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{"name": "Ugur's copy of Parameterisation of a curve - 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 Parameterisation of a curve - 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, $t\\mapsto\\pmatrix{t^\\var{a},\\simplify{{b}t}}$, defined with respect to the parameter $t$.</p>", "advice": "<p>a)</p>\n    <p>To find the coordinates of the point corresponding to $t=\\var{c}$, substitute $t=\\var{c}$ into the expression for the curve, i.e.</p>\n    <p>\\[\\pmatrix{x,y}=\\pmatrix{\\var{c}^\\var{a},\\var{b}\\times\\var{c}}=\\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}t^\\var{a},\\frac{\\mathrm{d}}{\\mathrm{d}t}\\simplify{{b}t}}=\\pmatrix{\\var{a}t^\\var{a-1},\\var{b}}.\\]</p>\n    <p>The tangent vector at $t=\\var{c}$ is found by substituting $t=\\var{c}$ into the tangent vector $\\boldsymbol{u}$, i.e.</p>\n    <p>\\[\\boldsymbol{u}\\vert_{t=\\var{c}}=\\pmatrix{\\var{a}\\times\\var{c}^\\var{a-1},\\var{b}}=\\pmatrix{\\var{dxdtc},\\var{b}}.\\]</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 $t=\\var{d}$, however, therefore</p>\n    <p>\\[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.}\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"speed": {"name": "speed", "group": "Ungrouped variables", "definition": "precround(sqrt(dxdtd^2+b^2),2)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "c^a", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..10)*sign(random(1,-1))", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "dxdtd": {"name": "dxdtd", "group": "Ungrouped variables", "definition": "a*d^(a-1)", "description": "", "templateType": "anything", "can_override": false}, "dxdtc": {"name": "dxdtc", "group": "Ungrouped variables", "definition": "a*c^(a-1)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "b*c", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d", "dxdtc", "dxdtd", "y", "x", "speed"], "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 $t=\\var{c}$.</p>\n      <p>$\\pmatrix{x,y}=($[[0]]$,$[[1]]$)$.</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 $t=\\var{c}$.</p>\n      <p>$\\boldsymbol{u}|_{t=\\var{c}}=($[[2]]$,$[[3]]$)$.</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, 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"showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b", "maxValue": "b", "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>Interpreting $t$ as time, and hence the tangent vector $\\boldsymbol{u}$ 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"contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}