Standard +0.8 This is a multi-part Further Maths question requiring Maclaurin series manipulation, successive differentiation of composite functions (ln and exponential), and L'Hôpital's rule application. Part (b) involves careful algebraic manipulation of derivatives, and part (c) requires combining both series to evaluate a limit—all standard FP3 techniques but requiring precision and multiple steps, placing it moderately above average difficulty.
Write down the expansion of \(\cos 4 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 4 }\). Give your answer in its simplest form.
Given that \(y = \ln \left( 2 - \mathrm { e } ^ { x } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
(You may leave your expression for \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) unsimplified.)
Hence, by using Maclaurin's theorem, show that the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \left( 2 - \mathrm { e } ^ { x } \right)\) are
$$- x - x ^ { 2 } - x ^ { 3 }$$
Find
$$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln \left( 2 - \mathrm { e } ^ { x } \right) } { 1 - \cos 4 x } \right]$$
5
\begin{enumerate}[label=(\alph*)]
\item Write down the expansion of $\cos 4 x$ in ascending powers of $x$ up to and including the term in $x ^ { 4 }$. Give your answer in its simplest form.
\item \begin{enumerate}[label=(\roman*)]
\item Given that $y = \ln \left( 2 - \mathrm { e } ^ { x } \right)$, find $\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$.\\
(You may leave your expression for $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ unsimplified.)
\item Hence, by using Maclaurin's theorem, show that the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln \left( 2 - \mathrm { e } ^ { x } \right)$ are
$$- x - x ^ { 2 } - x ^ { 3 }$$
\end{enumerate}\item Find
$$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln \left( 2 - \mathrm { e } ^ { x } \right) } { 1 - \cos 4 x } \right]$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2010 Q5 [13]}}