| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | Specimen |
| Marks | 10 |
| Topic | Taylor series |
| Type | Taylor series about x=1: differential equation with given conditions at x=1 |
| Difficulty | Standard +0.8 This is a multi-part Taylor series question requiring differentiation of a differential equation to find higher derivatives, then constructing the series. While systematic, it demands careful algebraic manipulation, understanding of implicit differentiation of DEs, and multiple computational steps. It's moderately challenging for Further Maths but follows a standard template once the method is recognized. |
| Spec | 4.08a Maclaurin series: find series for function |
**(i)**
Substituting $x = 1$, $\mathrm{f}(1) = 2$ and $\mathrm{f}'(1) = 3$ into (*) $\Rightarrow \mathrm{f}''(1) = 5$ **M1 A1**
**(ii)**
$\{x^2\mathrm{f}''(x) + 2x\mathrm{f}'(x)\} + \{(2x-1)\mathrm{f}''(x) + 2\mathrm{f}'(x)\} - 2\mathrm{f}'(x) = 3\mathrm{e}^{x-1}$ **M1**
Product Rule used twice; at least one bracket correct **A1**
Substituting $x=1$, $\mathrm{f}'(1) = 3$ and $\mathrm{f}''(1) = 5$ into this $\Rightarrow \mathrm{f}'''(1) = -12$ **ft** their $\mathrm{f}''(1)$ **M1 A1**
**(iii)**
$\mathrm{f}(x) = \mathrm{f}(1) + \mathrm{f}'(1)(x-1) + \frac{1}{2}\mathrm{f}''(1)(x-1)^2 + \frac{1}{6}\mathrm{f}'''(1)(x-1)^3 + \ldots$ **M1**
Use of the Taylor series
$= 2 + 3(x-1) + \frac{5}{2}(x-1)^2 - 2(x-1)^3 + \ldots$ — 1st two terms CAO; 2nd two terms **ft (i) & (ii)'s answers** **A1 A1**
**(iv)**
Substituting $x = 1.1 \Rightarrow \mathrm{f}(1.1) \approx 2.323$ to 3d.p. CAO **M1 A1**
**Total: 10 marks**8 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 $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$.\\
(i) Determine $f ^ { \prime \prime } ( 1 )$.\\
(ii) Differentiate (*) with respect to $x$ and hence evaluate $\mathrm { f } ^ { \prime \prime \prime } ( 1 )$.\\
(iii) Hence determine the Taylor series approximation for $\mathrm { f } ( x )$ about $x = 1$, up to and including the term in $( x - 1 ) ^ { 3 }$.\\
(iv) Deduce, to 3 decimal places, an approximation for $\mathrm { f } ( 1.1 )$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q8 [10]}}