// Numbas version: exam_results_page_options {"name": "Gemma's copy of Gemma's copy of Maclaurin series (first four terms) - exponential", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"failureRate": 1, "showCorrectAnswer": true, "checkingType": "absdiff", "variableReplacementStrategy": "originalfirst", "vsetRange": [0, 1], "checkingAccuracy": 0.001, "variableReplacements": [], "scripts": {}, "customMarkingAlgorithm": "", "prompt": "

Find the first four non-zero terms for the Maclaurin Series of $f(x)$.

Do not use factorials in your answer. For example, input 6 rather than 3!.

", "vsetRangePoints": 5, "checkVariableNames": false, "expectedVariableNames": [], "marks": 4, "showFeedbackIcon": true, "type": "jme", "unitTests": [], "answer": "1 + {a}x + {a^2}*x^2/2+{a^3}*x^3/6", "answerSimplification": "all,fractionNumbers,!collectNumbers", "showPreview": true, "extendBaseMarkingAlgorithm": true}], "advice": "

Recall that a Maclaurin series is simply a Taylor series about the point $x_0=0$.

The general expression for the Maclaurin series of a function $f(x)$ is

$f(x) = f(0) + f'(0)x + \\frac{f''(0)}{2!}x^2 + \\dots + \\frac{f^{n}(0)}{n!}x^n + \\dots$

For this example,
\\[\\begin{eqnarray*} f'(x)&=& \\var{a} e^{\\var{a}x}\\\\ f''(x)&=& \\var{a^2} e^{\\var{a}x}\\\\ f'''(x) &=& \\var{a^3} e^{\\var{a}x} \\end{eqnarray*}\\]

Evaluating at $x=0$ gives:
\\[\\begin{eqnarray*} f(0)&=& 1\\\\ f'(0)&=& \\var{a}\\\\ f''(0)&=& \\var{a^2} \\\\ f'''(0) &=& \\var{a^3}\\end{eqnarray*}\\]

Hence the first three non-zero terms of the Maclaurin series are:

\\[ 1 + \\var{a}x + \\frac{\\var{a^2}}{2}x^2+ \\frac{\\var{a^3}}{6} x^3\\]

For further information see Chapter 2 (Series) notes.

", "rulesets": {}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Find the first four non-zero terms in the Maclaurin series for $e^{ax}$.

"}, "preamble": {"css": "", "js": ""}, "extensions": [], "statement": "

Consider the function $f(x)=e^{\\var{a}x}$.

", "variables": {"a": {"definition": "random(2..9#1)", "group": "Ungrouped variables", "templateType": "randrange", "name": "a", "description": ""}, "number": {"definition": "6", "group": "Ungrouped variables", "templateType": "number", "name": "number", "description": ""}}, "name": "Gemma's copy of Gemma's copy of Maclaurin series (first four terms) - exponential", "variable_groups": [], "ungrouped_variables": ["a", "number"], "type": "question", "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}, {"name": "Antonia Wilmot-Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2410/"}, {"name": "Gemma Crowe", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2440/"}]}]}], "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}, {"name": "Antonia Wilmot-Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2410/"}, {"name": "Gemma Crowe", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2440/"}]}