// Numbas version: exam_results_page_options {"name": "Vector Calculus", "metadata": {"description": "

Questions on Vector Calculus

", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], [], [], []], "questions": [{"name": "Lengths of and distance between vectors, dot and cross products, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),3))", "description": "", "name": "a"}, "crossab": {"group": "Ungrouped variables", "templateType": "anything", "definition": "cross(a,b)", "description": "", "name": "crossab"}, "lena": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(abs(a),2)", "description": "", "name": "lena"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "vector(repeat(random(1..9)*sign(random(1,-1)),3))", "description": "", "name": "b"}, "dotab": {"group": "Ungrouped variables", "templateType": "anything", "definition": "dot(a,b)", "description": "", "name": "dotab"}, "sumab": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a+b", "description": "", "name": "sumab"}, "dist": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(abs(a-b),2)", "description": "", "name": "dist"}, "lenb": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(abs(b),2)", "description": "", "name": "lenb"}, "diffab": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a-b", "description": "", "name": "diffab"}}, "ungrouped_variables": ["a", "lenb", "lena", "b", "dist", "dotab", "diffab", "sumab", "crossab"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "lena+0.01", "minValue": "lena-0.01", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "lenb+0.01", "minValue": "lenb-0.01", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "

Their lengths: $a=\\lvert\\boldsymbol{a}\\rvert=$ [[0]], $b=\\lvert\\boldsymbol{b}\\rvert=$ [[1]]. (Enter your answers to 2d.p.)

The distance, $d=$ [[0]], between $\\boldsymbol{a}$ and $\\boldsymbol{b}$, assuming their common initial point is at the origin. (Enter your answer to 2d.p.)

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "sumab[0]", "minValue": "sumab[0]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "sumab[1]", "minValue": "sumab[1]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "sumab[2]", "minValue": "sumab[2]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "diffab[0]", "minValue": "diffab[0]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "diffab[1]", "minValue": "diffab[1]", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "diffab[2]", "minValue": "diffab[2]", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "

Their sum, $\\boldsymbol{a}+\\boldsymbol{b}=($[[0]]$,$[[1]]$,$[[2]]$)$, and difference, $\\boldsymbol{a}-\\boldsymbol{b}=($[[3]]$,$[[4]]$,$[[5]]$)$.

Their dot product $\\boldsymbol{a\\cdot b}=$ [[0]].

Their cross product $\\boldsymbol{a}\\times\\boldsymbol{b}=($[[0]]$,$[[1]]$,$[[2]]$)$.

", "showCorrectAnswer": true, "marks": 0}], "statement": "

Given the vectors $\\boldsymbol{a}=\\pmatrix{\\var{a[0]},\\var{a[1]},\\var{a[2]}}$ and $\\boldsymbol{b}=\\pmatrix{\\var{b[0]},\\var{b[1]},\\var{b[2]}}$ find:

", "tags": ["checked2015", "MAS1602", "MAS2104"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "

Calculations of the lengths of two 3D vectors, the distance between their terminal points, their sum, difference, and dot and cross products.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

For the general 3-component vectors $\\boldsymbol{a}=\\pmatrix{a_1,a_2,a_3}$ and $\\boldsymbol{b}=\\pmatrix{b_1,b_2,b_3}$, we have

Lengths: $a=\\lvert\\boldsymbol{a}\\rvert=\\sqrt{a_1^2+a_2^2+a_3^2}$ and $b=\\lvert\\boldsymbol{b}\\rvert=\\sqrt{b_1^2+b_2^2+b_3^2}$, which are scalar quantities.

Distance between the terminal points: $d=\\sqrt{(a_1-b_1)^2+(a_2-b_2)^2+(a_3-b_3)^2}$, which is a scalar quantity.

Sum $\\boldsymbol{a}+\\boldsymbol{b}=\\pmatrix{a_1+b_1,a_2+b_2,a_3+b_3}$ and difference $\\boldsymbol{a}-\\boldsymbol{b}=\\pmatrix{a_1-b_1,a_2-b_2,a_3-b_3}$, which are vector quantities.

Dot product: $\\boldsymbol{a\\cdot b}=a_1b_1+a_2b_2+a_3b_3$, which is a scalar quantity.

Cross product: $\\boldsymbol{a}\\times\\boldsymbol{b}=\\pmatrix{a_2b_3-a_3b_2,a_3b_1-a_1b_3,a_1b_2-a_2b_1}$, which is a vector quantity.

In this question, therefore, we have:

Lengths: $a=\\lvert\\boldsymbol{a}\\rvert=\\sqrt{\\var{a[0]^2}+\\var{a[1]^2}+\\var{a[2]^2}}=\\var{lena}$ and $b=\\lvert\\boldsymbol{b}\\rvert=\\sqrt{\\var{b[0]^2}+\\var{b[1]^2}+\\var{b[2]^2}}=\\var{lenb}$.

Distance between the terminal points: $d=\\sqrt{(\\simplify[std]{{a[0]}-{b[0]}})^2+(\\simplify[std]{{a[1]}-{b[1]}})^2+(\\simplify[std]{{a[2]}-{b[2]}})^2}=\\var{dist}$.

Sum $\\boldsymbol{a}+\\boldsymbol{b}=\\pmatrix{\\simplify[std]{{a[0]}+{b[0]}},\\simplify[std]{{a[1]}+{b[1]}},\\simplify[std]{{a[2]}+{b[2]}}}=\\pmatrix{\\var{sumab[0]},\\var{sumab[1]},\\var{sumab[2]}}$ and difference $\\boldsymbol{a}-\\boldsymbol{b}=\\pmatrix{\\simplify[std]{{a[0]}-{b[0]}},\\simplify[std]{{a[1]}-{b[1]}},\\simplify[std]{{a[2]}-{b[2]}}}=\\pmatrix{\\var{diffab[0]},\\var{diffab[1]},\\var{diffab[2]}}$.

Dot product: $\\boldsymbol{a\\cdot b}=(\\var{a[0]}\\times\\var{b[0]})+(\\var{a[1]}\\times\\var{b[1]})+(\\var{a[2]}\\times\\var{b[2]})=\\var{dotab}$.

Cross product: $\\boldsymbol{a}\\times\\boldsymbol{b}=\\pmatrix{\\simplify[std]{{a[1]*b[2]}-{a[2]*b[1]}},\\simplify[std]{{a[2]*b[0]}-{a[0]*b[2]}},\\simplify[std]{{a[0]*b[1]}-{a[1]*b[0]}}}=\\pmatrix{\\var{crossab[0]},\\var{crossab[1]},\\var{crossab[2]}}$.

"}, {"name": "Dot and cross product combinations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {}, "ungrouped_variables": [], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"displayType": "radiogroup", "layout": {"type": "all", "expression": ""}, "choices": ["

$\\boldsymbol{(A\\cdot B)\\cdot C}$

", "

$\\boldsymbol{(A\\cdot B)C}$

", "

$\\boldsymbol{(A\\cdot B)\\times C}$

", "

$\\boldsymbol{(A\\times B)\\times C}$

", "

$\\boldsymbol{(A\\times B)\\cdot C}$

"], "showCorrectAnswer": true, "matrix": [[0, 0, 0.4], [0, 0.4, 0], [0, 0, 0.4], [0, 0.4, 0], [0.4, 0, 0]], "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "marks": 0, "scripts": {}, "maxMarks": 0, "type": "m_n_x", "minMarks": 0, "shuffleAnswers": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "answers": ["

Scalar

", "

Vector

", "

Undefined

"], "warningType": "none"}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

Given the vectors $\\boldsymbol{A},\\;\\;\\boldsymbol{B}$ and $\\boldsymbol{C}$ in $3$ dimensional space, state whether the following quantities are scalars, vectors or undefined.

", "tags": ["checked2015", "cross product", "dot product", "inner product", "mas1602", "MAS2104", "scalar product", "scalars", "unused", "vector", "Vector", "vector product", "vectors"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t

15/07/2012:

\n \t\t

Added tags.

\n \t\t

16/07/2012:

\n \t\t

Added tags.

\n \t\t

\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "

Determine if various combinations of vectors are defined or not.

"}, "advice": "\n \n \n

1. $\\boldsymbol{(A\\cdot B)\\cdot C}$ is undefined as $\\boldsymbol{A\\cdot B}$ is a scalar and we cannot take the inner product of a scalar with the vector $\\boldsymbol{C}$.

\n \n \n \n

2. $\\boldsymbol{(A\\cdot B)C}$ is a vector and is a multiple of $\\boldsymbol{C}$ as $\\boldsymbol{A \\cdot B}$ is a scalar.

\n \n \n \n

3. $\\boldsymbol{(A\\cdot B)\\times C}$ is undefined as $\\boldsymbol{A\\cdot B}$ is a scalar and the cross product is only defined between vectors.

\n \n \n \n

4. $\\boldsymbol{(A\\times B)\\times C}$ is a vector as $\\boldsymbol{A \\times B}$ and $\\boldsymbol{C}$ are vectors and the cross product between vectors produces a vector.

\n \n \n \n

5. $\\boldsymbol{(A\\times B)\\cdot C}$ is a scalar as $\\boldsymbol{A \\times B}$ and $\\boldsymbol{C}$ are vectors and the inner or dot product is between vectors and produces a scalar.

\n \n \n "}, {"name": "Ugur's copy of Find gradient of scalar field,", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "tags": ["checked2015", "gradient", "nabla", "partial derivatives", "scalar field"], "metadata": {"description": "

Gradient of $f(x,y,z)$.

Should warn that multiplied terms need * to denote multiplication.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Find $\\boldsymbol{\\nabla}f$ for the following function $f(x,y,z)$.

", "advice": "

This question is simply an exercise in partial differentiation, using the fact that

\\[\\boldsymbol{\\nabla}f=\\pmatrix{\\frac{\\partial f}{\\partial x},\\frac{\\partial f}{\\partial y},\\frac{\\partial f}{\\partial z}}.\\]

Hence

\\[\\boldsymbol{\\nabla}f=\\pmatrix{\\simplify{{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}+{c1*d1*p7*(1-t)}*x^{p7-1}*y^{p8}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p10*t}*x^{p10-1}*y^{p11}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})},\\;\\simplify{{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}+{c1*d1*p8*(1-t)}*x^{p7}*y^{p8-1}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p11*t}*x^{p10}*y^{p11-1}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})},\\; \\simplify{{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}+{c1*d1*p9*(1-t)}*x^{p7}*y^{p8}*z^{p9-1}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p12*t}*x^{p10}*y^{p11}*z^{p12-1}*sin({e1}*x^{p10}*y^{p11}*z^{p12})}}.\\]

", "rulesets": {"std": ["all", "!noLeadingMinus", "!collectNumbers"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p4": {"name": "p4", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "", "templateType": "anything", "can_override": false}, "p12": {"name": "p12", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "p9": {"name": "p9", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "p3": {"name": "p3", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "", "templateType": "anything", "can_override": false}, "p8": {"name": "p8", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "p7": {"name": "p7", "group": "Ungrouped variables", "definition": "if(p8=0 and p9=0,1,random(0,1))", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "templateType": "anything", "can_override": false}, "p5": {"name": "p5", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "", "templateType": "anything", "can_override": false}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "templateType": "anything", "can_override": false}, "p11": {"name": "p11", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "p6": {"name": "p6", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "", "templateType": "anything", "can_override": false}, "p10": {"name": "p10", "group": "Ungrouped variables", "definition": "if(p11=0 and p12=0,1,random(0,1))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["p2", "p3", "p1", "p6", "p7", "p4", "p5", "p8", "p9", "a1", "p11", "p12", "t", "b1", "p10", "c1", "e1", "d1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$f(x,y,z)=\\simplify[std]{{a1}*x^{p1}*y^{p2}*z^{p3}+{b1}*x^{p4}*y^{p5}*z^{p6}+{c1}*({1-t}*sin({d1}*x^{p7}*y^{p8}*z^{p9})+{t}*cos({e1}*x^{p10}*y^{p11}*z^{p12}))}$.

$\\boldsymbol{\\nabla}f=($[[0]]$,$[[1]]$,$[[2]]$)$.

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}+{c1*d1*p7*(1-t)}*x^{p7-1}*y^{p8}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p10*t}*x^{p10-1}*y^{p11}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}+{c1*d1*p8*(1-t)}*x^{p7}*y^{p8-1}*z^{p9}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p11*t}*x^{p10}*y^{p11-1}*z^{p12}*sin({e1}*x^{p10}*y^{p11}*z^{p12})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}+{c1*d1*p9*(1-t)}*x^{p7}*y^{p8}*z^{p9-1}*cos({d1}*x^{p7}*y^{p8}*z^{p9})-{c1*e1*p12*t}*x^{p10}*y^{p11}*z^{p12-1}*sin({e1}*x^{p10}*y^{p11}*z^{p12})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ugur's copy of Find directional derivative of a scalar field,", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "tags": ["checked2015"], "metadata": {"description": "

Directional derivative of a scalar field.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

You are given the scalar field $f=\\simplify{{a1}*x^{p1}*y^{p2}*z^{p3}+{b1}*x^{p4}*y^{p5}*z^{p6}}$.

", "advice": "

Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity.

The gradient of $f$ is given by

\\[\\begin{align}\\boldsymbol{\\nabla}f&=\\pmatrix{\\frac{\\partial f}{\\partial x},\\frac{\\partial f}{\\partial y},\\frac{\\partial f}{\\partial z}}\\\\&=\\pmatrix{\\simplify{{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}},\\simplify{{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}},\\simplify{{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}}},\\end{align}\\]

by straight forward partial differentiation.

The gradient of $f$ at the point $\\boldsymbol{q}=\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}$ is found by substituting $\\boldsymbol{q}$ into $\\boldsymbol{\\nabla}f$, hence

\\[\\boldsymbol{\\nabla}f\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}=\\pmatrix{\\var{gradfq[0]},\\var{gradfq[1]},\\var{gradfq[2]}}.\\]

The unit vector $\\boldsymbol{\\hat{u}}$ in the direction of $\\boldsymbol{u}$ is given by

\\[\\begin{align}\\boldsymbol{\\hat{u}}=\\frac{\\boldsymbol{u}}{\\lvert\\boldsymbol{u}\\rvert}&=\\frac{1}{\\sqrt{(\\var{u[0]})^2+(\\var{u[1]})^2+(\\var{u[2]})^2}}\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}\\\\&=\\frac{1}{\\var{precround(lenu,3)}}\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}\\\\&=\\pmatrix{\\var{precround(uhat[0],3)},\\var{precround(uhat[1],3)},\\var{precround(uhat[2],3)}}.\\end{align}\\]

The directional derivative $\\frac{\\partial f}{\\partial\\boldsymbol{u}}$, of the scalar field $f$, in the direction of $\\boldsymbol{u}$, at the point $\\boldsymbol{q}$ is given by

\\[\\begin{align}\\frac{\\partial f}{\\partial\\boldsymbol{u}}\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}&=\\boldsymbol{\\hat{u}\\cdot\\nabla}f\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}\\\\&=\\pmatrix{\\var{precround(uhat[0],3)},\\var{precround(uhat[1],3)},\\var{precround(uhat[2],3)}}\\boldsymbol{\\cdot}\\pmatrix{\\var{gradfq[0]},\\var{gradfq[1]},\\var{gradfq[2]}}\\\\&=\\var{uhatdotgradfq}\\;\\text{to 3d.p., using the full calculator display for the answers in the previous part.}\\end{align}\\]

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p4": {"name": "p4", "group": "Ungrouped variables", "definition": "random(0..2 except p1)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "vector(repeat(random(1..2)*sign(random(-1,1)),3))", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(2..9)*sign(random(-1,1))", "description": "", "templateType": "anything", "can_override": false}, "u": {"name": "u", "group": "Ungrouped variables", "definition": "vector(repeat(random(1..9)*sign(random(-1,1)),3))", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(2..9)*sign(random(-1,1))", "description": "", "templateType": "anything", "can_override": false}, "lenu": {"name": "lenu", "group": "Ungrouped variables", "definition": "abs(u)", "description": "", "templateType": "anything", "can_override": false}, "p5": {"name": "p5", "group": "Ungrouped variables", "definition": "random(0..2 except p2)", "description": "", "templateType": "anything", "can_override": false}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything", "can_override": false}, "p3": {"name": "p3", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything", "can_override": false}, "uhat": {"name": "uhat", "group": "Ungrouped variables", "definition": "vector(u[0]/lenu,u[1]/lenu,u[2]/lenu)", "description": "", "templateType": "anything", "can_override": false}, "gradfq": {"name": "gradfq", "group": "Ungrouped variables", "definition": "vector(p1*a1*q[0]^(p1-1)*q[1]^p2*q[2]^p3+p4*b1*q[0]^(p4-1)*q[1]^p5*q[2]^p6,p2*a1*q[0]^p1*q[1]^(p2-1)*q[2]^p3+p5*b1*q[0]^p4*q[1]^(p5-1)*q[2]^p6,p3*a1*q[0]^p1*q[1]^p2*q[2]^(p3-1)+p6*b1*q[0]^p4*q[1]^p5*q[2]^(p6-1))", "description": "", "templateType": "anything", "can_override": false}, "p6": {"name": "p6", "group": "Ungrouped variables", "definition": "random(0..2 except p3)", "description": "", "templateType": "anything", "can_override": false}, "uhatdotgradfq": {"name": "uhatdotgradfq", "group": "Ungrouped variables", "definition": "precround(uhat[0]*gradfq[0]+uhat[1]*gradfq[1]+uhat[2]*gradfq[2],3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "b1", "gradfq", "lenu", "p1", "p2", "p3", "p4", "p5", "p6", "q", "u", "uhat", "uhatdotgradfq"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Calculate $\\boldsymbol{\\nabla}f$.

$\\boldsymbol{\\nabla}f=($[[0]]$,$[[1]]$,$[[2]]$)$.

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{p1*a1}*x^{p1-1}*y^{p2}*z^{p3}+{p4*b1}*x^{p4-1}*y^{p5}*z^{p6}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{p2*a1}*x^{p1}*y^{p2-1}*z^{p3}+{p5*b1}*x^{p4}*y^{p5-1}*z^{p6}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{p3*a1}*x^{p1}*y^{p2}*z^{p3-1}+{p6*b1}*x^{p4}*y^{p5}*z^{p6-1}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Calculate $\\boldsymbol{\\nabla}f$ at the point $\\boldsymbol{q}=\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}$.

$\\boldsymbol{\\nabla}f\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}=($[[0]]$,$[[1]]$,$[[2]]$)$.

Calculate the unit vector $\\boldsymbol{\\hat{u}}$ in the direction of $\\boldsymbol{u}=\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}$.

$\\boldsymbol{\\hat{u}}=($[[0]]$,$[[1]]$,$[[2]]$)$. (Enter your answers to 3d.p.)

Calculate the directional derivative $\\frac{\\partial f}{\\partial\\boldsymbol{u}}$, of the scalar field $f$, in the direction of $\\boldsymbol{u}$, at the point $\\boldsymbol{q}$.

$\\frac{\\partial f}{\\partial\\boldsymbol{u}}\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}=$ [[0]]. (Enter your answer to 3d.p., and be sure to use the full calculator display from any previous parts in calculating your answer.)

Divergence of vector fields.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

For each of the following vector fields $\\boldsymbol{F}$, find the divergence $\\boldsymbol{\\nabla\\cdot F}$.

", "advice": "

The divergence of a vector field $\\boldsymbol{u}=\\pmatrix{u_1,u_2,u_3}$ is given by

\\[\\boldsymbol{\\nabla\\cdot u}=\\frac{\\partial u_1}{\\partial x}+\\frac{\\partial u_2}{\\partial y}+\\frac{\\partial u_3}{\\partial z}.\\]

The variables $x$, $y$, and $z$ appear in a cyclical manner in each of the three components of $\\boldsymbol{u}$. Once you have calculated $\\frac{\\partial u_1}{\\partial x}$, you can use cyclic permutations to determine the other two derivatives. Hence

\\[\\begin{align}\\frac{\\partial u_1}{\\partial x}&=\\frac{\\partial}{\\partial x}\\left(\\simplify{({a1}*x+{b1}*y+{c1}*z)*({b1}*y-{c1}*z)}\\right)\\\\&=\\simplify{{a1}*({b1}*y-{c1}*z)},\\end{align}\\]

and so, cyclically permuting the variables,

\\[\\frac{\\partial u_2}{\\partial y}=\\simplify{{a1}*({b1}*z-{c1}*x)}\\]

and

\\[\\frac{\\partial u_3}{\\partial z}=\\simplify{{a1}*({b1}*x-{c1}*y)}.\\]

Finally, adding the components together gives the divergence

\\[\\boldsymbol{\\nabla\\cdot u}=\\simplify[std]{{a1}*({b1}*y-{c1}*z)+{a1}*({b1}*z-{c1}*x)+{a1}*({b1}*x-{c1}*y)}=\\simplify{{a1*b1-a1*c1}*x+{a1*b1-a1*c1}*y+{a1*b1-a1*c1}*z}.\\]

As in part a) the variables $x$, $y$, and $z$ appear cyclically in each component of $\\boldsymbol{u}$, so we only need calculate one derivative explicitly, then use cyclic permutations to calculate the other two. Hence

\\[\\begin{align}\\frac{\\partial u_1}{\\partial x}&=\\frac{\\partial}{\\partial x}\\left\\{\\left(x^\\var{p1}+y^\\var{p1}\\right)\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right)\\right\\}\\\\&=\\simplify{{p1}*x^{p1-1}}\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right),\\end{align}\\]

and by symmetry

\\[\\frac{\\partial u_2}{\\partial y}=\\simplify{{p1}*y^{p1-1}}\\left(\\simplify{{a2}*z^{p1-1}-{a2}*x^{p1-1}}\\right),\\]

and

\\[\\frac{\\partial u_3}{\\partial z}=\\simplify{{p1}*z^{p1-1}}\\left(\\simplify{{a2}*x^{p1-1}-{a2}*y^{p1-1}}\\right).\\]

Finally, adding the derivatives together gives the divergence

\\[\\boldsymbol{\\nabla\\cdot u}=\\simplify{{p1}*x^{p1-1}}\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right)+\\simplify{{p1}*y^{p1-1}}\\left(\\simplify{{a2}*z^{p1-1}-{a2}*x^{p1-1}}\\right)+\\simplify{{p1}*z^{p1-1}}\\left(\\simplify{{a2}*x^{p1-1}-{a2}*y^{p1-1}}\\right)=0.\\]

First note that $f_1$ does not depend on $z$, $f_2$ does not depend on $y$, and $f_3$ does not depend on $z$. This makes the differentiation very straight forward, and hence

\\[\\begin{align}\\boldsymbol{\\nabla\\cdot u}&=\\frac{\\partial}{\\partial x}\\left(\\simplify{{a3}*x}+f_1(y,z)\\right)+\\frac{\\partial}{\\partial y}\\left(\\simplify{{b3}*y}+f_2(x,z)\\right)+\\frac{\\partial}{\\partial z}\\left(\\simplify{{c3}*z}+f_3(x,y)\\right)\\\\&=\\simplify[all,!collectNumbers]{{a3}+{b3}+{c3}}\\\\&=\\var{a3+b3+c3}.\\end{align}\\]

$\\boldsymbol{F}=\\pmatrix{(\\simplify[std]{{a1}*x+{b1}*y+{c1}*z})(\\simplify[std]{{b1}*y-{c1}*z}),(\\simplify[std]{{a1}*y+{b1}*z+{c1}*x})(\\simplify[std]{{b1}*z-{c1}*x}),(\\simplify[std]{{a1}*z+{b1}*x+{c1}*y})(\\simplify[std]{{b1}*x-{c1}*y})}$.

$\\boldsymbol{\\nabla\\cdot F}=$ [[0]].

$\\boldsymbol{F}=\\pmatrix{\\left(x^\\var{p1}+y^\\var{p1}\\right)\\left(\\simplify{{a2}*y^{p1-1}-{a2}*z^{p1-1}}\\right),\\left(y^\\var{p1}+z^\\var{p1}\\right)\\left(\\simplify{{a2}*z^{p1-1}-{a2}*x^{p1-1}}\\right),\\left(z^\\var{p1}+x^\\var{p1}\\right)\\left(\\simplify{{a2}*x^{p1-1}-{a2}*y^{p1-1}}\\right)}$.

$\\boldsymbol{\\nabla\\cdot F}=$ [[0]].

$\\boldsymbol{F}=\\pmatrix{\\simplify{{a3}*x}+f_1(y,z),\\simplify{{b3}*y}+f_2(x,z),\\simplify{{c3}*z}+f_3(x,y)}$, for any general functions $f_1$, $f_2$, and $f_3$.

$\\boldsymbol{\\nabla\\cdot F}=$ [[0]].

Curl and divergence of a vector field. Determine whether the vector field is irrotational or solenoidal.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

For the vector field $\\boldsymbol{F}=\\pmatrix{\\simplify{{a1}*x^{p1}+{b1}*y^{p2}*z^{p3}},\\simplify{{c1}*y^{p4}+{d1}*x^{p5}*z^{p6}},\\simplify{{e1}*z^{p7}+{f1}*x^{p8}*y^{p9}}}$, calculate $\\boldsymbol{\\nabla}\\times\\boldsymbol{u}$ and $\\boldsymbol{\\nabla\\cdot u}$, and determine whether $\\boldsymbol{u}$ is irrotational or solenoidal, or both.

", "advice": "

The curl of a vector field $\\boldsymbol{F}=\\pmatrix{u_x,u_y,u_z}$ is given by

\\[\\boldsymbol{\\nabla}\\times\\boldsymbol{F}=\\pmatrix{\\frac{\\partial u_z}{\\partial y}-\\frac{\\partial u_y}{\\partial z},\\frac{\\partial u_x}{\\partial z}-\\frac{\\partial u_z}{\\partial x},\\frac{\\partial u_y}{\\partial x}-\\frac{\\partial u_x}{\\partial y}}.\\]

The divergence of the same vector field is given by

\\[\\boldsymbol{\\nabla\\cdot F}=\\frac{\\partial u_x}{\\partial x}+\\frac{\\partial u_y}{\\partial y}+\\frac{\\partial u_z}{\\partial z}.\\]

a)

By straightforward partial differentiation

\\[\\boldsymbol{\\nabla\\times F}=\\pmatrix{\\simplify{{f1*p9}*x^{p8}*y^{p9-1}+{-d1*p6}*x^{p5}*z^{p6-1}},\\simplify{{b1*p3}*y^{p2}*z^{p3 -1}+{-f1*p8}*x^{p8-1}*y^{p9}},\\simplify{{d1*p5}*x^{p5-1}*z^{p6}+{-b1*p2}*y^{p2-1}*z^{p3}}}.\\]

b)

Again, by partial differentiation

\\[\\boldsymbol{\\nabla\\cdot F}=\\simplify{{a1*p1}*x^{p1-1}+{c1*p4}*y^{p4-1}+{e1*p7}*z^{p7-1}}.\\]

A vector field is irrotational if its curl is equal to the zero vector; a vector field is solenoidal if its divergence is equal to zero.

c)

Since $\\boldsymbol{\\nabla}\\times\\boldsymbol{F}$ {irrequal} to the zero vector, the vector field {isirr}.

d)

Since $\\boldsymbol{\\nabla\\cdot F}$ {solequal} to zero, the vector field {issol}.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"isirr": {"name": "isirr", "group": "Ungrouped variables", "definition": "if(n=1,\"is irrotational\",\"is not irrotational\")", "description": "", "templateType": "anything", "can_override": false}, "p8": {"name": "p8", "group": "Ungrouped variables", "definition": "random(1..9 except p3)", "description": "", "templateType": "anything", "can_override": false}, "p7": {"name": "p7", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "notsolenoidal": {"name": "notsolenoidal", "group": "Ungrouped variables", "definition": "if(n=2,\"No\",\"Yes\")", "description": "", "templateType": "anything", "can_override": false}, "p9": {"name": "p9", "group": "Ungrouped variables", "definition": "random(1..9 except p6)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "if(n=2,0,random(1..9))", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "if(n=1,0,random(1..9))", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "if(n=1,0,random(1..9))", "description": "", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "if(n=2,0,random(1..9))", "description": "", "templateType": "anything", "can_override": false}, "p4": {"name": "p4", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "p6": {"name": "p6", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "if(n=2,0,random(1..9))", "description": "", "templateType": "anything", "can_override": false}, "p3": {"name": "p3", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "if(n=1,0,random(1..9))", "description": "", "templateType": "anything", "can_override": false}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "solequal": {"name": "solequal", "group": "Ungrouped variables", "definition": "if(n=2,\"is equal\",\"is not equal\")", "description": "", "templateType": "anything", "can_override": false}, "issol": {"name": "issol", "group": "Ungrouped variables", "definition": "if(n=2,\"is solenoidal\",\"is not solenoidal\")", "description": "", "templateType": "anything", "can_override": false}, "notirrotational": {"name": "notirrotational", "group": "Ungrouped variables", "definition": "if(n=1,\"No\",\"Yes\")", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything", "can_override": false}, "solenoidal": {"name": "solenoidal", "group": "Ungrouped variables", "definition": "if(n=2,\"Yes\",\"No\")", "description": "", "templateType": "anything", "can_override": false}, "irrotational": {"name": "irrotational", "group": "Ungrouped variables", "definition": "if(n=1,\"Yes\",\"No\")", "description": "", "templateType": "anything", "can_override": false}, "irrequal": {"name": "irrequal", "group": "Ungrouped variables", "definition": "if(n=1,\"is equal\",\"is not equal\")", "description": "", "templateType": "anything", "can_override": false}, "p5": {"name": "p5", "group": "Ungrouped variables", "definition": "random(1..9 except p2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f1", "irrequal", "isirr", "b1", "d1", "issol", "e1", "irrotational", "a1", "c1", "solenoidal", "p2", "p3", "solequal", "p1", "p6", "p7", "p4", "p5", "notsolenoidal", "p8", "p9", "notirrotational", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$\\boldsymbol{\\nabla}\\times\\boldsymbol{F}=($[[0]]$,$[[1]]$,$[[2]]$)$.

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{f1*p9}*x^{p8}*y^{p9-1}+{-d1*p6}*x^{p5}*z^{p6-1}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{b1*p3}*y^{p2}*z^{p3 -1}+{-f1*p8}*x^{p8-1}*y^{p9}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{d1*p5}*x^{p5-1}*z^{p6}+{-b1*p2}*y^{p2-1}*z^{p3}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$\\boldsymbol{\\nabla\\cdot F}=$ [[0]].

Is the vector field $\\boldsymbol{F}$ irrotational?

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["

{irrotational}

", "

{notirrotational}

"], "matrix": [1, 0], "distractors": ["", ""]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Is the vector field $\\boldsymbol{F}$ solenoidal?

", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["

{solenoidal}

", "

{notsolenoidal}

"], "matrix": [1, 0], "distractors": ["", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ugur's copy of Parametric representations of a curve - find tangent and length", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "tags": ["checked2015"], "metadata": {"description": "

Calculation of the length and alternative form of the parameteric representation of a curve, involving trigonometric functions.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

You are given the curve $\\mathcal C$ be the space curve parametrically given by $\\boldsymbol{r} = \\pmatrix{\\var{a}\\cos(\\simplify{{b}t}),\\var{-a}\\sin(\\simplify{{b}t})}$, where $t_1\\pi\\leqslant t\\leqslant t_2\\pi$.

", "advice": "

a) The tangent vector to the curve is given by $\\boldsymbol{u} = \\frac{\\mathrm{d}r}{\\mathrm{d}t} \\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}} =
\\pmatrix{\\simplify{{-a*b}sin({b}*t)}, \\simplify{{-a*b}*cos({b}*t)}} $ (note that, since the curve is not parametrised by arc length, $\\frac{\\mathrm{d}r}{\\mathrm{d}t}$ does not give the unit tangent!).

b) The length $s$ of the curve in the range $t_1\\pi\\leqslant t\\leqslant t_2\\pi$ is given by

\\[s = \\int_{t_1\\pi}^{t_2\\pi}{u\\mathrm{d}t},\\]

where $u^2=\\lvert\\boldsymbol{u}\\rvert^2=\\boldsymbol{u\\cdot u}$.

In this question, therefore, $\\boldsymbol{u}=\\pmatrix{\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left\\{\\var{a}\\cos(\\simplify{{b}*t})\\right\\},\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left\\{\\var{-a}\\sin(\\simplify{{b}*t})\\right\\}}=\\pmatrix{\\var{-a*b}\\sin(\\simplify{{b}*t}),\\var{-a*b}\\cos(\\simplify{{b}*t})}$, and so $u^2=\\var{(a*b)^2}$.

Then

\\[s=\\int_{t_1\\pi}^{t_2\\pi}{u\\mathrm{d}t}=\\var{a*b}\\int_{t_1\\pi}^{t_2\\pi}{\\mathrm{d}t}=\\var{a*b}(t_2-t_1)\\pi.\\]

Finally, substitute $t_1=-\\simplify{1/{t2}}$ and $t_2=\\simplify{1/{t2}}$ into this expression for $s$, to find the length of the curve over the given range of $t$.

Hence $s=\\simplify{{2*a*b}/{t2}}\\pi$.

c) An alternative parametric representation, using $s$ as the curve parameter is given by

\\[s=\\int_{t_1\\pi}^{t}{u\\mathrm{d}\\tau}=\\var{a*b}\\int_{t_1\\pi}^{t}{\\mathrm{d}\\tau}=\\var{a*b}(t-t_1\\pi).\\]

Now rearrange this expression for $t(s)$, so

\\[t(s)=\\frac{s}{\\var{a*b}}+t_1\\pi,\\]

and substitute into the original representation of the curve $\\mathcal C\\equiv\\pmatrix{\\var{a}\\cos(\\simplify{{b}t}),\\var{-a}\\sin(\\simplify{{b}t})}$ with $t_1\\pi\\leqslant t\\leqslant t_2\\pi$. Hence

\\[\\mathcal C \\equiv\\pmatrix{\\var{a}\\cos\\left(\\simplify{s/{a}}+\\var{b}t_1\\pi\\right),\\var{-a}\\sin\\left(\\simplify{s/{a}}+\\var{b}t_1\\pi\\right)},\\]

with $0\\leqslant s\\leqslant\\var{a*b}(t_2-t_1)\\pi$.

Finally, substitute $t_1=-\\simplify{1/{t2}}$ and $t_2=\\simplify{1/{t2}}$ into the above expressions, to find the specific parametric representation corresponding to the given range of t:

\\[\\mathcal C \\equiv\\pmatrix{\\var{a}\\cos\\left(\\simplify{s/{a}}-\\simplify{{b*pi}/{t2}}\\right),\\var{-a}\\sin\\left(\\simplify{s/{a}}-\\simplify{{b*pi}/{t2}}\\right)},\\]

with $0\\leqslant s\\leqslant \\simplify{{2*a*b}/{t2}}\\pi$.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"t2": {"name": "t2", "group": "Ungrouped variables", "definition": "random(1,2,4)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "t2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find the tangent vector $\\boldsymbol{u}$ to the curve.

$\\boldsymbol{u}=($[[0]]$,$[[1]]$)$. (Do not enter decimals in your answers.)

Find the length $s$ of the curve between $t_1=-\\simplify{1/{t2}}$ and $t_2=\\simplify{1/{t2}}$.

$s=$ [[0]]. (Enter your answer as a fractional multiple of $\\pi$. Do not enter decimals.)

Do not enter decimals in your answer.

"}, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find another parametric representation of the curve, again with $t_1=-\\simplify{1/{t2}}$ and $t_2=\\simplify{1/{t2}}$, using $s$ as the curve parameter, such that $0\\leqslant s\\leqslant \\simplify{{2*a*b}/{t2}}\\pi$.

$s\\rightarrow($[[0]]$,$[[1]]$)$. (Enter your answers as fractional multiples of $\\pi$. Do not enter decimals.)

Enter your answer as a fractional multiple of $\\pi$. Do not enter decimals.

"}, "valuegenerators": [{"name": "s", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{-a}*sin(s/{a}-{b}*pi/{t2})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "

Enter your answer as a fractional multiple of $\\pi$. Do not enter decimals.

"}, "valuegenerators": [{"name": "s", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ugur's copy of Find points of intersection, tangents, and angles between parametric curves", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}], "tags": ["checked2015", "intersection of curves", "parametric curves", "tangent vectors"], "metadata": {"description": "

Intersection points, tangent vectors, angles between pairs of curves, given in parametric form.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

The pair of curves

\\[\\begin{align}\\mathcal{C}_1&:t\\mapsto\\pmatrix{\\simplify{{a1}*t},\\simplify{{b1}*t},\\simplify{{c1}*t}},-\\infty\\leqslant t\\leqslant\\infty\\\\\\mathcal{C}_2&:\\tau\\mapsto\\pmatrix{\\simplify{{d1}*tau},\\simplify{{e1}*tau^2},\\simplify{{f1}*tau^3}},-\\infty\\leqslant \\tau\\leqslant\\infty\\end{align}\\]

intersect at two distinct points $\\boldsymbol{p}$ and $\\boldsymbol{q}$.

", "advice": "

Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity.

The two curves $\\mathcal{C}_1$ and $\\mathcal{C}_2$ intersect where

\\[\\begin{align}\\simplify{{a1}*t}&=\\simplify{{d1}*tau}\\tag{1},\\\\\\simplify{{b1}t}&=\\simplify{{e1}*tau^2},\\tag{2}\\\\\\simplify{{c1}*t}&=\\simplify{{f1}*tau^3}.\\tag{3}\\end{align}\\]

From equation (1)

\\[\\tau=\\frac{\\var{a1}}{\\var{d1}}t=\\simplify{{a1}/{d1}t},\\tag{4}\\]

which we substitute into equation (2) to determine that

\\[\\var{b1}t=\\var{e1}\\times\\left(\\simplify{{a1}/{d1}t}\\right)^2=\\simplify{{e1*a1^2}/{d1^2}t^2}.\\]

Then either $t=0$ or $t=\\simplify{{b1*d1^2}/{e1*a1^2}}$.

Substitute these two expressions into equation (4), then either $\\tau=0$ (when $t=0$), or $\\tau=\\simplify{{b1*d1}/{e1*a1}}$ (when $t=\\var{t}$).

(As a check, substitute these pairs of values into equation (3), to show that equality holds.)

To determine the intersection points $\\boldsymbol{p}$ and $\\boldsymbol{q}$, substitute the values of $t$ and $\\tau$ into either expression for the curves $\\mathcal{C}_1$ and $\\mathcal{C}_2$.

The point $\\boldsymbol{p}$ is given by the least value of $t$, which is $t=0$ (and correspondingly $\\tau=0$). The point $\\boldsymbol{p}$ is therefore $\\boldsymbol{p}=\\pmatrix{0,0,0}$.

The point $\\boldsymbol{q}$ is given by the greatest value of $t$, which is $t=\\var{t}$ (and correspondingly $\\tau=\\var{tau}$). The point $\\boldsymbol{q}$ is therefore $\\boldsymbol{q}=\\pmatrix{\\var{a1}\\times\\var{t},\\var{b1}\\times\\var{t},\\var{c1}\\times\\var{t}}=\\pmatrix{\\var{q[0]},\\var{q[1]},\\var{q[2]}}$.

In general, the tangent vector $\\boldsymbol{u}$, of a curve $t\\rightarrow\\pmatrix{x(t),y(t),z(t)}$, is given by $\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t},\\frac{\\mathrm{d}z}{\\mathrm{d}t}}$.

The tangent vector of the curve $\\mathcal{C}_1$ is therefore given by $\\boldsymbol{u}=\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}$, which is constant, and independent of $t$.

The tangent vector of $\\mathcal{C}_2$ is given by $\\pmatrix{\\var{d1},\\var{2*e1}\\tau,\\var{3*f1}\\tau^2}$, so the tangent vector at the point $\\boldsymbol{p}$ (where $\\tau=0$) is given by $\\boldsymbol{v}=\\pmatrix{\\var{v[0]},\\var{v[1]},\\var{v[2]}}$.

In a similar way, the tangent vector of $\\mathcal{C}_2$ at the point $\\boldsymbol{q}$ (where $\\tau=\\var{tau}$) is given by $\\boldsymbol{w}=\\pmatrix{\\var{w[0]},\\var{w[1]},\\var{w[2]}}$.

The angle $\\theta$ between any two vectors $\\boldsymbol{a}$ and $\\boldsymbol{b}$ can be calculated using

\\[\\cos(\\theta)=\\frac{\\boldsymbol{a\\cdot b}}{\\lvert\\boldsymbol{a}\\rvert\\lvert\\boldsymbol{b}\\rvert},\\]

where $\\lvert\\boldsymbol{x}\\rvert=\\sqrt{x_1^2+x_2^2+x_3^2}$ is the length of the vector $\\boldsymbol{x}$.

The angle $\\theta$ between the tangent vectors at the point $\\boldsymbol{p}$ is the angle between the vectors $\\boldsymbol{u}$ and $\\boldsymbol{v}$, so

\\[\\cos(\\theta)=\\frac{(\\var{u[0]}\\times\\var{v[0]})+(\\var{u[1]}\\times\\var{v[1]})+(\\var{u[2]}\\times\\var{v[2]})}{\\sqrt{(\\var{u[0]})^2+(\\var{u[1]})^2+(\\var{u[2]})^2}\\sqrt{(\\var{v[0]})^2+(\\var{v[1]})^2+(\\var{v[2]})^2}}=\\frac{\\var{dotuv}}{\\var{precround(lenu,4)}\\times\\var{precround(lenv,4)}}=\\var{precround(dotuv/(lenu*lenv),4)}\\;\\text{to 4d.p.}\\]

Then $\\theta=\\arccos(\\var{precround(dotuv/(lenu*lenv),4)})=\\var{theta}^\\circ$ to 2d.p.

In an identical way, the angle $\\phi$ between the tangent vectors at the point $\\boldsymbol{q}$ is the angle between the vectors $\\boldsymbol{u}$ and $\\boldsymbol{w}$, so $\\phi=\\var{phi}^\\circ$ to 2d.p.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"lenu": {"name": "lenu", "group": "Ungrouped variables", "definition": "abs(u)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "vector(a1*t,b1*t,c1*t)", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "vector(d1,0,0)", "description": "", "templateType": "anything", "can_override": false}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "vector(d1,2*e1*tau,3*f1*tau^2)", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "(e1*a1^2)*t/d1^2", "description": "", "templateType": "anything", "can_override": false}, "phi": {"name": "phi", "group": "Ungrouped variables", "definition": "precround(degrees(arccos(dotuw/(lenu*lenw))),2)", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "dotuv": {"name": "dotuv", "group": "Ungrouped variables", "definition": "dot(u,v)", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "dotuw": {"name": "dotuw", "group": "Ungrouped variables", "definition": "dot(u,w)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "(f1*d1*b1^2)/(a1*e1^2)", "description": "", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "lenw": {"name": "lenw", "group": "Ungrouped variables", "definition": "abs(w)", "description": "", "templateType": "anything", "can_override": false}, "theta": {"name": "theta", "group": "Ungrouped variables", "definition": "precround(degrees(arccos(dotuv/(lenu*lenv))),2)", "description": "", "templateType": "anything", "can_override": false}, "lenv": {"name": "lenv", "group": "Ungrouped variables", "definition": "abs(v)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "d1*random(-2..2 except 0)", "description": "", "templateType": "anything", "can_override": false}, "tau": {"name": "tau", "group": "Ungrouped variables", "definition": "a1*t/d1", "description": "", "templateType": "anything", "can_override": false}, "u": {"name": "u", "group": "Ungrouped variables", "definition": "vector(a1,b1,c1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f1", "phi", "lenu", "dotuw", "tau", "e1", "dotuv", "a1", "u", "t", "w", "v", "lenw", "lenv", "d1", "q", "theta", "c1", "b1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Enter the least value of $t$, and the corresponding value of $\\tau$, defining the first intersection point. Hence enter the values of the intersection point $\\boldsymbol{p}$ for these values of $t$ and $\\tau$.

$t=$ [[0]]; $\\tau=$ [[1]].

$\\boldsymbol{p}=($[[2]]$,$[[3]]$,$[[4]]$)$.

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Enter the greatest value of $t$, and the corresponding value of $\\tau$, defining the second intersection point. Hence enter the values of the intersection point $\\boldsymbol{q}$ for these values of $t$ and $\\tau$.

$t=$ [[0]]; $\\tau=$ [[1]].

$\\boldsymbol{q}=($[[2]]$,$[[3]]$,$[[4]]$)$.

", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "t", "maxValue": "t", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "tau", "maxValue": "tau", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "q[0]", "maxValue": "q[0]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "q[1]", "maxValue": "q[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "q[2]", "maxValue": "q[2]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find the tangent vector $\\boldsymbol{u}$ of the curve $\\mathcal{C}_1$.

$\\boldsymbol{u}=($[[0]]$,$[[1]]$,$[[2]]$)$.

Find the tangent vector $\\boldsymbol{v}$ of the curve $\\mathcal{C}_2$ at the point $\\boldsymbol{p}$.

$\\boldsymbol{v}=($[[0]]$,$[[1]]$,$[[2]]$)$.

Find the tangent vector $\\boldsymbol{w}$ of the curve $\\mathcal{C}_2$ at the point $\\boldsymbol{q}$.

$\\boldsymbol{w}=($[[0]]$,$[[1]]$,$[[2]]$)$.

Calculate the angle $\\theta$ (in degrees) between the tangent vectors of each curve, at the point $\\boldsymbol{p}$.

$\\theta=$ [[0]]$^\\circ$. (Enter your answer to 2d.p.)

Calculate the angle $\\phi$ (in degrees) between the tangent vectors of each curve, at the point $\\boldsymbol{q}$.

$\\phi=$ [[0]]$^\\circ$. (Enter your answer to 2d.p.)

Parametric form of a curve, cartesian points, tangent vector, and speed.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

You are given the following curve, $t\\mapsto\\pmatrix{\\simplify{{a1}*cos({b1}t)},\\simplify{{c1}*sin({d1}t)}}$, defined with respect to the parameter $t$.

", "advice": "

To find the coordinates of the point corresponding to $t=\\var{e1}$, substitute $t=\\var{e1}$ into the expression for the curve, i.e.

\\[\\pmatrix{x,y}=\\pmatrix{\\simplify{{a1}*cos({b1*e1})},\\simplify{{c1}*cos({d1*e1})}}=\\pmatrix{\\var{x},\\var{y}}.\\]

Differentiate each component of the vector in part a) to find the tangent vector $\\boldsymbol{u}$, i.e.

\\[\\boldsymbol{u}=\\pmatrix{\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\simplify{{a1}*cos({b1}*t)}\\right),\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\simplify{{c1}*sin({d1}*t)}\\right)}=\\pmatrix{\\simplify{{-a1*b1}*sin({b1}*t)},\\simplify{{c1*d1}*cos({d1}*t)}}.\\]

The tangent vector at $t=\\var{e1}$ is found by substituting $t=\\var{e1}$ into the tangent vector $\\boldsymbol{u}$, i.e.

\\[\\boldsymbol{u}\\vert_{t=\\var{e1}}=\\pmatrix{\\simplify{{-a1*b1}*sin({b1*e1})},\\simplify{{c1*d1}*cos({d1*e1})}}=\\pmatrix{\\var{dxdte1},\\var{dydte1}}.\\]

The velocity $u$ is given by $u=\\lvert\\boldsymbol{u}\\rvert=\\sqrt{\\left(\\frac{\\mathrm{d}x}{\\mathrm{d}t}\\right)^2+\\left(\\frac{\\mathrm{d}y}{\\mathrm{d}t}\\right)^2}$. We must calculate the speed at $t=\\var{f1}$, however, therefore

\\[u\\vert_{t=\\var{f1}}=\\sqrt{\\left(\\simplify{{-a1*b1}*sin({b1*f1})}\\right)^2+\\left(\\simplify{{c1*d1}*cos({d1*f1})}\\right)^2}=\\sqrt{\\var{dxdtf1}^2+\\var{dydtf1}^2}=\\var{speed} \\; \\text{to 3d.p.}\\]

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"speed": {"name": "speed", "group": "Ungrouped variables", "definition": "precround(sqrt(dxdtf1^2+dydtf1^2),3)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "precround(a1*cos(b1*e1),3)", "description": "", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "templateType": "anything", "can_override": false}, "dydte1": {"name": "dydte1", "group": "Ungrouped variables", "definition": "precround(c1*d1*cos(d1*e1),3)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "templateType": "anything", "can_override": false}, "dydtf1": {"name": "dydtf1", "group": "Ungrouped variables", "definition": "c1*d1*cos(d1*f1)", "description": "", "templateType": "anything", "can_override": false}, "dxdtf1": {"name": "dxdtf1", "group": "Ungrouped variables", "definition": "-a1*b1*sin(b1*f1)", "description": "", "templateType": "anything", "can_override": false}, "dxdte1": {"name": "dxdte1", "group": "Ungrouped variables", "definition": "precround(-a1*b1*sin(b1*e1),3)", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "precround(c1*sin(d1*e1),3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f1", "dxdte1", "dxdtf1", "dydte1", "dydtf1", "a1", "b1", "y", "x", "c1", "e1", "speed", "d1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find the coordinates $\\pmatrix{x,y}$ of the point corresponding to $t=\\var{e1}$.

$\\pmatrix{x,y}=($[[0]]$,$[[1]]$)$. (Enter your answers to 3d.p.)

Find the components of the tangent vector $\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}}$.

$\\boldsymbol{u}=($[[0]]$,$[[1]]$)$.

The components of the same tangent vector, given $t=\\var{e1}$.

$\\boldsymbol{u}|_{t=\\var{e1}}=($[[2]]$,$[[3]]$)$. (Enter your answers to 3d.p.)

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{-a1*b1}*sin({b1}*t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{c1*d1}*cos({d1}*t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "dxdte1-0.001", "maxValue": "dxdte1+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "dydte1-0.001", "maxValue": "dydte1+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Interpreting $t$ as time, and hence the tangent vector $\\boldsymbol{u}$ as velocity, find the speed $u=|\\boldsymbol{u}|$ at $t=\\var{f1}$.

$u=$ [[0]]. (Enter your answer to 3d.p.)

Parametric form of a curve, cartesian points, tangent vector, and speed.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

You are given the following curve, $t\\mapsto\\pmatrix{t^\\var{a},\\simplify{{b}t}}$, defined with respect to the parameter $t$.

", "advice": "

To find the coordinates of the point corresponding to $t=\\var{c}$, substitute $t=\\var{c}$ into the expression for the curve, i.e.

\\[\\pmatrix{x,y}=\\pmatrix{\\var{c}^\\var{a},\\var{b}\\times\\var{c}}=\\pmatrix{\\var{x},\\var{y}}.\\]

Differentiate each component of the vector in part a) to find the tangent vector $\\boldsymbol{u}$, i.e.

\\[\\boldsymbol{u}=\\pmatrix{\\frac{\\mathrm{d}}{\\mathrm{d}t}t^\\var{a},\\frac{\\mathrm{d}}{\\mathrm{d}t}\\simplify{{b}t}}=\\pmatrix{\\var{a}t^\\var{a-1},\\var{b}}.\\]

The tangent vector at $t=\\var{c}$ is found by substituting $t=\\var{c}$ into the tangent vector $\\boldsymbol{u}$, i.e.

\\[\\boldsymbol{u}\\vert_{t=\\var{c}}=\\pmatrix{\\var{a}\\times\\var{c}^\\var{a-1},\\var{b}}=\\pmatrix{\\var{dxdtc},\\var{b}}.\\]

\\[u\\vert_{t=\\var{d}}=\\sqrt{\\left(\\var{a}\\times\\var{d}^\\var{a-1}\\right)^2+\\var{b}^2}=\\sqrt{\\var{dxdtd^2}+\\var{b^2}}=\\var{speed} \\; \\text{to 2d.p.}\\]

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"speed": {"name": "speed", "group": "Ungrouped variables", "definition": "precround(sqrt(dxdtd^2+b^2),2)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "c^a", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..10)*sign(random(1,-1))", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "dxdtd": {"name": "dxdtd", "group": "Ungrouped variables", "definition": "a*d^(a-1)", "description": "", "templateType": "anything", "can_override": false}, "dxdtc": {"name": "dxdtc", "group": "Ungrouped variables", "definition": "a*c^(a-1)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "b*c", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d", "dxdtc", "dxdtd", "y", "x", "speed"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Find the coordinates $\\pmatrix{x,y}$ of the point corresponding to $t=\\var{c}$.

$\\pmatrix{x,y}=($[[0]]$,$[[1]]$)$.

Find the components of the tangent vector $\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}}$.

$\\boldsymbol{u}=($[[0]]$,$[[1]]$)$.

The components of the same tangent vector, given $t=\\var{c}$.

$\\boldsymbol{u}|_{t=\\var{c}}=($[[2]]$,$[[3]]$)$.

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}*t^{a-1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{b}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "dxdtc", "maxValue": "dxdtc", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b", "maxValue": "b", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Interpreting $t$ as time, and hence the tangent vector $\\boldsymbol{u}$ as velocity, find the speed $u=|\\boldsymbol{u}|$ at $t=\\var{d}$.

$u=$ [[0]]. (Enter your answer to 2d.p.)