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{"name": "Harry's copy  Applications of differentiation ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Harry's copy  Applications of differentiation ", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": ""}, "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>", "statement": "", "extensions": [], "variable_groups": [], "variables": {"c": {"group": "Ungrouped variables", "name": "c", "templateType": "anything", "definition": "random(2..7)", "description": ""}, "w": {"group": "Ungrouped variables", "name": "w", "templateType": "randrange", "definition": "random(2..5#0.1)", "description": ""}, "maximum": {"group": "Ungrouped variables", "name": "maximum", "templateType": "anything", "definition": "f<0", "description": "<p>Is the stationary point a maximum?</p>"}, "a": {"group": "Ungrouped variables", "name": "a", "templateType": "randrange", "definition": "random(0..10#0.5)", "description": ""}, "z": {"group": "Ungrouped variables", "name": "z", "templateType": "randrange", "definition": "random(20..30#0.5)", "description": ""}, "h": {"group": "Ungrouped variables", "name": "h", "templateType": "randrange", "definition": "random(0..5#0.5)", "description": ""}, "d": {"group": "Ungrouped variables", "name": "d", "templateType": "anything", "definition": "random(2..5)", "description": ""}, "t": {"group": "Ungrouped variables", "name": "t", "templateType": "randrange", "definition": "random(0..1#0.1)", "description": ""}, "f": {"group": "Ungrouped variables", "name": "f", "templateType": "randrange", "definition": "random(-10..10#1)", "description": ""}, "g": {"group": "Ungrouped variables", "name": "g", "templateType": "randrange", "definition": "random(-10..10#1)", "description": ""}, "b": {"group": "Ungrouped variables", "name": "b", "templateType": "anything", "definition": "random(2..5)", "description": ""}}, "type": "question", "showQuestionGroupNames": false, "preamble": {"js": "", "css": ""}, "rulesets": {"std": ["all", "fractionNumbers"]}, "ungrouped_variables": ["z", "c", "b", "d", "f", "w", "a", "g", "h", "t", "maximum"], "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>", "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacementStrategy": "originalfirst", "minValue": "2*a*d+b", "correctAnswerFraction": false, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "precisionType": "dp", "variableReplacements": [], "allowFractions": false, "scripts": {}, "showPrecisionHint": false, "maxValue": "2*a*d+b", "type": "numberentry", "precisionPartialCredit": 0, "marks": 1, "strictPrecision": false, "precision": "2", "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "answer": "2*{a}*x+{b}", "expectedvariablenames": [], "variableReplacements": [], "scripts": {}, "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "checkingtype": "absdiff", "marks": 1, "vsetrange": [0, 1], "showpreview": true, "checkvariablenames": false, "vsetrangepoints": 5}], "scripts": {}}, {"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>", "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacementStrategy": "originalfirst", "minValue": "-g/(2*f)", "correctAnswerFraction": true, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "precisionType": "dp", "variableReplacements": [], "allowFractions": true, "scripts": {}, "showPrecisionHint": false, "maxValue": "-g/(2*f)", "type": "numberentry", "precisionPartialCredit": 0, "marks": 1, "strictPrecision": false, "precision": "2", "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "minValue": "g^2/(4*f)-g^2/(2*f)+h", "correctAnswerFraction": true, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "precisionType": "dp", "variableReplacements": [], "allowFractions": true, "scripts": {}, "showPrecisionHint": false, "maxValue": "g^2/(4*f)-g^2/(2*f)+h", "type": "numberentry", "precisionPartialCredit": 0, "marks": 1, "strictPrecision": false, "precision": "2", "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "answer": "2*{f}*x+{g}", "expectedvariablenames": [], "variableReplacements": [], "scripts": {}, "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "checkingtype": "absdiff", "marks": 1, "vsetrange": [0, 1], "showpreview": true, "checkvariablenames": false, "vsetrangepoints": 5}, {"variableReplacementStrategy": "originalfirst", "answer": "2*{f}", "expectedvariablenames": [], "variableReplacements": [], "scripts": {}, "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "checkingtype": "absdiff", "marks": 1, "vsetrange": [0, 1], "showpreview": true, "checkvariablenames": false, "vsetrangepoints": 5}, {"minMarks": 0, "maxMarks": "0", "variableReplacementStrategy": "originalfirst", "shuffleChoices": false, "variableReplacements": [], "scripts": {}, "distractors": ["", ""], "matrix": ["if(maximum, 1, 0)", "if(maximum, 0, 1)"], "type": "1_n_2", "showCorrectAnswer": true, "marks": 0, "displayColumns": 0, "displayType": "radiogroup", "choices": ["<p>maximum</p>", "<p>minimum</p>"]}], "scripts": {}}, {"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>", "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacementStrategy": "originalfirst", "answer": "{z}-2*{w}*t", "expectedvariablenames": [], "variableReplacements": [], "scripts": {}, "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "checkingtype": "absdiff", "marks": 1, "vsetrange": [0, 1], "showpreview": true, "checkvariablenames": false, "vsetrangepoints": 5}, {"variableReplacementStrategy": "originalfirst", "minValue": "z^2/(4w)", "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "variableReplacements": [], "allowFractions": false, "scripts": {}, "showPrecisionHint": false, "maxValue": "z^2/(4w)", "type": "numberentry", "precisionPartialCredit": 0, "marks": 1, "strictPrecision": false, "precision": "2", "showCorrectAnswer": true}], "scripts": {}}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}