Use the first three non-zero terms of the Maclaurin series for $\cos x$ to find the Maclaurin series

Question

Use the first three non-zero terms of the Maclaurin series for $\cos x$ to find the Maclaurin series for $\mathrm{g}\left( x \right)$, where $\mathrm{g}\left( x \right)={{\cos }^{6}}x$, up to and including the term in ${{x}^{4}}$. [3] Use your answer to part (i) to give an approximation for $\displaystyle\int_{0}^{a}{\mathrm{g}\left( x \right)}\mathrm{ d}x$ in terms of $a$, and evaluate this approximation in the case where $a=\frac{1}{4}\pi $. [3] Use your calculator to find an accurate value for $\displaystyle\int_{0}^{\frac{1}{4}\pi }{\mathrm{g}\left( x \right)}\mathrm{ d}x$. Why is the approximation in part (ii)(a) not very good? [2]

Accepted Answer

(i) $1 - 3x^{2} + 4x^{4}$; (ii)(a) $0.540$; (b) $0.475$

N2011 P1 Q4

N2011 P1 Q4

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