2017 HCI P1 Q7
(i)
It is given that $\ln y=2\sin x$. Show that $\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}=-y\ln y+\frac{1}{y}{{\left( \frac{\text{d}y}{\text{d}x} \right)}^{2}}$.
[2]
(ii)
Find the first four terms of the Maclaurin series for $y$ in ascending powers of $x$.
[4]
(iii)
Using appropriate expansions from the List of Formulae (MF26), verify the expansion found in part (ii).
[2]
(iv)
Given that $x$ is sufficiently small for ${{x}^{4}}$ and higher powers of $x$ to be neglected, deduce an approximation for ${{\text{e}}^{\left( 2\sin x \right)-\ln \left( \sec x \right)}}$ in ascending powers of $x$.
[2]
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