// Numbas version: exam_results_page_options
{"name": "Parameterisation of a curve - find tangent and length", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"num3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "4*9*(a^2+t1*b^2)", "description": "", "name": "num3"}, "num4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "3*a", "description": "", "name": "num4"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "b"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((8/(27*4*b^2))*((9*a^2+((9*t2*4*b^2)/4))^(3/2)-(9*a^2+((9*t1*4*b^2)/4))^(3/2)),2)", "description": "", "name": "s"}, "t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "name": "t1"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)+t1", "description": "", "name": "t2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..2)", "description": "", "name": "a"}, "num1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "9*4*b^2", "description": "", "name": "num1"}, "num2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "27*4*b^2", "description": "", "name": "num2"}}, "ungrouped_variables": ["a", "num4", "b", "num1", "num2", "num3", "t2", "t1", "s"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Parameterisation of a curve - find tangent and length", "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "<p>Find the tangent vector $\\boldsymbol{u}$ to the curve.</p>\n        <p>$\\boldsymbol{u}=($[[0]]$,$[[1]]$)$.&nbsp;&nbsp;(Do not enter decimals in your answers.)</p>", "scripts": {}, "gaps": [{"answer": "{3*a}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answers.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{3*b}*t^(1/2)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answersimplification": "all", "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answers.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>The arc-length along the curve can be written in the form $s(t)=f(t)-f(t_1)$. &nbsp;Find $f(t)$.</p>\n<p>$f(t)=$ [[0]]. &nbsp;(Do not enter decimals in your answers.)</p>", "scripts": {}, "gaps": [{"answer": "2/{b^2}*({a^2}+({b^2})*t)^(3/2)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answersimplification": "all", "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": false, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": false, "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "<p>Find the total length of the curve, $S$, given $t_1=\\var{t1}$ and $t_2=\\var{t2}$.</p>\n<p>$S=$ [[0]]. &nbsp;(Enter your answer to 2d.p.)</p>", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "s+0.01", "minValue": "s-0.01", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>$s\\longmapsto\\pmatrix{\\simplify{{3a}/{b^2}}\\left\\{\\left[\\simplify{{b^2}s/2}+\\left(\\var{a^2+t1*b^2}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\\simplify{2/{b^2}}\\left\\{\\left[\\left(\\simplify{{b^2}s/2}+\\left(\\var{a^2+t1*b^2}\\right)^\\frac{3}{2}\\right)^\\frac{2}{3}-\\var{a^2}\\right]\\right\\}^\\frac{3}{2}}$</p>", "<p>$s\\longmapsto\\pmatrix{\\simplify{{a}/{3*b^2}}\\left\\{\\left[\\simplify{{a^2}s/2}+\\left(\\var{a^2+t1*b^2}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\\simplify{2/{27*b^2}}\\left\\{\\left[\\left(\\simplify{{a^2}s/2}+\\left(\\var{a^2+t1*b^2}\\right)^\\frac{3}{2}\\right)^\\frac{2}{3}-\\var{a^2}\\right]\\right\\}^\\frac{3}{2}}$</p>", "<p>$s\\longmapsto\\pmatrix{\\simplify{{3*a}}\\left\\{\\left[\\simplify{{a^2+t1*b^2}s/2}+\\left(\\var{b^2}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\\simplify{{2*b}}\\left\\{\\left[\\left(\\simplify{{a^2+t1*b^2}s/2}+\\left(\\var{b^2}\\right)^\\frac{3}{2}\\right)^\\frac{2}{3}-\\var{a^2}\\right]\\right\\}^\\frac{3}{2}}$</p>", "<p>$s\\longmapsto\\pmatrix{\\simplify{{3}/{a*b^2}}\\left\\{\\left[\\simplify{{b^2}s/2}+\\left(\\var{a^2}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2+t1*b^2}\\right\\},\\simplify{2/{a^3*b^2}}\\left\\{\\left[\\left(\\simplify{{b^2}s/2}+\\left(\\var{a^2}\\right)^\\frac{3}{2}\\right)^\\frac{2}{3}-\\var{a^2+t1*b^2}\\right]\\right\\}^\\frac{3}{2}}$</p>"], "showCorrectAnswer": true, "displayColumns": 0, "prompt": "<p>Which of the following corresponds to an alternative parametric representation of the curve, again with $t_1=\\var{t1}$ and $t_2=\\var{t2}$, using the arc-length $s$ as the curve parameter?</p>", "distractors": ["", "", "", ""], "variableReplacements": [], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "matrix": [1, 0, 0, 0], "marks": 0}], "statement": "<p>You are given the curve $t\\longmapsto\\pmatrix{\\var{3*a}t,\\var{2*b}t^\\frac{3}{2}}$, where $t_1\\leqslant t\\leqslant t_2$.</p>", "tags": ["checked2015", "MAS2104"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Calculation of the length and alternative form of the parameteric representation of a curve.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>The tangent vector to the curve $t\\longmapsto\\pmatrix{x,y}$ is given by $\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}}$.</p>\n<p>The length $s$ of the curve in the range $t_1\\leqslant t\\leqslant t_2$ is given by</p>\n<p>\\[s=\\int_{t_1}^{t_2}{u\\,\\mathrm{d}t},\\]</p>\n<p>where $u^2=\\lvert\\boldsymbol{u}\\rvert^2=\\boldsymbol{u\\cdot u}$.</p>\n<p>In this question, therefore, $\\boldsymbol{u}=\\pmatrix{\\frac{\\mathrm{d}}{\\mathrm{d}t}(\\var{3*a}t),\\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\var{2*b}t^\\frac{3}{2}\\right)}=\\pmatrix{\\var{3*a},\\simplify{{3*b}*t^(1/2)}}$, and $u^2=9\\left(\\var{a^2}+\\simplify{{b^2}*t}\\right)$.</p>\n<p>Then</p>\n<p>\\[\\begin{align}s=\\int_{t_1}^{t_2}{u\\,\\mathrm{d}t}&amp;=3\\int_{t_1}^{t_2}{\\sqrt{\\var{a^2}+\\simplify{{b^2}*t}}\\,\\mathrm{d}t}\\\\&amp;=\\simplify{2/{b^2}}\\left[\\left(\\var{a^2}+\\simplify{{b^2}*t}\\right)^\\frac{3}{2}\\right]_{t_1}^{t_2},\\end{align}\\]</p>\n<p>so $f(t)=\\simplify{2/{b^2}}\\left(\\var{a^2}+\\simplify{{b^2}*t}\\right)^\\frac{3}{2}$.</p>\n<p>Finally, substitute $t_1=\\var{t1}$ and $t_2=\\var{t2}$ into the expression for $s$ to find the length of the curve over the given range of $t$.</p>\n<p>Hence $s=\\simplify{2/{b^2}}\\left\\{\\left(\\var{a^2+t2*b^2}\\right)^\\frac{3}{2}-\\left(\\var{a^2+t1*b^2}\\right)^\\frac{3}{2}\\right\\}=\\var{s}$ to 2d.p.</p>\n<p>An alternative parametric representation, using $s$ as the curve parameter is given by</p>\n<p>\\[\\begin{align}s(t)=\\int_{t_1}^t{u\\,\\mathrm{d}\\tau}&amp;=3\\int_{t_1}^t{\\sqrt{\\var{a^2}+\\simplify{{b^2}*tau}}\\,\\mathrm{d}\\tau}\\\\&amp;=\\simplify{2/{b^2}}\\left[\\left(\\var{a^2}+\\simplify{{b^2}*tau}\\right)^\\frac{3}{2}\\right]_{t_1}^t\\\\&amp;=\\simplify{2/{b^2}}\\left\\{\\left(\\var{a^2}+\\simplify{{b^2}*t}\\right)^\\frac{3}{2}-\\left(\\var{a^2}+\\simplify{{b^2}*t_1}\\right)^\\frac{3}{2}\\right\\}.\\end{align}\\]</p>\n<p>Now rearrange this expression for $t(s)$, so</p>\n<p>\\[t(s)=\\simplify{1/{b^2}}\\left\\{\\left[\\simplify{{b^2}/2}s+\\left(\\var{a^2}+\\simplify{{b^2}*t_1}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\\]</p>\n<p>and substitute into the original representation of the curve $t\\longmapsto\\pmatrix{\\var{3*a}t,\\var{2*b}t^\\frac{3}{2}}$ with $t_1\\leqslant t\\leqslant t_2$. &nbsp;Hence</p>\n<p>\\[s\\longmapsto\\pmatrix{\\simplify{{3*a}/{b^2}}\\left\\{\\left[\\simplify{{b^2}/2}s+\\left(\\var{a^2}+\\simplify{{b^2}*t_1}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\\simplify{2/{b^2}}\\left\\{\\left[\\simplify{{b^2}/2}s+\\left(\\var{a^2}+\\simplify{{b^2}*t_1}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\}^\\frac{3}{2}}\\]</p>\n<p>with $0\\leqslant s\\leqslant\\simplify{2/{b^2}}\\left(\\left(\\var{a^2}+\\simplify{{b^2}*t_2}\\right)^\\frac{3}{2}-\\left(\\var{a^2}+\\simplify{{b^2}*t_1}\\right)^\\frac{3}{2}\\right)$.</p>\n<p>Finally, substitute $t_1=\\var{t1}$ and $t_2=\\var{t2}$ into the above expressions, to find the specific parametric representation corresponding to the given range of t:</p>\n<p>\\[s\\longmapsto\\pmatrix{\\simplify{{3*a}/{b^2}}\\left\\{\\left[\\simplify{{b^2}/2}s+\\left(\\var{a^2+b^2*t1}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\},\\simplify{2/{b^2}}\\left\\{\\left[\\simplify{{b^2}/2}s+\\left(\\var{a^2+b^2*t1}\\right)^\\frac{3}{2}\\right]^\\frac{2}{3}-\\var{a^2}\\right\\}^\\frac{3}{2}}\\]</p>\n<p>with $0\\leqslant s\\leqslant\\simplify{2/{b^2}}\\left(\\left(\\var{a^2+b^2*t2}\\right)^\\frac{3}{2}-\\left(\\var{a^2+b^2*t1}\\right)^\\frac{3}{2}\\right)$.</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/"}]}