9758/2019/P2/Q04

(i) Given that $\mathrm{f}(x)=\sec 2 x$, find $\mathrm{f}^{\prime}(x)$ and $\mathrm{f}^{\prime \prime}(x)$. Hence, or otherwise, find the Maclaurin series for $f(x)$, up to and including the term in $x^{2}$.

[5]

(ii) Use your series from part (i) to estimate

[2]

$\int_{0}^{0.02} \sec 2 x \mathrm{~d} x$, correct to 5 decimal places.

(iii) Use your calculator to find $\int_{0}^{0. [1] 02} \sec 2 x \mathrm{~d} x$, correct to 5 decimal places.

(ii) and (iii), and with reference to the value of $x$, comment on the accuracy of your approximations.
(v) Explain why a Maclaurin series for $\ [1] mathrm{g}(x)=\operatorname{cosec} 2 x$ cannot be found.