Find the coordinates $\\pmatrix{x,y}$ of the point corresponding to $t=\\var{c}$.
\n$\\pmatrix{x,y}=($[[0]]$,$[[1]]$)$.
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{a}*t^{a-1}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{b}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "dxdtc", "maxValue": "dxdtc", "marks": 1}, {"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "b", "maxValue": "b", "marks": 1}], "type": "gapfill", "prompt": "Find the components of the tangent vector $\\boldsymbol{u}\\equiv\\pmatrix{\\frac{\\mathrm{d}x}{\\mathrm{d}t},\\frac{\\mathrm{d}y}{\\mathrm{d}t}}$.
\n$\\boldsymbol{u}=($[[0]]$,$[[1]]$)$.
\nThe components of the same tangent vector, given $t=\\var{c}$.
\n$\\boldsymbol{u}|_{t=\\var{c}}=($[[2]]$,$[[3]]$)$.
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "minValue": "speed-0.01", "maxValue": "speed+0.01", "marks": 1}], "type": "gapfill", "prompt": "Interpreting $t$ as time, and hence the tangent vector $\\boldsymbol{u}$ as velocity, find the speed $u=|\\boldsymbol{u}|$ at $t=\\var{d}$.
\n$u=$ [[0]]. (Enter your answer to 2d.p.)
", "showCorrectAnswer": true, "marks": 0}], "statement": "You are given the following curve, $t\\rightarrow\\pmatrix{t^\\var{a},\\simplify{{b}t}}$, defined with respect to the parameter $t$.
", "tags": ["checked2015", "MAS1902", "MAS2104"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Parametric form of a curve, cartesian points, tangent vector, and speed.
"}, "functions": {}, "advice": "a)
\nTo find the coordinates of the point corresponding to $t=\\var{c}$, substitute $t=\\var{c}$ into the expression for the curve, i.e.
\n\\[\\pmatrix{x,y}=\\pmatrix{\\var{c}^\\var{a},\\var{b}\\times\\var{c}}=\\pmatrix{\\var{x},\\var{y}}.\\]
\n\n
b)
\nDifferentiate each component of the vector in part a) to find the tangent vector $\\boldsymbol{u}$, i.e.
\n\\[\\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}}.\\]
\nThe tangent vector at $t=\\var{c}$ is found by substituting $t=\\var{c}$ into the tangent vector $\\boldsymbol{u}$, i.e.
\n\\[\\boldsymbol{u}\\vert_{t=\\var{c}}=\\pmatrix{\\var{a}\\times\\var{c}^\\var{a-1},\\var{b}}=\\pmatrix{\\var{dxdtc},\\var{b}}.\\]
\n\n
c)
\nThe 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 $t=\\var{d}$, however, therefore
\n\\[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.}\\]
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