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{"name": "Differentiation: First Principles 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: First Principles 2", "tags": [], "metadata": {"description": "<p>Calculating the derivative of a function of the form $f(x)=ax^3$ from first principles.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Differentiate $f(x)=\\simplify{{a}*x^3}$ from first principles.</p>", "advice": "<p>For a function $f(x)$, its derivative is defined as \\[\\frac{df}{dx}=\\frac{f(x+h) - f(x)}{h} \\quad\\text{in the limit as $h$ tends to $0$.}\\]</p>\n<p>This is written \\[ \\frac{df}{dx}= \\lim_\\limits{h\\to 0}\\frac{f(x+h) - f(x)}{h}.\\]</p>\n<p>So, for the function $f(x)=\\simplify{{a}*x^3}$,</p>\n<p>\\begin{split}\\frac{df}{dx}&amp;=\\lim_\\limits{h\\to 0} \\frac{f(x+h)-f(x)}{h}\\\\ &amp;=\\lim_\\limits{h\\to 0} \\frac{[ \\simplify{{a}*(x+h)^3}] - [\\simplify{{a}*x^3}]}{h}\\\\&nbsp;&amp;=\\lim_\\limits{h\\to 0} \\frac{\\simplify{{a}*[x^3+3*x^2*h+3x*h^2+h^3]}-\\simplify{{a}*x^3}}{h}\\\\ &amp;=\\lim_\\limits{h\\to 0}\\frac{\\simplify{{a}*x^3}+\\simplify{3{a}x^2h}+\\simplify{3{a}x*h^2}+\\simplify{{a}*h^3}-\\simplify{{a}*x^3}}{h}\\\\ &amp;=\\lim_\\limits{h\\to 0} \\simplify{3{a}x^2}+\\simplify{3{a}*x*h}+\\simplify{{a}h^2}\\\\ &amp;=\\simplify{3{a}x^2}\\end{split}</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 9#1)", "description": "<p>x^3 coefficient</p>", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\frac{df}{dx}= $[[0]]</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0.", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{3*a}*x^2", "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": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}