// Numbas version: exam_results_page_options
{"name": "Differentiation: Polynomial Functions - Max Growth", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: Polynomial Functions - Max Growth", "tags": [], "metadata": {"description": "<p>Calculating the gradient of a quadratic equation at a specific point and finding the stationary point (maximum) in a contextualised problem.&nbsp;</p>", "licence": "None specified"}, "statement": "<p>The discharge of water into a stream, $D$ dm$^3/$s, affects the growth of the fish in the stream.</p>\n<p>The average length of the fish, $x$ cm, is modelled by the curve:</p>\n<p>$ x=\\var{a}D^2+\\var{b}D+\\var{c} $</p>", "advice": "<p>a) We first need to find differentiate the formula of the curve in respect of D.</p>\n<p>\\[ \\frac{dx}{dD} = 2 \\times (\\var{a}) D+ \\var{b}+0 \\\\ \\implies \\frac{dx}{dD}= \\simplify{2*{a}}D+\\var{b} \\]</p>\n<p>Now to find the gradient at the point where $D=\\var{d_1}$ we need to substitute the value in the equation we found in the previous step.</p>\n<p>\\[ \\frac{dx}{dD}=\\simplify{2*{a}}\\times \\var{d_1} +\\var{b} \\\\ \\implies \\frac{dx}{dD}= \\simplify{2*{a}*{d_1}}+\\var{b} \\\\ \\implies \\frac{dx}{dD}= \\simplify{2*{a}*{d_1}+{b}} \\]</p>\n<p></p>\n<p>b) The discharge that results in the maximum average length of the fish can be calculated by solving the equation:&nbsp;</p>\n<p>\\[ \\frac{dx}{dD}=0\\]</p>\n<p>Therefore:</p>\n<p>\\[ \\simplify{2*{a}}D+\\var{b}=0 \\\\ \\implies \\simplify{2*{a}}D= - \\var{b} \\\\ \\implies D = \\frac{-\\var{b}}{\\simplify{2*{a}}} \\\\ \\implies D= \\simplify{{-{b}}/{2*{a}}} \\]</p>\n<p>Or, if rounded,&nbsp;</p>\n<p>\\[ D=\\var{ans_b_rounded}\\]</p>\n<p>c) Now that we know that $D=\\var{ans_b_rounded}$ is the discharge that results in the maximum average length we can substitute this value in the original equation of the curve and calculate the maximum length.</p>\n<p>So,&nbsp;</p>\n<p>\\[ x_{max}= \\var{a} \\times \\var{ans_b_rounded}^2 + \\var{b} \\times \\var{ans_b_rounded} + \\var{c} \\\\ \\implies x_{max}=\\var{a} \\times \\var{ans_b_rounded^2}+ \\var{b} \\times \\var{ans_b_rounded}+\\var{c} \\\\ \\implies x_{max}=\\simplify{{a*ans_b_rounded^2}}+ \\simplify{{b*ans_b_rounded}}+\\var{c} \\\\ \\implies x_{max}=\\simplify{{a*ans_b_rounded^2+b*ans_b_rounded+c}}\\]</p>\n<p>So, $x_{max}=\\var{ans_c_rounded}$.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Problem posing", "definition": "-random ( 50 .. 200)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Problem posing", "definition": "random(50..200)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": 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"absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the discharge which results in the largest average length of the fish.</p>\n<p><em>Give your answer rounded to 2 decimal places.</em></p>", "answer": "{ans_b_rounded}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the maximum average length of the fish.</p>\n<p><em>Give your answer rounded to 2 decimal places.</em></p>", "answer": "{ans_c_rounded}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": 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