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{"name": "Christian's copy of Applications of differentiation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "variables": {"z": {"templateType": "randrange", "name": "z", "description": "", "definition": "random(20 .. 30#0.5)", "group": "Ungrouped variables"}, "h": {"templateType": "randrange", "name": "h", "description": "", "definition": "random(0 .. 5#0.5)", "group": "Ungrouped variables"}, "g": {"templateType": "randrange", "name": "g", "description": "", "definition": "random(-10 .. 10#1)", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "name": "c", "description": "", "definition": "random(2..7)", "group": "Ungrouped variables"}, "maximum": {"templateType": "anything", "name": "maximum", "description": "<p>Is the stationary point a maximum?</p>", "definition": "f<0", "group": "Ungrouped variables"}, "d": {"templateType": "anything", "name": "d", "description": "", "definition": "random(2..5)", "group": "Ungrouped variables"}, "f": {"templateType": "anything", "name": "f", "description": "", "definition": "random(-10..10 except 0)", "group": "Ungrouped variables"}, "t": {"templateType": "randrange", "name": "t", "description": "", "definition": "random(0 .. 1#0.1)", "group": "Ungrouped variables"}, "b": {"templateType": "anything", "name": "b", "description": "", "definition": "random(2..5)", "group": "Ungrouped variables"}, "a": {"templateType": "randrange", "name": "a", "description": "", "definition": "random(0 .. 10#0.5)", "group": "Ungrouped variables"}, "w": {"templateType": "randrange", "name": "w", "description": "", "definition": "random(2 .. 5#0.1)", "group": "Ungrouped variables"}}, "variablesTest": {"condition": "isclose(-g/(2f),2/3)", "maxRuns": "500"}, "rulesets": {"std": ["all", "fractionNumbers"]}, "extensions": [], "name": "Christian's copy of Applications of differentiation", "variable_groups": [], "tags": [], "ungrouped_variables": ["z", "c", "b", "d", "f", "w", "a", "g", "h", "t", "maximum"], "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>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": ""}, "statement": "", "preamble": {"js": "", "css": ""}, "parts": [{"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>", "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerStyle": "plain", "scripts": {}, "customName": "", "precision": "2", "showFeedbackIcon": true, "showPrecisionHint": false, "customMarkingAlgorithm": "", "unitTests": [], "variableReplacements": [], "useCustomName": false, "strictPrecision": false, "correctAnswerFraction": false, "mustBeReduced": false, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "mustBeReducedPC": 0, "maxValue": "2*a*d+b", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "marks": 1, "precisionPartialCredit": 0, "allowFractions": false, "minValue": "2*a*d+b", "type": "numberentry"}, {"scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "vsetRangePoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 1, "checkVariableNames": false, "answer": "2*{a}*x+{b}", "customMarkingAlgorithm": "", "checkingAccuracy": 0.001, "vsetRange": [0, 1], "unitTests": [], "failureRate": 1, "checkingType": "absdiff", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "type": "jme", "showPreview": true, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false, "customMarkingAlgorithm": "", "unitTests": [], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "type": "gapfill"}, {"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>", "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "gaps": [{"correctAnswerStyle": "plain", "scripts": {}, "customName": "", "precision": "2", "showFeedbackIcon": true, "showPrecisionHint": false, "customMarkingAlgorithm": "", "unitTests": [], "variableReplacements": [], "useCustomName": false, "strictPrecision": false, "correctAnswerFraction": 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"type": "gapfill"}, {"prompt": "<p>An unpowered missile is launched vertically from the ground.</p>\n<p>At a time $t$ seconds after the instant of projection, its height, $y$ metres, above the ground is given by the formula</p>\n<p>\\[ y=\\var{z}t-\\var{w}t^2. \\]</p>\n<p>Calculate the maximum height reached by the missile.</p>\n<p>Firstly, differentiate.</p>\n<p>$\\displaystyle \\frac{dy}{dt}=$&nbsp;[[0]]</p>\n<p>Now use this result and your knowledge of differentiation to find the maximum height of the missile, rounding your answer to $2$ decimal places.</p>\n<p>$y=$&nbsp;[[1]]</p>", "scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 0, "gaps": [{"scripts": {}, "customName": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "vsetRangePoints": 5, "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 1, "checkVariableNames": false, "answer": "{z}-2*{w}*t", 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