// Numbas version: exam_results_page_options
{"name": "Andrew's copy of Curve Sketching using Calculus", "feedback": {"allowrevealanswer": true, "showtotalmark": true, "advicethreshold": 0, "intro": "", "feedbackmessages": [], "showanswerstate": true, "showactualmark": true}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "allQuestions": true, "shuffleQuestions": true, "percentPass": "60", "duration": 0, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "metadata": {"description": "<p>Critical points, absolute minimum, local maximum and minimum points, increasing and decreasing, concavity</p>\n<p>rebel</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "exam", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-shuffled", "pickQuestions": 0, "questions": [{"name": "Critical Points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "j"], "tags": ["rebel", "Rebel", "rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>$f(x) = \\var{a}x^2 + \\var{b}x$.</p>\n<p>Critical value $x = $ [[0]].</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "-{b}/(2{a})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$f(x) = \\frac{1}{3}x^3 \\simplify{-({c}+{d})/2x^2 + {c}{d}x}$</p>\n<p>Smaller critical value, $x = $[[0]]. Larger critical value, $x = $ [[1]].&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{c}", "minValue": "{c}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{d}", "minValue": "{d}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$f(p) = \\frac{p-1}{p^2-p+1}$</p>\n<p>Smaller critical value $p = $ [[0]]. Larger critical value $p = $ [[1]].</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "0", "minValue": "0", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "2", "minValue": "2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$g(y) = y^\\var{j}e^{\\simplify{-({j}+1)}y}$</p>\n<p>Smaller critical value, $y = $ [[0]]. Larger critical value, $y = $ [[1]].</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "0", "minValue": "0", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{j}/({j}+1)", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Find the critical numbers of the functions:</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..8 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(1..3) + c", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "j": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "j", "description": ""}}, "metadata": {"description": "<p>Critical Points</p>\n<p>rebel rebelmaths</p>\n<p>Rebel</p>\n<p></p>", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Absolute Minima", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}], "functions": {}, "ungrouped_variables": [], "tags": ["rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>$f(x) = 3x^2-12x+5$ on $[0,3]$</p>\n<p>Absolute minimum at $x = $ [[0]].</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "2", "minValue": "2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$f(x) = (x^2-1)^3$ on $[-1,2]$.</p>\n<p>Absolute minimum at $x = $ [[0]].</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "0", "minValue": "0", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$f(x) = \\ln(x^2+x+1)$ on $[-1,1]$.</p>\n<p>Absolute minimum at $x = $ [[0]].</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "-1/2", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$f(x) = x\\sqrt{4-x^2}$.</p>\n<p>Absolute minimum at $x = $ [[0]].</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "-1", "minValue": "-1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Find the absolute minimum value of $f$ on the given interval.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {}, "metadata": {"description": "<p>calculus min minimum</p>\n<p>rebel</p>\n<p>rebelmaths</p>\n<p></p>", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": " Optimisation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}], "functions": {}, "ungrouped_variables": ["c", "b", "d"], "tags": ["rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"prompt": "<p>Find two numbers whose difference is {b} and whose product is minimum.</p>\n<p>Smaller number : [[0]]. Larger number: [[1]].</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "-{b}/2", "minValue": "-{b}/2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{b}/2", "minValue": "{b}/2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Find two positive numbers whose product is $\\simplify{{c}^2}$ and whose sum is minimum.</p>\n<p>Smaller number: [[0]]. Larger number: [[1]].</p>\n<p>(If the numbers are the same size still be sure to fill in both boxes.)</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{c}", "minValue": "{c}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{c}", "minValue": "{c}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>The sum of two positive numbers is {d}. What is the smallest possible value for the sum of their squares?</p>", "allowFractions": false, "variableReplacements": [], "maxValue": "{d}^2/2", "minValue": "{d}^2/2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"c": {"definition": "random(5..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "2*random(10..30)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "2*random(3..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"description": "<p>Optimisation using calculus</p>\n<p>rebel</p>\n<p>rebelmaths</p>\n<p>Rebel</p>\n<p></p>", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Shapes of Graphs", "extensions": [], "custom_part_types": [], "resources": [["question-resources/graph1.pdf", "/srv/numbas/media/question-resources/graph1.pdf"], ["question-resources/graph1_1.jpg", "/srv/numbas/media/question-resources/graph1_1.jpg"], ["question-resources/graph2.jpg", "/srv/numbas/media/question-resources/graph2.jpg"], ["question-resources/graph3.jpg", "/srv/numbas/media/question-resources/graph3.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}], "functions": {}, "ungrouped_variables": [], "tags": ["rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"displayColumns": 0, "prompt": "<p><img src=\"resources/question-resources/graph1_1.jpg\"/></p>\n<p>The function is&nbsp;</p>", "matrix": [0, "1", 0], "shuffleChoices": false, "variableReplacements": [], "choices": ["<p>Decreasing only</p>", "<p>Increasing only</p>", "<p>Both increasing and decreasing</p>"], "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "distractors": ["", "", ""], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"displayColumns": 0, "prompt": "<p><img src=\"resources/question-resources/graph2.jpg\"/></p>\n<p>On the interval [-1,0] the function is:&nbsp;</p>", "matrix": [0, 0, "1"], "shuffleChoices": false, "variableReplacements": [], "choices": ["<p>Increasing only</p>", "<p>Decreasing only</p>", "<p>Both increasing and decreasing</p>"], "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "distractors": ["", "", ""], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"displayColumns": 0, "prompt": "<p><img src=\"resources/question-resources/graph3.jpg\"/></p>\n<p>On the interval [0,1] the derivative of the function is&nbsp;</p>", "matrix": [0, "1", 0], "shuffleChoices": false, "variableReplacements": [], "choices": ["<p>Positive</p>", "<p>Negative</p>", "<p>Cannot tell</p>"], "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "distractors": ["", "", ""], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {}, "metadata": {"description": "<p>using calculus</p>\n<p>rebel</p>\n<p>rebelmaths</p>\n<p>Rebel</p>", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Concavity", "extensions": [], "custom_part_types": [], "resources": [["question-resources/graph4.jpg", "/srv/numbas/media/question-resources/graph4.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b"], "tags": ["rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"displayColumns": 0, "prompt": "<p><img src=\"resources/question-resources/graph4.jpg\"/></p>\n<p>The function graphed above is:</p>", "matrix": ["1", 0, 0], "shuffleChoices": false, "variableReplacements": [], "choices": ["<p>Concave upward</p>", "<p>Concave downward</p>", "<p>Cannot tell</p>"], "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "distractors": ["", "", ""], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}, {"displayColumns": 0, "prompt": "<p>The graph of the function $f(x) = -\\var{a}x^2-\\var{b}x+\\var{c}$ is&nbsp;</p>\n<p></p>", "matrix": [0, "1", 0], "shuffleChoices": false, "variableReplacements": [], "choices": ["<p>Concave upward</p>", "<p>Concave downward</p>", "<p>Cannot tell</p>"], "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "distractors": ["", "", ""], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": 0}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..9 except a except b)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..9 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}}, "metadata": {"description": "<p>Concavity using Calculus</p>\n<p>rebel</p>\n<p>rebelmaths</p>\n<p>Rebel</p>", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Applications of differentiation ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "rulesets": {"std": ["all", "fractionNumbers"]}, "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": "", "ungrouped_variables": ["z", "c", "b", "d", "f", "w", "a", "g", "h", "t", "maximum"], "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "variables": {"g": {"definition": "random(-10 .. 10#1)", "group": "Ungrouped variables", "name": "g", "description": "", "templateType": "randrange"}, "w": {"definition": "random(2 .. 5#0.1)", "group": "Ungrouped variables", "name": "w", "description": "", "templateType": "randrange"}, "t": {"definition": "random(0 .. 1#0.1)", "group": "Ungrouped variables", "name": "t", "description": "", "templateType": "randrange"}, "c": {"definition": "random(2..7)", "group": "Ungrouped variables", "name": "c", "description": "", "templateType": "anything"}, "b": {"definition": "random(2..5)", "group": "Ungrouped variables", "name": "b", "description": "", "templateType": "anything"}, "f": {"definition": "random(-10 .. 10#1)", "group": "Ungrouped variables", "name": "f", "description": "", "templateType": "randrange"}, "z": {"definition": "random(20 .. 30#0.5)", "group": "Ungrouped variables", "name": "z", "description": "", "templateType": "randrange"}, "a": {"definition": "random(0 .. 10#0.5)", "group": "Ungrouped variables", "name": "a", "description": "", "templateType": "randrange"}, "h": {"definition": "random(0 .. 5#0.5)", "group": "Ungrouped variables", "name": "h", "description": "", "templateType": "randrange"}, "d": {"definition": "random(2..5)", "group": "Ungrouped variables", "name": "d", "description": "", "templateType": "anything"}, "maximum": {"definition": "f<0", "group": "Ungrouped variables", "name": "maximum", "description": "<p>Is the stationary point a maximum?</p>", "templateType": "anything"}}, "parts": [{"marks": 0, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "showFeedbackIcon": true, "variableReplacements": [], "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>", "scripts": {}, "customName": "", "gaps": [{"variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "strictPrecision": false, "minValue": "2*a*d+b", "maxValue": "2*a*d+b", "precisionType": "dp", "scripts": {}, "mustBeReduced": false, "customName": "", "unitTests": [], "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "marks": 1, "useCustomName": false, "mustBeReducedPC": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "allowFractions": false, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "type": "numberentry", "precision": "2"}, {"marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "answer": "2*{a}*x+{b}", "showFeedbackIcon": true, "vsetRange": [0, 1], "variableReplacements": [], "unitTests": [], "showPreview": true, "valuegenerators": [{"value": "", "name": "x"}], "scripts": {}, "checkingAccuracy": 0.001, "customName": "", "checkingType": "absdiff", "customMarkingAlgorithm": "", "failureRate": 1, "type": "jme", "checkVariableNames": false, "showCorrectAnswer": true}], "customMarkingAlgorithm": "", "type": "gapfill", "showCorrectAnswer": true}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "showFeedbackIcon": true, "variableReplacements": [], "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>", "scripts": {}, "customName": "", "gaps": [{"variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "strictPrecision": false, "minValue": "-g/(2*f)", "maxValue": "-g/(2*f)", "precisionType": "dp", "scripts": {}, "mustBeReduced": false, "customName": "", "unitTests": [], "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "marks": 1, "useCustomName": false, "mustBeReducedPC": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "allowFractions": false, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "type": "numberentry", "precision": "2"}, {"variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "strictPrecision": false, "minValue": "g^2/(4*f)-g^2/(2*f)+h", "maxValue": "g^2/(4*f)-g^2/(2*f)+h", "precisionType": "dp", "scripts": {}, "mustBeReduced": false, "customName": "", "unitTests": [], "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "marks": 1, "useCustomName": false, "mustBeReducedPC": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "allowFractions": false, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "type": "numberentry", "precision": "2"}, {"marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "answer": "2*{f}*x+{g}", "showFeedbackIcon": true, "vsetRange": [0, 1], "variableReplacements": [], "unitTests": [], "showPreview": true, "valuegenerators": [{"value": "", "name": "x"}], "scripts": {}, "checkingAccuracy": 0.001, "customName": "", "checkingType": "absdiff", "customMarkingAlgorithm": "", "failureRate": 1, "type": "jme", "checkVariableNames": false, "showCorrectAnswer": true}, {"marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "answer": "2*{f}", "showFeedbackIcon": true, "vsetRange": [0, 1], "variableReplacements": [], "unitTests": [], "showPreview": true, "valuegenerators": [], "scripts": {}, "checkingAccuracy": 0.001, "customName": "", "checkingType": "absdiff", "customMarkingAlgorithm": "", "failureRate": 1, "type": "jme", "checkVariableNames": false, "showCorrectAnswer": true}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "displayColumns": 0, "showFeedbackIcon": true, "shuffleChoices": false, "variableReplacements": [], "unitTests": [], "maxMarks": "0", "matrix": ["if(maximum, 1, 0)", "if(maximum, 0, 1)"], "showCellAnswerState": true, "distractors": ["", ""], "scripts": {}, "choices": ["<p>maximum</p>", "<p>minimum</p>"], "customName": "", "minMarks": 0, "customMarkingAlgorithm": "", "displayType": "radiogroup", "type": "1_n_2", "showCorrectAnswer": true}], "customMarkingAlgorithm": "", "type": "gapfill", "showCorrectAnswer": true}, {"marks": 0, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "extendBaseMarkingAlgorithm": true, "sortAnswers": false, "showFeedbackIcon": true, "variableReplacements": [], "unitTests": [], "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": "", "gaps": [{"marks": 1, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "vsetRangePoints": 5, "extendBaseMarkingAlgorithm": true, "answer": "{z}-2*{w}*t", "showFeedbackIcon": true, "vsetRange": [0, 1], "variableReplacements": [], "unitTests": [], "showPreview": true, "valuegenerators": [{"value": "", "name": "t"}], "scripts": {}, "checkingAccuracy": 0.001, "customName": "", "checkingType": "absdiff", "customMarkingAlgorithm": "", "failureRate": 1, "type": "jme", "checkVariableNames": false, "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "strictPrecision": false, "minValue": "z^2/(4w)", "maxValue": "z^2/(4w)", "precisionType": "dp", "scripts": {}, "mustBeReduced": false, "customName": "", "unitTests": [], "correctAnswerStyle": "plain", "customMarkingAlgorithm": "", "marks": 1, "useCustomName": false, "mustBeReducedPC": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "allowFractions": false, "precisionMessage": "You have not given your answer to the correct precision.", "type": "numberentry", "precision": "2"}], "customMarkingAlgorithm": "", "type": "gapfill", "showCorrectAnswer": true}], "type": "question"}, {"name": "Differentiation Power Rule:  3 parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}], "rulesets": {}, "variablesTest": {"maxRuns": "100", "condition": ""}, "metadata": {"licence": "All rights reserved", "description": ""}, "preamble": {"css": "", "js": ""}, "tags": [], "statement": "<p><strong></strong></p>\n<p><strong>Differentiate</strong> each of the following functions:</p>\n<p></p>\n<p></p>", "functions": {}, "ungrouped_variables": ["bc", "dc", "ec", "fc", "ap", "bp", "cp", "dp", "h", "k", "l", "g"], "variable_groups": [], "advice": "<p style=\"text-align: left;\"><strong>The Power Rule</strong>: &nbsp;If $y=x^n$, &nbsp; &nbsp; $\\frac{dy}{dx}=nx^{n-1}$ where $n$ is a rational number.</p>\n<p style=\"text-align: left;\">A special case of this is:</p>\n<p style=\"text-align: left;\"><strong>The Constant Rule</strong>: &nbsp;If $y=a$, where $a$ is a constant then &nbsp; $\\frac{dy}{dx}=0$&nbsp;</p>\n<p></p>\n<p style=\"text-align: left;\">For example differentiate:&nbsp;&nbsp; $y = {\\var{bc}}x^\\var{bp}$</p>\n<p>Using the rule above we must differentiate by multiplying by the original power and then reducing it by 1 as follows:</p>\n<p></p>\n<p style=\"text-align: left;\"><strong>$\\displaystyle\\frac{dy}{dx}$&nbsp; =&nbsp; ${\\var{bc}}$$({\\var{bp}}x^{\\var{bp-1}})$&nbsp; =&nbsp; $\\simplify{{bc*bp}x^{bp-1}}$</strong></p>\n<p></p>", "variables": {"l": {"description": "", "templateType": "randrange", "group": "Ungrouped variables", "name": "l", "definition": "random(2 .. 10#1)"}, "g": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "definition": "random(2..10)"}, "dc": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "dc", "definition": "random(-6..-2)"}, "bc": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "bc", "definition": "random(2..8)"}, "k": {"description": "", "templateType": "randrange", "group": "Ungrouped variables", "name": "k", "definition": "random(2 .. 10#2)"}, "dp": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "dp", "definition": "2"}, "bp": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "bp", "definition": "4"}, "h": {"description": "", "templateType": "randrange", "group": "Ungrouped variables", "name": "h", "definition": "random(1 .. 10#1)"}, "fc": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "fc", "definition": "random(-5..5 except 0 except 1 except -1)"}, "ap": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "ap", "definition": "5"}, "ec": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "ec", "definition": "random(-10..-5)"}, "cp": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "name": "cp", "definition": "3"}}, "parts": [{"scripts": {}, "type": "gapfill", "gaps": [{"valuegenerators": [{"name": "x", "value": ""}], "type": "jme", "vsetRangePoints": 5, "showCorrectAnswer": true, "failureRate": 1, "checkingAccuracy": 0.001, "marks": "1", "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "customName": "", "showPreview": true, "variableReplacements": [], "scripts": {}, "vsetRange": [0, 1], "answer": "{bc}*{dp}x^{dp-1}+{ec}", "useCustomName": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "checkVariableNames": false, "checkingType": "absdiff"}], "showCorrectAnswer": true, "unitTests": [], "useCustomName": false, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "prompt": "<p>$y=\\simplify{{bc}x^{dp}+{ec}x+{fc}}$</p>\n<p>$\\displaystyle \\frac{dy}{dx}=$ [[0]]</p>", "variableReplacements": [], "marks": 0, "customName": "", "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": ""}, {"scripts": {}, "type": "gapfill", "gaps": [{"valuegenerators": [{"name": "x", "value": ""}], "type": "jme", "vsetRangePoints": 5, "showCorrectAnswer": true, "answerSimplification": "all", "failureRate": 1, "checkingAccuracy": 0.001, "marks": 1, "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "customName": "", "showPreview": true, "variableReplacements": [], "scripts": {}, "vsetRange": [0, 1], "answer": "{dc}*{ap}*x^{ap-1}+{cp}/{fc}*x^{cp-1}", "useCustomName": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "checkVariableNames": false, "checkingType": "absdiff"}], "showCorrectAnswer": true, "unitTests": [], "useCustomName": false, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "prompt": "<p>$y=\\simplify{{dc}x^{ap}+{1}/{fc}x^{cp}}$</p>\n<p></p>\n<p>$\\displaystyle \\frac{dy}{dx}=$ [[0]]</p>", "variableReplacements": [], "marks": 0, "customName": "", "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": ""}, {"scripts": {}, "type": "gapfill", "gaps": [{"valuegenerators": [{"name": "x", "value": ""}], "type": "jme", "vsetRangePoints": 5, "showCorrectAnswer": true, "answerSimplification": "all", "failureRate": 1, "checkingAccuracy": 0.001, "marks": "1", "adaptiveMarkingPenalty": 0, "showFeedbackIcon": true, "customName": "", "showPreview": true, "variableReplacements": [], "scripts": {}, "vsetRange": [0, 1], "answer": "{fc*-bp}*x^{-bp-1}+{-ap}/{ec}*x^{-ap-1}", "useCustomName": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "checkVariableNames": false, "checkingType": "absdiff"}], "showCorrectAnswer": true, "unitTests": [], "useCustomName": false, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "prompt": "<p>$y=\\simplify{{fc}x^{-bp}+{1}/{ec}x^{-ap}}$</p>\n<p></p>\n<p>$\\displaystyle \\frac{dy}{dx}=$ [[0]]</p>\n<p></p>\n<p></p>", "variableReplacements": [], "marks": 0, "customName": "", "showFeedbackIcon": true, "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": ""}]}, {"name": "Vicky's copy of Differentiation 17 - Applications", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Vicky Hall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/659/"}], "functions": {}, "ungrouped_variables": ["cts", "ct"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>This section draws on the skills learnt the previous parts of the&nbsp;<em>'Differentiation'</em> series of questions, and some logical thinking about the physics of the problem.</p>\n<p>The steps within the part walk through the&nbsp;process.</p>", "rulesets": {}, "parts": [{"stepsPenalty": "2", "prompt": "<p>$y=\\var{ct}t-\\var{cts}t^2$</p>\n<p>Firstly, differentiate.</p>\n<p>$\\frac{dy}{dt}=$&nbsp;[[0]]</p>\n<p>Now use this result and your knowledge of differentiation to find the maximum height of the missile, rounding to the nearest whole number.</p>\n<p>$y=$&nbsp;[[1]]</p>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>The missile will follow a parabolic curve when height is plotted against time.</p>\n<p>It takes the same amount of time to reach its maximum as it does to fall back down.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}, {"vsetrangepoints": 5, "prompt": "<p>The maximum height will be at the stationary point.</p>\n<p>The stationary point can be found by equating $\\frac{dy}{dt}$ to $0$.</p>\n<p>$\\frac{dy}{dt}=0=$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{ct}-2{cts}t", "marks": "0", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"prompt": "<p>Therefore, $t=$</p>", "allowFractions": true, "variableReplacements": [], "maxValue": "{ct}/(2{cts})", "minValue": "{ct}/(2{cts})", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": "0", "type": "numberentry"}], "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{ct}-2{cts}t", "marks": "1", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": true, "variableReplacements": [], "maxValue": "{ct}({ct}/(2{cts}))-{cts}({ct}/(2{cts}))^2", "strictPrecision": false, "minValue": "{ct}({ct}/(2{cts}))-{cts}({ct}/(2{cts}))^2", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": "50", "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "0", "scripts": {}, "marks": "3", "showPrecisionHint": false, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<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{ct}t-\\var{cts}t^2. \\]</p>\n<p><strong>Calculate the maximum height reached by the missile.</strong></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"cts": {"definition": "random(3..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "cts", "description": "<p>coeff of t^2</p>"}, "ct": {"definition": "siground(random(30..90),1)", "templateType": "anything", "group": "Ungrouped variables", "name": "ct", "description": "<p>coeff of t</p>"}}, "metadata": {"description": "<p>Real life problems with differentiation</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Andrew's copy of CF Maths Portfolio - Differentiation 12 - Product Rule (with Chain Rule)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andrew Dunbar", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/770/"}], "functions": {}, "ungrouped_variables": ["c", "p"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>Patience and careful substitution is all that is necessary to differentiate these types of functions.</p>\n<p>Take the function:&nbsp;&nbsp;&nbsp; $y=4x^2(5x^3+2)^4$</p>\n<p>First label the two separate parts which would be differentiated with the <strong>product rule</strong>, namely:</p>\n<p>$u=4x^2\\;\\;\\;\\text{and}\\;\\;\\;v=(5x^3+2)^4$</p>\n<p>We also know that, by the product rule:</p>\n<p>$\\frac{dy}{dx}=u\\frac{dv}{dx}+v\\frac{du}{dx}$</p>\n<p>Looking at the substitution for $u$, the derivate can be easily found with the power rule:</p>\n<p>$\\frac{du}{dx}=(4\\times2)x^{2-1}=8x^1=8x$</p>\n<p>So far, from&nbsp;the product rule,&nbsp;we have:</p>\n<p>$\\frac{dy}{dx}=(4x^2)\\frac{dv}{dx}+(5x^3+2)^4(8x)$</p>\n<p></p>\n<p>Looking at the expression for $v=(5x^3+2)^4$ we can see that finding&nbsp;$\\frac{dv}{dx}$, however, requires the <strong>chain rule</strong>.</p>\n<p>Our usual&nbsp;subsitution letter $u$ has already been assigned an expression in the product rule part. This time, we can use $t$.</p>\n<p>By the chain rule:</p>\n<p>$\\frac{dv}{dx}=\\frac{dv}{dt}\\times\\frac{dt}{dx}$</p>\n<p>Let $t=5x^3+2$</p>\n<p>This means that: $v=t^4\\;\\;\\;\\therefore\\;\\;\\;\\frac{dv}{dt}=4t^3$</p>\n<p>Also, $t=5x^3+2\\;\\;\\;\\therefore\\;\\;\\;\\frac{dt}{dx}=15x^2$</p>\n<p>It follows that: $\\frac{dv}{dx}=\\frac{dv}{dt}\\times\\frac{dt}{dx}=4t^3\\times15x^2=60x^2t^3$</p>\n<p>Subsituting back the original expression for $t$:</p>\n<p>$\\frac{dv}{dx}=60x^2(5x^3+2)^3$</p>\n<p>We can now input our expression for $\\frac{dv}{dx}$ into the original product rule section.</p>\n<p></p>\n<p>$\\frac{dy}{dx}=(4x^2)\\frac{dv}{dx}+(5x^3+2)^4(8x)\\;\\;\\;\\therefore\\;\\;\\;\\frac{dy}{dx}=(4x^2)(60x^2(5x^3+2)^3)+(5x^3+2)^4(8x)$</p>\n<p>This is simplified further:</p>\n<p>$\\frac{dy}{dx}=240x^4(5x^3+2)^3+8x(5x^3+2)^4=8x(5x^3+2)^3\\left(30x^3+(5x^3+2)\\right)$</p>\n<p></p>\n<p>Finally:</p>\n<p>$\\frac{dy}{dx}=8x(5x^3+2)^3(35x^3+2)$</p>\n<p></p>\n<p></p>", "rulesets": {}, "parts": [{"vsetrangepoints": 5, "prompt": "<p>$\\simplify{{c[1]}x^3({c[2]}x+{c[3]})^{p[1]}}$</p>", "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "3{c[1]}x^2*({c[2]}x+{c[3]})^{p[1]}+{p[1]}{c[2]}{c[1]}x^3*({c[2]}x+{c[3]})^({p[1]}-1)", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>$\\simplify{{c[4]}x^{p[2]}({c[5]}x^2+{c[6]})^{p[3]}}$</p>", "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{p[2]}{c[4]}x^({p[2]}-1)*({c[5]}x^2+{c[6]})^{p[3]}+2{p[3]}{c[4]}{c[5]}x^({p[2]}+1)*({c[5]}x^2+{c[6]})^({p[3]}-1)", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "statement": "<p><strong>Differentiate the following expressions with respect to $x$ using the both product rule and the chain rule.</strong></p>\n<p><strong></strong>Simplify your answers as much as possible.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"p": {"definition": "repeat(random(2..6),10)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "c": {"definition": "repeat(random(-9..9 except 0),13)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}}, "metadata": {"description": "<p>Using the chain rule within product rule problems.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "contributors": [{"name": "Andrew Dunbar", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/770/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/graph1.pdf", "/srv/numbas/media/question-resources/graph1.pdf"], ["question-resources/graph1_1.jpg", "/srv/numbas/media/question-resources/graph1_1.jpg"], ["question-resources/graph2.jpg", "/srv/numbas/media/question-resources/graph2.jpg"], ["question-resources/graph3.jpg", "/srv/numbas/media/question-resources/graph3.jpg"], ["question-resources/graph4.jpg", "/srv/numbas/media/question-resources/graph4.jpg"]]}