// Numbas version: exam_results_page_options
{"name": "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": [{"variable_groups": [], "variables": {"speed": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(sqrt(dxdtf1^2+dydtf1^2),3)", "description": "", "name": "speed"}, "x": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(a1*cos(b1*e1),3)", "description": "", "name": "x"}, "e1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "e1"}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "f1"}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "b1"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "a1"}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "d1"}, "dydte1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(c1*d1*cos(d1*e1),3)", "description": "", "name": "dydte1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(1,-1))", "description": "", "name": "c1"}, "dydtf1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "c1*d1*cos(d1*f1)", "description": "", "name": "dydtf1"}, "dxdtf1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "-a1*b1*sin(b1*f1)", "description": "", "name": "dxdtf1"}, "dxdte1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(-a1*b1*sin(b1*e1),3)", "description": "", "name": "dxdte1"}, "y": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(c1*sin(d1*e1),3)", "description": "", "name": "y"}}, "ungrouped_variables": ["f1", "dxdte1", "dxdtf1", "dydte1", "dydtf1", "a1", "b1", "y", "x", "c1", "e1", "speed", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Parametric curve - find tangent and coordinates at given point, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "x", "minValue": "x", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "y", "minValue": "y", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "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>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{-a1*b1}*sin({b1}*t)", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{c1*d1}*cos({d1}*t)", "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "dxdte1+0.001", "minValue": "dxdte1-0.001", "showCorrectAnswer": true, "marks": 1}, {"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "dydte1+0.001", "minValue": "dydte1-0.001", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "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>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "speed+0.001", "minValue": "speed-0.001", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "<p>Interpreting $t$ as time, and hence the tangent vector $\\boldsymbol{u}$ as velocity, find the speed $u=|\\boldsymbol{u}|$ at $t=\\var{f1}$.</p>\n      <p>$u=$ [[0]]. &nbsp;(Enter your answer to 3d.p.)</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>You are given the following curve, $t\\rightarrow\\pmatrix{\\simplify{{a1}*cos({b1}t)},\\simplify{{c1}*sin({d1}t)}}$, defined with respect to the parameter $t$.</p>", "tags": ["checked2015", "MAS1902", "MAS2104"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Parametric form of a curve, cartesian points, tangent vector, and speed.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}