R{f}\\,dA=$ [ANS]", "answer_v3": [ "y", "x", "0", "3", "x", "3*x", "x", "y", "x", "y", "0", "3", "y/3", "y", "3", "9", "y/3", "3", "[(3*3-1)*e^9/2+1/2]*(3-1)" ], "answer_type_v3": [ "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "NV", "EX", "EX", "NV", "NV", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0376", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over general regions", "level": "3", "keywords": [ "Multivariable", "Double Integral", "Surface Area", "Surface Integral" ], "problem_v1": "Evaluate the integral with respect to surface area $ \\int\\!\\int_T 18x \\, dA$, where $T$ is the part of the plane $x+y+5 z=8$ in the first octant.\n$ \\int\\!\\int_T 18x \\, dA=$ [ANS]", "answer_v1": [ "3*sqrt(27)*8^3/5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the integral with respect to surface area $ \\int\\!\\int_T 12x \\, dA$, where $T$ is the part of the plane $x+y+2 z=9$ in the first octant.\n$ \\int\\!\\int_T 12x \\, dA=$ [ANS]", "answer_v2": [ "2*sqrt(6)*9^3/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the integral with respect to surface area $ \\int\\!\\int_T 12x \\, dA$, where $T$ is the part of the plane $x+y+3 z=8$ in the first octant.\n$ \\int\\!\\int_T 12x \\, dA=$ [ANS]", "answer_v3": [ "2*sqrt(11)*8^3/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0377", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over general regions", "level": "2", "keywords": [ "Double Integral" ], "problem_v1": "Suppose $R$ is the triangle with vertices $(-1, 0), (0,1),$ and $(1,0)$.\n(a) As an iterated integral, $ \\iint\\limits_R (8x+6 y)^2 \\, dA=\\int_A^B \\!\\! \\int_C^D (8x+6 y)^2 \\, dx \\, dy$ with limits of integration\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\n(b) Evaluate the integral in part (a). Hint: substitution may make the integral easier. Integral=[ANS]", "answer_v1": [ "0", "1", "y-1", "1-y", "(8^2+6^2)/6" ], "answer_type_v1": [ "NV", "NV", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Suppose $R$ is the triangle with vertices $(-1, 0), (0,1),$ and $(1,0)$.\n(a) As an iterated integral, $ \\iint\\limits_R (2x+9 y)^2 \\, dA=\\int_A^B \\!\\! \\int_C^D (2x+9 y)^2 \\, dx \\, dy$ with limits of integration\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\n(b) Evaluate the integral in part (a). Hint: substitution may make the integral easier. Integral=[ANS]", "answer_v2": [ "0", "1", "y-1", "1-y", "(2^2+9^2)/6" ], "answer_type_v2": [ "NV", "NV", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Suppose $R$ is the triangle with vertices $(-1, 0), (0,1),$ and $(1,0)$.\n(a) As an iterated integral, $ \\iint\\limits_R (4x+6 y)^2 \\, dA=\\int_A^B \\!\\! \\int_C^D (4x+6 y)^2 \\, dx \\, dy$ with limits of integration\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\n(b) Evaluate the integral in part (a). Hint: substitution may make the integral easier. Integral=[ANS]", "answer_v3": [ "0", "1", "y-1", "1-y", "(4^2+6^2)/6" ], "answer_type_v3": [ "NV", "NV", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0379", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals over general regions", "level": "2", "keywords": [ "Double Integral" ], "problem_v1": "Decide, without calculation, if each of the integrals below are positive, negative, or zero. Let D be the region inside the unit circle centered at the origin. Let T, B, R, and L denote the regions enclosed by the top half, the bottom half, the right half, and the left half of unit circle, respectively.\n[ANS] 1. $ \\iint\\limits_L x \\cos(y) \\, dA$\n[ANS] 2. $ \\iint\\limits_B x \\cos(y) \\, dA$\n[ANS] 3. $ \\iint\\limits_R x \\cos(y) \\, dA$\n[ANS] 4. $ \\iint\\limits_T x \\cos(y) \\, dA$\n[ANS] 5. $ \\iint\\limits_D x \\cos(y) \\, dA$", "answer_v1": [ "NEGATIVE", "ZERO", "POSITIVE", "Zero", "Zero" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ] ], "problem_v2": "Decide, without calculation, if each of the integrals below are positive, negative, or zero. Let D be the region inside the unit circle centered at the origin. Let T, B, R, and L denote the regions enclosed by the top half, the bottom half, the right half, and the left half of unit circle, respectively.\n[ANS] 1. $ \\iint\\limits_T x \\cos(y) \\, dA$\n[ANS] 2. $ \\iint\\limits_B x \\cos(y) \\, dA$\n[ANS] 3. $ \\iint\\limits_D x \\cos(y) \\, dA$\n[ANS] 4. $ \\iint\\limits_L x \\cos(y) \\, dA$\n[ANS] 5. $ \\iint\\limits_R x \\cos(y) \\, dA$", "answer_v2": [ "ZERO", "ZERO", "ZERO", "Negative", "Positive" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ] ], "problem_v3": "Decide, without calculation, if each of the integrals below are positive, negative, or zero. Let D be the region inside the unit circle centered at the origin. Let T, B, R, and L denote the regions enclosed by the top half, the bottom half, the right half, and the left half of unit circle, respectively.\n[ANS] 1. $ \\iint\\limits_R x \\cos(y) \\, dA$\n[ANS] 2. $ \\iint\\limits_B x \\cos(y) \\, dA$\n[ANS] 3. $ \\iint\\limits_D x \\cos(y) \\, dA$\n[ANS] 4. $ \\iint\\limits_T x \\cos(y) \\, dA$\n[ANS] 5. $ \\iint\\limits_L x \\cos(y) \\, dA$", "answer_v3": [ "POSITIVE", "ZERO", "ZERO", "Zero", "Negative" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ], [ "Positive", "Negative", "Zero" ] ] }, { "id": "Calculus_-_multivariable_0380", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "3", "keywords": [ "polar", "integral", "Cartesian" ], "problem_v1": "Convert the integral I=\\int_0^{4/\\sqrt{2}} \\int_y^{\\sqrt{16-y^2}} e^{7x^{2}+7y^{2}} \\,dx\\,dy to polar coordinates, getting \\int_C^D \\int_A^B h(r,\\theta)\\,dr\\,d\\theta, where $h(r,\\theta)=$ [ANS]\n$A=$ [ANS]\n$B=$ [ANS]\n$C=$ [ANS]\n$D=$ [ANS]\nand then evaluate the resulting integral to get $I=$ [ANS].", "answer_v1": [ "r*e^(7*r^2)", "0", "4", "0", "pi/4", "(pi/56)*(e^(112) - 1)" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Convert the integral I=\\int_0^{6/\\sqrt{2}} \\int_y^{\\sqrt{36-y^2}} e^{x^{2}+y^{2}} \\,dx\\,dy to polar coordinates, getting \\int_C^D \\int_A^B h(r,\\theta)\\,dr\\,d\\theta, where $h(r,\\theta)=$ [ANS]\n$A=$ [ANS]\n$B=$ [ANS]\n$C=$ [ANS]\n$D=$ [ANS]\nand then evaluate the resulting integral to get $I=$ [ANS].", "answer_v2": [ "r*e^(1*r^2)", "0", "6", "0", "pi/4", "(pi/8)*(e^(36) - 1)" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Convert the integral I=\\int_0^{4/\\sqrt{2}} \\int_y^{\\sqrt{16-y^2}} e^{3x^{2}+3y^{2}} \\,dx\\,dy to polar coordinates, getting \\int_C^D \\int_A^B h(r,\\theta)\\,dr\\,d\\theta, where $h(r,\\theta)=$ [ANS]\n$A=$ [ANS]\n$B=$ [ANS]\n$C=$ [ANS]\n$D=$ [ANS]\nand then evaluate the resulting integral to get $I=$ [ANS].", "answer_v3": [ "r*e^(3*r^2)", "0", "4", "0", "pi/4", "(pi/24)*(e^(48) - 1)" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0381", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Sketch the region of integration and evaluate by changing to polar coordinates:\n\\int_{8}^{16}\\int_0^{f(x)} \\frac{1}{\\sqrt{x^2+y^2}} \\,dy\\,dx For $f(x)=\\sqrt{16x-x^2}$ Answer: [ANS]", "answer_v1": [ "4.26272" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sketch the region of integration and evaluate by changing to polar coordinates:\n\\int_{2}^{4}\\int_0^{f(x)} \\frac{1}{\\sqrt{x^2+y^2}} \\,dy\\,dx For $f(x)=\\sqrt{4x-x^2}$ Answer: [ANS]", "answer_v2": [ "1.06568" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sketch the region of integration and evaluate by changing to polar coordinates:\n\\int_{4}^{8}\\int_0^{f(x)} \\frac{1}{\\sqrt{x^2+y^2}} \\,dy\\,dx For $f(x)=\\sqrt{8x-x^2}$ Answer: [ANS]", "answer_v3": [ "2.13136" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0382", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Sketch the region of integration and evaluate by changing to polar coordinates: $\\int_0^{1/2}\\int_{\\sqrt{3}x}^{\\sqrt{1-x^2}} 24x\\,dy\\,dx=$ [ANS]", "answer_v1": [ "1.0718" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sketch the region of integration and evaluate by changing to polar coordinates: $\\int_0^{1/2}\\int_{\\sqrt{3}x}^{\\sqrt{1-x^2}} 6x\\,dy\\,dx=$ [ANS]", "answer_v2": [ "0.267949" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sketch the region of integration and evaluate by changing to polar coordinates: $\\int_0^{1/2}\\int_{\\sqrt{3}x}^{\\sqrt{1-x^2}} 12x\\,dy\\,dx=$ [ANS]", "answer_v3": [ "0.535898" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0383", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Sketch the $\\mathcal{D}$ indicated and integrate $f(x,y)$ over $\\mathcal{D}$ using polar coordinates. $f(x,y)=14xy$ ; $\\qquad x\\ge 0,\\quad y\\ge 0,\\quad x^2+y^2\\le 36$ Answer: [ANS]", "answer_v1": [ "2268" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sketch the $\\mathcal{D}$ indicated and integrate $f(x,y)$ over $\\mathcal{D}$ using polar coordinates. $f(x,y)=2xy$ ; $\\qquad x\\ge 0,\\quad y\\ge 0,\\quad x^2+y^2\\le 81$ Answer: [ANS]", "answer_v2": [ "1640.25" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sketch the $\\mathcal{D}$ indicated and integrate $f(x,y)$ over $\\mathcal{D}$ using polar coordinates. $f(x,y)=6xy$ ; $\\qquad x\\ge 0,\\quad y\\ge 0,\\quad x^2+y^2\\le 36$ Answer: [ANS]", "answer_v3": [ "972" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0384", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "2", "keywords": [ "double", "integral", "polar", "Multiple Integral", "Polar Coordinates", "calculus", "double integral", "polar coordinates" ], "problem_v1": "Using polar coordinates, evaluate the integral $ \\int \\!\\! \\int_{R} \\sin (x^2+y^2) dA$ where R is the region $16 \\leq x^2+y^2 \\leq 49$. [ANS]", "answer_v1": [ "-3.95291531517344" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Using polar coordinates, evaluate the integral $ \\int \\!\\! \\int_{R} \\sin (x^2+y^2) dA$ where R is the region $1 \\leq x^2+y^2 \\leq 81$. [ANS]", "answer_v2": [ "-0.742621220432357" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Using polar coordinates, evaluate the integral $ \\int \\!\\! \\int_{R} \\sin (x^2+y^2) dA$ where R is the region $4 \\leq x^2+y^2 \\leq 64$. [ANS]", "answer_v3": [ "-3.28453779374447" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0385", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "4", "keywords": [ "double", "integral", "polar", "integral" ], "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_-_multivariable_0386", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "5", "keywords": [ "double", "integral", "polar' 'volume", "Multiple Integral", "Polar Coordinates", "polar", "volume" ], "problem_v1": "Use the polar coordinates to find the volume of a sphere of radius 8. [ANS]", "answer_v1": [ "2144.66058485063" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the polar coordinates to find the volume of a sphere of radius 1. [ANS]", "answer_v2": [ "4.18879020478639" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the polar coordinates to find the volume of a sphere of radius 4. [ANS]", "answer_v3": [ "268.082573106329" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0387", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "3", "keywords": [ "double", "integral", "polar", "improper", "double", "polar", "Multiple Integral", "Polar Coordinates", "Improper", "calculus", "polar coordinates", "double integral" ], "problem_v1": "A. Using polar coordinates, evaluate the improper integral $ \\int \\!\\! \\int_{R^2} e^{-8 (x^2+y^2)} \\ dx \\ dy$. [ANS]\nB. Use part A to evaluate the improper integral $ \\int_{-\\infty}^{\\infty} e^{-8x^2} \\ dx$. [ANS]", "answer_v1": [ "0.392699081698724", "0.62665706865775" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "A. Using polar coordinates, evaluate the improper integral $ \\int \\!\\! \\int_{R^2} e^{-1 (x^2+y^2)} \\ dx \\ dy$. [ANS]\nB. Use part A to evaluate the improper integral $ \\int_{-\\infty}^{\\infty} e^{-1x^2} \\ dx$. [ANS]", "answer_v2": [ "3.14159265358979", "1.77245385090552" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "A. Using polar coordinates, evaluate the improper integral $ \\int \\!\\! \\int_{R^2} e^{-4 (x^2+y^2)} \\ dx \\ dy$. [ANS]\nB. Use part A to evaluate the improper integral $ \\int_{-\\infty}^{\\infty} e^{-4x^2} \\ dx$. [ANS]", "answer_v3": [ "0.785398163397448", "0.886226925452758" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0388", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "2", "keywords": [ "polar coordinates", "integral", "calculus" ], "problem_v1": "(a) Graph $r=1/(8\\cos\\theta)$ for $-\\pi/2\\le\\theta\\le\\pi/2$ and $r=1$. Then write an iterated integral in polar coordinates representing the area inside the curve $r=1$ and to the right of $r=1/(8\\cos\\theta)$. (Use $t$ for $\\theta$ in your work.) With $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], and $d=$ [ANS], area=$\\int_a^b\\int_c^d\\,$ [ANS] $d$ [ANS] $d$ [ANS]\n(b) Evaluate your integral to find the area. area=[ANS]", "answer_v1": [ "-[acos(1/8)]", "acos(1/8)", "1/[8*cos(t)]", "1", "r", "r", "t", "acos(1/8)-[tan(acos(1/8))]/(8^2)" ], "answer_type_v1": [ "NV", "NV", "EX", "NV", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "(a) Graph $r=1/(2\\cos\\theta)$ for $-\\pi/2\\le\\theta\\le\\pi/2$ and $r=1$. Then write an iterated integral in polar coordinates representing the area inside the curve $r=1$ and to the right of $r=1/(2\\cos\\theta)$. (Use $t$ for $\\theta$ in your work.) With $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], and $d=$ [ANS], area=$\\int_a^b\\int_c^d\\,$ [ANS] $d$ [ANS] $d$ [ANS]\n(b) Evaluate your integral to find the area. area=[ANS]", "answer_v2": [ "-[acos(1/2)]", "acos(1/2)", "1/[2*cos(t)]", "1", "r", "r", "t", "acos(1/2)-[tan(acos(1/2))]/(2^2)" ], "answer_type_v2": [ "NV", "NV", "EX", "NV", "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "(a) Graph $r=1/(4\\cos\\theta)$ for $-\\pi/2\\le\\theta\\le\\pi/2$ and $r=1$. Then write an iterated integral in polar coordinates representing the area inside the curve $r=1$ and to the right of $r=1/(4\\cos\\theta)$. (Use $t$ for $\\theta$ in your work.) With $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], and $d=$ [ANS], area=$\\int_a^b\\int_c^d\\,$ [ANS] $d$ [ANS] $d$ [ANS]\n(b) Evaluate your integral to find the area. area=[ANS]", "answer_v3": [ "-[acos(1/4)]", "acos(1/4)", "1/[4*cos(t)]", "1", "r", "r", "t", "acos(1/4)-[tan(acos(1/4))]/(4^2)" ], "answer_type_v3": [ "NV", "NV", "EX", "NV", "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0389", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "2", "keywords": [ "polar coordinates", "integral", "calculus" ], "problem_v1": "Convert the integral \\int_0^{\\sqrt{8}}\\int_{-x}^x\\,dy\\,dx to polar coordinates and evaluate it (use $t$ for $\\theta$): With $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS], $\\int_0^{\\sqrt{8}}\\int_{-x}^x\\,dy\\,dx=\\int_{a}^{b}\\int_{c}^{d}$ [ANS] $dr\\,dt$\n$=\\int_{a}^{b}$ [ANS] $\\,dt$ $=$ [ANS] $\\bigg|_{a}^{b}$ $=$ [ANS].", "answer_v1": [ "-pi/4", "pi/4", "0", "[sqrt(8)]/[cos(t)]", "r", "4/([cos(t)]^2)", "4*tan(t)", "8" ], "answer_type_v1": [ "NV", "NV", "NV", "EX", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Convert the integral \\int_0^{\\sqrt{2}}\\int_{-x}^x\\,dy\\,dx to polar coordinates and evaluate it (use $t$ for $\\theta$): With $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS], $\\int_0^{\\sqrt{2}}\\int_{-x}^x\\,dy\\,dx=\\int_{a}^{b}\\int_{c}^{d}$ [ANS] $dr\\,dt$\n$=\\int_{a}^{b}$ [ANS] $\\,dt$ $=$ [ANS] $\\bigg|_{a}^{b}$ $=$ [ANS].", "answer_v2": [ "-pi/4", "pi/4", "0", "[sqrt(2)]/[cos(t)]", "r", "1/([cos(t)]^2)", "1*tan(t)", "2" ], "answer_type_v2": [ "NV", "NV", "NV", "EX", "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Convert the integral \\int_0^{\\sqrt{4}}\\int_{-x}^x\\,dy\\,dx to polar coordinates and evaluate it (use $t$ for $\\theta$): With $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS], $\\int_0^{\\sqrt{4}}\\int_{-x}^x\\,dy\\,dx=\\int_{a}^{b}\\int_{c}^{d}$ [ANS] $dr\\,dt$\n$=\\int_{a}^{b}$ [ANS] $\\,dt$ $=$ [ANS] $\\bigg|_{a}^{b}$ $=$ [ANS].", "answer_v3": [ "-pi/4", "pi/4", "0", "[sqrt(4)]/[cos(t)]", "r", "2/([cos(t)]^2)", "2*tan(t)", "4" ], "answer_type_v3": [ "NV", "NV", "NV", "EX", "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0390", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "2", "keywords": [ "polar coordinates", "integral", "calculus" ], "problem_v1": "Evaluate the integral $\\int\\!\\!\\!\\int_R (x^{2}-2y^{2}) \\, dA$, where $R$ is the first quadrant region between the circles of radius 4 and radius 7 centred at the origin. $\\int\\!\\!\\!\\int_R (x^{2}-2y^{2}) \\, dA=$ [ANS]", "answer_v1": [ "-1*(7^4-4^4)*(2-1)*pi/16" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the integral $\\int\\!\\!\\!\\int_R (x^{2}-2y^{2}) \\, dA$, where $R$ is the first quadrant region between the circles of radius 1 and radius 2 centred at the origin. $\\int\\!\\!\\!\\int_R (x^{2}-2y^{2}) \\, dA=$ [ANS]", "answer_v2": [ "-1*(2^4-1^4)*(2-1)*pi/16" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the integral $\\int\\!\\!\\!\\int_R (x^{2}-2y^{2}) \\, dA$, where $R$ is the first quadrant region between the circles of radius 2 and radius 5 centred at the origin. $\\int\\!\\!\\!\\int_R (x^{2}-2y^{2}) \\, dA=$ [ANS]", "answer_v3": [ "-1*(5^4-2^4)*(2-1)*pi/16" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0391", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Double integrals in polar", "level": "3", "keywords": [ "Double Integrals", "Iterated Integrals", "Polar Coordinates" ], "problem_v1": "Sketch the region of integration for the following integral. $ \\int_{0}^{\\pi/4} \\int_{0}^{6/\\cos(\\theta)} f(r,\\theta) \\, r \\, dr \\, d\\theta$ The region of integration is bounded by [ANS] A. $y=0, y=x, \\mbox{and} x=6$ B. $y=0, x=\\sqrt{36-y^2}, \\mbox{and} y=6$ C. $y=0, y=\\sqrt{36-x^2}, \\mbox{and} x=6$ D. $y=0, y=x, \\mbox{and} y=6$ E. None of the above", "answer_v1": [ "A" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Sketch the region of integration for the following integral. $ \\int_{0}^{\\pi/4} \\int_{0}^{3/\\cos(\\theta)} f(r,\\theta) \\, r \\, dr \\, d\\theta$ The region of integration is bounded by [ANS] A. $y=0, y=\\sqrt{9-x^2}, \\mbox{and} x=3$ B. $y=0, x=\\sqrt{9-y^2}, \\mbox{and} y=3$ C. $y=0, y=x, \\mbox{and} y=3$ D. $y=0, y=x, \\mbox{and} x=3$ E. None of the above", "answer_v2": [ "D" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Sketch the region of integration for the following integral. $ \\int_{0}^{\\pi/4} \\int_{0}^{4/\\cos(\\theta)} f(r,\\theta) \\, r \\, dr \\, d\\theta$ The region of integration is bounded by [ANS] A. $y=0, y=\\sqrt{16-x^2}, \\mbox{and} x=4$ B. $y=0, y=x, \\mbox{and} x=4$ C. $y=0, x=\\sqrt{16-y^2}, \\mbox{and} y=4$ D. $y=0, y=x, \\mbox{and} y=4$ E. None of the above", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Calculus_-_multivariable_0392", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "calculus", "triple integral", "iterated" ], "problem_v1": "Find the limits of integration ly, uy, lx, ux, lz, uz (some of which will involve variables $x, y, z$) so that \\int_{\\text{lz}}^{\\text{uz}} \\int_{\\text{lx}}^{\\text{ux}} \\int_{\\text{ly}}^{\\text{uy}} \\,dy\\,dx\\,dz represents the volume of the region in the first octant that is bounded by the 3 coordinate planes and the plane x+7y+6z=42. ly=[ANS]\nuy=[ANS]\nlx=[ANS]\nux=[ANS]\nlz=[ANS]\nuz=[ANS]", "answer_v1": [ "0", "(42-x-6*z)/7", "0", "42-6*z", "0", "42/6" ], "answer_type_v1": [ "NV", "EX", "NV", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find the limits of integration ly, uy, lx, ux, lz, uz (some of which will involve variables $x, y, z$) so that \\int_{\\text{lz}}^{\\text{uz}} \\int_{\\text{lx}}^{\\text{ux}} \\int_{\\text{ly}}^{\\text{uy}} \\,dy\\,dx\\,dz represents the volume of the region in the first octant that is bounded by the 3 coordinate planes and the plane x+y+9z=9. ly=[ANS]\nuy=[ANS]\nlx=[ANS]\nux=[ANS]\nlz=[ANS]\nuz=[ANS]", "answer_v2": [ "0", "(9-x-9*z)/1", "0", "9-9*z", "0", "9/9" ], "answer_type_v2": [ "NV", "EX", "NV", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find the limits of integration ly, uy, lx, ux, lz, uz (some of which will involve variables $x, y, z$) so that \\int_{\\text{lz}}^{\\text{uz}} \\int_{\\text{lx}}^{\\text{ux}} \\int_{\\text{ly}}^{\\text{uy}} \\,dy\\,dx\\,dz represents the volume of the region in the first octant that is bounded by the 3 coordinate planes and the plane x+3y+6z=18. ly=[ANS]\nuy=[ANS]\nlx=[ANS]\nux=[ANS]\nlz=[ANS]\nuz=[ANS]", "answer_v3": [ "0", "(18-x-6*z)/3", "0", "18-6*z", "0", "18/6" ], "answer_type_v3": [ "NV", "EX", "NV", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0393", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Integrate $f(x,y,z)=16xz$ over the region in the first octant $(x,y,z\\ge 0)$ above the parabolic cylinder $z=y^2$ and below the paraboloid $z=8-2x^2-y^2$. Answer: [ANS]", "answer_v1": [ "390.095" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Integrate $f(x,y,z)=4xz$ over the region in the first octant $(x,y,z\\ge 0)$ above the parabolic cylinder $z=y^2$ and below the paraboloid $z=8-2x^2-y^2$. Answer: [ANS]", "answer_v2": [ "97.5238" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Integrate $f(x,y,z)=8xz$ over the region in the first octant $(x,y,z\\ge 0)$ above the parabolic cylinder $z=y^2$ and below the paraboloid $z=8-2x^2-y^2$. Answer: [ANS]", "answer_v3": [ "195.048" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0394", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Find the volume of the solid in $\\mathbf{R}^3$ bounded by $y=x^2$, $x=y^2$, $z=x+y+30$, and $z=0$. $\\mathrm{V}=$ [ANS]", "answer_v1": [ "10.3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the solid in $\\mathbf{R}^3$ bounded by $y=x^2$, $x=y^2$, $z=x+y+3$, and $z=0$. $\\mathrm{V}=$ [ANS]", "answer_v2": [ "1.3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the solid in $\\mathbf{R}^3$ bounded by $y=x^2$, $x=y^2$, $z=x+y+12$, and $z=0$. $\\mathrm{V}=$ [ANS]", "answer_v3": [ "4.3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0395", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Evaluate $\\int\\!\\!\\int\\!\\!\\int _{\\mathcal{W}} f(x,y,z)\\,dV$ for the function $f$ and region $\\mathcal{W}$ specified:\nf(x,y,z)=42(x+y)\\quad \\mathcal{W}: y\\le z \\le x, 0\\le y\\le x, 0\\le x\\le 1 $\\int\\!\\!\\int\\!\\!\\int _{\\mathcal{W}} (42(x+y)) \\,dV=$ [ANS]", "answer_v1": [ "7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate $\\int\\!\\!\\int\\!\\!\\int _{\\mathcal{W}} f(x,y,z)\\,dV$ for the function $f$ and region $\\mathcal{W}$ specified:\nf(x,y,z)=6(x+y)\\quad \\mathcal{W}: y\\le z \\le x, 0\\le y\\le x, 0\\le x\\le 1 $\\int\\!\\!\\int\\!\\!\\int _{\\mathcal{W}} (6(x+y)) \\,dV=$ [ANS]", "answer_v2": [ "1" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate $\\int\\!\\!\\int\\!\\!\\int _{\\mathcal{W}} f(x,y,z)\\,dV$ for the function $f$ and region $\\mathcal{W}$ specified:\nf(x,y,z)=18(x+y)\\quad \\mathcal{W}: y\\le z \\le x, 0\\le y\\le x, 0\\le x\\le 1 $\\int\\!\\!\\int\\!\\!\\int _{\\mathcal{W}} (18(x+y)) \\,dV=$ [ANS]", "answer_v3": [ "3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0396", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Consider the triple integral $\\int\\!\\!\\int\\!\\!\\int _{\\mathcal{W}} {xyz}^2 \\,dV$, where $\\mathcal{W}$ is the region bounded by z=25-{y}^2,\\quad z=0,\\quad y=6x,\\quad x=0,\\quad y \\ge 0 \\text{.} Write the triple integral as an iterated integral in the order $dz\\,dx\\,dy$, and describe the region of integration: [ANS] $\\le x \\le$ [ANS] [ANS] $\\le y \\le$ [ANS] [ANS] $\\le z \\le$ [ANS]", "answer_v1": [ "0", "y/6", "0", "5", "0", "25-y^2" ], "answer_type_v1": [ "NV", "EX", "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Consider the triple integral $\\int\\!\\!\\int\\!\\!\\int _{\\mathcal{W}} {xyz}^2 \\,dV$, where $\\mathcal{W}$ is the region bounded by z=4-{y}^2,\\quad z=0,\\quad y=9x,\\quad x=0,\\quad y \\ge 0 \\text{.} Write the triple integral as an iterated integral in the order $dz\\,dx\\,dy$, and describe the region of integration: [ANS] $\\le x \\le$ [ANS] [ANS] $\\le y \\le$ [ANS] [ANS] $\\le z \\le$ [ANS]", "answer_v2": [ "0", "y/9", "0", "2", "0", "4-y^2" ], "answer_type_v2": [ "NV", "EX", "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Consider the triple integral $\\int\\!\\!\\int\\!\\!\\int _{\\mathcal{W}} {xyz}^2 \\,dV$, where $\\mathcal{W}$ is the region bounded by z=9-{y}^2,\\quad z=0,\\quad y=6x,\\quad x=0,\\quad y \\ge 0 \\text{.} Write the triple integral as an iterated integral in the order $dz\\,dx\\,dy$, and describe the region of integration: [ANS] $\\le x \\le$ [ANS] [ANS] $\\le y \\le$ [ANS] [ANS] $\\le z \\le$ [ANS]", "answer_v3": [ "0", "y/6", "0", "3", "0", "9-y^2" ], "answer_type_v3": [ "NV", "EX", "NV", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0397", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "Multivariable", "Triple Integral", "Multiple Integral" ], "problem_v1": "Evaluate the triple integral \\int \\!\\! \\int \\!\\! \\int_{\\mathbf{E}} xyz \\, dV where E is the solid: $0 \\leq z \\leq 8, \\ \\ 0 \\leq y \\leq z, \\ \\ 0 \\leq x \\leq y$. [ANS]", "answer_v1": [ "5461.33333333333" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the triple integral \\int \\!\\! \\int \\!\\! \\int_{\\mathbf{E}} xyz \\, dV where E is the solid: $0 \\leq z \\leq 1, \\ \\ 0 \\leq y \\leq z, \\ \\ 0 \\leq x \\leq y$. [ANS]", "answer_v2": [ "0.0208333333333333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the triple integral \\int \\!\\! \\int \\!\\! \\int_{\\mathbf{E}} xyz \\, dV where E is the solid: $0 \\leq z \\leq 4, \\ \\ 0 \\leq y \\leq z, \\ \\ 0 \\leq x \\leq y$. [ANS]", "answer_v3": [ "85.3333333333333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0398", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "Multivariable", "Triple Integral", "volume", "Multiple Integral", "Volume", "triple integral' 'volume" ], "problem_v1": "Find the volume of the solid enclosed by the paraboloids $z=16 \\left(x^{2}+y^{2} \\right)$ and $z=18-16 \\left(x^{2}+y^{2} \\right)$. [ANS]", "answer_v1": [ "15.9043128087983" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the solid enclosed by the paraboloids $z=1 \\left(x^{2}+y^{2} \\right)$ and $z=32-1 \\left(x^{2}+y^{2} \\right)$. [ANS]", "answer_v2": [ "804.247719318987" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the solid enclosed by the paraboloids $z=4 \\left(x^{2}+y^{2} \\right)$ and $z=18-4 \\left(x^{2}+y^{2} \\right)$. [ANS]", "answer_v3": [ "63.6172512351933" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0399", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "4", "keywords": [ "triple integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Write a triple integral, including limits of integration, that gives the volume between $4x+3y+z=2$ and $6x+6y+z=2$ and above $x+y\\le 1, x\\ge 0, y\\ge 0$. volume=$\\int_a^b\\int_c^d\\int_e^f$ [ANS] $d$ [ANS] $d$ [ANS] $d$ [ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], $e=$ [ANS], and $f=$ [ANS]. (Note: values for all answer blanks must be supplied for this problem to be able to check the answers provided.)", "answer_v1": [ "1", "z", "y", "x", "0", "1", "0", "1-x", "2-6*x-6*y", "2-4*x-3*y" ], "answer_type_v1": [ "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Write a triple integral, including limits of integration, that gives the volume between $x+5y+z=8$ and $2x+7y+z=8$ and above $x+y\\le 1, x\\ge 0, y\\ge 0$. volume=$\\int_a^b\\int_c^d\\int_e^f$ [ANS] $d$ [ANS] $d$ [ANS] $d$ [ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], $e=$ [ANS], and $f=$ [ANS]. (Note: values for all answer blanks must be supplied for this problem to be able to check the answers provided.)", "answer_v2": [ "1", "z", "y", "x", "0", "1", "0", "1-x", "8-2*x-7*y", "8-1*x-5*y" ], "answer_type_v2": [ "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Write a triple integral, including limits of integration, that gives the volume between $2x+4y+z=1$ and $3x+6y+z=1$ and above $x+y\\le 2, x\\ge 0, y\\ge 0$. volume=$\\int_a^b\\int_c^d\\int_e^f$ [ANS] $d$ [ANS] $d$ [ANS] $d$ [ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], $e=$ [ANS], and $f=$ [ANS]. (Note: values for all answer blanks must be supplied for this problem to be able to check the answers provided.)", "answer_v3": [ "1", "z", "y", "x", "0", "2", "0", "2-x", "1-3*x-6*y", "1-2*x-4*y" ], "answer_type_v3": [ "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0400", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "triple integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Find the mass of the solid bounded by the $xy$-plane, $yz$-plane, $xz$-plane, and the plane $(x/4)+(y/3)+(z/12)=1$, if the density of the solid is given by $\\delta (x,y,z)=x+3 y$. mass=[ANS]", "answer_v1": [ "4*4*3*3*(4+3*3)/24" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the mass of the solid bounded by the $xy$-plane, $yz$-plane, $xz$-plane, and the plane $(x/2)+(y/4)+(z/8)=1$, if the density of the solid is given by $\\delta (x,y,z)=x+2 y$. mass=[ANS]", "answer_v2": [ "2*2*4*4*(2+4*2)/24" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the mass of the solid bounded by the $xy$-plane, $yz$-plane, $xz$-plane, and the plane $(x/2)+(y/3)+(z/6)=1$, if the density of the solid is given by $\\delta (x,y,z)=x+2 y$. mass=[ANS]", "answer_v3": [ "2*2*3*3*(2+3*2)/24" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0401", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "triple integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Find the volume of the region bounded by $z=x^2$, $0 \\le x \\le 7$, and the planes $y=0$, $y=5,$ and $z=0$. volume=[ANS]", "answer_v1": [ "7^3*5/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the region bounded by $z=x^2$, $0 \\le x \\le 2$, and the planes $y=0$, $y=8,$ and $z=0$. volume=[ANS]", "answer_v2": [ "2^3*8/3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the region bounded by $z=x^2$, $0 \\le x \\le 4$, and the planes $y=0$, $y=5,$ and $z=0$. volume=[ANS]", "answer_v3": [ "4^3*5/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0402", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "triple integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Find the volume of the pyramid with base in the plane $z=-6$ and sides formed by the three planes $y=0$ and $y-x=4$ and $2x+y+z=4$. volume=[ANS]", "answer_v1": [ "(4+2*4--6)^3/(18*2)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the pyramid with base in the plane $z=-10$ and sides formed by the three planes $y=0$ and $y-x=5$ and $x+2y+z=5$. volume=[ANS]", "answer_v2": [ "(5+1*5--10)^3/(18*1)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the pyramid with base in the plane $z=-9$ and sides formed by the three planes $y=0$ and $y-x=4$ and $x+2y+z=4$. volume=[ANS]", "answer_v3": [ "(4+1*4--9)^3/(18*1)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0403", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "triple integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Find the triple integral of the function $f(x,y,z)$ over the region $W$, if $f(x,y,z)=x^{2}+7y^{2}+z$ and $W$ is the rectangular box $1 \\le x \\le 2$, $1 \\le y \\le 3$, $0 \\le z \\le 2$. triple integral=[ANS]", "answer_v1": [ "1/6*(2-1)*(3-1)*(2-0)*[2*(1*1+1*2+2*2)+2*7*(1*1+1*3+3*3)+3*1*(2+0)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the triple integral of the function $f(x,y,z)$ over the region $W$, if $f(x,y,z)=x^{2}+y^{2}+z$ and $W$ is the rectangular box $-1 \\le x \\le 0$, $0 \\le y \\le 1$, $2 \\le z \\le 3$. triple integral=[ANS]", "answer_v2": [ "1/6*(0--1)*(1-0)*(3-2)*[2*(-1*-1+-1*0+0*0)+2*1*(0*0+0*1+1*1)+3*1*(3+2)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the triple integral of the function $f(x,y,z)$ over the region $W$, if $f(x,y,z)=x^{2}+3y^{2}+z$ and $W$ is the rectangular box $0 \\le x \\le 2$, $1 \\le y \\le 4$, $-1 \\le z \\le 2$. triple integral=[ANS]", "answer_v3": [ "1/6*(2-0)*(4-1)*(2--1)*[2*(0*0+0*2+2*2)+2*3*(1*1+1*4+4*4)+3*1*(2+-1)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0404", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "4", "keywords": [ "triple integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "(a) What is the equation of the plane passing through the points $(3,0,0)$, $(0,2,0),$ and $(0,0,3)$? $z=$ [ANS]\n(b) Find the volume of the region bounded by this plane and the planes $x=0$, $y=0,$ and $z=0$. volume=[ANS]", "answer_v1": [ "3-3*x/3-3*y/2", "3*2*3/6" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "(a) What is the equation of the plane passing through the points $(1,0,0)$, $(0,3,0),$ and $(0,0,1)$? $z=$ [ANS]\n(b) Find the volume of the region bounded by this plane and the planes $x=0$, $y=0,$ and $z=0$. volume=[ANS]", "answer_v2": [ "1-1*x/1-1*y/3", "1*3*1/6" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "(a) What is the equation of the plane passing through the points $(1,0,0)$, $(0,2,0),$ and $(0,0,1)$? $z=$ [ANS]\n(b) Find the volume of the region bounded by this plane and the planes $x=0$, $y=0,$ and $z=0$. volume=[ANS]", "answer_v3": [ "1-1*x/1-1*y/2", "1*2*1/6" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0405", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "triple integral" ], "problem_v1": "Evaluate $ \\iiint_B y e^{-xy} dV$ where $B$ is the box determined by $0 \\le x \\le 4$, $0 \\le y \\le 3$, and $0 \\le z \\le 4$. The value is [ANS].", "answer_v1": [ "11.0000061442124" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate $ \\iiint_B y e^{-xy} dV$ where $B$ is the box determined by $0 \\le x \\le 1$, $0 \\le y \\le 5$, and $0 \\le z \\le 1$. The value is [ANS].", "answer_v2": [ "4.00673794699909" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate $ \\iiint_B y e^{-xy} dV$ where $B$ is the box determined by $0 \\le x \\le 2$, $0 \\le y \\le 4$, and $0 \\le z \\le 2$. The value is [ANS].", "answer_v3": [ "7.0003354626279" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0406", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "Triple Integral" ], "problem_v1": "Find the triple integral of the function $f(x,y,z)=x^{3}\\cos\\!\\left(y+z\\right)$ over the cube $5 \\leq x \\leq 7$, $0 \\leq y \\leq \\pi$, $0 \\leq z \\leq \\pi.$ [ANS]", "answer_v1": [ "-1776" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the triple integral of the function $f(x,y,z)=x^{2}\\cos\\!\\left(y+z\\right)$ over the cube $2 \\leq x \\leq 5$, $0 \\leq y \\leq \\pi$, $0 \\leq z \\leq \\pi.$ [ANS]", "answer_v2": [ "-156" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the triple integral of the function $f(x,y,z)=x^{2}\\cos\\!\\left(y+z\\right)$ over the cube $3 \\leq x \\leq 5$, $0 \\leq y \\leq \\pi$, $0 \\leq z \\leq \\pi.$ [ANS]", "answer_v3": [ "-130.667" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0407", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "Triple Integrals" ], "problem_v1": "Suppose $R$ is the solid bounded by the plane $z=5x$, the surface $z=x^2$, and the planes $y=0$ and $y=4$. Write an iterated integral in the form below to find the volume of the solid $R$.\n$ \\iiint\\limits_R f(x,y,z) \\, dV=\\int_A^B \\!\\! \\int_C^D \\!\\! \\int_E^F$ [ANS] $\\, dz \\, dy \\, dx$ with limits of integration\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\nE=[ANS]\nF=[ANS]", "answer_v1": [ "1", "0", "5", "0", "4", "x^2", "5*x" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Suppose $R$ is the solid bounded by the plane $z=2x$, the surface $z=x^2$, and the planes $y=0$ and $y=5$. Write an iterated integral in the form below to find the volume of the solid $R$.\n$ \\iiint\\limits_R f(x,y,z) \\, dV=\\int_A^B \\!\\! \\int_C^D \\!\\! \\int_E^F$ [ANS] $\\, dz \\, dy \\, dx$ with limits of integration\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\nE=[ANS]\nF=[ANS]", "answer_v2": [ "1", "0", "2", "0", "5", "x^2", "2*x" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Suppose $R$ is the solid bounded by the plane $z=3x$, the surface $z=x^2$, and the planes $y=0$ and $y=4$. Write an iterated integral in the form below to find the volume of the solid $R$.\n$ \\iiint\\limits_R f(x,y,z) \\, dV=\\int_A^B \\!\\! \\int_C^D \\!\\! \\int_E^F$ [ANS] $\\, dz \\, dy \\, dx$ with limits of integration\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\nE=[ANS]\nF=[ANS]", "answer_v3": [ "1", "0", "3", "0", "4", "x^2", "3*x" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0408", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "calculus", "iterated integral", "volume" ], "problem_v1": "Use a triple integral to find the volume of the solid bounded by the parabolic cylinder $y=8x^2$ and the planes $z=0, z=6$ and $y=6$. [ANS]", "answer_v1": [ "41.569219381653" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use a triple integral to find the volume of the solid bounded by the parabolic cylinder $y=2x^2$ and the planes $z=0, z=9$ and $y=2$. [ANS]", "answer_v2": [ "24" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use a triple integral to find the volume of the solid bounded by the parabolic cylinder $y=4x^2$ and the planes $z=0, z=6$ and $y=3$. [ANS]", "answer_v3": [ "20.7846096908265" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0409", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "calculus", "iterated integral", "volume" ], "problem_v1": "Use a triple integral to find the volume of the solid enclosed by the paraboloid $x=7 y^2+7 z^2$ and the plane $x=7$. [ANS]", "answer_v1": [ "10.9955742875643" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use a triple integral to find the volume of the solid enclosed by the paraboloid $x=y^2+z^2$ and the plane $x=1$. [ANS]", "answer_v2": [ "1.5707963267949" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use a triple integral to find the volume of the solid enclosed by the paraboloid $x=3 y^2+3 z^2$ and the plane $x=3$. [ANS]", "answer_v3": [ "4.71238898038469" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0410", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "4", "keywords": [ "calculus", "iterated integral" ], "problem_v1": "Express the integral $ \\iiint_E f(x,y,z) dV$ as an iterated integral in six different ways, where E is the solid bounded by $z=0, z=5 y$ and $x^2=49-y$. 1. $ \\int_a^b \\int_{g_1(x)}^{g_2(x)} \\int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dy dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]\n$h_1(x,y)=$ [ANS] $h_2(x,y)=$ [ANS]\n2. $ \\int_a^b \\int_{g_1(y)}^{g_2(y)} \\int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dx dy$ $a=$ [ANS] $b=$ [ANS]\n$g_1(y)=$ [ANS] $g_2(y)=$ [ANS]\n$h_1(x,y)=$ [ANS] $h_2(x,y)=$ [ANS]\n3. $ \\int_a^b \\int_{g_1(z)}^{g_2(z)} \\int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dy dz$ $a=$ [ANS] $b=$ [ANS]\n$g_1(z)=$ [ANS] $g_2(z)=$ [ANS]\n$h_1(y,z)=$ [ANS] $h_2(y,z)=$ [ANS]\n4. $ \\int_a^b \\int_{g_1(y)}^{g_2(y)} \\int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dz dy$ $a=$ [ANS] $b=$ [ANS]\n$g_1(y)=$ [ANS] $g_2(y)=$ [ANS]\n$h_1(y,z)=$ [ANS] $h_2(y,z)=$ [ANS]\n5. $ \\int_a^b \\int_{g_1(x)}^{g_2(x)} \\int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dz dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]\n$h_1(x,z)=$ [ANS] $h_2(x,z)=$ [ANS]\n6. $ \\int_a^b \\int_{g_1(z)}^{g_2(z)} \\int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dx dz$ $a=$ [ANS] $b=$ [ANS]\n$g_1(z)=$ [ANS] $g_2(z)=$ [ANS]\n$h_1(x,z)=$ [ANS] $h_2(x,z)=$ [ANS]", "answer_v1": [ "-7", "7", "0", "49 - x^2", "0", "5*y", "0", "49", "-sqrt(49-y)", "sqrt(49-y)", "0", "5*y", "0", "245", "z/5", "49", "-sqrt(49-y)", "sqrt(49-y)", "0", "49", "0", "5*y", "-sqrt(49-y)", "sqrt(49-y)", "-7", "7", "0", "5*(49-x^2)", "z/5", "49 - x^2", "0", "245", "-sqrt(49-z/5)", "sqrt(49-z/5)", "z/5", "49-x^2" ], "answer_type_v1": [ "NV", "NV", "NV", "EX", "NV", "EX", "NV", "NV", "EX", "EX", "NV", "EX", "NV", "NV", "EX", "NV", "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Express the integral $ \\iiint_E f(x,y,z) dV$ as an iterated integral in six different ways, where E is the solid bounded by $z=0, z=8 y$ and $x^2=1-y$. 1. $ \\int_a^b \\int_{g_1(x)}^{g_2(x)} \\int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dy dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]\n$h_1(x,y)=$ [ANS] $h_2(x,y)=$ [ANS]\n2. $ \\int_a^b \\int_{g_1(y)}^{g_2(y)} \\int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dx dy$ $a=$ [ANS] $b=$ [ANS]\n$g_1(y)=$ [ANS] $g_2(y)=$ [ANS]\n$h_1(x,y)=$ [ANS] $h_2(x,y)=$ [ANS]\n3. $ \\int_a^b \\int_{g_1(z)}^{g_2(z)} \\int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dy dz$ $a=$ [ANS] $b=$ [ANS]\n$g_1(z)=$ [ANS] $g_2(z)=$ [ANS]\n$h_1(y,z)=$ [ANS] $h_2(y,z)=$ [ANS]\n4. $ \\int_a^b \\int_{g_1(y)}^{g_2(y)} \\int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dz dy$ $a=$ [ANS] $b=$ [ANS]\n$g_1(y)=$ [ANS] $g_2(y)=$ [ANS]\n$h_1(y,z)=$ [ANS] $h_2(y,z)=$ [ANS]\n5. $ \\int_a^b \\int_{g_1(x)}^{g_2(x)} \\int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dz dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]\n$h_1(x,z)=$ [ANS] $h_2(x,z)=$ [ANS]\n6. $ \\int_a^b \\int_{g_1(z)}^{g_2(z)} \\int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dx dz$ $a=$ [ANS] $b=$ [ANS]\n$g_1(z)=$ [ANS] $g_2(z)=$ [ANS]\n$h_1(x,z)=$ [ANS] $h_2(x,z)=$ [ANS]", "answer_v2": [ "-1", "1", "0", "1 - x^2", "0", "8*y", "0", "1", "-sqrt(1-y)", "sqrt(1-y)", "0", "8*y", "0", "8", "z/8", "1", "-sqrt(1-y)", "sqrt(1-y)", "0", "1", "0", "8*y", "-sqrt(1-y)", "sqrt(1-y)", "-1", "1", "0", "8*(1-x^2)", "z/8", "1 - x^2", "0", "8", "-sqrt(1-z/8)", "sqrt(1-z/8)", "z/8", "1-x^2" ], "answer_type_v2": [ "NV", "NV", "NV", "EX", "NV", "EX", "NV", "NV", "EX", "EX", "NV", "EX", "NV", "NV", "EX", "NV", "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Express the integral $ \\iiint_E f(x,y,z) dV$ as an iterated integral in six different ways, where E is the solid bounded by $z=0, z=5 y$ and $x^2=9-y$. 1. $ \\int_a^b \\int_{g_1(x)}^{g_2(x)} \\int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dy dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]\n$h_1(x,y)=$ [ANS] $h_2(x,y)=$ [ANS]\n2. $ \\int_a^b \\int_{g_1(y)}^{g_2(y)} \\int_{h_1(x,y)}^{h_2(x,y)}f(x,y,z) dz dx dy$ $a=$ [ANS] $b=$ [ANS]\n$g_1(y)=$ [ANS] $g_2(y)=$ [ANS]\n$h_1(x,y)=$ [ANS] $h_2(x,y)=$ [ANS]\n3. $ \\int_a^b \\int_{g_1(z)}^{g_2(z)} \\int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dy dz$ $a=$ [ANS] $b=$ [ANS]\n$g_1(z)=$ [ANS] $g_2(z)=$ [ANS]\n$h_1(y,z)=$ [ANS] $h_2(y,z)=$ [ANS]\n4. $ \\int_a^b \\int_{g_1(y)}^{g_2(y)} \\int_{h_1(y,z)}^{h_2(y,z)}f(x,y,z) dx dz dy$ $a=$ [ANS] $b=$ [ANS]\n$g_1(y)=$ [ANS] $g_2(y)=$ [ANS]\n$h_1(y,z)=$ [ANS] $h_2(y,z)=$ [ANS]\n5. $ \\int_a^b \\int_{g_1(x)}^{g_2(x)} \\int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dz dx$ $a=$ [ANS] $b=$ [ANS]\n$g_1(x)=$ [ANS] $g_2(x)=$ [ANS]\n$h_1(x,z)=$ [ANS] $h_2(x,z)=$ [ANS]\n6. $ \\int_a^b \\int_{g_1(z)}^{g_2(z)} \\int_{h_1(x,z)}^{h_2(x,z)}f(x,y,z) dy dx dz$ $a=$ [ANS] $b=$ [ANS]\n$g_1(z)=$ [ANS] $g_2(z)=$ [ANS]\n$h_1(x,z)=$ [ANS] $h_2(x,z)=$ [ANS]", "answer_v3": [ "-3", "3", "0", "9 - x^2", "0", "5*y", "0", "9", "-sqrt(9-y)", "sqrt(9-y)", "0", "5*y", "0", "45", "z/5", "9", "-sqrt(9-y)", "sqrt(9-y)", "0", "9", "0", "5*y", "-sqrt(9-y)", "sqrt(9-y)", "-3", "3", "0", "5*(9-x^2)", "z/5", "9 - x^2", "0", "45", "-sqrt(9-z/5)", "sqrt(9-z/5)", "z/5", "9-x^2" ], "answer_type_v3": [ "NV", "NV", "NV", "EX", "NV", "EX", "NV", "NV", "EX", "EX", "NV", "EX", "NV", "NV", "EX", "NV", "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0411", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "calculus", "iterated integral" ], "problem_v1": "Evaluate the triple integral $ \\iiint_{E} x^6 e^y \\, dV$ where $E$ is bounded by the parabolic cylinder $z=49-y^2$ and the planes $z=0, x=7,$ and $x=-7$. [ANS]", "answer_v1": [ "3096430499.96684" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the triple integral $ \\iiint_{E} x^8 e^y \\, dV$ where $E$ is bounded by the parabolic cylinder $z=1-y^2$ and the planes $z=0, x=1,$ and $x=-1$. [ANS]", "answer_v2": [ "0.327003947707949" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the triple integral $ \\iiint_{E} x^6 e^y \\, dV$ where $E$ is bounded by the parabolic cylinder $z=9-y^2$ and the planes $z=0, x=3,$ and $x=-3$. [ANS]", "answer_v3": [ "50451.2433006314" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0412", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Triple integrals", "level": "3", "keywords": [ "calculus", "double integral" ], "problem_v1": "Let \\text{sgn}(a):=\\begin{cases}-1, & a<0 \\\\ 0, & a=0 \\\\ 1, & 0 1/6$: probability=[ANS]\n(b) $x < \\frac{1}{6} +y$: probability=[ANS]", "answer_v1": [ "1-2/(3*6)-1/(3*6*6)", "1-[4/9-1/6+2/(3*6*6)-1/(9*6^3)]" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $x$ and $y$ have joint density function p(x,y)=\\begin{cases} {2 \\over 3}(x+2y)\\quad&\\mbox{for} 0 \\le x \\le 1, 0 \\le y \\le 1,\\\\ 0 & \\mbox{otherwise.} \\end{cases} Find the probability that\n(a) $x > 1/2$: probability=[ANS]\n(b) $x < \\frac{1}{2} +y$: probability=[ANS]", "answer_v2": [ "1-2/(3*2)-1/(3*2*2)", "1-[4/9-1/2+2/(3*2*2)-1/(9*2^3)]" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $x$ and $y$ have joint density function p(x,y)=\\begin{cases} {2 \\over 3}(x+2y)\\quad&\\mbox{for} 0 \\le x \\le 1, 0 \\le y \\le 1,\\\\ 0 & \\mbox{otherwise.} \\end{cases} Find the probability that\n(a) $x > 1/3$: probability=[ANS]\n(b) $x < \\frac{1}{3} +y$: probability=[ANS]", "answer_v3": [ "1-2/(3*3)-1/(3*3*3)", "1-[4/9-1/3+2/(3*3*3)-1/(9*3^3)]" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0463", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "5", "keywords": [ "probability distributions", "integration", "multivariable", "functions" ], "problem_v1": "Two independent random numbers $x$ and $y$ between 0 and 1 have joint density function p(x,y)=\\begin{cases} 1 & \\mbox{if} 0 \\le x,y \\le 1 \\\\ 0 & \\mbox{otherwise}. \\end{cases} This problem concerns the weighted average $z=(x+5 y)/6$, which has a one-variable probability density function of its own.\n(a) Find $F(t)$, the probability that $z\\le t$. Treat each of the following cases separately (note that $F(t)$ is the cumulative distribution function of $z$). $t\\le 0$: $F(t)=$ [ANS]\n$0< t\\le \\frac{1}{6} $: $F(t)=$ [ANS]\n$ \\frac{1}{6} < t\\le \\frac{5}{6} $: $F(t)=$ [ANS]\n$ \\frac{5}{6} < t\\le 1$: $F(t)=$ [ANS]\n$1 < t$: $F(t)=$ [ANS]\n(b) Similarly find the probability density function $f(t)$ of $z$. $t\\le 0$: $f(t)=$ [ANS]\n$0< t\\le \\frac{1}{6} $: $f(t)=$ [ANS]\n$ \\frac{1}{6} < t\\le \\frac{5}{6} $: $f(t)=$ [ANS]\n$ \\frac{5}{6} < t\\le 1$: $f(t)=$ [ANS]\n$1 < t$: $f(t)=$ [ANS]\n(c) Graph your function $f(t)$ and use it and the function $p(x,y)$ to fill in the following: $x$ and $y$ are [ANS] $z$ is most likely [ANS]", "answer_v1": [ "0", "1/10*6^2*t^2", "6/5*t-1/10", "1-1/10*(6-6*t)^2", "1", "0", "6^2*t/5", "6/5", "6/5*(6-6*t)", "0", "equally likely to be near any of these values", "more likely to be near 1/2" ], "answer_type_v1": [ "NV", "EX", "EX", "EX", "NV", "NV", "EX", "NV", "EX", "NV", "MCS", "MCS" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [ "more likely to be near 0", "more likely to be near 1/2", "more likely to be near 1", "equally likely to be near any of these values" ], [ "more likely to be near 0", "more likely to be near 1/2", "more likely to be near 1", "equally likely to be near any of these values" ] ], "problem_v2": "Two independent random numbers $x$ and $y$ between 0 and 1 have joint density function p(x,y)=\\begin{cases} 1 & \\mbox{if} 0 \\le x,y \\le 1 \\\\ 0 & \\mbox{otherwise}. \\end{cases} This problem concerns the weighted average $z=(x+2 y)/3$, which has a one-variable probability density function of its own.\n(a) Find $F(t)$, the probability that $z\\le t$. Treat each of the following cases separately (note that $F(t)$ is the cumulative distribution function of $z$). $t\\le 0$: $F(t)=$ [ANS]\n$0< t\\le \\frac{1}{3} $: $F(t)=$ [ANS]\n$ \\frac{1}{3} < t\\le \\frac{2}{3} $: $F(t)=$ [ANS]\n$ \\frac{2}{3} < t\\le 1$: $F(t)=$ [ANS]\n$1 < t$: $F(t)=$ [ANS]\n(b) Similarly find the probability density function $f(t)$ of $z$. $t\\le 0$: $f(t)=$ [ANS]\n$0< t\\le \\frac{1}{3} $: $f(t)=$ [ANS]\n$ \\frac{1}{3} < t\\le \\frac{2}{3} $: $f(t)=$ [ANS]\n$ \\frac{2}{3} < t\\le 1$: $f(t)=$ [ANS]\n$1 < t$: $f(t)=$ [ANS]\n(c) Graph your function $f(t)$ and use it and the function $p(x,y)$ to fill in the following: $x$ and $y$ are [ANS] $z$ is most likely [ANS]", "answer_v2": [ "0", "1/4*3^2*t^2", "3/2*t-1/4", "1-1/4*(3-3*t)^2", "1", "0", "3^2*t/2", "3/2", "3/2*(3-3*t)", "0", "equally likely to be near any of these values", "more likely to be near 1/2" ], "answer_type_v2": [ "NV", "EX", "EX", "EX", "NV", "NV", "EX", "NV", "EX", "NV", "MCS", "MCS" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [ "more likely to be near 0", "more likely to be near 1/2", "more likely to be near 1", "equally likely to be near any of these values" ], [ "more likely to be near 0", "more likely to be near 1/2", "more likely to be near 1", "equally likely to be near any of these values" ] ], "problem_v3": "Two independent random numbers $x$ and $y$ between 0 and 1 have joint density function p(x,y)=\\begin{cases} 1 & \\mbox{if} 0 \\le x,y \\le 1 \\\\ 0 & \\mbox{otherwise}. \\end{cases} This problem concerns the weighted average $z=(x+3 y)/4$, which has a one-variable probability density function of its own.\n(a) Find $F(t)$, the probability that $z\\le t$. Treat each of the following cases separately (note that $F(t)$ is the cumulative distribution function of $z$). $t\\le 0$: $F(t)=$ [ANS]\n$0< t\\le \\frac{1}{4} $: $F(t)=$ [ANS]\n$ \\frac{1}{4} < t\\le \\frac{3}{4} $: $F(t)=$ [ANS]\n$ \\frac{3}{4} < t\\le 1$: $F(t)=$ [ANS]\n$1 < t$: $F(t)=$ [ANS]\n(b) Similarly find the probability density function $f(t)$ of $z$. $t\\le 0$: $f(t)=$ [ANS]\n$0< t\\le \\frac{1}{4} $: $f(t)=$ [ANS]\n$ \\frac{1}{4} < t\\le \\frac{3}{4} $: $f(t)=$ [ANS]\n$ \\frac{3}{4} < t\\le 1$: $f(t)=$ [ANS]\n$1 < t$: $f(t)=$ [ANS]\n(c) Graph your function $f(t)$ and use it and the function $p(x,y)$ to fill in the following: $x$ and $y$ are [ANS] $z$ is most likely [ANS]", "answer_v3": [ "0", "1/6*4^2*t^2", "4/3*t-1/6", "1-1/6*(4-4*t)^2", "1", "0", "4^2*t/3", "4/3", "4/3*(4-4*t)", "0", "equally likely to be near any of these values", "more likely to be near 1/2" ], "answer_type_v3": [ "NV", "EX", "EX", "EX", "NV", "NV", "EX", "NV", "EX", "NV", "MCS", "MCS" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [ "more likely to be near 0", "more likely to be near 1/2", "more likely to be near 1", "equally likely to be near any of these values" ], [ "more likely to be near 0", "more likely to be near 1/2", "more likely to be near 1", "equally likely to be near any of these values" ] ] }, { "id": "Calculus_-_multivariable_0464", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "4", "keywords": [ "probability distributions", "integration", "multivariable", "functions" ], "problem_v1": "Let $p$ be the joint density function such that $p(x,y)= \\frac{1}{81} xy$ in $R$, the rectangle $0 \\le x \\le 6, 0\\le y \\le 3$, and $p(x,y)=0$ outside $R$. Find the fraction of the population satisfying the constraint $x \\ge y$ fraction=[ANS]", "answer_v1": [ "1-3^2/(2*36)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $p$ be the joint density function such that $p(x,y)= \\frac{1}{225} xy$ in $R$, the rectangle $0 \\le x \\le 6, 0\\le y \\le 5$, and $p(x,y)=0$ outside $R$. Find the fraction of the population satisfying the constraint $x \\ge y$ fraction=[ANS]", "answer_v2": [ "1-5^2/(2*36)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $p$ be the joint density function such that $p(x,y)= \\frac{1}{144} xy$ in $R$, the rectangle $0 \\le x \\le 6, 0\\le y \\le 4$, and $p(x,y)=0$ outside $R$. Find the fraction of the population satisfying the constraint $x \\ge y$ fraction=[ANS]", "answer_v3": [ "1-4^2/(2*36)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0465", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "2", "keywords": [ "iterated integral", "double integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "Find the volume of the region under the graph of $f(x,y)=3x+y+1$ and above the region $y^2\\le x$, $0\\le x\\le 16$. volume=[ANS]", "answer_v1": [ "4*3*4^5/5+4*4^3/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the region under the graph of $f(x,y)=5x+y+1$ and above the region $y^2\\le x$, $0\\le x\\le 4$. volume=[ANS]", "answer_v2": [ "4*5*2^5/5+4*2^3/3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the region under the graph of $f(x,y)=4x+y+1$ and above the region $y^2\\le x$, $0\\le x\\le 4$. volume=[ANS]", "answer_v3": [ "4*4*2^5/5+4*2^3/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0466", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "2", "keywords": [ "iterated integral", "double integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "The region $W$ lies below the surface $f(x,y)=6 e^{-(x-1)^2-y^2}$ and above the disk $x^2+y^2\\le 16$ in the $xy$-plane.\n(a) Think about what the contours of $f$ look like. You may want to using $f(x,y)=1$ as an example. Sketch a rough contour diagram on a separate sheet of paper. (b) Write an integral giving the area of the cross-section of $W$ in the plane $x=1$. Area=$\\int_a^b$ [ANS] $d$ [ANS], where $a=$ [ANS] and $b=$ [ANS]\n(c) Use your work from (b) to write an iterated double integral giving the volume of $W$, using the work from (b) to inform the construction of the inside integral. Volume=$\\int_a^b\\int_c^d$ [ANS] $d$ [ANS] $d$ [ANS], where $a=$ [ANS], $b=$ [ANS] $c=$ [ANS] and $d=$ [ANS]", "answer_v1": [ "6*e^(-y^2)", "y", "-3.87298", "3.87298", "6*e^[-(x-1)^2-y^2]", "y", "x", "-4", "4", "-[sqrt(16-x^2)]", "sqrt(16-x^2)" ], "answer_type_v1": [ "EX", "EX", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "The region $W$ lies below the surface $f(x,y)=3 e^{-(x-1)^2-y^2}$ and above the disk $x^2+y^2\\le 4$ in the $xy$-plane.\n(a) Think about what the contours of $f$ look like. You may want to using $f(x,y)=1$ as an example. Sketch a rough contour diagram on a separate sheet of paper. (b) Write an integral giving the area of the cross-section of $W$ in the plane $x=1$. Area=$\\int_a^b$ [ANS] $d$ [ANS], where $a=$ [ANS] and $b=$ [ANS]\n(c) Use your work from (b) to write an iterated double integral giving the volume of $W$, using the work from (b) to inform the construction of the inside integral. Volume=$\\int_a^b\\int_c^d$ [ANS] $d$ [ANS] $d$ [ANS], where $a=$ [ANS], $b=$ [ANS] $c=$ [ANS] and $d=$ [ANS]", "answer_v2": [ "3*e^(-y^2)", "y", "-1.73205", "1.73205", "3*e^[-(x-1)^2-y^2]", "y", "x", "-2", "2", "-[sqrt(4-x^2)]", "sqrt(4-x^2)" ], "answer_type_v2": [ "EX", "EX", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "The region $W$ lies below the surface $f(x,y)=3 e^{-(x-1)^2-y^2}$ and above the disk $x^2+y^2\\le 16$ in the $xy$-plane.\n(a) Think about what the contours of $f$ look like. You may want to using $f(x,y)=1$ as an example. Sketch a rough contour diagram on a separate sheet of paper. (b) Write an integral giving the area of the cross-section of $W$ in the plane $x=1$. Area=$\\int_a^b$ [ANS] $d$ [ANS], where $a=$ [ANS] and $b=$ [ANS]\n(c) Use your work from (b) to write an iterated double integral giving the volume of $W$, using the work from (b) to inform the construction of the inside integral. Volume=$\\int_a^b\\int_c^d$ [ANS] $d$ [ANS] $d$ [ANS], where $a=$ [ANS], $b=$ [ANS] $c=$ [ANS] and $d=$ [ANS]", "answer_v3": [ "3*e^(-y^2)", "y", "-3.87298", "3.87298", "3*e^[-(x-1)^2-y^2]", "y", "x", "-4", "4", "-[sqrt(16-x^2)]", "sqrt(16-x^2)" ], "answer_type_v3": [ "EX", "EX", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0467", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "3", "keywords": [ "iterated integral", "double integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "The function $f(x,y)=ax+by$ has an average value of $40$ on the rectangle $0 \\le x \\le 4$, $0 \\le y \\le 5$.\n(a) What can you say about the constants $a$ and $b$? [ANS]\n(Give your answer as an equation that describes the values of a and b.) (Give your answer as an equation that describes the values of a and b.) (b) Find two different choices for $f$ that have average value $40$ on the rectangle. Both answers must be correct to receive credit. $f=$ [ANS], or $f=$ [ANS].", "answer_v1": [ "4*a+5*b = 80", "2*40*y/5", "2*40*x/4" ], "answer_type_v1": [ "EQ", "EX", "EX" ], "options_v1": [ [], [], [] ], "problem_v2": "The function $f(x,y)=ax+by$ has an average value of $25$ on the rectangle $0 \\le x \\le 1$, $0 \\le y \\le 7$.\n(a) What can you say about the constants $a$ and $b$? [ANS]\n(Give your answer as an equation that describes the values of a and b.) (Give your answer as an equation that describes the values of a and b.) (b) Find two different choices for $f$ that have average value $25$ on the rectangle. Both answers must be correct to receive credit. $f=$ [ANS], or $f=$ [ANS].", "answer_v2": [ "1*a+7*b = 50", "2*25*y/7", "2*25*x/1" ], "answer_type_v2": [ "EQ", "EX", "EX" ], "options_v2": [ [], [], [] ], "problem_v3": "The function $f(x,y)=ax+by$ has an average value of $25$ on the rectangle $0 \\le x \\le 2$, $0 \\le y \\le 5$.\n(a) What can you say about the constants $a$ and $b$? [ANS]\n(Give your answer as an equation that describes the values of a and b.) (Give your answer as an equation that describes the values of a and b.) (b) Find two different choices for $f$ that have average value $25$ on the rectangle. Both answers must be correct to receive credit. $f=$ [ANS], or $f=$ [ANS].", "answer_v3": [ "2*a+5*b = 50", "2*25*y/5", "2*25*x/2" ], "answer_type_v3": [ "EQ", "EX", "EX" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0468", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "4", "keywords": [ "surface area" ], "problem_v1": "Find the area cut out of the cylinder $x^2+z^2=64$ by the cylinder $x^2+y^2=64$. [ANS]", "answer_v1": [ "512" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area cut out of the cylinder $x^2+z^2=1$ by the cylinder $x^2+y^2=1$. [ANS]", "answer_v2": [ "8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area cut out of the cylinder $x^2+z^2=16$ by the cylinder $x^2+y^2=16$. [ANS]", "answer_v3": [ "128" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0469", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "4", "keywords": [ "mass' 'application", "integral", "density", "physics", "mass", "inertia" ], "problem_v1": "A lamina occupies the part of the disk $x^2+y^2 \\leq 16$ in the first quadrant and the density at each point is given by the function $\\rho(x,y)=3(x^2+y^2)$. A. What is the total mass? [ANS]\nB. What is the moment about the x-axis? [ANS]\nC. What is the moment about the y-axis? [ANS]\nD. Where is the center of mass? ([ANS], [ANS]) E. What is the moment of inertia about the origin? [ANS]", "answer_v1": [ "301.59289474462", "614.4", "614.4", "2.03718327157626", "2.03718327157626", "3216.99087727595" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "A lamina occupies the part of the disk $x^2+y^2 \\leq 1$ in the first quadrant and the density at each point is given by the function $\\rho(x,y)=5(x^2+y^2)$. A. What is the total mass? [ANS]\nB. What is the moment about the x-axis? [ANS]\nC. What is the moment about the y-axis? [ANS]\nD. Where is the center of mass? ([ANS], [ANS]) E. What is the moment of inertia about the origin? [ANS]", "answer_v2": [ "1.96349540849362", "1", "1", "0.509295817894065", "0.509295817894065", "1.30899693899575" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "A lamina occupies the part of the disk $x^2+y^2 \\leq 4$ in the first quadrant and the density at each point is given by the function $\\rho(x,y)=4(x^2+y^2)$. A. What is the total mass? [ANS]\nB. What is the moment about the x-axis? [ANS]\nC. What is the moment about the y-axis? [ANS]\nD. Where is the center of mass? ([ANS], [ANS]) E. What is the moment of inertia about the origin? [ANS]", "answer_v3": [ "25.1327412287183", "25.6", "25.6", "1.01859163578813", "1.01859163578813", "67.0206432765822" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0470", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "4", "keywords": [ "electric' 'charge' 'application", "Multiple Integral", "Electrical Charge", "integral", "density", "physics" ], "problem_v1": "Electric charge is distributed over the disk $x^2+y^2 \\leq 16$ so that the charge density at (x,y) is $\\sigma(x,y)=12+x^2+y^2$ coulombs per square meter. Find the total charge on the disk. [ANS]", "answer_v1": [ "1005.30964914873" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Electric charge is distributed over the disk $x^2+y^2 \\leq 2$ so that the charge density at (x,y) is $\\sigma(x,y)=19+x^2+y^2$ coulombs per square meter. Find the total charge on the disk. [ANS]", "answer_v2": [ "125.663706143592" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Electric charge is distributed over the disk $x^2+y^2 \\leq 7$ so that the charge density at (x,y) is $\\sigma(x,y)=13+x^2+y^2$ coulombs per square meter. Find the total charge on the disk. [ANS]", "answer_v3": [ "362.853951489621" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0471", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "4", "keywords": [ "average' 'value' 'application" ], "problem_v1": "Find the average value of the function $f \\left(x, y, z \\right)=x^{2}+y^{2}+z^{2}$ over the rectangular prism $0 \\leq x \\leq 4$, $0 \\leq y \\leq 3$, $0 \\leq z \\leq 4$ [ANS]", "answer_v1": [ "13.6666666666667" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the average value of the function $f \\left(x, y, z \\right)=x^{2}+y^{2}+z^{2}$ over the rectangular prism $0 \\leq x \\leq 1$, $0 \\leq y \\leq 5$, $0 \\leq z \\leq 1$ [ANS]", "answer_v2": [ "9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the average value of the function $f \\left(x, y, z \\right)=x^{2}+y^{2}+z^{2}$ over the rectangular prism $0 \\leq x \\leq 2$, $0 \\leq y \\leq 4$, $0 \\leq z \\leq 2$ [ANS]", "answer_v3": [ "8" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0472", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "2", "keywords": [ "double integral' 'iterated integral", "integral' 'iterated' 'multivariable", "Iterated", "Integral", "Multivariable" ], "problem_v1": "Consider the solid that lies above the square (in the xy-plane) $R=[0, 2] \\times [0, 2]$, and below the elliptic paraboloid $z=81-x^{2}-3y^2$. (A) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the lower left hand corners. [ANS]\n(B) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the upper right hand corners.. [ANS]\n(C) What is the average of the two answers from (A) and (B)? [ANS]\n(D) Using iterated integrals, compute the exact value of the volume. [ANS]", "answer_v1": [ "316", "284", "300", "302.666666666667" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the solid that lies above the square (in the xy-plane) $R=[0, 2] \\times [0, 2]$, and below the elliptic paraboloid $z=25-x^{2}-4y^2$. (A) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the lower left hand corners. [ANS]\n(B) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the upper right hand corners.. [ANS]\n(C) What is the average of the two answers from (A) and (B)? [ANS]\n(D) Using iterated integrals, compute the exact value of the volume. [ANS]", "answer_v2": [ "90", "50", "70", "73.3333333333333" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the solid that lies above the square (in the xy-plane) $R=[0, 2] \\times [0, 2]$, and below the elliptic paraboloid $z=36-x^{2}-3y^2$. (A) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the lower left hand corners. [ANS]\n(B) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the upper right hand corners.. [ANS]\n(C) What is the average of the two answers from (A) and (B)? [ANS]\n(D) Using iterated integrals, compute the exact value of the volume. [ANS]", "answer_v3": [ "136", "104", "120", "122.666666666667" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0473", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "3", "keywords": [ "calculus", "parametric", "surface area", "integral' 'surface area" ], "problem_v1": "A torus of radius 8 (and cross-sectional radius 1) can be represented parametrically by the function $\\mathbf{r}: D \\to \\mathbb{R}^3$: \\mathbf{r}(\\theta, \\phi)=((8+\\cos\\phi)\\cos\\theta,(8+\\cos\\phi)\\sin\\theta, \\sin\\phi) where D is the rectangle given by $0\\le \\theta \\le 2\\pi, \\ 0\\le \\phi \\le 2\\pi$. The surface area of the torus is [ANS]", "answer_v1": [ "315.827340834859" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A torus of radius 2 (and cross-sectional radius 1) can be represented parametrically by the function $\\mathbf{r}: D \\to \\mathbb{R}^3$: \\mathbf{r}(\\theta, \\phi)=((2+\\cos\\phi)\\cos\\theta,(2+\\cos\\phi)\\sin\\theta, \\sin\\phi) where D is the rectangle given by $0\\le \\theta \\le 2\\pi, \\ 0\\le \\phi \\le 2\\pi$. The surface area of the torus is [ANS]", "answer_v2": [ "78.9568352087149" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A torus of radius 4 (and cross-sectional radius 1) can be represented parametrically by the function $\\mathbf{r}: D \\to \\mathbb{R}^3$: \\mathbf{r}(\\theta, \\phi)=((4+\\cos\\phi)\\cos\\theta,(4+\\cos\\phi)\\sin\\theta, \\sin\\phi) where D is the rectangle given by $0\\le \\theta \\le 2\\pi, \\ 0\\le \\phi \\le 2\\pi$. The surface area of the torus is [ANS]", "answer_v3": [ "157.91367041743" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0474", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "2", "keywords": [ "calculus", "parametric", "surface area" ], "problem_v1": "If a parametric surface given by $\\mathbf{r_{1}}(u, v)=f(u, v)\\mathbf{i}+g(u, v)\\mathbf{j}+h(u, v)\\mathbf{k}$ and $-4 \\leq u \\leq 4,-3 \\leq v \\leq 3$, has surface area equal to 4, what is the surface area of the parametric surface given by $\\mathbf{r_{2}}(u, v)=4\\mathbf{r_{1}}(u, v)$ with $-4 \\leq u \\leq 4,-3 \\leq v \\leq 3$? [ANS]", "answer_v1": [ "64" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If a parametric surface given by $\\mathbf{r_{1}}(u, v)=f(u, v)\\mathbf{i}+g(u, v)\\mathbf{j}+h(u, v)\\mathbf{k}$ and $-1 \\leq u \\leq 1,-5 \\leq v \\leq 5$, has surface area equal to 1, what is the surface area of the parametric surface given by $\\mathbf{r_{2}}(u, v)=3\\mathbf{r_{1}}(u, v)$ with $-1 \\leq u \\leq 1,-5 \\leq v \\leq 5$? [ANS]", "answer_v2": [ "9" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If a parametric surface given by $\\mathbf{r_{1}}(u, v)=f(u, v)\\mathbf{i}+g(u, v)\\mathbf{j}+h(u, v)\\mathbf{k}$ and $-2 \\leq u \\leq 2,-4 \\leq v \\leq 4$, has surface area equal to 2, what is the surface area of the parametric surface given by $\\mathbf{r_{2}}(u, v)=4\\mathbf{r_{1}}(u, v)$ with $-2 \\leq u \\leq 2,-4 \\leq v \\leq 4$? [ANS]", "answer_v3": [ "32" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0475", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "2", "keywords": [ "calculus", "average value", "iterated integral" ], "problem_v1": "Find the average value of $f(x,y)=7 e^{y} \\sqrt{x+e^{y}}$ over the rectangle $R=[0,6] \\times [0,5]$. Average value=[ANS]", "answer_v1": [ "1731.01138413411" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the average value of $f(x,y)=2 e^{y} \\sqrt{x+e^{y}}$ over the rectangle $R=[0,2] \\times [0,8]$. Average value=[ANS]", "answer_v2": [ "27138.9633516093" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the average value of $f(x,y)=4 e^{y} \\sqrt{x+e^{y}}$ over the rectangle $R=[0,3] \\times [0,5]$. Average value=[ANS]", "answer_v3": [ "976.752963321842" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0476", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "2", "keywords": [ "calculus", "average value", "iterated integral" ], "problem_v1": "Find the average value of $f(x,y)=5x^4 y^4$ over the rectangle R with vertices $(-2,0), (-2,5), (2,0), (2,5)$. Average value=[ANS]", "answer_v1": [ "2000" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the average value of $f(x,y)=2x^6 y^2$ over the rectangle R with vertices $(-6,0), (-6,3), (6,0), (6,3)$. Average value=[ANS]", "answer_v2": [ "39990.8571428571" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the average value of $f(x,y)=3x^4 y^3$ over the rectangle R with vertices $(-2,0), (-2,4), (2,0), (2,4)$. Average value=[ANS]", "answer_v3": [ "153.6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0477", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "5", "keywords": [ "area", "ellipse" ], "problem_v1": "Find the area of the ellipse given by $ \\frac{x^2}{64} + \\frac{y^2}{36} =1$. Area=[ANS].", "answer_v1": [ "150.79644737231" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the area of the ellipse given by $ \\frac{x^2}{4} + \\frac{y^2}{81} =1$. Area=[ANS].", "answer_v2": [ "56.5486677646163" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the area of the ellipse given by $ \\frac{x^2}{16} + \\frac{y^2}{36} =1$. Area=[ANS].", "answer_v3": [ "75.398223686155" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0478", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of double integrals", "level": "3", "keywords": [], "problem_v1": "Consider an empty thin ice cream cone standing with its tip at the origin. The height of the cone is 8 and the radius of the top is 6. Find the center of gravity of the cone by following the steps below. Assume the density of the cone is constant 1. a. We parametrize the cone $S$ with $s(r,t)=(r\\cos(t),r\\sin(t),$ [ANS] $)$ where $0\\le t \\le 2\\pi$ and $0\\le r \\le 6$. b. The partial derivatives are $s_r(r,t)=$ ([ANS], [ANS], [ANS]) and $s_t(r,t)=$ ([ANS], [ANS], [ANS]). c. The cross product is $s_r(r,t)\\times s_t(r,t)=$ ([ANS], [ANS], [ANS]). d. The norm of the cross product is $\\| s_r(r,t)\\times s_t(r,t) \\|=$ [ANS]. e. $m=\\int\\int_S 1 dS=$ [ANS]\nf. $m_z=\\int\\int_S z dS=$ [ANS]\ng. The center of gravity is $(0,0,\\bar z)$ where $\\bar z= \\frac{m_z}{m} =$ [ANS].", "answer_v1": [ "8*r/6", "cos(t)", "sin(t)", "1.33333", "-r*sin(t)", "r*cos(t)", "0", "-8*r*cos(t)/6", "-8*r*sin(t)/6", "r", "r*sqrt(8^2+6^2)/6", "pi*6*sqrt(8^2+6^2)", "2*8/3*pi*6*sqrt(8^2+6^2)", "5.33333" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "EX", "EX", "NV", "EX", "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Consider an empty thin ice cream cone standing with its tip at the origin. The height of the cone is 2 and the radius of the top is 9. Find the center of gravity of the cone by following the steps below. Assume the density of the cone is constant 1. a. We parametrize the cone $S$ with $s(r,t)=(r\\cos(t),r\\sin(t),$ [ANS] $)$ where $0\\le t \\le 2\\pi$ and $0\\le r \\le 9$. b. The partial derivatives are $s_r(r,t)=$ ([ANS], [ANS], [ANS]) and $s_t(r,t)=$ ([ANS], [ANS], [ANS]). c. The cross product is $s_r(r,t)\\times s_t(r,t)=$ ([ANS], [ANS], [ANS]). d. The norm of the cross product is $\\| s_r(r,t)\\times s_t(r,t) \\|=$ [ANS]. e. $m=\\int\\int_S 1 dS=$ [ANS]\nf. $m_z=\\int\\int_S z dS=$ [ANS]\ng. The center of gravity is $(0,0,\\bar z)$ where $\\bar z= \\frac{m_z}{m} =$ [ANS].", "answer_v2": [ "2*r/9", "cos(t)", "sin(t)", "0.222222", "-r*sin(t)", "r*cos(t)", "0", "-2*r*cos(t)/9", "-2*r*sin(t)/9", "r", "r*sqrt(2^2+9^2)/9", "pi*9*sqrt(2^2+9^2)", "2*2/3*pi*9*sqrt(2^2+9^2)", "1.33333" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "EX", "EX", "NV", "EX", "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Consider an empty thin ice cream cone standing with its tip at the origin. The height of the cone is 4 and the radius of the top is 6. Find the center of gravity of the cone by following the steps below. Assume the density of the cone is constant 1. a. We parametrize the cone $S$ with $s(r,t)=(r\\cos(t),r\\sin(t),$ [ANS] $)$ where $0\\le t \\le 2\\pi$ and $0\\le r \\le 6$. b. The partial derivatives are $s_r(r,t)=$ ([ANS], [ANS], [ANS]) and $s_t(r,t)=$ ([ANS], [ANS], [ANS]). c. The cross product is $s_r(r,t)\\times s_t(r,t)=$ ([ANS], [ANS], [ANS]). d. The norm of the cross product is $\\| s_r(r,t)\\times s_t(r,t) \\|=$ [ANS]. e. $m=\\int\\int_S 1 dS=$ [ANS]\nf. $m_z=\\int\\int_S z dS=$ [ANS]\ng. The center of gravity is $(0,0,\\bar z)$ where $\\bar z= \\frac{m_z}{m} =$ [ANS].", "answer_v3": [ "4*r/6", "cos(t)", "sin(t)", "0.666667", "-r*sin(t)", "r*cos(t)", "0", "-4*r*cos(t)/6", "-4*r*sin(t)/6", "r", "r*sqrt(4^2+6^2)/6", "pi*6*sqrt(4^2+6^2)", "2*4/3*pi*6*sqrt(4^2+6^2)", "2.66667" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "EX", "EX", "NV", "EX", "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0479", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of triple integrals", "level": "", "keywords": [], "problem_v1": "Find the volume of the ellipsoid given by $ \\frac{x^2}{49} + \\frac{y^2}{25} + \\frac{z^2}{36} =1$.\nVolume=[ANS].", "answer_v1": [ "7*5*6*4/3*pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the volume of the ellipsoid given by $ \\frac{x^2}{4} + \\frac{y^2}{49} + \\frac{z^2}{9} =1$.\nVolume=[ANS].", "answer_v2": [ "2*7*3*4/3*pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the volume of the ellipsoid given by $ \\frac{x^2}{16} + \\frac{y^2}{25} + \\frac{z^2}{9} =1$.\nVolume=[ANS].", "answer_v3": [ "4*5*3*4/3*pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0480", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of triple integrals", "level": "3", "keywords": [ "Multivariable", "Triple Integral' 'mass", "mass' 'application", "Multiple Integral", "Density", "Mass" ], "problem_v1": "Find the mass of the rectangular prism $0 \\leq x \\leq 4, \\ \\ 0 \\leq y \\leq 3, \\ \\ 0 \\leq z \\leq 3$, with density function $\\rho \\left(x, y, z \\right)=x$. [ANS]", "answer_v1": [ "72" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the mass of the rectangular prism $0 \\leq x \\leq 1, \\ \\ 0 \\leq y \\leq 4, \\ \\ 0 \\leq z \\leq 1$, with density function $\\rho \\left(x, y, z \\right)=x$. [ANS]", "answer_v2": [ "2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the mass of the rectangular prism $0 \\leq x \\leq 2, \\ \\ 0 \\leq y \\leq 3, \\ \\ 0 \\leq z \\leq 2$, with density function $\\rho \\left(x, y, z \\right)=x$. [ANS]", "answer_v3": [ "12" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0481", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of triple integrals", "level": "2", "keywords": [ "Multivariable", "Triple Integral", "Spherical" ], "problem_v1": "A spherical solid, centered at the origin, has radius 64 and mass density $\\delta(x,y,z)=70-\\left(x^2+y^2+z^2\\right)$. Find its mass.\n$\\begin{array}{ccccccccccccc}\\hline \\int \\!\\! \\int \\!\\!\\int \\, \\delta(x,y,z)\\, dV=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\hspace{5pt}d\\rho d\\theta\\ d\\phi=& & [ANS] \\\\ \\hline \\end{array}$\nFor your answers $\\theta$=theta, $\\rho$=rho, $\\phi$=phi.", "answer_v1": [ "0", "pi", "0", "2*pi", "0", "64", "(70-rho^2)*rho^2*sin(phi)", "4*pi*(70*64^3/3-64^5/5)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "A spherical solid, centered at the origin, has radius 1 and mass density $\\delta(x,y,z)=9-\\left(x^2+y^2+z^2\\right)$. Find its mass.\n$\\begin{array}{ccccccccccccc}\\hline \\int \\!\\! \\int \\!\\!\\int \\, \\delta(x,y,z)\\, dV=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\hspace{5pt}d\\rho d\\theta\\ d\\phi=& & [ANS] \\\\ \\hline \\end{array}$\nFor your answers $\\theta$=theta, $\\rho$=rho, $\\phi$=phi.", "answer_v2": [ "0", "pi", "0", "2*pi", "0", "1", "(9-rho^2)*rho^2*sin(phi)", "4*pi*(9*1^3/3-1^5/5)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "A spherical solid, centered at the origin, has radius 16 and mass density $\\delta(x,y,z)=22-\\left(x^2+y^2+z^2\\right)$. Find its mass.\n$\\begin{array}{ccccccccccccc}\\hline \\int \\!\\! \\int \\!\\!\\int \\, \\delta(x,y,z)\\, dV=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\hspace{5pt}d\\rho d\\theta\\ d\\phi=& & [ANS] \\\\ \\hline \\end{array}$\nFor your answers $\\theta$=theta, $\\rho$=rho, $\\phi$=phi.", "answer_v3": [ "0", "pi", "0", "2*pi", "0", "16", "(22-rho^2)*rho^2*sin(phi)", "4*pi*(22*16^3/3-16^5/5)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0482", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of triple integrals", "level": "5", "keywords": [ "triple integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "The moment of inertia of a solid body about an axis in 3-space relates the angular acceleration about this axis to torque (force twisting the body). The moments of inertia about the coordinate axes of a body of constant density and mass $m$ occupying a region $W$ of volume $V$ are defined to be I_x= \\frac{m}{V} \\int_W (y^2+z^2) \\,dV\\qquad I_y= \\frac{m}{V} \\int_W (x^2+z^2) \\,dV\\qquad I_z= \\frac{m}{V} \\int_W (x^2+y^2) \\,dV Use these definitions to find the moment of inertia about the $z$-axis of the rectangular solid of mass $144$ given by $0 \\le x \\le 4$, $0 \\le y \\le 3$, $0 \\le z \\le 4$. $I_x=$ [ANS]\n$I_y=$ [ANS]\n$I_z=$ [ANS]", "answer_v1": [ "3*4*3*4*(3*3+4*4)/3", "3*4*3*4*(4*4+4*4)/3", "3*4*3*4*(4*4+3*3)/3" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The moment of inertia of a solid body about an axis in 3-space relates the angular acceleration about this axis to torque (force twisting the body). The moments of inertia about the coordinate axes of a body of constant density and mass $m$ occupying a region $W$ of volume $V$ are defined to be I_x= \\frac{m}{V} \\int_W (y^2+z^2) \\,dV\\qquad I_y= \\frac{m}{V} \\int_W (x^2+z^2) \\,dV\\qquad I_z= \\frac{m}{V} \\int_W (x^2+y^2) \\,dV Use these definitions to find the moment of inertia about the $z$-axis of the rectangular solid of mass $10$ given by $0 \\le x \\le 1$, $0 \\le y \\le 5$, $0 \\le z \\le 1$. $I_x=$ [ANS]\n$I_y=$ [ANS]\n$I_z=$ [ANS]", "answer_v2": [ "2*1*5*1*(5*5+1*1)/3", "2*1*5*1*(1*1+1*1)/3", "2*1*5*1*(1*1+5*5)/3" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The moment of inertia of a solid body about an axis in 3-space relates the angular acceleration about this axis to torque (force twisting the body). The moments of inertia about the coordinate axes of a body of constant density and mass $m$ occupying a region $W$ of volume $V$ are defined to be I_x= \\frac{m}{V} \\int_W (y^2+z^2) \\,dV\\qquad I_y= \\frac{m}{V} \\int_W (x^2+z^2) \\,dV\\qquad I_z= \\frac{m}{V} \\int_W (x^2+y^2) \\,dV Use these definitions to find the moment of inertia about the $z$-axis of the rectangular solid of mass $32$ given by $0 \\le x \\le 2$, $0 \\le y \\le 4$, $0 \\le z \\le 2$. $I_x=$ [ANS]\n$I_y=$ [ANS]\n$I_z=$ [ANS]", "answer_v3": [ "2*2*4*2*(4*4+2*2)/3", "2*2*4*2*(2*2+2*2)/3", "2*2*4*2*(2*2+4*4)/3" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0483", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of triple integrals", "level": "5", "keywords": [ "triple integral", "definite integrals", "functions", "multivariable" ], "problem_v1": "The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density $\\rho(x,y,z)$ at the point $(x,y,z)$ and occupies a region $W$, then the coordinates $(\\overline{x},\\overline{y},\\overline{z})$ of the center of mass are given by \\overline{x}= \\frac{1}{m} \\int_W x\\rho \\, dV \\quad \\overline{y}= \\frac{1}{m} \\int_W y\\rho \\, dV \\quad \\overline{z}= \\frac{1}{m} \\int_W z\\rho \\, dV, where $m=\\int_W \\rho\\,dV$ is the total mass of the body. Consider a solid is bounded below by the square $z=0$, $0 \\le x \\le 4$, $0 \\le y \\le 3$ and above by the surface $z=x+y+4$. Let the density of the solid be 1 g/cm ${}^3$, with $x,y,z$ measured in cm. Find each of the following: The mass of the solid=[ANS] g\n$\\overline x$ for the solid=[ANS] cm\n$\\overline y$ for the solid=[ANS] cm\n$\\overline z$ for the solid=[ANS] cm", "answer_v1": [ "90", "2.17778", "1.6", "3.88889" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density $\\rho(x,y,z)$ at the point $(x,y,z)$ and occupies a region $W$, then the coordinates $(\\overline{x},\\overline{y},\\overline{z})$ of the center of mass are given by \\overline{x}= \\frac{1}{m} \\int_W x\\rho \\, dV \\quad \\overline{y}= \\frac{1}{m} \\int_W y\\rho \\, dV \\quad \\overline{z}= \\frac{1}{m} \\int_W z\\rho \\, dV, where $m=\\int_W \\rho\\,dV$ is the total mass of the body. Consider a solid is bounded below by the square $z=0$, $0 \\le x \\le 1$, $0 \\le y \\le 5$ and above by the surface $z=x+y+1$. Let the density of the solid be 1 g/cm ${}^3$, with $x,y,z$ measured in cm. Find each of the following: The mass of the solid=[ANS] g\n$\\overline x$ for the solid=[ANS] cm\n$\\overline y$ for the solid=[ANS] cm\n$\\overline z$ for the solid=[ANS] cm", "answer_v2": [ "20", "0.520833", "3.02083", "2.27083" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "The motion of a solid object can be analyzed by thinking of the mass as concentrated at a single point, the center of mass. If the object has density $\\rho(x,y,z)$ at the point $(x,y,z)$ and occupies a region $W$, then the coordinates $(\\overline{x},\\overline{y},\\overline{z})$ of the center of mass are given by \\overline{x}= \\frac{1}{m} \\int_W x\\rho \\, dV \\quad \\overline{y}= \\frac{1}{m} \\int_W y\\rho \\, dV \\quad \\overline{z}= \\frac{1}{m} \\int_W z\\rho \\, dV, where $m=\\int_W \\rho\\,dV$ is the total mass of the body. Consider a solid is bounded below by the square $z=0$, $0 \\le x \\le 2$, $0 \\le y \\le 4$ and above by the surface $z=x+y+2$. Let the density of the solid be 1 g/cm ${}^3$, with $x,y,z$ measured in cm. Find each of the following: The mass of the solid=[ANS] g\n$\\overline x$ for the solid=[ANS] cm\n$\\overline y$ for the solid=[ANS] cm\n$\\overline z$ for the solid=[ANS] cm", "answer_v3": [ "40", "1.06667", "2.26667", "2.66667" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0485", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of triple integrals", "level": "4", "keywords": [ "coordinates", "cylindrical", "spherical", "integrals", "calculus" ], "problem_v1": "The density, $\\delta$, of the cylinder $x^2+y^2\\le 16$, $0\\le z\\le 4$ varies with the distance, $r$, from the $z$-axis: \\delta=4+r\\mbox{g}/\\mbox{cm}^3. Find the mass of the cylinder, assuming $x,y,z$ are in cm. mass=[ANS] g", "answer_v1": [ "1340.41" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The density, $\\delta$, of the cylinder $x^2+y^2\\le 1$, $0\\le z\\le 5$ varies with the distance, $r$, from the $z$-axis: \\delta=1+r\\mbox{g}/\\mbox{cm}^3. Find the mass of the cylinder, assuming $x,y,z$ are in cm. mass=[ANS] g", "answer_v2": [ "26.1799" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The density, $\\delta$, of the cylinder $x^2+y^2\\le 4$, $0\\le z\\le 4$ varies with the distance, $r$, from the $z$-axis: \\delta=2+r\\mbox{g}/\\mbox{cm}^3. Find the mass of the cylinder, assuming $x,y,z$ are in cm. mass=[ANS] g", "answer_v3": [ "167.552" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0486", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of triple integrals", "level": "4", "keywords": [ "integral' 'surface area" ], "problem_v1": "The cylinder $x^2+y^2=16$ divides the sphere $x^2+y^2+z^2=64$ into two regions $I$ (for the region inside the cylinder), and $O$ (for the region outside the cylinder). Find the ratio of the areas $A(O)/A(I)$. [ANS]", "answer_v1": [ "6.46410161513775" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The cylinder $x^2+y^2=1$ divides the sphere $x^2+y^2+z^2=49$ into two regions $I$ (for the region inside the cylinder), and $O$ (for the region outside the cylinder). Find the ratio of the areas $A(O)/A(I)$. [ANS]", "answer_v2": [ "96.497422611928" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The cylinder $x^2+y^2=4$ divides the sphere $x^2+y^2+z^2=36$ into two regions $I$ (for the region inside the cylinder), and $O$ (for the region outside the cylinder). Find the ratio of the areas $A(O)/A(I)$. [ANS]", "answer_v3": [ "16.4852813742386" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0487", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of triple integrals", "level": "3", "keywords": [ "Triple Integrals" ], "problem_v1": "Set up a triple integral to find the mass of the solid tetrahedron bounded by the xy-plane, the yz-plane, the xz-plane, and the plane $x/6+y/5+z/30=1$, if the density function is given by $\\delta(x,y,z)=x+y$. Write an iterated integral in the form below to find the mass of the solid.\n$ \\iiint\\limits_R f(x,y,z) \\, dV=\\int_A^B \\!\\! \\int_C^D \\!\\! \\int_E^F$ [ANS] $\\, dz \\, dy \\, dx$ with limits of integration\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\nE=[ANS]\nF=[ANS]", "answer_v1": [ "x+y", "0", "6", "0", "5-0.833333*x", "0", "30-5*x-6*y" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "EX", "NV", "EX" ], "options_v1": [ [], [], [], [], [], [], [] ], "problem_v2": "Set up a triple integral to find the mass of the solid tetrahedron bounded by the xy-plane, the yz-plane, the xz-plane, and the plane $x/3+y/2+z/6=1$, if the density function is given by $\\delta(x,y,z)=x+y$. Write an iterated integral in the form below to find the mass of the solid.\n$ \\iiint\\limits_R f(x,y,z) \\, dV=\\int_A^B \\!\\! \\int_C^D \\!\\! \\int_E^F$ [ANS] $\\, dz \\, dy \\, dx$ with limits of integration\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\nE=[ANS]\nF=[ANS]", "answer_v2": [ "x+y", "0", "3", "0", "2-0.666667*x", "0", "6-2*x-3*y" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "EX", "NV", "EX" ], "options_v2": [ [], [], [], [], [], [], [] ], "problem_v3": "Set up a triple integral to find the mass of the solid tetrahedron bounded by the xy-plane, the yz-plane, the xz-plane, and the plane $x/4+y/3+z/12=1$, if the density function is given by $\\delta(x,y,z)=x+y$. Write an iterated integral in the form below to find the mass of the solid.\n$ \\iiint\\limits_R f(x,y,z) \\, dV=\\int_A^B \\!\\! \\int_C^D \\!\\! \\int_E^F$ [ANS] $\\, dz \\, dy \\, dx$ with limits of integration\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\nE=[ANS]\nF=[ANS]", "answer_v3": [ "x+y", "0", "4", "0", "3-0.75*x", "0", "12-3*x-4*y" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "EX", "NV", "EX" ], "options_v3": [ [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0488", "subject": "Calculus_-_multivariable", "topic": "Integration of multivariable functions", "subtopic": "Applications of triple integrals", "level": "4", "keywords": [ "sage", "sagelet", "integration", "multiple" ], "problem_v1": "def record_answer(ansList): html(''\\%(ansList,))\n# Sage Python Code def dlist(vs): vs=vs.split() return [\" \".join(\"d\\%s\" \\% v for v in a) for a in Arrangements(vs, len(vs))]\nvar('x,y,z')\n@interact(layout={\"bottom\": [[\"top0\", \"top1\", \"top2\",\"spacer\"], [\"int0\", \"int1\", \"int2\", \"dV\",], [\"bot0\", \"bot1\", \"bot2\",\"auto_update\"],], \"top\": [[\"plot_axes\", \"plot_origin\", \"opacity\"],],})\n## The nicely formatted interact was taken from a talk at SAGE EDU 5\ndef _(top0=input_box(label=\"\", default=3, type=str, width=4), int0=text_control(r\"\"\"\\(\\color{red}{\\int} \\)\"\"\"), bot0=input_box(label=\"\", default=1, type=str, width=4), top1=input_box(label=\"\", default=3, type=str, width=8), int1=text_control(r\"\"\"\\(\\color{green}{\\int} \\)\"\"\"), bot1=input_box(label=\"\", default=1, type=str, width=8), top2=input_box(label=\"\", default=3, type=str, width=12), int2=text_control(r\"\"\"\\(\\color{blue}{\\int} \\)\"\"\"), bot2=input_box(label=\"\", default=1, type=str, width=12), spacer=text_control(\" \"), dV=selector(flatten(map(dlist, [\"x y z\", \"r theta z\", \"rho theta phi\"])), label=\"\", default=\"dz dy dx\"), opacity=slider(range(101), label=\"Opacity:\", default=75), plot_axes=checkbox(label=\" Axes:\", default=True), plot_origin=checkbox(label=\" Origin:\", default=True), auto_update=True): f=1+0*x opacity/=100 vs=[d[1:] for d in dV.split()] vs.reverse() V=map(SR, vs) v0, v1, v2=(SR.var(v) for v in vs) # While ln=log, symbolically they are treated differently limits=[eval(\"[v\\%d, SR(str(bot\\%d).replace('ln', 'log')), SR(str(top\\%d).replace('ln', 'log'))]\" \\% (i, i, i)) for i in range(3)] f=SR(str(f).replace('ln', 'log')) compute=true; if compute: integrals=[] differentials=[] for v, a, b in limits: integrals.append(r\"\\int_{\\%s}^{\\%s} \" \\% (latex(a), latex(b))) differentials.insert(0, r\"\\, d \\%s\" \\% latex(v)) try: assume(a < v) assume(v < b) except ValueError: pass expression=[r\"\\(\"] integrand=f try: for i in [3, 2, 1]: expression.extend(integrals[:i]) expression.append(latex(integrand)) expression.extend(differentials[-i:]) expression.append(\"=\") v, a, b=limits[i-1] try: integrand=integral(integrand, v, a, b) except: integrand=integral(integrand, v) integrand=(integrand.subs({v: b})-integrand.subs({v: a})).simplify_full() # Don't add these at once in case of exceptions from n(). expression.extend([latex(integrand), r\"\\approx\"]) expression.append(latex(integrand.n())) except: expression.append(r\"\\dots?\") forget() expression.append(r\"\\)\") pretty_print(html(\"

The integral is as follows:

\")) pretty_print(html(\" \".join(expression))) pretty_print(html(\"

\")) if \"x\" in vs: def to_xyz(*V): return [V[vs.index(v)] for v in \"xyz\"] elif \"r\" in vs: def to_xyz(*V): r, theta, z=[V[vs.index(v)] for v in [\"r\", \"theta\", \"z\"]] return [r*cos(theta), r*sin(theta), z] else: def to_xyz(*V): rho, phi, theta=[V[vs.index(v)] for v in [\"rho\", \"phi\", \"theta\"]] return [rho*sin(phi)*cos(theta), rho*sin(phi)*sin(theta), rho*cos(phi)] try: u=SR.var(\"u\") v=SR.var(\"v\") p=Graphics() V1=(1-u)*limits[1][1]+u*limits[1][2] for i in [1,2]: V2=limits[2][i].subs({v0: v0, v1: V1}) p+=parametric_plot3d(to_xyz(v0, V1, V2), limits[0], (u, 0, 1), opacity=opacity, color=\"blue\") for i in [1,2]: V1=limits[1][i] V2=(1-u)*limits[2][1].subs({v1: V1})+u*limits[2][2].subs({v1: V1}) p+=parametric_plot3d(to_xyz(v0, V1, V2), limits[0], (u, 0, 1), opacity=opacity, color=\"green\", boundary_style={\"color\": \"green\", \"thickness\": 5}) for i in [1,2]: V0=limits[0][i] V1=(1-u)*limits[1][1].subs({v0: V0})+u*limits[1][2].subs({v0: V0}) V2=(1-v)*limits[2][1].subs({v0: V0, v1: V1})+v*limits[2][2].subs({v0: V0, v1: V1}) p+=parametric_plot3d(to_xyz(V0, V1, V2), (u, 0, 1), (v, 0, 1), opacity=opacity, color=\"red\", boundary_style={\"color\": \"red\", \"thickness\": 10}) if plot_origin: p+=point3d((0, 0, 0), size=25, color=\"yellow\") if plot_axes: bb=p.bounding_box() for i, v in enumerate(\"XYZ\"): start=min(bb[0][i], 0) end=max(bb[1][i], 0) length=end-start p+=line([[0]*i+[start-0.08*length]+[0]*(2-i), [0]*i+[end+0.08*length]+[0]*(2-i)], thickness=3, color=\"brown\") p+=text3d(v, [0]*i+[end+0.1*length]+[0]*(2-i)) p.show(viewer='threejs', frame=False, aspect_ratio=1) except ValueError: pretty_print(html(r\"\"\"Cannot draw the plot, is the integral valid?..\"\"\"))\nrecord_answer((integrand))\n\\$(function () {sagecell.makeSagecell({inputLocation: '#sagecell', template: sagecell.templates.minimal, autoeval: true, linked: true, evalButtonText: 'Restart the Interactive Cell'});}); Using the interactive tool above, determine the order of integration and endpoints that give the volume of the region bounded by:\n$x$ between $-2$ and $(y^2)$ $y$ between $0$ and $4$ $z$ between $(y+x+4)$ and $(-y)$. When you think you have things set up correctly, go ahead and click below to check your work.", "answer_v1": [ "371.733" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "def record_answer(ansList): html(''\\%(ansList,))\n# Sage Python Code def dlist(vs): vs=vs.split() return [\" \".join(\"d\\%s\" \\% v for v in a) for a in Arrangements(vs, len(vs))]\nvar('x,y,z')\n@interact(layout={\"bottom\": [[\"top0\", \"top1\", \"top2\",\"spacer\"], [\"int0\", \"int1\", \"int2\", \"dV\",], [\"bot0\", \"bot1\", \"bot2\",\"auto_update\"],], \"top\": [[\"plot_axes\", \"plot_origin\", \"opacity\"],],})\n## The nicely formatted interact was taken from a talk at SAGE EDU 5\ndef _(top0=input_box(label=\"\", default=3, type=str, width=4), int0=text_control(r\"\"\"\\(\\color{red}{\\int} \\)\"\"\"), bot0=input_box(label=\"\", default=1, type=str, width=4), top1=input_box(label=\"\", default=3, type=str, width=8), int1=text_control(r\"\"\"\\(\\color{green}{\\int} \\)\"\"\"), bot1=input_box(label=\"\", default=1, type=str, width=8), top2=input_box(label=\"\", default=3, type=str, width=12), int2=text_control(r\"\"\"\\(\\color{blue}{\\int} \\)\"\"\"), bot2=input_box(label=\"\", default=1, type=str, width=12), spacer=text_control(\" \"), dV=selector(flatten(map(dlist, [\"x y z\", \"r theta z\", \"rho theta phi\"])), label=\"\", default=\"dz dy dx\"), opacity=slider(range(101), label=\"Opacity:\", default=75), plot_axes=checkbox(label=\" Axes:\", default=True), plot_origin=checkbox(label=\" Origin:\", default=True), auto_update=True): f=1+0*x opacity/=100 vs=[d[1:] for d in dV.split()] vs.reverse() V=map(SR, vs) v0, v1, v2=(SR.var(v) for v in vs) # While ln=log, symbolically they are treated differently limits=[eval(\"[v\\%d, SR(str(bot\\%d).replace('ln', 'log')), SR(str(top\\%d).replace('ln', 'log'))]\" \\% (i, i, i)) for i in range(3)] f=SR(str(f).replace('ln', 'log')) compute=true; if compute: integrals=[] differentials=[] for v, a, b in limits: integrals.append(r\"\\int_{\\%s}^{\\%s} \" \\% (latex(a), latex(b))) differentials.insert(0, r\"\\, d \\%s\" \\% latex(v)) try: assume(a < v) assume(v < b) except ValueError: pass expression=[r\"\\(\"] integrand=f try: for i in [3, 2, 1]: expression.extend(integrals[:i]) expression.append(latex(integrand)) expression.extend(differentials[-i:]) expression.append(\"=\") v, a, b=limits[i-1] try: integrand=integral(integrand, v, a, b) except: integrand=integral(integrand, v) integrand=(integrand.subs({v: b})-integrand.subs({v: a})).simplify_full() # Don't add these at once in case of exceptions from n(). expression.extend([latex(integrand), r\"\\approx\"]) expression.append(latex(integrand.n())) except: expression.append(r\"\\dots?\") forget() expression.append(r\"\\)\") pretty_print(html(\"

The integral is as follows:

\")) pretty_print(html(\" \".join(expression))) pretty_print(html(\"

\")) if \"x\" in vs: def to_xyz(*V): return [V[vs.index(v)] for v in \"xyz\"] elif \"r\" in vs: def to_xyz(*V): r, theta, z=[V[vs.index(v)] for v in [\"r\", \"theta\", \"z\"]] return [r*cos(theta), r*sin(theta), z] else: def to_xyz(*V): rho, phi, theta=[V[vs.index(v)] for v in [\"rho\", \"phi\", \"theta\"]] return [rho*sin(phi)*cos(theta), rho*sin(phi)*sin(theta), rho*cos(phi)] try: u=SR.var(\"u\") v=SR.var(\"v\") p=Graphics() V1=(1-u)*limits[1][1]+u*limits[1][2] for i in [1,2]: V2=limits[2][i].subs({v0: v0, v1: V1}) p+=parametric_plot3d(to_xyz(v0, V1, V2), limits[0], (u, 0, 1), opacity=opacity, color=\"blue\") for i in [1,2]: V1=limits[1][i] V2=(1-u)*limits[2][1].subs({v1: V1})+u*limits[2][2].subs({v1: V1}) p+=parametric_plot3d(to_xyz(v0, V1, V2), limits[0], (u, 0, 1), opacity=opacity, color=\"green\", boundary_style={\"color\": \"green\", \"thickness\": 5}) for i in [1,2]: V0=limits[0][i] V1=(1-u)*limits[1][1].subs({v0: V0})+u*limits[1][2].subs({v0: V0}) V2=(1-v)*limits[2][1].subs({v0: V0, v1: V1})+v*limits[2][2].subs({v0: V0, v1: V1}) p+=parametric_plot3d(to_xyz(V0, V1, V2), (u, 0, 1), (v, 0, 1), opacity=opacity, color=\"red\", boundary_style={\"color\": \"red\", \"thickness\": 10}) if plot_origin: p+=point3d((0, 0, 0), size=25, color=\"yellow\") if plot_axes: bb=p.bounding_box() for i, v in enumerate(\"XYZ\"): start=min(bb[0][i], 0) end=max(bb[1][i], 0) length=end-start p+=line([[0]*i+[start-0.08*length]+[0]*(2-i), [0]*i+[end+0.08*length]+[0]*(2-i)], thickness=3, color=\"brown\") p+=text3d(v, [0]*i+[end+0.1*length]+[0]*(2-i)) p.show(viewer='threejs', frame=False, aspect_ratio=1) except ValueError: pretty_print(html(r\"\"\"Cannot draw the plot, is the integral valid?..\"\"\"))\nrecord_answer((integrand))\n\\$(function () {sagecell.makeSagecell({inputLocation: '#sagecell', template: sagecell.templates.minimal, autoeval: true, linked: true, evalButtonText: 'Restart the Interactive Cell'});}); Using the interactive tool above, determine the order of integration and endpoints that give the volume of the region bounded by:\n$z$ between $-1$ and $(x^2)$ $x$ between $0$ and $2$ $y$ between $(x+z+1)$ and $(-x)$. When you think you have things set up correctly, go ahead and click below to check your work.", "answer_v2": [ "18.8667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "def record_answer(ansList): html(''\\%(ansList,))\n# Sage Python Code def dlist(vs): vs=vs.split() return [\" \".join(\"d\\%s\" \\% v for v in a) for a in Arrangements(vs, len(vs))]\nvar('x,y,z')\n@interact(layout={\"bottom\": [[\"top0\", \"top1\", \"top2\",\"spacer\"], [\"int0\", \"int1\", \"int2\", \"dV\",], [\"bot0\", \"bot1\", \"bot2\",\"auto_update\"],], \"top\": [[\"plot_axes\", \"plot_origin\", \"opacity\"],],})\n## The nicely formatted interact was taken from a talk at SAGE EDU 5\ndef _(top0=input_box(label=\"\", default=3, type=str, width=4), int0=text_control(r\"\"\"\\(\\color{red}{\\int} \\)\"\"\"), bot0=input_box(label=\"\", default=1, type=str, width=4), top1=input_box(label=\"\", default=3, type=str, width=8), int1=text_control(r\"\"\"\\(\\color{green}{\\int} \\)\"\"\"), bot1=input_box(label=\"\", default=1, type=str, width=8), top2=input_box(label=\"\", default=3, type=str, width=12), int2=text_control(r\"\"\"\\(\\color{blue}{\\int} \\)\"\"\"), bot2=input_box(label=\"\", default=1, type=str, width=12), spacer=text_control(\" \"), dV=selector(flatten(map(dlist, [\"x y z\", \"r theta z\", \"rho theta phi\"])), label=\"\", default=\"dz dy dx\"), opacity=slider(range(101), label=\"Opacity:\", default=75), plot_axes=checkbox(label=\" Axes:\", default=True), plot_origin=checkbox(label=\" Origin:\", default=True), auto_update=True): f=1+0*x opacity/=100 vs=[d[1:] for d in dV.split()] vs.reverse() V=map(SR, vs) v0, v1, v2=(SR.var(v) for v in vs) # While ln=log, symbolically they are treated differently limits=[eval(\"[v\\%d, SR(str(bot\\%d).replace('ln', 'log')), SR(str(top\\%d).replace('ln', 'log'))]\" \\% (i, i, i)) for i in range(3)] f=SR(str(f).replace('ln', 'log')) compute=true; if compute: integrals=[] differentials=[] for v, a, b in limits: integrals.append(r\"\\int_{\\%s}^{\\%s} \" \\% (latex(a), latex(b))) differentials.insert(0, r\"\\, d \\%s\" \\% latex(v)) try: assume(a < v) assume(v < b) except ValueError: pass expression=[r\"\\(\"] integrand=f try: for i in [3, 2, 1]: expression.extend(integrals[:i]) expression.append(latex(integrand)) expression.extend(differentials[-i:]) expression.append(\"=\") v, a, b=limits[i-1] try: integrand=integral(integrand, v, a, b) except: integrand=integral(integrand, v) integrand=(integrand.subs({v: b})-integrand.subs({v: a})).simplify_full() # Don't add these at once in case of exceptions from n(). expression.extend([latex(integrand), r\"\\approx\"]) expression.append(latex(integrand.n())) except: expression.append(r\"\\dots?\") forget() expression.append(r\"\\)\") pretty_print(html(\"

The integral is as follows:

\")) pretty_print(html(\" \".join(expression))) pretty_print(html(\"

\")) if \"x\" in vs: def to_xyz(*V): return [V[vs.index(v)] for v in \"xyz\"] elif \"r\" in vs: def to_xyz(*V): r, theta, z=[V[vs.index(v)] for v in [\"r\", \"theta\", \"z\"]] return [r*cos(theta), r*sin(theta), z] else: def to_xyz(*V): rho, phi, theta=[V[vs.index(v)] for v in [\"rho\", \"phi\", \"theta\"]] return [rho*sin(phi)*cos(theta), rho*sin(phi)*sin(theta), rho*cos(phi)] try: u=SR.var(\"u\") v=SR.var(\"v\") p=Graphics() V1=(1-u)*limits[1][1]+u*limits[1][2] for i in [1,2]: V2=limits[2][i].subs({v0: v0, v1: V1}) p+=parametric_plot3d(to_xyz(v0, V1, V2), limits[0], (u, 0, 1), opacity=opacity, color=\"blue\") for i in [1,2]: V1=limits[1][i] V2=(1-u)*limits[2][1].subs({v1: V1})+u*limits[2][2].subs({v1: V1}) p+=parametric_plot3d(to_xyz(v0, V1, V2), limits[0], (u, 0, 1), opacity=opacity, color=\"green\", boundary_style={\"color\": \"green\", \"thickness\": 5}) for i in [1,2]: V0=limits[0][i] V1=(1-u)*limits[1][1].subs({v0: V0})+u*limits[1][2].subs({v0: V0}) V2=(1-v)*limits[2][1].subs({v0: V0, v1: V1})+v*limits[2][2].subs({v0: V0, v1: V1}) p+=parametric_plot3d(to_xyz(V0, V1, V2), (u, 0, 1), (v, 0, 1), opacity=opacity, color=\"red\", boundary_style={\"color\": \"red\", \"thickness\": 10}) if plot_origin: p+=point3d((0, 0, 0), size=25, color=\"yellow\") if plot_axes: bb=p.bounding_box() for i, v in enumerate(\"XYZ\"): start=min(bb[0][i], 0) end=max(bb[1][i], 0) length=end-start p+=line([[0]*i+[start-0.08*length]+[0]*(2-i), [0]*i+[end+0.08*length]+[0]*(2-i)], thickness=3, color=\"brown\") p+=text3d(v, [0]*i+[end+0.1*length]+[0]*(2-i)) p.show(viewer='threejs', frame=False, aspect_ratio=1) except ValueError: pretty_print(html(r\"\"\"Cannot draw the plot, is the integral valid?..\"\"\"))\nrecord_answer((integrand))\n\\$(function () {sagecell.makeSagecell({inputLocation: '#sagecell', template: sagecell.templates.minimal, autoeval: true, linked: true, evalButtonText: 'Restart the Interactive Cell'});}); Using the interactive tool above, determine the order of integration and endpoints that give the volume of the region bounded by:\n$x$ between $-2$ and $(z^2)$ $z$ between $0$ and $2$ $y$ between $(z+x+2)$ and $(-z)$. When you think you have things set up correctly, go ahead and click below to check your work.", "answer_v3": [ "28.5333" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0489", "subject": "Calculus_-_multivariable", "topic": "Vector fields", "subtopic": "Graphs, flows lines, and level surfaces", "level": "1", "keywords": [ "calculus" ], "problem_v1": "Compute and sketch the vector assigned to the points $P=(0,5,2)$ and $Q=(7,1,0)$ by the vector field $\\mathbf{F}=\\left$. $\\mathbf{F}(P)=$ [ANS]\n$\\mathbf{F}(Q)=$ [ANS]", "answer_v1": [ "(0,4,0)", "(7,0,7)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Compute and sketch the vector assigned to the points $P=(0,-9,9)$ and $Q=(3,1,0)$ by the vector field $\\mathbf{F}=\\left$. $\\mathbf{F}(P)=$ [ANS]\n$\\mathbf{F}(Q)=$ [ANS]", "answer_v2": [ "(0,81,0)", "(3,0,3)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Compute and sketch the vector assigned to the points $P=(0,-4,2)$ and $Q=(4,1,0)$ by the vector field $\\mathbf{F}=\\left$. $\\mathbf{F}(P)=$ [ANS]\n$\\mathbf{F}(Q)=$ [ANS]", "answer_v3": [ "(0,4,0)", "(4,0,4)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0490", "subject": "Calculus_-_multivariable", "topic": "Vector fields", "subtopic": "Graphs, flows lines, and level surfaces", "level": "3", "keywords": [ "vector' 'multivariable' 'surfaces' curves", "vector", "field", "level", "surfaces", "curves" ], "problem_v1": "Match the following vector fields with the verbal descriptions of the level curves or level surfaces to which they are perpendicular by putting the letter of the verbal description to the left of the number of the vector field. [ANS] 1. $\\mathbf F=x\\mathbf i+y\\mathbf j-\\mathbf k$ [ANS] 2. $\\mathbf F=2x\\mathbf i+y\\mathbf j+z\\mathbf k$ [ANS] 3. $\\mathbf F=2\\mathbf i+\\mathbf j$ [ANS] 4. $\\mathbf F=2\\mathbf i+\\mathbf j+\\mathbf k$ [ANS] 5. $\\mathbf F=x\\mathbf i+y\\mathbf j+z\\mathbf k$ [ANS] 6. $\\mathbf F=x\\mathbf i+y\\mathbf j$ [ANS] 7. $\\mathbf F=2x\\mathbf i+y\\mathbf j$ [ANS] 8. $\\mathbf F=x\\mathbf i+y\\mathbf j-z\\mathbf k$ [ANS] 9. $\\mathbf F=y\\mathbf i+x\\mathbf j$ [ANS] 10. $\\mathbf F=x\\mathbf i-y\\mathbf j$ [ANS] 11. $\\mathbf F=-y\\mathbf i+x\\mathbf j$\nA. paraboloids B. circles C. hyperboloids D. lines E. planes F. spheres G. ellipses H. hyperbolas I. ellipsoids", "answer_v1": [ "A", "I", "D", "E", "F", "B", "G", "C", "H", "H", "D" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ], "problem_v2": "Match the following vector fields with the verbal descriptions of the level curves or level surfaces to which they are perpendicular by putting the letter of the verbal description to the left of the number of the vector field. [ANS] 1. $\\mathbf F=x\\mathbf i+y\\mathbf j-\\mathbf k$ [ANS] 2. $\\mathbf F=x\\mathbf i+y\\mathbf j+z\\mathbf k$ [ANS] 3. $\\mathbf F=y\\mathbf i+x\\mathbf j$ [ANS] 4. $\\mathbf F=-y\\mathbf i+x\\mathbf j$ [ANS] 5. $\\mathbf F=2\\mathbf i+\\mathbf j+\\mathbf k$ [ANS] 6. $\\mathbf F=2\\mathbf i+\\mathbf j$ [ANS] 7. $\\mathbf F=2x\\mathbf i+y\\mathbf j+z\\mathbf k$ [ANS] 8. $\\mathbf F=2x\\mathbf i+y\\mathbf j$ [ANS] 9. $\\mathbf F=x\\mathbf i-y\\mathbf j$ [ANS] 10. $\\mathbf F=x\\mathbf i+y\\mathbf j$ [ANS] 11. $\\mathbf F=x\\mathbf i+y\\mathbf j-z\\mathbf k$\nA. circles B. paraboloids C. hyperboloids D. spheres E. ellipses F. planes G. hyperbolas H. lines I. ellipsoids", "answer_v2": [ "B", "D", "G", "H", "F", "H", "I", "E", "G", "A", "C" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ], "problem_v3": "Match the following vector fields with the verbal descriptions of the level curves or level surfaces to which they are perpendicular by putting the letter of the verbal description to the left of the number of the vector field. [ANS] 1. $\\mathbf F=y\\mathbf i+x\\mathbf j$ [ANS] 2. $\\mathbf F=2\\mathbf i+\\mathbf j+\\mathbf k$ [ANS] 3. $\\mathbf F=2x\\mathbf i+y\\mathbf j+z\\mathbf k$ [ANS] 4. $\\mathbf F=x\\mathbf i+y\\mathbf j+z\\mathbf k$ [ANS] 5. $\\mathbf F=-y\\mathbf i+x\\mathbf j$ [ANS] 6. $\\mathbf F=2x\\mathbf i+y\\mathbf j$ [ANS] 7. $\\mathbf F=2\\mathbf i+\\mathbf j$ [ANS] 8. $\\mathbf F=x\\mathbf i+y\\mathbf j$ [ANS] 9. $\\mathbf F=x\\mathbf i+y\\mathbf j-z\\mathbf k$ [ANS] 10. $\\mathbf F=x\\mathbf i-y\\mathbf j$ [ANS] 11. $\\mathbf F=x\\mathbf i+y\\mathbf j-\\mathbf k$\nA. paraboloids B. spheres C. ellipses D. ellipsoids E. hyperboloids F. lines G. circles H. planes I. hyperbolas", "answer_v3": [ "I", "H", "D", "B", "F", "C", "F", "G", "E", "I", "A" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ], [ "A", "B", "C", "D", "E", "F", "G", "H", "I" ] ] }, { "id": "Calculus_-_multivariable_0491", "subject": "Calculus_-_multivariable", "topic": "Vector fields", "subtopic": "Graphs, flows lines, and level surfaces", "level": "4", "keywords": [ "vector field", "multivariable", "calculus" ], "problem_v1": "Consider the vector field $\\vec v=-5x\\vec i+5 y\\vec j$. On a sheet of paper, sketch the vector field and the flow.\n(a) After sketching the vector field and flow, find the system of differential equations associated with the vector field: $x'=$ [ANS]\n$y'=$ [ANS]\n(b) Solve the system you found above to find the flow $x(t)=$ [ANS]\n$y(t)=$ [ANS]\n(Your solution for $x$ should involve a constant $a$, and for $y$ a constant $b$.)", "answer_v1": [ "-5*x", "5*y", "a*e^(-5*t)", "b*e^(5*t)" ], "answer_type_v1": [ "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the vector field $\\vec v=-2x\\vec i-2 y\\vec j$. On a sheet of paper, sketch the vector field and the flow.\n(a) After sketching the vector field and flow, find the system of differential equations associated with the vector field: $x'=$ [ANS]\n$y'=$ [ANS]\n(b) Solve the system you found above to find the flow $x(t)=$ [ANS]\n$y(t)=$ [ANS]\n(Your solution for $x$ should involve a constant $a$, and for $y$ a constant $b$.)", "answer_v2": [ "-2*x", "-2*y", "a*e^(-2*t)", "b*e^(-2*t)" ], "answer_type_v2": [ "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the vector field $\\vec v=-3x\\vec i+3 y\\vec j$. On a sheet of paper, sketch the vector field and the flow.\n(a) After sketching the vector field and flow, find the system of differential equations associated with the vector field: $x'=$ [ANS]\n$y'=$ [ANS]\n(b) Solve the system you found above to find the flow $x(t)=$ [ANS]\n$y(t)=$ [ANS]\n(Your solution for $x$ should involve a constant $a$, and for $y$ a constant $b$.)", "answer_v3": [ "-3*x", "3*y", "a*e^(-3*t)", "b*e^(3*t)" ], "answer_type_v3": [ "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0492", "subject": "Calculus_-_multivariable", "topic": "Vector fields", "subtopic": "Graphs, flows lines, and level surfaces", "level": "3", "keywords": [ "vector field", "multivariable", "calculus" ], "problem_v1": "Consider the vector field $\\vec v=-y\\,\\vec i+6x\\,\\vec j$ and function $f(x,y)=6x^{2}+y^{2}$. In this problem we show that the flow lines of the vector field are level curves of the function $f$.\n(a) Suppose that $\\vec r(t)=x(t)\\,\\vec i+y(t)\\,\\vec j$ is a flow line of $\\vec v$. Let $g(t)=f(\\vec r(t))$. If $\\vec r$ is a level curve of $f(x,y)$, that is $g'(t)$? $g'(t)=$ [ANS]\n(b) Use the definition of $f$ to find $g'$. (Note that $x$ and $y$ are functions of $t$, so that your expression should involve factors of $x$, $y$, $x'$ and $y'$ ; enter $x$, $y$, $x'$ and $y'$ in your answer rather than $x(t)$, $y(t)$, $x'(t)$ and $y'(t)$.) $g'=$ [ANS]\n(c) Knowing that $\\vec r$ is a flow line of the vector field $\\vec v$, what are $x'$ and $y'$? $x'=$ [ANS]\n$y'=$ [ANS]\n(d) Substituting these into your result from (b), what do you get? $g'=$ [ANS]\n(Note that this confirms our expectation from\n(a), showing that the flow lines are level curves of $f$.)", "answer_v1": [ "0", "2*6*x*x", "-y", "6*x", "2*-1*6*x*y+2*1*6*y*x" ], "answer_type_v1": [ "NV", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Consider the vector field $\\vec v=-y\\,\\vec i+2x\\,\\vec j$ and function $f(x,y)=2x^{2}+y^{2}$. In this problem we show that the flow lines of the vector field are level curves of the function $f$.\n(a) Suppose that $\\vec r(t)=x(t)\\,\\vec i+y(t)\\,\\vec j$ is a flow line of $\\vec v$. Let $g(t)=f(\\vec r(t))$. If $\\vec r$ is a level curve of $f(x,y)$, that is $g'(t)$? $g'(t)=$ [ANS]\n(b) Use the definition of $f$ to find $g'$. (Note that $x$ and $y$ are functions of $t$, so that your expression should involve factors of $x$, $y$, $x'$ and $y'$ ; enter $x$, $y$, $x'$ and $y'$ in your answer rather than $x(t)$, $y(t)$, $x'(t)$ and $y'(t)$.) $g'=$ [ANS]\n(c) Knowing that $\\vec r$ is a flow line of the vector field $\\vec v$, what are $x'$ and $y'$? $x'=$ [ANS]\n$y'=$ [ANS]\n(d) Substituting these into your result from (b), what do you get? $g'=$ [ANS]\n(Note that this confirms our expectation from\n(a), showing that the flow lines are level curves of $f$.)", "answer_v2": [ "0", "2*2*x*x", "-y", "2*x", "2*-1*2*x*y+2*1*2*y*x" ], "answer_type_v2": [ "NV", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Consider the vector field $\\vec v=-y\\,\\vec i+3x\\,\\vec j$ and function $f(x,y)=3x^{2}+y^{2}$. In this problem we show that the flow lines of the vector field are level curves of the function $f$.\n(a) Suppose that $\\vec r(t)=x(t)\\,\\vec i+y(t)\\,\\vec j$ is a flow line of $\\vec v$. Let $g(t)=f(\\vec r(t))$. If $\\vec r$ is a level curve of $f(x,y)$, that is $g'(t)$? $g'(t)=$ [ANS]\n(b) Use the definition of $f$ to find $g'$. (Note that $x$ and $y$ are functions of $t$, so that your expression should involve factors of $x$, $y$, $x'$ and $y'$ ; enter $x$, $y$, $x'$ and $y'$ in your answer rather than $x(t)$, $y(t)$, $x'(t)$ and $y'(t)$.) $g'=$ [ANS]\n(c) Knowing that $\\vec r$ is a flow line of the vector field $\\vec v$, what are $x'$ and $y'$? $x'=$ [ANS]\n$y'=$ [ANS]\n(d) Substituting these into your result from (b), what do you get? $g'=$ [ANS]\n(Note that this confirms our expectation from\n(a), showing that the flow lines are level curves of $f$.)", "answer_v3": [ "0", "2*3*x*x", "-y", "3*x", "2*-1*3*x*y+2*1*3*y*x" ], "answer_type_v3": [ "NV", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0493", "subject": "Calculus_-_multivariable", "topic": "Vector fields", "subtopic": "Graphs, flows lines, and level surfaces", "level": "2", "keywords": [ "vector field", "multivariable", "calculus" ], "problem_v1": "Write a formula for a two-dimensional vector field which has all vectors of length 7 and perpendicular to the position vector at that point. $\\vec V=$ [ANS]", "answer_v1": [ "7*y/[sqrt(x^2+y^2)]i-7*x/[sqrt(x^2+y^2)]j" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Write a formula for a two-dimensional vector field which has all vectors of length 1 and perpendicular to the position vector at that point. $\\vec V=$ [ANS]", "answer_v2": [ "(1*y/[sqrt(x^2+y^2)],-1*x/[sqrt(x^2+y^2)])" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Write a formula for a two-dimensional vector field which has all vectors of length 3 and perpendicular to the position vector at that point. $\\vec V=$ [ANS]", "answer_v3": [ "(3*y/[sqrt(x^2+y^2)],-3*x/[sqrt(x^2+y^2)])" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0495", "subject": "Calculus_-_multivariable", "topic": "Vector fields", "subtopic": "Graphs, flows lines, and level surfaces", "level": "5", "keywords": [ "Vector Field" ], "problem_v1": "Find a formula $\\vec{F}=\\langle\\ F_1(x,y),\\ F_2(x,y)\\ \\rangle$ for the vector field in the plane that has the properties that $\\vec{F}(0,0)=\\langle 0,0 \\rangle$ and that at any other point $(a,b) \\ne (0,0)$ the vector field $\\vec{F}$ is tangent to the circle $x^2+y^2=a^2+b^2$ and points in the counterclockwise direction with magnitude $\\| \\vec{F}(a,b) \\|=5 \\sqrt{a^2+b^2}$.\n$\\vec{F}=$ [ANS]", "answer_v1": [ "((-5)*y,5*x)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Find a formula $\\vec{F}=\\langle\\ F_1(x,y),\\ F_2(x,y)\\ \\rangle$ for the vector field in the plane that has the properties that $\\vec{F}(0,0)=\\langle 0,0 \\rangle$ and that at any other point $(a,b) \\ne (0,0)$ the vector field $\\vec{F}$ is tangent to the circle $x^2+y^2=a^2+b^2$ and points in the counterclockwise direction with magnitude $\\| \\vec{F}(a,b) \\|=2 \\sqrt{a^2+b^2}$.\n$\\vec{F}=$ [ANS]", "answer_v2": [ "((-2)*y,2*x)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Find a formula $\\vec{F}=\\langle\\ F_1(x,y),\\ F_2(x,y)\\ \\rangle$ for the vector field in the plane that has the properties that $\\vec{F}(0,0)=\\langle 0,0 \\rangle$ and that at any other point $(a,b) \\ne (0,0)$ the vector field $\\vec{F}$ is tangent to the circle $x^2+y^2=a^2+b^2$ and points in the counterclockwise direction with magnitude $\\| \\vec{F}(a,b) \\|=3 \\sqrt{a^2+b^2}$.\n$\\vec{F}=$ [ANS]", "answer_v3": [ "((-3)*y,3*x)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0496", "subject": "Calculus_-_multivariable", "topic": "Vector fields", "subtopic": "Graphs, flows lines, and level surfaces", "level": "3", "keywords": [ "Vector Field" ], "problem_v1": "Suppose the vector field $\\vec{F}$ takes the values shown in the table below. Sketch this vector field on a piece of paper and find a formula for it.\n$\\vec{F}=$ [ANS]\nValues of $\\vec{F}$\n$\\begin{array}{cccccc}\\hline y=2 & \\left<-10,-10\\right> & \\left<-10,-5\\right> & \\left<-10,0\\right> & \\left<-10,5\\right> & \\left<-10,10\\right> \\\\ \\hline y=1 & \\left<-5,-10\\right> & \\left<-5,-5\\right> & \\left<-5,0\\right> & \\left<-5,5\\right> & \\left<-5,10\\right> \\\\ \\hline y=0 & \\left<0,-10\\right> & \\left<0,-5\\right> & \\left<0,0\\right> & \\left<0,5\\right> & \\left<0,10\\right> \\\\ \\hline y=-1 & \\left<5,-10\\right> & \\left<5,-5\\right> & \\left<5,0\\right> & \\left<5,5\\right> & \\left<5,10\\right> \\\\ \\hline y=-2 & \\left<10,-10\\right> & \\left<10,-5\\right> & \\left<10,0\\right> & \\left<10,5\\right> & \\left<10,10\\right> \\\\ \\hline & x=-2 & x=-1 & x=0 & x=1 & x=2 \\\\ \\hline \\end{array}$", "answer_v1": [ "(-5*y,5*x)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Suppose the vector field $\\vec{F}$ takes the values shown in the table below. Sketch this vector field on a piece of paper and find a formula for it.\n$\\vec{F}=$ [ANS]\nValues of $\\vec{F}$\n$\\begin{array}{cccccc}\\hline y=2 & \\left<-4,-4\\right> & \\left<-4,-2\\right> & \\left<-4,0\\right> & \\left<-4,2\\right> & \\left<-4,4\\right> \\\\ \\hline y=1 & \\left<-2,-4\\right> & \\left<-2,-2\\right> & \\left<-2,0\\right> & \\left<-2,2\\right> & \\left<-2,4\\right> \\\\ \\hline y=0 & \\left<0,-4\\right> & \\left<0,-2\\right> & \\left<0,0\\right> & \\left<0,2\\right> & \\left<0,4\\right> \\\\ \\hline y=-1 & \\left<2,-4\\right> & \\left<2,-2\\right> & \\left<2,0\\right> & \\left<2,2\\right> & \\left<2,4\\right> \\\\ \\hline y=-2 & \\left<4,-4\\right> & \\left<4,-2\\right> & \\left<4,0\\right> & \\left<4,2\\right> & \\left<4,4\\right> \\\\ \\hline & x=-2 & x=-1 & x=0 & x=1 & x=2 \\\\ \\hline \\end{array}$", "answer_v2": [ "(-2*y,2*x)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Suppose the vector field $\\vec{F}$ takes the values shown in the table below. Sketch this vector field on a piece of paper and find a formula for it.\n$\\vec{F}=$ [ANS]\nValues of $\\vec{F}$\n$\\begin{array}{cccccc}\\hline y=2 & \\left<-6,-6\\right> & \\left<-6,-3\\right> & \\left<-6,0\\right> & \\left<-6,3\\right> & \\left<-6,6\\right> \\\\ \\hline y=1 & \\left<-3,-6\\right> & \\left<-3,-3\\right> & \\left<-3,0\\right> & \\left<-3,3\\right> & \\left<-3,6\\right> \\\\ \\hline y=0 & \\left<0,-6\\right> & \\left<0,-3\\right> & \\left<0,0\\right> & \\left<0,3\\right> & \\left<0,6\\right> \\\\ \\hline y=-1 & \\left<3,-6\\right> & \\left<3,-3\\right> & \\left<3,0\\right> & \\left<3,3\\right> & \\left<3,6\\right> \\\\ \\hline y=-2 & \\left<6,-6\\right> & \\left<6,-3\\right> & \\left<6,0\\right> & \\left<6,3\\right> & \\left<6,6\\right> \\\\ \\hline & x=-2 & x=-1 & x=0 & x=1 & x=2 \\\\ \\hline \\end{array}$", "answer_v3": [ "(-3*y,3*x)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0497", "subject": "Calculus_-_multivariable", "topic": "Vector fields", "subtopic": "Graphs, flows lines, and level surfaces", "level": "3", "keywords": [ "Vector Field" ], "problem_v1": "Sketch the vector field $ \\vec{F}(x,y)=- \\frac{\\vec{r}}{||\\,\\vec{r} \\,||^3}$ in the plane, where $\\vec{r}=\\langle x, y \\rangle.$ Select all that apply. [ANS] A. All the vectors point toward the origin. B. The length of each vector is 1. C. The vectors increase in length as you move away from the origin. D. All the vectors point away from the origin. E. The vectors decrease in length as you move away from the origin.", "answer_v1": [ "AE" ], "answer_type_v1": [ "MCM" ], "options_v1": [ [ "A", "B", "C", "D", "E" ] ], "problem_v2": "Sketch the vector field $ \\vec{F}(x,y)=- \\frac{\\vec{r}}{||\\,\\vec{r} \\,||^3}$ in the plane, where $\\vec{r}=\\langle x, y \\rangle.$ Select all that apply. [ANS] A. The vectors decrease in length as you move away from the origin. B. The length of each vector is 1. C. The vectors increase in length as you move away from the origin. D. All the vectors point toward the origin. E. All the vectors point away from the origin.", "answer_v2": [ "AD" ], "answer_type_v2": [ "MCM" ], "options_v2": [ [ "A", "B", "C", "D", "E" ] ], "problem_v3": "Sketch the vector field $ \\vec{F}(x,y)=- \\frac{\\vec{r}}{||\\,\\vec{r} \\,||^3}$ in the plane, where $\\vec{r}=\\langle x, y \\rangle.$ Select all that apply. [ANS] A. All the vectors point away from the origin. B. The vectors decrease in length as you move away from the origin. C. The length of each vector is 1. D. The vectors increase in length as you move away from the origin. E. All the vectors point toward the origin.", "answer_v3": [ "BE" ], "answer_type_v3": [ "MCM" ], "options_v3": [ [ "A", "B", "C", "D", "E" ] ] }, { "id": "Calculus_-_multivariable_0499", "subject": "Calculus_-_multivariable", "topic": "Vector fields", "subtopic": "Graphs, flows lines, and level surfaces", "level": "3", "keywords": [ "calculus", "vector field", "flow line" ], "problem_v1": "The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus, the vectors in a vector field are tangent to the flow lines. Consider the vector field $\\mathbf{F}(x,y,z)=\\langle 5y, 5x, 2z \\rangle$. Show that $\\mathbf{r}(t)=\\langle e^{5t}+e^{-5t}, e^{5t}-e^{-5t}, e^{2t}\\rangle$ is a flowline for the vector field $\\mathbf{F}$. That is, verify that (I trust you here) $\\mathbf{r'}(t)=\\mathbf{F}(\\mathbf{r}(t))=\\langle$ [ANS], [ANS], [ANS] $\\rangle$. Now consider the curve $\\mathbf{r}(t)=\\langle \\cos(5t), \\sin(5t), e^{2t}\\rangle$. It is not a flowline of the vector field $\\mathbf{F}$, but of a vector field $\\mathbf{G}$ which differs in definition from $\\mathbf{F}$ only slightly. $\\mathbf{G}(x,y,z)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$.", "answer_v1": [ "5 * (exp(5 * t) - exp(- 5 * t))", "5 * (exp(5 * t) + exp(- 5 * t))", "2 * exp(2 * t)", "- 5 * y", "5 * x", "2 * z" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus, the vectors in a vector field are tangent to the flow lines. Consider the vector field $\\mathbf{F}(x,y,z)=\\langle-8y,-8x, 8z \\rangle$. Show that $\\mathbf{r}(t)=\\langle e^{-8t}+e^{8t}, e^{-8t}-e^{8t}, e^{8t}\\rangle$ is a flowline for the vector field $\\mathbf{F}$. That is, verify that (I trust you here) $\\mathbf{r'}(t)=\\mathbf{F}(\\mathbf{r}(t))=\\langle$ [ANS], [ANS], [ANS] $\\rangle$. Now consider the curve $\\mathbf{r}(t)=\\langle \\cos(-8t), \\sin(-8t), e^{8t}\\rangle$. It is not a flowline of the vector field $\\mathbf{F}$, but of a vector field $\\mathbf{G}$ which differs in definition from $\\mathbf{F}$ only slightly. $\\mathbf{G}(x,y,z)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$.", "answer_v2": [ "-8 * (exp(-8 * t) - exp(- -8 * t))", "-8 * (exp(-8 * t) + exp(- -8 * t))", "8 * exp(8 * t)", "- -8 * y", "-8 * x", "8 * z" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus, the vectors in a vector field are tangent to the flow lines. Consider the vector field $\\mathbf{F}(x,y,z)=\\langle-4y,-4x, 2z \\rangle$. Show that $\\mathbf{r}(t)=\\langle e^{-4t}+e^{4t}, e^{-4t}-e^{4t}, e^{2t}\\rangle$ is a flowline for the vector field $\\mathbf{F}$. That is, verify that (I trust you here) $\\mathbf{r'}(t)=\\mathbf{F}(\\mathbf{r}(t))=\\langle$ [ANS], [ANS], [ANS] $\\rangle$. Now consider the curve $\\mathbf{r}(t)=\\langle \\cos(-4t), \\sin(-4t), e^{2t}\\rangle$. It is not a flowline of the vector field $\\mathbf{F}$, but of a vector field $\\mathbf{G}$ which differs in definition from $\\mathbf{F}$ only slightly. $\\mathbf{G}(x,y,z)=\\langle$ [ANS], [ANS], [ANS] $\\rangle$.", "answer_v3": [ "-4 * (exp(-4 * t) - exp(- -4 * t))", "-4 * (exp(-4 * t) + exp(- -4 * t))", "2 * exp(2 * t)", "- -4 * y", "-4 * x", "2 * z" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0501", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Let $f(x,y,z)=x+yz$ and let $\\mathcal{C}$ be the line segment from $P=(0,0,0)$ to $(8,6,6)$.\n(a) Calculate $f(\\mathbf{c}(t))$ and $\\,ds=\\Vert \\mathbf{c}'(t)\\Vert\\,dt$ for the parametrization $\\mathbf{c}(t)=(8 t,6 t,6 t)$ for $0\\le t \\le 1$. (b) Evaluate $\\int_{\\mathcal{C}} f(x,y,z)\\,ds$.\n(a) $f(\\mathbf{c}(t))=$ [ANS], $\\,ds=$ [ANS] $\\,dt$ (b) $\\int_{\\mathcal{C}} f(x,y,z)\\,ds=$ [ANS]", "answer_v1": [ "8*t+36*t^2", "11.6619", "186.59" ], "answer_type_v1": [ "EX", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $f(x,y,z)=x+yz$ and let $\\mathcal{C}$ be the line segment from $P=(0,0,0)$ to $(2,9,2)$.\n(a) Calculate $f(\\mathbf{c}(t))$ and $\\,ds=\\Vert \\mathbf{c}'(t)\\Vert\\,dt$ for the parametrization $\\mathbf{c}(t)=(2 t,9 t,2 t)$ for $0\\le t \\le 1$. (b) Evaluate $\\int_{\\mathcal{C}} f(x,y,z)\\,ds$.\n(a) $f(\\mathbf{c}(t))=$ [ANS], $\\,ds=$ [ANS] $\\,dt$ (b) $\\int_{\\mathcal{C}} f(x,y,z)\\,ds=$ [ANS]", "answer_v2": [ "2*t+18*t^2", "9.43398", "66.0379" ], "answer_type_v2": [ "EX", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $f(x,y,z)=x+yz$ and let $\\mathcal{C}$ be the line segment from $P=(0,0,0)$ to $(4,6,3)$.\n(a) Calculate $f(\\mathbf{c}(t))$ and $\\,ds=\\Vert \\mathbf{c}'(t)\\Vert\\,dt$ for the parametrization $\\mathbf{c}(t)=(4 t,6 t,3 t)$ for $0\\le t \\le 1$. (b) Evaluate $\\int_{\\mathcal{C}} f(x,y,z)\\,ds$.\n(a) $f(\\mathbf{c}(t))=$ [ANS], $\\,ds=$ [ANS] $\\,dt$ (b) $\\int_{\\mathcal{C}} f(x,y,z)\\,ds=$ [ANS]", "answer_v3": [ "4*t+18*t^2", "7.81025", "62.482" ], "answer_type_v3": [ "EX", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0502", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Compute the line integral of the vector field $\\mathbf{F}=\\left< 3zy^{-1}, 4x,-y\\right>$ over the path $\\mathbf{c}(t)=(e^t,e^t,t)$ for $-7\\le t \\le 7$ $\\int_{\\mathcal{C}} \\mathbf{F} \\cdot \\,d\\mathbf{\\,s}=$ [ANS]", "answer_v1": [ "2.40411E+6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the line integral of the vector field $\\mathbf{F}=\\left< 3zy^{-1}, 4x,-y\\right>$ over the path $\\mathbf{c}(t)=(e^t,e^t,t)$ for $-1\\le t \\le 1$ $\\int_{\\mathcal{C}} \\mathbf{F} \\cdot \\,d\\mathbf{\\,s}=$ [ANS]", "answer_v2": [ "12.157" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the line integral of the vector field $\\mathbf{F}=\\left< 3zy^{-1}, 4x,-y\\right>$ over the path $\\mathbf{c}(t)=(e^t,e^t,t)$ for $-3\\le t \\le 3$ $\\int_{\\mathcal{C}} \\mathbf{F} \\cdot \\,d\\mathbf{\\,s}=$ [ANS]", "answer_v3": [ "786.817" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0503", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Compute the line integral of the vector field $\\mathbf{F}=\\left< 6 y,-6x\\right>$ over the circle $x^2+y^2=49$ oriented clockwise $\\int_{\\mathcal{C}} \\mathbf{F} \\cdot \\,d\\mathbf{\\,s}=$ [ANS]", "answer_v1": [ "1847.26" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the line integral of the vector field $\\mathbf{F}=\\left< 9 y,-9x\\right>$ over the circle $x^2+y^2=1$ oriented clockwise $\\int_{\\mathcal{C}} \\mathbf{F} \\cdot \\,d\\mathbf{\\,s}=$ [ANS]", "answer_v2": [ "56.5487" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the line integral of the vector field $\\mathbf{F}=\\left< 6 y,-6x\\right>$ over the circle $x^2+y^2=9$ oriented clockwise $\\int_{\\mathcal{C}} \\mathbf{F} \\cdot \\,d\\mathbf{\\,s}=$ [ANS]", "answer_v3": [ "339.292" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0504", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Use a CAS to calculate $\\int_{\\mathcal{C}} \\left< e^{x-y},e^{x+y} \\right>\\cdot \\,d\\mathbf{\\,s}$ to four decimal places, where $\\mathcal{C}$ is the curve $y=\\sin x$ for $0\\le x \\le \\frac{\\pi}{8} $, oriented from left to right. Answer: [ANS]", "answer_v1": [ "0.970327" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use a CAS to calculate $\\int_{\\mathcal{C}} \\left< e^{x-y},e^{x+y} \\right>\\cdot \\,d\\mathbf{\\,s}$ to four decimal places, where $\\mathcal{C}$ is the curve $y=\\sin x$ for $0\\le x \\le \\frac{\\pi}{3} $, oriented from left to right. Answer: [ANS]", "answer_v2": [ "3.5682" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use a CAS to calculate $\\int_{\\mathcal{C}} \\left< e^{x-y},e^{x+y} \\right>\\cdot \\,d\\mathbf{\\,s}$ to four decimal places, where $\\mathcal{C}$ is the curve $y=\\sin x$ for $0\\le x \\le \\frac{\\pi}{5} $, oriented from left to right. Answer: [ANS]", "answer_v3": [ "1.76965" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0505", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Compute the line integral of the scalar function $f(x,y,z)=2x^2+8z$ over the curve $\\mathbf{c}(t)=(e^t,t^2,t) \\text{,} \\quad 0\\le t\\le 7$ $\\int_{\\mathcal{C}} f(x,y,z) \\,ds=$ [ANS]", "answer_v1": [ "8.79427E+8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the line integral of the scalar function $f(x,y,z)=2x^2+8z$ over the curve $\\mathbf{c}(t)=(e^t,t^2,t) \\text{,} \\quad 0\\le t\\le 1$ $\\int_{\\mathcal{C}} f(x,y,z) \\,ds=$ [ANS]", "answer_v2": [ "27.1858" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the line integral of the scalar function $f(x,y,z)=2x^2+8z$ over the curve $\\mathbf{c}(t)=(e^t,t^2,t) \\text{,} \\quad 0\\le t\\le 3$ $\\int_{\\mathcal{C}} f(x,y,z) \\,ds=$ [ANS]", "answer_v3": [ "6160.12" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0506", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Compute the line integral of the scalar function $f(x,y)=\\sqrt{1+9xy}$ over the curve $y=x^3$ for $0\\le x \\le 7$ $\\int_{\\mathcal{C}} f(x,y) \\,ds=$ [ANS]", "answer_v1": [ "30259.6" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the line integral of the scalar function $f(x,y)=\\sqrt{1+9xy}$ over the curve $y=x^3$ for $0\\le x \\le 1$ $\\int_{\\mathcal{C}} f(x,y) \\,ds=$ [ANS]", "answer_v2": [ "2.8" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the line integral of the scalar function $f(x,y)=\\sqrt{1+9xy}$ over the curve $y=x^3$ for $0\\le x \\le 3$ $\\int_{\\mathcal{C}} f(x,y) \\,ds=$ [ANS]", "answer_v3": [ "440.4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0507", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "multivariable", "vector", "line integral", "work", "Vector Fields", "Line Integral" ], "problem_v1": "Let $\\bf F$ be the radial force field $\\mathbf{F}=x\\mathbf i+y\\mathbf j$. Find the work done by this force along the following two curves, both which go from (0, 0) to (8, 64). (Compare your answers!) A. If $C_1$ is the parabola: $x=t, \\ y=t^2, \\ 0 \\leq t \\leq 8$, then $ \\int_{C_1} \\mathbf{F} \\cdot \\, d\\mathbf{r}=$ [ANS]\nB. If $C_2$ is the straight line segment: $x=8 t^2, \\ y=64 t^2, \\ 0 \\leq t \\leq 1$, then $ \\int_{C_2} \\mathbf F \\cdot \\, d\\mathbf{r}=$ [ANS]", "answer_v1": [ "2080", "2080" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $\\bf F$ be the radial force field $\\mathbf{F}=x\\mathbf i+y\\mathbf j$. Find the work done by this force along the following two curves, both which go from (0, 0) to (1, 1). (Compare your answers!) A. If $C_1$ is the parabola: $x=t, \\ y=t^2, \\ 0 \\leq t \\leq 1$, then $ \\int_{C_1} \\mathbf{F} \\cdot \\, d\\mathbf{r}=$ [ANS]\nB. If $C_2$ is the straight line segment: $x=1 t^2, \\ y=1 t^2, \\ 0 \\leq t \\leq 1$, then $ \\int_{C_2} \\mathbf F \\cdot \\, d\\mathbf{r}=$ [ANS]", "answer_v2": [ "1", "1" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $\\bf F$ be the radial force field $\\mathbf{F}=x\\mathbf i+y\\mathbf j$. Find the work done by this force along the following two curves, both which go from (0, 0) to (4, 16). (Compare your answers!) A. If $C_1$ is the parabola: $x=t, \\ y=t^2, \\ 0 \\leq t \\leq 4$, then $ \\int_{C_1} \\mathbf{F} \\cdot \\, d\\mathbf{r}=$ [ANS]\nB. If $C_2$ is the straight line segment: $x=4 t^2, \\ y=16 t^2, \\ 0 \\leq t \\leq 1$, then $ \\int_{C_2} \\mathbf F \\cdot \\, d\\mathbf{r}=$ [ANS]", "answer_v3": [ "136", "136" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0508", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "integrals", "parametric curve", "multivariable", "calculus" ], "problem_v1": "Find $\\int_C\\vec F\\cdot d\\vec r$ for $\\vec F=7 y\\vec i-(\\sin y)\\vec j$ on the curve counterclockwise around the unit circle $C$ starting at the point $(1,0).$ $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "-1*7*pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $\\int_C\\vec F\\cdot d\\vec r$ for $\\vec F=2 y\\vec i-(\\sin y)\\vec j$ on the curve counterclockwise around the unit circle $C$ starting at the point $(1,0).$ $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "-1*2*pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $\\int_C\\vec F\\cdot d\\vec r$ for $\\vec F=4 y\\vec i-(\\sin y)\\vec j$ on the curve counterclockwise around the unit circle $C$ starting at the point $(1,0).$ $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "-1*4*pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0509", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "integrals", "parametric curve", "multivariable", "calculus" ], "problem_v1": "Find $\\int_C\\vec F\\cdot d\\vec r$ for $\\vec F=3 y\\,\\vec i+x\\,\\vec j+z\\,\\vec k$ if $C$ is the helix $x=\\cos t, y=\\sin t, z=t$, for $0\\le t\\le 4\\pi$. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "2*pi*(4*1*pi-3+1)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $\\int_C\\vec F\\cdot d\\vec r$ for $\\vec F=-5 y\\,\\vec i+5x\\,\\vec j-4 z\\,\\vec k$ if $C$ is the helix $x=\\cos t, y=\\sin t, z=t$, for $0\\le t\\le 4\\pi$. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "2*pi*(4*-4*pi--5+5)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $\\int_C\\vec F\\cdot d\\vec r$ for $\\vec F=-2 y\\,\\vec i+x\\,\\vec j-2 z\\,\\vec k$ if $C$ is the helix $x=\\cos t, y=\\sin t, z=t$, for $0\\le t\\le 4\\pi$. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "2*pi*(4*-2*pi--2+1)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0510", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "integrals", "parametric curve", "multivariable", "calculus" ], "problem_v1": "Curves $C_1$ and $C_2$ are parametrized as follows: C_1 \\mbox{is} (x(t),y(t))=(t,0)\\quad\\mbox{for}-1\\le t\\le1 and C_2 \\mbox{is} (x(t),y(t))=(\\cos t,\\sin t) \\quad\\mbox{for} 0 \\le t\\le \\pi. Sketch, on a separate sheet of paper, the curves $C_1$ and $C_2$ with arrows showing their orientation. Next, suppose that $\\vec F=5x\\,\\vec i+(5x+6 y)\\,\\vec j$. Calculate $\\int_C\\vec F\\cdot d\\vec r$, where $C$ is the curve given by $C=C_1+C_2$. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "5*pi/2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Curves $C_1$ and $C_2$ are parametrized as follows: C_1 \\mbox{is} (x(t),y(t))=(0,t)\\quad\\mbox{for}-1\\le t\\le1 and C_2 \\mbox{is} (x(t),y(t))=(\\cos t,\\sin t) \\quad\\mbox{for} \\pi/2 \\le t\\le 3\\pi/2. Sketch, on a separate sheet of paper, the curves $C_1$ and $C_2$ with arrows showing their orientation. Next, suppose that $\\vec F=(7x+2 y)\\,\\vec i+4 y\\,\\vec j$. Calculate $\\int_C\\vec F\\cdot d\\vec r$, where $C$ is the curve given by $C=C_1+C_2$. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "-1*2*pi/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Curves $C_1$ and $C_2$ are parametrized as follows: C_1 \\mbox{is} (x(t),y(t))=(0,t)\\quad\\mbox{for}-1\\le t\\le1 and C_2 \\mbox{is} (x(t),y(t))=(\\cos t,\\sin t) \\quad\\mbox{for} \\pi/2 \\le t\\le 3\\pi/2. Sketch, on a separate sheet of paper, the curves $C_1$ and $C_2$ with arrows showing their orientation. Next, suppose that $\\vec F=(5x+3 y)\\,\\vec i+5 y\\,\\vec j$. Calculate $\\int_C\\vec F\\cdot d\\vec r$, where $C$ is the curve given by $C=C_1+C_2$. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "-1*3*pi/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0511", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "integrals", "parametric curve", "multivariable", "calculus" ], "problem_v1": "Evaluate the line integral $\\int_C y\\, dx+x\\, dy$ where $C$ is the parameterized path $x=t^{3}$, $y=t^{2}$, $2\\le t \\le 7$. $\\int_C y\\, dx+x\\, dy=$ [ANS]", "answer_v1": [ "7^5-2^5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the line integral $\\int_C y\\, dx+x\\, dy$ where $C$ is the parameterized path $x=t^{2}$, $y=t^{3}$, $3\\le t \\le 5$. $\\int_C y\\, dx+x\\, dy=$ [ANS]", "answer_v2": [ "5^5-3^5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the line integral $\\int_C y\\, dx+x\\, dy$ where $C$ is the parameterized path $x=t^{2}$, $y=t^{3}$, $2\\le t \\le 5$. $\\int_C y\\, dx+x\\, dy=$ [ANS]", "answer_v3": [ "5^5-2^5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0512", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "2", "keywords": [ "line integral", "integrals", "parametric curve", "multivariable", "calculus" ], "problem_v1": "If $C$ is the line segment from $(5,4)$ to, $(0,0)$, find the value of the line integral: $\\int_C (7 y^2\\,\\vec i+3x\\,\\vec j)\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "-3*5*4/2-7*5*4^2/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $C$ is the line segment from $(2,2)$ to, $(0,0)$, find the value of the line integral: $\\int_C (2 y^2\\,\\vec i+5x\\,\\vec j)\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "-5*2*2/2-2*2*2^2/3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $C$ is the line segment from $(3,3)$ to, $(0,0)$, find the value of the line integral: $\\int_C (4 y^2\\,\\vec i+4x\\,\\vec j)\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "-4*3*3/2-4*3*3^2/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0513", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "4", "keywords": [ "line integral", "integrals", "parametric curve", "multivariable", "calculus" ], "problem_v1": "If $\\int_C \\vec F \\cdot d\\vec r=\\int_C (3x+y^{3})\\,dx+(x^{2} y)\\,dy$, what is $\\vec F$? $\\vec F=$ [ANS]", "answer_v1": [ "(3*x+y^3)i+x^2*yj" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "If $\\int_C \\vec F \\cdot d\\vec r=\\int_C (-5x+5 y^{2})\\,dx+(x^{3} y)\\,dy$, what is $\\vec F$? $\\vec F=$ [ANS]", "answer_v2": [ "(5*y^2-5*x,x^3*y)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "If $\\int_C \\vec F \\cdot d\\vec r=\\int_C (-2x+y^{3})\\,dx+(x^{2} y)\\,dy$, what is $\\vec F$? $\\vec F=$ [ANS]", "answer_v3": [ "(y^3-2*x,x^2*y)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0514", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "4", "keywords": [ "line integral", "integrals", "parametric curve", "multivariable", "calculus" ], "problem_v1": "Suppose $C$ is the line segment from the point $(0,0)$ to the point $(3,21)$ and $\\vec{F}=2xy\\,\\vec i+2x\\,\\vec j$.\n(a) Is $\\int_C \\vec{F} \\cdot d\\vec{r}$ greater than, less than, or equal to zero? [ANS] (Be sure you are able to give a geometric explanation for your answer.) (Be sure you are able to give a geometric explanation for your answer.) (b) A parameterization of $C$ is $(x(t),y(t))=(t,7 t)$ for $0 \\leq t \\leq 3$. Use this to compute $\\int_C \\vec{F} \\cdot d\\vec{r}$. $\\int_C \\vec{F} \\cdot d\\vec{r}=$ [ANS]\n(c) Suppose a particle leaves the point $(0,0)$, moves along the line toward the point $(3,21)$, stops before reaching it and backs up, stops again and reverses direction, then completes its journey to the endpoint. All travel takes place along the line segment joining the point $(0,0)$ to the point $(3,21)$. Call this path $C'$. How is $\\int_{C'} \\vec{F} \\cdot d\\vec{r}$ related to $\\int_{C} \\vec{F} \\cdot d\\vec{r}$? $\\int_{C'} \\vec{F} \\cdot d\\vec{r}$ [ANS] $\\int_{C} \\vec{F} \\cdot d\\vec{r}$ (Be sure you can explain why this is.) (Be sure you can explain why this is.) (d) A parameterization for a path like $C'$ is given by (x(t),y(t))=\\left( \\frac{1}{27} \\left(16\\,t^3-72\\,t^2+99\\,t\\right), \\frac{7}{27} \\left(16\\,t^3-72\\,t^2+99\\,t\\right))\\right). For what value of $t$ is this parameterization at $(0,0)$? $t=$ [ANS]\nat $(3,21)$? $t=$ [ANS]\nWhat equation must $(x(t),y(t))$ satisfy? [ANS]\n(Enter your equation in terms of $x$ and $y$: thus, if $y(t)=3x(t)^2-6$, you would enter $y=3x^2-6$.). By hand, on a separate sheet of paper, verify that this equation is in fact satisfied by this parameterization. At what values of $t$ does the particle change direction? At $t=$ [ANS] and [ANS]\n(e) Set up and find $\\int_{C'} \\vec{F} \\cdot d\\vec{r}$ using the parameterization in part (d). With $a=$ [ANS] and $b=$ [ANS], $\\int_{C'} \\vec{F} \\cdot d\\vec{r}=\\int_a^b$ [ANS] $dt=$ [ANS]\n(Note whether you get the same answer as you did in part (b).)", "answer_v1": [ "greater than zero", "2*7*3^3/3+2*7*3^2/2", "is equal to", "0", "3", "y = 7*x", "3/2-3*sqrt(3)/12", "3/2+3*sqrt(3)/12", "0", "3", "1/(27*3^6)*7*t*(768*t^4-1920*t^3*3+1856*t^2*3^2-792*t*3^3+121*3^4)*[3*2*3^2+2*t*(16*t^2-24*t*3+11*3^2)]", "2*7*3^3/3+2*7*3^2/2" ], "answer_type_v1": [ "MCS", "NV", "MCS", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v1": [ [ "greater than zero", "less than zero", "equal to zero" ], [], [ "is greater than", "is less than", "is equal to" ], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Suppose $C$ is the line segment from the point $(0,0)$ to the point $(3,6)$ and $\\vec{F}=-5xy\\,\\vec i-2x\\,\\vec j$.\n(a) Is $\\int_C \\vec{F} \\cdot d\\vec{r}$ greater than, less than, or equal to zero? [ANS] (Be sure you are able to give a geometric explanation for your answer.) (Be sure you are able to give a geometric explanation for your answer.) (b) A parameterization of $C$ is $(x(t),y(t))=(t,2 t)$ for $0 \\leq t \\leq 3$. Use this to compute $\\int_C \\vec{F} \\cdot d\\vec{r}$. $\\int_C \\vec{F} \\cdot d\\vec{r}=$ [ANS]\n(c) Suppose a particle leaves the point $(0,0)$, moves along the line toward the point $(3,6)$, stops before reaching it and backs up, stops again and reverses direction, then completes its journey to the endpoint. All travel takes place along the line segment joining the point $(0,0)$ to the point $(3,6)$. Call this path $C'$. How is $\\int_{C'} \\vec{F} \\cdot d\\vec{r}$ related to $\\int_{C} \\vec{F} \\cdot d\\vec{r}$? $\\int_{C'} \\vec{F} \\cdot d\\vec{r}$ [ANS] $\\int_{C} \\vec{F} \\cdot d\\vec{r}$ (Be sure you can explain why this is.) (Be sure you can explain why this is.) (d) A parameterization for a path like $C'$ is given by (x(t),y(t))=\\left( \\frac{1}{27} \\left(16\\,t^3-72\\,t^2+99\\,t\\right), \\frac{2}{27} \\left(16\\,t^3-72\\,t^2+99\\,t\\right))\\right). For what value of $t$ is this parameterization at $(0,0)$? $t=$ [ANS]\nat $(3,6)$? $t=$ [ANS]\nWhat equation must $(x(t),y(t))$ satisfy? [ANS]\n(Enter your equation in terms of $x$ and $y$: thus, if $y(t)=3x(t)^2-6$, you would enter $y=3x^2-6$.). By hand, on a separate sheet of paper, verify that this equation is in fact satisfied by this parameterization. At what values of $t$ does the particle change direction? At $t=$ [ANS] and [ANS]\n(e) Set up and find $\\int_{C'} \\vec{F} \\cdot d\\vec{r}$ using the parameterization in part (d). With $a=$ [ANS] and $b=$ [ANS], $\\int_{C'} \\vec{F} \\cdot d\\vec{r}=\\int_a^b$ [ANS] $dt=$ [ANS]\n(Note whether you get the same answer as you did in part (b).)", "answer_v2": [ "less than zero", "-5*2*3^3/3+-2*2*3^2/2", "is equal to", "0", "3", "y = 2*x", "3/2-3*sqrt(3)/12", "3/2+3*sqrt(3)/12", "0", "3", "1/(27*3^6)*2*t*(768*t^4-1920*t^3*3+1856*t^2*3^2-792*t*3^3+121*3^4)*[3*-2*3^2+-5*t*(16*t^2-24*t*3+11*3^2)]", "-5*2*3^3/3+-2*2*3^2/2" ], "answer_type_v2": [ "MCS", "NV", "MCS", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v2": [ [ "greater than zero", "less than zero", "equal to zero" ], [], [ "is greater than", "is less than", "is equal to" ], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Suppose $C$ is the line segment from the point $(0,0)$ to the point $(3,12)$ and $\\vec{F}=2xy\\,\\vec i+2x\\,\\vec j$.\n(a) Is $\\int_C \\vec{F} \\cdot d\\vec{r}$ greater than, less than, or equal to zero? [ANS] (Be sure you are able to give a geometric explanation for your answer.) (Be sure you are able to give a geometric explanation for your answer.) (b) A parameterization of $C$ is $(x(t),y(t))=(t,4 t)$ for $0 \\leq t \\leq 3$. Use this to compute $\\int_C \\vec{F} \\cdot d\\vec{r}$. $\\int_C \\vec{F} \\cdot d\\vec{r}=$ [ANS]\n(c) Suppose a particle leaves the point $(0,0)$, moves along the line toward the point $(3,12)$, stops before reaching it and backs up, stops again and reverses direction, then completes its journey to the endpoint. All travel takes place along the line segment joining the point $(0,0)$ to the point $(3,12)$. Call this path $C'$. How is $\\int_{C'} \\vec{F} \\cdot d\\vec{r}$ related to $\\int_{C} \\vec{F} \\cdot d\\vec{r}$? $\\int_{C'} \\vec{F} \\cdot d\\vec{r}$ [ANS] $\\int_{C} \\vec{F} \\cdot d\\vec{r}$ (Be sure you can explain why this is.) (Be sure you can explain why this is.) (d) A parameterization for a path like $C'$ is given by (x(t),y(t))=\\left( \\frac{1}{27} \\left(16\\,t^3-72\\,t^2+99\\,t\\right), \\frac{4}{27} \\left(16\\,t^3-72\\,t^2+99\\,t\\right))\\right). For what value of $t$ is this parameterization at $(0,0)$? $t=$ [ANS]\nat $(3,12)$? $t=$ [ANS]\nWhat equation must $(x(t),y(t))$ satisfy? [ANS]\n(Enter your equation in terms of $x$ and $y$: thus, if $y(t)=3x(t)^2-6$, you would enter $y=3x^2-6$.). By hand, on a separate sheet of paper, verify that this equation is in fact satisfied by this parameterization. At what values of $t$ does the particle change direction? At $t=$ [ANS] and [ANS]\n(e) Set up and find $\\int_{C'} \\vec{F} \\cdot d\\vec{r}$ using the parameterization in part (d). With $a=$ [ANS] and $b=$ [ANS], $\\int_{C'} \\vec{F} \\cdot d\\vec{r}=\\int_a^b$ [ANS] $dt=$ [ANS]\n(Note whether you get the same answer as you did in part (b).)", "answer_v3": [ "greater than zero", "2*4*3^3/3+2*4*3^2/2", "is equal to", "0", "3", "y = 4*x", "3/2-3*sqrt(3)/12", "3/2+3*sqrt(3)/12", "0", "3", "1/(27*3^6)*4*t*(768*t^4-1920*t^3*3+1856*t^2*3^2-792*t*3^3+121*3^4)*[3*2*3^2+2*t*(16*t^2-24*t*3+11*3^2)]", "2*4*3^3/3+2*4*3^2/2" ], "answer_type_v3": [ "MCS", "NV", "MCS", "NV", "NV", "EX", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v3": [ [ "greater than zero", "less than zero", "equal to zero" ], [], [ "is greater than", "is less than", "is equal to" ], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0515", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "4", "keywords": [ "line integral", "integrals", "parametric curve", "multivariable", "calculus" ], "problem_v1": "Let $\\vec F=-5 y\\vec i+5x\\vec j$ and let $C$ be the unit circle oriented counterclockwise, parameterized with the parameter $t$.\n(a) Parameterize $C$. $x(t)=$ [ANS]\n$y(t)=$ [ANS]\nwith [ANS] $\\le t\\le$ [ANS]\n(Note that you have to provide answers for all of these blanks for it to be possible to evaluate whether your parameterization is correct.) (Note that you have to provide answers for all of these blanks for it to be possible to evaluate whether your parameterization is correct.) (b) With the parameterization in (a), what is a vector $\\vec v(t)$ that is tangent to the circle $C$? $\\vec v(t)=$ [ANS] $\\vec i+$ [ANS] $\\vec j$ (c) With the same parameterization, what is $F$ on $C$? $\\vec F=$ [ANS] $\\vec i+$ [ANS] $\\vec j$ (Note that your results from (c) and (d) show that $\\vec F$ is tangent to $C$ at all points.) (d) Find $\\Vert\\vec{F}\\Vert$ on $C$: $\\Vert\\vec{F}\\Vert=$ [ANS]\n(e) Find $\\int_C\\vec F\\cdot d\\vec r$. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n(Note how is this related to the length of the curve $C$ (that is, the circumference of the circle).)", "answer_v1": [ "cos(t)", "sin(t)", "0", "2*pi", "-[sin(t)]", "cos(t)", "-5*sin(t)", "5*cos(t)", "5", "2*5*pi" ], "answer_type_v1": [ "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $\\vec F=-2 y\\vec i+2x\\vec j$ and let $C$ be the unit circle oriented counterclockwise, parameterized with the parameter $t$.\n(a) Parameterize $C$. $x(t)=$ [ANS]\n$y(t)=$ [ANS]\nwith [ANS] $\\le t\\le$ [ANS]\n(Note that you have to provide answers for all of these blanks for it to be possible to evaluate whether your parameterization is correct.) (Note that you have to provide answers for all of these blanks for it to be possible to evaluate whether your parameterization is correct.) (b) With the parameterization in (a), what is a vector $\\vec v(t)$ that is tangent to the circle $C$? $\\vec v(t)=$ [ANS] $\\vec i+$ [ANS] $\\vec j$ (c) With the same parameterization, what is $F$ on $C$? $\\vec F=$ [ANS] $\\vec i+$ [ANS] $\\vec j$ (Note that your results from (c) and (d) show that $\\vec F$ is tangent to $C$ at all points.) (d) Find $\\Vert\\vec{F}\\Vert$ on $C$: $\\Vert\\vec{F}\\Vert=$ [ANS]\n(e) Find $\\int_C\\vec F\\cdot d\\vec r$. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n(Note how is this related to the length of the curve $C$ (that is, the circumference of the circle).)", "answer_v2": [ "cos(t)", "sin(t)", "0", "2*pi", "-[sin(t)]", "cos(t)", "-2*sin(t)", "2*cos(t)", "2", "2*2*pi" ], "answer_type_v2": [ "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $\\vec F=-3 y\\vec i+3x\\vec j$ and let $C$ be the unit circle oriented counterclockwise, parameterized with the parameter $t$.\n(a) Parameterize $C$. $x(t)=$ [ANS]\n$y(t)=$ [ANS]\nwith [ANS] $\\le t\\le$ [ANS]\n(Note that you have to provide answers for all of these blanks for it to be possible to evaluate whether your parameterization is correct.) (Note that you have to provide answers for all of these blanks for it to be possible to evaluate whether your parameterization is correct.) (b) With the parameterization in (a), what is a vector $\\vec v(t)$ that is tangent to the circle $C$? $\\vec v(t)=$ [ANS] $\\vec i+$ [ANS] $\\vec j$ (c) With the same parameterization, what is $F$ on $C$? $\\vec F=$ [ANS] $\\vec i+$ [ANS] $\\vec j$ (Note that your results from (c) and (d) show that $\\vec F$ is tangent to $C$ at all points.) (d) Find $\\Vert\\vec{F}\\Vert$ on $C$: $\\Vert\\vec{F}\\Vert=$ [ANS]\n(e) Find $\\int_C\\vec F\\cdot d\\vec r$. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n(Note how is this related to the length of the curve $C$ (that is, the circumference of the circle).)", "answer_v3": [ "cos(t)", "sin(t)", "0", "2*pi", "-[sin(t)]", "cos(t)", "-3*sin(t)", "3*cos(t)", "3", "2*3*pi" ], "answer_type_v3": [ "EX", "EX", "NV", "NV", "EX", "EX", "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0516", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "integrals", "parametric curve", "multivariable", "calculus" ], "problem_v1": "Find $\\int_C((x^2+4 y)\\vec i+2 y^3\\vec j)\\cdot d\\vec r$ where $C$ consists of the three line segments from $(4,0,0)$ to $(4,3,0)$ to $(0,3,0)$ to $(0,3,4)$. $\\int_C((x^2+4 y)\\vec i+2 y^3\\vec j)\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "2*3^4/4-4^3/3-4*4*3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $\\int_C((x^2+2 y)\\vec i+5 y^3\\vec j)\\cdot d\\vec r$ where $C$ consists of the three line segments from $(1,0,0)$ to $(1,5,0)$ to $(0,5,0)$ to $(0,5,1)$. $\\int_C((x^2+2 y)\\vec i+5 y^3\\vec j)\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "5*5^4/4-1^3/3-2*1*5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $\\int_C((x^2+3 y)\\vec i+2 y^3\\vec j)\\cdot d\\vec r$ where $C$ consists of the three line segments from $(2,0,0)$ to $(2,4,0)$ to $(0,4,0)$ to $(0,4,2)$. $\\int_C((x^2+3 y)\\vec i+2 y^3\\vec j)\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "2*4^4/4-2^3/3-3*2*4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0517", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "2", "keywords": [ "line integral", "integral", "multivariable", "calculus" ], "problem_v1": "Calculate the line integral of the vector field $\\vec F=3\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j}$ along the line from the point $\\left(4,0\\right)$ to the point $\\left(12,0\\right)$. The line integral=[ANS]", "answer_v1": [ "(12-4)*3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the line integral of the vector field $\\vec F=-5\\,\\mathit{\\vec i}+5\\,\\mathit{\\vec j}$ along the line from the point $\\left(0,1\\right)$ to the point $\\left(0,6\\right)$. The line integral=[ANS]", "answer_v2": [ "(6-1)*5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the line integral of the vector field $\\vec F=-2\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j}$ along the line from the point $\\left(2,0\\right)$ to the point $\\left(9,0\\right)$. The line integral=[ANS]", "answer_v3": [ "(9-2)*-2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0518", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "integral", "multivariable", "calculus" ], "problem_v1": "If $C$ is the $y$-axis from the origin to the point $\\left(0,8,0\\right)$, then $\\int_{C}(2 \\vec i-2 \\vec j)\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "8*-2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $C$ is the $z$-axis from the origin to the point $\\left(0,0,8\\right)$, then $\\int_{C}(-2 \\vec j+5 \\vec k)\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "8*5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $C$ is the $z$-axis from the origin to the point $\\left(0,0,8\\right)$, then $\\int_{C}(1 \\vec j-3 \\vec k)\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "8*-3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0520", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "integral", "multivariable", "calculus" ], "problem_v1": "Let $C$ be the straight path from $(0, 0)$ to $(8, 8)$ and let $\\vec F=(y-x+3)\\vec i+(\\sin(y-x)+3)\\vec j$.\n(a) At each point of $C$, what angle does $\\vec F$ make with a tangent vector to $C$? angle=[ANS]\n(Give your answer in radians.) (Give your answer in radians.) (b) Find the magnitude $\\Vert \\vec F\\Vert$ at each point of $C$. $\\Vert \\vec F\\Vert=$ [ANS]\nEvaluate $\\int_C \\vec F \\cdot d\\vec r$. $\\int_C \\vec F \\cdot d\\vec r=$ [ANS]", "answer_v1": [ "0", "sqrt(18)", "3*8*2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $C$ be the straight path from $(0, 0)$ to $(1, 1)$ and let $\\vec F=(y-x-5)\\vec i+(\\sin(y-x)-5)\\vec j$.\n(a) At each point of $C$, what angle does $\\vec F$ make with a tangent vector to $C$? angle=[ANS]\n(Give your answer in radians.) (Give your answer in radians.) (b) Find the magnitude $\\Vert \\vec F\\Vert$ at each point of $C$. $\\Vert \\vec F\\Vert=$ [ANS]\nEvaluate $\\int_C \\vec F \\cdot d\\vec r$. $\\int_C \\vec F \\cdot d\\vec r=$ [ANS]", "answer_v2": [ "pi", "sqrt(50)", "-10" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $C$ be the straight path from $(0, 0)$ to $(4, 4)$ and let $\\vec F=(y-x-4)\\vec i+(\\sin(y-x)-4)\\vec j$.\n(a) At each point of $C$, what angle does $\\vec F$ make with a tangent vector to $C$? angle=[ANS]\n(Give your answer in radians.) (Give your answer in radians.) (b) Find the magnitude $\\Vert \\vec F\\Vert$ at each point of $C$. $\\Vert \\vec F\\Vert=$ [ANS]\nEvaluate $\\int_C \\vec F \\cdot d\\vec r$. $\\int_C \\vec F \\cdot d\\vec r=$ [ANS]", "answer_v3": [ "pi", "sqrt(32)", "-32" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0521", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "arc length" ], "problem_v1": "If $C$ is the part of the circle $ \\left( \\frac{x}{5} \\right)^2+\\left( \\frac{y}{5} \\right)^2=1$ in the first quadrant, find the following line integral with respect to arc length.\n$ \\int_C (6x-7y) ds=$ [ANS]", "answer_v1": [ "6*5^2-7*5^2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $C$ is the part of the circle $ \\left( \\frac{x}{2} \\right)^2+\\left( \\frac{y}{2} \\right)^2=1$ in the first quadrant, find the following line integral with respect to arc length.\n$ \\int_C (9x-3y) ds=$ [ANS]", "answer_v2": [ "9*2^2-3*2^2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $C$ is the part of the circle $ \\left( \\frac{x}{3} \\right)^2+\\left( \\frac{y}{3} \\right)^2=1$ in the first quadrant, find the following line integral with respect to arc length.\n$ \\int_C (6x-4y) ds=$ [ANS]", "answer_v3": [ "6*3^2-4*3^2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0522", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "arc length" ], "problem_v1": "Find the line integral with respect to arc length $ \\int_C (6x+7 y) ds$, where $C$ is the line segment in the $xy$-plane with endpoints $P=\\left(7,0\\right)$ and $Q=\\left(0,6\\right)$.\n(a) Find a vector parametric equation $\\vec{r}(t)$ for the line segment $C$ so that points $P$ and $Q$ correspond to $t=0$ and $t=1$, respectively. $\\vec{r}(t)=$ [ANS]\n(b) Using the parametrization in part (a), the line integral with respect to arc length is $ \\int_C (6x+7 y) ds=\\int_a^b$ [ANS] $dt$ with limits of integration $a=$ [ANS] and $b=$ [ANS]\n(c) Evaluate the line integral with respect to arc length in part (b). $ \\int_C (6x+7 y) ds=$ [ANS]", "answer_v1": [ "(-7t+7,6t)", "[6*(7-7*t)+7*6*t]*sqrt(7^2+6^2)", "0", "1", "387.221" ], "answer_type_v1": [ "OL", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Find the line integral with respect to arc length $ \\int_C (3x+4 y) ds$, where $C$ is the line segment in the $xy$-plane with endpoints $P=\\left(2,0\\right)$ and $Q=\\left(0,9\\right)$.\n(a) Find a vector parametric equation $\\vec{r}(t)$ for the line segment $C$ so that points $P$ and $Q$ correspond to $t=0$ and $t=1$, respectively. $\\vec{r}(t)=$ [ANS]\n(b) Using the parametrization in part (a), the line integral with respect to arc length is $ \\int_C (3x+4 y) ds=\\int_a^b$ [ANS] $dt$ with limits of integration $a=$ [ANS] and $b=$ [ANS]\n(c) Evaluate the line integral with respect to arc length in part (b). $ \\int_C (3x+4 y) ds=$ [ANS]", "answer_v2": [ "(-2t+2,9t)", "[3*(2-2*t)+4*9*t]*sqrt(2^2+9^2)", "0", "1", "193.61" ], "answer_type_v2": [ "OL", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Find the line integral with respect to arc length $ \\int_C (4x+6 y) ds$, where $C$ is the line segment in the $xy$-plane with endpoints $P=\\left(4,0\\right)$ and $Q=\\left(0,6\\right)$.\n(a) Find a vector parametric equation $\\vec{r}(t)$ for the line segment $C$ so that points $P$ and $Q$ correspond to $t=0$ and $t=1$, respectively. $\\vec{r}(t)=$ [ANS]\n(b) Using the parametrization in part (a), the line integral with respect to arc length is $ \\int_C (4x+6 y) ds=\\int_a^b$ [ANS] $dt$ with limits of integration $a=$ [ANS] and $b=$ [ANS]\n(c) Evaluate the line integral with respect to arc length in part (b). $ \\int_C (4x+6 y) ds=$ [ANS]", "answer_v3": [ "(-4t+4,6t)", "[4*(4-4*t)+6*6*t]*sqrt(4^2+6^2)", "0", "1", "187.489" ], "answer_type_v3": [ "OL", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0523", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "2", "keywords": [ "Line integrals", "Path integrals" ], "problem_v1": "Sketch the vector field $\\vec{F}(x,y)=x \\vec{i}+y \\vec{j}$ and calculate the line integral of $\\vec{F}$ along the line segment from $\\left(6,5\\right)$ to $\\left(6,8\\right).$ [ANS]", "answer_v1": [ "19.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Sketch the vector field $\\vec{F}(x,y)=x \\vec{i}+y \\vec{j}$ and calculate the line integral of $\\vec{F}$ along the line segment from $\\left(2,2\\right)$ to $\\left(6,2\\right).$ [ANS]", "answer_v2": [ "16" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Sketch the vector field $\\vec{F}(x,y)=x \\vec{i}+y \\vec{j}$ and calculate the line integral of $\\vec{F}$ along the line segment from $\\left(3,3\\right)$ to $\\left(3,6\\right).$ [ANS]", "answer_v3": [ "13.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0524", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "Line integrals", "Path integrals" ], "problem_v1": "Suppose $\\vec{F}(x,y)=\\langle 2y,-\\sin(y) \\rangle$ and $C$ is the circle of radius $8$ centered at the origin oriented counterclockwise.\n(a) Find a vector parametric equation $\\vec{r}(t)$ for the circle $C$ that starts at the point $(8,0)$ and travels around the circle once counterclockwise for $0 \\leq t \\leq 2\\pi$. $\\vec{r}(t)=$ [ANS]\n(b) Using your parametrization in part (a), set up an integral for calculating the circulation of $\\vec{F}$ around $C$. $ \\int_C \\vec{F} \\cdot d \\vec{r}=\\int_a^b \\vec{F}(\\vec{r}(t)) \\cdot \\vec{r}\\,'(t) \\, dt=\\int_a^b$ [ANS] $dt$ with limits of integration $a=$ [ANS] and $b=$ [ANS]\n(c) Find the circulation of $\\vec{F}$ around $C$. Circulation=[ANS]", "answer_v1": [ "(8*cos(t),8*sin(t))", "-(128*[sin(t)]^2+8*cos(t)*sin(8*sin(t)))", "0", "2*pi", "-128*pi" ], "answer_type_v1": [ "OL", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Suppose $\\vec{F}(x,y)=\\langle 2y,-\\sin(y) \\rangle$ and $C$ is the circle of radius $3$ centered at the origin oriented counterclockwise.\n(a) Find a vector parametric equation $\\vec{r}(t)$ for the circle $C$ that starts at the point $(3,0)$ and travels around the circle once counterclockwise for $0 \\leq t \\leq 2\\pi$. $\\vec{r}(t)=$ [ANS]\n(b) Using your parametrization in part (a), set up an integral for calculating the circulation of $\\vec{F}$ around $C$. $ \\int_C \\vec{F} \\cdot d \\vec{r}=\\int_a^b \\vec{F}(\\vec{r}(t)) \\cdot \\vec{r}\\,'(t) \\, dt=\\int_a^b$ [ANS] $dt$ with limits of integration $a=$ [ANS] and $b=$ [ANS]\n(c) Find the circulation of $\\vec{F}$ around $C$. Circulation=[ANS]", "answer_v2": [ "(3*cos(t),3*sin(t))", "-(18*[sin(t)]^2+3*cos(t)*sin(3*sin(t)))", "0", "2*pi", "-18*pi" ], "answer_type_v2": [ "OL", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Suppose $\\vec{F}(x,y)=\\langle 2y,-\\sin(y) \\rangle$ and $C$ is the circle of radius $5$ centered at the origin oriented counterclockwise.\n(a) Find a vector parametric equation $\\vec{r}(t)$ for the circle $C$ that starts at the point $(5,0)$ and travels around the circle once counterclockwise for $0 \\leq t \\leq 2\\pi$. $\\vec{r}(t)=$ [ANS]\n(b) Using your parametrization in part (a), set up an integral for calculating the circulation of $\\vec{F}$ around $C$. $ \\int_C \\vec{F} \\cdot d \\vec{r}=\\int_a^b \\vec{F}(\\vec{r}(t)) \\cdot \\vec{r}\\,'(t) \\, dt=\\int_a^b$ [ANS] $dt$ with limits of integration $a=$ [ANS] and $b=$ [ANS]\n(c) Find the circulation of $\\vec{F}$ around $C$. Circulation=[ANS]", "answer_v3": [ "(5*cos(t),5*sin(t))", "-(50*[sin(t)]^2+5*cos(t)*sin(5*sin(t)))", "0", "2*pi", "-50*pi" ], "answer_type_v3": [ "OL", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0525", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "2", "keywords": [ "Line integrals", "Path integrals" ], "problem_v1": "Find the line integral of $\\vec{F}(x,y,z)=x^3\\vec{i}+y^2\\vec{j}+z\\vec{k}$ along the line segment $C$ from the origin to the point $(4,3,7)$.\n$ \\int_C \\vec{F} \\cdot d\\vec{r}$=[ANS]", "answer_v1": [ "4^4/4+3^3/3+7^2/2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the line integral of $\\vec{F}(x,y,z)=x^3\\vec{i}+y^2\\vec{j}+z\\vec{k}$ along the line segment $C$ from the origin to the point $(2,5,6)$.\n$ \\int_C \\vec{F} \\cdot d\\vec{r}$=[ANS]", "answer_v2": [ "2^4/4+5^3/3+6^2/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the line integral of $\\vec{F}(x,y,z)=x^3\\vec{i}+y^2\\vec{j}+z\\vec{k}$ along the line segment $C$ from the origin to the point $(2,4,7)$.\n$ \\int_C \\vec{F} \\cdot d\\vec{r}$=[ANS]", "answer_v3": [ "2^4/4+4^3/3+7^2/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0526", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "Line integrals", "Path integrals" ], "problem_v1": "Suppose $\\vec{F}(x,y)=x^2 \\vec{i}+y^2 \\vec{j}$ and $C$ is the line segment from point $P=\\left(3,1\\right)$ to $Q=\\left(4,3\\right)$.\n(a) Find a vector parametric equation $\\vec{r}(t)$ for the line segment $C$ so that points $P$ and $Q$ correspond to $t=0$ and $t=1$, respectively. $\\vec{r}(t)=$ [ANS]\n(b) Using the parametrization in part (a), the line integral of $\\vec{F}$ along $C$ is $ \\int_C \\vec{F} \\cdot d \\vec{r}=\\int_a^b \\vec{F}(\\vec{r}(t)) \\cdot \\vec{r}\\,'(t) \\, dt=\\int_a^b$ [ANS] $dt$ with limits of integration $a=$ [ANS] and $b=$ [ANS]\n(c) Evaluate the line integral in part (b). [ANS]", "answer_v1": [ "(t+3,2t+1)", "(3+t)^2+2*(1+2*t)^2", "0", "1", "21" ], "answer_type_v1": [ "OL", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Suppose $\\vec{F}(x,y)=x^2 \\vec{i}+y^2 \\vec{j}$ and $C$ is the line segment from point $P=\\left(-5,5\\right)$ to $Q=\\left(-9,3\\right)$.\n(a) Find a vector parametric equation $\\vec{r}(t)$ for the line segment $C$ so that points $P$ and $Q$ correspond to $t=0$ and $t=1$, respectively. $\\vec{r}(t)=$ [ANS]\n(b) Using the parametrization in part (a), the line integral of $\\vec{F}$ along $C$ is $ \\int_C \\vec{F} \\cdot d \\vec{r}=\\int_a^b \\vec{F}(\\vec{r}(t)) \\cdot \\vec{r}\\,'(t) \\, dt=\\int_a^b$ [ANS] $dt$ with limits of integration $a=$ [ANS] and $b=$ [ANS]\n(c) Evaluate the line integral in part (b). [ANS]", "answer_v2": [ "(-4t-5,-2t+5)", "-(4*[-(5+4*t)]^2+2*(5-2*t)^2)", "0", "1", "-234" ], "answer_type_v2": [ "OL", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Suppose $\\vec{F}(x,y)=x^2 \\vec{i}+y^2 \\vec{j}$ and $C$ is the line segment from point $P=\\left(-2,1\\right)$ to $Q=\\left(-4,2\\right)$.\n(a) Find a vector parametric equation $\\vec{r}(t)$ for the line segment $C$ so that points $P$ and $Q$ correspond to $t=0$ and $t=1$, respectively. $\\vec{r}(t)=$ [ANS]\n(b) Using the parametrization in part (a), the line integral of $\\vec{F}$ along $C$ is $ \\int_C \\vec{F} \\cdot d \\vec{r}=\\int_a^b \\vec{F}(\\vec{r}(t)) \\cdot \\vec{r}\\,'(t) \\, dt=\\int_a^b$ [ANS] $dt$ with limits of integration $a=$ [ANS] and $b=$ [ANS]\n(c) Evaluate the line integral in part (b). [ANS]", "answer_v3": [ "(-2t-2,t+1)", "(1+t)^2-2*[-(2+2*t)]^2", "0", "1", "-16.3333" ], "answer_type_v3": [ "OL", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0527", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "calculus", "line integral", "vector' 'line' 'integral" ], "problem_v1": "Evaluate the line integral $\\int_C \\mathbf{F}\\cdot d\\mathbf{r}$, where $\\mathbf{F}(x,y,z)=3x\\mathbf{i}+y\\mathbf{j}+z\\mathbf{k}$ and C is given by the vector function $\\mathbf{r}(t)=\\langle \\sin t, \\cos t, t \\rangle, \\quad \\ 0\\le t \\le 3\\pi/2.$ [ANS]", "answer_v1": [ "12.1033049512255" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the line integral $\\int_C \\mathbf{F}\\cdot d\\mathbf{r}$, where $\\mathbf{F}(x,y,z)=-5x\\mathbf{i}+5y\\mathbf{j}-4z\\mathbf{k}$ and C is given by the vector function $\\mathbf{r}(t)=\\langle \\sin t, \\cos t, t \\rangle, \\quad \\ 0\\le t \\le 3\\pi/2.$ [ANS]", "answer_v2": [ "-49.4132198049021" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the line integral $\\int_C \\mathbf{F}\\cdot d\\mathbf{r}$, where $\\mathbf{F}(x,y,z)=-2x\\mathbf{i}+y\\mathbf{j}-2z\\mathbf{k}$ and C is given by the vector function $\\mathbf{r}(t)=\\langle \\sin t, \\cos t, t \\rangle, \\quad \\ 0\\le t \\le 3\\pi/2.$ [ANS]", "answer_v3": [ "-23.7066099024511" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0528", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "calculus", "line integral" ], "problem_v1": "Evaluate the line integral $ \\int_C x^3 z \\; ds$, where C is the line segment from $(0,7,5)$ to $(5,6,3)$. [ANS]", "answer_v1": [ "581.955217349239" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the line integral $ \\int_C x^2 z \\; ds$, where C is the line segment from $(0,1,8)$ to $(2,3,3)$. [ANS]", "answer_v2": [ "32.5525216637155" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the line integral $ \\int_C x^3 z \\; ds$, where C is the line segment from $(0,3,5)$ to $(3,5,2)$. [ANS]", "answer_v3": [ "82.3167965849012" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0529", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [ "calculus", "line integral" ], "problem_v1": "Evaluate the line integral $ \\int_C 6x y^4 \\; ds$, where C is the right half of the circle $x^2+y^2=25$. [ANS]", "answer_v1": [ "37500" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the line integral $ \\int_C 8x y^2 \\; ds$, where C is the right half of the circle $x^2+y^2=4$. [ANS]", "answer_v2": [ "85.3333333333333" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the line integral $ \\int_C 6x y^2 \\; ds$, where C is the right half of the circle $x^2+y^2=9$. [ANS]", "answer_v3": [ "324" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0530", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Line integrals", "level": "3", "keywords": [], "problem_v1": "Let $C \\subseteq \\mathbb{R}^2$ be the circle with radius $8$ centered at the origin. Let $F: \\mathbb{R}^2\\to\\mathbb{R}^2$ be the vector field defined by $F(x,y)=(12x,13)$. Find the flux of $F$ coming out of the circle through the curve $C$. $\\int_C F\\cdot N=$ [ANS]", "answer_v1": [ "768*pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $C \\subseteq \\mathbb{R}^2$ be the circle with radius $2$ centered at the origin. Let $F: \\mathbb{R}^2\\to\\mathbb{R}^2$ be the vector field defined by $F(x,y)=(18x,4)$. Find the flux of $F$ coming out of the circle through the curve $C$. $\\int_C F\\cdot N=$ [ANS]", "answer_v2": [ "72*pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $C \\subseteq \\mathbb{R}^2$ be the circle with radius $4$ centered at the origin. Let $F: \\mathbb{R}^2\\to\\mathbb{R}^2$ be the vector field defined by $F(x,y)=(12x,7)$. Find the flux of $F$ coming out of the circle through the curve $C$. $\\int_C F\\cdot N=$ [ANS]", "answer_v3": [ "192*pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0531", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Conservative vector fields", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Determine whether the vector field is conservative and, if so, find the general potential function.\n\\mathbf{F}=\\left< \\cos z,2 y^{15},-x\\sin z\\right> $\\varphi=$ [ANS] $+c$ Note: if the vector field is not conservative, write \"DNE\".", "answer_v1": [ "x*cos(z)+y^16/8" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Determine whether the vector field is conservative and, if so, find the general potential function.\n\\mathbf{F}=\\left< \\cos z,2 y^{1},-x\\sin z\\right> $\\varphi=$ [ANS] $+c$ Note: if the vector field is not conservative, write \"DNE\".", "answer_v2": [ "x*cos(z)+y^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Determine whether the vector field is conservative and, if so, find the general potential function.\n\\mathbf{F}=\\left< \\cos z,2 y^{7},-x\\sin z\\right> $\\varphi=$ [ANS] $+c$ Note: if the vector field is not conservative, write \"DNE\".", "answer_v3": [ "x*cos(z)+y^8/4" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0532", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Conservative vector fields", "level": "3", "keywords": [ "vector' 'multivariable' 'curl' 'divergence", "Multivariable", "Vector", "Field", "Curl", "Potential", "Conservative", "Vector Fields", "Line Integral", "Conservative", "calculus", "vector field", "convervative" ], "problem_v1": "For each of the following vector fields F, decide whether it is conservative or not by computing curl F. Type in a potential function f (that is, $\\nabla f=\\mathbf{F}$). If it is not conservative, type N. A. $\\mathbf{F} \\left(x, y \\right)=\\left(8x+1 y \\right) \\mathbf{i}+\\left(1x+10 y \\right) \\mathbf{j}$ $f \\left(x, y \\right)=$ [ANS]\nB. $\\mathbf{F} \\left(x, y \\right)=4 y \\mathbf{i}+5x \\mathbf{j}$ $f \\left(x, y \\right)=$ [ANS]\nC. $\\mathbf{F} \\left(x, y, z \\right)=4x \\mathbf{i}+5 y \\mathbf{j}+\\mathbf{k}$ $f \\left(x, y, z \\right)=$ [ANS]\nD. $\\mathbf{F} \\left(x, y \\right)=\\left(4 \\sin y \\right) \\mathbf{i}+\\left(2 y+4x \\cos y \\right) \\mathbf{j}$ $f \\left(x, y \\right)=$ [ANS]\nE. $\\mathbf{F} \\left(x, y, z \\right)=4x^{2} \\mathbf{i}+1 y^{2} \\mathbf{j}+5 z^{2} \\mathbf{k}$ $f \\left(x, y, z \\right)=$ [ANS]\nNote: Your answers should be either expressions of x, y and z (e.g. \"3xy+2yz\"), or the letter \"N\"", "answer_v1": [ "4*x^2+1*x*y+5*y^2", "N", "4*x^2/2+(4+1)*y^2/2+z", "4*x*sin(y)+1*y^2", "1/3*(4*x^3+1*y^3+5*z^3)" ], "answer_type_v1": [ "EX", "OE", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "For each of the following vector fields F, decide whether it is conservative or not by computing curl F. Type in a potential function f (that is, $\\nabla f=\\mathbf{F}$). If it is not conservative, type N. A. $\\mathbf{F} \\left(x, y \\right)=\\left(-14x+7 y \\right) \\mathbf{i}+\\left(7x+4 y \\right) \\mathbf{j}$ $f \\left(x, y \\right)=$ [ANS]\nB. $\\mathbf{F} \\left(x, y \\right)=-7 y \\mathbf{i}-6x \\mathbf{j}$ $f \\left(x, y \\right)=$ [ANS]\nC. $\\mathbf{F} \\left(x, y, z \\right)=-7x \\mathbf{i}-6 y \\mathbf{j}+\\mathbf{k}$ $f \\left(x, y, z \\right)=$ [ANS]\nD. $\\mathbf{F} \\left(x, y \\right)=\\left(-7 \\sin y \\right) \\mathbf{i}+\\left(14 y-7x \\cos y \\right) \\mathbf{j}$ $f \\left(x, y \\right)=$ [ANS]\nE. $\\mathbf{F} \\left(x, y, z \\right)=-7x^{2} \\mathbf{i}+7 y^{2} \\mathbf{j}+2 z^{2} \\mathbf{k}$ $f \\left(x, y, z \\right)=$ [ANS]\nNote: Your answers should be either expressions of x, y and z (e.g. \"3xy+2yz\"), or the letter \"N\"", "answer_v2": [ "-7*x^2+7*x*y+2*y^2", "N", "-7*x^2/2+(-7+1)*y^2/2+z", "-7*x*sin(y)+7*y^2", "1/3*(-7*x^3+7*y^3+2*z^3)" ], "answer_type_v2": [ "EX", "OE", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "For each of the following vector fields F, decide whether it is conservative or not by computing curl F. Type in a potential function f (that is, $\\nabla f=\\mathbf{F}$). If it is not conservative, type N. A. $\\mathbf{F} \\left(x, y \\right)=\\left(-6x+2 y \\right) \\mathbf{i}+\\left(2x+6 y \\right) \\mathbf{j}$ $f \\left(x, y \\right)=$ [ANS]\nB. $\\mathbf{F} \\left(x, y \\right)=-3 y \\mathbf{i}-2x \\mathbf{j}$ $f \\left(x, y \\right)=$ [ANS]\nC. $\\mathbf{F} \\left(x, y, z \\right)=-3x \\mathbf{i}-2 y \\mathbf{j}+\\mathbf{k}$ $f \\left(x, y, z \\right)=$ [ANS]\nD. $\\mathbf{F} \\left(x, y \\right)=\\left(-3 \\sin y \\right) \\mathbf{i}+\\left(4 y-3x \\cos y \\right) \\mathbf{j}$ $f \\left(x, y \\right)=$ [ANS]\nE. $\\mathbf{F} \\left(x, y, z \\right)=-3x^{2} \\mathbf{i}+2 y^{2} \\mathbf{j}+3 z^{2} \\mathbf{k}$ $f \\left(x, y, z \\right)=$ [ANS]\nNote: Your answers should be either expressions of x, y and z (e.g. \"3xy+2yz\"), or the letter \"N\"", "answer_v3": [ "-3*x^2+2*x*y+3*y^2", "N", "-3*x^2/2+(-3+1)*y^2/2+z", "-3*x*sin(y)+2*y^2", "1/3*(-3*x^3+2*y^3+3*z^3)" ], "answer_type_v3": [ "EX", "OE", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0534", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Conservative vector fields", "level": "3", "keywords": [ "integral", "vector", "conservative", "vector field" ], "problem_v1": "Let $\\mathbf{F}=9xy\\boldsymbol{i}+6y^{2}\\boldsymbol{j}$ be a vector field in the plane, and $C$ the path $y=6x^2$ joining (0,0) to (1,6) in the plane.\nA. Evaluate $\\int_C \\mathbf{F}\\cdot d\\mathbf{r}$ [ANS]\nB. Does the integral in part (A) depend on the path joining (0,0) to (1,6)? [ANS] (y/n)", "answer_v1": [ "445.5", "Y" ], "answer_type_v1": [ "NV", "TF" ], "options_v1": [ [], [] ], "problem_v2": "Let $\\mathbf{F}=3xy\\boldsymbol{i}+9y^{2}\\boldsymbol{j}$ be a vector field in the plane, and $C$ the path $y=3x^2$ joining (0,0) to (1,3) in the plane.\nA. Evaluate $\\int_C \\mathbf{F}\\cdot d\\mathbf{r}$ [ANS]\nB. Does the integral in part (A) depend on the path joining (0,0) to (1,3)? [ANS] (y/n)", "answer_v2": [ "83.25", "Y" ], "answer_type_v2": [ "NV", "TF" ], "options_v2": [ [], [] ], "problem_v3": "Let $\\mathbf{F}=5xy\\boldsymbol{i}+6y^{2}\\boldsymbol{j}$ be a vector field in the plane, and $C$ the path $y=4x^2$ joining (0,0) to (1,4) in the plane.\nA. Evaluate $\\int_C \\mathbf{F}\\cdot d\\mathbf{r}$ [ANS]\nB. Does the integral in part (A) depend on the path joining (0,0) to (1,4)? [ANS] (y/n)", "answer_v3": [ "133", "Y" ], "answer_type_v3": [ "NV", "TF" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0535", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Conservative vector fields", "level": "3", "keywords": [ "gradient", "vector", "conservative", "independence", "field" ], "problem_v1": "Show that $\\int_{\\left(5,2\\right)}^{\\left(12,10\\right)}fdx+gdy=\\int_{\\left(5,2\\right)}^{\\left(12,10\\right)}-\\left(12x^{2}y^{4}dx+16y^{3}x^{3}dy\\right)$ is independent of path: $\\frac {\\partial f}{\\partial y}$=[ANS]\n$\\frac {\\partial g}{\\partial x}$=[ANS]\n$\\int_{\\left(5,2\\right)}^{\\left(12,10\\right)}-\\left(12x^{2}y^{4}dx+16y^{3}x^{3}dy\\right)$=[ANS]", "answer_v1": [ "-(4*3*x^2*4*y^3)", "-(4*4*y^3*3*x^2)", "-(6.912E+7)--8000" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Show that $\\int_{\\left(-9,9\\right)}^{\\left(-7,13\\right)}fdx+gdy=\\int_{\\left(-9,9\\right)}^{\\left(-7,13\\right)}27x^{2}y^{2}dx+18yx^{3}dy$ is independent of path: $\\frac {\\partial f}{\\partial y}$=[ANS]\n$\\frac {\\partial g}{\\partial x}$=[ANS]\n$\\int_{\\left(-9,9\\right)}^{\\left(-7,13\\right)}27x^{2}y^{2}dx+18yx^{3}dy$=[ANS]", "answer_v2": [ "9*3*x^2*2*y", "9*2*y*3*x^2", "-521703--531441" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Show that $\\int_{\\left(-4,2\\right)}^{\\left(-1,8\\right)}fdx+gdy=\\int_{\\left(-4,2\\right)}^{\\left(-1,8\\right)}-\\left(18x^{2}y^{5}dx+30y^{4}x^{3}dy\\right)$ is independent of path: $\\frac {\\partial f}{\\partial y}$=[ANS]\n$\\frac {\\partial g}{\\partial x}$=[ANS]\n$\\int_{\\left(-4,2\\right)}^{\\left(-1,8\\right)}-\\left(18x^{2}y^{5}dx+30y^{4}x^{3}dy\\right)$=[ANS]", "answer_v3": [ "-(6*3*x^2*5*y^4)", "-(6*5*y^4*3*x^2)", "196608-12288" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0536", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Conservative vector fields", "level": "3", "keywords": [ "gradient", "vector", "conservative", "independence", "field" ], "problem_v1": "Show that $\\int_{\\left(3,1,-2\\right)}^{\\left(1,2,1\\right)}f\\,dx+g\\,dy+h\\,dz=\\int_{\\left(3,1,-2\\right)}^{\\left(1,2,1\\right)}2xy^{2}z^{3}dx+2yx^{2}z^{3}dy+3z^{2}y^{2}x^{2}dz$ is independent of path: $\\frac {\\partial h}{\\partial y}$=[ANS]\n$\\frac {\\partial g}{\\partial z}$=[ANS]\n$\\frac {\\partial f}{\\partial z}$=[ANS]\n$\\frac {\\partial h}{\\partial x}$=[ANS]\n$\\frac {\\partial g}{\\partial x}$=[ANS]\n$\\frac {\\partial f}{\\partial y}$=[ANS]\nTherefore curl F=[ANS]\n$\\int_{\\left(3,1,-2\\right)}^{\\left(1,2,1\\right)}2xy^{2}z^{3}dx+2yx^{2}z^{3}dy+3z^{2}y^{2}x^{2}dz$=[ANS]", "answer_v1": [ "3*z^2*2*y*x^2", "2*y*x^2*3*z^2", "2*x*y^2*3*z^2", "3*z^2*y^2*2*x", "2*y*2*x*z^3", "2*x*2*y*z^3", "(0,0,0)", "4--72" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "OL", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Show that $\\int_{\\left(-3,1,2\\right)}^{\\left(-2,-5,-1\\right)}f\\,dx+g\\,dy+h\\,dz=\\int_{\\left(-3,1,2\\right)}^{\\left(-2,-5,-1\\right)}6xy^{2}z^{3}dx+6yx^{2}z^{3}dy+9z^{2}y^{2}x^{2}dz$ is independent of path: $\\frac {\\partial h}{\\partial y}$=[ANS]\n$\\frac {\\partial g}{\\partial z}$=[ANS]\n$\\frac {\\partial f}{\\partial z}$=[ANS]\n$\\frac {\\partial h}{\\partial x}$=[ANS]\n$\\frac {\\partial g}{\\partial x}$=[ANS]\n$\\frac {\\partial f}{\\partial y}$=[ANS]\nTherefore curl F=[ANS]\n$\\int_{\\left(-3,1,2\\right)}^{\\left(-2,-5,-1\\right)}6xy^{2}z^{3}dx+6yx^{2}z^{3}dy+9z^{2}y^{2}x^{2}dz$=[ANS]", "answer_v2": [ "3*3*z^2*2*y*x^2", "3*2*y*x^2*3*z^2", "3*2*x*y^2*3*z^2", "3*3*z^2*y^2*2*x", "3*2*y*2*x*z^3", "3*2*x*2*y*z^3", "(0,0,0)", "-300-216" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX", "OL", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Show that $\\int_{\\left(-2,-2,-3\\right)}^{\\left(1,1,-2\\right)}f\\,dx+g\\,dy+h\\,dz=\\int_{\\left(-2,-2,-3\\right)}^{\\left(1,1,-2\\right)}9x^{2}y^{3}z^{2}dx+9y^{2}x^{3}z^{2}dy+6zy^{3}x^{3}dz$ is independent of path: $\\frac {\\partial h}{\\partial y}$=[ANS]\n$\\frac {\\partial g}{\\partial z}$=[ANS]\n$\\frac {\\partial f}{\\partial z}$=[ANS]\n$\\frac {\\partial h}{\\partial x}$=[ANS]\n$\\frac {\\partial g}{\\partial x}$=[ANS]\n$\\frac {\\partial f}{\\partial y}$=[ANS]\nTherefore curl F=[ANS]\n$\\int_{\\left(-2,-2,-3\\right)}^{\\left(1,1,-2\\right)}9x^{2}y^{3}z^{2}dx+9y^{2}x^{3}z^{2}dy+6zy^{3}x^{3}dz$=[ANS]", "answer_v3": [ "3*2*z*3*y^2*x^3", "3*3*y^2*x^3*2*z", "3*3*x^2*y^3*2*z", "3*2*z*y^3*3*x^2", "3*3*y^2*3*x^2*z^2", "3*3*x^2*3*y^2*z^2", "(0,0,0)", "12-1728" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "OL", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0537", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Conservative vector fields", "level": "2", "keywords": [ "line integral", "fundamental theorem of calculus", "gradient", "vector field" ], "problem_v1": "For each of the following decide whether the vector field could be a gradient vector field. Be sure that you can justify your answer.\n(a) $\\vec F(x,y,z)=\\langle 3y, 3z, 3x\\rangle$ $\\vec F(x,y,z)$ is [ANS] (b) $\\vec F(x,y,z)=x\\sqrt{4x^2+y^2+z^2}\\,\\vec i+y\\sqrt{4x^2+y^2+z^2}\\,\\vec j+z\\sqrt{4x^2+y^2+z^2}\\,\\vec k$ $\\vec F(x,y,z)$ is [ANS] (c) $\\vec F(x,y,z)=4x\\,\\vec i+4 y\\,\\vec j+4 z\\,\\vec k$ $\\vec F(x,y,z)$ is [ANS] (d) $\\vec F(x,y,z)=\\langle 4 y z, 4x z, 4x y\\rangle$ $\\vec F(x,y,z)$ is [ANS]", "answer_v1": [ "not a gradient vector field", "not a gradient vector field", "a gradient vector field", "a gradient vector field" ], "answer_type_v1": [ "MCS", "MCS", "MCS", "MCS" ], "options_v1": [ [ "a gradient vector field", "not a gradient vector field" ], [ "a gradient vector field", "not a gradient vector field" ], [ "a gradient vector field", "not a gradient vector field" ], [ "a gradient vector field", "not a gradient vector field" ] ], "problem_v2": "For each of the following decide whether the vector field could be a gradient vector field. Be sure that you can justify your answer.\n(a) $\\vec F(x,y,z)=\\langle y, z, x\\rangle$ $\\vec F(x,y,z)$ is [ANS] (b) $\\vec F(x,y,z)=\\langle 2 y z, 2x z, 2x y\\rangle$ $\\vec F(x,y,z)$ is [ANS] (c) $\\vec F(x,y,z)=3 y\\,\\vec i+3x\\,\\vec j$ $\\vec F(x,y,z)$ is [ANS] (d) $\\vec F(x,y,z)= \\frac{5x}{\\sqrt{x^2+y^2+z^2} }\\,\\vec i+ \\frac{5 y}{\\sqrt{x^2+y^2+z^2} }\\,\\vec j+ \\frac{5 z}{\\sqrt{x^2+y^2+z^2} }\\,\\vec k+$ $\\vec F(x,y,z)$ is [ANS]", "answer_v2": [ "not a gradient vector field", "a gradient vector field", "a gradient vector field", "a gradient vector field" ], "answer_type_v2": [ "MCS", "MCS", "MCS", "MCS" ], "options_v2": [ [ "a gradient vector field", "not a gradient vector field" ], [ "a gradient vector field", "not a gradient vector field" ], [ "a gradient vector field", "not a gradient vector field" ], [ "a gradient vector field", "not a gradient vector field" ] ], "problem_v3": "For each of the following decide whether the vector field could be a gradient vector field. Be sure that you can justify your answer.\n(a) $\\vec F(x,y,z)=x\\sqrt{2x^2+y^2+z^2}\\,\\vec i+y\\sqrt{2x^2+y^2+z^2}\\,\\vec j+z\\sqrt{2x^2+y^2+z^2}\\,\\vec k$ $\\vec F(x,y,z)$ is [ANS] (b) $\\vec F(x,y,z)=\\langle 3 y z, 3x z, 3x y\\rangle$ $\\vec F(x,y,z)$ is [ANS] (c) $\\vec F(x,y,z)= \\frac{2 z}{\\sqrt{x^2+z^2} }\\,\\vec i+ \\frac{2 y}{\\sqrt{x^2+z^2} }\\,\\vec j+ \\frac{2x}{\\sqrt{x^2+z^2} }\\,\\vec k$ $\\vec F(x,y,z)$ is [ANS] (d) $\\vec F(x,y,z)=\\langle 5z, 0, 5x\\rangle$ $\\vec F(x,y,z)$ is [ANS]", "answer_v3": [ "not a gradient vector field", "a gradient vector field", "not a gradient vector field", "a gradient vector field" ], "answer_type_v3": [ "MCS", "MCS", "MCS", "MCS" ], "options_v3": [ [ "a gradient vector field", "not a gradient vector field" ], [ "a gradient vector field", "not a gradient vector field" ], [ "a gradient vector field", "not a gradient vector field" ], [ "a gradient vector field", "not a gradient vector field" ] ] }, { "id": "Calculus_-_multivariable_0538", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Conservative vector fields", "level": "3", "keywords": [ "greens theorem", "line integral", "gradient", "vector field", "integral", "calculus" ], "problem_v1": "Find $f$ if $\\nabla f=8xy\\,\\vec i+(4x^{2}+12y^{3})\\,\\vec j$ $f=$ [ANS]", "answer_v1": [ "4*x^2*y+3*y^4+K" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Find $f$ if $\\nabla f=2xy\\,\\vec i+(x^{2}+10y)\\,\\vec j$ $f=$ [ANS]", "answer_v2": [ "1*x^2*y+5*y^2+K" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Find $f$ if $\\nabla f=4xy\\,\\vec i+(2x^{2}+12y^{2})\\,\\vec j$ $f=$ [ANS]", "answer_v3": [ "2*x^2*y+4*y^3+K" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0540", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Conservative vector fields", "level": "3", "keywords": [ "Greens theorem", "path-dependent vector fields", "non-conservative vector fields" ], "problem_v1": "Are the following statements true or false?\n[ANS] 1. If $\\int_C \\vec{F} \\cdot d\\vec{r} \\not=0$ for some closed path $C$, then $\\vec{F}$ is path dependent. [ANS] 2. If the vector fields $\\vec{F}$ and $\\vec{G}$ have $\\int_C \\vec{F} \\cdot d\\vec{r}=\\int_C \\vec{G} \\cdot d\\vec{r}$ for a particular path $C$, then $\\vec{F}=\\vec{G}$. [ANS] 3. The circulation of any vector field $\\vec{F}$ around any closed curve $C$ is zero. [ANS] 4. If $\\vec{F}$ is path-independent and $C$ is any closed curve, then $\\int_C \\vec{F} \\cdot d\\vec{r}=0$. [ANS] 5. If $\\vec{F}=\\nabla f$, then $\\vec{F}$ is conservative. [ANS] 6. If $\\int_C \\vec{F} \\cdot d\\vec{r}=0$ for one particular closed path, then $\\vec{F}$ is path-independent. [ANS] 7. $\\int_C \\vec{F} \\cdot d\\vec{r}=f(Q)-f(P)$ whenever $P$ and $Q$ are the endpoints of the curve $C$. [ANS] 8. If $\\vec{F}$ is path-independent, then there is a potential function for $\\vec{F}$.", "answer_v1": [ "TRUE", "FALSE", "FALSE", "True", "True", "False", "False", "True" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v2": "Are the following statements true or false?\n[ANS] 1. If $\\vec{F}=\\nabla f$, then $\\vec{F}$ is conservative. [ANS] 2. If the vector fields $\\vec{F}$ and $\\vec{G}$ have $\\int_C \\vec{F} \\cdot d\\vec{r}=\\int_C \\vec{G} \\cdot d\\vec{r}$ for a particular path $C$, then $\\vec{F}=\\vec{G}$. [ANS] 3. $\\int_C \\vec{F} \\cdot d\\vec{r}=f(Q)-f(P)$ whenever $P$ and $Q$ are the endpoints of the curve $C$. [ANS] 4. If $\\vec{F}$ is path-independent, then there is a potential function for $\\vec{F}$. [ANS] 5. If $\\int_C \\vec{F} \\cdot d\\vec{r} \\not=0$ for some closed path $C$, then $\\vec{F}$ is path dependent. [ANS] 6. If $\\vec{F}$ is path-independent and $C$ is any closed curve, then $\\int_C \\vec{F} \\cdot d\\vec{r}=0$. [ANS] 7. If $\\int_C \\vec{F} \\cdot d\\vec{r}=0$ for one particular closed path, then $\\vec{F}$ is path-independent. [ANS] 8. The circulation of any vector field $\\vec{F}$ around any closed curve $C$ is zero.", "answer_v2": [ "TRUE", "FALSE", "FALSE", "True", "True", "True", "False", "False" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v3": "Are the following statements true or false?\n[ANS] 1. If $\\int_C \\vec{F} \\cdot d\\vec{r}=0$ for one particular closed path, then $\\vec{F}$ is path-independent. [ANS] 2. If the vector fields $\\vec{F}$ and $\\vec{G}$ have $\\int_C \\vec{F} \\cdot d\\vec{r}=\\int_C \\vec{G} \\cdot d\\vec{r}$ for a particular path $C$, then $\\vec{F}=\\vec{G}$. [ANS] 3. If $\\vec{F}$ is path-independent and $C$ is any closed curve, then $\\int_C \\vec{F} \\cdot d\\vec{r}=0$. [ANS] 4. The circulation of any vector field $\\vec{F}$ around any closed curve $C$ is zero. [ANS] 5. $\\int_C \\vec{F} \\cdot d\\vec{r}=f(Q)-f(P)$ whenever $P$ and $Q$ are the endpoints of the curve $C$. [ANS] 6. If $\\int_C \\vec{F} \\cdot d\\vec{r} \\not=0$ for some closed path $C$, then $\\vec{F}$ is path dependent. [ANS] 7. If $\\vec{F}=\\nabla f$, then $\\vec{F}$ is conservative. [ANS] 8. If $\\vec{F}$ is path-independent, then there is a potential function for $\\vec{F}$.", "answer_v3": [ "FALSE", "FALSE", "TRUE", "False", "False", "True", "True", "True" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ] }, { "id": "Calculus_-_multivariable_0541", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Applications of line integrals", "level": "4", "keywords": [ "calculus" ], "problem_v1": "Calculate the total mass of a circular piece of wire of radius $8$ cm centered at the origin whose mass density is $\\rho(x,y)=x^2~\\mathrm{g/cm}$. Answer: [ANS] $\\mathrm{g}$", "answer_v1": [ "1608.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the total mass of a circular piece of wire of radius $2$ cm centered at the origin whose mass density is $\\rho(x,y)=x^2~\\mathrm{g/cm}$. Answer: [ANS] $\\mathrm{g}$", "answer_v2": [ "25.1327" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the total mass of a circular piece of wire of radius $4$ cm centered at the origin whose mass density is $\\rho(x,y)=x^2~\\mathrm{g/cm}$. Answer: [ANS] $\\mathrm{g}$", "answer_v3": [ "201.062" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0542", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Applications of line integrals", "level": "3", "keywords": [ "vector' 'line integral' 'multivariable' 'work", "Vector Fields", "Line Integral", "Work", "vector", "line", "integral", "vector' 'line' 'integral' 'work" ], "problem_v1": "Find the work done by the force field $\\mathbf{F}(x, y, z)=5x\\mathbf i+5y\\mathbf j+6\\mathbf k$ on a particle that moves along the helix $\\mathbf{r}(t)=6 \\cos(t)\\mathbf i+6 \\sin(t)\\mathbf j+5t\\mathbf k, 0 \\leq t \\leq 2\\pi$. [ANS]", "answer_v1": [ "188.495559215388" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the work done by the force field $\\mathbf{F}(x, y, z)=2x\\mathbf i+2y\\mathbf j+3\\mathbf k$ on a particle that moves along the helix $\\mathbf{r}(t)=1 \\cos(t)\\mathbf i+1 \\sin(t)\\mathbf j+7t\\mathbf k, 0 \\leq t \\leq 2\\pi$. [ANS]", "answer_v2": [ "131.946891450771" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the work done by the force field $\\mathbf{F}(x, y, z)=2x\\mathbf i+2y\\mathbf j+4\\mathbf k$ on a particle that moves along the helix $\\mathbf{r}(t)=3 \\cos(t)\\mathbf i+3 \\sin(t)\\mathbf j+5t\\mathbf k, 0 \\leq t \\leq 2\\pi$. [ANS]", "answer_v3": [ "125.663706143592" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0543", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Applications of line integrals", "level": "3", "keywords": [ "vector' 'line integral' 'multivariable' 'mass", "vector", "wire", "line", "integral", "Vector Fields", "Line Integral", "Mass", "Center of Mass", "Moment of Inertia" ], "problem_v1": "Consider a wire in the shape of a helix $\\mathbf{r}(t)=6 \\cos t\\mathbf i+6 \\sin t\\mathbf j+5 t\\mathbf k, 0 \\leq t \\leq 2\\pi$ with constant density function $\\rho(x, y, z)=1$. A. Determine the mass of the wire: [ANS]\nB. Determine the coordinates of the center of mass: ([ANS], [ANS], [ANS]) C. Determine the moment of inertia about the z-axis: [ANS]", "answer_v1": [ "49.0732460090608", "0", "0", "15.707963267949", "1766.63685632619" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Consider a wire in the shape of a helix $\\mathbf{r}(t)=1 \\cos t\\mathbf i+1 \\sin t\\mathbf j+7 t\\mathbf k, 0 \\leq t \\leq 2\\pi$ with constant density function $\\rho(x, y, z)=1$. A. Determine the mass of the wire: [ANS]\nB. Determine the coordinates of the center of mass: ([ANS], [ANS], [ANS]) C. Determine the moment of inertia about the z-axis: [ANS]", "answer_v2": [ "44.4288293815837", "0", "0", "21.9911485751286", "44.4288293815837" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Consider a wire in the shape of a helix $\\mathbf{r}(t)=3 \\cos t\\mathbf i+3 \\sin t\\mathbf j+5 t\\mathbf k, 0 \\leq t \\leq 2\\pi$ with constant density function $\\rho(x, y, z)=1$. A. Determine the mass of the wire: [ANS]\nB. Determine the coordinates of the center of mass: ([ANS], [ANS], [ANS]) C. Determine the moment of inertia about the z-axis: [ANS]", "answer_v3": [ "36.636951272563", "0", "0", "15.707963267949", "329.732561453067" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0544", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Applications of line integrals", "level": "3", "keywords": [ "vector' 'integral' 'multivariable' 'mass", "Vector Fields", "Line Integral", "Parametric", "Mass", "vector", "mass", "wire", "line", "integral" ], "problem_v1": "Compute the total mass of a wire bent in a quarter circle with parametric equations: $x=8 \\cos t, \\ y=8 \\sin t, \\ 0 \\leq t \\leq \\frac{\\pi}{2} $ and density function $\\rho(x, y)=x^2+y^2$. [ANS]", "answer_v1": [ "804.247719318987" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the total mass of a wire bent in a quarter circle with parametric equations: $x=1 \\cos t, \\ y=1 \\sin t, \\ 0 \\leq t \\leq \\frac{\\pi}{2} $ and density function $\\rho(x, y)=x^2+y^2$. [ANS]", "answer_v2": [ "1.5707963267949" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the total mass of a wire bent in a quarter circle with parametric equations: $x=4 \\cos t, \\ y=4 \\sin t, \\ 0 \\leq t \\leq \\frac{\\pi}{2} $ and density function $\\rho(x, y)=x^2+y^2$. [ANS]", "answer_v3": [ "100.530964914873" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0545", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Applications of line integrals", "level": "3", "keywords": [ "Multivariable", "Vector", "Work", "Vector Fields", "Line Integral", "Work" ], "problem_v1": "If C is the curve given by $\\mathbf{r} \\left(t \\right)=\\left(1+4 \\sin t \\right) \\mathbf{i}+\\left(1+3 \\sin^{2} t \\right) \\mathbf{j}+\\left(1+4 \\sin^{3} t \\right) \\mathbf{k}$, $0 \\leq t \\leq \\frac{\\pi}{2} $ and F is the radial vector field $\\mathbf{F} \\left(x, y, z \\right)=x \\mathbf{i}+y \\mathbf{j}+z \\mathbf{k}$, compute the work done by F on a particle moving along C. [ANS]", "answer_v1": [ "31.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If C is the curve given by $\\mathbf{r} \\left(t \\right)=\\left(1+1 \\sin t \\right) \\mathbf{i}+\\left(1+5 \\sin^{2} t \\right) \\mathbf{j}+\\left(1+1 \\sin^{3} t \\right) \\mathbf{k}$, $0 \\leq t \\leq \\frac{\\pi}{2} $ and F is the radial vector field $\\mathbf{F} \\left(x, y, z \\right)=x \\mathbf{i}+y \\mathbf{j}+z \\mathbf{k}$, compute the work done by F on a particle moving along C. [ANS]", "answer_v2": [ "20.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If C is the curve given by $\\mathbf{r} \\left(t \\right)=\\left(1+2 \\sin t \\right) \\mathbf{i}+\\left(1+4 \\sin^{2} t \\right) \\mathbf{j}+\\left(1+2 \\sin^{3} t \\right) \\mathbf{k}$, $0 \\leq t \\leq \\frac{\\pi}{2} $ and F is the radial vector field $\\mathbf{F} \\left(x, y, z \\right)=x \\mathbf{i}+y \\mathbf{j}+z \\mathbf{k}$, compute the work done by F on a particle moving along C. [ANS]", "answer_v3": [ "20" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0546", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Applications of line integrals", "level": "4", "keywords": [ "vector' 'multivariable' 'green's' 'work" ], "problem_v1": "Find the stream function for the vector field $F=- \\frac{16y^{3}}{16} \\boldsymbol{i}- \\frac{16x^{3}}{16} \\boldsymbol{j}$: [ANS]\nIf C is a curve from the point P=$\\left(5,4\\right)$ to the point Q=$\\left(5,5\\right)$, then the flux across C is given by: [ANS]", "answer_v1": [ "x^4/4-y^4/4", "-92.25" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the stream function for the vector field $F=- \\frac{49y^{6}}{49} \\boldsymbol{i}- \\frac{16x^{3}}{16} \\boldsymbol{j}$: [ANS]\nIf C is a curve from the point P=$\\left(2,6\\right)$ to the point Q=$\\left(2,3\\right)$, then the flux across C is given by: [ANS]", "answer_v2": [ "x^4/4-y^7/7", "39678.4" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the stream function for the vector field $F=- \\frac{16y^{3}}{16} \\boldsymbol{i}- \\frac{16x^{3}}{16} \\boldsymbol{j}$: [ANS]\nIf C is a curve from the point P=$\\left(3,5\\right)$ to the point Q=$\\left(3,4\\right)$, then the flux across C is given by: [ANS]", "answer_v3": [ "x^4/4-y^4/4", "92.25" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0547", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Applications of line integrals", "level": "5", "keywords": [ "line integral", "integral", "multivariable", "calculus" ], "problem_v1": "A horizontal square has sides of 1400 km running north-south and east-west. A wind blows from the west and decreases in magnitude toward the north at a rate of 8 m/s for every 700 km. Compute the circulation of the wind counterclockwise around the square. circulation=[ANS] m^2/s", "answer_v1": [ "2.24E+7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A horizontal square has sides of 400 km running north-south and east-west. A wind blows from the west and increases in magnitude toward the north at a rate of 4 m/s for every 200 km. Compute the circulation of the wind counterclockwise around the square. circulation=[ANS] m^2/s", "answer_v2": [ "-3.2E+6" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A horizontal square has sides of 800 km running north-south and east-west. A wind blows from the west and increases in magnitude toward the north at a rate of 6 m/s for every 400 km. Compute the circulation of the wind counterclockwise around the square. circulation=[ANS] m^2/s", "answer_v3": [ "-9.6E+6" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0548", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Applications of line integrals", "level": "5", "keywords": [ "line integral", "integral", "multivariable", "calculus" ], "problem_v1": "In this problem, use the fact that the force of gravity on a particle of mass $m$ at the point with position vector $\\vec r$ is \\vec F=- \\frac{GMm\\vec r}{{\\Vert\\vec r\\Vert} ^3} where $G$ is a constant and $M$ is the mass of the earth. Calculate the work done by the force of gravity on a particle of mass $m$ as it moves radially from 8000 km to 9600 km from the center of the earth. Work=[ANS]\n(Your answer may involve the constants $m$, $M$ and $G$.)", "answer_v1": [ "G*M*m*(1/9600-1/8000)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "In this problem, use the fact that the force of gravity on a particle of mass $m$ at the point with position vector $\\vec r$ is \\vec F=- \\frac{GMm\\vec r}{{\\Vert\\vec r\\Vert} ^3} where $G$ is a constant and $M$ is the mass of the earth. Calculate the work done by the force of gravity on a particle of mass $m$ as it moves radially from 5000 km to 7000 km from the center of the earth. Work=[ANS]\n(Your answer may involve the constants $m$, $M$ and $G$.)", "answer_v2": [ "G*M*m*(1/7000-1/5000)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "In this problem, use the fact that the force of gravity on a particle of mass $m$ at the point with position vector $\\vec r$ is \\vec F=- \\frac{GMm\\vec r}{{\\Vert\\vec r\\Vert} ^3} where $G$ is a constant and $M$ is the mass of the earth. Calculate the work done by the force of gravity on a particle of mass $m$ as it moves radially from 6000 km to 7600 km from the center of the earth. Work=[ANS]\n(Your answer may involve the constants $m$, $M$ and $G$.)", "answer_v3": [ "G*M*m*(1/7600-1/6000)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0549", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Applications of line integrals", "level": "2", "keywords": [], "problem_v1": "Consider a piece of wire with uniform density. It is the quarter of a circle in the first quadrant. The circle is centered at the origin and has radius 8. Find the centroid $(\\bar x, \\bar y)$ of the wire. $\\bar x=\\bar y=$ [ANS]", "answer_v1": [ "16/pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Consider a piece of wire with uniform density. It is the quarter of a circle in the first quadrant. The circle is centered at the origin and has radius 2. Find the centroid $(\\bar x, \\bar y)$ of the wire. $\\bar x=\\bar y=$ [ANS]", "answer_v2": [ "4/pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Consider a piece of wire with uniform density. It is the quarter of a circle in the first quadrant. The circle is centered at the origin and has radius 4. Find the centroid $(\\bar x, \\bar y)$ of the wire. $\\bar x=\\bar y=$ [ANS]", "answer_v3": [ "8/pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0550", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Compute the divergence of the vector field:\n\\mathbf{F}=\\left< 7x-2zx^{6},6z-xy,z^{7}x^{2}\\right> $\\text{div} (\\mathbf{F})=$ [ANS]", "answer_v1": [ "7-12*z*x^5-x+7*z^6*x^2" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Compute the divergence of the vector field:\n\\mathbf{F}=\\left< x-2zx^{9},2z-xy,z^{4}x^{2}\\right> $\\text{div} (\\mathbf{F})=$ [ANS]", "answer_v2": [ "1-18*z*x^8-x+4*z^3*x^2" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Compute the divergence of the vector field:\n\\mathbf{F}=\\left< 3x-2zx^{6},3z-xy,z^{5}x^{2}\\right> $\\text{div} (\\mathbf{F})=$ [ANS]", "answer_v3": [ "3-12*z*x^5-x+5*z^4*x^2" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0551", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Let $\\mathbf{F}=\\left$. Calculate $\\text{curl}(\\mathbf{F})$. $\\text{curl}(\\mathbf{F})=$ [ANS]", "answer_v1": [ "(0,8*[cos(x)]^7*sin(x),6*[sin(x)]^5*cos(x)-6*e^(6*y))" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Let $\\mathbf{F}=\\left$. Calculate $\\text{curl}(\\mathbf{F})$. $\\text{curl}(\\mathbf{F})=$ [ANS]", "answer_v2": [ "(0,2*cos(x)*sin(x),9*[sin(x)]^8*cos(x)-3*e^(3*y))" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Let $\\mathbf{F}=\\left$. Calculate $\\text{curl}(\\mathbf{F})$. $\\text{curl}(\\mathbf{F})=$ [ANS]", "answer_v3": [ "(0,4*[cos(x)]^3*sin(x),6*[sin(x)]^5*cos(x)-4*e^(4*y))" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0552", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "multivariable", "Vector", "Divergence", "Curl", "Field", "Vector Fields", "Div", "Curl" ], "problem_v1": "Let $\\mathbf{F}=\\left(8 y z \\right) \\mathbf{i}+\\left(6x z \\right) \\mathbf{j}+\\left(7x y \\right) \\mathbf{k}$. Compute the following: A. div $\\mathbf{F}=$ [ANS]\nB. curl $\\mathbf{F}=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$ C. div curl $\\mathbf{F}=$ [ANS]\nNote: Your answers should be expressions of x, y and/or z; e.g. \"3xy\" or \"z\" or \"5\"", "answer_v1": [ "0", "(7 - 6)*x", "(8 - 7)*y", "(6 - 8)*z", "0" ], "answer_type_v1": [ "NV", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Let $\\mathbf{F}=\\left(1 y z \\right) \\mathbf{i}+\\left(10x z \\right) \\mathbf{j}+\\left(2x y \\right) \\mathbf{k}$. Compute the following: A. div $\\mathbf{F}=$ [ANS]\nB. curl $\\mathbf{F}=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$ C. div curl $\\mathbf{F}=$ [ANS]\nNote: Your answers should be expressions of x, y and/or z; e.g. \"3xy\" or \"z\" or \"5\"", "answer_v2": [ "0", "(2 - 10)*x", "(1 - 2)*y", "(10 - 1)*z", "0" ], "answer_type_v2": [ "NV", "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Let $\\mathbf{F}=\\left(4 y z \\right) \\mathbf{i}+\\left(7x z \\right) \\mathbf{j}+\\left(3x y \\right) \\mathbf{k}$. Compute the following: A. div $\\mathbf{F}=$ [ANS]\nB. curl $\\mathbf{F}=$ [ANS] $\\mathbf{i}+$ [ANS] $\\mathbf{j}+$ [ANS] $\\mathbf{k}$ C. div curl $\\mathbf{F}=$ [ANS]\nNote: Your answers should be expressions of x, y and/or z; e.g. \"3xy\" or \"z\" or \"5\"", "answer_v3": [ "0", "(3 - 7)*x", "(4 - 3)*y", "(7 - 4)*z", "0" ], "answer_type_v3": [ "NV", "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0553", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "4", "keywords": [ "vector' 'multivariable" ], "problem_v1": "A vector field gives a geographical description of the flow of money in a society. In the neighborhood of a political convention, the divergence of this vector field is: [ANS] A. zero B. positive C. negative", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C" ] ], "problem_v2": "A vector field gives a geographical description of the flow of money in a society. In the neighborhood of a political convention, the divergence of this vector field is: [ANS] A. negative B. positive C. zero", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C" ] ], "problem_v3": "A vector field gives a geographical description of the flow of money in a society. In the neighborhood of a political convention, the divergence of this vector field is: [ANS] A. zero B. negative C. positive", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C" ] ] }, { "id": "Calculus_-_multivariable_0554", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "divergence", "vector field", "multivariable", "calculus" ], "problem_v1": "The flux of $\\vec F$ out of a small sphere of radius $0.005$ centered at $(3, 4, 4)$, is $0.01$. Estimate:\n(a) $\\mbox{div} \\vec F$ at $(3, 4, 4)$: $\\mbox{div} \\vec F \\approx$ [ANS]\n(b) The flux of $\\vec F$ out of a sphere of radius $0.01$ centered at $(3, 4, 4)$. flux $\\approx$ [ANS]", "answer_v1": [ "3*0.01/(4*pi*0.005^3)", "0.01*8" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The flux of $\\vec F$ out of a small sphere of radius $0.1$ centered at $(5, 1, 2)$, is $0.025$. Estimate:\n(a) $\\mbox{div} \\vec F$ at $(5, 1, 2)$: $\\mbox{div} \\vec F \\approx$ [ANS]\n(b) The flux of $\\vec F$ out of a sphere of radius $0.2$ centered at $(5, 1, 2)$. flux $\\approx$ [ANS]", "answer_v2": [ "3*0.025/(4*pi*0.1^3)", "0.025*8" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The flux of $\\vec F$ out of a small sphere of radius $0.05$ centered at $(4, 2, 3)$, is $0.0075$. Estimate:\n(a) $\\mbox{div} \\vec F$ at $(4, 2, 3)$: $\\mbox{div} \\vec F \\approx$ [ANS]\n(b) The flux of $\\vec F$ out of a sphere of radius $0.1$ centered at $(4, 2, 3)$. flux $\\approx$ [ANS]", "answer_v3": [ "3*0.0075/(4*pi*0.05^3)", "0.0075*8" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0555", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "divergence", "vector field", "multivariable", "calculus" ], "problem_v1": "Let $\\vec F(x,y,z)=6x^2\\vec i-\\cos(xz)(\\vec i+\\vec k)$. Calulate the divergence: $\\mbox{div}\\vec F(x,y,z)=$ [ANS]", "answer_v1": [ "2*6*x+z*sin(x*z)+x*sin(x*z)" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Let $\\vec F(x,y,z)=x^2\\vec i-\\cos(xy)(\\vec i+\\vec j)$. Calulate the divergence: $\\mbox{div}\\vec F(x,y,z)=$ [ANS]", "answer_v2": [ "2*1*x+y*sin(x*y)+x*sin(x*y)" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Let $\\vec F(x,y,z)=3x^2\\vec i-\\cos(xy)(\\vec i+\\vec j)$. Calulate the divergence: $\\mbox{div}\\vec F(x,y,z)=$ [ANS]", "answer_v3": [ "2*3*x+y*sin(x*y)+x*sin(x*y)" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0556", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "3", "keywords": [ "divergence", "vector field", "multivariable", "calculus" ], "problem_v1": "Find the divergence of the vector field $\\vec F=\\vec a\\times \\vec r$, if $\\vec r=x\\,\\mathit{\\vec i}+y\\,\\mathit{\\vec j}+z\\,\\mathit{\\vec k}$ and $\\vec a=7\\,\\mathit{\\vec i}+6\\,\\mathit{\\vec j}+6\\,\\mathit{\\vec k}$, by first calculating the cross-product. $\\vec F=$ [ANS]\nso that $\\mbox{div}\\vec F=$ [ANS]", "answer_v1": [ "(6*z-6*y)i+(6*x-7*z)j+(7*y-6*x)k", "0" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the divergence of the vector field $\\vec F=\\vec a\\times \\vec r$, if $\\vec r=x\\,\\mathit{\\vec i}+y\\,\\mathit{\\vec j}+z\\,\\mathit{\\vec k}$ and $\\vec a=\\,\\mathit{\\vec i}+9\\,\\mathit{\\vec j}+2\\,\\mathit{\\vec k}$, by first calculating the cross-product. $\\vec F=$ [ANS]\nso that $\\mbox{div}\\vec F=$ [ANS]", "answer_v2": [ "(9*z-2*y)i+(2*x-1*z)j+(1*y-9*x)k", "0" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the divergence of the vector field $\\vec F=\\vec a\\times \\vec r$, if $\\vec r=x\\,\\mathit{\\vec i}+y\\,\\mathit{\\vec j}+z\\,\\mathit{\\vec k}$ and $\\vec a=3\\,\\mathit{\\vec i}+6\\,\\mathit{\\vec j}+3\\,\\mathit{\\vec k}$, by first calculating the cross-product. $\\vec F=$ [ANS]\nso that $\\mbox{div}\\vec F=$ [ANS]", "answer_v3": [ "(6*z-3*y)i+(3*x-3*z)j+(3*y-6*x)k", "0" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0557", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "divergence", "vector field", "multivariable", "calculus" ], "problem_v1": "Consider $ \\mbox{div}\\,\\left( \\frac{y\\vec i-x\\vec j}{(x^2+y^2)^{4} }\\right)$.\n(a) Is this a vector or a scalar? [ANS] (b) Calculate it: $\\mbox{div}\\,\\left( \\frac{y\\vec i-x\\vec j}{(x^2+y^2)^{4} }\\right)=$ [ANS]+[ANS]=[ANS]", "answer_v1": [ "scalar", "-2*4*x*y/[(x^2+y^2)^(4+1)]", "2*4*x*y/[(x^2+y^2)^(4+1)]", "0" ], "answer_type_v1": [ "MCS", "EX", "EX", "NV" ], "options_v1": [ [ "vector", "scalar" ], [], [], [] ], "problem_v2": "Consider $ \\mbox{div}\\,\\left( \\frac{y\\vec i-x\\vec j}{(x^2+y^2)} \\right)$.\n(a) Is this a vector or a scalar? [ANS] (b) Calculate it: $\\mbox{div}\\,\\left( \\frac{y\\vec i-x\\vec j}{(x^2+y^2)} \\right)=$ [ANS]+[ANS]=[ANS]", "answer_v2": [ "scalar", "-2*1*x*y/[(x^2+y^2)^(1+1)]", "2*1*x*y/[(x^2+y^2)^(1+1)]", "0" ], "answer_type_v2": [ "MCS", "EX", "EX", "NV" ], "options_v2": [ [ "vector", "scalar" ], [], [], [] ], "problem_v3": "Consider $ \\mbox{div}\\,\\left( \\frac{y\\vec i-x\\vec j}{(x^2+y^2)^{2} }\\right)$.\n(a) Is this a vector or a scalar? [ANS] (b) Calculate it: $\\mbox{div}\\,\\left( \\frac{y\\vec i-x\\vec j}{(x^2+y^2)^{2} }\\right)=$ [ANS]+[ANS]=[ANS]", "answer_v3": [ "scalar", "-2*2*x*y/[(x^2+y^2)^(2+1)]", "2*2*x*y/[(x^2+y^2)^(2+1)]", "0" ], "answer_type_v3": [ "MCS", "EX", "EX", "NV" ], "options_v3": [ [ "vector", "scalar" ], [], [], [] ] }, { "id": "Calculus_-_multivariable_0558", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "5", "keywords": [ "divergence", "vector field", "multivariable", "calculus" ], "problem_v1": "Due to roadwork, the traffic on a highway slows linearly from 70 miles/hour to 15 miles/hour over 2500 foot stretch of road, then crawls along at 15 miles/hour for 6000 feet then speeds back up linearly to 70 miles/hour in the next 1250 feet, after which it moves steadily at 70 miles/hour.\n(a) Sketch a velocity vector field for the traffic flow. Then write a formula for the velocity vector $\\vec v$ (in mi/hr) as a function of the distance $x$ feet along the road from the initial point of slowdown. Take the direction of motion to be $\\vec i$ and consider the various sections of the road separately. (For each, write the vector $\\vec v$ as a two-dimensional vector; assume that cars do not move perpendicular to the direction of the road.) For $0 \\le x <$ [ANS], $\\vec v=$ [ANS]\nFor [ANS] $\\le x <$ [ANS], $\\vec v=$ [ANS]\nFor [ANS] $\\le x <$ [ANS], $\\vec v=$ [ANS]\nFor [ANS] $\\le x$, $\\vec v=$ [ANS]\n(b) Find each of the following values of the divergence. For each, include. $\\mbox{div}\\vec v$ at $x=1000$: $\\mbox{div}\\vec v=$ [ANS] mi/hr*ft\n$\\mbox{div}\\vec v$ at $x=6000$: $\\mbox{div}\\vec v=$ [ANS] mi/hr*ft\n$\\mbox{div}\\vec v$ at $x=9500$: $\\mbox{div}\\vec v=$ [ANS] mi/hr*ft", "answer_v1": [ "2500", "(70-55*x/2500)i", "2500", "2500+6000", "15i", "2500+6000", "2500+6000+1250", "[15+55*(x-8500)/1250]i", "2500+6000+1250", "70i", "-0.022", "0", "0.044" ], "answer_type_v1": [ "NV", "EX", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Due to roadwork, the traffic on a highway slows linearly from 50 miles/hour to 20 miles/hour over 1500 foot stretch of road, then crawls along at 20 miles/hour for 4500 feet then speeds back up linearly to 50 miles/hour in the next 2000 feet, after which it moves steadily at 50 miles/hour.\n(a) Sketch a velocity vector field for the traffic flow. Then write a formula for the velocity vector $\\vec v$ (in mi/hr) as a function of the distance $x$ feet along the road from the initial point of slowdown. Take the direction of motion to be $\\vec i$ and consider the various sections of the road separately. (For each, write the vector $\\vec v$ as a two-dimensional vector; assume that cars do not move perpendicular to the direction of the road.) For $0 \\le x <$ [ANS], $\\vec v=$ [ANS]\nFor [ANS] $\\le x <$ [ANS], $\\vec v=$ [ANS]\nFor [ANS] $\\le x <$ [ANS], $\\vec v=$ [ANS]\nFor [ANS] $\\le x$, $\\vec v=$ [ANS]\n(b) Find each of the following values of the divergence. For each, include. $\\mbox{div}\\vec v$ at $x=500$: $\\mbox{div}\\vec v=$ [ANS] mi/hr*ft\n$\\mbox{div}\\vec v$ at $x=2500$: $\\mbox{div}\\vec v=$ [ANS] mi/hr*ft\n$\\mbox{div}\\vec v$ at $x=7000$: $\\mbox{div}\\vec v=$ [ANS] mi/hr*ft", "answer_v2": [ "1500", "(50-30*x/1500)i", "1500", "1500+4500", "20i", "1500+4500", "1500+4500+2000", "[20+30*(x-6000)/2000]i", "1500+4500+2000", "50i", "-0.02", "0", "0.015" ], "answer_type_v2": [ "NV", "EX", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Due to roadwork, the traffic on a highway slows linearly from 55 miles/hour to 15 miles/hour over 2000 foot stretch of road, then crawls along at 15 miles/hour for 5000 feet then speeds back up linearly to 55 miles/hour in the next 1250 feet, after which it moves steadily at 55 miles/hour.\n(a) Sketch a velocity vector field for the traffic flow. Then write a formula for the velocity vector $\\vec v$ (in mi/hr) as a function of the distance $x$ feet along the road from the initial point of slowdown. Take the direction of motion to be $\\vec i$ and consider the various sections of the road separately. (For each, write the vector $\\vec v$ as a two-dimensional vector; assume that cars do not move perpendicular to the direction of the road.) For $0 \\le x <$ [ANS], $\\vec v=$ [ANS]\nFor [ANS] $\\le x <$ [ANS], $\\vec v=$ [ANS]\nFor [ANS] $\\le x <$ [ANS], $\\vec v=$ [ANS]\nFor [ANS] $\\le x$, $\\vec v=$ [ANS]\n(b) Find each of the following values of the divergence. For each, include. $\\mbox{div}\\vec v$ at $x=1000$: $\\mbox{div}\\vec v=$ [ANS] mi/hr*ft\n$\\mbox{div}\\vec v$ at $x=6500$: $\\mbox{div}\\vec v=$ [ANS] mi/hr*ft\n$\\mbox{div}\\vec v$ at $x=8250$: $\\mbox{div}\\vec v=$ [ANS] mi/hr*ft", "answer_v3": [ "2000", "(55-40*x/2000)i", "2000", "2000+5000", "15i", "2000+5000", "2000+5000+1250", "[15+40*(x-7000)/1250]i", "2000+5000+1250", "55i", "-0.02", "0", "0.032" ], "answer_type_v3": [ "NV", "EX", "NV", "NV", "EX", "NV", "NV", "EX", "NV", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0559", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "3", "keywords": [ "divergence", "vector field", "multivariable", "calculus" ], "problem_v1": "Let $\\vec{F}=(6 z+6)\\vec i+7 z\\vec j+(3 z+3)\\vec k$, and let the point $P=(a,b,c)$, where $a$, $b$ and $c$ are constants. In this problem we will calculate $\\mbox{div}\\vec F$ in two different ways, first by using the geometric definition and second by using partial derivatives.\n(a) Consider a (three-dimensional) box with four of its corners at $(a,b,c)$, $(a+w,b,c)$, $(a,b+w,c)$ and $(a,b,c+w)$, where $w$ is a constant edge length. Find the flux through the box. flux=[ANS]\nThus, we have $\\mbox{div}\\vec F(x,y,z)=\\lim\\limits_{w\\to 0} \\left(\\right.$ [ANS]/[ANS] $\\left.\\right)=$ [ANS]\n(b) Next, find the divergence using partial derivatives: $\\mbox{div}\\vec F(x,y,z)=$ [ANS]", "answer_v1": [ "3*w^3", "3*w^3", "w^3", "3", "3" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Let $\\vec{F}=(9 z+2)\\vec i+4 z\\vec j+(9 z+3)\\vec k$, and let the point $P=(a,b,c)$, where $a$, $b$ and $c$ are constants. In this problem we will calculate $\\mbox{div}\\vec F$ in two different ways, first by using the geometric definition and second by using partial derivatives.\n(a) Consider a (three-dimensional) box with four of its corners at $(a,b,c)$, $(a+w,b,c)$, $(a,b+w,c)$ and $(a,b,c+w)$, where $w$ is a constant edge length. Find the flux through the box. flux=[ANS]\nThus, we have $\\mbox{div}\\vec F(x,y,z)=\\lim\\limits_{w\\to 0} \\left(\\right.$ [ANS]/[ANS] $\\left.\\right)=$ [ANS]\n(b) Next, find the divergence using partial derivatives: $\\mbox{div}\\vec F(x,y,z)=$ [ANS]", "answer_v2": [ "9*w^3", "9*w^3", "w^3", "9", "9" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Let $\\vec{F}=(6 z+3)\\vec i+5 z\\vec j+(2 z+4)\\vec k$, and let the point $P=(a,b,c)$, where $a$, $b$ and $c$ are constants. In this problem we will calculate $\\mbox{div}\\vec F$ in two different ways, first by using the geometric definition and second by using partial derivatives.\n(a) Consider a (three-dimensional) box with four of its corners at $(a,b,c)$, $(a+w,b,c)$, $(a,b+w,c)$ and $(a,b,c+w)$, where $w$ is a constant edge length. Find the flux through the box. flux=[ANS]\nThus, we have $\\mbox{div}\\vec F(x,y,z)=\\lim\\limits_{w\\to 0} \\left(\\right.$ [ANS]/[ANS] $\\left.\\right)=$ [ANS]\n(b) Next, find the divergence using partial derivatives: $\\mbox{div}\\vec F(x,y,z)=$ [ANS]", "answer_v3": [ "2*w^3", "2*w^3", "w^3", "2", "2" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0560", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "curl", "vector field", "multivariable", "calculus" ], "problem_v1": "Compute the curl of the vector field $\\vec F=4z\\,\\mathit{\\vec i}+y\\,\\mathit{\\vec j}+2x\\,\\mathit{\\vec k}$. curl=[ANS]", "answer_v1": [ "2j" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Compute the curl of the vector field $\\vec F=-7z\\,\\mathit{\\vec i}+7y\\,\\mathit{\\vec j}-6x\\,\\mathit{\\vec k}$. curl=[ANS]", "answer_v2": [ "-j" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Compute the curl of the vector field $\\vec F=-3z\\,\\mathit{\\vec i}+2y\\,\\mathit{\\vec j}-4x\\,\\mathit{\\vec k}$. curl=[ANS]", "answer_v3": [ "j" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0561", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "4", "keywords": [ "curl", "vector field", "multivariable", "calculus" ], "problem_v1": "Three small squares, $S_1, S_2,$ and $S_3$, each with side 0.1 and centered at the point $(5, 5, 6)$, lie parallel to the $xy$-, $yz$-and $xz$-planes, respectively. The squares are oriented counterclockwise when viewed from the positive $z$-, $x$-, and $y$-axes, respectively. A vector field $\\vec G$ has circulation around $S_1$ of $-2$, around $S_2$ of $-2$, and around $S_3$ of $0.01$. Estimate curl $\\vec G$ at the point $(5, 5, 6)$ $\\mbox{curl}\\vec G \\approx$ [ANS]", "answer_v1": [ "-200i+j-200k" ], "answer_type_v1": [ "EX" ], "options_v1": [ [] ], "problem_v2": "Three small squares, $S_1, S_2,$ and $S_3$, each with side 0.01 and centered at the point $(8, 2, 3)$, lie parallel to the $xy$-, $yz$-and $xz$-planes, respectively. The squares are oriented counterclockwise when viewed from the positive $z$-, $x$-, and $y$-axes, respectively. A vector field $\\vec G$ has circulation around $S_1$ of $0.05$, around $S_2$ of $-0.0002$, and around $S_3$ of $-0.03$. Estimate curl $\\vec G$ at the point $(8, 2, 3)$ $\\mbox{curl}\\vec G \\approx$ [ANS]", "answer_v2": [ "-2i-300j+500k" ], "answer_type_v2": [ "EX" ], "options_v2": [ [] ], "problem_v3": "Three small squares, $S_1, S_2,$ and $S_3$, each with side 0.01 and centered at the point $(5, 3, 5)$, lie parallel to the $xy$-, $yz$-and $xz$-planes, respectively. The squares are oriented counterclockwise when viewed from the positive $z$-, $x$-, and $y$-axes, respectively. A vector field $\\vec G$ has circulation around $S_1$ of $-0.0003$, around $S_2$ of $-0.02$, and around $S_3$ of $0.03$. Estimate curl $\\vec G$ at the point $(5, 3, 5)$ $\\mbox{curl}\\vec G \\approx$ [ANS]", "answer_v3": [ "-200i+300j-3k" ], "answer_type_v3": [ "EX" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0562", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "3", "keywords": [ "divergence theorem", "stokes theorem", "divergence", "curl", "gradient vector field" ], "problem_v1": "Express $\\left<7x+6y,6x+7y,0\\right>$ as the sum of a curl free vector field and a divergence free vector field. $\\left<7x+6y,6x+7y,0\\right>=$ [ANS]+[ANS], where the first vector in the sum is curl free and the second is divergence free.\n(For this problem, enter your vectors with angle-bracket notation: $$, not in $ijk$-notation.)", "answer_v1": [ "(7*x,7*y,0)", "(6*y,6*x,0)" ], "answer_type_v1": [ "OL", "OL" ], "options_v1": [ [], [] ], "problem_v2": "Express $\\left<2x+8y,3x+4y,0\\right>$ as the sum of a curl free vector field and a divergence free vector field. $\\left<2x+8y,3x+4y,0\\right>=$ [ANS]+[ANS], where the first vector in the sum is curl free and the second is divergence free.\n(For this problem, enter your vectors with angle-bracket notation: $$, not in $ijk$-notation.)", "answer_v2": [ "(2*x,4*y,0)", "(8*y,3*x,0)" ], "answer_type_v2": [ "OL", "OL" ], "options_v2": [ [], [] ], "problem_v3": "Express $\\left<4x+6y,3x+5y,0\\right>$ as the sum of a curl free vector field and a divergence free vector field. $\\left<4x+6y,3x+5y,0\\right>=$ [ANS]+[ANS], where the first vector in the sum is curl free and the second is divergence free.\n(For this problem, enter your vectors with angle-bracket notation: $$, not in $ijk$-notation.)", "answer_v3": [ "(4*x,5*y,0)", "(6*y,3*x,0)" ], "answer_type_v3": [ "OL", "OL" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0563", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "divergence theorem", "stokes theorem", "divergence", "curl", "gradient vector field" ], "problem_v1": "For each of the following vector fields, determine whether a vector potential exists. If so, find one.\nFor this problem, enter your vectors with angle-bracket notation: $(a, b, c)$, not in $ijk$-notation.\n(a) $\\vec F=7y^{2}\\,\\mathit{\\vec i}+5z^{2}\\,\\mathit{\\vec j}+5x^{2}\\,\\mathit{\\vec k}$ $\\vec F$ [ANS] $\\vec H$, $\\vec H=$ [ANS]\n(If there is no potential function, enter none for the function.) (b) $\\vec F=7x^{2}\\,\\mathit{\\vec i}+5y^{2}\\,\\mathit{\\vec j}+5z^{2}\\,\\mathit{\\vec k}$ $\\vec F$ [ANS] $\\vec H$, $\\vec H=$ [ANS]\n(If there is no potential function, enter none for the function.)", "answer_v1": [ "has a vector potential", "-5*x^2*yi-7*y^2*zj-5*x*z^2k", "does not have a vector potential", "none" ], "answer_type_v1": [ "MCS", "EX", "MCS", "OE" ], "options_v1": [ [ "has a vector potential", "does not have a vector potential" ], [], [ "has a vector potential", "does not have a vector potential" ], [] ], "problem_v2": "For each of the following vector fields, determine whether a vector potential exists. If so, find one.\nFor this problem, enter your vectors with angle-bracket notation: $(a, b, c)$, not in $ijk$-notation.\n(a) $\\vec F=x^{2}\\,\\mathit{\\vec i}+8y^{2}\\,\\mathit{\\vec j}+2z^{2}\\,\\mathit{\\vec k}$ $\\vec F$ [ANS] $\\vec H$, $\\vec H=$ [ANS]\n(If there is no potential function, enter none for the function.) (b) $\\vec F=y^{2}\\,\\mathit{\\vec i}+8z^{2}\\,\\mathit{\\vec j}+2x^{2}\\,\\mathit{\\vec k}$ $\\vec F$ [ANS] $\\vec H$, $\\vec H=$ [ANS]\n(If there is no potential function, enter none for the function.)", "answer_v2": [ "does not have a vector potential", "none", "has a vector potential", "-2*x^2*yi-y^2*zj-8*x*z^2k" ], "answer_type_v2": [ "MCS", "TF", "MCS", "EX" ], "options_v2": [ [ "has a vector potential", "does not have a vector potential" ], [], [ "has a vector potential", "does not have a vector potential" ], [] ], "problem_v3": "For each of the following vector fields, determine whether a vector potential exists. If so, find one.\nFor this problem, enter your vectors with angle-bracket notation: $(a, b, c)$, not in $ijk$-notation.\n(a) $\\vec F=3y^{2}\\,\\mathit{\\vec i}+5z^{2}\\,\\mathit{\\vec j}+3x^{2}\\,\\mathit{\\vec k}$ $\\vec F$ [ANS] $\\vec H$, $\\vec H=$ [ANS]\n(If there is no potential function, enter none for the function.) (b) $\\vec F=3x^{2}\\,\\mathit{\\vec i}+5y^{2}\\,\\mathit{\\vec j}+3z^{2}\\,\\mathit{\\vec k}$ $\\vec F$ [ANS] $\\vec H$, $\\vec H=$ [ANS]\n(If there is no potential function, enter none for the function.)", "answer_v3": [ "has a vector potential", "-3*x^2*yi-3*y^2*zj-5*x*z^2k", "does not have a vector potential", "none" ], "answer_type_v3": [ "MCS", "EX", "MCS", "OE" ], "options_v3": [ [ "has a vector potential", "does not have a vector potential" ], [], [ "has a vector potential", "does not have a vector potential" ], [] ] }, { "id": "Calculus_-_multivariable_0564", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "divergence theorem", "stokes theorem", "divergence", "curl", "gradient vector field" ], "problem_v1": "For each of the following vector fields, find its curl and determine if it is a gradient field.\n(a) $\\vec F=4\\!\\left(xy+z^{2}\\right)\\,\\mathit{\\vec i}+8\\!\\left(x^{2}+yz\\right)\\,\\mathit{\\vec j}+8\\!\\left(xz+y^{2}\\right)\\,\\mathit{\\vec k}$: $\\mbox{curl}\\vec F=$ [ANS]\n$\\vec F$ [ANS] (b) $\\vec G=\\left(8xy+3x^{3}\\right)\\,\\mathit{\\vec i}+\\left(4x^{2}+z^{2}\\right)\\,\\mathit{\\vec j}+\\left(2yz-4z\\right)\\,\\mathit{\\vec k}$: $\\mbox{curl}\\vec G=$ [ANS]\n$\\vec G$ [ANS] (c) $\\vec H=\\left(8xz+y^{2}\\right)\\,\\mathit{\\vec i}+2xy\\,\\mathit{\\vec j}+4x^{2}\\,\\mathit{\\vec k}$: $\\mbox{curl}\\vec H=$ [ANS]\n$\\vec H$ [ANS]", "answer_v1": [ "8*yi+12*xk", "is not a gradient field", "0", "is a gradient field", "0", "is a gradient field" ], "answer_type_v1": [ "EX", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v1": [ [], [ "is a gradient field", "is not a gradient field" ], [], [ "is a gradient field", "is not a gradient field" ], [], [ "is a gradient field", "is not a gradient field" ] ], "problem_v2": "For each of the following vector fields, find its curl and determine if it is a gradient field.\n(a) $\\vec F=\\left(2xz+y^{2}\\right)\\,\\mathit{\\vec i}+2xy\\,\\mathit{\\vec j}+x^{2}\\,\\mathit{\\vec k}$: $\\mbox{curl}\\vec F=$ [ANS]\n$\\vec F$ [ANS] (b) $\\vec G=\\left(xy+yz\\right)\\,\\mathit{\\vec i}+\\left(5x^{2}+z^{2}\\right)\\,\\mathit{\\vec j}+xz\\,\\mathit{\\vec k}$: $\\mbox{curl}\\vec G=$ [ANS]\n$\\vec G$ [ANS] (c) $\\vec H=\\left(2xy+5x^{3}\\right)\\,\\mathit{\\vec i}+\\left(x^{2}+z^{2}\\right)\\,\\mathit{\\vec j}+\\left(2yz-z\\right)\\,\\mathit{\\vec k}$: $\\mbox{curl}\\vec H=$ [ANS]\n$\\vec H$ [ANS]", "answer_v2": [ "0", "is a gradient field", "-2*zi+(y-z)j+(9*x-z)k", "is not a gradient field", "0", "is a gradient field" ], "answer_type_v2": [ "NV", "MCS", "EX", "MCS", "NV", "MCS" ], "options_v2": [ [], [ "is a gradient field", "is not a gradient field" ], [], [ "is a gradient field", "is not a gradient field" ], [], [ "is a gradient field", "is not a gradient field" ] ], "problem_v3": "For each of the following vector fields, find its curl and determine if it is a gradient field.\n(a) $\\vec F=2yz\\,\\mathit{\\vec i}+\\left(z^{2}-2xz\\right)\\,\\mathit{\\vec j}+\\left(2xy+2yz\\right)\\,\\mathit{\\vec k}$: $\\mbox{curl}\\vec F=$ [ANS]\n$\\vec F$ [ANS] (b) $\\vec G=\\left(4xy+4x^{3}\\right)\\,\\mathit{\\vec i}+\\left(2x^{2}+z^{2}\\right)\\,\\mathit{\\vec j}+\\left(2yz-2z\\right)\\,\\mathit{\\vec k}$: $\\mbox{curl}\\vec G=$ [ANS]\n$\\vec G$ [ANS] (c) $\\vec H=\\left(4xz+y^{2}\\right)\\,\\mathit{\\vec i}+2xy\\,\\mathit{\\vec j}+2x^{2}\\,\\mathit{\\vec k}$: $\\mbox{curl}\\vec H=$ [ANS]\n$\\vec H$ [ANS]", "answer_v3": [ "4*xi-4*zk", "is not a gradient field", "0", "is a gradient field", "0", "is a gradient field" ], "answer_type_v3": [ "EX", "MCS", "NV", "MCS", "NV", "MCS" ], "options_v3": [ [], [ "is a gradient field", "is not a gradient field" ], [], [ "is a gradient field", "is not a gradient field" ], [], [ "is a gradient field", "is not a gradient field" ] ] }, { "id": "Calculus_-_multivariable_0565", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "curl' 'divergence" ], "problem_v1": "Let $F(x,y,z)=(5xz^2, 2xyz, 2xy^3z)$ be a vector field and $f(x,y,z)=x^3 y^2 z$. $\\nabla f=($ [ANS], [ANS], [ANS] $)$. $\\nabla \\times F=($ [ANS], [ANS], [ANS] $)$. $F \\times \\nabla f=($ [ANS], [ANS], [ANS] $)$. $F \\cdot \\nabla f=$ [ANS].", "answer_v1": [ "3*x*x*y*y*z", "2*x^3 * y * z", "x^3 * y^2", "3*2*x*y*y*z - 2*x*y", "2*5*x*z - 2*y^3 * z", "2*y*z", "2*x^4 * y^3 * z - 2*2 * x^4 * y^4 *z^2", "3*2*x^3 * y**5 * z^2 - 5*x^4 * y^2 * z^2", "2*5*x^4 * y * z^3 - 3*2*x^3 * y^3 * z^2", "3*5*x^3 * y^2 * z^3 + 2*2* x^4 * y^2 * z^2 + 2* x^4 * y**5 * z" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $F(x,y,z)=(-8xz^2, 8xyz,-7xy^3z)$ be a vector field and $f(x,y,z)=x^3 y^2 z$. $\\nabla f=($ [ANS], [ANS], [ANS] $)$. $\\nabla \\times F=($ [ANS], [ANS], [ANS] $)$. $F \\times \\nabla f=($ [ANS], [ANS], [ANS] $)$. $F \\cdot \\nabla f=$ [ANS].", "answer_v2": [ "3*x*x*y*y*z", "2*x^3 * y * z", "x^3 * y^2", "3*-7*x*y*y*z - 8*x*y", "2*-8*x*z - -7*y^3 * z", "8*y*z", "8*x^4 * y^3 * z - 2*-7 * x^4 * y^4 *z^2", "3*-7*x^3 * y**5 * z^2 - -8*x^4 * y^2 * z^2", "2*-8*x^4 * y * z^3 - 3*8*x^3 * y^3 * z^2", "3*-8*x^3 * y^2 * z^3 + 2*8* x^4 * y^2 * z^2 + -7* x^4 * y**5 * z" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $F(x,y,z)=(-4xz^2, 2xyz,-4xy^3z)$ be a vector field and $f(x,y,z)=x^3 y^2 z$. $\\nabla f=($ [ANS], [ANS], [ANS] $)$. $\\nabla \\times F=($ [ANS], [ANS], [ANS] $)$. $F \\times \\nabla f=($ [ANS], [ANS], [ANS] $)$. $F \\cdot \\nabla f=$ [ANS].", "answer_v3": [ "3*x*x*y*y*z", "2*x^3 * y * z", "x^3 * y^2", "3*-4*x*y*y*z - 2*x*y", "2*-4*x*z - -4*y^3 * z", "2*y*z", "2*x^4 * y^3 * z - 2*-4 * x^4 * y^4 *z^2", "3*-4*x^3 * y**5 * z^2 - -4*x^4 * y^2 * z^2", "2*-4*x^4 * y * z^3 - 3*2*x^3 * y^3 * z^2", "3*-4*x^3 * y^2 * z^3 + 2*2* x^4 * y^2 * z^2 + -4* x^4 * y**5 * z" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0566", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "curl' 'divergence" ], "problem_v1": "Consider the vector field F(x,y,z)=(5yz, 2xz, 2xy). Find the divergence and curl of $F$.\n$\\textrm{div}(F)=\\nabla \\cdot F=$ [ANS]. $\\textrm{curl}(F)=\\nabla \\times F=($ [ANS], [ANS], [ANS] $)$.", "answer_v1": [ "0", "(2 - 2)*x", "(5 - 2)*y", "(2 - 5)*z" ], "answer_type_v1": [ "NV", "EX", "EX", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Consider the vector field F(x,y,z)=(-8yz, 8xz,-7xy). Find the divergence and curl of $F$.\n$\\textrm{div}(F)=\\nabla \\cdot F=$ [ANS]. $\\textrm{curl}(F)=\\nabla \\times F=($ [ANS], [ANS], [ANS] $)$.", "answer_v2": [ "0", "(-7 - 8)*x", "(-8 - -7)*y", "(8 - -8)*z" ], "answer_type_v2": [ "NV", "EX", "EX", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Consider the vector field F(x,y,z)=(-4yz, 2xz,-4xy). Find the divergence and curl of $F$.\n$\\textrm{div}(F)=\\nabla \\cdot F=$ [ANS]. $\\textrm{curl}(F)=\\nabla \\times F=($ [ANS], [ANS], [ANS] $)$.", "answer_v3": [ "0", "(-4 - 2)*x", "(-4 - -4)*y", "(2 - -4)*z" ], "answer_type_v3": [ "NV", "EX", "EX", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0567", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "3", "keywords": [ "Divergence" ], "problem_v1": "Suppose $\\vec{F}(x,y,z)=4 \\boldsymbol{i}+4 y \\boldsymbol{j}+4 z^2 \\boldsymbol{k}.$\n(a) Find the flux of $\\vec{F}$ through the cube in the first octant with edge length $c$, one corner at the origin, and edges along the axes. The cube is oriented outward.\nFlux through the front face (the plane $x=c$) is [ANS]\nFlux through the back face (the plane $x=0$) is [ANS]\nFlux through the right face (the plane $y=c$) is [ANS]\nFlux through the left face (the plane $y=0$) is [ANS]\nFlux through the top face (the plane $z=c$) is [ANS]\nFlux through the bottom face (the plane $z=0$) is [ANS]\nTotal flux through the cube=[ANS]\n(b) Use the geometric definition of divergence and the total flux to find $\\mathrm{div}(\\vec{F})$ at the origin. $\\mathrm{div}(\\vec{F}(0,0,0))= \\lim_{c \\to 0}$ [ANS]=[ANS]", "answer_v1": [ "4*c^2", "-4*c^2", "4*c^3", "0", "4*c^4", "0", "4*c^3+4*c^4", "4+4*c", "4" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "EX", "NV", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Suppose $\\vec{F}(x,y,z)=1 \\boldsymbol{i}+5 y \\boldsymbol{j}+2 z^2 \\boldsymbol{k}.$\n(a) Find the flux of $\\vec{F}$ through the cube in the first octant with edge length $c$, one corner at the origin, and edges along the axes. The cube is oriented outward.\nFlux through the front face (the plane $x=c$) is [ANS]\nFlux through the back face (the plane $x=0$) is [ANS]\nFlux through the right face (the plane $y=c$) is [ANS]\nFlux through the left face (the plane $y=0$) is [ANS]\nFlux through the top face (the plane $z=c$) is [ANS]\nFlux through the bottom face (the plane $z=0$) is [ANS]\nTotal flux through the cube=[ANS]\n(b) Use the geometric definition of divergence and the total flux to find $\\mathrm{div}(\\vec{F})$ at the origin. $\\mathrm{div}(\\vec{F}(0,0,0))= \\lim_{c \\to 0}$ [ANS]=[ANS]", "answer_v2": [ "1*c^2", "-1*c^2", "5*c^3", "0", "2*c^4", "0", "5*c^3+2*c^4", "5+2*c", "5" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "EX", "NV", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Suppose $\\vec{F}(x,y,z)=2 \\boldsymbol{i}+4 y \\boldsymbol{j}+3 z^2 \\boldsymbol{k}.$\n(a) Find the flux of $\\vec{F}$ through the cube in the first octant with edge length $c$, one corner at the origin, and edges along the axes. The cube is oriented outward.\nFlux through the front face (the plane $x=c$) is [ANS]\nFlux through the back face (the plane $x=0$) is [ANS]\nFlux through the right face (the plane $y=c$) is [ANS]\nFlux through the left face (the plane $y=0$) is [ANS]\nFlux through the top face (the plane $z=c$) is [ANS]\nFlux through the bottom face (the plane $z=0$) is [ANS]\nTotal flux through the cube=[ANS]\n(b) Use the geometric definition of divergence and the total flux to find $\\mathrm{div}(\\vec{F})$ at the origin. $\\mathrm{div}(\\vec{F}(0,0,0))= \\lim_{c \\to 0}$ [ANS]=[ANS]", "answer_v3": [ "2*c^2", "-2*c^2", "4*c^3", "0", "3*c^4", "0", "4*c^3+3*c^4", "4+3*c", "4" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "EX", "NV", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0568", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "Divergence" ], "problem_v1": "A vector field $\\vec{F}$ has the property that the flux of $\\vec{F}$ out of a small cube of side length $0.04$ centered about the point $(1,2,-2)$ is $0.0025.$ Estimate $\\mathrm{div}(\\vec{F})$ at the point $(1,2,-2).$\n$\\mathrm{div}(\\vec{F} (1,2,-2)) \\approx$ [ANS]", "answer_v1": [ "39.0625" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A vector field $\\vec{F}$ has the property that the flux of $\\vec{F}$ out of a small cube of side length $0.01$ centered about the point $(-4,-2,5)$ is $0.0025.$ Estimate $\\mathrm{div}(\\vec{F})$ at the point $(-4,-2,5).$\n$\\mathrm{div}(\\vec{F} (-4,-2,5)) \\approx$ [ANS]", "answer_v2": [ "2500" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A vector field $\\vec{F}$ has the property that the flux of $\\vec{F}$ out of a small cube of side length $0.02$ centered about the point $(-2,1,-3)$ is $0.0025.$ Estimate $\\mathrm{div}(\\vec{F})$ at the point $(-2,1,-3).$\n$\\mathrm{div}(\\vec{F} (-2,1,-3)) \\approx$ [ANS]", "answer_v3": [ "312.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0569", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "2", "keywords": [ "Divergence" ], "problem_v1": "Find the divergence of each of the following vector fields.\n(a) $\\mathrm{div} (\\,-3x\\boldsymbol{i}-3y\\boldsymbol{j} \\,)=$ [ANS]\n(b) $\\mathrm{div} (\\, 3x\\boldsymbol{i}-3y\\boldsymbol{j} \\,)=$ [ANS]\n(c) $\\mathrm{div} (\\,-4y\\boldsymbol{i}+4x\\boldsymbol{j} \\,)=$ [ANS]\n(d) $\\mathrm{div} (\\, 5x\\boldsymbol{i}+4y\\boldsymbol{j} \\,)=$ [ANS]\n(e) $\\mathrm{div} (\\, \\left(3x^{2}+4y^{2}\\right)\\boldsymbol{i}+4xy\\boldsymbol{j} \\,)=$ [ANS]", "answer_v1": [ "-6", "0", "0", "9", "10*x" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Find the divergence of each of the following vector fields.\n(a) $\\mathrm{div} (\\, 2x\\boldsymbol{i}+2y\\boldsymbol{j} \\,)=$ [ANS]\n(b) $\\mathrm{div} (\\, 5x\\boldsymbol{i}-5y\\boldsymbol{j} \\,)=$ [ANS]\n(c) $\\mathrm{div} (\\,-4y\\boldsymbol{i}+2x\\boldsymbol{j} \\,)=$ [ANS]\n(d) $\\mathrm{div} (\\,-2x\\boldsymbol{i}-4y\\boldsymbol{j} \\,)=$ [ANS]\n(e) $\\mathrm{div} (\\, \\left(4x^{2}+4y^{2}\\right)\\boldsymbol{i}+5xy\\boldsymbol{j} \\,)=$ [ANS]", "answer_v2": [ "4", "0", "0", "-6", "13*x" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Find the divergence of each of the following vector fields.\n(a) $\\mathrm{div} (\\, \\left(5x^{2}+3y^{2}\\right)\\boldsymbol{i}+2xy\\boldsymbol{j} \\,)=$ [ANS]\n(b) $\\mathrm{div} (\\,-4y\\boldsymbol{i}+5x\\boldsymbol{j} \\,)=$ [ANS]\n(c) $\\mathrm{div} (\\,-3x\\boldsymbol{i}-3y\\boldsymbol{j} \\,)=$ [ANS]\n(d) $\\mathrm{div} (\\, 3x\\boldsymbol{i}+3y\\boldsymbol{j} \\,)=$ [ANS]\n(e) $\\mathrm{div} (\\, 2x\\boldsymbol{i}-2y\\boldsymbol{j} \\,)=$ [ANS]", "answer_v3": [ "12*x", "0", "-6", "6", "0" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0571", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "4", "keywords": [ "Curl", "Curl of a vector field" ], "problem_v1": "Three small circles $C_1$, $C_2$, and $C_3$, each with radius $0.3$ and centered at the origin are in the xy-, yz-, and xz-planes, respectively. The circles are oriented counterclockwise when viewed from the positive z-, x-, and y-axes, respectively. A vector field $\\vec{F}$ has circulation around $C_1$ of $0.03 \\pi$, around $C_2$ of $0.2 \\pi$, and around $C_3$ of $4 \\pi$. Estimate $\\mathrm{curl}(\\vec{F})$ at the origin.\n$\\mathrm{curl}(\\vec{F}(0,0,0)) \\approx$ [ANS]", "answer_v1": [ "(2.22222,44.4444,0.333333)" ], "answer_type_v1": [ "OL" ], "options_v1": [ [] ], "problem_v2": "Three small circles $C_1$, $C_2$, and $C_3$, each with radius $0.1$ and centered at the origin are in the xy-, yz-, and xz-planes, respectively. The circles are oriented counterclockwise when viewed from the positive z-, x-, and y-axes, respectively. A vector field $\\vec{F}$ has circulation around $C_1$ of $0.04 \\pi$, around $C_2$ of $0.5 \\pi$, and around $C_3$ of $3 \\pi$. Estimate $\\mathrm{curl}(\\vec{F})$ at the origin.\n$\\mathrm{curl}(\\vec{F}(0,0,0)) \\approx$ [ANS]", "answer_v2": [ "(50,300,4)" ], "answer_type_v2": [ "OL" ], "options_v2": [ [] ], "problem_v3": "Three small circles $C_1$, $C_2$, and $C_3$, each with radius $0.5$ and centered at the origin are in the xy-, yz-, and xz-planes, respectively. The circles are oriented counterclockwise when viewed from the positive z-, x-, and y-axes, respectively. A vector field $\\vec{F}$ has circulation around $C_1$ of $0.04 \\pi$, around $C_2$ of $0.5 \\pi$, and around $C_3$ of $5 \\pi$. Estimate $\\mathrm{curl}(\\vec{F})$ at the origin.\n$\\mathrm{curl}(\\vec{F}(0,0,0)) \\approx$ [ANS]", "answer_v3": [ "(2,20,0.16)" ], "answer_type_v3": [ "OL" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0572", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Curl and divergence", "level": "3", "keywords": [ "Curl", "Curl of a vector field" ], "problem_v1": "A smooth vector field $\\vec{G}$ has $\\mathrm{curl}(\\vec{G}(0,0,0))=8 \\vec{i}-6 \\vec{j}+7 \\vec{k}$. Estimate the circulation around a circle of radius $0.03$ centered at the origin in each of the following planes.\n(a) The xy-plane, oriented counterclockwise when viewed from the positive z-axis. Circulation $\\approx$ [ANS]\n(b) The yz-plane, oriented counterclockwise when viewed from the positive x-axis. Circulation $\\approx$ [ANS]\n(c) The xz-plane, oriented counterclockwise when viewed from the positive y-axis. Circulation $\\approx$ [ANS]", "answer_v1": [ "7*pi*0.03^2", "8*pi*0.03^2", "-6*pi*0.03^2" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "A smooth vector field $\\vec{G}$ has $\\mathrm{curl}(\\vec{G}(0,0,0))=2 \\vec{i}-9 \\vec{j}+3 \\vec{k}$. Estimate the circulation around a circle of radius $0.04$ centered at the origin in each of the following planes.\n(a) The xy-plane, oriented counterclockwise when viewed from the positive z-axis. Circulation $\\approx$ [ANS]\n(b) The yz-plane, oriented counterclockwise when viewed from the positive x-axis. Circulation $\\approx$ [ANS]\n(c) The xz-plane, oriented counterclockwise when viewed from the positive y-axis. Circulation $\\approx$ [ANS]", "answer_v2": [ "3*pi*0.04^2", "2*pi*0.04^2", "-9*pi*0.04^2" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "A smooth vector field $\\vec{G}$ has $\\mathrm{curl}(\\vec{G}(0,0,0))=4 \\vec{i}-6 \\vec{j}+3 \\vec{k}$. Estimate the circulation around a circle of radius $0.04$ centered at the origin in each of the following planes.\n(a) The xy-plane, oriented counterclockwise when viewed from the positive z-axis. Circulation $\\approx$ [ANS]\n(b) The yz-plane, oriented counterclockwise when viewed from the positive x-axis. Circulation $\\approx$ [ANS]\n(c) The xz-plane, oriented counterclockwise when viewed from the positive y-axis. Circulation $\\approx$ [ANS]", "answer_v3": [ "3*pi*0.04^2", "4*pi*0.04^2", "-6*pi*0.04^2" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0574", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of scalar fields", "level": "4", "keywords": [ "calculus" ], "problem_v1": "Show that $\\Phi(u,v)=(8 u+6,u-v,15 u+v)$ parametrizes the plane $2x-y-z=12$. Then:\n(a) Calculate ${\\mathbf{T}}_u$, ${\\mathbf{T}}_v$, and ${\\mathbf{n}}(u,v)$. (b) Find the area of $\\mathcal{S}=\\Phi(\\mathcal{D})$, where $\\mathcal{D}=(u,v): 0\\le u\\le 6, 0\\le v \\le 7$. (c) Express $f(x,y,z)=yz$ in terms of $u$ and $v$ and evaluate $\\iint_{\\mathcal{S}} f(x,y,z)\\,d \\mathcal{S}$.\n(a) ${\\mathbf{T}}_u=$ [ANS], ${\\mathbf{T}}_v=$ [ANS], ${\\mathbf{n}}(u,v)=$ [ANS]\n(b) $Area(S)=$ [ANS]\n(c) $\\iint_{\\mathcal{S}} f(x,y,z)\\,d \\mathcal{S}=$ [ANS]", "answer_v1": [ "(8,1,15)", "(0,-1,1)", "(16,-8,-8)", "823.029", "13717.1" ], "answer_type_v1": [ "OL", "OL", "OL", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Show that $\\Phi(u,v)=(2 u+9,u-v,3 u+v)$ parametrizes the plane $2x-y-z=18$. Then:\n(a) Calculate ${\\mathbf{T}}_u$, ${\\mathbf{T}}_v$, and ${\\mathbf{n}}(u,v)$. (b) Find the area of $\\mathcal{S}=\\Phi(\\mathcal{D})$, where $\\mathcal{D}=(u,v): 0\\le u\\le 3, 0\\le v \\le 4$. (c) Express $f(x,y,z)=yz$ in terms of $u$ and $v$ and evaluate $\\iint_{\\mathcal{S}} f(x,y,z)\\,d \\mathcal{S}$.\n(a) ${\\mathbf{T}}_u=$ [ANS], ${\\mathbf{T}}_v=$ [ANS], ${\\mathbf{n}}(u,v)=$ [ANS]\n(b) $Area(S)=$ [ANS]\n(c) $\\iint_{\\mathcal{S}} f(x,y,z)\\,d \\mathcal{S}=$ [ANS]", "answer_v2": [ "(2,1,3)", "(0,-1,1)", "(4,-2,-2)", "58.7878", "-137.171" ], "answer_type_v2": [ "OL", "OL", "OL", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Show that $\\Phi(u,v)=(4 u+6,u-v,7 u+v)$ parametrizes the plane $2x-y-z=12$. Then:\n(a) Calculate ${\\mathbf{T}}_u$, ${\\mathbf{T}}_v$, and ${\\mathbf{n}}(u,v)$. (b) Find the area of $\\mathcal{S}=\\Phi(\\mathcal{D})$, where $\\mathcal{D}=(u,v): 0\\le u\\le 4, 0\\le v \\le 6$. (c) Express $f(x,y,z)=yz$ in terms of $u$ and $v$ and evaluate $\\iint_{\\mathcal{S}} f(x,y,z)\\,d \\mathcal{S}$.\n(a) ${\\mathbf{T}}_u=$ [ANS], ${\\mathbf{T}}_v=$ [ANS], ${\\mathbf{n}}(u,v)=$ [ANS]\n(b) $Area(S)=$ [ANS]\n(c) $\\iint_{\\mathcal{S}} f(x,y,z)\\,d \\mathcal{S}=$ [ANS]", "answer_v3": [ "(4,1,7)", "(0,-1,1)", "(8,-4,-4)", "235.151", "-2508.28" ], "answer_type_v3": [ "OL", "OL", "OL", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0575", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of scalar fields", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Calculate $\\iint_{\\mathcal{S}} f(x,y,z)\\,dS$ For y=7-z^2 \\text{,} \\qquad 0\\le x, z\\le 7 \\text{;}\\qquad f(x,y,z)=z $\\iint_{\\mathcal{S}} f(x,y,z)\\,dS=$ [ANS]", "answer_v1": [ "1612.35" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate $\\iint_{\\mathcal{S}} f(x,y,z)\\,dS$ For y=1-z^2 \\text{,} \\qquad 0\\le x, z\\le 9 \\text{;}\\qquad f(x,y,z)=z $\\iint_{\\mathcal{S}} f(x,y,z)\\,dS=$ [ANS]", "answer_v2": [ "4393.52" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate $\\iint_{\\mathcal{S}} f(x,y,z)\\,dS$ For y=3-z^2 \\text{,} \\qquad 0\\le x, z\\le 8 \\text{;}\\qquad f(x,y,z)=z $\\iint_{\\mathcal{S}} f(x,y,z)\\,dS=$ [ANS]", "answer_v3": [ "2746.02" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0576", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of scalar fields", "level": "4", "keywords": [ "calculus" ], "problem_v1": "Find the surface area of the portion $S$ of the cone $z^2=x^2+y^2$, where $z\\ge 0$, contained within the cylinder $y^2+z^2\\le 64$. $\\mathrm{Area}(S)=$ [ANS]", "answer_v1": [ "201.062" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the surface area of the portion $S$ of the cone $z^2=x^2+y^2$, where $z\\ge 0$, contained within the cylinder $y^2+z^2\\le 4$. $\\mathrm{Area}(S)=$ [ANS]", "answer_v2": [ "12.5664" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the surface area of the portion $S$ of the cone $z^2=x^2+y^2$, where $z\\ge 0$, contained within the cylinder $y^2+z^2\\le 16$. $\\mathrm{Area}(S)=$ [ANS]", "answer_v3": [ "50.2655" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0577", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of scalar fields", "level": "2", "keywords": [ "vector' 'double integral' 'multivariable", "Surface Integral", "Double Integral", "Double Integral", "Multivariable", "Geometry" ], "problem_v1": "Evaluate $ \\int\\!\\!\\int_{S} \\sqrt{1+x^{2}+y^{2}} \\: dS$ where $S$ is the helicoid: $\\mathbf{r}(u, v)=u\\cos(v)\\mathbf{i}+u\\sin(v)\\mathbf{j}+v\\mathbf{k}$, with $0 \\leq u \\leq 4, 0 \\leq v \\leq 3\\pi$ [ANS]", "answer_v1": [ "238.761041672824" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate $ \\int\\!\\!\\int_{S} \\sqrt{1+x^{2}+y^{2}} \\: dS$ where $S$ is the helicoid: $\\mathbf{r}(u, v)=u\\cos(v)\\mathbf{i}+u\\sin(v)\\mathbf{j}+v\\mathbf{k}$, with $0 \\leq u \\leq 1, 0 \\leq v \\leq 5\\pi$ [ANS]", "answer_v2": [ "20.943951023932" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate $ \\int\\!\\!\\int_{S} \\sqrt{1+x^{2}+y^{2}} \\: dS$ where $S$ is the helicoid: $\\mathbf{r}(u, v)=u\\cos(v)\\mathbf{i}+u\\sin(v)\\mathbf{j}+v\\mathbf{k}$, with $0 \\leq u \\leq 2, 0 \\leq v \\leq 4\\pi$ [ANS]", "answer_v3": [ "58.6430628670095" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0578", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of scalar fields", "level": "4", "keywords": [ "Multivariable", "Double Integral", "Surface Area" ], "problem_v1": "The vector equation $\\mathbf{r} \\left(u, v \\right)=u \\cos v \\mathbf{i}+u \\sin v \\mathbf{j}+v \\mathbf{k}$, $0 \\leq v \\leq 8 \\pi$, $0 \\leq u \\leq 1$, describes a helicoid (spiral ramp). What is the surface area? [ANS]", "answer_v1": [ "28.8471988968282" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The vector equation $\\mathbf{r} \\left(u, v \\right)=u \\cos v \\mathbf{i}+u \\sin v \\mathbf{j}+v \\mathbf{k}$, $0 \\leq v \\leq 1 \\pi$, $0 \\leq u \\leq 1$, describes a helicoid (spiral ramp). What is the surface area? [ANS]", "answer_v2": [ "3.60589986210352" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The vector equation $\\mathbf{r} \\left(u, v \\right)=u \\cos v \\mathbf{i}+u \\sin v \\mathbf{j}+v \\mathbf{k}$, $0 \\leq v \\leq 4 \\pi$, $0 \\leq u \\leq 1$, describes a helicoid (spiral ramp). What is the surface area? [ANS]", "answer_v3": [ "14.4235994484141" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0579", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of scalar fields", "level": "4", "keywords": [ "vector' 'double integral' 'multivariable' 'surface integral" ], "problem_v1": "Evaluate the surface integral $\\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS$ where $f=x+y$ and $\\sigma$ is the portion of the plane $z=5-4x-4 y$ in the first octant. Then\n$\\begin{array}{c}\\hline \\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS=\\int_a^b\\int_c^d F(u,v)\\,du\\,dv \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccc}\\hline & & & &=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & du\\,dv \\\\ \\hline \\end{array}$=[ANS]", "answer_v1": [ "0", "1.25", "0", "(-1)*v+1.25", "(u+v)*5.74456", "3.73995" ], "answer_type_v1": [ "NV", "NV", "NV", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Evaluate the surface integral $\\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS$ where $f=x+y$ and $\\sigma$ is the portion of the plane $z=2-6x-1 y$ in the first octant. Then\n$\\begin{array}{c}\\hline \\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS=\\int_a^b\\int_c^d F(u,v)\\,du\\,dv \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccc}\\hline & & & &=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & du\\,dv \\\\ \\hline \\end{array}$=[ANS]", "answer_v2": [ "0", "2", "0", "(-0.166667)*v+0.333333", "(u+v)*6.16441", "1.59818" ], "answer_type_v2": [ "NV", "NV", "NV", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Evaluate the surface integral $\\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS$ where $f=x+y$ and $\\sigma$ is the portion of the plane $z=3-4x-2 y$ in the first octant. Then\n$\\begin{array}{c}\\hline \\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS=\\int_a^b\\int_c^d F(u,v)\\,du\\,dv \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccc}\\hline & & & &=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & du\\,dv \\\\ \\hline \\end{array}$=[ANS]", "answer_v3": [ "0", "1.5", "0", "(-0.5)*v+0.75", "(u+v)*4.58258", "1.93327" ], "answer_type_v3": [ "NV", "NV", "NV", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0580", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of scalar fields", "level": "3", "keywords": [ "vector' 'double integral' 'multivariable' 'surface integral" ], "problem_v1": "Suppose that surface $\\sigma$ is parameterized by $r(u,v)=\\left$, $0\\leq u\\leq 5$ and $0 \\leq v\\leq \\frac {2\\pi}{5}$ and $f(x,y,z)=x^{2}+y^{2}+z^{2}$. Set up the surface integral (you don't need to evaluate it).\n$\\begin{array}{c}\\hline \\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS=\\int\\limits_R\\!\\!\\int f\\left(x(u,v),y(u,v),z(u,v)\\right)\\left\\Vert \\frac{\\partial r}{\\partial u} \\times \\frac{\\partial r}{\\partial v} \\right\\Vert dA \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccccccc}\\hline & & & &=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\Bigg\\Vert & & [ANS] & & \\Bigg\\Vert du\\,dv \\\\ \\hline \\end{array}$", "answer_v1": [ "0", "1.25664", "0", "5", "[u*cos(5*v)]^2+[u*sin(5*v)]^2+v^2", "(sin(5*v),-[cos(5*v)],5*u)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "EX", "OL" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Suppose that surface $\\sigma$ is parameterized by $r(u,v)=\\left$, $0\\leq u\\leq 7$ and $0 \\leq v\\leq \\frac {1\\pi}{1}$ and $f(x,y,z)=x^{2}+y^{2}+z^{2}$. Set up the surface integral (you don't need to evaluate it).\n$\\begin{array}{c}\\hline \\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS=\\int\\limits_R\\!\\!\\int f\\left(x(u,v),y(u,v),z(u,v)\\right)\\left\\Vert \\frac{\\partial r}{\\partial u} \\times \\frac{\\partial r}{\\partial v} \\right\\Vert dA \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccccccc}\\hline & & & &=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\Bigg\\Vert & & [ANS] & & \\Bigg\\Vert du\\,dv \\\\ \\hline \\end{array}$", "answer_v2": [ "0", "3.14159", "0", "7", "[u*cos(2*v)]^2+[u*sin(2*v)]^2+v^2", "(sin(2*v),-[cos(2*v)],2*u)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "EX", "OL" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Suppose that surface $\\sigma$ is parameterized by $r(u,v)=\\left$, $0\\leq u\\leq 6$ and $0 \\leq v\\leq \\frac {2\\pi}{3}$ and $f(x,y,z)=x^{2}+y^{2}+z^{2}$. Set up the surface integral (you don't need to evaluate it).\n$\\begin{array}{c}\\hline \\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS=\\int\\limits_R\\!\\!\\int f\\left(x(u,v),y(u,v),z(u,v)\\right)\\left\\Vert \\frac{\\partial r}{\\partial u} \\times \\frac{\\partial r}{\\partial v} \\right\\Vert dA \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccccccc}\\hline & & & &=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\Bigg\\Vert & & [ANS] & & \\Bigg\\Vert du\\,dv \\\\ \\hline \\end{array}$", "answer_v3": [ "0", "2.0944", "0", "6", "[u*cos(3*v)]^2+[u*sin(3*v)]^2+v^2", "(sin(3*v),-[cos(3*v)],3*u)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "EX", "OL" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0581", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of scalar fields", "level": "3", "keywords": [ "vector' 'double integral' 'multivariable' 'surface integral" ], "problem_v1": "Find the surface integral $\\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS$ where $f(x,y,z)=\\left(x^{2}+y^{2}\\right)z$ and $\\sigma$ is the sphere $x^2+y^2+z^2=25$ above $z=3$. Parameterize the surface integral\n$\\begin{array}{c}\\hline \\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS=\\int\\limits_R\\!\\!\\int f\\left(x(\\theta,\\phi),y(\\theta,\\phi),z(\\theta,\\phi)\\right)\\left\\Vert \\frac{\\partial r}{\\partial \\theta} \\times \\frac{\\partial r}{\\partial \\phi} \\right\\Vert dA \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccccccc}\\hline & & & &=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\Bigg\\Vert & & [ANS] & & \\Bigg\\Vert d\\theta\\, d\\phi \\\\ \\hline \\end{array}$\nNote: For $\\theta$ type theta and for $\\phi$ type phi.", "answer_v1": [ "0", "0.927295", "0", "6.28319", "([5*cos(theta)*sin(phi)]^2+[5*sin(theta)*sin(phi)]^2)*5*cos(phi)", "(-25*cos(theta)*[sin(phi)]^2,-25*sin(theta)*[sin(phi)]^2,-25*sin(phi)*cos(phi))" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "EX", "OL" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find the surface integral $\\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS$ where $f(x,y,z)=\\left(x^{2}+y^{2}\\right)z$ and $\\sigma$ is the sphere $x^2+y^2+z^2=4$ above $z=1$. Parameterize the surface integral\n$\\begin{array}{c}\\hline \\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS=\\int\\limits_R\\!\\!\\int f\\left(x(\\theta,\\phi),y(\\theta,\\phi),z(\\theta,\\phi)\\right)\\left\\Vert \\frac{\\partial r}{\\partial \\theta} \\times \\frac{\\partial r}{\\partial \\phi} \\right\\Vert dA \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccccccc}\\hline & & & &=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\Bigg\\Vert & & [ANS] & & \\Bigg\\Vert d\\theta\\, d\\phi \\\\ \\hline \\end{array}$\nNote: For $\\theta$ type theta and for $\\phi$ type phi.", "answer_v2": [ "0", "1.0472", "0", "6.28319", "([2*cos(theta)*sin(phi)]^2+[2*sin(theta)*sin(phi)]^2)*2*cos(phi)", "(-4*cos(theta)*[sin(phi)]^2,-4*sin(theta)*[sin(phi)]^2,-4*sin(phi)*cos(phi))" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "EX", "OL" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find the surface integral $\\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS$ where $f(x,y,z)=\\left(x^{2}+y^{2}\\right)z$ and $\\sigma$ is the sphere $x^2+y^2+z^2=9$ above $z=2$. Parameterize the surface integral\n$\\begin{array}{c}\\hline \\int\\limits_\\sigma\\!\\!\\int f(x,y,z)\\,dS=\\int\\limits_R\\!\\!\\int f\\left(x(\\theta,\\phi),y(\\theta,\\phi),z(\\theta,\\phi)\\right)\\left\\Vert \\frac{\\partial r}{\\partial \\theta} \\times \\frac{\\partial r}{\\partial \\phi} \\right\\Vert dA \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccccccccc}\\hline & & & &=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\Bigg\\Vert & & [ANS] & & \\Bigg\\Vert d\\theta\\, d\\phi \\\\ \\hline \\end{array}$\nNote: For $\\theta$ type theta and for $\\phi$ type phi.", "answer_v3": [ "0", "0.841069", "0", "6.28319", "([3*cos(theta)*sin(phi)]^2+[3*sin(theta)*sin(phi)]^2)*3*cos(phi)", "(-9*cos(theta)*[sin(phi)]^2,-9*sin(theta)*[sin(phi)]^2,-9*sin(phi)*cos(phi))" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "EX", "OL" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0582", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of scalar fields", "level": "3", "keywords": [ "Multivariable", "Double Integral", "Surface Area", "Surface Integral" ], "problem_v1": "(a) Find a vector parametric equation for the ellipse that lies on the plane $z-\\left(3x+y\\right)=5$ and inside the cylinder $x^2+y^2=16$.\n$\\boldsymbol{\\vec{r}}(u,v)=$ [ANS] for $0 \\leq u \\leq 4$ and $0 \\leq v \\leq 2\\pi$.\n(b) $\\vec{dA}= \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v=$ [ANS]\n(c) $ dA=\\|\\vec{dA}\\|=\\left\\| \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v \\right\\|=$ [ANS]\n(d) Set up and evaluate a double integral for the surface area of the ellipse.\nSurface area=[ANS]", "answer_v1": [ "(u*cos(v),u*sin(v),3*u*cos(v)+u*sin(v)+5)", "(u*-3,-1,1)", "u*sqrt(3^2+1^2+1)", "pi*4^2*sqrt(3^2+1^2+1)" ], "answer_type_v1": [ "OL", "OL", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "(a) Find a vector parametric equation for the ellipse that lies on the plane $5x-5y+z=-7$ and inside the cylinder $x^2+y^2=16$.\n$\\boldsymbol{\\vec{r}}(u,v)=$ [ANS] for $0 \\leq u \\leq 4$ and $0 \\leq v \\leq 2\\pi$.\n(b) $\\vec{dA}= \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v=$ [ANS]\n(c) $ dA=\\|\\vec{dA}\\|=\\left\\| \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v \\right\\|=$ [ANS]\n(d) Set up and evaluate a double integral for the surface area of the ellipse.\nSurface area=[ANS]", "answer_v2": [ "(u*cos(v),u*sin(v),5*u*sin(v)-5*u*cos(v)-7)", "(u*-(-5),-5,1)", "u*sqrt((-5)^2+5^2+1)", "pi*4^2*sqrt((-5)^2+5^2+1)" ], "answer_type_v2": [ "OL", "OL", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "(a) Find a vector parametric equation for the ellipse that lies on the plane $2x-y+z=-5$ and inside the cylinder $x^2+y^2=36$.\n$\\boldsymbol{\\vec{r}}(u,v)=$ [ANS] for $0 \\leq u \\leq 6$ and $0 \\leq v \\leq 2\\pi$.\n(b) $\\vec{dA}= \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v=$ [ANS]\n(c) $ dA=\\|\\vec{dA}\\|=\\left\\| \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v \\right\\|=$ [ANS]\n(d) Set up and evaluate a double integral for the surface area of the ellipse.\nSurface area=[ANS]", "answer_v3": [ "(u*cos(v),u*sin(v),u*sin(v)-2*u*cos(v)-5)", "(u*-(-2),-1,1)", "u*sqrt((-2)^2+1^2+1)", "pi*6^2*sqrt((-2)^2+1^2+1)" ], "answer_type_v3": [ "OL", "OL", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0583", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of scalar fields", "level": "3", "keywords": [ "Multivariable", "Double Integral", "Surface Area", "Surface Integral" ], "problem_v1": "(a) Find a vector parametric equation for the part of the plane $z=3x+y+85$ that lies above $\\lbrack 0, 4 \\rbrack \\times \\lbrack 0, 3 \\rbrack$.\n$\\boldsymbol{\\vec{r}}(u,v)=$ [ANS] for $0 \\leq u \\leq 4$ and $0 \\leq v \\leq 3$.\n(b) $\\vec{dA}= \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial u} \\times \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial v} =$ [ANS]\n(c) $dA=\\| \\vec{dA} \\|= \\left\\| \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial u} \\times \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial v} \\right\\|=$ [ANS]\n(d) Set up and evaluate a double integral for the surface area of the part of the plane $z=3x+y+85$ that lies above the square $\\lbrack 0, 4 \\rbrack \\times \\lbrack 0, 3 \\rbrack$.\nSurface area=[ANS]\n(e) A region $R$ in the $xy$-plane has area $2.3$. What is the area of the part of the plane $z=3x+y+85$ that lies above and/or below the region $R$?\nSurface area=[ANS]\n(f) Would the method you used to answer part (e) work for a function that is not a plane? [ANS]", "answer_v1": [ "(u,v,3*u+v+85)", "(-3,-1,1)", "sqrt(3^2+1^2+1)", "4*3*sqrt(3^2+1^2+1)", "2.3*sqrt(3^2+1^2+1)", "No" ], "answer_type_v1": [ "OL", "OL", "NV", "NV", "NV", "TF" ], "options_v1": [ [], [], [], [], [], [ "Yes", "No" ] ], "problem_v2": "(a) Find a vector parametric equation for the part of the plane $z=5y-5x+66$ that lies above $\\lbrack 0, 3 \\rbrack \\times \\lbrack 0, 5 \\rbrack$.\n$\\boldsymbol{\\vec{r}}(u,v)=$ [ANS] for $0 \\leq u \\leq 3$ and $0 \\leq v \\leq 5$.\n(b) $\\vec{dA}= \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial u} \\times \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial v} =$ [ANS]\n(c) $dA=\\| \\vec{dA} \\|= \\left\\| \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial u} \\times \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial v} \\right\\|=$ [ANS]\n(d) Set up and evaluate a double integral for the surface area of the part of the plane $z=5y-5x+66$ that lies above the square $\\lbrack 0, 3 \\rbrack \\times \\lbrack 0, 5 \\rbrack$.\nSurface area=[ANS]\n(e) A region $R$ in the $xy$-plane has area $2.3$. What is the area of the part of the plane $z=5y-5x+66$ that lies above and/or below the region $R$?\nSurface area=[ANS]\n(f) Would the method you used to answer part (e) work for a function that is not a plane? [ANS]", "answer_v2": [ "(u,v,5*v-5*u+66)", "(5,-5,1)", "sqrt((-5)^2+5^2+1)", "3*5*sqrt((-5)^2+5^2+1)", "2.3*sqrt((-5)^2+5^2+1)", "No" ], "answer_type_v2": [ "OL", "OL", "NV", "NV", "NV", "TF" ], "options_v2": [ [], [], [], [], [], [ "Yes", "No" ] ], "problem_v3": "(a) Find a vector parametric equation for the part of the plane $z=y-2x+71$ that lies above $\\lbrack 0, 4 \\rbrack \\times \\lbrack 0, 2 \\rbrack$.\n$\\boldsymbol{\\vec{r}}(u,v)=$ [ANS] for $0 \\leq u \\leq 4$ and $0 \\leq v \\leq 2$.\n(b) $\\vec{dA}= \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial u} \\times \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial v} =$ [ANS]\n(c) $dA=\\| \\vec{dA} \\|= \\left\\| \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial u} \\times \\frac{\\partial \\boldsymbol{\\vec{r}}}{\\partial v} \\right\\|=$ [ANS]\n(d) Set up and evaluate a double integral for the surface area of the part of the plane $z=y-2x+71$ that lies above the square $\\lbrack 0, 4 \\rbrack \\times \\lbrack 0, 2 \\rbrack$.\nSurface area=[ANS]\n(e) A region $R$ in the $xy$-plane has area $2.3$. What is the area of the part of the plane $z=y-2x+71$ that lies above and/or below the region $R$?\nSurface area=[ANS]\n(f) Would the method you used to answer part (e) work for a function that is not a plane? [ANS]", "answer_v3": [ "(u,v,v-2*u+71)", "(2,-1,1)", "sqrt((-2)^2+1^2+1)", "4*2*sqrt((-2)^2+1^2+1)", "2.3*sqrt((-2)^2+1^2+1)", "No" ], "answer_type_v3": [ "OL", "OL", "NV", "NV", "NV", "TF" ], "options_v3": [ [], [], [], [], [], [ "Yes", "No" ] ] }, { "id": "Calculus_-_multivariable_0584", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of scalar fields", "level": "3", "keywords": [ "Multivariable", "Double Integral", "Surface Area", "Surface Integral" ], "problem_v1": "Find the surface area of the helicoid (i.e., spiral ramp) given by the vector parametric equation\n\\boldsymbol{\\vec{r}} \\left(u, v \\right)=\\langle u \\cos v, u \\sin v, v \\rangle, \\ \\ \\ 0 \\leq u \\leq 1, \\ \\ \\ 0 \\leq v \\leq 8 \\pi.\n(a) Find $\\| \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v \\|$ and simplify your answer as much as possible.\n$\\| \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v \\|=$ [ANS]\n(b) Set up and evaluate a double integral for the surface area of this helicoid. (When you answer part (a) correctly, a hint will appear here with an integral formula.)\nSurface area=[ANS]", "answer_v1": [ "sqrt(1+u^2)", "8*pi/2*[sqrt(2)+ln(1+sqrt(2))]" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Find the surface area of the helicoid (i.e., spiral ramp) given by the vector parametric equation\n\\boldsymbol{\\vec{r}} \\left(u, v \\right)=\\langle u \\cos v, u \\sin v, v \\rangle, \\ \\ \\ 0 \\leq u \\leq 1, \\ \\ \\ 0 \\leq v \\leq 2 \\pi.\n(a) Find $\\| \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v \\|$ and simplify your answer as much as possible.\n$\\| \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v \\|=$ [ANS]\n(b) Set up and evaluate a double integral for the surface area of this helicoid. (When you answer part (a) correctly, a hint will appear here with an integral formula.)\nSurface area=[ANS]", "answer_v2": [ "sqrt(1+u^2)", "2*pi/2*[sqrt(2)+ln(1+sqrt(2))]" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Find the surface area of the helicoid (i.e., spiral ramp) given by the vector parametric equation\n\\boldsymbol{\\vec{r}} \\left(u, v \\right)=\\langle u \\cos v, u \\sin v, v \\rangle, \\ \\ \\ 0 \\leq u \\leq 1, \\ \\ \\ 0 \\leq v \\leq 4 \\pi.\n(a) Find $\\| \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v \\|$ and simplify your answer as much as possible.\n$\\| \\boldsymbol{\\vec{r}}_u \\times \\boldsymbol{\\vec{r}}_v \\|=$ [ANS]\n(b) Set up and evaluate a double integral for the surface area of this helicoid. (When you answer part (a) correctly, a hint will appear here with an integral formula.)\nSurface area=[ANS]", "answer_v3": [ "sqrt(1+u^2)", "4*pi/2*[sqrt(2)+ln(1+sqrt(2))]" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0585", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Compute the surface integral over the given oriented surface: $\\mathbf{F}=\\left< 0,0,e^{y+z}\\right> \\text{,}\\quad$ boundary of the cube $0\\le x, y, z\\le 8 \\text{,} \\quad$ outward-pointing normal $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]", "answer_v1": [ "7.10412E+7" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the surface integral over the given oriented surface: $\\mathbf{F}=\\left< 0,0,e^{y+z}\\right> \\text{,}\\quad$ boundary of the cube $0\\le x, y, z\\le 2 \\text{,} \\quad$ outward-pointing normal $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]", "answer_v2": [ "81.6401" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the surface integral over the given oriented surface: $\\mathbf{F}=\\left< 0,0,e^{y+z}\\right> \\text{,}\\quad$ boundary of the cube $0\\le x, y, z\\le 4 \\text{,} \\quad$ outward-pointing normal $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]", "answer_v3": [ "11491" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0586", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Compute the surface integral over the given oriented surface: $\\mathbf{F}=y^{8} \\mathbf{i}+6 \\mathbf{j}-x \\mathbf{k},\\quad$ portion of the plane $x+y+z=1$ in the octant $x, y, z \\ge 0,\\quad$ downward-pointing normal $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]", "answer_v1": [ "-2.84444" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the surface integral over the given oriented surface: $\\mathbf{F}=y^{2} \\mathbf{i}+9 \\mathbf{j}-x \\mathbf{k},\\quad$ portion of the plane $x+y+z=1$ in the octant $x, y, z \\ge 0,\\quad$ downward-pointing normal $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]", "answer_v2": [ "-4.41667" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the surface integral over the given oriented surface: $\\mathbf{F}=y^{4} \\mathbf{i}+6 \\mathbf{j}-x \\mathbf{k},\\quad$ portion of the plane $x+y+z=1$ in the octant $x, y, z \\ge 0,\\quad$ downward-pointing normal $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]", "answer_v3": [ "-2.86667" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0587", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Let ${\\mathbf{e}}_{\\mathbf{r}}$ be the unit radial vector and $r=\\sqrt{x^2+y^2+z^2}$. Calculate the integral of $\\mathbf{F}=e^{-r} {\\mathbf{e}}_{\\mathbf{r}}$ over:\n(a) The upper-hemisphere of $x^2+y^2+z^2=64$, outward-pointing normal (b) The region on the sphere of radius $r=6$ centered at the origin that lies inside the first octant $x,y,z \\ge 0$\n(a) $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]\n(b) $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]", "answer_v1": [ "0.134898", "0.14017" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let ${\\mathbf{e}}_{\\mathbf{r}}$ be the unit radial vector and $r=\\sqrt{x^2+y^2+z^2}$. Calculate the integral of $\\mathbf{F}=e^{-r} {\\mathbf{e}}_{\\mathbf{r}}$ over:\n(a) The upper-hemisphere of $x^2+y^2+z^2=4$, outward-pointing normal (b) The region on the sphere of radius $r=9$ centered at the origin that lies inside the first octant $x,y,z \\ge 0$\n(a) $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]\n(b) $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]", "answer_v2": [ "3.40135", "0.015702" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let ${\\mathbf{e}}_{\\mathbf{r}}$ be the unit radial vector and $r=\\sqrt{x^2+y^2+z^2}$. Calculate the integral of $\\mathbf{F}=e^{-r} {\\mathbf{e}}_{\\mathbf{r}}$ over:\n(a) The upper-hemisphere of $x^2+y^2+z^2=16$, outward-pointing normal (b) The region on the sphere of radius $r=6$ centered at the origin that lies inside the first octant $x,y,z \\ge 0$\n(a) $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]\n(b) $\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]", "answer_v3": [ "1.84129", "0.14017" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0588", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "4", "keywords": [ "calculus" ], "problem_v1": "Let $\\mathbf{v}=\\left< 8x,0,6 z \\right>$ be the velocity field (in $\\mathrm{ft/s}$) of a fluid in $\\mathbf{R}^3$. Calculate the flow rate (in $\\mathrm{ft^3/s}$) through the upper hemisphere of the sphere $x^2+y^2+z^2=1$ $(z\\ge 0)$. $\\iint_S \\mathbf{v} \\cdot d\\mathbf{S}=$ [ANS] ${\\mathrm{ft}}^3/\\mathrm{s}$", "answer_v1": [ "29.3215" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $\\mathbf{v}=\\left< 2x,0,9 z \\right>$ be the velocity field (in $\\mathrm{ft/s}$) of a fluid in $\\mathbf{R}^3$. Calculate the flow rate (in $\\mathrm{ft^3/s}$) through the upper hemisphere of the sphere $x^2+y^2+z^2=1$ $(z\\ge 0)$. $\\iint_S \\mathbf{v} \\cdot d\\mathbf{S}=$ [ANS] ${\\mathrm{ft}}^3/\\mathrm{s}$", "answer_v2": [ "23.0383" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $\\mathbf{v}=\\left< 4x,0,6 z \\right>$ be the velocity field (in $\\mathrm{ft/s}$) of a fluid in $\\mathbf{R}^3$. Calculate the flow rate (in $\\mathrm{ft^3/s}$) through the upper hemisphere of the sphere $x^2+y^2+z^2=1$ $(z\\ge 0)$. $\\iint_S \\mathbf{v} \\cdot d\\mathbf{S}=$ [ANS] ${\\mathrm{ft}}^3/\\mathrm{s}$", "answer_v3": [ "20.944" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0589", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "vector' 'multivariable' 'surface' 'heat flow", "Surface Integral" ], "problem_v1": "The temperature $u$ in a star of conductivity 5 is inversely proportional to the distance from the center: $u= \\frac{5}{\\sqrt{x^{2} +y^{2}+z^{2}}}$. If the star is a sphere of radius 6, find the rate of heat flow outward across the surface of the star. [ANS]", "answer_v1": [ "314.159265358979" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "The temperature $u$ in a star of conductivity 7 is inversely proportional to the distance from the center: $u= \\frac{2}{\\sqrt{x^{2} +y^{2}+z^{2}}}$. If the star is a sphere of radius 1, find the rate of heat flow outward across the surface of the star. [ANS]", "answer_v2": [ "175.929188601028" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "The temperature $u$ in a star of conductivity 5 is inversely proportional to the distance from the center: $u= \\frac{2}{\\sqrt{x^{2} +y^{2}+z^{2}}}$. If the star is a sphere of radius 3, find the rate of heat flow outward across the surface of the star. [ANS]", "answer_v3": [ "125.663706143592" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0590", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "vector' 'integral' 'multivariable' 'flux", "Surface Integral", "Flux" ], "problem_v1": "Let S be the part of the plane $4\\!x+3\\!y+z=3$ which lies in the first octant, oriented upward. Find the flux of the vector field $\\mathbf{F}=3\\mathbf{i}+2\\mathbf{j}+2\\mathbf{k}$ across the surface S. [ANS]", "answer_v1": [ "7.5" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let S be the part of the plane $1\\!x+4\\!y+z=1$ which lies in the first octant, oriented upward. Find the flux of the vector field $\\mathbf{F}=2\\mathbf{i}+4\\mathbf{j}+2\\mathbf{k}$ across the surface S. [ANS]", "answer_v2": [ "2.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let S be the part of the plane $2\\!x+3\\!y+z=2$ which lies in the first octant, oriented upward. Find the flux of the vector field $\\mathbf{F}=3\\mathbf{i}+1\\mathbf{j}+2\\mathbf{k}$ across the surface S. [ANS]", "answer_v3": [ "3.66666666666667" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0591", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "vector' 'double integral' 'multivariable' 'flux", "Flux Integral", "Double Integral", "calculus", "Surface Integral", "Flux Integral" ], "problem_v1": "Suppose $\\bf{F}$ is a radial force field, $S_{1}$ is a sphere of radius $8$ centered at the origin, and the flux integral $ \\int\\!\\!\\int_{S_{1}} \\mathbf{F} \\cdot d\\mathbf{S}=6$. Let $S_{2}$ be a sphere of radius $48$ centered at the origin, and consider the flux integral $ \\int\\!\\!\\int_{S_{2}} \\mathbf{F} \\cdot d\\mathbf{S}$. (A) If the magnitude of $\\mathbf{F}$ is inversely proportional to the square of the distance from the origin,what is the value of $ \\int\\!\\!\\int_{S_{2}} \\mathbf{F} \\cdot d\\mathbf{S}$? [ANS]\n(B) If the magnitude of $\\bf{F}$ is inversely proportional to the cube of the distance from the origin, what is the value of $ \\int\\!\\!\\int_{S_{2}} \\mathbf{F} \\cdot d\\mathbf{S}$? [ANS]", "answer_v1": [ "6", "1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Suppose $\\bf{F}$ is a radial force field, $S_{1}$ is a sphere of radius $2$ centered at the origin, and the flux integral $ \\int\\!\\!\\int_{S_{1}} \\mathbf{F} \\cdot d\\mathbf{S}=9$. Let $S_{2}$ be a sphere of radius $6$ centered at the origin, and consider the flux integral $ \\int\\!\\!\\int_{S_{2}} \\mathbf{F} \\cdot d\\mathbf{S}$. (A) If the magnitude of $\\mathbf{F}$ is inversely proportional to the square of the distance from the origin,what is the value of $ \\int\\!\\!\\int_{S_{2}} \\mathbf{F} \\cdot d\\mathbf{S}$? [ANS]\n(B) If the magnitude of $\\bf{F}$ is inversely proportional to the cube of the distance from the origin, what is the value of $ \\int\\!\\!\\int_{S_{2}} \\mathbf{F} \\cdot d\\mathbf{S}$? [ANS]", "answer_v2": [ "9", "3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Suppose $\\bf{F}$ is a radial force field, $S_{1}$ is a sphere of radius $4$ centered at the origin, and the flux integral $ \\int\\!\\!\\int_{S_{1}} \\mathbf{F} \\cdot d\\mathbf{S}=6$. Let $S_{2}$ be a sphere of radius $16$ centered at the origin, and consider the flux integral $ \\int\\!\\!\\int_{S_{2}} \\mathbf{F} \\cdot d\\mathbf{S}$. (A) If the magnitude of $\\mathbf{F}$ is inversely proportional to the square of the distance from the origin,what is the value of $ \\int\\!\\!\\int_{S_{2}} \\mathbf{F} \\cdot d\\mathbf{S}$? [ANS]\n(B) If the magnitude of $\\bf{F}$ is inversely proportional to the cube of the distance from the origin, what is the value of $ \\int\\!\\!\\int_{S_{2}} \\mathbf{F} \\cdot d\\mathbf{S}$? [ANS]", "answer_v3": [ "6", "1.5" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0592", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "3", "keywords": [ "Vector", "Surface Integral", "Field", "Flux" ], "problem_v1": "F=$4x\\boldsymbol{i}+3y\\boldsymbol{j}$ and $\\sigma$ is the cube with opposite corners at (0,0,0) and (5,5,5), oriented outwards. Find the flux of the flow field F across $\\sigma$: [ANS]", "answer_v1": [ "875" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "F=$x\\boldsymbol{i}+5y\\boldsymbol{j}$ and $\\sigma$ is the cube with opposite corners at (0,0,0) and (2,2,2), oriented outwards. Find the flux of the flow field F across $\\sigma$: [ANS]", "answer_v2": [ "48" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "F=$2x\\boldsymbol{i}+4y\\boldsymbol{j}$ and $\\sigma$ is the cube with opposite corners at (0,0,0) and (3,3,3), oriented outwards. Find the flux of the flow field F across $\\sigma$: [ANS]", "answer_v3": [ "162" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0593", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "flux integral", "surface integral", "multivariable", "integral", "calculus" ], "problem_v1": "Compute the flux of the vector field $\\vec F=8x^2y^2z\\vec k$ through the surface $S$ which is the cone $\\sqrt{x^2+y^2}=z$, with $0\\le z\\le R$, oriented downward.\n(a) Parameterize the cone using cylindrical coordinates (write $\\theta$ as theta). $x(r,\\theta)=$ [ANS]\n$y(r,\\theta)=$ [ANS]\n$z(r,\\theta)=$ [ANS]\nwith [ANS] $\\le r\\le$ [ANS]\nand [ANS] $\\le \\theta \\le$ [ANS]\n(b) With this parameterization, what is $d\\vec A$? $d\\vec A$=[ANS]\n(c) Find the flux of $\\vec F$ through $S$. flux=[ANS]", "answer_v1": [ "r*cos(theta)", "r*sin(theta)", "r", "0", "R", "0", "2*pi", "r*cos(theta)i+r*sin(theta)j-rk*dr*dtheta", "-8*pi*R^7/28" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "EX", "NV", "NV", "EX", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Compute the flux of the vector field $\\vec F=2x^2y^2z\\vec k$ through the surface $S$ which is the cone $\\sqrt{x^2+y^2}=z$, with $0\\le z\\le R$, oriented downward.\n(a) Parameterize the cone using cylindrical coordinates (write $\\theta$ as theta). $x(r,\\theta)=$ [ANS]\n$y(r,\\theta)=$ [ANS]\n$z(r,\\theta)=$ [ANS]\nwith [ANS] $\\le r\\le$ [ANS]\nand [ANS] $\\le \\theta \\le$ [ANS]\n(b) With this parameterization, what is $d\\vec A$? $d\\vec A$=[ANS]\n(c) Find the flux of $\\vec F$ through $S$. flux=[ANS]", "answer_v2": [ "r*cos(theta)", "r*sin(theta)", "r", "0", "R", "0", "2*pi", "r*cos(theta)i+r*sin(theta)j-rk*dr*dtheta", "-2*pi*R^7/28" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "EX", "NV", "NV", "EX", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Compute the flux of the vector field $\\vec F=4x^2y^2z\\vec k$ through the surface $S$ which is the cone $\\sqrt{x^2+y^2}=z$, with $0\\le z\\le R$, oriented downward.\n(a) Parameterize the cone using cylindrical coordinates (write $\\theta$ as theta). $x(r,\\theta)=$ [ANS]\n$y(r,\\theta)=$ [ANS]\n$z(r,\\theta)=$ [ANS]\nwith [ANS] $\\le r\\le$ [ANS]\nand [ANS] $\\le \\theta \\le$ [ANS]\n(b) With this parameterization, what is $d\\vec A$? $d\\vec A$=[ANS]\n(c) Find the flux of $\\vec F$ through $S$. flux=[ANS]", "answer_v3": [ "r*cos(theta)", "r*sin(theta)", "r", "0", "R", "0", "2*pi", "r*cos(theta)i+r*sin(theta)j-rk*dr*dtheta", "-4*pi*R^7/28" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "EX", "NV", "NV", "EX", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0594", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "flux integral", "surface integral", "multivariable", "integral", "calculus" ], "problem_v1": "$\\vec F=x\\vec i$ through the surface $S$ oriented downward and parameterized for $0\\le s\\le 3, 0\\le t\\le \\pi/6$ by x=e^s, \\quad y=\\cos(3 t),\\quad z=7 s. flux=[ANS]", "answer_v1": [ "7*(e^3-1)" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "$\\vec F=x\\vec i$ through the surface $S$ oriented downward and parameterized for $0\\le s\\le 1, 0\\le t\\le \\pi/8$ by x=e^s, \\quad y=\\cos(4 t),\\quad z=2 s. flux=[ANS]", "answer_v2": [ "2*(e^1-1)" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "$\\vec F=x\\vec i$ through the surface $S$ oriented downward and parameterized for $0\\le s\\le 2, 0\\le t\\le \\pi/6$ by x=e^s, \\quad y=\\cos(3 t),\\quad z=4 s. flux=[ANS]", "answer_v3": [ "4*(e^2-1)" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0595", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "flux integral", "surface integral", "multivariable", "integral", "calculus" ], "problem_v1": "Compute the flux of $\\vec F=x\\vec i+y\\vec j+z\\vec k$ through the curved surface of the cylinder $x^2+y^2=16$ bounded below by the plane $x+y+z=2$, above by the plane $x+y+z=7$, and oriented away from the $z$-axis. flux=[ANS]", "answer_v1": [ "160*pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the flux of $\\vec F=x\\vec i+y\\vec j+z\\vec k$ through the curved surface of the cylinder $x^2+y^2=1$ bounded below by the plane $x+y+z=3$, above by the plane $x+y+z=5$, and oriented away from the $z$-axis. flux=[ANS]", "answer_v2": [ "4*pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the flux of $\\vec F=x\\vec i+y\\vec j+z\\vec k$ through the curved surface of the cylinder $x^2+y^2=4$ bounded below by the plane $x+y+z=2$, above by the plane $x+y+z=5$, and oriented away from the $z$-axis. flux=[ANS]", "answer_v3": [ "24*pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0596", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "flux integral", "surface integral", "multivariable", "integral", "calculus" ], "problem_v1": "Compute the flux of the vector field $\\vec F=z\\vec k$ through the parameterized surface $S$, which is oriented toward the $z$-axis and given, for $0\\leq s \\leq 4, \\ \\ 0\\leq t\\leq 4$, by x=4 s+3 t, \\quad y=4 s-3 t, \\quad z=s^2+t^2. flux=[ANS]", "answer_v1": [ "2*4*3*4*4*(4*4+4*4)/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the flux of the vector field $\\vec F=z\\vec k$ through the parameterized surface $S$, which is oriented toward the $z$-axis and given, for $0\\leq s \\leq 1, \\ \\ 0\\leq t\\leq 2$, by x=s+5 t, \\quad y=s-5 t, \\quad z=s^2+t^2. flux=[ANS]", "answer_v2": [ "2*1*5*1*2*(1*1+2*2)/3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the flux of the vector field $\\vec F=z\\vec k$ through the parameterized surface $S$, which is oriented toward the $z$-axis and given, for $0\\leq s \\leq 2, \\ \\ 0\\leq t\\leq 3$, by x=2 s+4 t, \\quad y=2 s-4 t, \\quad z=s^2+t^2. flux=[ANS]", "answer_v3": [ "2*2*4*2*3*(2*2+3*3)/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0597", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "flux integral", "surface integral", "multivariable", "integral", "calculus" ], "problem_v1": "A rectangular channel of width 7 and depth 6 meters lies in the $\\vec j$ direction. At a point $d_1$ meters from one side and $d_2$ meters from the other side, the velocity vector of fluid in the channel is $\\vec v=4 d_1 d_2\\vec j$ meters/sec. Find the flux through a rectangle stretching the full width and depth of the channel, and perpendicular to the flow. flux=[ANS] m^3/s", "answer_v1": [ "1372" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A rectangular channel of width 1 and depth 9 meters lies in the $\\vec j$ direction. At a point $d_1$ meters from one side and $d_2$ meters from the other side, the velocity vector of fluid in the channel is $\\vec v=d_1 d_2\\vec j$ meters/sec. Find the flux through a rectangle stretching the full width and depth of the channel, and perpendicular to the flow. flux=[ANS] m^3/s", "answer_v2": [ "1.5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A rectangular channel of width 3 and depth 6 meters lies in the $\\vec j$ direction. At a point $d_1$ meters from one side and $d_2$ meters from the other side, the velocity vector of fluid in the channel is $\\vec v=2 d_1 d_2\\vec j$ meters/sec. Find the flux through a rectangle stretching the full width and depth of the channel, and perpendicular to the flow. flux=[ANS] m^3/s", "answer_v3": [ "54" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0598", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "flux integral", "surface integral", "integral", "multivariable", "calculus" ], "problem_v1": "Let $S$ be the cube with side length 8, faces parallel to the coordinate planes, and centered at the origin.\n(a) Calculate the total flux of the constant vector field $\\vec v=\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j}+2\\,\\mathit{\\vec k}$ out of $S$ by computing the flux through each face separately. flux through the face at $x=4$: [ANS]\nflux through the face at $y=4$: [ANS]\nflux through the face at $z=4$: [ANS]\nflux through the face at $x=-4$: [ANS]\nflux through the face at $y=-4$: [ANS]\nflux through the face at $z=-4$: [ANS]\nThus the total flux is flux=[ANS]\nCalculate the flux out of $S$ for any constant vector field $\\vec v=a\\vec i+b\\vec j+c\\vec k$. flux=[ANS]\n(Be sure that you can explain why your answers to parts\n(a) and (b) make sense.) (Be sure that you can explain why your answers to parts\n(a) and (b) make sense.)", "answer_v1": [ "1*64", "1*64", "2*64", "-1*1*64", "-1*1*64", "-1*2*64", "0", "0" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let $S$ be the cube with side length 2, faces parallel to the coordinate planes, and centered at the origin.\n(a) Calculate the total flux of the constant vector field $\\vec v=5\\,\\mathit{\\vec i}-4\\,\\mathit{\\vec j}-2\\,\\mathit{\\vec k}$ out of $S$ by computing the flux through each face separately. flux through the face at $x=1$: [ANS]\nflux through the face at $y=1$: [ANS]\nflux through the face at $z=1$: [ANS]\nflux through the face at $x=-1$: [ANS]\nflux through the face at $y=-1$: [ANS]\nflux through the face at $z=-1$: [ANS]\nThus the total flux is flux=[ANS]\nCalculate the flux out of $S$ for any constant vector field $\\vec v=a\\vec i+b\\vec j+c\\vec k$. flux=[ANS]\n(Be sure that you can explain why your answers to parts\n(a) and (b) make sense.) (Be sure that you can explain why your answers to parts\n(a) and (b) make sense.)", "answer_v2": [ "5*4", "-4*4", "-2*4", "-1*5*4", "-1*-4*4", "-1*-2*4", "0", "0" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let $S$ be the cube with side length 4, faces parallel to the coordinate planes, and centered at the origin.\n(a) Calculate the total flux of the constant vector field $\\vec v=\\,\\mathit{\\vec i}-2\\,\\mathit{\\vec j}+\\,\\mathit{\\vec k}$ out of $S$ by computing the flux through each face separately. flux through the face at $x=2$: [ANS]\nflux through the face at $y=2$: [ANS]\nflux through the face at $z=2$: [ANS]\nflux through the face at $x=-2$: [ANS]\nflux through the face at $y=-2$: [ANS]\nflux through the face at $z=-2$: [ANS]\nThus the total flux is flux=[ANS]\nCalculate the flux out of $S$ for any constant vector field $\\vec v=a\\vec i+b\\vec j+c\\vec k$. flux=[ANS]\n(Be sure that you can explain why your answers to parts\n(a) and (b) make sense.) (Be sure that you can explain why your answers to parts\n(a) and (b) make sense.)", "answer_v3": [ "1*16", "-2*16", "1*16", "-1*1*16", "-1*-2*16", "-1*1*16", "0", "0" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0599", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "flux integral", "surface integral", "integral", "multivariable", "calculus" ], "problem_v1": "Calculate the flux integral $\\int_S(7\\vec i+7\\vec j+7\\vec k)\\cdot d\\vec A$ where $S$ is a disk of radius 4 in the plane $x+y+z=3$, oriented upward. $\\int_S(7\\vec i+7\\vec j+7\\vec k)\\cdot d\\vec A=$ [ANS]", "answer_v1": [ "sqrt(147)*pi*16" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the flux integral $\\int_S(2\\vec i+2\\vec j+2\\vec k)\\cdot d\\vec A$ where $S$ is a disk of radius 2 in the plane $x+y+z=5$, oriented upward. $\\int_S(2\\vec i+2\\vec j+2\\vec k)\\cdot d\\vec A=$ [ANS]", "answer_v2": [ "sqrt(12)*pi*4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the flux integral $\\int_S(4\\vec i+4\\vec j+4\\vec k)\\cdot d\\vec A$ where $S$ is a disk of radius 3 in the plane $x+y+z=4$, oriented upward. $\\int_S(4\\vec i+4\\vec j+4\\vec k)\\cdot d\\vec A=$ [ANS]", "answer_v3": [ "sqrt(48)*pi*9" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0600", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "flux integral", "surface integral", "integral", "multivariable", "calculus" ], "problem_v1": "Calculate the flux of the vector field $\\vec F=-\\,\\mathit{\\vec i}-\\,\\mathit{\\vec j}+3\\,\\mathit{\\vec k}$ through a cube of side 2 with sides parallel to the axes. flux=[ANS]\nSuppose that one face of the cube is at $y=0$ and the opposite face is at $y=2$. If the face at $y=0$ is removed, what is the flux through the resulting surface? flux=[ANS]", "answer_v1": [ "0", "2*2*-1" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Calculate the flux of the vector field $\\vec F=-5\\,\\mathit{\\vec i}+5\\,\\mathit{\\vec j}+5\\,\\mathit{\\vec k}$ through a cube of side 2 with sides parallel to the axes. flux=[ANS]\nSuppose that one face of the cube is at $x=0$ and the opposite face is at $x=2$. If the face at $x=0$ is removed, what is the flux through the resulting surface? flux=[ANS]", "answer_v2": [ "0", "2*2*-5" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Calculate the flux of the vector field $\\vec F=-2\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j}+\\,\\mathit{\\vec k}$ through a cube of side 3 with sides parallel to the axes. flux=[ANS]\nSuppose that one face of the cube is at $z=0$ and the opposite face is at $z=3$. If the face at $z=0$ is removed, what is the flux through the resulting surface? flux=[ANS]", "answer_v3": [ "0", "3*3*1" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0601", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "flux integral", "surface integral", "integral", "multivariable", "calculus" ], "problem_v1": "Find the flux of the constant vector field $\\vec v=3\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j}+\\,\\mathit{\\vec k}$ through a square plate of area 16 in the $xy$-plane oriented in the positive $z$-direction. flux=[ANS]", "answer_v1": [ "16*1" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the flux of the constant vector field $\\vec v=-5\\,\\mathit{\\vec i}+5\\,\\mathit{\\vec j}-4\\,\\mathit{\\vec k}$ through a square plate of area 9 in the $zx$-plane oriented in the positive $y$-direction. flux=[ANS]", "answer_v2": [ "9*5" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the flux of the constant vector field $\\vec v=-2\\,\\mathit{\\vec i}+\\,\\mathit{\\vec j}-2\\,\\mathit{\\vec k}$ through a square plate of area 16 in the $xy$-plane oriented in the positive $z$-direction. flux=[ANS]", "answer_v3": [ "16*-2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0602", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "flux integral", "surface integral", "integral", "multivariable", "calculus" ], "problem_v1": "The flux of the constant vector field $a\\vec i+b\\vec j+c\\vec k$ through the square of side 7 in the plane $z=6$, oriented in the positive $z$-direction, is 49. Which of the constants $a$, $b$, $c$ can be determined from the information given? Give the value(s).\n(If the value for a given constant is not determined, enter undetermined for your answer, below.) $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]", "answer_v1": [ "undetermined", "undetermined", "1" ], "answer_type_v1": [ "OE", "OE", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The flux of the constant vector field $a\\vec i+b\\vec j+c\\vec k$ through the square of side 2 in the plane $y=3$, oriented in the positive $y$-direction, is 20. Which of the constants $a$, $b$, $c$ can be determined from the information given? Give the value(s).\n(If the value for a given constant is not determined, enter undetermined for your answer, below.) $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]", "answer_v2": [ "undetermined", "5", "undetermined" ], "answer_type_v2": [ "OE", "NV", "OE" ], "options_v2": [ [], [], [] ], "problem_v3": "The flux of the constant vector field $a\\vec i+b\\vec j+c\\vec k$ through the square of side 4 in the plane $y=4$, oriented in the positive $y$-direction, is 16. Which of the constants $a$, $b$, $c$ can be determined from the information given? Give the value(s).\n(If the value for a given constant is not determined, enter undetermined for your answer, below.) $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]", "answer_v3": [ "undetermined", "1", "undetermined" ], "answer_type_v3": [ "OE", "NV", "OE" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0603", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "flux integral", "surface integral", "integral", "calculus" ], "problem_v1": "Compute the flux of the vector field $\\vec{F}=4 z\\vec k$ through $S$, the upper hemisphere of radius 6 centered at the origin, oriented outward. flux=[ANS]", "answer_v1": [ "2*pi*4*6^3/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Compute the flux of the vector field $\\vec{F}=z\\vec k$ through $S$, the upper hemisphere of radius 8 centered at the origin, oriented outward. flux=[ANS]", "answer_v2": [ "2*pi*1*8^3/3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Compute the flux of the vector field $\\vec{F}=2 z\\vec k$ through $S$, the upper hemisphere of radius 6 centered at the origin, oriented outward. flux=[ANS]", "answer_v3": [ "2*pi*2*6^3/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0604", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "integral' 'double integral' 'surface' 'flux" ], "problem_v1": "Let $M$ be the closed surface that consists of the hemisphere M_1: x^2+y^2+z^2=1,\\quad z \\ge 0, and its base M_2: x^2+y^2 \\le 1,\\quad z=0\\,. Let $\\mathbf{E}$ be the electric field defined by $\\mathbf{E}=(16x, 16 y, 16 z)$. Find the electric flux across $M$. Write the integral over the hemisphere using spherical coordinates, and use the outward pointing normal. \\iint_{M_1} \\mathbf{E} \\cdot d\\mathbf{S}=\\int_a^b\\int_c^d f(\\theta, \\phi)\\,d\\theta\\,d\\phi, where\n$a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS],\nUsing $\\verb+t+$ for $\\theta$ and $\\verb+p+$ for $\\phi$, $f(\\theta, \\phi)=$ [ANS]\n$\\iint_{M_1} \\mathbf{E} \\cdot d\\mathbf{S}=$ [ANS]\n$\\iint_{M_2} \\mathbf{E} \\cdot d\\mathbf{S}=$ [ANS], so\n$\\iint_{M} \\mathbf{E} \\cdot d\\mathbf{S}=$ [ANS].", "answer_v1": [ "0", "1.5707963267949", "0", "6.28318530717959", "16*(sin(p)^3 + sin(p)*cos(p)^2)", "100.530964914873", "0", "100.530964914873" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [] ], "problem_v2": "Let $M$ be the closed surface that consists of the hemisphere M_1: x^2+y^2+z^2=1,\\quad z \\ge 0, and its base M_2: x^2+y^2 \\le 1,\\quad z=0\\,. Let $\\mathbf{E}$ be the electric field defined by $\\mathbf{E}=(3x, 3 y, 3 z)$. Find the electric flux across $M$. Write the integral over the hemisphere using spherical coordinates, and use the outward pointing normal. \\iint_{M_1} \\mathbf{E} \\cdot d\\mathbf{S}=\\int_a^b\\int_c^d f(\\theta, \\phi)\\,d\\theta\\,d\\phi, where\n$a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS],\nUsing $\\verb+t+$ for $\\theta$ and $\\verb+p+$ for $\\phi$, $f(\\theta, \\phi)=$ [ANS]\n$\\iint_{M_1} \\mathbf{E} \\cdot d\\mathbf{S}=$ [ANS]\n$\\iint_{M_2} \\mathbf{E} \\cdot d\\mathbf{S}=$ [ANS], so\n$\\iint_{M} \\mathbf{E} \\cdot d\\mathbf{S}=$ [ANS].", "answer_v2": [ "0", "1.5707963267949", "0", "6.28318530717959", "3*(sin(p)^3 + sin(p)*cos(p)^2)", "18.8495559215388", "0", "18.8495559215388" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [] ], "problem_v3": "Let $M$ be the closed surface that consists of the hemisphere M_1: x^2+y^2+z^2=1,\\quad z \\ge 0, and its base M_2: x^2+y^2 \\le 1,\\quad z=0\\,. Let $\\mathbf{E}$ be the electric field defined by $\\mathbf{E}=(7x, 7 y, 7 z)$. Find the electric flux across $M$. Write the integral over the hemisphere using spherical coordinates, and use the outward pointing normal. \\iint_{M_1} \\mathbf{E} \\cdot d\\mathbf{S}=\\int_a^b\\int_c^d f(\\theta, \\phi)\\,d\\theta\\,d\\phi, where\n$a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS],\nUsing $\\verb+t+$ for $\\theta$ and $\\verb+p+$ for $\\phi$, $f(\\theta, \\phi)=$ [ANS]\n$\\iint_{M_1} \\mathbf{E} \\cdot d\\mathbf{S}=$ [ANS]\n$\\iint_{M_2} \\mathbf{E} \\cdot d\\mathbf{S}=$ [ANS], so\n$\\iint_{M} \\mathbf{E} \\cdot d\\mathbf{S}=$ [ANS].", "answer_v3": [ "0", "1.5707963267949", "0", "6.28318530717959", "7*(sin(p)^3 + sin(p)*cos(p)^2)", "43.9822971502571", "0", "43.9822971502571" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0605", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "3", "keywords": [ "integral' 'double integral' 'surface area" ], "problem_v1": "Evaluate $ \\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}$, where $M$ is the surface $x^2+y^2+12 z^2=1, z \\le 0$ and $\\mathbf{F}=y\\mathbf{i}-x\\mathbf{j}+zx^3y^2\\mathbf{k}$. $ \\nabla \\times \\mathbf{F}=($ [ANS], [ANS], [ANS] $)$. Write the surface $M$ as a function $z=f(x,y)$. Then $ \\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}=\\int_a^b\\int_{g(x)}^{h(x)} V(x,y) dy\\,dx$ $a=$ [ANS], $b=$ [ANS]\n$g(x)=$ [ANS], $h(x)=$ [ANS]\n$V(x,y)=$ [ANS]\nThe value of $ \\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}=$ [ANS].", "answer_v1": [ "2*z*x^3 *y", "-3*z*x^2 * y^2", "-2", "-1", "1", "-sqrt(1-x*x)", "sqrt(1-x*x)", "2*x^4 * y/12 - 3*x^2 * y^3/12 - 2", "-6.28318530717959" ], "answer_type_v1": [ "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Evaluate $ \\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}$, where $M$ is the surface $x^2+y^2+3 z^2=1, z \\le 0$ and $\\mathbf{F}=y\\mathbf{i}-x\\mathbf{j}+zx^3y^2\\mathbf{k}$. $ \\nabla \\times \\mathbf{F}=($ [ANS], [ANS], [ANS] $)$. Write the surface $M$ as a function $z=f(x,y)$. Then $ \\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}=\\int_a^b\\int_{g(x)}^{h(x)} V(x,y) dy\\,dx$ $a=$ [ANS], $b=$ [ANS]\n$g(x)=$ [ANS], $h(x)=$ [ANS]\n$V(x,y)=$ [ANS]\nThe value of $ \\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}=$ [ANS].", "answer_v2": [ "2*z*x^3 *y", "-3*z*x^2 * y^2", "-2", "-1", "1", "-sqrt(1-x*x)", "sqrt(1-x*x)", "2*x^4 * y/3 - 3*x^2 * y^3/3 - 2", "-6.28318530717959" ], "answer_type_v2": [ "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Evaluate $ \\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}$, where $M$ is the surface $x^2+y^2+6 z^2=1, z \\le 0$ and $\\mathbf{F}=y\\mathbf{i}-x\\mathbf{j}+zx^3y^2\\mathbf{k}$. $ \\nabla \\times \\mathbf{F}=($ [ANS], [ANS], [ANS] $)$. Write the surface $M$ as a function $z=f(x,y)$. Then $ \\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}=\\int_a^b\\int_{g(x)}^{h(x)} V(x,y) dy\\,dx$ $a=$ [ANS], $b=$ [ANS]\n$g(x)=$ [ANS], $h(x)=$ [ANS]\n$V(x,y)=$ [ANS]\nThe value of $ \\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}=$ [ANS].", "answer_v3": [ "2*z*x^3 *y", "-3*z*x^2 * y^2", "-2", "-1", "1", "-sqrt(1-x*x)", "sqrt(1-x*x)", "2*x^4 * y/6 - 3*x^2 * y^3/6 - 2", "-6.28318530717959" ], "answer_type_v3": [ "EX", "EX", "NV", "NV", "NV", "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0606", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "integral' 'double integral' 'surface integral' 'flux" ], "problem_v1": "Find the flux of $\\mathbf{F}(x,y,z)=(3xy^2, 3x^2y, z^3)$ out of the sphere of radius 7 centered at the origin. Hint: Use spherical coordinates and be mindful of the orientation. The flux is given by the integral: $7^5 \\int_a^b\\int_c^d f(\\theta,\\phi) d\\theta\\, d\\phi$ where: $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], and $f(\\theta,\\phi)=$ [ANS]\n(use variables \"t\" for theta and \"p\" for phi). The value of the integral is [ANS].", "answer_v1": [ "0", "3.14159265358979", "0", "6.28318530717959", "6*cos(t)^2 * sin(t)^2 * sin(p)**5 + cos(p)^4 * sin(p)", "126721.794549321" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "Find the flux of $\\mathbf{F}(x,y,z)=(3xy^2, 3x^2y, z^3)$ out of the sphere of radius 1 centered at the origin. Hint: Use spherical coordinates and be mindful of the orientation. The flux is given by the integral: $1^5 \\int_a^b\\int_c^d f(\\theta,\\phi) d\\theta\\, d\\phi$ where: $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], and $f(\\theta,\\phi)=$ [ANS]\n(use variables \"t\" for theta and \"p\" for phi). The value of the integral is [ANS].", "answer_v2": [ "0", "3.14159265358979", "0", "6.28318530717959", "6*cos(t)^2 * sin(t)^2 * sin(p)**5 + cos(p)^4 * sin(p)", "7.5398223686155" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "Find the flux of $\\mathbf{F}(x,y,z)=(3xy^2, 3x^2y, z^3)$ out of the sphere of radius 3 centered at the origin. Hint: Use spherical coordinates and be mindful of the orientation. The flux is given by the integral: $3^5 \\int_a^b\\int_c^d f(\\theta,\\phi) d\\theta\\, d\\phi$ where: $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], and $f(\\theta,\\phi)=$ [ANS]\n(use variables \"t\" for theta and \"p\" for phi). The value of the integral is [ANS].", "answer_v3": [ "0", "3.14159265358979", "0", "6.28318530717959", "6*cos(t)^2 * sin(t)^2 * sin(p)**5 + cos(p)^4 * sin(p)", "1832.17683557357" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0607", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "Flux integrals", "Surface integrals" ], "problem_v1": "(a) Set up a double integral for calculating the flux of the vector field $\\vec{F}(x,y,z)=x \\vec{i}+y \\vec{j}$ through the open-ended circular cylinder of radius $8$ and height $8$ with its base on the xy-plane and centered about the positive z-axis, oriented away from the z-axis. If necessary, enter $\\theta$ as theta. theta.\nFlux=$ \\int_{A}^{B} \\!\\! \\int_{C}^{D}$ [ANS] $dz \\, d\\theta$\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\n(b) Evaluate the integral. Flux=$ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}$=[ANS]", "answer_v1": [ "8^2", "0", "2*pi", "0", "8", "2*pi*8^2*8" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [] ], "problem_v2": "(a) Set up a double integral for calculating the flux of the vector field $\\vec{F}(x,y,z)=x \\vec{i}+y \\vec{j}$ through the open-ended circular cylinder of radius $12$ and height $3$ with its base on the xy-plane and centered about the positive z-axis, oriented away from the z-axis. If necessary, enter $\\theta$ as theta. theta.\nFlux=$ \\int_{A}^{B} \\!\\! \\int_{C}^{D}$ [ANS] $dz \\, d\\theta$\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\n(b) Evaluate the integral. Flux=$ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}$=[ANS]", "answer_v2": [ "12^2", "0", "2*pi", "0", "3", "2*pi*12^2*3" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [] ], "problem_v3": "(a) Set up a double integral for calculating the flux of the vector field $\\vec{F}(x,y,z)=x \\vec{i}+y \\vec{j}$ through the open-ended circular cylinder of radius $9$ and height $5$ with its base on the xy-plane and centered about the positive z-axis, oriented away from the z-axis. If necessary, enter $\\theta$ as theta. theta.\nFlux=$ \\int_{A}^{B} \\!\\! \\int_{C}^{D}$ [ANS] $dz \\, d\\theta$\nA=[ANS]\nB=[ANS]\nC=[ANS]\nD=[ANS]\n(b) Evaluate the integral. Flux=$ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}$=[ANS]", "answer_v3": [ "9^2", "0", "2*pi", "0", "5", "2*pi*9^2*5" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0609", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "Flux integrals", "Surface integrals" ], "problem_v1": "Calculate the flux of the vector field $\\vec{F}(x,y,z)=(e^{xy}+8 z+4) \\vec{i}+(e^{xy}+4 z+8) \\vec{j}+(8 z+e^{xy}) \\vec{k}$ through the square of side length $6$ with one vertex at the origin, one edge along the positive y-axis, one edge in the xz-plane with $x \\geq 0$ and $z \\geq 0$, oriented downward with normal $\\vec{n}=\\vec{i}-\\vec{k}$.\nFlux=$ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}$=[ANS]", "answer_v1": [ "4*6^2/[sqrt(2)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the flux of the vector field $\\vec{F}(x,y,z)=(e^{xy}+2 z+6) \\vec{i}+(e^{xy}+6 z+2) \\vec{j}+(2 z+e^{xy}) \\vec{k}$ through the square of side length $3$ with one vertex at the origin, one edge along the positive y-axis, one edge in the xz-plane with $x \\geq 0$ and $z \\geq 0$, oriented downward with normal $\\vec{n}=\\vec{i}-\\vec{k}$.\nFlux=$ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}$=[ANS]", "answer_v2": [ "6*3^2/[sqrt(2)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the flux of the vector field $\\vec{F}(x,y,z)=(e^{xy}+4 z+5) \\vec{i}+(e^{xy}+5 z+4) \\vec{j}+(4 z+e^{xy}) \\vec{k}$ through the square of side length $4$ with one vertex at the origin, one edge along the positive y-axis, one edge in the xz-plane with $x \\geq 0$ and $z \\geq 0$, oriented downward with normal $\\vec{n}=\\vec{i}-\\vec{k}$.\nFlux=$ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}$=[ANS]", "answer_v3": [ "5*4^2/[sqrt(2)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0610", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "3", "keywords": [ "Flux integrals", "Surface integrals" ], "problem_v1": "Are the following statements true or false?\n[ANS] 1. If $S$ is an open-ended circular cylinder centered about the z-axis, oriented away from the z-axis, and $\\vec{F}=\\langle x,y,0 \\rangle$, then the flux of $\\vec{F}$ through $S$ is positive. [ANS] 2. If $S$ is the unit sphere centered at the origin, oriented outward and the flux integral $ \\iint_S \\vec{F} \\cdot d\\vec{A}$ is zero, then $\\vec{F}=\\vec{0}$. [ANS] 3. If $S$ is the unit sphere centered at the origin, oriented outward and $\\vec{F}=x \\vec{i}+y \\vec{j}+z\\vec{k}=\\vec{r}$, then the flux integral $ \\iint_S \\vec{F} \\cdot d\\vec{A}$ is positive. [ANS] 4. If $S$ is an open-ended circular cylinder centered about the z-axis, oriented away from the z-axis, and $\\vec{F}=\\langle 3,-2,6 \\rangle$, then the flux of $\\vec{F}$ through $S$ is zero. [ANS] 5. If $S$ is the cube bounded by the six planes $x=\\pm 3$, $y=\\pm 3$, $z=\\pm 3$, oriented outward, and $\\vec{F}=2 \\vec{i}-\\vec{k}$, then the flux of $\\vec{F}$ through $S$ is zero. [ANS] 6. The area vector of a flat, oriented surface is parallel to the surface. [ANS] 7. If $S$ is the unit sphere centered at the origin, oriented outward and the flux integral $ \\iint_S \\vec{F} \\cdot d\\vec{A}$ is zero, then $\\vec{F}(x,y,z)$ is perpendicular to $\\vec{r}=\\langle x,y,z \\rangle$ at every point of $S$. [ANS] 8. If $S_1$ is a rectangle with area 1 and $S_2$ is a rectangle with area 2, then $ 2 \\iint_{S_1} \\vec{F} \\cdot d\\vec{A}=\\iint_{S_2} \\vec{F} \\cdot d\\vec{A}$", "answer_v1": [ "TRUE", "FALSE", "TRUE", "True", "True", "False", "False", "False" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v2": "Are the following statements true or false?\n[ANS] 1. If $S$ is the cube bounded by the six planes $x=\\pm 3$, $y=\\pm 3$, $z=\\pm 3$, oriented outward, and $\\vec{F}=2 \\vec{i}-\\vec{k}$, then the flux of $\\vec{F}$ through $S$ is zero. [ANS] 2. If $S$ is the unit sphere centered at the origin, oriented outward and the flux integral $ \\iint_S \\vec{F} \\cdot d\\vec{A}$ is zero, then $\\vec{F}=\\vec{0}$. [ANS] 3. If $S$ is the unit sphere centered at the origin, oriented outward and the flux integral $ \\iint_S \\vec{F} \\cdot d\\vec{A}$ is zero, then $\\vec{F}(x,y,z)$ is perpendicular to $\\vec{r}=\\langle x,y,z \\rangle$ at every point of $S$. [ANS] 4. If $S_1$ is a rectangle with area 1 and $S_2$ is a rectangle with area 2, then $ 2 \\iint_{S_1} \\vec{F} \\cdot d\\vec{A}=\\iint_{S_2} \\vec{F} \\cdot d\\vec{A}$ [ANS] 5. If $S$ is an open-ended circular cylinder centered about the z-axis, oriented away from the z-axis, and $\\vec{F}=\\langle x,y,0 \\rangle$, then the flux of $\\vec{F}$ through $S$ is positive. [ANS] 6. If $S$ is an open-ended circular cylinder centered about the z-axis, oriented away from the z-axis, and $\\vec{F}=\\langle 3,-2,6 \\rangle$, then the flux of $\\vec{F}$ through $S$ is zero. [ANS] 7. The area vector of a flat, oriented surface is parallel to the surface. [ANS] 8. If $S$ is the unit sphere centered at the origin, oriented outward and $\\vec{F}=x \\vec{i}+y \\vec{j}+z\\vec{k}=\\vec{r}$, then the flux integral $ \\iint_S \\vec{F} \\cdot d\\vec{A}$ is positive.", "answer_v2": [ "TRUE", "FALSE", "FALSE", "False", "True", "True", "False", "True" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ], "problem_v3": "Are the following statements true or false?\n[ANS] 1. The area vector of a flat, oriented surface is parallel to the surface. [ANS] 2. If $S$ is the unit sphere centered at the origin, oriented outward and the flux integral $ \\iint_S \\vec{F} \\cdot d\\vec{A}$ is zero, then $\\vec{F}=\\vec{0}$. [ANS] 3. If $S$ is an open-ended circular cylinder centered about the z-axis, oriented away from the z-axis, and $\\vec{F}=\\langle 3,-2,6 \\rangle$, then the flux of $\\vec{F}$ through $S$ is zero. [ANS] 4. If $S$ is the unit sphere centered at the origin, oriented outward and $\\vec{F}=x \\vec{i}+y \\vec{j}+z\\vec{k}=\\vec{r}$, then the flux integral $ \\iint_S \\vec{F} \\cdot d\\vec{A}$ is positive. [ANS] 5. If $S$ is the unit sphere centered at the origin, oriented outward and the flux integral $ \\iint_S \\vec{F} \\cdot d\\vec{A}$ is zero, then $\\vec{F}(x,y,z)$ is perpendicular to $\\vec{r}=\\langle x,y,z \\rangle$ at every point of $S$. [ANS] 6. If $S$ is an open-ended circular cylinder centered about the z-axis, oriented away from the z-axis, and $\\vec{F}=\\langle x,y,0 \\rangle$, then the flux of $\\vec{F}$ through $S$ is positive. [ANS] 7. If $S$ is the cube bounded by the six planes $x=\\pm 3$, $y=\\pm 3$, $z=\\pm 3$, oriented outward, and $\\vec{F}=2 \\vec{i}-\\vec{k}$, then the flux of $\\vec{F}$ through $S$ is zero. [ANS] 8. If $S_1$ is a rectangle with area 1 and $S_2$ is a rectangle with area 2, then $ 2 \\iint_{S_1} \\vec{F} \\cdot d\\vec{A}=\\iint_{S_2} \\vec{F} \\cdot d\\vec{A}$", "answer_v3": [ "FALSE", "FALSE", "TRUE", "True", "False", "True", "True", "False" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ], [ "True", "False" ] ] }, { "id": "Calculus_-_multivariable_0611", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "Flux integrals", "Surface integrals" ], "problem_v1": "Calculate the flux of the vector field $\\vec{F}(x,y,z)=\\cos(x^2+y^2) \\vec{k}$ through the disk $x^2+y^2 \\leq 36$ oriented upward in the plane $z=3$.\nFlux=[ANS]", "answer_v1": [ "-3.11577" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the flux of the vector field $\\vec{F}(x,y,z)=\\cos(x^2+y^2) \\vec{k}$ through the disk $x^2+y^2 \\leq 9$ oriented upward in the plane $z=5$.\nFlux=[ANS]", "answer_v2": [ "1.29471" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the flux of the vector field $\\vec{F}(x,y,z)=\\cos(x^2+y^2) \\vec{k}$ through the disk $x^2+y^2 \\leq 16$ oriented upward in the plane $z=4$.\nFlux=[ANS]", "answer_v3": [ "-0.904475" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0612", "subject": "Calculus_-_multivariable", "topic": "Vector calculus", "subtopic": "Surface integrals of vector fields", "level": "2", "keywords": [ "Flux integrals", "Surface integrals" ], "problem_v1": "Calculate the flux of the vector field $\\vec{F}(x,y,z)=(7x+8) \\vec{i}$ through a disk of radius $7$ centered at the origin in the yz-plane, oriented in the negative x-direction.\nFlux=[ANS]", "answer_v1": [ "-pi*8*7^2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the flux of the vector field $\\vec{F}(x,y,z)=(4x+2) \\vec{i}$ through a disk of radius $9$ centered at the origin in the yz-plane, oriented in the negative x-direction.\nFlux=[ANS]", "answer_v2": [ "-pi*2*9^2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the flux of the vector field $\\vec{F}(x,y,z)=(6x+4) \\vec{i}$ through a disk of radius $7$ centered at the origin in the yz-plane, oriented in the negative x-direction.\nFlux=[ANS]", "answer_v3": [ "-pi*4*7^2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0613", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "potential" ], "problem_v1": "Show \\mathbf{F}(x,y)=(6x y^3+7)\\mathbf{i}+(9x^2 y^2+2e^{2y})\\mathbf{j} is conservative by finding a potential function $f$ for $\\mathbf{F}$, and use $f$ to compute $ \\int_C \\mathbf{F}\\cdot d\\mathbf{r}$, where $C$ is the curve given by \\mathbf{r}(t)=(2\\sin^{7} t) \\mathbf{i}+\\left( \\frac{2t}{\\pi} \\sin^{10} (5t) \\right) \\mathbf{j} for $0 \\le t \\le \\pi/2$. $f(x,y)=$ [ANS]\n$ \\int_C \\mathbf{F}\\cdot d\\mathbf{r}=$ [ANS]", "answer_v1": [ "7*x+3*x^2*y^3+exp(2*y)", "32.3891" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Show \\mathbf{F}(x,y)=(8x y^3+1)\\mathbf{i}+(12x^2 y^2+2e^{2y})\\mathbf{j} is conservative by finding a potential function $f$ for $\\mathbf{F}$, and use $f$ to compute $ \\int_C \\mathbf{F}\\cdot d\\mathbf{r}$, where $C$ is the curve given by \\mathbf{r}(t)=(2\\sin^{3} t) \\mathbf{i}+\\left( \\frac{2t}{\\pi} \\sin^{6} (5t) \\right) \\mathbf{j} for $0 \\le t \\le \\pi/2$. $f(x,y)=$ [ANS]\n$ \\int_C \\mathbf{F}\\cdot d\\mathbf{r}=$ [ANS]", "answer_v2": [ "1*x+4*x^2*y^3+exp(2*y)", "24.3891" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Show \\mathbf{F}(x,y)=(6x y^3+3)\\mathbf{i}+(9x^2 y^2+2e^{2y})\\mathbf{j} is conservative by finding a potential function $f$ for $\\mathbf{F}$, and use $f$ to compute $ \\int_C \\mathbf{F}\\cdot d\\mathbf{r}$, where $C$ is the curve given by \\mathbf{r}(t)=(2\\sin^{5} t) \\mathbf{i}+\\left( \\frac{2t}{\\pi} \\sin^{8} (5t) \\right) \\mathbf{j} for $0 \\le t \\le \\pi/2$. $f(x,y)=$ [ANS]\n$ \\int_C \\mathbf{F}\\cdot d\\mathbf{r}=$ [ANS]", "answer_v3": [ "3*x+3*x^2*y^3+exp(2*y)", "24.3891" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0614", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Line integrals", "level": "2", "keywords": [ "line integral", "fundamental theorem of calculus", "gradient", "vector field" ], "problem_v1": "Let $\\vec r=x\\,\\vec i+y\\,\\vec j+z\\,\\vec k$ and $\\vec a=7\\,\\vec i+5\\,\\vec j+5\\,\\vec k$.\n(a) Find $\\nabla (\\vec r\\cdot \\vec a)$. $\\nabla (\\vec r\\cdot \\vec a)=$ [ANS]\n(b) Let $C$ be a path from the origin to the point with position vector $\\vec{r_0}=a\\,\\vec i+b\\,\\vec j+c\\,\\vec k$. Find $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r$. $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r=$ [ANS]\n(c) If $\\|\\vec r_0\\|=16$, what is the maximum possible value of $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r$? (Be sure you can explain why your answer is correct.) maximum value of $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "7i+5j+5k", "7*a+5*b+5*c", "16*sqrt(7^2+5^2+5^2)" ], "answer_type_v1": [ "EX", "EX", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $\\vec r=x\\,\\vec i+y\\,\\vec j+z\\,\\vec k$ and $\\vec a=\\,\\vec i+8\\,\\vec j+2\\,\\vec k$.\n(a) Find $\\nabla (\\vec r\\cdot \\vec a)$. $\\nabla (\\vec r\\cdot \\vec a)=$ [ANS]\n(b) Let $C$ be a path from the origin to the point with position vector $\\vec{r_0}=a\\,\\vec i+b\\,\\vec j+c\\,\\vec k$. Find $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r$. $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r=$ [ANS]\n(c) If $\\|\\vec r_0\\|=10$, what is the maximum possible value of $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r$? (Be sure you can explain why your answer is correct.) maximum value of $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "i+8j+2k", "1*a+8*b+2*c", "10*sqrt(1^2+8^2+2^2)" ], "answer_type_v2": [ "EX", "EX", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $\\vec r=x\\,\\vec i+y\\,\\vec j+z\\,\\vec k$ and $\\vec a=3\\,\\vec i+5\\,\\vec j+3\\,\\vec k$.\n(a) Find $\\nabla (\\vec r\\cdot \\vec a)$. $\\nabla (\\vec r\\cdot \\vec a)=$ [ANS]\n(b) Let $C$ be a path from the origin to the point with position vector $\\vec{r_0}=a\\,\\vec i+b\\,\\vec j+c\\,\\vec k$. Find $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r$. $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r=$ [ANS]\n(c) If $\\|\\vec r_0\\|=14$, what is the maximum possible value of $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r$? (Be sure you can explain why your answer is correct.) maximum value of $\\int_C\\nabla(\\vec r\\cdot\\vec a)\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "3i+5j+3k", "3*a+5*b+3*c", "14*sqrt(3^2+5^2+3^2)" ], "answer_type_v3": [ "EX", "EX", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0615", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Line integrals", "level": "2", "keywords": [ "line integral", "fundamental theorem of calculus", "gradient", "vector field" ], "problem_v1": "Let $\\vec F=7 y\\vec i+8x\\vec j$, $\\phi=2x^3+7x y$, and $h=y-3x^2$.\n(a) Find each of the following: $\\vec F-\\nabla \\phi=$ [ANS]\n$\\nabla h=$ [ANS]\nHow are $\\vec F-\\nabla \\phi$ and $\\nabla h$ related? $\\vec F-\\nabla \\phi=$ [ANS] $\\nabla h$\n(Note that this shows that $\\vec F-\\nabla \\phi$ is parallel to $\\nabla h$.) (b) Use $\\phi$ and the Fundamental Theorem of Calculus for Line Integrals to evaluate $\\int_C \\vec F\\cdot d\\vec r$, where $C$ is the oriented path on a contour of $h$ from $P(0,6)$ to $Q(5,81)$. $\\int_C \\vec F\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "-6*x^2i+xj", "-6*xi+j", "x", "6*5^3/3+7*5*81" ], "answer_type_v1": [ "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $\\vec F=y\\vec i+2x\\vec j$, $\\phi= \\frac{8}{3} x^3+x y$, and $h=y-4x^2$.\n(a) Find each of the following: $\\vec F-\\nabla \\phi=$ [ANS]\n$\\nabla h=$ [ANS]\nHow are $\\vec F-\\nabla \\phi$ and $\\nabla h$ related? $\\vec F-\\nabla \\phi=$ [ANS] $\\nabla h$\n(Note that this shows that $\\vec F-\\nabla \\phi$ is parallel to $\\nabla h$.) (b) Use $\\phi$ and the Fundamental Theorem of Calculus for Line Integrals to evaluate $\\int_C \\vec F\\cdot d\\vec r$, where $C$ is the oriented path on a contour of $h$ from $P(0,2)$ to $Q(3,38)$. $\\int_C \\vec F\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "-8*x^2i+xj", "-8*xi+j", "x", "8*3^3/3+1*3*38" ], "answer_type_v2": [ "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $\\vec F=3 y\\vec i+4x\\vec j$, $\\phi=2x^3+3x y$, and $h=y-3x^2$.\n(a) Find each of the following: $\\vec F-\\nabla \\phi=$ [ANS]\n$\\nabla h=$ [ANS]\nHow are $\\vec F-\\nabla \\phi$ and $\\nabla h$ related? $\\vec F-\\nabla \\phi=$ [ANS] $\\nabla h$\n(Note that this shows that $\\vec F-\\nabla \\phi$ is parallel to $\\nabla h$.) (b) Use $\\phi$ and the Fundamental Theorem of Calculus for Line Integrals to evaluate $\\int_C \\vec F\\cdot d\\vec r$, where $C$ is the oriented path on a contour of $h$ from $P(0,4)$ to $Q(4,52)$. $\\int_C \\vec F\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "-6*x^2i+xj", "-6*xi+j", "x", "6*4^3/3+3*4*52" ], "answer_type_v3": [ "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0616", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "fundamental theorem of calculus", "gradient", "vector field" ], "problem_v1": "Let ${\\nabla f}=-8xe^{-x^{2}}\\sin\\!\\left(3y\\right)\\,\\vec i+12e^{-x^{2}}\\cos\\!\\left(3y\\right)\\, \\vec j$. Find the change in $f$ between $(0,0)$ and $(1,\\pi/2)$ in two ways.\n(a) First, find the change by computing the line integral $\\int_C \\nabla f\\cdot d\\vec r$, where $C$ is a curve connecting $(0,0)$ and $(1,\\pi/2)$. The simplest curve is the line segment joining these points. Parameterize it: with $0\\le t\\le 1$, $\\vec r(t)=$ [ANS] $\\vec i+$ [ANS] $\\vec j$ So that $\\int_C\\nabla f\\cdot d\\vec r=\\int_0^1$ [ANS] $dt$ Note that this isn't a very pleasant integral to evaluate by hand (though we could easily find a numerical estimate for it). It's easier to find $\\int_C\\nabla f\\cdot d\\vec r$ as the sum $\\int_{C_1}\\nabla f\\cdot d\\vec r+\\int_{C_2}\\nabla f\\cdot d\\vec r$, where $C_1$ is the line segment from $(0,0)$ to $(1,0)$ and $C_2$ is the line segment from $(1,0)$ to $(1,\\pi/2)$. Calculate these integrals to find the change in $f$. $\\int_{C_1}\\nabla f\\cdot d\\vec r=$ [ANS]\n$\\int_{C_2}\\nabla f\\cdot d\\vec r=$ [ANS]\nSo that the change in $f=\\int_C\\nabla f\\cdot\\vec r=\\int_{C_1}\\nabla f\\cdot d\\vec r+\\int_{C_2}\\nabla f\\cdot d\\vec r=$ [ANS]\n(b) By computing values of $f$. To do this, First find $f(x,y)=$ [ANS]\nThus $f(0,0)=$ [ANS] and $f(1,\\pi/2)=$ [ANS], and the change in $f$ is [ANS].", "answer_v1": [ "t", "pi*t/2", "3*4*pi/2*e^(-t^2)*cos(3*pi*t/2)-8*t*e^(-t^2)*sin(3*pi*t/2)", "0", "4*e^{-1}*sin(3*pi/2)", "4*e^{-1}*sin(3*pi/2)", "4*e^(-x^2)*sin(3*y)", "0", "-1*4*e^{-1}", "4*e^{-1}*sin(3*pi/2)-0" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let ${\\nabla f}=-2xe^{-x^{2}}\\sin\\!\\left(5y\\right)\\,\\vec i+5e^{-x^{2}}\\cos\\!\\left(5y\\right)\\, \\vec j$. Find the change in $f$ between $(0,0)$ and $(1,\\pi/2)$ in two ways.\n(a) First, find the change by computing the line integral $\\int_C \\nabla f\\cdot d\\vec r$, where $C$ is a curve connecting $(0,0)$ and $(1,\\pi/2)$. The simplest curve is the line segment joining these points. Parameterize it: with $0\\le t\\le 1$, $\\vec r(t)=$ [ANS] $\\vec i+$ [ANS] $\\vec j$ So that $\\int_C\\nabla f\\cdot d\\vec r=\\int_0^1$ [ANS] $dt$ Note that this isn't a very pleasant integral to evaluate by hand (though we could easily find a numerical estimate for it). It's easier to find $\\int_C\\nabla f\\cdot d\\vec r$ as the sum $\\int_{C_1}\\nabla f\\cdot d\\vec r+\\int_{C_2}\\nabla f\\cdot d\\vec r$, where $C_1$ is the line segment from $(0,0)$ to $(1,0)$ and $C_2$ is the line segment from $(1,0)$ to $(1,\\pi/2)$. Calculate these integrals to find the change in $f$. $\\int_{C_1}\\nabla f\\cdot d\\vec r=$ [ANS]\n$\\int_{C_2}\\nabla f\\cdot d\\vec r=$ [ANS]\nSo that the change in $f=\\int_C\\nabla f\\cdot\\vec r=\\int_{C_1}\\nabla f\\cdot d\\vec r+\\int_{C_2}\\nabla f\\cdot d\\vec r=$ [ANS]\n(b) By computing values of $f$. To do this, First find $f(x,y)=$ [ANS]\nThus $f(0,0)=$ [ANS] and $f(1,\\pi/2)=$ [ANS], and the change in $f$ is [ANS].", "answer_v2": [ "t", "pi*t/2", "5*pi/2*e^(-t^2)*cos(5*pi*t/2)-2*t*e^(-t^2)*sin(5*pi*t/2)", "0", "1*e^{-1}*sin(5*pi/2)", "1*e^{-1}*sin(5*pi/2)", "e^(-x^2)*sin(5*y)", "0", "1*1*e^{-1}", "1*e^{-1}*sin(5*pi/2)-0" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let ${\\nabla f}=-4xe^{-x^{2}}\\sin\\!\\left(3y\\right)\\,\\vec i+6e^{-x^{2}}\\cos\\!\\left(3y\\right)\\, \\vec j$. Find the change in $f$ between $(0,0)$ and $(1,\\pi/2)$ in two ways.\n(a) First, find the change by computing the line integral $\\int_C \\nabla f\\cdot d\\vec r$, where $C$ is a curve connecting $(0,0)$ and $(1,\\pi/2)$. The simplest curve is the line segment joining these points. Parameterize it: with $0\\le t\\le 1$, $\\vec r(t)=$ [ANS] $\\vec i+$ [ANS] $\\vec j$ So that $\\int_C\\nabla f\\cdot d\\vec r=\\int_0^1$ [ANS] $dt$ Note that this isn't a very pleasant integral to evaluate by hand (though we could easily find a numerical estimate for it). It's easier to find $\\int_C\\nabla f\\cdot d\\vec r$ as the sum $\\int_{C_1}\\nabla f\\cdot d\\vec r+\\int_{C_2}\\nabla f\\cdot d\\vec r$, where $C_1$ is the line segment from $(0,0)$ to $(1,0)$ and $C_2$ is the line segment from $(1,0)$ to $(1,\\pi/2)$. Calculate these integrals to find the change in $f$. $\\int_{C_1}\\nabla f\\cdot d\\vec r=$ [ANS]\n$\\int_{C_2}\\nabla f\\cdot d\\vec r=$ [ANS]\nSo that the change in $f=\\int_C\\nabla f\\cdot\\vec r=\\int_{C_1}\\nabla f\\cdot d\\vec r+\\int_{C_2}\\nabla f\\cdot d\\vec r=$ [ANS]\n(b) By computing values of $f$. To do this, First find $f(x,y)=$ [ANS]\nThus $f(0,0)=$ [ANS] and $f(1,\\pi/2)=$ [ANS], and the change in $f$ is [ANS].", "answer_v3": [ "t", "pi*t/2", "3*2*pi/2*e^(-t^2)*cos(3*pi*t/2)-4*t*e^(-t^2)*sin(3*pi*t/2)", "0", "2*e^{-1}*sin(3*pi/2)", "2*e^{-1}*sin(3*pi/2)", "2*e^(-x^2)*sin(3*y)", "0", "-1*2*e^{-1}", "2*e^{-1}*sin(3*pi/2)-0" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "EX", "EX", "EX", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0617", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Line integrals", "level": "2", "keywords": [ "line integral", "fundamental theorem of calculus", "gradient", "vector field" ], "problem_v1": "If $\\vec F=\\nabla (4x^{2}+3y^{4})$, find $\\int_C\\vec{F}\\cdot d\\vec r$ where $C$ is the quarter of the circle $x^2+y^2=9$ in the first quadrant, oriented counterclockwise. $\\int_C\\vec{F}\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "207" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $\\vec F=\\nabla (x^{2}+5y^{3})$, find $\\int_C\\vec{F}\\cdot d\\vec r$ where $C$ is the quarter of the circle $x^2+y^2=4$ in the first quadrant, oriented counterclockwise. $\\int_C\\vec{F}\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "36" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $\\vec F=\\nabla (2x^{2}+4y^{3})$, find $\\int_C\\vec{F}\\cdot d\\vec r$ where $C$ is the quarter of the circle $x^2+y^2=9$ in the first quadrant, oriented counterclockwise. $\\int_C\\vec{F}\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "90" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0618", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Line integrals", "level": "2", "keywords": [ "line integral", "fundamental theorem of calculus", "gradient", "vector field" ], "problem_v1": "The domain of $f(x,y)$ is the $xy$-plane, and values of $f$ are given in the table below.\n$\\begin{array}{cccccc}\\hline y\\backslash x & 0 & 1 & 2 & 3 & 4 \\\\ \\hline 0 & 70 & 70 & 72 & 74 & 76 \\\\ \\hline 1 & 71 & 71 & 72 & 72 & 73 \\\\ \\hline 2 & 69 & 68 & 68 & 69 & 69 \\\\ \\hline 3 & 66 & 63 & 61 & 60 & 61 \\\\ \\hline 4 & 64 & 63 & 64 & 63 & 62 \\\\ \\hline \\end{array}$\nFind $\\int_C \\mbox{grad} f\\cdot d\\vec{r}$, where $C$ is\n(a) A line from $(2,0)$ to $(3,1)$. $\\int_C \\mbox{grad} f\\cdot d\\vec{r}=$ [ANS]\n(b) A circle of radius $1$ centered at $(2,3)$ traversed counterclockwise. $\\int_C \\mbox{grad} f\\cdot d\\vec{r}=$ [ANS]", "answer_v1": [ "72-72", "0" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "The domain of $f(x,y)$ is the $xy$-plane, and values of $f$ are given in the table below.\n$\\begin{array}{cccccc}\\hline y\\backslash x & 0 & 1 & 2 & 3 & 4 \\\\ \\hline 0 & 45 & 45 & 46 & 45 & 44 \\\\ \\hline 1 & 42 & 40 & 39 & 38 & 38 \\\\ \\hline 2 & 41 & 42 & 43 & 44 & 45 \\\\ \\hline 3 & 42 & 43 & 44 & 44 & 44 \\\\ \\hline 4 & 39 & 36 & 34 & 33 & 32 \\\\ \\hline \\end{array}$\nFind $\\int_C \\mbox{grad} f\\cdot d\\vec{r}$, where $C$ is\n(a) A line from $(0,4)$ to $(2,2)$. $\\int_C \\mbox{grad} f\\cdot d\\vec{r}=$ [ANS]\n(b) A circle of radius $1$ centered at $(3,3)$ traversed counterclockwise. $\\int_C \\mbox{grad} f\\cdot d\\vec{r}=$ [ANS]", "answer_v2": [ "43-39", "0" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "The domain of $f(x,y)$ is the $xy$-plane, and values of $f$ are given in the table below.\n$\\begin{array}{cccccc}\\hline y\\backslash x & 0 & 1 & 2 & 3 & 4 \\\\ \\hline 0 & 55 & 56 & 56 & 56 & 57 \\\\ \\hline 1 & 58 & 61 & 64 & 64 & 64 \\\\ \\hline 2 & 56 & 54 & 53 & 54 & 53 \\\\ \\hline 3 & 57 & 58 & 59 & 59 & 61 \\\\ \\hline 4 & 56 & 57 & 56 & 57 & 57 \\\\ \\hline \\end{array}$\nFind $\\int_C \\mbox{grad} f\\cdot d\\vec{r}$, where $C$ is\n(a) A line from $(0,2)$ to $(4,4)$. $\\int_C \\mbox{grad} f\\cdot d\\vec{r}=$ [ANS]\n(b) A circle of radius $1$ centered at $(1,2)$ traversed counterclockwise. $\\int_C \\mbox{grad} f\\cdot d\\vec{r}=$ [ANS]", "answer_v3": [ "57-56", "0" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0619", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Line integrals", "level": "3", "keywords": [ "line integral", "fundamental theorem of calculus", "gradient", "vector field" ], "problem_v1": "For the vector field $\\vec G=(ye^{xy}+4\\cos\\!\\left(4x+y\\right))\\,\\vec i+(xe^{xy}+\\cos\\!\\left(4x+y\\right))\\,\\vec j$, find the line integral of $\\vec G$ along the curve $C$ from the origin along the $x$-axis to the point $(4,0)$ and then counterclockwise around the circumference of the circle $x^2+y^2=16$ to the point $(4/\\sqrt{2},4/\\sqrt{2})$. $\\int_C\\vec G\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "2980.96" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "For the vector field $\\vec G=(ye^{xy}+\\cos\\!\\left(x+y\\right))\\,\\vec i+(xe^{xy}+\\cos\\!\\left(x+y\\right))\\,\\vec j$, find the line integral of $\\vec G$ along the curve $C$ from the origin along the $x$-axis to the point $(5,0)$ and then counterclockwise around the circumference of the circle $x^2+y^2=25$ to the point $(5/\\sqrt{2},5/\\sqrt{2})$. $\\int_C\\vec G\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "268337" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "For the vector field $\\vec G=(ye^{xy}+2\\cos\\!\\left(2x+y\\right))\\,\\vec i+(xe^{xy}+\\cos\\!\\left(2x+y\\right))\\,\\vec j$, find the line integral of $\\vec G$ along the curve $C$ from the origin along the $x$-axis to the point $(4,0)$ and then counterclockwise around the circumference of the circle $x^2+y^2=16$ to the point $(4/\\sqrt{2},4/\\sqrt{2})$. $\\int_C\\vec G\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "2980.77" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0620", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Line integrals", "level": "4", "keywords": [ "line integral", "fundamental theorem of calculus", "gradient", "vector field" ], "problem_v1": "The path $C$ is a line segment of length 39 in the plane starting at $(4,4)$. For $f(x,y)=5x+12 y$, consider \\int_C \\nabla f \\cdot d\\vec r.\n(a) Where should the other end of the line segment $C$ be placed to maximize the value of the integral? At $x=$ [ANS], $y=$ [ANS]\n(b) What is the maximum value of the integral? maximum value=[ANS]", "answer_v1": [ "4+3*5", "4+3*12", "39*13" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "The path $C$ is a line segment of length 5 in the plane starting at $(1,2)$. For $f(x,y)=3x+4 y$, consider \\int_C \\nabla f \\cdot d\\vec r.\n(a) Where should the other end of the line segment $C$ be placed to maximize the value of the integral? At $x=$ [ANS], $y=$ [ANS]\n(b) What is the maximum value of the integral? maximum value=[ANS]", "answer_v2": [ "1+1*3", "2+1*4", "5*5" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "The path $C$ is a line segment of length 15 in the plane starting at $(2,3)$. For $f(x,y)=4x+3 y$, consider \\int_C \\nabla f \\cdot d\\vec r.\n(a) Where should the other end of the line segment $C$ be placed to maximize the value of the integral? At $x=$ [ANS], $y=$ [ANS]\n(b) What is the maximum value of the integral? maximum value=[ANS]", "answer_v3": [ "2+3*4", "3+3*3", "15*5" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0621", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Line integrals", "level": "2", "keywords": [ "line integral", "fundamental theorem of calculus", "gradient", "vector field" ], "problem_v1": "If $\\vec F$ is a path-independent vector field, with $\\int_{(0,0)}^{(0,1)}\\vec F \\cdot d\\vec r=5.5$ and $\\int_{(0,1)}^{(1,1)}\\vec F \\cdot d\\vec r=3.2$ and $\\int_{(0,0)}^{(1,0)}\\vec F \\cdot d\\vec r=4.3$, find \\int_{(1,0)}^{(1,1)}\\vec F \\cdot d\\vec r. $\\int_{(1,0)}^{(1,1)}\\vec F \\cdot d\\vec r=$ [ANS]", "answer_v1": [ "5.5+3.2-4.3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "If $\\vec F$ is a path-independent vector field, with $\\int_{(0,0)}^{(1,0)}\\vec F \\cdot d\\vec r=4.1$ and $\\int_{(1,0)}^{(1,1)}\\vec F \\cdot d\\vec r=3.9$ and $\\int_{(0,0)}^{(0,1)}\\vec F \\cdot d\\vec r=3.3$, find \\int_{(0,1)}^{(1,1)}\\vec F \\cdot d\\vec r. $\\int_{(0,1)}^{(1,1)}\\vec F \\cdot d\\vec r=$ [ANS]", "answer_v2": [ "4.1+3.9-3.3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "If $\\vec F$ is a path-independent vector field, with $\\int_{(0,0)}^{(0,1)}\\vec F \\cdot d\\vec r=4.6$ and $\\int_{(0,1)}^{(1,1)}\\vec F \\cdot d\\vec r=3.2$ and $\\int_{(0,0)}^{(1,0)}\\vec F \\cdot d\\vec r=3.5$, find \\int_{(1,0)}^{(1,1)}\\vec F \\cdot d\\vec r. $\\int_{(1,0)}^{(1,1)}\\vec F \\cdot d\\vec r=$ [ANS]", "answer_v3": [ "4.6+3.2-3.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0622", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Line integrals", "level": "2", "keywords": [ "vector' 'line' 'integral' 'gradient", "Vector Fields", "Line Integral" ], "problem_v1": "Suppose that $\\nabla f(x,y,z)=2xyze^{x^2}\\mathbf{i}+ze^{x^2}\\mathbf{j}+ye^{x^2}\\mathbf{k}.$ If $f(0,0,0)=1$, find $f(3,3,6)$. Hint: As a first step, define a path from (0,0,0) to (3, 3, 6) and compute a line integral. [ANS]", "answer_v1": [ "145856.510696357" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose that $\\nabla f(x,y,z)=2xyze^{x^2}\\mathbf{i}+ze^{x^2}\\mathbf{j}+ye^{x^2}\\mathbf{k}.$ If $f(0,0,0)=-4$, find $f(1,1,9)$. Hint: As a first step, define a path from (0,0,0) to (1, 1, 9) and compute a line integral. [ANS]", "answer_v2": [ "20.4645364561314" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose that $\\nabla f(x,y,z)=2xyze^{x^2}\\mathbf{i}+ze^{x^2}\\mathbf{j}+ye^{x^2}\\mathbf{k}.$ If $f(0,0,0)=-2$, find $f(1,1,6)$. Hint: As a first step, define a path from (0,0,0) to (1, 1, 6) and compute a line integral. [ANS]", "answer_v3": [ "14.3096909707543" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0623", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Line integrals", "level": "2", "keywords": [ "calculus", "vector field", "conservative", "line integral", "vector' 'line integral' 'multivariable", "Vector Fields", "Line Integral", "Conservative", "Multivariable", "Vector" ], "problem_v1": "Consider the vector field $\\mathbf{F} \\left(x, y, z \\right)=\\left(3 z+4 y\\right) \\mathbf{i}+\\left(4 z+4x \\right) \\mathbf{j}+\\left(4 y+3x \\right) \\mathbf{k}$. a) Find a function $f$ such that $\\mathbf{F}=\\nabla f$ and $f(0,0,0)=0$. $f(x,y,z)=$ [ANS]\nb) Suppose C is any curve from $\\left(0, 0, 0 \\right)$ to $\\left(1, 1, 1 \\right).$ Use part a) to compute the line integral $\\int_{C} \\mathbf{F} \\cdot d\\mathbf{r}$. [ANS]", "answer_v1": [ " 3*z*x +4*y*x +4*z*y", "11" ], "answer_type_v1": [ "EX", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Consider the vector field $\\mathbf{F} \\left(x, y, z \\right)=\\left(5 z+y\\right) \\mathbf{i}+\\left(z+x \\right) \\mathbf{j}+\\left(y+5x \\right) \\mathbf{k}$. a) Find a function $f$ such that $\\mathbf{F}=\\nabla f$ and $f(0,0,0)=0$. $f(x,y,z)=$ [ANS]\nb) Suppose C is any curve from $\\left(0, 0, 0 \\right)$ to $\\left(1, 1, 1 \\right).$ Use part a) to compute the line integral $\\int_{C} \\mathbf{F} \\cdot d\\mathbf{r}$. [ANS]", "answer_v2": [ " 5*z*x +1*y*x +1*z*y", "7" ], "answer_type_v2": [ "EX", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Consider the vector field $\\mathbf{F} \\left(x, y, z \\right)=\\left(4 z+2 y\\right) \\mathbf{i}+\\left(2 z+2x \\right) \\mathbf{j}+\\left(2 y+4x \\right) \\mathbf{k}$. a) Find a function $f$ such that $\\mathbf{F}=\\nabla f$ and $f(0,0,0)=0$. $f(x,y,z)=$ [ANS]\nb) Suppose C is any curve from $\\left(0, 0, 0 \\right)$ to $\\left(1, 1, 1 \\right).$ Use part a) to compute the line integral $\\int_{C} \\mathbf{F} \\cdot d\\mathbf{r}$. [ANS]", "answer_v3": [ " 4*z*x +2*y*x +2*z*y", "8" ], "answer_type_v3": [ "EX", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0624", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Use the equation A=\\frac{1}{2} \\int_{\\mathcal{C}} x \\,dy-y \\,dx to calculate the area of the circle of radius $8$ centered at the origin. $A=$ [ANS]", "answer_v1": [ "201.062" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the equation A=\\frac{1}{2} \\int_{\\mathcal{C}} x \\,dy-y \\,dx to calculate the area of the circle of radius $2$ centered at the origin. $A=$ [ANS]", "answer_v2": [ "12.5664" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the equation A=\\frac{1}{2} \\int_{\\mathcal{C}} x \\,dy-y \\,dx to calculate the area of the circle of radius $4$ centered at the origin. $A=$ [ANS]", "answer_v3": [ "50.2655" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0625", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "2", "keywords": [ "vector' 'multivariable' 'green's", "Vector", "Line Integral", "Greens Theorem", "Vector Fields", "Green's Theorem", "integral' 'Greens" ], "problem_v1": "Let C be the positively oriented circle $x^{2}+y^{2}=1$. Use Green's Theorem to evaluate the line integral $\\int_{C} 16 y \\, dx+12x \\, dy$. [ANS]", "answer_v1": [ "-12.5663706143592" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let C be the positively oriented circle $x^{2}+y^{2}=1$. Use Green's Theorem to evaluate the line integral $\\int_{C} 2 y \\, dx+19x \\, dy$. [ANS]", "answer_v2": [ "53.4070751110265" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let C be the positively oriented circle $x^{2}+y^{2}=1$. Use Green's Theorem to evaluate the line integral $\\int_{C} 7 y \\, dx+13x \\, dy$. [ANS]", "answer_v3": [ "18.8495559215388" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0626", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "2", "keywords": [ "vector' 'multivariable' 'green's", "Vector Fields", "Green's Theorem", "Vector", "Greens Theorem", "Line Integral", "Circulation Integral", "Field" ], "problem_v1": "Let $\\mathbf{F}=-4 y \\mathbf{i}+3x \\mathbf{j}$. Use the tangential vector form of Green's Theorem to compute the circulation integral $\\int_{C} \\mathbf{F} \\cdot d\\mathbf{r}$ where C is the positively oriented circle $x^{2}+y^{2}=16$. [ANS]", "answer_v1": [ "351.858377202057" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $\\mathbf{F}=-1 y \\mathbf{i}+5x \\mathbf{j}$. Use the tangential vector form of Green's Theorem to compute the circulation integral $\\int_{C} \\mathbf{F} \\cdot d\\mathbf{r}$ where C is the positively oriented circle $x^{2}+y^{2}=1$. [ANS]", "answer_v2": [ "18.8495559215388" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $\\mathbf{F}=-2 y \\mathbf{i}+4x \\mathbf{j}$. Use the tangential vector form of Green's Theorem to compute the circulation integral $\\int_{C} \\mathbf{F} \\cdot d\\mathbf{r}$ where C is the positively oriented circle $x^{2}+y^{2}=4$. [ANS]", "answer_v3": [ "75.398223686155" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0628", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "3", "keywords": [ "vector", "multivariable", "Green's Theorem", "work" ], "problem_v1": "Find the work done by the vector field $\\left<5x+yx,x^{2}+5\\right>$ on a particle moving along the boundary of the rectangle $0 \\leq x \\leq 5, 0 \\leq y \\leq 4$ in the counterclockwise direction. (The force is measured in newtons, length in meters, work in joules=(newton-meters).) W=[ANS] joules", "answer_v1": [ "50" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the work done by the vector field $\\left<2x+yx,x^{2}+3\\right>$ on a particle moving along the boundary of the rectangle $0 \\leq x \\leq 2, 0 \\leq y \\leq 6$ in the counterclockwise direction. (The force is measured in newtons, length in meters, work in joules=(newton-meters).) W=[ANS] joules", "answer_v2": [ "12" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the work done by the vector field $\\left<3x+yx,x^{2}+4\\right>$ on a particle moving along the boundary of the rectangle $0 \\leq x \\leq 3, 0 \\leq y \\leq 5$ in the counterclockwise direction. (The force is measured in newtons, length in meters, work in joules=(newton-meters).) W=[ANS] joules", "answer_v3": [ "22.5" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0629", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "2", "keywords": [ "greens theorem", "line integral", "gradient", "vector field", "integral", "calculus" ], "problem_v1": "Calculate $\\int_C \\left(7 (x^2-y)\\vec i+6(y^2+x)\\vec j\\right) \\cdot d\\vec r$ if:\n(a) $C$ is the circle $(x-5)^2+(y-6)^2=4$ oriented counterclockwise. $\\int_C \\left(7 (x^2-y)\\vec i+6(y^2+x)\\vec j\\right) \\cdot d\\vec r=$ [ANS]\n(b) $C$ is the circle $(x-a)^2+(y-b)^2=R^2$ in the $xy$-plane oriented counterclockwise. $\\int_C \\left(7 (x^2-y)\\vec i+6(y^2+x)\\vec j\\right) \\cdot d\\vec r=$ [ANS]", "answer_v1": [ "(6+7)*pi*4", "(6+7)*pi*R^2" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "Calculate $\\int_C \\left(2 (x^2-y)\\vec i+8(y^2+x)\\vec j\\right) \\cdot d\\vec r$ if:\n(a) $C$ is the circle $(x-2)^2+(y-3)^2=25$ oriented counterclockwise. $\\int_C \\left(2 (x^2-y)\\vec i+8(y^2+x)\\vec j\\right) \\cdot d\\vec r=$ [ANS]\n(b) $C$ is the circle $(x-a)^2+(y-b)^2=R^2$ in the $xy$-plane oriented counterclockwise. $\\int_C \\left(2 (x^2-y)\\vec i+8(y^2+x)\\vec j\\right) \\cdot d\\vec r=$ [ANS]", "answer_v2": [ "(8+2)*pi*25", "(8+2)*pi*R^2" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "Calculate $\\int_C \\left(4 (x^2-y)\\vec i+6(y^2+x)\\vec j\\right) \\cdot d\\vec r$ if:\n(a) $C$ is the circle $(x-3)^2+(y-5)^2=4$ oriented counterclockwise. $\\int_C \\left(4 (x^2-y)\\vec i+6(y^2+x)\\vec j\\right) \\cdot d\\vec r=$ [ANS]\n(b) $C$ is the circle $(x-a)^2+(y-b)^2=R^2$ in the $xy$-plane oriented counterclockwise. $\\int_C \\left(4 (x^2-y)\\vec i+6(y^2+x)\\vec j\\right) \\cdot d\\vec r=$ [ANS]", "answer_v3": [ "(6+4)*pi*4", "(6+4)*pi*R^2" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0630", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "2", "keywords": [ "greens theorem", "line integral", "gradient", "vector field", "integral", "calculus" ], "problem_v1": "Let $\\vec F=12x e^{y}\\,\\vec i+6x^2 e^{y}\\,\\vec j$ and $\\vec G=12(x-y)\\,\\vec i+6(x+y)\\,\\vec j$. Let $C$ be the path consisting of lines from $(0,0)$ to $(7,0)$ to $(7,3)$ to $(0,0)$. Find each of the following integrals exactly:\n(a) $\\int_{C}\\vec F\\cdot d\\vec r=$ [ANS]\n(b) $\\int_{C}\\vec G\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "0", "18*7*3/2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $\\vec F=8x e^{y}\\,\\vec i+4x^2 e^{y}\\,\\vec j$ and $\\vec G=8(x-y)\\,\\vec i+4(x+y)\\,\\vec j$. Let $C$ be the path consisting of lines from $(0,0)$ to $(3,0)$ to $(3,4)$ to $(0,0)$. Find each of the following integrals exactly:\n(a) $\\int_{C}\\vec F\\cdot d\\vec r=$ [ANS]\n(b) $\\int_{C}\\vec G\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "0", "12*3*4/2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $\\vec F=10x e^{y}\\,\\vec i+5x^2 e^{y}\\,\\vec j$ and $\\vec G=10(x-y)\\,\\vec i+5(x+y)\\,\\vec j$. Let $C$ be the path consisting of lines from $(0,0)$ to $(4,0)$ to $(4,3)$ to $(0,0)$. Find each of the following integrals exactly:\n(a) $\\int_{C}\\vec F\\cdot d\\vec r=$ [ANS]\n(b) $\\int_{C}\\vec G\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "0", "15*4*3/2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0631", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "2", "keywords": [ "greens theorem", "line integral", "gradient", "vector field", "integral", "calculus" ], "problem_v1": "Use Green's Theorem to calculate the circulation of $\\vec F=4xy\\,\\vec i$ around the rectangle $0\\le x\\le 7$, $0\\le y\\le 5$, oriented counterclockwise. circulation=[ANS]", "answer_v1": [ "-4*7^2*5/2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Green's Theorem to calculate the circulation of $\\vec F=xy\\,\\vec i$ around the rectangle $0\\le x\\le 1$, $0\\le y\\le 8$, oriented counterclockwise. circulation=[ANS]", "answer_v2": [ "-1*1^2*8/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Green's Theorem to calculate the circulation of $\\vec F=2xy\\,\\vec i$ around the rectangle $0\\le x\\le 3$, $0\\le y\\le 5$, oriented counterclockwise. circulation=[ANS]", "answer_v3": [ "-2*3^2*5/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0632", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "2", "keywords": [ "greens theorem", "line integral", "gradient", "vector field", "integral", "calculus" ], "problem_v1": "(a) Show that each of the vector fields $\\vec F=4y\\,\\vec i+4x\\,\\vec j$, $\\vec G= \\frac{3 y}{x^2+y^2} \\,\\vec i+ \\frac{-3x}{x^2+y^2} \\,\\vec j$, and $\\vec H= \\frac{4x}{\\sqrt{x^2+y^2} }\\,\\vec i+ \\frac{4 y}{\\sqrt{x^2+y^2} }\\,\\vec j$ are gradient vector fields on some domain (not necessarily the whole plane) by finding a potential function for each. For $\\vec F$, a potential function is $f(x,y)=$ [ANS]\nFor $\\vec G$, a potential function is $g(x,y)=$ [ANS]\nFor $\\vec H$, a potential function is $h(x,y)=$ [ANS]\n(b) Find the line integrals of $\\vec F,\\vec G,\\vec H$ around the curve $C$ given to be the unit circle in the $xy$-plane, centered at the origin, and traversed counterclockwise. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n$\\int_C\\vec G\\cdot d\\vec r=$ [ANS]\n$\\int_C\\vec H\\cdot d\\vec r=$ [ANS]\n(c) For which of the three vector fields can Green's Theorem be used to calculate the line integral in part (b)? It may be used for [ANS] (Be sure that you are able to explain why or why not.) (Be sure that you are able to explain why or why not.)", "answer_v1": [ "4*x*y", "3*atan(x/y)", "4*sqrt(x^2+y^2)", "0", "-2*3*pi", "0", "only F" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "NV", "NV", "MCS" ], "options_v1": [ [], [], [], [], [], [], [ "only F", "only G", "only H", "F and G", "F and H", "G and H", "F", "G and H", "G and H" ] ], "problem_v2": "(a) Show that each of the vector fields $\\vec F=y\\,\\vec i+x\\,\\vec j$, $\\vec G= \\frac{5 y}{x^2+y^2} \\,\\vec i+ \\frac{-5x}{x^2+y^2} \\,\\vec j$, and $\\vec H= \\frac{x}{\\sqrt{x^2+y^2} }\\,\\vec i+ \\frac{y}{\\sqrt{x^2+y^2} }\\,\\vec j$ are gradient vector fields on some domain (not necessarily the whole plane) by finding a potential function for each. For $\\vec F$, a potential function is $f(x,y)=$ [ANS]\nFor $\\vec G$, a potential function is $g(x,y)=$ [ANS]\nFor $\\vec H$, a potential function is $h(x,y)=$ [ANS]\n(b) Find the line integrals of $\\vec F,\\vec G,\\vec H$ around the curve $C$ given to be the unit circle in the $xy$-plane, centered at the origin, and traversed counterclockwise. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n$\\int_C\\vec G\\cdot d\\vec r=$ [ANS]\n$\\int_C\\vec H\\cdot d\\vec r=$ [ANS]\n(c) For which of the three vector fields can Green's Theorem be used to calculate the line integral in part (b)? It may be used for [ANS] (Be sure that you are able to explain why or why not.) (Be sure that you are able to explain why or why not.)", "answer_v2": [ "x*y", "5*atan(x/y)", "sqrt(x^2+y^2)", "0", "-2*5*pi", "0", "only F" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "NV", "NV", "MCS" ], "options_v2": [ [], [], [], [], [], [], [ "only F", "only G", "only H", "F and G", "F and H", "G and H", "F", "G and H", "G and H" ] ], "problem_v3": "(a) Show that each of the vector fields $\\vec F=2y\\,\\vec i+2x\\,\\vec j$, $\\vec G= \\frac{4 y}{x^2+y^2} \\,\\vec i+ \\frac{-4x}{x^2+y^2} \\,\\vec j$, and $\\vec H= \\frac{2x}{\\sqrt{x^2+y^2} }\\,\\vec i+ \\frac{2 y}{\\sqrt{x^2+y^2} }\\,\\vec j$ are gradient vector fields on some domain (not necessarily the whole plane) by finding a potential function for each. For $\\vec F$, a potential function is $f(x,y)=$ [ANS]\nFor $\\vec G$, a potential function is $g(x,y)=$ [ANS]\nFor $\\vec H$, a potential function is $h(x,y)=$ [ANS]\n(b) Find the line integrals of $\\vec F,\\vec G,\\vec H$ around the curve $C$ given to be the unit circle in the $xy$-plane, centered at the origin, and traversed counterclockwise. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n$\\int_C\\vec G\\cdot d\\vec r=$ [ANS]\n$\\int_C\\vec H\\cdot d\\vec r=$ [ANS]\n(c) For which of the three vector fields can Green's Theorem be used to calculate the line integral in part (b)? It may be used for [ANS] (Be sure that you are able to explain why or why not.) (Be sure that you are able to explain why or why not.)", "answer_v3": [ "2*x*y", "4*atan(x/y)", "2*sqrt(x^2+y^2)", "0", "-2*4*pi", "0", "only F" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "NV", "NV", "MCS" ], "options_v3": [ [], [], [], [], [], [], [ "only F", "only G", "only H", "F and G", "F and H", "G and H", "F", "G and H", "G and H" ] ] }, { "id": "Calculus_-_multivariable_0633", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "2", "keywords": [ "greens theorem", "line integral", "gradient", "vector field", "integral", "calculus" ], "problem_v1": "Let $\\vec F=(6x^2 y+2 y^3+6 e^x)\\,\\vec i+(7 e^{y^2}+216x)\\,\\vec j$. Consider the line integral of $\\vec F$ around the circle of radius $a$, centered at the origin and traversed counterclockwise.\n(a) Find the line integral for $a=1$. line integral=[ANS]\n(b) For which value of $a$ is the line integral a maximum? $a=$ [ANS]\n(Be sure you can explain why your answer gives the correct maximum.) (Be sure you can explain why your answer gives the correct maximum.)", "answer_v1": [ "2*pi*(216/2-6/4)", "6" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Let $\\vec F=(9x^2 y+3 y^3+3 e^x)\\,\\vec i+(4 e^{y^2}+81x)\\,\\vec j$. Consider the line integral of $\\vec F$ around the circle of radius $a$, centered at the origin and traversed counterclockwise.\n(a) Find the line integral for $a=1$. line integral=[ANS]\n(b) For which value of $a$ is the line integral a maximum? $a=$ [ANS]\n(Be sure you can explain why your answer gives the correct maximum.) (Be sure you can explain why your answer gives the correct maximum.)", "answer_v2": [ "2*pi*(81/2-9/4)", "3" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Let $\\vec F=(6x^2 y+2 y^3+3 e^x)\\,\\vec i+(5 e^{y^2}+96x)\\,\\vec j$. Consider the line integral of $\\vec F$ around the circle of radius $a$, centered at the origin and traversed counterclockwise.\n(a) Find the line integral for $a=1$. line integral=[ANS]\n(b) For which value of $a$ is the line integral a maximum? $a=$ [ANS]\n(Be sure you can explain why your answer gives the correct maximum.) (Be sure you can explain why your answer gives the correct maximum.)", "answer_v3": [ "2*pi*(96/2-6/4)", "4" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0634", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "2", "keywords": [ "greens theorem", "line integral", "gradient", "vector field", "integral", "calculus" ], "problem_v1": "Let $\\vec F=7\\!\\left(x+y\\right)\\,\\vec i+5\\sin\\!\\left(y\\right)\\,\\vec j$. Find the line integral of $\\vec F$ around the perimeter of the rectangle with corners $(3, 0)$, $(3, 4)$, $(-1, 4)$, $(-1, 0)$, traversed in that order. line integral=[ANS]", "answer_v1": [ "-7*16" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $\\vec F=2\\!\\left(x+y\\right)\\,\\vec i+8\\sin\\!\\left(y\\right)\\,\\vec j$. Find the line integral of $\\vec F$ around the perimeter of the rectangle with corners $(5, 0)$, $(5, 4)$, $(-2, 4)$, $(-2, 0)$, traversed in that order. line integral=[ANS]", "answer_v2": [ "-2*28" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $\\vec F=4\\!\\left(x+y\\right)\\,\\vec i+5\\sin\\!\\left(y\\right)\\,\\vec j$. Find the line integral of $\\vec F$ around the perimeter of the rectangle with corners $(2, 0)$, $(2, 4)$, $(-2, 4)$, $(-2, 0)$, traversed in that order. line integral=[ANS]", "answer_v3": [ "-4*16" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0635", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "2", "keywords": [ "integral' 'Greens' 'area" ], "problem_v1": "Use Green's theorem to compute the area of one petal of the 24-leafed rose defined by $r=16 \\sin(12 \\theta)$. It may be useful for recall that the area of a region D enclosed by a curve C can be expressed as $A= \\frac{1}{2} \\int_C x\\,dy-y\\,dx$. [ANS]", "answer_v1": [ "16.7551608191456" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Green's theorem to compute the area of one petal of the 40-leafed rose defined by $r=3 \\sin(20 \\theta)$. It may be useful for recall that the area of a region D enclosed by a curve C can be expressed as $A= \\frac{1}{2} \\int_C x\\,dy-y\\,dx$. [ANS]", "answer_v2": [ "0.353429173528852" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Green's theorem to compute the area of one petal of the 28-leafed rose defined by $r=7 \\sin(14 \\theta)$. It may be useful for recall that the area of a region D enclosed by a curve C can be expressed as $A= \\frac{1}{2} \\int_C x\\,dy-y\\,dx$. [ANS]", "answer_v3": [ "2.74889357189107" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0636", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "2", "keywords": [ "integral' 'Greens' 'double integral' 'ellipse" ], "problem_v1": "Use Green's theorem to compute the area inside the ellipse $ \\frac{x^2}{16^2} + \\frac{y^2}{13^2} =1$. That is use the fact that the area can be written as\n\\iint_D dx\\,dy=\\iint_D ( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} ) dx \\,dy=\\int_{\\partial D}P dx+Qdy for appropriately chosen $P$ and $Q$. The area is [ANS].", "answer_v1": [ "653.451271946677" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Green's theorem to compute the area inside the ellipse $ \\frac{x^2}{3^2} + \\frac{y^2}{19^2} =1$. That is use the fact that the area can be written as\n\\iint_D dx\\,dy=\\iint_D ( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} ) dx \\,dy=\\int_{\\partial D}P dx+Qdy for appropriately chosen $P$ and $Q$. The area is [ANS].", "answer_v2": [ "179.070781254618" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Green's theorem to compute the area inside the ellipse $ \\frac{x^2}{7^2} + \\frac{y^2}{13^2} =1$. That is use the fact that the area can be written as\n\\iint_D dx\\,dy=\\iint_D ( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} ) dx \\,dy=\\int_{\\partial D}P dx+Qdy for appropriately chosen $P$ and $Q$. The area is [ANS].", "answer_v3": [ "285.884931476671" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0637", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "3", "keywords": [ "Line integrals", "Path integrals" ], "problem_v1": "Suppose $\\vec{F}(x,y)=\\langle 2y,-\\sin(y) \\rangle$. Find a the circulation of $\\vec{F}$ around a circle of radius $6$ centered at the origin and oriented counterclockwise.\nCirculation=[ANS]", "answer_v1": [ "-72*pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose $\\vec{F}(x,y)=\\langle 2y,-\\sin(y) \\rangle$. Find a the circulation of $\\vec{F}$ around a circle of radius $3$ centered at the origin and oriented counterclockwise.\nCirculation=[ANS]", "answer_v2": [ "-18*pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose $\\vec{F}(x,y)=\\langle 2y,-\\sin(y) \\rangle$. Find a the circulation of $\\vec{F}$ around a circle of radius $4$ centered at the origin and oriented counterclockwise.\nCirculation=[ANS]", "answer_v3": [ "-32*pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0638", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Green's theorem", "level": "3", "keywords": [], "problem_v1": "Let $F(x,y)=(8x, 6 y)$. Find the area of the closed, bounded region $R\\subseteq \\mathbb{R}^2$ if the flux of $F$ through the boundary $\\partial R$ of $R$ is $\\int_{\\partial R}F\\cdot N\\; ds=6$. area $(R)=$ [ANS]", "answer_v1": [ "0.428571" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $F(x,y)=(2x, 9 y)$. Find the area of the closed, bounded region $R\\subseteq \\mathbb{R}^2$ if the flux of $F$ through the boundary $\\partial R$ of $R$ is $\\int_{\\partial R}F\\cdot N\\; ds=3$. area $(R)=$ [ANS]", "answer_v2": [ "0.272727" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $F(x,y)=(4x, 6 y)$. Find the area of the closed, bounded region $R\\subseteq \\mathbb{R}^2$ if the flux of $F$ through the boundary $\\partial R$ of $R$ is $\\int_{\\partial R}F\\cdot N\\; ds=4$. area $(R)=$ [ANS]", "answer_v3": [ "0.4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0639", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "2", "keywords": [ "calculus" ], "problem_v1": "Let $I$ be the flux of $\\mathbf{G}=\\left<8 e^y,6x^{5}e^{x^{6}},0\\right>$ through the upper hemisphere $\\mathcal{S}$ of the unit sphere.\n(a) Find a vector field $\\mathbf{A}$ such that $\\text{curl}(\\mathbf{A})=\\mathbf{G}$. (b) Calculate the circulation of $\\mathbf{A}$ around $\\partial \\mathcal{S}$. (c) Compute $I$, the flux of $\\mathbf{G}$ through $\\mathcal{S}$.\n(a) $\\mathbf{A}=$ [ANS]\n(b) $\\int_{\\mathcal{C}} \\mathbf{A}\\cdot d\\mathbf{s}=$ [ANS]\n(c) $I=$ [ANS]", "answer_v1": [ "(0,0,8*e^y-e^(x^6))", "0", "0" ], "answer_type_v1": [ "OL", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $I$ be the flux of $\\mathbf{G}=\\left<2 e^y,9x^{8}e^{x^{9}},0\\right>$ through the upper hemisphere $\\mathcal{S}$ of the unit sphere.\n(a) Find a vector field $\\mathbf{A}$ such that $\\text{curl}(\\mathbf{A})=\\mathbf{G}$. (b) Calculate the circulation of $\\mathbf{A}$ around $\\partial \\mathcal{S}$. (c) Compute $I$, the flux of $\\mathbf{G}$ through $\\mathcal{S}$.\n(a) $\\mathbf{A}=$ [ANS]\n(b) $\\int_{\\mathcal{C}} \\mathbf{A}\\cdot d\\mathbf{s}=$ [ANS]\n(c) $I=$ [ANS]", "answer_v2": [ "(0,0,2*e^y-e^(x^9))", "0", "0" ], "answer_type_v2": [ "OL", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $I$ be the flux of $\\mathbf{G}=\\left<4 e^y,6x^{5}e^{x^{6}},0\\right>$ through the upper hemisphere $\\mathcal{S}$ of the unit sphere.\n(a) Find a vector field $\\mathbf{A}$ such that $\\text{curl}(\\mathbf{A})=\\mathbf{G}$. (b) Calculate the circulation of $\\mathbf{A}$ around $\\partial \\mathcal{S}$. (c) Compute $I$, the flux of $\\mathbf{G}$ through $\\mathcal{S}$.\n(a) $\\mathbf{A}=$ [ANS]\n(b) $\\int_{\\mathcal{C}} \\mathbf{A}\\cdot d\\mathbf{s}=$ [ANS]\n(c) $I=$ [ANS]", "answer_v3": [ "(0,0,4*e^y-e^(x^6))", "0", "0" ], "answer_type_v3": [ "OL", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0640", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "2", "keywords": [ "vector' 'double integral' 'multivariable' 'stokes", "Stokes Theorem", "Stokes", "Double Integral", "Multivariable", "Curl", "calculus" ], "problem_v1": "Use Stokes' theorem to evaluate $ \\iint_{S} (\\nabla\\times \\mathbf{F}) \\cdot d\\mathbf{S}$ where $\\mathbf{F}(x, y, z)=-15 yz\\mathbf{i}+15xz\\mathbf{j}+12 (x^{2}+y^{2})z\\mathbf{k}$ and S is the part of the paraboloid $z=x^{2}+y^{2}$ that lies inside the cylinder $x^{2}+y^{2}=1$, oriented upward. [ANS]", "answer_v1": [ "94.2477796076938" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use Stokes' theorem to evaluate $ \\iint_{S} (\\nabla\\times \\mathbf{F}) \\cdot d\\mathbf{S}$ where $\\mathbf{F}(x, y, z)=-2 yz\\mathbf{i}+2xz\\mathbf{j}+18 (x^{2}+y^{2})z\\mathbf{k}$ and S is the part of the paraboloid $z=x^{2}+y^{2}$ that lies inside the cylinder $x^{2}+y^{2}=1$, oriented upward. [ANS]", "answer_v2": [ "12.5663706143592" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use Stokes' theorem to evaluate $ \\iint_{S} (\\nabla\\times \\mathbf{F}) \\cdot d\\mathbf{S}$ where $\\mathbf{F}(x, y, z)=-6 yz\\mathbf{i}+6xz\\mathbf{j}+12 (x^{2}+y^{2})z\\mathbf{k}$ and S is the part of the paraboloid $z=x^{2}+y^{2}$ that lies inside the cylinder $x^{2}+y^{2}=1$, oriented upward. [ANS]", "answer_v3": [ "37.6991118430775" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0641", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "4", "keywords": [ "Vector", "Surface Integral", "Field", "Stokes" ], "problem_v1": "Let $\\sigma$ be the surface $8x+4 y+4 z=7$ in the first octant, oriented upwards. Let C be the oriented boundary of $\\sigma$. Compute the work done in moving a unit mass particle around the boundary of $\\sigma$ through the vector field $F=\\left(8x-6y\\right)\\boldsymbol{i}+\\left(6y-7z\\right)\\boldsymbol{j}+\\left(7z-8x\\right)\\boldsymbol{k}$ using line integrals, and using Stokes' Theorem. Assume mass is measured in kg, length in meters, and force in Newtons (1 nt=1kg-m). LINE INTEGRALS Parameterize the boundary of $\\sigma$ positively using the standard form, tv+P with $0\\leq t \\leq 1$, starting with the segment in the xy plane. $C_1$ (the edge in the xy plane) is parameterized by [ANS]\n$C_2$ (the edge following $C_1$) is parameterized by [ANS]\n$C_3$ (the last edge) is parameterized by [ANS]\n$\\int_{C_1}F\\cdot dr=$ [ANS]\n$\\int_{C_2}F\\cdot dr=$ [ANS]\n$\\int_{C_3}F\\cdot dr=$ [ANS]\n$\\int_{C}F\\cdot dr=$ [ANS]\nSTOKES' THEOREM $\\sigma$ may be parameterized by $r(x,y)=(x,y,f(x,y))=$ [ANS]\n$\\mbox{curl} F=$ [ANS]\n$ \\frac{\\partial r}{\\partial x} \\times \\frac{\\partial r}{\\partial y} =$ [ANS]\n$\\begin{array}{ccccccccc}\\hline \\iint\\limits_\\sigma (\\mbox{curl} F)\\cdot {\\bf n}\\hspace{1pt} dS=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\hspace{3pt} dy\\,dx \\\\ \\hline \\end{array}$\n$\\phantom{\\iint\\limits_\\sigma (\\mbox{curl} F)\\cdot {\\bf n}\\hspace{1pt} dS}=$ [ANS]", "answer_v1": [ "(0.875-7*t/8)i+7*t/4j", "(1.75-1.75*t)j+7*t/4k", "7*t/8i+(1.75-7*t/4)k", "10.7188", "12.25", "-1.53125", "21.4375", "xi+yj+(7-8*x-4*y)/4k", "7i+8j+6k", "2i+j+k", "0", "0.875", "0", "(-2)*x+1.75", "28", "21.4375" ], "answer_type_v1": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Let $\\sigma$ be the surface $4x+10 y+4 z=3$ in the first octant, oriented upwards. Let C be the oriented boundary of $\\sigma$. Compute the work done in moving a unit mass particle around the boundary of $\\sigma$ through the vector field $F=\\left(x-10y\\right)\\boldsymbol{i}+\\left(10y-2z\\right)\\boldsymbol{j}+\\left(2z-x\\right)\\boldsymbol{k}$ using line integrals, and using Stokes' Theorem. Assume mass is measured in kg, length in meters, and force in Newtons (1 nt=1kg-m). LINE INTEGRALS Parameterize the boundary of $\\sigma$ positively using the standard form, tv+P with $0\\leq t \\leq 1$, starting with the segment in the xy plane. $C_1$ (the edge in the xy plane) is parameterized by [ANS]\n$C_2$ (the edge following $C_1$) is parameterized by [ANS]\n$C_3$ (the last edge) is parameterized by [ANS]\n$\\int_{C_1}F\\cdot dr=$ [ANS]\n$\\int_{C_2}F\\cdot dr=$ [ANS]\n$\\int_{C_3}F\\cdot dr=$ [ANS]\n$\\int_{C}F\\cdot dr=$ [ANS]\nSTOKES' THEOREM $\\sigma$ may be parameterized by $r(x,y)=(x,y,f(x,y))=$ [ANS]\n$\\mbox{curl} F=$ [ANS]\n$ \\frac{\\partial r}{\\partial x} \\times \\frac{\\partial r}{\\partial y} =$ [ANS]\n$\\begin{array}{ccccccccc}\\hline \\iint\\limits_\\sigma (\\mbox{curl} F)\\cdot {\\bf n}\\hspace{1pt} dS=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\hspace{3pt} dy\\,dx \\\\ \\hline \\end{array}$\n$\\phantom{\\iint\\limits_\\sigma (\\mbox{curl} F)\\cdot {\\bf n}\\hspace{1pt} dS}=$ [ANS]", "answer_v2": [ "(0.75-3*t/4)i+3*t/10j", "(0.3-0.3*t)j+3*t/4k", "3*t/4i+(0.75-3*t/4)k", "1.29375", "0.3375", "0", "1.63125", "xi+yj+(3-4*x-10*y)/4k", "2i+j+10k", "i+2.5j+k", "0", "0.75", "0", "(-0.4)*x+0.3", "14.5", "1.63125" ], "answer_type_v2": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Let $\\sigma$ be the surface $6x+3 y+4 z=9$ in the first octant, oriented upwards. Let C be the oriented boundary of $\\sigma$. Compute the work done in moving a unit mass particle around the boundary of $\\sigma$ through the vector field $F=\\left(4x-7y\\right)\\boldsymbol{i}+\\left(7y-3z\\right)\\boldsymbol{j}+\\left(3z-4x\\right)\\boldsymbol{k}$ using line integrals, and using Stokes' Theorem. Assume mass is measured in kg, length in meters, and force in Newtons (1 nt=1kg-m). LINE INTEGRALS Parameterize the boundary of $\\sigma$ positively using the standard form, tv+P with $0\\leq t \\leq 1$, starting with the segment in the xy plane. $C_1$ (the edge in the xy plane) is parameterized by [ANS]\n$C_2$ (the edge following $C_1$) is parameterized by [ANS]\n$C_3$ (the last edge) is parameterized by [ANS]\n$\\int_{C_1}F\\cdot dr=$ [ANS]\n$\\int_{C_2}F\\cdot dr=$ [ANS]\n$\\int_{C_3}F\\cdot dr=$ [ANS]\n$\\int_{C}F\\cdot dr=$ [ANS]\nSTOKES' THEOREM $\\sigma$ may be parameterized by $r(x,y)=(x,y,f(x,y))=$ [ANS]\n$\\mbox{curl} F=$ [ANS]\n$ \\frac{\\partial r}{\\partial x} \\times \\frac{\\partial r}{\\partial y} =$ [ANS]\n$\\begin{array}{ccccccccc}\\hline \\iint\\limits_\\sigma (\\mbox{curl} F)\\cdot {\\bf n}\\hspace{1pt} dS=& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & \\hspace{3pt} dy\\,dx \\\\ \\hline \\end{array}$\n$\\phantom{\\iint\\limits_\\sigma (\\mbox{curl} F)\\cdot {\\bf n}\\hspace{1pt} dS}=$ [ANS]", "answer_v3": [ "(1.5-9*t/6)i+9*t/3j", "(3-3*t)j+9*t/4k", "9*t/6i+(2.25-9*t/4)k", "42.75", "-13.7812", "3.65625", "32.625", "xi+yj+(9-6*x-3*y)/4k", "3i+4j+7k", "1.5i+0.75j+k", "0", "1.5", "0", "(-2)*x+3", "14.5", "32.625" ], "answer_type_v3": [ "EX", "EX", "EX", "NV", "NV", "NV", "NV", "EX", "EX", "EX", "NV", "NV", "NV", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0642", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "2", "keywords": [ "stokes theorem", "circulation integral", "curl", "multivariable", "calculus" ], "problem_v1": "Let $\\vec F=\\left(6z+6x^{4}\\right)\\,\\mathit{\\vec i}+\\left(5y+5z+5\\sin\\!\\left(y^{4}\\right)\\right)\\,\\mathit{\\vec j}+\\left(6x+5y+5e^{z^{4}}\\right)\\,\\mathit{\\vec k}$.\n(a) Find $\\mbox{curl}\\vec F$. $\\mbox{curl}\\vec F=$ [ANS]\n(b) What does your answer to part (a) tell you about $\\int_C\\vec F\\cdot d\\vec r$ where $C$ is the circle $(x-15)^2+(y-15)^2=1$ in the $xy$-plane, oriented clockwise? $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n(c) If $C$ is any closed curve, what can you say about $\\int_C \\vec F\\cdot d\\vec r$? $\\int_C \\vec F\\cdot d\\vec r=$ [ANS]\n(d) Now let $C$ be the half circle $(x-15)^2+(y-15)^2=1$ in the $xy$-plane with $y > 15$, traversed from $(16, 15)$ to $(14, 15)$. Find $\\int_C \\vec F\\cdot d\\vec r$ by using your result from (c) and considering $C$ plus the line segment connecting the endpoints of $C$. $\\int_C \\vec F\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "0", "0", "0", "6*[14^(4+1)-16^(4+1)]/(4+1)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Let $\\vec F=\\left(2z+2x^{3}\\right)\\,\\mathit{\\vec i}+\\left(7y+2z+2\\sin\\!\\left(y^{3}\\right)\\right)\\,\\mathit{\\vec j}+\\left(2x+2y+7e^{z^{3}}\\right)\\,\\mathit{\\vec k}$.\n(a) Find $\\mbox{curl}\\vec F$. $\\mbox{curl}\\vec F=$ [ANS]\n(b) What does your answer to part (a) tell you about $\\int_C\\vec F\\cdot d\\vec r$ where $C$ is the circle $(x-35)^2+(y-15)^2=1$ in the $xy$-plane, oriented clockwise? $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n(c) If $C$ is any closed curve, what can you say about $\\int_C \\vec F\\cdot d\\vec r$? $\\int_C \\vec F\\cdot d\\vec r=$ [ANS]\n(d) Now let $C$ be the half circle $(x-35)^2+(y-15)^2=1$ in the $xy$-plane with $y > 15$, traversed from $(36, 15)$ to $(34, 15)$. Find $\\int_C \\vec F\\cdot d\\vec r$ by using your result from (c) and considering $C$ plus the line segment connecting the endpoints of $C$. $\\int_C \\vec F\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "0", "0", "0", "2*[34^(3+1)-36^(3+1)]/(3+1)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Let $\\vec F=\\left(3z+3x^{3}\\right)\\,\\mathit{\\vec i}+\\left(5y+3z+3\\sin\\!\\left(y^{3}\\right)\\right)\\,\\mathit{\\vec j}+\\left(3x+3y+5e^{z^{3}}\\right)\\,\\mathit{\\vec k}$.\n(a) Find $\\mbox{curl}\\vec F$. $\\mbox{curl}\\vec F=$ [ANS]\n(b) What does your answer to part (a) tell you about $\\int_C\\vec F\\cdot d\\vec r$ where $C$ is the circle $(x-10)^2+(y-15)^2=1$ in the $xy$-plane, oriented clockwise? $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n(c) If $C$ is any closed curve, what can you say about $\\int_C \\vec F\\cdot d\\vec r$? $\\int_C \\vec F\\cdot d\\vec r=$ [ANS]\n(d) Now let $C$ be the half circle $(x-10)^2+(y-15)^2=1$ in the $xy$-plane with $y > 15$, traversed from $(11, 15)$ to $(9, 15)$. Find $\\int_C \\vec F\\cdot d\\vec r$ by using your result from (c) and considering $C$ plus the line segment connecting the endpoints of $C$. $\\int_C \\vec F\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "0", "0", "0", "3*[9^(3+1)-11^(3+1)]/(3+1)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0643", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "2", "keywords": [ "stokes theorem", "circulation integral", "curl", "multivariable", "calculus" ], "problem_v1": "Find $\\int_C\\vec F\\cdot d\\vec r$ where $C$ is a circle of radius $3$ in the plane $x+y+z=6$, centered at $(3,3,0)$ and oriented clockwise when viewed from the origin, if $\\vec F=2y\\vec i-2x\\vec j+3\\!\\left(y-x\\right)\\vec k$ $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "(2*3-2-2)*pi*3*3/[sqrt(3)]" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find $\\int_C\\vec F\\cdot d\\vec r$ where $C$ is a circle of radius $1$ in the plane $x+y+z=9$, centered at $(1,2,6)$ and oriented clockwise when viewed from the origin, if $\\vec F=5y\\vec i-2x\\vec j+(y-x)\\vec k$ $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "(2*1-5-2)*pi*1*1/[sqrt(3)]" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find $\\int_C\\vec F\\cdot d\\vec r$ where $C$ is a circle of radius $1$ in the plane $x+y+z=6$, centered at $(2,3,1)$ and oriented clockwise when viewed from the origin, if $\\vec F=2y\\vec i-2x\\vec j+5\\!\\left(y-x\\right)\\vec k$ $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "(2*5-2-2)*pi*1*1/[sqrt(3)]" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0644", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "2", "keywords": [ "stokes theorem", "circulation integral", "curl", "multivariable", "calculus" ], "problem_v1": "If $\\mbox{curl}\\vec F=(x^2+z^2)\\vec j+7\\vec k$, find $\\int_C\\vec F\\cdot d\\vec r$, where $C$ is a circle of radius 3, centered at the origin, with\n(a) $C$ in the $xy$-plane, oriented counterclockwise when viewed from above. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n(b) $C$ in the $xz$-plane, oriented counterclockwise when viewed from the positive $y$-axis. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v1": [ "7*9*pi", "9*9*pi/2" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "If $\\mbox{curl}\\vec F=(x^2+z^2)\\vec j+2\\vec k$, find $\\int_C\\vec F\\cdot d\\vec r$, where $C$ is a circle of radius 5, centered at the origin, with\n(a) $C$ in the $xy$-plane, oriented counterclockwise when viewed from above. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n(b) $C$ in the $xz$-plane, oriented counterclockwise when viewed from the positive $y$-axis. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v2": [ "2*25*pi", "25*25*pi/2" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "If $\\mbox{curl}\\vec F=(x^2+z^2)\\vec j+4\\vec k$, find $\\int_C\\vec F\\cdot d\\vec r$, where $C$ is a circle of radius 4, centered at the origin, with\n(a) $C$ in the $xy$-plane, oriented counterclockwise when viewed from above. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\n(b) $C$ in the $xz$-plane, oriented counterclockwise when viewed from the positive $y$-axis. $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]", "answer_v3": [ "4*16*pi", "16*16*pi/2" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0645", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "2", "keywords": [ "stokes theorem", "circulation integral", "curl", "multivariable", "calculus" ], "problem_v1": "Calculate the circulation, $\\int_C\\vec F\\cdot d\\vec r$, in two ways, directly and using Stokes' Theorem. The vector field $\\vec F=7 y\\vec i-7x\\vec j$ and $C$ is the boundary of $S$, the part of the surface $z=9-x^2-y^2$ above the $xy$-plane, oriented upward. Note that $C$ is a circle in the $xy$-plane. Find a $\\vec r(t)$ that parameterizes this curve. $\\vec r(t)=$ [ANS], with [ANS] $\\le t\\le$ [ANS]\n(Note that answers must be provided for all three of these answer blanks to be able to determine correctness of the parameterization.) (Note that answers must be provided for all three of these answer blanks to be able to determine correctness of the parameterization.) With this parameterization, the circulation integral is $\\int_C\\vec F\\cdot d\\vec r=\\int_a^b$ [ANS] $dt$, where $a$ and $b$ are the endpoints you gave above. Evaluate your integral to find the circulation: $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\nUsing Stokes' Theorem, we equate $\\int_C\\vec F\\cdot d\\vec r=\\int_S\\mbox{curl}\\vec F\\cdot d\\vec A$. Find $\\mbox{curl}\\vec F=$ [ANS]. Noting that the surface is given by $z=9-x^2-y^2$, find $d\\vec A=$ [ANS] $dy\\,dx$. With $R$ giving the region in the $xy$-plane enclosed by the surface, this gives $\\int_S \\mbox{curl}\\vec F\\cdot d\\vec A=\\int_{R}$ [ANS] $dy\\,dx$. Evaluate this integral to find the circulation: $\\int_C \\vec F\\cdot d\\vec r=\\int_{S} \\mbox{curl}\\vec F\\cdot d\\vec A=$ [ANS].", "answer_v1": [ "3*cos(t)i+3*sin(t)j", "0", "2*pi", "-9*7", "-7*9*2*pi", "-14k", "2*xi+2*yj+k", "-2*7", "-7*9*2*pi" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV", "EX", "EX", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Calculate the circulation, $\\int_C\\vec F\\cdot d\\vec r$, in two ways, directly and using Stokes' Theorem. The vector field $\\vec F=2 y\\vec i-2x\\vec j$ and $C$ is the boundary of $S$, the part of the surface $z=16-x^2-y^2$ above the $xy$-plane, oriented upward. Note that $C$ is a circle in the $xy$-plane. Find a $\\vec r(t)$ that parameterizes this curve. $\\vec r(t)=$ [ANS], with [ANS] $\\le t\\le$ [ANS]\n(Note that answers must be provided for all three of these answer blanks to be able to determine correctness of the parameterization.) (Note that answers must be provided for all three of these answer blanks to be able to determine correctness of the parameterization.) With this parameterization, the circulation integral is $\\int_C\\vec F\\cdot d\\vec r=\\int_a^b$ [ANS] $dt$, where $a$ and $b$ are the endpoints you gave above. Evaluate your integral to find the circulation: $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\nUsing Stokes' Theorem, we equate $\\int_C\\vec F\\cdot d\\vec r=\\int_S\\mbox{curl}\\vec F\\cdot d\\vec A$. Find $\\mbox{curl}\\vec F=$ [ANS]. Noting that the surface is given by $z=16-x^2-y^2$, find $d\\vec A=$ [ANS] $dy\\,dx$. With $R$ giving the region in the $xy$-plane enclosed by the surface, this gives $\\int_S \\mbox{curl}\\vec F\\cdot d\\vec A=\\int_{R}$ [ANS] $dy\\,dx$. Evaluate this integral to find the circulation: $\\int_C \\vec F\\cdot d\\vec r=\\int_{S} \\mbox{curl}\\vec F\\cdot d\\vec A=$ [ANS].", "answer_v2": [ "4*cos(t)i+4*sin(t)j", "0", "2*pi", "-16*2", "-2*16*2*pi", "-4k", "2*xi+2*yj+k", "-2*2", "-2*16*2*pi" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV", "EX", "EX", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Calculate the circulation, $\\int_C\\vec F\\cdot d\\vec r$, in two ways, directly and using Stokes' Theorem. The vector field $\\vec F=4 y\\vec i-4x\\vec j$ and $C$ is the boundary of $S$, the part of the surface $z=9-x^2-y^2$ above the $xy$-plane, oriented upward. Note that $C$ is a circle in the $xy$-plane. Find a $\\vec r(t)$ that parameterizes this curve. $\\vec r(t)=$ [ANS], with [ANS] $\\le t\\le$ [ANS]\n(Note that answers must be provided for all three of these answer blanks to be able to determine correctness of the parameterization.) (Note that answers must be provided for all three of these answer blanks to be able to determine correctness of the parameterization.) With this parameterization, the circulation integral is $\\int_C\\vec F\\cdot d\\vec r=\\int_a^b$ [ANS] $dt$, where $a$ and $b$ are the endpoints you gave above. Evaluate your integral to find the circulation: $\\int_C\\vec F\\cdot d\\vec r=$ [ANS]\nUsing Stokes' Theorem, we equate $\\int_C\\vec F\\cdot d\\vec r=\\int_S\\mbox{curl}\\vec F\\cdot d\\vec A$. Find $\\mbox{curl}\\vec F=$ [ANS]. Noting that the surface is given by $z=9-x^2-y^2$, find $d\\vec A=$ [ANS] $dy\\,dx$. With $R$ giving the region in the $xy$-plane enclosed by the surface, this gives $\\int_S \\mbox{curl}\\vec F\\cdot d\\vec A=\\int_{R}$ [ANS] $dy\\,dx$. Evaluate this integral to find the circulation: $\\int_C \\vec F\\cdot d\\vec r=\\int_{S} \\mbox{curl}\\vec F\\cdot d\\vec A=$ [ANS].", "answer_v3": [ "3*cos(t)i+3*sin(t)j", "0", "2*pi", "-9*4", "-4*9*2*pi", "-8k", "2*xi+2*yj+k", "-2*4", "-4*9*2*pi" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV", "EX", "EX", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0646", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "2", "keywords": [ "stokes theorem", "circulation integral", "curl", "multivariable", "calculus" ], "problem_v1": "Evaluate $\\int_C (-5 z\\vec i+5 y\\vec j+5x\\vec k)\\cdot d\\vec{r}$, where $C$ is a circle of radius $4$, parallel to the $xz$-plane and around the positive $y$-axis with counterclockwise orientation when viewed from a point on the $y$-axis far from the origin. $\\int_C (-5 z\\vec i+5 y\\vec j+5x\\vec k)\\cdot d\\vec{r}=$ [ANS]", "answer_v1": [ "-1*(5+5)*pi*16" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate $\\int_C (-7 z\\vec i+7 y\\vec j+2x\\vec k)\\cdot d\\vec{r}$, where $C$ is a circle of radius $2$, parallel to the $xz$-plane and around the positive $y$-axis with counterclockwise orientation when viewed from a point on the $y$-axis far from the origin. $\\int_C (-7 z\\vec i+7 y\\vec j+2x\\vec k)\\cdot d\\vec{r}=$ [ANS]", "answer_v2": [ "-1*(7+2)*pi*4" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate $\\int_C (-5 z\\vec i+5 y\\vec j+2x\\vec k)\\cdot d\\vec{r}$, where $C$ is a circle of radius $2$, parallel to the $xz$-plane and around the positive $y$-axis with counterclockwise orientation when viewed from a point on the $y$-axis far from the origin. $\\int_C (-5 z\\vec i+5 y\\vec j+2x\\vec k)\\cdot d\\vec{r}=$ [ANS]", "answer_v3": [ "-1*(5+2)*pi*4" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0647", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "2", "keywords": [ "Stokes", "Stokes theorem" ], "problem_v1": "Evaluate the circulation of $\\vec{G}=xy \\vec{i}+z\\vec{j}+6 y\\vec{k}$ around a square of side $7$, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis.\nCirculation=$ \\int_C \\vec{F} \\cdot d\\vec{r}$=[ANS]", "answer_v1": [ "(6-1)*7^2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate the circulation of $\\vec{G}=xy \\vec{i}+z\\vec{j}+2 y\\vec{k}$ around a square of side $9$, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis.\nCirculation=$ \\int_C \\vec{F} \\cdot d\\vec{r}$=[ANS]", "answer_v2": [ "(2-1)*9^2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate the circulation of $\\vec{G}=xy \\vec{i}+z\\vec{j}+3 y\\vec{k}$ around a square of side $7$, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis.\nCirculation=$ \\int_C \\vec{F} \\cdot d\\vec{r}$=[ANS]", "answer_v3": [ "(3-1)*7^2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0648", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "2", "keywords": [ "Stokes", "Stokes theorem" ], "problem_v1": "Let $\\vec{F}=\\langle a x+b y+8 z, \\, x+c z, \\, 7 y+m x \\rangle$ where $a, b, c, m$ are constants.\n(a) Suppose the flux of $\\vec{F}$ through any closed surface is zero. Which of the constants can be determined? Select all that apply. [ANS] A. The constant $a$ B. The constant $b$ C. The constant $c$ D. The constant $m$\n(b) Suppose instead that the circulation around any closed curve is zero. Then $b$=[ANS]\n$c$=[ANS]\n$m$=[ANS]", "answer_v1": [ "A", "1", "7", "8" ], "answer_type_v1": [ "MCS", "NV", "NV", "NV" ], "options_v1": [ [ "A", "B", "C", "D" ], [], [], [] ], "problem_v2": "Let $\\vec{F}=\\langle a x+b y+2 z, \\, x+c z, \\, 10 y+m x \\rangle$ where $a, b, c, m$ are constants.\n(a) Suppose the flux of $\\vec{F}$ through any closed surface is zero. Which of the constants can be determined? Select all that apply. [ANS] A. The constant $a$ B. The constant $b$ C. The constant $c$ D. The constant $m$\n(b) Suppose instead that the circulation around any closed curve is zero. Then $b$=[ANS]\n$c$=[ANS]\n$m$=[ANS]", "answer_v2": [ "A", "1", "10", "2" ], "answer_type_v2": [ "MCS", "NV", "NV", "NV" ], "options_v2": [ [ "A", "B", "C", "D" ], [], [], [] ], "problem_v3": "Let $\\vec{F}=\\langle a x+b y+4 z, \\, x+c z, \\, 7 y+m x \\rangle$ where $a, b, c, m$ are constants.\n(a) Suppose the flux of $\\vec{F}$ through any closed surface is zero. Which of the constants can be determined? Select all that apply. [ANS] A. The constant $a$ B. The constant $b$ C. The constant $c$ D. The constant $m$\n(b) Suppose instead that the circulation around any closed curve is zero. Then $b$=[ANS]\n$c$=[ANS]\n$m$=[ANS]", "answer_v3": [ "A", "1", "7", "4" ], "answer_type_v3": [ "MCS", "NV", "NV", "NV" ], "options_v3": [ [ "A", "B", "C", "D" ], [], [], [] ] }, { "id": "Calculus_-_multivariable_0649", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "2", "keywords": [ "calculus", "stokes", "line integral", "double integral", "vector" ], "problem_v1": "Use Stokes' Theorem to evaluate $ \\iint_M (\\nabla \\times \\mathbf{F}) \\cdot d\\mathbf{S}$ where $M$ is the hemisphere $x^2+y^2+z^2=16, x \\ge 0$, with the normal in the direction of the positive x direction, and $\\mathbf{F}=\\langle x^8, 0, y^2 \\rangle$. Begin by writing down the \"standard\" parametrization of $\\partial M$ as a function of the angle $\\theta$ (denoted by \"t\" in your answer) $x=$ [ANS], $y=$ [ANS], $z=$ [ANS]. $\\int_{\\partial M} \\mathbf{F}\\cdot d\\mathbf{s}=\\int_0^{2\\pi}f(\\theta)\\,d\\theta$, where $f(\\theta)=$ [ANS] (use \"t\" for theta). The value of the integral is [ANS].", "answer_v1": [ "0", "4*cos(t)", "4*sin(t)", "4**(2+1)*cos(t)**(2+1)", "0" ], "answer_type_v1": [ "NV", "EX", "EX", "EX", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Use Stokes' Theorem to evaluate $ \\iint_M (\\nabla \\times \\mathbf{F}) \\cdot d\\mathbf{S}$ where $M$ is the hemisphere $x^2+y^2+z^2=4, x \\ge 0$, with the normal in the direction of the positive x direction, and $\\mathbf{F}=\\langle x^2, 0, y^2 \\rangle$. Begin by writing down the \"standard\" parametrization of $\\partial M$ as a function of the angle $\\theta$ (denoted by \"t\" in your answer) $x=$ [ANS], $y=$ [ANS], $z=$ [ANS]. $\\int_{\\partial M} \\mathbf{F}\\cdot d\\mathbf{s}=\\int_0^{2\\pi}f(\\theta)\\,d\\theta$, where $f(\\theta)=$ [ANS] (use \"t\" for theta). The value of the integral is [ANS].", "answer_v2": [ "0", "2*cos(t)", "2*sin(t)", "2**(2+1)*cos(t)**(2+1)", "0" ], "answer_type_v2": [ "NV", "EX", "EX", "EX", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Use Stokes' Theorem to evaluate $ \\iint_M (\\nabla \\times \\mathbf{F}) \\cdot d\\mathbf{S}$ where $M$ is the hemisphere $x^2+y^2+z^2=9, x \\ge 0$, with the normal in the direction of the positive x direction, and $\\mathbf{F}=\\langle x^4, 0, y^2 \\rangle$. Begin by writing down the \"standard\" parametrization of $\\partial M$ as a function of the angle $\\theta$ (denoted by \"t\" in your answer) $x=$ [ANS], $y=$ [ANS], $z=$ [ANS]. $\\int_{\\partial M} \\mathbf{F}\\cdot d\\mathbf{s}=\\int_0^{2\\pi}f(\\theta)\\,d\\theta$, where $f(\\theta)=$ [ANS] (use \"t\" for theta). The value of the integral is [ANS].", "answer_v3": [ "0", "3*cos(t)", "3*sin(t)", "3**(2+1)*cos(t)**(2+1)", "0" ], "answer_type_v3": [ "NV", "EX", "EX", "EX", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0650", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Stokes' theorem", "level": "2", "keywords": [ "calculus", "stokes", "line integral", "Stokes' 'double integral' 'vector" ], "problem_v1": "Verify Stokes' theorem for the helicoid $\\Psi(r,\\theta)=\\langle r\\cos \\theta, r\\sin \\theta, \\theta \\rangle$ where $(r,\\theta)$ lies in the rectangle $[0,1] \\times [0,\\pi/2]$, and $\\mathbf{F}$ is the vector field $\\mathbf{F}=\\langle 8 z, 6x, 6 y \\rangle$. First, compute the surface integral: $\\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}=\\int_a^b\\int_c^d f(r, \\theta) dr\\,d\\theta$, where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], and $f(r, \\theta)=$ [ANS] (use \"t\" for theta). Finally, the value of the surface integral is [ANS]. Next compute the line integral on that part of the boundary from $(1,0,0)$ to $(0,1,\\pi/2)$. $\\int_C \\mathbf{F}\\cdot d\\mathbf{r}=\\int_a^b g(\\theta)\\,d\\theta$, where $a=$ [ANS], $b=$ [ANS], and $g(\\theta)=$ [ANS] (use \"t\" for theta).", "answer_v1": [ "0", "1.5707963267949", "0", "1", "6*sin(t) - 8*cos(t) + 6*r", "2.71238898038469", "0", "1.5707963267949", "-8*t*sin(t) + 6*cos(t)^2 + 6*sin(t)" ], "answer_type_v1": [ "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [], [], [], [], [], [] ], "problem_v2": "Verify Stokes' theorem for the helicoid $\\Psi(r,\\theta)=\\langle r\\cos \\theta, r\\sin \\theta, \\theta \\rangle$ where $(r,\\theta)$ lies in the rectangle $[0,1] \\times [0,\\pi/2]$, and $\\mathbf{F}$ is the vector field $\\mathbf{F}=\\langle 2 z, 9x, 3 y \\rangle$. First, compute the surface integral: $\\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}=\\int_a^b\\int_c^d f(r, \\theta) dr\\,d\\theta$, where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], and $f(r, \\theta)=$ [ANS] (use \"t\" for theta). Finally, the value of the surface integral is [ANS]. Next compute the line integral on that part of the boundary from $(1,0,0)$ to $(0,1,\\pi/2)$. $\\int_C \\mathbf{F}\\cdot d\\mathbf{r}=\\int_a^b g(\\theta)\\,d\\theta$, where $a=$ [ANS], $b=$ [ANS], and $g(\\theta)=$ [ANS] (use \"t\" for theta).", "answer_v2": [ "0", "1.5707963267949", "0", "1", "3*sin(t) - 2*cos(t) + 9*r", "8.06858347057704", "0", "1.5707963267949", "-2*t*sin(t) + 9*cos(t)^2 + 3*sin(t)" ], "answer_type_v2": [ "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [], [], [], [], [], [] ], "problem_v3": "Verify Stokes' theorem for the helicoid $\\Psi(r,\\theta)=\\langle r\\cos \\theta, r\\sin \\theta, \\theta \\rangle$ where $(r,\\theta)$ lies in the rectangle $[0,1] \\times [0,\\pi/2]$, and $\\mathbf{F}$ is the vector field $\\mathbf{F}=\\langle 4 z, 6x, 4 y \\rangle$. First, compute the surface integral: $\\iint_M (\\nabla \\times \\mathbf{F})\\cdot d\\mathbf{S}=\\int_a^b\\int_c^d f(r, \\theta) dr\\,d\\theta$, where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], and $f(r, \\theta)=$ [ANS] (use \"t\" for theta). Finally, the value of the surface integral is [ANS]. Next compute the line integral on that part of the boundary from $(1,0,0)$ to $(0,1,\\pi/2)$. $\\int_C \\mathbf{F}\\cdot d\\mathbf{r}=\\int_a^b g(\\theta)\\,d\\theta$, where $a=$ [ANS], $b=$ [ANS], and $g(\\theta)=$ [ANS] (use \"t\" for theta).", "answer_v3": [ "0", "1.5707963267949", "0", "1", "4*sin(t) - 4*cos(t) + 6*r", "4.71238898038469", "0", "1.5707963267949", "-4*t*sin(t) + 6*cos(t)^2 + 4*sin(t)" ], "answer_type_v3": [ "NV", "NV", "NV", "NV", "EX", "NV", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0651", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Verify the Divergence Theorem for the vector field and region: $\\mathbf{F}=\\left< 6x,6z,7y \\right>$ and the region $x^2+y^2\\le 1$, $0\\le z \\le 8$ $\\iint_{\\mathcal{S}} \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]\n$\\iiint_{\\mathcal{R}} \\text{div} (\\mathbf{F}) \\,dV=$ [ANS]", "answer_v1": [ "150.796", "150.796" ], "answer_type_v1": [ "NV", "NV" ], "options_v1": [ [], [] ], "problem_v2": "Verify the Divergence Theorem for the vector field and region: $\\mathbf{F}=\\left< 9x,3z,4y \\right>$ and the region $x^2+y^2\\le 1$, $0\\le z \\le 2$ $\\iint_{\\mathcal{S}} \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]\n$\\iiint_{\\mathcal{R}} \\text{div} (\\mathbf{F}) \\,dV=$ [ANS]", "answer_v2": [ "56.5487", "56.5487" ], "answer_type_v2": [ "NV", "NV" ], "options_v2": [ [], [] ], "problem_v3": "Verify the Divergence Theorem for the vector field and region: $\\mathbf{F}=\\left< 6x,4z,6y \\right>$ and the region $x^2+y^2\\le 1$, $0\\le z \\le 4$ $\\iint_{\\mathcal{S}} \\mathbf{F} \\cdot d\\mathbf{S}=$ [ANS]\n$\\iiint_{\\mathcal{R}} \\text{div} (\\mathbf{F}) \\,dV=$ [ANS]", "answer_v3": [ "75.3982", "75.3982" ], "answer_type_v3": [ "NV", "NV" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0652", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "calculus" ], "problem_v1": "Use the Divergence Theorem to evaluate the surface integral $\\iint_{\\mathcal{S}} \\mathbf{F}\\cdot d\\mathbf{S}$. $\\mathbf{F}=\\left<6x+y, z,6 z-x\\right>$, $\\mathcal{S}$ is the boundary of the region between the paraboloid $z=64-x^2-y^2$ and the $xy$-plane. $\\iint_{\\mathcal{S}} \\mathbf{F}\\cdot d\\mathbf{S}=$ [ANS]", "answer_v1": [ "77207.8" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the Divergence Theorem to evaluate the surface integral $\\iint_{\\mathcal{S}} \\mathbf{F}\\cdot d\\mathbf{S}$. $\\mathbf{F}=\\left<9x+y, z,3 z-x\\right>$, $\\mathcal{S}$ is the boundary of the region between the paraboloid $z=4-x^2-y^2$ and the $xy$-plane. $\\iint_{\\mathcal{S}} \\mathbf{F}\\cdot d\\mathbf{S}=$ [ANS]", "answer_v2": [ "301.593" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the Divergence Theorem to evaluate the surface integral $\\iint_{\\mathcal{S}} \\mathbf{F}\\cdot d\\mathbf{S}$. $\\mathbf{F}=\\left<6x+y, z,4 z-x\\right>$, $\\mathcal{S}$ is the boundary of the region between the paraboloid $z=16-x^2-y^2$ and the $xy$-plane. $\\iint_{\\mathcal{S}} \\mathbf{F}\\cdot d\\mathbf{S}=$ [ANS]", "answer_v3": [ "4021.24" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0653", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "2", "keywords": [ "multivariable", "Vector Fields", "Vector", "Field", "Line Integral", "Flux Integral", "integral", "Greens" ], "problem_v1": "Let $\\mathbf{F}=4x \\mathbf{i}+3 y \\mathbf{j}$ and let n be the outward unit normal vector to the positively oriented circle $x^{2}+y^{2}=16$. Compute the flux integral $\\int_{C} \\mathbf{F \\cdot n} \\, ds$. [ANS]", "answer_v1": [ "351.858377202057" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Let $\\mathbf{F}=1x \\mathbf{i}+5 y \\mathbf{j}$ and let n be the outward unit normal vector to the positively oriented circle $x^{2}+y^{2}=1$. Compute the flux integral $\\int_{C} \\mathbf{F \\cdot n} \\, ds$. [ANS]", "answer_v2": [ "18.8495559215388" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Let $\\mathbf{F}=2x \\mathbf{i}+4 y \\mathbf{j}$ and let n be the outward unit normal vector to the positively oriented circle $x^{2}+y^{2}=4$. Compute the flux integral $\\int_{C} \\mathbf{F \\cdot n} \\, ds$. [ANS]", "answer_v3": [ "75.398223686155" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0654", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "2", "keywords": [ "multivariable", "Surface Integral", "Vector", "Field" ], "problem_v1": "A fluid has density 4 and velocity field $\\mathbf{v}=x\\boldsymbol{j}-y\\boldsymbol{i}+3z\\boldsymbol{k}$. Find the rate of flow outward through the sphere $x^{2}+y^{2}+z^{2}=16$: [ANS]", "answer_v1": [ "4*3*4*16*4*pi/3" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "A fluid has density 1 and velocity field $\\mathbf{v}=x\\boldsymbol{j}-y\\boldsymbol{i}+5z\\boldsymbol{k}$. Find the rate of flow outward through the sphere $x^{2}+y^{2}+z^{2}=1$: [ANS]", "answer_v2": [ "4*5*1*1*1*pi/3" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "A fluid has density 2 and velocity field $\\mathbf{v}=x\\boldsymbol{j}-y\\boldsymbol{i}+4z\\boldsymbol{k}$. Find the rate of flow outward through the sphere $x^{2}+y^{2}+z^{2}=4$: [ANS]", "answer_v3": [ "4*4*2*4*2*pi/3" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0655", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "2", "keywords": [ "vector' 'divergence" ], "problem_v1": "In springtime, the average value over time of the divergence of the vector field which represents air flow is: [ANS] A. zero B. negative C. positive", "answer_v1": [ "C" ], "answer_type_v1": [ "MCS" ], "options_v1": [ [ "A", "B", "C" ] ], "problem_v2": "In springtime, the average value over time of the divergence of the vector field which represents air flow is: [ANS] A. positive B. negative C. zero", "answer_v2": [ "A" ], "answer_type_v2": [ "MCS" ], "options_v2": [ [ "A", "B", "C" ] ], "problem_v3": "In springtime, the average value over time of the divergence of the vector field which represents air flow is: [ANS] A. zero B. positive C. negative", "answer_v3": [ "B" ], "answer_type_v3": [ "MCS" ], "options_v3": [ [ "A", "B", "C" ] ] }, { "id": "Calculus_-_multivariable_0656", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "2", "keywords": [ "vector' 'double integral' 'multivariable' 'flux' 'divergence", "Divergence theorem", "Flux", "Gauss's Theorem", "Flux" ], "problem_v1": "Use the divergence theorem to find the outward flux of the vector field $\\mathbf{F}(x, y, z)=4\\!x^{2}\\mathbf{i}+3\\!y^{2}\\mathbf{j}+4\\!z^{2}\\mathbf{k}$ across the boundary of the rectangular prism: $0 \\leq x \\leq 4, 0 \\leq y \\leq 2, 0 \\leq z \\leq 2$. [ANS]", "answer_v1": [ "480" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Use the divergence theorem to find the outward flux of the vector field $\\mathbf{F}(x, y, z)=1\\!x^{2}\\mathbf{i}+5\\!y^{2}\\mathbf{j}+1\\!z^{2}\\mathbf{k}$ across the boundary of the rectangular prism: $0 \\leq x \\leq 2, 0 \\leq y \\leq 5, 0 \\leq z \\leq 2$. [ANS]", "answer_v2": [ "580" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Use the divergence theorem to find the outward flux of the vector field $\\mathbf{F}(x, y, z)=2\\!x^{2}\\mathbf{i}+4\\!y^{2}\\mathbf{j}+2\\!z^{2}\\mathbf{k}$ across the boundary of the rectangular prism: $0 \\leq x \\leq 3, 0 \\leq y \\leq 2, 0 \\leq z \\leq 2$. [ANS]", "answer_v3": [ "216" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0657", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "vector", "multivariable", "green's", "flux" ], "problem_v1": "Find the flux through the boundary of the rectangle $0 \\leq x \\leq 5, 0 \\leq y \\leq 4$ for fluid flowing along the vector field $\\left$. Flux=[ANS]", "answer_v1": [ "12499.9" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the flux through the boundary of the rectangle $0 \\leq x \\leq 2, 0 \\leq y \\leq 6$ for fluid flowing along the vector field $\\left$. Flux=[ANS]", "answer_v2": [ "45.7296" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the flux through the boundary of the rectangle $0 \\leq x \\leq 3, 0 \\leq y \\leq 5$ for fluid flowing along the vector field $\\left$. Flux=[ANS]", "answer_v3": [ "405.687" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0658", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "vector", "double integral", "multivariable", "flux", "divergence" ], "problem_v1": "Compute the outward flux of the vector field $\\mathbf{F}(x, y, z)=4\\!x\\mathbf{i}+3\\!y\\mathbf{j}+4\\!z\\mathbf{k}$ across the boundary of the right cylinder with radius 6 with bottom edge at height z=-3 and upper edge at z=4. Note: The vectors in this field point outwards from the origin, so we would expect the flux across each face of the cylinder to be positive. Part 1-Using a Surface Integral First we parameterize the three faces of the cylinder. Note: in the following type $\\theta$ as theta. The bottom face: $\\sigma_1(r,\\theta)=(x_1(r,\\theta),y_1(r,\\theta),z_1(r,\\theta)$ where $0\\leq\\theta\\leq 2\\pi$, $0\\leq r\\leq 6$ and $x_1(r,\\theta)=$ [ANS]\n$y_1(r,\\theta)=$ [ANS]\n$z_1(r,\\theta)=-3$ The top face: $\\sigma_2(r,\\theta)=(x_2(r,\\theta),y_2(r,\\theta),z_2(r,\\theta)$ where $0\\leq\\theta\\leq 2\\pi$, $0\\leq r\\leq 6$ and $x_2(r,\\theta)=$ [ANS]\n$y_2(r,\\theta)=$ [ANS]\n$z_2(r,\\theta)=4$ The lateral face: $\\sigma_3(\\theta,z)=(x_3(\\theta,z),y_3(\\theta,z),z_3(\\theta,z)$ where $0\\leq\\theta\\leq 2\\pi$, $-3\\leq z\\leq 4$ and $x_3(\\theta)=$ [ANS]\n$y_3(\\theta)=$ [ANS]\n$z_3(z)=$ [ANS]\nThen (mind the orientation) $\\int\\int_\\sigma F\\cdot n dS=\\int_0^{2\\pi}\\int_0^{6}F(\\sigma_1)\\cdot\\left( \\frac{\\partial \\sigma_1}{\\partial \\theta} \\times \\frac{\\partial \\sigma_1}{\\partial r} \\right) dr\\,d\\theta$ $+\\int_0^{2\\pi}\\int_0^{6} F(\\sigma_2)\\cdot\\left( \\frac{\\partial \\sigma_2}{\\partial r} \\times \\frac{\\partial \\sigma_2}{\\partial \\theta} \\right) dr\\,d\\theta$ $+\\int_0^{2\\pi}\\int_{-3}^{4} F(\\sigma_3)\\cdot\\left( \\frac{\\partial \\sigma_3}{\\partial z} \\times \\frac{\\partial \\sigma_3}{\\partial \\theta} \\right) dz\\,d\\theta$ $=$ [ANS]+[ANS]+[ANS]\n$=$ [ANS]\nPart 2-Using the Divergence Theorem\n$\\begin{array}{c}\\hline \\int\\int_\\sigma F\\cdot n \\,dS=\\int\\int\\int_G {\\rm div} F\\,dV \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccc}\\hline =& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & dz\\,dr\\,d\\theta \\\\ \\hline \\end{array}$\n$=$ [ANS]", "answer_v1": [ "r*cos(theta)", "r*sin(theta)", "r*cos(theta)", "r*sin(theta)", "6*cos(theta)", "6*sin(theta)", "z", "-4*pi*6^2*-3", "4*pi*6^2*4", "pi*6^2*(4+3)*(4--3)", "1357.17+1809.56+5541.77", "0", "2*pi", "0", "6", "-3", "4", "r*11", "1357.17+1809.56+5541.77" ], "answer_type_v1": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Compute the outward flux of the vector field $\\mathbf{F}(x, y, z)=1\\!x\\mathbf{i}+5\\!y\\mathbf{j}+1\\!z\\mathbf{k}$ across the boundary of the right cylinder with radius 4 with bottom edge at height z=-5 and upper edge at z=7. Note: The vectors in this field point outwards from the origin, so we would expect the flux across each face of the cylinder to be positive. Part 1-Using a Surface Integral First we parameterize the three faces of the cylinder. Note: in the following type $\\theta$ as theta. The bottom face: $\\sigma_1(r,\\theta)=(x_1(r,\\theta),y_1(r,\\theta),z_1(r,\\theta)$ where $0\\leq\\theta\\leq 2\\pi$, $0\\leq r\\leq 4$ and $x_1(r,\\theta)=$ [ANS]\n$y_1(r,\\theta)=$ [ANS]\n$z_1(r,\\theta)=-5$ The top face: $\\sigma_2(r,\\theta)=(x_2(r,\\theta),y_2(r,\\theta),z_2(r,\\theta)$ where $0\\leq\\theta\\leq 2\\pi$, $0\\leq r\\leq 4$ and $x_2(r,\\theta)=$ [ANS]\n$y_2(r,\\theta)=$ [ANS]\n$z_2(r,\\theta)=7$ The lateral face: $\\sigma_3(\\theta,z)=(x_3(\\theta,z),y_3(\\theta,z),z_3(\\theta,z)$ where $0\\leq\\theta\\leq 2\\pi$, $-5\\leq z\\leq 7$ and $x_3(\\theta)=$ [ANS]\n$y_3(\\theta)=$ [ANS]\n$z_3(z)=$ [ANS]\nThen (mind the orientation) $\\int\\int_\\sigma F\\cdot n dS=\\int_0^{2\\pi}\\int_0^{4}F(\\sigma_1)\\cdot\\left( \\frac{\\partial \\sigma_1}{\\partial \\theta} \\times \\frac{\\partial \\sigma_1}{\\partial r} \\right) dr\\,d\\theta$ $+\\int_0^{2\\pi}\\int_0^{4} F(\\sigma_2)\\cdot\\left( \\frac{\\partial \\sigma_2}{\\partial r} \\times \\frac{\\partial \\sigma_2}{\\partial \\theta} \\right) dr\\,d\\theta$ $+\\int_0^{2\\pi}\\int_{-5}^{7} F(\\sigma_3)\\cdot\\left( \\frac{\\partial \\sigma_3}{\\partial z} \\times \\frac{\\partial \\sigma_3}{\\partial \\theta} \\right) dz\\,d\\theta$ $=$ [ANS]+[ANS]+[ANS]\n$=$ [ANS]\nPart 2-Using the Divergence Theorem\n$\\begin{array}{c}\\hline \\int\\int_\\sigma F\\cdot n \\,dS=\\int\\int\\int_G {\\rm div} F\\,dV \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccc}\\hline =& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & dz\\,dr\\,d\\theta \\\\ \\hline \\end{array}$\n$=$ [ANS]", "answer_v2": [ "r*cos(theta)", "r*sin(theta)", "r*cos(theta)", "r*sin(theta)", "4*cos(theta)", "4*sin(theta)", "z", "-1*pi*4^2*-5", "1*pi*4^2*7", "pi*4^2*(1+5)*(7--5)", "251.327+351.858+3619.11", "0", "2*pi", "0", "4", "-5", "7", "r*7", "251.327+351.858+3619.11" ], "answer_type_v2": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Compute the outward flux of the vector field $\\mathbf{F}(x, y, z)=2\\!x\\mathbf{i}+4\\!y\\mathbf{j}+2\\!z\\mathbf{k}$ across the boundary of the right cylinder with radius 5 with bottom edge at height z=-2 and upper edge at z=4. Note: The vectors in this field point outwards from the origin, so we would expect the flux across each face of the cylinder to be positive. Part 1-Using a Surface Integral First we parameterize the three faces of the cylinder. Note: in the following type $\\theta$ as theta. The bottom face: $\\sigma_1(r,\\theta)=(x_1(r,\\theta),y_1(r,\\theta),z_1(r,\\theta)$ where $0\\leq\\theta\\leq 2\\pi$, $0\\leq r\\leq 5$ and $x_1(r,\\theta)=$ [ANS]\n$y_1(r,\\theta)=$ [ANS]\n$z_1(r,\\theta)=-2$ The top face: $\\sigma_2(r,\\theta)=(x_2(r,\\theta),y_2(r,\\theta),z_2(r,\\theta)$ where $0\\leq\\theta\\leq 2\\pi$, $0\\leq r\\leq 5$ and $x_2(r,\\theta)=$ [ANS]\n$y_2(r,\\theta)=$ [ANS]\n$z_2(r,\\theta)=4$ The lateral face: $\\sigma_3(\\theta,z)=(x_3(\\theta,z),y_3(\\theta,z),z_3(\\theta,z)$ where $0\\leq\\theta\\leq 2\\pi$, $-2\\leq z\\leq 4$ and $x_3(\\theta)=$ [ANS]\n$y_3(\\theta)=$ [ANS]\n$z_3(z)=$ [ANS]\nThen (mind the orientation) $\\int\\int_\\sigma F\\cdot n dS=\\int_0^{2\\pi}\\int_0^{5}F(\\sigma_1)\\cdot\\left( \\frac{\\partial \\sigma_1}{\\partial \\theta} \\times \\frac{\\partial \\sigma_1}{\\partial r} \\right) dr\\,d\\theta$ $+\\int_0^{2\\pi}\\int_0^{5} F(\\sigma_2)\\cdot\\left( \\frac{\\partial \\sigma_2}{\\partial r} \\times \\frac{\\partial \\sigma_2}{\\partial \\theta} \\right) dr\\,d\\theta$ $+\\int_0^{2\\pi}\\int_{-2}^{4} F(\\sigma_3)\\cdot\\left( \\frac{\\partial \\sigma_3}{\\partial z} \\times \\frac{\\partial \\sigma_3}{\\partial \\theta} \\right) dz\\,d\\theta$ $=$ [ANS]+[ANS]+[ANS]\n$=$ [ANS]\nPart 2-Using the Divergence Theorem\n$\\begin{array}{c}\\hline \\int\\int_\\sigma F\\cdot n \\,dS=\\int\\int\\int_G {\\rm div} F\\,dV \\\\ \\hline \\end{array}$\n$\\begin{array}{ccccccccccc}\\hline =& & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] \\int [ANS] & & [ANS] & & dz\\,dr\\,d\\theta \\\\ \\hline \\end{array}$\n$=$ [ANS]", "answer_v3": [ "r*cos(theta)", "r*sin(theta)", "r*cos(theta)", "r*sin(theta)", "5*cos(theta)", "5*sin(theta)", "z", "-2*pi*5^2*-2", "2*pi*5^2*4", "pi*5^2*(2+4)*(4--2)", "314.159+628.319+2827.43", "0", "2*pi", "0", "5", "-2", "4", "r*8", "314.159+628.319+2827.43" ], "answer_type_v3": [ "EX", "EX", "EX", "EX", "EX", "EX", "EX", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "EX", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0659", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "divergence theorem", "divergence", "surface integral", "multivariable", "calculus" ], "problem_v1": "Use the Divergence Theorem to calculate the flux of the vector fields $\\vec F_1=-2z\\vec i+2x\\vec k$ and $\\vec F_2=-2z\\vec i+(y+5)\\vec j+2x\\vec k$ through the surface $S$ given by the sphere of radius $a$ centered at the origin with outwards orientation. Be sure that you are able to explain your answers geometrically. With $W$ giving the interior of the sphere, $\\int_S\\vec F_1\\cdot d\\vec A=\\int_W$ [ANS] $dV=$ [ANS]\nand $\\int_S\\vec F_2\\cdot d\\vec A=\\int_W$ [ANS] $dV=$ [ANS]", "answer_v1": [ "0", "0", "1", "4*1*pi*a^3/3" ], "answer_type_v1": [ "NV", "NV", "NV", "EX" ], "options_v1": [ [], [], [], [] ], "problem_v2": "Use the Divergence Theorem to calculate the flux of the vector fields $\\vec F_1=-8z\\vec i+8x\\vec k$ and $\\vec F_2=-8z\\vec i+(7y+2)\\vec j+8x\\vec k$ through the surface $S$ given by the sphere of radius $a$ centered at the origin with outwards orientation. Be sure that you are able to explain your answers geometrically. With $W$ giving the interior of the sphere, $\\int_S\\vec F_1\\cdot d\\vec A=\\int_W$ [ANS] $dV=$ [ANS]\nand $\\int_S\\vec F_2\\cdot d\\vec A=\\int_W$ [ANS] $dV=$ [ANS]", "answer_v2": [ "0", "0", "7", "4*7*pi*a^3/3" ], "answer_type_v2": [ "NV", "NV", "NV", "EX" ], "options_v2": [ [], [], [], [] ], "problem_v3": "Use the Divergence Theorem to calculate the flux of the vector fields $\\vec F_1=-6z\\vec i+6x\\vec k$ and $\\vec F_2=-6z\\vec i+(2y+3)\\vec j+6x\\vec k$ through the surface $S$ given by the sphere of radius $a$ centered at the origin with outwards orientation. Be sure that you are able to explain your answers geometrically. With $W$ giving the interior of the sphere, $\\int_S\\vec F_1\\cdot d\\vec A=\\int_W$ [ANS] $dV=$ [ANS]\nand $\\int_S\\vec F_2\\cdot d\\vec A=\\int_W$ [ANS] $dV=$ [ANS]", "answer_v3": [ "0", "0", "2", "4*2*pi*a^3/3" ], "answer_type_v3": [ "NV", "NV", "NV", "EX" ], "options_v3": [ [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0660", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "divergence theorem", "divergence", "surface integral", "multivariable", "calculus" ], "problem_v1": "Compute the flux integral $\\int_S\\vec F\\cdot d\\vec A$ in two ways, directly and using the Divergence Theorem. $S$ is the surface of the box with faces $x=3,x=5,y=0,y=2,z=0,z=3$, closed and oriented outward, and $\\vec{F}=2x^{2}\\vec i+2y^{2}\\vec j+3z^{2}\\vec k$. Using the Divergence Theorem, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d\\int_p^q\\,$ [ANS] $\\,dz\\,dy\\,dx=$ [ANS], where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], $p=$ [ANS] and $q=$ [ANS]. Next, calculating directly, we have $\\int_S\\vec F\\cdot d\\vec A=$ (the sum of the flux through each of the six faces of the box). Calculating the flux through each face separately, we have: On $x=5$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dy$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $x=3$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dy$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $y=2$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $y=0$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $z=3$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dy\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. And on $z=0$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dy\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. Thus, summing these, we have $\\int_S\\vec F\\cdot d\\vec A=$ [ANS]", "answer_v1": [ "2*2*x+2*2*y+2*3*z", "(5-3)*(2-0)*(3-0)*[2*(5+3)+2*(2+0)+3*(3+0)]", "3", "5", "0", "2", "0", "3", "50", "2*5*5*(2-0)*(3-0)", "0", "2", "0", "3", "-18", "-2*3*3*(2-0)*(3-0)", "0", "2", "0", "3", "8", "2*2*2*(5-3)*(3-0)", "3", "5", "0", "3", "0", "-2*0*0*(5-3)*(3-0)", "3", "5", "0", "3", "27", "3*3*3*(5-3)*(2-0)", "3", "5", "0", "2", "0", "-3*0*0*(5-3)*(2-0)", "3", "5", "0", "2", "(5-3)*(2-0)*(3-0)*[2*(5+3)+2*(2+0)+3*(3+0)]" ], "answer_type_v1": [ "EX", "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", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v2": "Compute the flux integral $\\int_S\\vec F\\cdot d\\vec A$ in two ways, directly and using the Divergence Theorem. $S$ is the surface of the box with faces $x=0,x=3,y=1,y=3,z=0,z=3$, closed and oriented outward, and $\\vec{F}=2x^{2}\\vec i+y^{2}\\vec j+2z^{2}\\vec k$. Using the Divergence Theorem, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d\\int_p^q\\,$ [ANS] $\\,dz\\,dy\\,dx=$ [ANS], where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], $p=$ [ANS] and $q=$ [ANS]. Next, calculating directly, we have $\\int_S\\vec F\\cdot d\\vec A=$ (the sum of the flux through each of the six faces of the box). Calculating the flux through each face separately, we have: On $x=3$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dy$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $x=0$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dy$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $y=3$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $y=1$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $z=3$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dy\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. And on $z=0$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dy\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. Thus, summing these, we have $\\int_S\\vec F\\cdot d\\vec A=$ [ANS]", "answer_v2": [ "2*2*x+2*1*y+2*2*z", "(3-0)*(3-1)*(3-0)*[2*(3+0)+1*(3+1)+2*(3+0)]", "0", "3", "1", "3", "0", "3", "18", "2*3*3*(3-1)*(3-0)", "1", "3", "0", "3", "0", "-2*0*0*(3-1)*(3-0)", "1", "3", "0", "3", "9", "1*3*3*(3-0)*(3-0)", "0", "3", "0", "3", "-1", "-1*1*1*(3-0)*(3-0)", "0", "3", "0", "3", "18", "2*3*3*(3-0)*(3-1)", "0", "3", "1", "3", "0", "-2*0*0*(3-0)*(3-1)", "0", "3", "1", "3", "(3-0)*(3-1)*(3-0)*[2*(3+0)+1*(3+1)+2*(3+0)]" ], "answer_type_v2": [ "EX", "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", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ], "problem_v3": "Compute the flux integral $\\int_S\\vec F\\cdot d\\vec A$ in two ways, directly and using the Divergence Theorem. $S$ is the surface of the box with faces $x=1,x=3,y=0,y=1,z=0,z=2$, closed and oriented outward, and $\\vec{F}=2x^{2}\\vec i+2y^{2}\\vec j+5z^{2}\\vec k$. Using the Divergence Theorem, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d\\int_p^q\\,$ [ANS] $\\,dz\\,dy\\,dx=$ [ANS], where $a=$ [ANS], $b=$ [ANS], $c=$ [ANS], $d=$ [ANS], $p=$ [ANS] and $q=$ [ANS]. Next, calculating directly, we have $\\int_S\\vec F\\cdot d\\vec A=$ (the sum of the flux through each of the six faces of the box). Calculating the flux through each face separately, we have: On $x=3$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dy$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $x=1$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dy$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $y=1$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $y=0$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dz\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. On $z=2$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dy\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. And on $z=0$, $\\int_S\\vec F\\cdot d\\vec A=\\int_a^b\\int_c^d$ [ANS] $dy\\,dx$=[ANS]\nwhere $a=$ [ANS], $b=$ [ANS], $c=$ [ANS] and $d=$ [ANS]. Thus, summing these, we have $\\int_S\\vec F\\cdot d\\vec A=$ [ANS]", "answer_v3": [ "2*2*x+2*2*y+2*5*z", "(3-1)*(1-0)*(2-0)*[2*(3+1)+2*(1+0)+5*(2+0)]", "1", "3", "0", "1", "0", "2", "18", "2*3*3*(1-0)*(2-0)", "0", "1", "0", "2", "-2", "-2*1*1*(1-0)*(2-0)", "0", "1", "0", "2", "2", "2*1*1*(3-1)*(2-0)", "1", "3", "0", "2", "0", "-2*0*0*(3-1)*(2-0)", "1", "3", "0", "2", "20", "5*2*2*(3-1)*(1-0)", "1", "3", "0", "1", "0", "-5*0*0*(3-1)*(1-0)", "1", "3", "0", "1", "(3-1)*(1-0)*(2-0)*[2*(3+1)+2*(1+0)+5*(2+0)]" ], "answer_type_v3": [ "EX", "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", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0661", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "2", "keywords": [ "divergence theorem", "divergence", "surface integral", "multivariable", "calculus" ], "problem_v1": "Find the flux of the vector field $\\vec J=6x^2 y\\, \\vec i+3x\\, \\vec k$ out of the closed box $0\\le x\\le 7$, $0\\le y\\le 6$, $0\\le z\\le 6$. flux=[ANS]", "answer_v1": [ "6*7*6*6*7*6/2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the flux of the vector field $\\vec J=3x^2 y\\, \\vec i+8 z\\, \\vec j$ out of the closed box $0\\le x\\le 2$, $0\\le y\\le 8$, $0\\le z\\le 3$. flux=[ANS]", "answer_v2": [ "3*2*8*3*2*8/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the flux of the vector field $\\vec J=5 y^2 z\\, \\vec j+2x\\, \\vec k$ out of the closed box $0\\le x\\le 4$, $0\\le y\\le 6$, $0\\le z\\le 3$. flux=[ANS]", "answer_v3": [ "5*4*6*3*6*3/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0662", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "2", "keywords": [ "divergence theorem", "divergence", "surface integral", "multivariable", "calculus" ], "problem_v1": "A cone has its tip at the point $(0, 0, 8)$ and its base the disk $D, x^2+y^2\\le 4$, in the plane $z=1$. The surface of the cone is the curved and slanted face, $S$, oriented upward, and the flat base, $D$, oriented downward. The flux of the constant vector field $\\vec F=a\\vec i+b\\vec j+c\\vec k$ through $S$ is given by \\int_S\\vec F\\cdot d\\vec A=4.36. What is $\\int_D\\vec F\\cdot d\\vec A$? $\\int_D\\vec F\\cdot d\\vec A=$ [ANS]\n(Enter indeterminate if it is not possible to find a value given the information provided.) Supposed we instead consider the vector field $\\vec F=a\\vec i+b\\vec j+c z \\vec k$. If we again know \\int_S\\vec F\\cdot d\\vec A=4.36. What is $\\int_D\\vec F\\cdot d\\vec A$ in this case? $\\int_D\\vec F\\cdot d\\vec A=$ [ANS]\n(Again, enter indeterminate if it is not possible to find a value given the information provided.)", "answer_v1": [ "-4.36", "pi*4*(8-1)*c/3-4.36" ], "answer_type_v1": [ "NV", "EX" ], "options_v1": [ [], [] ], "problem_v2": "A cone has its tip at the point $(0, 0, 5)$ and its base the disk $D, x^2+y^2\\le 9$, in the plane $z=1$. The surface of the cone is the curved and slanted face, $S$, oriented upward, and the flat base, $D$, oriented downward. The flux of the constant vector field $\\vec F=a\\vec i+b\\vec j+c\\vec k$ through $S$ is given by \\int_S\\vec F\\cdot d\\vec A=1.17. What is $\\int_D\\vec F\\cdot d\\vec A$? $\\int_D\\vec F\\cdot d\\vec A=$ [ANS]\n(Enter indeterminate if it is not possible to find a value given the information provided.) Supposed we instead consider the vector field $\\vec F=a\\vec i+b\\vec j+c z \\vec k$. If we again know \\int_S\\vec F\\cdot d\\vec A=1.17. What is $\\int_D\\vec F\\cdot d\\vec A$ in this case? $\\int_D\\vec F\\cdot d\\vec A=$ [ANS]\n(Again, enter indeterminate if it is not possible to find a value given the information provided.)", "answer_v2": [ "-1.17", "pi*9*(5-1)*c/3-1.17" ], "answer_type_v2": [ "NV", "EX" ], "options_v2": [ [], [] ], "problem_v3": "A cone has its tip at the point $(0, 0, 6)$ and its base the disk $D, x^2+y^2\\le 4$, in the plane $z=1$. The surface of the cone is the curved and slanted face, $S$, oriented upward, and the flat base, $D$, oriented downward. The flux of the constant vector field $\\vec F=a\\vec i+b\\vec j+c\\vec k$ through $S$ is given by \\int_S\\vec F\\cdot d\\vec A=2.27. What is $\\int_D\\vec F\\cdot d\\vec A$? $\\int_D\\vec F\\cdot d\\vec A=$ [ANS]\n(Enter indeterminate if it is not possible to find a value given the information provided.) Supposed we instead consider the vector field $\\vec F=a\\vec i+b\\vec j+c z \\vec k$. If we again know \\int_S\\vec F\\cdot d\\vec A=2.27. What is $\\int_D\\vec F\\cdot d\\vec A$ in this case? $\\int_D\\vec F\\cdot d\\vec A=$ [ANS]\n(Again, enter indeterminate if it is not possible to find a value given the information provided.)", "answer_v3": [ "-2.27", "pi*4*(6-1)*c/3-2.27" ], "answer_type_v3": [ "NV", "EX" ], "options_v3": [ [], [] ] }, { "id": "Calculus_-_multivariable_0663", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "2", "keywords": [ "divergence theorem", "divergence", "surface integral", "multivariable", "calculus" ], "problem_v1": "Let $\\mbox{div}\\vec F=4(7-x)$ and $0\\le a, b, c\\le 13$.\n(a) Find the flux of $\\vec F$ out of the rectangular solid $0\\le x\\le a$, $0\\le y\\le b$, $0\\le z\\le c$. flux=[ANS]\n(b) For what values of $a$, $b$, $c$ is the flux largest? $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]\n(c) What is that largest flux? flux=[ANS]", "answer_v1": [ "4*7*a*b*c-2*a^2*b*c", "7", "13", "13", "4*7*7*13^2-2*7^2*13^2" ], "answer_type_v1": [ "EX", "NV", "NV", "NV", "NV" ], "options_v1": [ [], [], [], [], [] ], "problem_v2": "Let $\\mbox{div}\\vec F=6(2-x)$ and $0\\le a, b, c\\le 10$.\n(a) Find the flux of $\\vec F$ out of the rectangular solid $0\\le x\\le a$, $0\\le y\\le b$, $0\\le z\\le c$. flux=[ANS]\n(b) For what values of $a$, $b$, $c$ is the flux largest? $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]\n(c) What is that largest flux? flux=[ANS]", "answer_v2": [ "6*2*a*b*c-3*a^2*b*c", "2", "10", "10", "6*2*2*10^2-3*2^2*10^2" ], "answer_type_v2": [ "EX", "NV", "NV", "NV", "NV" ], "options_v2": [ [], [], [], [], [] ], "problem_v3": "Let $\\mbox{div}\\vec F=4(4-x)$ and $0\\le a, b, c\\le 11$.\n(a) Find the flux of $\\vec F$ out of the rectangular solid $0\\le x\\le a$, $0\\le y\\le b$, $0\\le z\\le c$. flux=[ANS]\n(b) For what values of $a$, $b$, $c$ is the flux largest? $a=$ [ANS]\n$b=$ [ANS]\n$c=$ [ANS]\n(c) What is that largest flux? flux=[ANS]", "answer_v3": [ "4*4*a*b*c-2*a^2*b*c", "4", "11", "11", "4*4*4*11^2-2*4^2*11^2" ], "answer_type_v3": [ "EX", "NV", "NV", "NV", "NV" ], "options_v3": [ [], [], [], [], [] ] }, { "id": "Calculus_-_multivariable_0664", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "divergence theorem", "divergence", "surface integral", "multivariable", "calculus" ], "problem_v1": "Suppose $\\mbox{div}\\vec F=xyz^{2}$.\n(a) Find $\\mbox{div}\\vec F$ at the point $(3,4,4)$. [Note: You are given $\\mbox{div}\\vec F$, not $\\vec F$.] $\\mbox{div}\\vec F\\bigg|_{(3,4,4)}=$ [ANS]\n(b) Using your answer to part (a), but no other information about the vector field $\\vec F$, estimate the flux out of a small box of side $0.1$ centered at the point $(3,4,4)$ and with edges parallel to the axes. $\\mbox{flux} \\approx$ [ANS]\n(c) Without computing the vector field $\\vec F$, calculate the exact flux out of the box. $\\mbox{flux}=$ [ANS]", "answer_v1": [ "192", "192*0.1^3", "0.1^3*3*4*(0.1^2/12+4^2)" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $\\mbox{div}\\vec F=xyz^{2}$.\n(a) Find $\\mbox{div}\\vec F$ at the point $(5,1,2)$. [Note: You are given $\\mbox{div}\\vec F$, not $\\vec F$.] $\\mbox{div}\\vec F\\bigg|_{(5,1,2)}=$ [ANS]\n(b) Using your answer to part (a), but no other information about the vector field $\\vec F$, estimate the flux out of a small box of side $0.2$ centered at the point $(5,1,2)$ and with edges parallel to the axes. $\\mbox{flux} \\approx$ [ANS]\n(c) Without computing the vector field $\\vec F$, calculate the exact flux out of the box. $\\mbox{flux}=$ [ANS]", "answer_v2": [ "20", "20*0.2^3", "0.2^3*5*1*(0.2^2/12+2^2)" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $\\mbox{div}\\vec F=xyz^{2}$.\n(a) Find $\\mbox{div}\\vec F$ at the point $(4,2,3)$. [Note: You are given $\\mbox{div}\\vec F$, not $\\vec F$.] $\\mbox{div}\\vec F\\bigg|_{(4,2,3)}=$ [ANS]\n(b) Using your answer to part (a), but no other information about the vector field $\\vec F$, estimate the flux out of a small box of side $0.1$ centered at the point $(4,2,3)$ and with edges parallel to the axes. $\\mbox{flux} \\approx$ [ANS]\n(c) Without computing the vector field $\\vec F$, calculate the exact flux out of the box. $\\mbox{flux}=$ [ANS]", "answer_v3": [ "72", "72*0.1^3", "0.1^3*4*2*(0.1^2/12+3^2)" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0665", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "5", "keywords": [ "divergence theorem", "divergence", "surface integral", "multivariable", "calculus" ], "problem_v1": "As a result of radioactive decay, heat is generated uniformly throughout the interior of the earth at a rate of around $30$ watts per cubic kilometer. (A watt is a rate of heat production.) The heat then flows to the earth's surface where it is lost to space. Let $\\vec F(x,y,z)$ denote the rate of flow of heat measured in watts per square kilometer. By definition, the flux of $\\vec F$ across a surface is the quantity of heat flowing through the surface per unit of time.\n(a) Suppose that the actual heat generation is $32 \\mbox{W}/\\mbox{km}^3$. What is the value of $\\mbox{div}\\vec F$? $\\mbox{div}\\vec F=$ [ANS] W/km^3. (b) Assume the heat flows outward symmetrically. Verify that $\\vec F=\\alpha\\vec r$, where $\\vec r=x\\vec i+y\\vec j+z\\vec k$ and $\\alpha$ is a suitable constant, satisfies the given conditions. Find $\\alpha$. $\\alpha=$ [ANS] W/km^3. (c) Let $T(x,y,z)$ denote the temperature inside the earth. Heat flows according to the equation $\\vec F=-k\\,\\mbox{grad}\\,T$, where $k$ is a constant. If $T$ is in ${}^\\circ$ C, then $k=32,000^\\circ\\mbox{C}/\\mbox{km}$. Assuming the earth is a sphere with radius $6400$ km and surface temperature $20^\\circ$ C, what is the temperature at the center? $T=$ [ANS] (degrees C)", "answer_v1": [ "32", "10.6667", "20+6400^2/6000" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "As a result of radioactive decay, heat is generated uniformly throughout the interior of the earth at a rate of around $30$ watts per cubic kilometer. (A watt is a rate of heat production.) The heat then flows to the earth's surface where it is lost to space. Let $\\vec F(x,y,z)$ denote the rate of flow of heat measured in watts per square kilometer. By definition, the flux of $\\vec F$ across a surface is the quantity of heat flowing through the surface per unit of time.\n(a) Suppose that the actual heat generation is $27 \\mbox{W}/\\mbox{km}^3$. What is the value of $\\mbox{div}\\vec F$? $\\mbox{div}\\vec F=$ [ANS] W/km^3. (b) Assume the heat flows outward symmetrically. Verify that $\\vec F=\\alpha\\vec r$, where $\\vec r=x\\vec i+y\\vec j+z\\vec k$ and $\\alpha$ is a suitable constant, satisfies the given conditions. Find $\\alpha$. $\\alpha=$ [ANS] W/km^3. (c) Let $T(x,y,z)$ denote the temperature inside the earth. Heat flows according to the equation $\\vec F=-k\\,\\mbox{grad}\\,T$, where $k$ is a constant. If $T$ is in ${}^\\circ$ C, then $k=27,000^\\circ\\mbox{C}/\\mbox{km}$. Assuming the earth is a sphere with radius $6400$ km and surface temperature $20^\\circ$ C, what is the temperature at the center? $T=$ [ANS] (degrees C)", "answer_v2": [ "27", "9", "20+6400^2/6000" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "As a result of radioactive decay, heat is generated uniformly throughout the interior of the earth at a rate of around $30$ watts per cubic kilometer. (A watt is a rate of heat production.) The heat then flows to the earth's surface where it is lost to space. Let $\\vec F(x,y,z)$ denote the rate of flow of heat measured in watts per square kilometer. By definition, the flux of $\\vec F$ across a surface is the quantity of heat flowing through the surface per unit of time.\n(a) Suppose that the actual heat generation is $29 \\mbox{W}/\\mbox{km}^3$. What is the value of $\\mbox{div}\\vec F$? $\\mbox{div}\\vec F=$ [ANS] W/km^3. (b) Assume the heat flows outward symmetrically. Verify that $\\vec F=\\alpha\\vec r$, where $\\vec r=x\\vec i+y\\vec j+z\\vec k$ and $\\alpha$ is a suitable constant, satisfies the given conditions. Find $\\alpha$. $\\alpha=$ [ANS] W/km^3. (c) Let $T(x,y,z)$ denote the temperature inside the earth. Heat flows according to the equation $\\vec F=-k\\,\\mbox{grad}\\,T$, where $k$ is a constant. If $T$ is in ${}^\\circ$ C, then $k=29,000^\\circ\\mbox{C}/\\mbox{km}$. Assuming the earth is a sphere with radius $6400$ km and surface temperature $20^\\circ$ C, what is the temperature at the center? $T=$ [ANS] (degrees C)", "answer_v3": [ "29", "9.66667", "20+6400^2/6000" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0666", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "divergence theorem", "divergence", "surface integral", "multivariable", "calculus" ], "problem_v1": "(a) Let $\\mbox{div}\\vec F=x^2+y^2+z^2+7$. Calculate $\\int_{S_1}\\vec F\\cdot d\\vec A$ where $S_1$ is the sphere of radius 1 centered at the origin. $\\int_{S_1}\\vec F\\cdot d\\vec A=$ [ANS]\n(b) Let $S_2$ be the sphere of radius 3 centered at the origin; let $S_3$ be the sphere of radius 5 centered at the origin; let $S_4$ be the box of side 10 centered at the origin with edges parallel to the axes. Without calculating them, arrange the following integrals in increasing order: A=\\int_{S_2}\\vec F\\cdot d\\vec A,\\quad B=\\int_{S_3}\\vec F\\cdot d\\vec A, \\quad C=\\int_{S_4}\\vec F\\cdot d\\vec A. [ANS] $<$ [ANS] $<$ [ANS]", "answer_v1": [ "4*pi*(1/5+7/3)", "A", "B", "C" ], "answer_type_v1": [ "NV", "MCS", "MCS", "MCS" ], "options_v1": [ [], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v2": "(a) Let $\\mbox{div}\\vec F=x^2+y^2+z^2+1$. Calculate $\\int_{S_1}\\vec F\\cdot d\\vec A$ where $S_1$ is the sphere of radius 1 centered at the origin. $\\int_{S_1}\\vec F\\cdot d\\vec A=$ [ANS]\n(b) Let $S_2$ be the sphere of radius 4 centered at the origin; let $S_3$ be the sphere of radius 5 centered at the origin; let $S_4$ be the box of side 8 centered at the origin with edges parallel to the axes. Without calculating them, arrange the following integrals in increasing order: A=\\int_{S_2}\\vec F\\cdot d\\vec A,\\quad B=\\int_{S_3}\\vec F\\cdot d\\vec A, \\quad C=\\int_{S_4}\\vec F\\cdot d\\vec A. [ANS] $<$ [ANS] $<$ [ANS]", "answer_v2": [ "4*pi*(1/5+1/3)", "A", "C", "B" ], "answer_type_v2": [ "NV", "MCS", "MCS", "MCS" ], "options_v2": [ [], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ], "problem_v3": "(a) Let $\\mbox{div}\\vec F=x^2+y^2+z^2+3$. Calculate $\\int_{S_1}\\vec F\\cdot d\\vec A$ where $S_1$ is the sphere of radius 1 centered at the origin. $\\int_{S_1}\\vec F\\cdot d\\vec A=$ [ANS]\n(b) Let $S_2$ be the sphere of radius 3 centered at the origin; let $S_3$ be the sphere of radius 4 centered at the origin; let $S_4$ be the box of side 8 centered at the origin with edges parallel to the axes. Without calculating them, arrange the following integrals in increasing order: A=\\int_{S_2}\\vec F\\cdot d\\vec A,\\quad B=\\int_{S_3}\\vec F\\cdot d\\vec A, \\quad C=\\int_{S_4}\\vec F\\cdot d\\vec A. [ANS] $<$ [ANS] $<$ [ANS]", "answer_v3": [ "4*pi*(1/5+3/3)", "A", "B", "C" ], "answer_type_v3": [ "NV", "MCS", "MCS", "MCS" ], "options_v3": [ [], [ "A", "B", "C" ], [ "A", "B", "C" ], [ "A", "B", "C" ] ] }, { "id": "Calculus_-_multivariable_0667", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "divergence theorem", "divergence", "surface integral", "multivariable", "calculus" ], "problem_v1": "Find the flux of $\\vec F=4 z\\,\\vec i+5 y^2\\,\\vec j+4x\\,\\vec k$ out of the closed cone $y=\\sqrt{x^2+z^2}$, with $0\\le y\\le 3$. flux=[ANS]", "answer_v1": [ "5*3^4*pi/2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the flux of $\\vec F=2x^2\\,\\vec i+5 z\\,\\vec j+2 y\\,\\vec k$ out of the closed cone $x=\\sqrt{y^2+z^2}$, with $0\\le x\\le 2$. flux=[ANS]", "answer_v2": [ "2*2^4*pi/2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the flux of $\\vec F=4 z\\,\\vec i+3 y^2\\,\\vec j+3x\\,\\vec k$ out of the closed cone $y=\\sqrt{x^2+z^2}$, with $0\\le y\\le 2$. flux=[ANS]", "answer_v3": [ "3*2^4*pi/2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0668", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "Gauss' 'double integral", "Divergence Theorem", "Gauss's Theorem" ], "problem_v1": "Evaluate $\\iint_{\\partial W} \\mathbf{F} \\cdot d\\mathbf{S}$ where $\\mathbf{F}=(x^2+y, z^2, e^y-z)$ and $W$ is the solid rectangular box whose sides are bounded by the coordinate planes, and the planes $x=8,\\ y=6,\\ z=7$. [ANS]", "answer_v1": [ "2352" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Evaluate $\\iint_{\\partial W} \\mathbf{F} \\cdot d\\mathbf{S}$ where $\\mathbf{F}=(x^2+y, z^2, e^y-z)$ and $W$ is the solid rectangular box whose sides are bounded by the coordinate planes, and the planes $x=1,\\ y=10,\\ z=2$. [ANS]", "answer_v2": [ "0" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Evaluate $\\iint_{\\partial W} \\mathbf{F} \\cdot d\\mathbf{S}$ where $\\mathbf{F}=(x^2+y, z^2, e^y-z)$ and $W$ is the solid rectangular box whose sides are bounded by the coordinate planes, and the planes $x=4,\\ y=7,\\ z=3$. [ANS]", "answer_v3": [ "252" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0669", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "4", "keywords": [ "Gauss' 'double integral", "Divergence Theorem", "Gauss's Theorem" ], "problem_v1": "Let $\\mathbf{F}=(y^2+z^3, x^3+z^2, xz)$. Evaluate $\\iint_{\\partial W}\\mathbf{F}\\cdot d \\mathbf{S}$ for each of the following regions $W$: A. $x^2+y^2 \\le z \\le 8$ [ANS]\nB. $x^2+y^2 \\le z \\le 8, \\ x \\ge 0$ [ANS]\nC. $x^2+y^2 \\le z \\le 8, \\ x \\le 0$ [ANS]", "answer_v1": [ "0", "48.2718229290017", "-48.2718229290017" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Let $\\mathbf{F}=(y^2+z^3, x^3+z^2, xz)$. Evaluate $\\iint_{\\partial W}\\mathbf{F}\\cdot d \\mathbf{S}$ for each of the following regions $W$: A. $x^2+y^2 \\le z \\le 2$ [ANS]\nB. $x^2+y^2 \\le z \\le 2, \\ x \\ge 0$ [ANS]\nC. $x^2+y^2 \\le z \\le 2, \\ x \\le 0$ [ANS]", "answer_v2": [ "0", "1.5084944665313", "-1.5084944665313" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Let $\\mathbf{F}=(y^2+z^3, x^3+z^2, xz)$. Evaluate $\\iint_{\\partial W}\\mathbf{F}\\cdot d \\mathbf{S}$ for each of the following regions $W$: A. $x^2+y^2 \\le z \\le 4$ [ANS]\nB. $x^2+y^2 \\le z \\le 4, \\ x \\ge 0$ [ANS]\nC. $x^2+y^2 \\le z \\le 4, \\ x \\le 0$ [ANS]", "answer_v3": [ "0", "8.53333333333333", "-8.53333333333333" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0670", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "Gauss" ], "problem_v1": "Find the outward flux of the vector field $\\mathbf{F}=(x^3, y^3, z^2)$ across the surface of the region that is enclosed by the circular cylinder $x^2+y^2=49$ and the planes $z=0$ and $z=6$. [ANS]", "answer_v1": [ "73428.4450923542" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Find the outward flux of the vector field $\\mathbf{F}=(x^3, y^3, z^2)$ across the surface of the region that is enclosed by the circular cylinder $x^2+y^2=1$ and the planes $z=0$ and $z=9$. [ANS]", "answer_v2": [ "296.880505764235" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Find the outward flux of the vector field $\\mathbf{F}=(x^3, y^3, z^2)$ across the surface of the region that is enclosed by the circular cylinder $x^2+y^2=9$ and the planes $z=0$ and $z=6$. [ANS]", "answer_v3": [ "3308.09706423005" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0671", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "4", "keywords": [ "Divergence Theorem" ], "problem_v1": "Are the following statements true or false?\n[ANS] 1. If $\\vec{F}$ is a vector field in 3-dimensional space satisfying $\\mathrm{div}(\\vec{F})=1$, and $S$ is a closed surface oriented outward, then $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}$ is equal to the volume enclosed by $S.$ [ANS] 2. If $\\vec{F}$ and $\\vec{G}$ are vector fields satisfying $\\mathrm{div}(\\vec{F})=\\mathrm{div}(\\vec{G})$, then $\\vec{F}=\\vec{G}.$ [ANS] 3. If $\\vec{F}$ is a vector field in 3-dimensional space, and $W$ is a solid region with boundary surface $S$, then $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}=\\iiint\\limits_W \\mathrm{div}(\\vec{F}) \\, dV.$ [ANS] 4. If $\\vec{F}$ is a vector field in 3-dimensional space, then $\\mathrm{grad}(\\mathrm{div} (\\vec{F}))=\\vec{0}.$ [ANS] 5. If $ \\iint\\limits_S \\vec{F} \\cdot d \\vec{A}=12$ and $S$ is a flat disk of area $4\\pi$, then $\\mathrm{div}(\\vec{F})=3/\\pi$.", "answer_v1": [ "T", "F", "T", "F", "F" ], "answer_type_v1": [ "TF", "TF", "TF", "TF", "TF" ], "options_v1": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v2": "Are the following statements true or false?\n[ANS] 1. If $\\vec{F}$ is a vector field in 3-dimensional space, then $\\mathrm{grad}(\\mathrm{div} (\\vec{F}))=\\vec{0}.$ [ANS] 2. If $\\vec{F}$ and $\\vec{G}$ are vector fields satisfying $\\mathrm{div}(\\vec{F})=\\mathrm{div}(\\vec{G})$, then $\\vec{F}=\\vec{G}.$ [ANS] 3. If $ \\iint\\limits_S \\vec{F} \\cdot d \\vec{A}=12$ and $S$ is a flat disk of area $4\\pi$, then $\\mathrm{div}(\\vec{F})=3/\\pi$. [ANS] 4. If $\\vec{F}$ is a vector field in 3-dimensional space satisfying $\\mathrm{div}(\\vec{F})=1$, and $S$ is a closed surface oriented outward, then $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}$ is equal to the volume enclosed by $S.$ [ANS] 5. If $\\vec{F}$ is a vector field in 3-dimensional space, and $W$ is a solid region with boundary surface $S$, then $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}=\\iiint\\limits_W \\mathrm{div}(\\vec{F}) \\, dV.$", "answer_v2": [ "F", "F", "F", "T", "T" ], "answer_type_v2": [ "TF", "TF", "TF", "TF", "TF" ], "options_v2": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ], "problem_v3": "Are the following statements true or false?\n[ANS] 1. If $\\vec{F}$ is a vector field in 3-dimensional space, and $W$ is a solid region with boundary surface $S$, then $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}=\\iiint\\limits_W \\mathrm{div}(\\vec{F}) \\, dV.$ [ANS] 2. If $\\vec{F}$ and $\\vec{G}$ are vector fields satisfying $\\mathrm{div}(\\vec{F})=\\mathrm{div}(\\vec{G})$, then $\\vec{F}=\\vec{G}.$ [ANS] 3. If $ \\iint\\limits_S \\vec{F} \\cdot d \\vec{A}=12$ and $S$ is a flat disk of area $4\\pi$, then $\\mathrm{div}(\\vec{F})=3/\\pi$. [ANS] 4. If $\\vec{F}$ is a vector field in 3-dimensional space, then $\\mathrm{grad}(\\mathrm{div} (\\vec{F}))=\\vec{0}.$ [ANS] 5. If $\\vec{F}$ is a vector field in 3-dimensional space satisfying $\\mathrm{div}(\\vec{F})=1$, and $S$ is a closed surface oriented outward, then $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}$ is equal to the volume enclosed by $S.$", "answer_v3": [ "T", "F", "F", "F", "T" ], "answer_type_v3": [ "TF", "TF", "TF", "TF", "TF" ], "options_v3": [ [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ], [ "T", "F" ] ] }, { "id": "Calculus_-_multivariable_0672", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "3", "keywords": [ "Divergence Theorem" ], "problem_v1": "Suppose $\\vec{F}(x,y,z)=\\langle x, y, 5 z \\rangle$. Let $W$ be the solid bounded by the paraboloid $z=x^2+y^2$ and the plane $z=16.$ Let $S$ be the closed boundary of $W$ oriented outward.\n(a) Use the divergence theorem to find the flux of $\\vec{F}$ through $S.$ $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}=$ [ANS]\n(b) Find the flux of $\\vec{F}$ out the bottom of $S$ (the truncated paraboloid) and the top of $S$ (the disk). Flux out the bottom=[ANS]\nFlux out the top=[ANS]", "answer_v1": [ "896*pi", "-384*pi", "1280*pi" ], "answer_type_v1": [ "NV", "NV", "NV" ], "options_v1": [ [], [], [] ], "problem_v2": "Suppose $\\vec{F}(x,y,z)=\\langle x, y, 3 z \\rangle$. Let $W$ be the solid bounded by the paraboloid $z=x^2+y^2$ and the plane $z=25.$ Let $S$ be the closed boundary of $W$ oriented outward.\n(a) Use the divergence theorem to find the flux of $\\vec{F}$ through $S.$ $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}=$ [ANS]\n(b) Find the flux of $\\vec{F}$ out the bottom of $S$ (the truncated paraboloid) and the top of $S$ (the disk). Flux out the bottom=[ANS]\nFlux out the top=[ANS]", "answer_v2": [ "1562.5*pi", "-312.5*pi", "1875*pi" ], "answer_type_v2": [ "NV", "NV", "NV" ], "options_v2": [ [], [], [] ], "problem_v3": "Suppose $\\vec{F}(x,y,z)=\\langle x, y, 3 z \\rangle$. Let $W$ be the solid bounded by the paraboloid $z=x^2+y^2$ and the plane $z=16.$ Let $S$ be the closed boundary of $W$ oriented outward.\n(a) Use the divergence theorem to find the flux of $\\vec{F}$ through $S.$ $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}=$ [ANS]\n(b) Find the flux of $\\vec{F}$ out the bottom of $S$ (the truncated paraboloid) and the top of $S$ (the disk). Flux out the bottom=[ANS]\nFlux out the top=[ANS]", "answer_v3": [ "640*pi", "-128*pi", "768*pi" ], "answer_type_v3": [ "NV", "NV", "NV" ], "options_v3": [ [], [], [] ] }, { "id": "Calculus_-_multivariable_0673", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "4", "keywords": [ "Divergence Theorem" ], "problem_v1": "Suppose $\\mathrm{div}(\\vec{F}(3, 1, 1))=-160.$ Estimate the flux of $\\vec{F}$ out of a small, outward-oriented box of side length $0.3$ centered at the point $(3,1,1)$ with edges parallel to the coordinate axes.\nFlux $\\approx$ [ANS]", "answer_v1": [ "-4.32" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose $\\mathrm{div}(\\vec{F}(-5, 5,-4))=160.$ Estimate the flux of $\\vec{F}$ out of a small, outward-oriented box of side length $0.2$ centered at the point $(-5,5,-4)$ with edges parallel to the coordinate axes.\nFlux $\\approx$ [ANS]", "answer_v2": [ "1.28" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose $\\mathrm{div}(\\vec{F}(-2, 1,-2))=-170.$ Estimate the flux of $\\vec{F}$ out of a small, outward-oriented box of side length $0.3$ centered at the point $(-2,1,-2)$ with edges parallel to the coordinate axes.\nFlux $\\approx$ [ANS]", "answer_v3": [ "-4.59" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0674", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "2", "keywords": [ "Divergence Theorem" ], "problem_v1": "Suppose $\\vec{F}$ is a vector field with $\\mathrm{div}(\\vec{F}(x,y,z))=5.$ Use the divergence theorem to calculate the flux of the vector field $\\vec{F}$ out of the closed, outward-oriented cylindrical surface $S$ of height $6$ and radius $4$ that is centered about the z-axis with its base in the xy-plane. $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}=$ [ANS]", "answer_v1": [ "480*pi" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Suppose $\\vec{F}$ is a vector field with $\\mathrm{div}(\\vec{F}(x,y,z))=-9.$ Use the divergence theorem to calculate the flux of the vector field $\\vec{F}$ out of the closed, outward-oriented cylindrical surface $S$ of height $3$ and radius $5$ that is centered about the z-axis with its base in the xy-plane. $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}=$ [ANS]", "answer_v2": [ "-675*pi" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Suppose $\\vec{F}$ is a vector field with $\\mathrm{div}(\\vec{F}(x,y,z))=-4.$ Use the divergence theorem to calculate the flux of the vector field $\\vec{F}$ out of the closed, outward-oriented cylindrical surface $S$ of height $4$ and radius $4$ that is centered about the z-axis with its base in the xy-plane. $ \\iint\\limits_S \\vec{F} \\cdot d\\vec{A}=$ [ANS]", "answer_v3": [ "-256*pi" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] }, { "id": "Calculus_-_multivariable_0676", "subject": "Calculus_-_multivariable", "topic": "Fundamental theorems", "subtopic": "Divergence theorem", "level": "2", "keywords": [ "Flux integrals", "Surface integrals" ], "problem_v1": "Calculate the flux of the vector field $\\vec{F}(\\vec{r})=8 \\vec{r}$, where $\\vec{r}=\\langle x, y, z \\rangle$, through a sphere of radius $5$ centered at the origin, oriented outward.\nFlux=[ANS]", "answer_v1": [ "40*4*pi*5^2" ], "answer_type_v1": [ "NV" ], "options_v1": [ [] ], "problem_v2": "Calculate the flux of the vector field $\\vec{F}(\\vec{r})=2 \\vec{r}$, where $\\vec{r}=\\langle x, y, z \\rangle$, through a sphere of radius $7$ centered at the origin, oriented outward.\nFlux=[ANS]", "answer_v2": [ "14*4*pi*7^2" ], "answer_type_v2": [ "NV" ], "options_v2": [ [] ], "problem_v3": "Calculate the flux of the vector field $\\vec{F}(\\vec{r})=4 \\vec{r}$, where $\\vec{r}=\\langle x, y, z \\rangle$, through a sphere of radius $6$ centered at the origin, oriented outward.\nFlux=[ANS]", "answer_v3": [ "24*4*pi*6^2" ], "answer_type_v3": [ "NV" ], "options_v3": [ [] ] } ]