Given that $\mathrm{f}\left( x \right)=\sec 2x$, find $\mathrm{f}'\left( x \right)$ and \(\

Question

Given that $\mathrm{f}\left( x \right)=\sec 2x$, find $\mathrm{f}'\left( x \right)$ and $\mathrm{f}''\left( x \right)$. Hence, or otherwise, find the Maclaurin series for $\mathrm{f}\left( x \right)$, up to and including the term in ${{x}^{2}}$.[5] Use your series from part (i) to estimate $\int_{0}^{0.02} \sec 2x \mathrm{d}x$, correct to $5$ decimal places. [2] Use your calculator to find $\int_{0}^{0.02} \sec 2x \mathrm{d}x$, correct to $5$ decimal places. [1] Comparing your answers to parts (ii) and (iii), and with reference to the value of $x$, comment on the accuracy of your approximations. [2] Explain why a Maclaurin series for $\mathrm{g}\left( x \right)=\operatorname{cosec}2x$ cannot be found.[1]

Accepted Answer

(i) $1 + 2x^2 + \dots$ (ii) $0.02001$ ($5$ d.p.) (iii) $0.02001$ ($5$ d.p.) (v) $\mathrm{g}(x)$ undefined at $x = 0$

N2019 P2 Q4

N2019 P2 Q4

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