| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2019 |
| Session | Specimen |
| Marks | 2 |
| Topic | Taylor series |
| Type | Taylor series about x=1: differential equation with given conditions at x=1 |
| Difficulty | Challenging +1.2 This is a structured multi-part question on Taylor series requiring differentiation of a differential equation to find successive derivatives at x=1. While it involves several steps and careful algebraic manipulation, the method is entirely standard and procedural—substitute given values, differentiate the DE, evaluate at x=1, then apply the Taylor series formula. No novel insight or problem-solving is required, making it moderately above average difficulty for Further Maths students. |
| Spec | 4.08a Maclaurin series: find series for function |
**(a)** Substituting $x=1$, $f(1)=2$ and $f'(1)=3$ into $(*)$ $\Rightarrow f''(1) = 5$ **M1A1**
**Total: 2**
**(b)** Product Rule used twice; at least one bracket correct **M1**
$\{x^2f'''(x) + 2xf''(x)\} + \{(2x-1)f''(x) + 2f'(x)\} - 2f'(x) = 3e^{x-1}$ **A1**
Substituting $x=1$, $f'(1)=3$ and $f''(1)=5$ into this $\Rightarrow f'''(1) = -12$
**ft** their $f''(1)$ **M1A1** (M1A1ft)
**Total: 4**
**(c)** $f(x) = f(1) + f'(1)(x-1) + \frac{1}{2}f''(1)(x-1)^2 + \frac{1}{6}f'''(1)(x-1)^3 + \ldots$
Use of the Taylor series **M1**
$= 2 + 3(x-1) + \frac{5}{2}(x-1)^2 - 2(x-1)^3 + \ldots$ 1st two terms CAO;
2nd two terms **ft (a)** & **(b)**'s answers **A1A1** (A1A1ft)
**Total: 3**
**(d)** Substituting $x = 1.1 \Rightarrow f(1.1) \approx 2.323$ to 3d.p. CAO **M1A1**
**Total: 2**7 The function f satisfies the differential equation
$$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$
and the conditions $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$.
\begin{enumerate}[label=(\alph*)]
\item Determine $\mathrm { f } ^ { \prime \prime } ( 1 )$.
\item Differentiate (*) with respect to $x$ and hence evaluate $\mathrm { f } ^ { \prime \prime \prime } ( 1 )$.
\item Hence determine the Taylor series approximation for $\mathrm { f } ( x )$ about $x = 1$, up to and including the term in $( x - 1 ) ^ { 3 }$.
\item Deduce, to 3 decimal places, an approximation for $\mathrm { f } ( 1.1 )$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2019 Q7 [2]}}