| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Taylor series |
| Type | Maclaurin series for ln(trigonometric expressions) |
| Difficulty | Challenging +1.3 This is a Further Maths question requiring systematic differentiation and Maclaurin series construction. Parts (a)-(c) involve routine calculus with some algebraic manipulation to express higher derivatives in terms of earlier ones. Part (d) requires evaluating derivatives at x=0 and assembling the series, which is methodical but not trivial. The question is structured to guide students through the process, making it more accessible than an unscaffolded series derivation, but still requires careful execution across multiple steps. |
| Spec | 1.07l Derivative of ln(x): and related functions4.08a Maclaurin series: find series for function |
6 It is given that $y = \ln ( 1 + \sin x )$.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item Show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \mathrm { e } ^ { - y }$.
\item Express $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$ in terms of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\mathrm { e } ^ { - y }$.
\item Hence, by using Maclaurin's theorem, find the first four non-zero terms in the expansion, in ascending powers of $x$, of $\ln ( 1 + \sin x )$.
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2012 Q6 [11]}}