// Numbas version: exam_results_page_options
{"name": "Stationary Points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Stationary Points", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "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>", "rulesets": {"std": ["all", "fractionNumbers"]}, "extensions": [], "variables": {"t": {"name": "t", "group": "Ungrouped variables", "definition": "random(0 .. 1#0.1)", "description": "", "templateType": "randrange"}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random(0 .. 5#0.5)", "description": "", "templateType": "randrange"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(0 .. 10#0.5)", "description": "", "templateType": "randrange"}, "maximum": {"name": "maximum", "group": "Ungrouped variables", "definition": "f<0", "description": "<p>Is the stationary point a maximum?</p>", "templateType": "anything"}, "w": {"name": "w", "group": "Ungrouped variables", "definition": "random(2 .. 5#0.1)", "description": "", "templateType": "randrange"}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "random(-10 .. 10#1)", "description": "", "templateType": "randrange"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "z": {"name": "z", "group": "Ungrouped variables", "definition": "random(20 .. 30#0.5)", "description": "", "templateType": "randrange"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..7)", "description": "", "templateType": "anything"}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(-10 .. 10#1)", "description": "", "templateType": "randrange"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["z", "c", "b", "d", "f", "w", "a", "g", "h", "t", "maximum"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "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>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "2*a*d+b", "maxValue": "2*a*d+b", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "2*{a}*x+{b}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "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>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "-g/(2*f)", "maxValue": "-g/(2*f)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "g^2/(4*f)-g^2/(2*f)+h", "maxValue": "g^2/(4*f)-g^2/(2*f)+h", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "2*{f}*x+{g}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "2*{f}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minMarks": 0, "maxMarks": "0", "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>maximum</p>", "<p>minimum</p>"], "matrix": ["if(maximum, 1, 0)", "if(maximum, 0, 1)"], "distractors": ["", ""]}], "sortAnswers": false}], "contributors": [{"name": "Matthew Mears", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1688/"}]}]}], "contributors": [{"name": "Matthew Mears", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1688/"}]}