<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">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 ${{x}^{4}}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use your answer to part (i) to give an approximation for $\int_{0}^{a}{\mathrm{g}\left( x \right)}\mathrm{ d}x$ in terms of $a$, and evaluate this approximation in the case where $a=\frac{1}{4}\pi $.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Use your calculator to find an accurate value for $\int_{0}^{\frac{1}{4}\pi }{\mathrm{g}\left( x \right)}\mathrm{ d}x$. Why is the approximation in part (ii)(a) not very good?</span></div><p style="text-align: right;">[2]</p></li></ol></li></ol>

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 ${{x}^{4}}$.

Use your answer to part (i) to give an approximation for $\displaystyle\displaystyle \int_{0}^{a}{\mathrm{g}\left( x \right)}\mathrm{ d}x$ in terms of $a$, and evaluate this approximation in the case where $a=\frac{1}{4}\pi $.

Use your calculator to find an accurate value for $\displaystyle\displaystyle \int_{0}^{\frac{1}{4}\pi }{\mathrm{g}\left( x \right)}\mathrm{ d}x$. Why is the approximation in part (ii)(a) not very good?

N2011 P1 Q4

N2011 P1 Q4

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