| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2021 |
| Session | November |
| Marks | 11 |
| Topic | Taylor series |
| Type | Direct multiplication of series |
| Difficulty | Standard +0.3 This is a straightforward Further Maths calculus question requiring repeated differentiation using the chain rule, evaluation at x=0 for Maclaurin coefficients, and multiplication of two series expansions. While it involves multiple steps and careful algebra, all techniques are standard and the question provides clear scaffolding. The chain rule applications and series multiplication are routine for FM students, making this slightly easier than average overall. |
| Spec | 4.08a Maclaurin series: find series for function4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
In this question you must show all stages of your working.
The function $f$ is defined by $f(x) = (1 + 2x)^{\frac{1}{2}}$.
\begin{enumerate}[label=(\alph*)]
\item Find $f'''(x)$ (i.e. the third derivative of $f$) showing all your intermediate steps. Hence, find the Maclaurin series for $f(x)$ up to and including the $x^3$ term.
[8 marks]
\item Use the expansion of $e^x$ together with the result in part (a) to show that, up to and including the $x^3$ term,
$$e^x(1 + 2x)^{\frac{1}{2}} = 1 + 2x + x^2 + kx^3,$$
where $k$ is a rational number to be found.
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2021 Q8 [11]}}