Standard +0.8 This is a substantial Further Maths question requiring multiple Taylor series techniques: standard expansions (part a), differentiation with chain/product rules to find higher derivatives (part b), applying Maclaurin's theorem systematically, and using series to evaluate a complex limit (part c). While the individual components are standard FP3 material, the multi-part structure, the need to find third derivatives using implicit relationships, and the final limit requiring careful series substitution and simplification make this moderately challenging even for Further Maths students.
Write down the expansions in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\) of:
\(\cos x + \sin x\);
\(\quad \ln ( 1 + 3 x )\).
It is given that \(y = \mathrm { e } ^ { \tan x }\).
Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = ( 1 + \tan x ) ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x }\).
Find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) when \(x = 0\).
Hence, by using Maclaurin's theorem, show that the first four terms in the expansion, in ascending powers of \(x\), of \(\mathrm { e } ^ { \tan x }\) are
$$1 + x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 }$$
Find
$$\lim _ { x \rightarrow 0 } \left[ \frac { \mathrm { e } ^ { \tan x } - ( \cos x + \sin x ) } { x \ln ( 1 + 3 x ) } \right]$$
7
\begin{enumerate}[label=(\alph*)]
\item Write down the expansions in ascending powers of $x$ up to and including the term in $x ^ { 3 }$ of:
\begin{enumerate}[label=(\roman*)]
\item $\cos x + \sin x$;
\item $\quad \ln ( 1 + 3 x )$.
\end{enumerate}\item It is given that $y = \mathrm { e } ^ { \tan x }$.
\begin{enumerate}[label=(\roman*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = ( 1 + \tan x ) ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x }$.
\item Find the value of $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ when $x = 0$.
\item Hence, by using Maclaurin's theorem, show that the first four terms in the expansion, in ascending powers of $x$, of $\mathrm { e } ^ { \tan x }$ are
$$1 + x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 }$$
\end{enumerate}\item Find
$$\lim _ { x \rightarrow 0 } \left[ \frac { \mathrm { e } ^ { \tan x } - ( \cos x + \sin x ) } { x \ln ( 1 + 3 x ) } \right]$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2011 Q7 [14]}}