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{"name": "Parametric Differentiation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Parametric Differentiation", "tags": [], "metadata": {"description": "Calculate dy/dx for a curve defined parametrically", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>A curve is defined by the parametric equations $x=\\var{a}+\\var{b}t, y = \\var{c}+\\var{d}t^2$</p>", "advice": "<p>We have&nbsp;$x=\\var{a}+\\var{b}t, y = \\var{c}+\\var{d}t^2$</p>\n<p>(i) To find $\\frac{dx}{dt}$ we simply differentiate&nbsp;$x=\\var{a}+\\var{b}t$ with respect to $t$ to obtain&nbsp;$\\frac{dx}{dt}=\\var{b}$</p>\n<p></p>\n<p>(ii) To find $\\frac{dy}{dt}$ we simply differentiate&nbsp;$y=\\var{c}+\\var{d}t^2$ with respect to $t$ to obtain&nbsp;$\\frac{dy}{dt}=2\\times\\var{d}t=\\var{2d}t$</p>\n<p></p>\n<p>(iii) Since&nbsp;$\\frac{dy}{dx}=\\frac{dy}{dt}/\\frac{dx}{dt}$ we have&nbsp;$\\frac{dy}{dx}=\\frac{\\var{2d}t}{\\var{b}}=\\var{2d/b}t$</p>\n<p></p>\n<p>(iv)&nbsp;When&nbsp;$x=\\var{x}$, we can use $x=\\var{a}+\\var{b}t$ to obtain&nbsp;$\\var{x}=\\var{a}+\\var{b}t$, hence $t = \\frac{\\var{x}-\\var{a}}{\\var{b}}=\\var{(x-a)/b}$</p>\n<p>Substituting this value of $t$ into&nbsp;$\\frac{dy}{dx}=\\var{2d/b}t$ gives&nbsp;$\\frac{dy}{dx}=\\var{2d/b}\\times \\var{(x-a)/b}=\\var{2d*(x-a)/b^2}$</p>", "rulesets": {}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2,4,5)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange"}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(-4 .. 4#1)", "description": "", "templateType": "randrange"}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "(x-a)/b", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "x", "t"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>(i) Find $\\frac{dx}{dt}$&nbsp; &nbsp; [[0]]</p>\n<p>(ii)&nbsp;Find&nbsp;$\\frac{dy}{dt}$&nbsp; &nbsp; [[1]]</p>\n<p>(iii) Hence find&nbsp;$\\frac{dy}{dx}$ in terms of $t$&nbsp; &nbsp; [[2]]</p>\n<p>(iv) Hence find&nbsp;$\\frac{dy}{dx}$ when $x =&nbsp; \\var{x}$&nbsp; &nbsp;[[3]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{b}", "maxValue": "{b}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "2*{d}t", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{2d/b}*t", "answerSimplification": "basic,fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "t", "value": ""}]}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "{2d*(x-a)/b^2}", "maxValue": "{2d*(x-a)/b^2}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "contributors": [{"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}