// Numbas version: exam_results_page_options
{"name": "Applications of differentiation ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": ""}, "name": "Applications of differentiation ", "tags": [], "functions": {}, "parts": [{"marks": 0, "gaps": [{"marks": 1, "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "strictPrecision": false, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "allowFractions": false, "precisionType": "dp", "showFeedbackIcon": true, "scripts": {}, "precisionPartialCredit": 0, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "precision": "2", "minValue": "2*a*d+b", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "2*a*d+b", "notationStyles": ["plain", "en", "si-en"]}, {"marks": 1, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "variableReplacements": [], "showCorrectAnswer": true, "expectedvariablenames": [], "showFeedbackIcon": true, "answer": "2*{a}*x+{b}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme", "checkingaccuracy": 0.001, "showpreview": true, "scripts": {}, "vsetrange": [0, 1]}], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "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": {}, "showFeedbackIcon": true, "showCorrectAnswer": true}, {"marks": 0, "gaps": [{"marks": 1, "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "strictPrecision": false, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "allowFractions": false, "precisionType": "dp", "showFeedbackIcon": true, "scripts": {}, "precisionPartialCredit": 0, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "precision": "2", "minValue": "-g/(2*f)", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "-g/(2*f)", "notationStyles": ["plain", "en", "si-en"]}, {"marks": 1, "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "strictPrecision": false, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "allowFractions": false, "precisionType": "dp", "showFeedbackIcon": true, "scripts": {}, "precisionPartialCredit": 0, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "precision": "2", "minValue": "g^2/(4*f)-g^2/(2*f)+h", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "g^2/(4*f)-g^2/(2*f)+h", "notationStyles": ["plain", "en", "si-en"]}, {"marks": 1, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "variableReplacements": [], "showCorrectAnswer": true, "expectedvariablenames": [], "showFeedbackIcon": true, "answer": "2*{f}*x+{g}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme", "checkingaccuracy": 0.001, "showpreview": true, "scripts": {}, "vsetrange": [0, 1]}, {"marks": 1, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "variableReplacements": [], "showCorrectAnswer": true, "expectedvariablenames": [], "showFeedbackIcon": true, "answer": "2*{f}", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme", "checkingaccuracy": 0.001, "showpreview": true, "scripts": {}, "vsetrange": [0, 1]}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "distractors": ["", ""], "minMarks": 0, "displayType": "radiogroup", "choices": ["<p>maximum</p>", "<p>minimum</p>"], "showFeedbackIcon": true, "maxMarks": "0", "matrix": ["if(maximum, 1, 0)", "if(maximum, 0, 1)"], "shuffleChoices": false, "type": "1_n_2", "displayColumns": 0, "showCorrectAnswer": true, "scripts": {}}], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "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": {}, "showFeedbackIcon": true, "showCorrectAnswer": true}, {"marks": 0, "gaps": [{"marks": 1, "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "variableReplacements": [], "showCorrectAnswer": true, "expectedvariablenames": [], "showFeedbackIcon": true, "answer": "{z}-2*{w}*t", "checkingtype": "absdiff", "checkvariablenames": false, "type": "jme", "checkingaccuracy": 0.001, "showpreview": true, "scripts": {}, "vsetrange": [0, 1]}, {"marks": 1, "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "strictPrecision": false, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "allowFractions": false, "precisionType": "dp", "showFeedbackIcon": true, "scripts": {}, "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "2", "minValue": "z^2/(4w)", "type": "numberentry", "correctAnswerFraction": false, "maxValue": "z^2/(4w)", "notationStyles": ["plain", "en", "si-en"]}], "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "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": {}, "showFeedbackIcon": true, "showCorrectAnswer": true}], "rulesets": {"std": ["all", "fractionNumbers"]}, "ungrouped_variables": ["z", "c", "b", "d", "f", "w", "a", "g", "h", "t", "maximum"], "variables": {"a": {"definition": "random(0..10#0.5)", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "name": "a"}, "w": {"definition": "random(2..5#0.1)", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "name": "w"}, "f": {"definition": "random(-21..21#2)", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "name": "f"}, "h": {"definition": "random(0..5#0.5)", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "name": "h"}, "b": {"definition": "random(2..5)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "b"}, "d": {"definition": "random(2..5)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "d"}, "g": {"definition": "random(-10..10#1)", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "name": "g"}, "z": {"definition": "random(20..30#0.5)", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "name": "z"}, "t": {"definition": "random(0..1#0.1)", "templateType": "randrange", "description": "", "group": "Ungrouped variables", "name": "t"}, "maximum": {"definition": "f<0", "templateType": "anything", "description": "<p>Is the stationary point a maximum?</p>", "group": "Ungrouped variables", "name": "maximum"}, "c": {"definition": "random(2..7)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "c"}}, "statement": "", "variable_groups": [], "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>", "preamble": {"css": "", "js": ""}, "type": "question", "contributors": [{"name": "Nigel Atkins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/275/"}]}]}], "contributors": [{"name": "Nigel Atkins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/275/"}]}