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{"name": "Ida'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": [{"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>", "variable_groups": [], "statement": "", "type": "question", "functions": {}, "rulesets": {"std": ["all", "fractionNumbers"]}, "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "tags": [], "parts": [{"variableReplacementStrategy": "originalfirst", "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>", "variableReplacements": [], "gaps": [{"variableReplacements": [], "maxValue": "2*a*d+b", "type": "numberentry", "minValue": "2*a*d+b", "strictPrecision": false, "showCorrectAnswer": true, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precisionType": "dp", "allowFractions": false, "precision": "2", "marks": 1, "scripts": {}, "correctAnswerFraction": false}, {"checkvariablenames": false, "variableReplacements": [], "expectedvariablenames": [], "type": "jme", "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst", "scripts": {}, "answer": "2*{a}*x+{b}", "marks": 1, "checkingaccuracy": 0.001, "checkingtype": "absdiff"}], "type": "gapfill", "marks": 0, "scripts": {}, "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "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>", "variableReplacements": [], "gaps": [{"variableReplacements": [], "maxValue": "-g/(2*f)", "type": "numberentry", "minValue": "-g/(2*f)", "strictPrecision": false, "showCorrectAnswer": true, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precisionType": "dp", "allowFractions": true, "precision": "2", "marks": 1, "scripts": {}, "correctAnswerFraction": true}, {"variableReplacements": [], "maxValue": "g^2/(4*f)-g^2/(2*f)+h", "type": "numberentry", "minValue": "g^2/(4*f)-g^2/(2*f)+h", "strictPrecision": false, "showCorrectAnswer": true, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precisionType": "dp", "allowFractions": true, "precision": "2", "marks": 1, "scripts": {}, "correctAnswerFraction": true}, {"checkvariablenames": false, "variableReplacements": [], "expectedvariablenames": [], "type": "jme", "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst", "scripts": {}, "answer": "2*{f}*x+{g}", "marks": 1, "checkingaccuracy": 0.001, "checkingtype": "absdiff"}, {"checkvariablenames": false, "variableReplacements": [], "expectedvariablenames": [], "type": "jme", "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst", "scripts": {}, "answer": "2*{f}", "marks": 1, "checkingaccuracy": 0.001, "checkingtype": "absdiff"}, {"variableReplacements": [], "shuffleChoices": false, "displayType": "radiogroup", "distractors": ["", ""], "type": "1_n_2", "showCorrectAnswer": true, "choices": ["<p>maximum</p>", "<p>minimum</p>"], "variableReplacementStrategy": "originalfirst", "scripts": {}, "matrix": ["if(maximum, 1, 0)", "if(maximum, 0, 1)"], "maxMarks": "0", "displayColumns": 0, "marks": 0, "minMarks": 0}], "type": "gapfill", "marks": 0, "scripts": {}, "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "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>", "variableReplacements": [], "gaps": [{"checkvariablenames": false, "variableReplacements": [], "expectedvariablenames": [], "type": "jme", "showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "vsetrange": [0, 1], "variableReplacementStrategy": "originalfirst", "scripts": {}, "answer": "{z}-2*{w}*t", "marks": 1, "checkingaccuracy": 0.001, "checkingtype": "absdiff"}, {"variableReplacements": [], "maxValue": "z^2/(4w)", "type": "numberentry", "minValue": "z^2/(4w)", "strictPrecision": false, "showCorrectAnswer": true, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, 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"description": ""}, "preamble": {"js": "", "css": ""}, "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [], "name": ""}], "name": "Ida's copy of Applications of differentiation ", "contributors": [{"name": "Ida Landg\u00e4rds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2336/"}]}]}], "contributors": [{"name": "Ida Landg\u00e4rds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2336/"}]}