| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Taylor series |
| Type | Differential equation given |
| Difficulty | Challenging +1.2 This is a structured Taylor series question with clear guidance through each step. Part (a) is routine differentiation using the product rule. Parts (b) and (c) involve standard techniques: substituting x=0 to find successive derivatives, then constructing the series and evaluating it. While it requires careful algebraic manipulation and understanding of Taylor series methodology, the question provides scaffolding and follows a predictable pattern typical of FP2 Taylor series problems. It's moderately harder than average A-level due to being Further Maths content, but straightforward within that context. |
| Spec | 4.08a Maclaurin series: find series for function |
$$(x^2 + 1)\frac{d^2y}{dx^2} = 2y^2 + (1 - 2x)\frac{dy}{dx}$$ (I)
\begin{enumerate}[label=(\alph*)]
\item By differentiating equation (I) with respect to $x$, show that
$$(x^2 + 1)\frac{d^3y}{dx^3} = (1 - 4x)\frac{d^2y}{dx^2} + (4y - 2)\frac{dy}{dx}.$$ (3)
Given that $y = 1$ and $\frac{dy}{dx} = 1$ at $x = 0$,
\item find the series solution for $y$, in ascending powers of $x$, up to and including the term in $x_3$.(4)
\item Use your series to estimate the value of $y$ at $x = -0.5$, giving your answer to two decimal places.(1)
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2008 Q9}}