| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Inverse functions (inverse trig/hyperbolic) |
| Difficulty | Standard +0.3 This is a straightforward Maclaurin series question requiring routine differentiation of arctan using the chain rule, evaluation at x=0, and substitution into the standard formula. The derivatives are mechanical with no conceptual challenges, making it slightly easier than average for FP2 but still requiring careful algebraic manipulation. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.07l Derivative of ln(x): and related functions4.08a Maclaurin series: find series for function |
It is given that $f(x) = \tan^{-1}(1 + x)$.
\begin{enumerate}[label=(\roman*)]
\item Find $f(0)$ and $f'(0)$, and show that $f''(0) = -\frac{1}{2}$. [4]
\item Hence find the Maclaurin series for $f(x)$ up to and including the term in $x^2$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR FP2 2010 Q2 [6]}}