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{"name": "nuExam13 - Differentiation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["z", "c", "b", "d", "f", "w", "a", "g", "h", "t", "maximum"], "variables": {"maximum": {"templateType": "anything", "group": "Ungrouped variables", "description": "<p>Is the stationary point a maximum?</p>", "definition": "f<0", "name": "maximum"}, "a": {"templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(0 .. 10#0.5)", "name": "a"}, "h": {"templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(0 .. 5#0.5)", "name": "h"}, "g": {"templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(-10 .. 10#1)", "name": "g"}, "t": {"templateType": "randrange", 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"extensions": [], "functions": {}, "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {"std": ["all", "fractionNumbers"]}, "variablesTest": {"maxRuns": "500", "condition": "isclose(-g/(2f),2/3)"}, "parts": [{"gaps": [{"scripts": {}, "type": "numberentry", "useCustomName": false, "precisionType": "dp", "precisionPartialCredit": 0, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "marks": 1, "allowFractions": false, "customName": "", "mustBeReduced": false, "correctAnswerStyle": "plain", "minValue": "2*a*d+b", "customMarkingAlgorithm": "", "showPrecisionHint": false, "showCorrectAnswer": true, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "maxValue": "2*a*d+b", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "mustBeReducedPC": 0, "unitTests": [], "precision": "2"}, {"checkingType": "absdiff", "vsetRange": [0, 1], "scripts": {}, "showCorrectAnswer": true, "useCustomName": false, "customMarkingAlgorithm": "", "valuegenerators": [{"value": "", "name": "x"}], "answer": "2*{a}*x+{b}", "type": "jme", "showPreview": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "failureRate": 1, "variableReplacements": [], "showFeedbackIcon": true, "checkVariableNames": false, "checkingAccuracy": 0.001, "vsetRangePoints": 5, "marks": 1, "customName": "", "unitTests": []}], "sortAnswers": false, "scripts": {}, "showCorrectAnswer": true, "useCustomName": false, "customMarkingAlgorithm": "", "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>Find the gradient of the curve $y$ at the point $x=\\var{d}$, giving your answer to $2$ decimal places if necessary.</p>\n<p>\\[ y = \\simplify{ {a}*x^2 + {b}x + {c}} \\]</p>\n<p>Firstly, differentiate.</p>\n<p>$\\displaystyle \\frac{dy}{dx}=$&nbsp;[[1]]</p>\n<p>Gradient at $x=\\var{d}\\;$ is&nbsp;[[0]]</p>", "marks": 0, "customName": "", "unitTests": []}, {"gaps": [{"scripts": {}, "type": "numberentry", "useCustomName": false, "precisionType": "dp", "precisionPartialCredit": 0, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "marks": 1, "allowFractions": false, "customName": "", "mustBeReduced": false, "correctAnswerStyle": "plain", "minValue": "-g/(2*f)", "customMarkingAlgorithm": "", "showPrecisionHint": false, "showCorrectAnswer": true, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "maxValue": "-g/(2*f)", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "mustBeReducedPC": 0, "unitTests": [], "precision": "2"}, {"scripts": {}, "type": "numberentry", 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"showCellAnswerState": true, "minMarks": 0, "maxMarks": "0", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "choices": ["<p>maximum</p>", "<p>minimum</p>"], "variableReplacements": [], "showFeedbackIcon": true, "matrix": ["if(maximum, 1, 0)", "if(maximum, 0, 1)"], "displayColumns": 0, "marks": 0, "customName": "", "unitTests": []}], "sortAnswers": false, "scripts": {}, "showCorrectAnswer": true, "useCustomName": false, "customMarkingAlgorithm": "", "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>Find the coordinates of the turning&nbsp;point of the function below and state whether it is a maximum or a minimum point. Give your answers to $2$ decimal places where necessary.</p>\n<p>$y=\\simplify {{f}x^2+{g}x+{h}}$</p>\n<p>Firstly, find the first and second derivatives $y$.</p>\n<p>$\\displaystyle \\frac{dy}{dx}=$&nbsp;[[2]]</p>\n<p>$\\displaystyle \\frac{d^2y}{dx^2}=$&nbsp;[[3]]</p>\n<p></p>\n<p>Secondly, find $x$ such that&nbsp;$\\displaystyle \\frac{dy}{dx}=0$.</p>\n<p>$x$-coordinate of the turning&nbsp;point $=$&nbsp;[[0]]</p>\n<p>$y$-coordinate of the turning&nbsp;point $=$&nbsp;[[1]]</p>\n<p>The turning point is a [[4]]</p>\n<p></p>\n<p></p>", "marks": 0, "customName": "", "unitTests": []}], "variable_groups": [], "tags": [], "advice": "<p><strong>Parts A and B</strong></p>\n<p>Here, the question takes you throught the stages needed to find the solution. The reason we differentiate is that the derivative of a function tells us its gradient at a given point, and we want to find where the function has gradient zero because when the gradient is zero we either have a maximum or a minimum point.</p>\n<p><strong>Part C</strong></p>\n<p>The first part of this question is similar to parts A and B. The tricky bit is the second part! You need to work out the value of $t$ that produces the maximum piont but that is not the final answer - you need to use that value of $t$ to find the maximum height, which you do by substituting $t$ into the original function to find $y$.</p>", "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}