| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2018 |
| Session | December |
| Marks | 7 |
| Topic | Taylor series |
| Type | Maclaurin series for ln(a+bx) |
| Difficulty | Standard +0.3 This is a straightforward Maclaurin series question requiring routine differentiation of ln(2+x) and evaluation at x=0. While it's Further Maths content, the execution is mechanical with no problem-solving insight needed—just apply the chain rule twice and substitute into the standard Maclaurin formula. Slightly above average difficulty due to being FM content, but otherwise standard textbook fare. |
| Spec | 4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f(x) = \ln(2+x) \Rightarrow f'(x) = \frac{1}{2+x} \Rightarrow f'(0) = \frac{1}{2}\) | M1, A1 | Differentiation |
| (b) \(f'(x) = \frac{1}{2+x} \Rightarrow f''(x) = -\frac{1}{(2+x)^2}\); \(\Rightarrow f''(0) = \frac{-1}{(2+0)^2} = -\frac{1}{4}\) | M1, A1 | Differentiation; AG |
| (c) \(f(0) = \ln 2\); \(f(x) = f(0) + xf'(0) + \frac{x^2}{2}f''(0) = \ln 2 + \frac{1}{2}x - \frac{1}{8}x^2\) | B1, B1, B1FT | soi; 2 two terms correct; 3rd term also correct. FT their values |
**(a)** $f(x) = \ln(2+x) \Rightarrow f'(x) = \frac{1}{2+x} \Rightarrow f'(0) = \frac{1}{2}$ | M1, A1 | Differentiation
**(b)** $f'(x) = \frac{1}{2+x} \Rightarrow f''(x) = -\frac{1}{(2+x)^2}$; $\Rightarrow f''(0) = \frac{-1}{(2+0)^2} = -\frac{1}{4}$ | M1, A1 | Differentiation; AG
**(c)** $f(0) = \ln 2$; $f(x) = f(0) + xf'(0) + \frac{x^2}{2}f''(0) = \ln 2 + \frac{1}{2}x - \frac{1}{8}x^2$ | B1, B1, B1FT | soi; 2 two terms correct; 3rd term also correct. FT their values
---3 You are given that $\mathrm { f } ( x ) = \ln ( 2 + x )$.
\begin{enumerate}[label=(\alph*)]
\item Determine the exact value of $\mathrm { f } ^ { \prime } ( 0 )$.
\item Show that $\mathrm { f } ^ { \prime \prime } ( 0 ) = - \frac { 1 } { 4 }$.
\item Hence write down the first three terms of the Maclaurin series for $\mathrm { f } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2018 Q3 [7]}}