| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| 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-step Further Maths question requiring systematic differentiation of a differential equation to find Taylor coefficients. While the technique is methodical (substitute x=1 after each differentiation), it requires careful algebraic manipulation and understanding of implicit differentiation of DEs. The conceptual demand is moderate but execution requires precision across multiple parts. |
| Spec | 4.08a Maclaurin series: find series for function4.10e Second order non-homogeneous: complementary + particular integral |
**(i)** Subst$^{\mathrm{g}}$. $x = 1$, $\mathrm{f}(1) = 2$ and $\mathrm{f}'(1) = 3$ into $(*)$ $\Rightarrow$ $\mathrm{f}''(1) = 5$ **M1 A1**
**[2]**
**(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**
Subst$^{\mathrm{g}}$. $x = 1$, $\mathrm{f}'(1) = 3$ and $\mathrm{f}''(1) = 5$ into this $\Rightarrow \mathrm{f}'''(1) = -12$ **ft** their $\mathrm{f}''(1)$ **M1 A1**
**[4]**
**(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**; **A1 A1**
2nd two terms **ft (i) & (ii)'s answers**
**[3]**
**(iv)** Subst$^{\mathrm{g}}$. $x = 1.1$ $\Rightarrow$ $\mathrm{f}(1.1) \approx 2.323$ to 3d.p. **cso** (i.e. exactly this answer) **M1 A1**
**[2]**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 2012 Q8 [11]}}