// Numbas version: exam_results_page_options {"name": "Differentiation: First Principles 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: First Principles 3", "tags": [], "metadata": {"description": "

Calculating the derivative of a function of the form $f(x)=ax^2+bx+c$ from first principles.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Differentiate $f(x)=\\simplify{{a}*x^2+{b}*x+{c}}$ from first principles.

", "advice": "

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 $0$.}\\]

This is written \\[ \\frac{df}{dx}= \\lim_\\limits{h\\to 0}\\frac{f(x+h) - f(x)}{h}.\\]

So, for the function $f(x)=\\simplify{{a}*x^2+{b}x+{c}}$,

\\begin{split}\\frac{df}{dx}&=\\lim_\\limits{h\\to 0} \\frac{f(x+h)-f(x)}{h}\\\\ &=\\lim_\\limits{h\\to 0} \\frac{[\\simplify{{a}*(x+h)^2} + \\simplify{{b}*(x+h)} + \\var{c}] - [\\simplify{{a}*x^2} +\\simplify{{b}*x} +\\var{c}]}{h}\\\\ &=\\lim_\\limits{h\\to 0} \\frac{\\simplify{{a}*x^2}+\\var{2*a}xh+\\simplify{{a}*h^2}+\\simplify{{b}*x}+\\simplify{{b}*h}+\\var{c}-\\simplify{{a}*x^2}-\\simplify{{b}*x}-\\var{c}}{h}\\\\ &=\\lim_\\limits{h\\to 0} \\frac{\\var{2*a}xh+\\simplify{{a}*h^2}+\\simplify{{b}*h}}{h}\\\\ &=\\lim_\\limits{h\\to 0} \\var{2*a}x+\\simplify{{a}*h}+\\var{b}\\\\ &=\\var{2*a}x+\\var{b}\\end{split}

", "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": "

x^2 coefficient

", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1 .. 9#1)", "description": "

x coefficient

", "templateType": "randrange", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1 .. 9#1)", "description": "

Constant coefficient

", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c"], "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": "

$\\frac{df}{dx}= $[[0]]

", "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": "{2*a}x+{b}", "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/"}]}