// Numbas version: exam_results_page_options
{"name": "Cartesian parameterisation of surface, normal vector, and magnitude", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "name": "a", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "name": "t", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "name": "d", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "name": "c", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "b", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "t"], "name": "Cartesian parameterisation of surface, normal vector, and magnitude", "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Express the surface in the form $z=g(x,y)$.</p>\n        <p>$z=g(x,y)=$ [[0]]. &nbsp;(Do not enter decimals in your answer.)</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "({d}/{a^2})*x^2+({d}/{c^2})*y^2", "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "notallowed": {"showStrings": false, "message": "<p>Do not enter decimals in your answer.</p>", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p><strong>Using the above Cartesian form</strong>&nbsp;$z=g(x,y)$&nbsp;of the surface you have just found, find a normal vector $\\boldsymbol{n}$ with $z$-component equal to $1$.</p>\n        <p>$\\boldsymbol{n}=($[[0]]$,$[[1]]$,$[[2]]$)$. &nbsp;(Do not enter decimals in your answer.)</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "({-2d}/{a^2})*x", "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "notallowed": {"showStrings": false, "message": "<p>Do not enter decimals in your answer.</p>", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"answer": "({-2d}/{c^2})*y", "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "notallowed": {"showStrings": false, "message": "<p>Do not enter decimals in your answer.</p>", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}, {"answer": "1", "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "notallowed": {"showStrings": false, "message": "<p>Do not enter decimals in your answer.</p>", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Calculate the magnitude $|\\boldsymbol{n}|$ of the normal vector $\\boldsymbol{n}$.</p>\n        <p>$|\\boldsymbol{n}|=$ [[0]]. &nbsp;(Do not enter decimals in your answer.)</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "sqrt(({4*d^2}/{a^4})x^2+({4*d^2}/{c^4})y^2+1)", "failureRate": 1, "customMarkingAlgorithm": "", "answerSimplification": "all", "showPreview": true, "notallowed": {"showStrings": false, "message": "<p>Do not enter decimals in your answer.</p>", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "<p>You are given the following surface, defined in parametric form</p>\n    <p>\\[\\pmatrix{u,v}\\rightarrow\\pmatrix{\\simplify{{a}*v*({1-t}*cos({b}*u)+{t}*sin({b}*u))},\\simplify{{c}*v*({1-t}*sin({b}*u)+{t}*cos({b}*u))},\\simplify{{d}*v^2}}, \\quad 0\\leqslant u\\leqslant 2\\pi, 0\\leqslant v\\leqslant 1\\]</p>", "tags": ["checked2015", "normals", "parametric form", "surfaces"], "rulesets": {}, "extensions": [], "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Cartesian form of the parametric representation of a surface, normal vector, and magnitude.</p>\n<p>Accuracy for part c) should be made more stringent as can be marked correct for an incorrect answer. Use a different sample range rather than 0 to 1 would help as would setting accuracy to something less than 0.001.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The given surface has components</p>\n    <p>\\[\\begin{align}x&amp;=\\simplify{{a}*v*({1-t}*cos({b}*u)+{t}*sin({b}*u))},\\\\y&amp;=\\simplify{{c}*v*({1-t}*sin({b}*u)+{t}*cos({b}*u))},\\\\z&amp;=\\simplify{{d}*v^2}.\\end{align}\\]</p>\n    <p>Then</p>\n    <p>\\[\\simplify{(x/{a})^2}+\\simplify{(y/{c})^2}=(\\simplify{v*({1-t}*cos({b}*u)+{t}*sin({b}*u))})^2+(\\simplify{v*({1-t}*sin({b}*u)+{t}*cos({b}*u))})^2=v^2,\\]</p>\n    <p>but we know that $z=\\simplify{{d}*v^2}$, so</p>\n    <p>\\[\\simplify{(x/{a})^2}+\\simplify{(y/{c})^2}=\\simplify{z/{d}},\\]</p>\n    <p>and hence</p>\n    <p>\\[z=\\simplify{{d}/{a^2}x^2+{d}/{c^2}y^2}.\\]</p>\n    <p>&nbsp;</p>\n    <p>Given the above form for $z=g(x,y)$, a normal vector $\\boldsymbol{n}$, with positive $z$-component is</p>\n    <p>\\[\\boldsymbol{n}=\\pmatrix{-\\frac{\\partial g}{\\partial x},-\\frac{\\partial g}{\\partial y},1}.\\]</p>\n    <p>In this case</p>\n    <p>\\[\\boldsymbol{n}=\\pmatrix{\\simplify{{-2d}/{a^2}}x,\\simplify{{-2d}/{c^2}}y,1}\\]</p>\n    <p>by straightforward partial differentiation.</p>\n    <p>&nbsp;</p>\n    <p>The magnitude $\\lvert\\boldsymbol{n}\\rvert$ of the normal vector $\\boldsymbol{n}$ is given by $\\lvert\\boldsymbol{n}\\rvert=\\sqrt{n_1^2+n_2^2+n_3^2}$, and hence</p>\n    <p>\\[\\lvert\\boldsymbol{n}\\rvert=\\sqrt{\\simplify{{4*d^2}/{a^4}}x^2+\\simplify{{4*d^2}/{c^4}}y^2+1}.\\]</p>", "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/"}]}]}], "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/"}]}