// Numbas version: exam_results_page_options {"name": "Maclaurin series (first three terms) - sine", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"css": "", "js": ""}, "parts": [{"checkVariableNames": false, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "type": "jme", "showFeedbackIcon": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "showPreview": true, "prompt": "

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

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

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

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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} \\cos{\\var{a}x}\\\\ f''(x)&=& -\\var{a^2}\\sin{\\var{a}x}\\\\ f'''(x) &=& -\\var{a^3}\\cos{\\var{a}x} \\\\ f^{(iv)}(x) &=& \\var{a^4}\\sin\\var{a}x \\\\ f^{(v)}(x) &=& \\var{a^5} \\cos\\var{a} x\\end{eqnarray*} \\]

Evaluating at $x=0$ gives:
\\[\\begin{eqnarray*} f(0)&=& 0\\\\ f'(0)&=& \\var{a}\\\\ f''(0)&=& 0 \\\\ f'''(0) &=&-\\var{a^3} \\\\ f^{(iv)}(0) &=& 0 \\\\ f^{(v)}(0) &=& \\var{a^5}\\end{eqnarray*}\\]

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

\\[ \\var{a}x - \\frac{\\var{a^3}}{6}x^3 + \\frac{\\var{a^5}}{120}x^5 \\]

For further information see Chapter 2 (Series) notes.

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Find the first three non-zero terms in the Maclaurin series for $\\sin(x)$.

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