| Exam Board | AQA |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Taylor series |
| Type | Maclaurin series for composite exponential/root functions |
| Difficulty | Standard +0.8 This is a multi-step Further Maths question requiring implicit differentiation, repeated differentiation of a composite function, and application of Maclaurin's theorem. While the techniques are standard for FP3, the algebraic manipulation and bookkeeping across three derivatives is non-trivial, placing it moderately above average difficulty. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates4.08a Maclaurin series: find series for function |
6 It is given that $y = ( 4 + \sin x ) ^ { \frac { 1 } { 2 } }$.
\begin{enumerate}[label=(\alph*)]
\item Express $y \frac { \mathrm {~d} y } { \mathrm {~d} x }$ in terms of $\cos x$.
\item Find the value of $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ when $x = 0$.
\item Hence, by using Maclaurin's theorem, find the first four terms in the expansion, in ascending powers of $x$, of $( 4 + \sin x ) ^ { \frac { 1 } { 2 } }$.\\
(2 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA FP3 2013 Q6 [9]}}