// Numbas version: exam_results_page_options
{"name": "Ugur's copy of Parametric curve - find tangent and coordinates at given point,", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Ugur's copy of Parametric curve - find tangent and coordinates at given point,", "tags": ["checked2015"], "metadata": {"description": "<p>Parametric form of a curve, cartesian points, tangent vector, and speed.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>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 $t$.</p>", "advice": "<p>a)</p>\n    <p>To find the coordinates of the point corresponding to $t=\\var{e1}$, substitute $t=\\var{e1}$ into the expression for the curve, i.e.</p>\n    <p>\\[\\pmatrix{x,y}=\\pmatrix{\\simplify{{a1}*cos({b1*e1})},\\simplify{{c1}*cos({d1*e1})}}=\\pmatrix{\\var{x},\\var{y}}.\\]</p>\n    <p>&nbsp;</p>\n    <p>b)</p>\n    <p>Differentiate each component of the vector in part a) to find the tangent vector $\\boldsymbol{u}$, i.e.</p>\n    <p>\\[\\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)}}.\\]</p>\n    <p>The tangent vector at $t=\\var{e1}$ is found by substituting $t=\\var{e1}$ into the tangent vector $\\boldsymbol{u}$, i.e.</p>\n    <p>\\[\\boldsymbol{u}\\vert_{t=\\var{e1}}=\\pmatrix{\\simplify{{-a1*b1}*sin({b1*e1})},\\simplify{{c1*d1}*cos({d1*e1})}}=\\pmatrix{\\var{dxdte1},\\var{dydte1}}.\\]</p>\n    <p>&nbsp;</p>\n    <p>c)</p>\n    <p>The velocity $u$ is given by $u=\\lvert\\boldsymbol{u}\\rvert=\\sqrt{\\left(\\frac{\\mathrm{d}x}{\\mathrm{d}t}\\right)^2+\\left(\\frac{\\mathrm{d}y}{\\mathrm{d}t}\\right)^2}$. &nbsp;We must calculate the speed at $t=\\var{f1}$, however, therefore</p>\n    <p>\\[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.}\\]</p>", "rulesets": {}, "extensions": [], "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": "<p>Find the coordinates $\\pmatrix{x,y}$ of the point corresponding to $t=\\var{e1}$.</p>\n        <p>$\\pmatrix{x,y}=($[[0]]$,$[[1]]$)$. &nbsp;(Enter your answers to 3d.p.)</p>", "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": "x", "maxValue": "x", "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": "y", "maxValue": "y", "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": "<p>Find the components of the tangent vector $\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}}$.</p>\n        <p>$\\boldsymbol{u}=($[[0]]$,$[[1]]$)$.</p>\n        <p>The components of the same tangent vector, given $t=\\var{e1}$.</p>\n        <p>$\\boldsymbol{u}|_{t=\\var{e1}}=($[[2]]$,$[[3]]$)$. &nbsp;(Enter your answers&nbsp;to 3d.p.)</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", 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"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, 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