| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | Specimen |
| 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 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 proper application of Taylor series formula about x=1. More challenging than routine Taylor series questions but follows a clear algorithmic path once the method is recognized. |
| Spec | 4.08a Maclaurin series: find series for function4.10a General/particular solutions: of differential equations |
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 $f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3$.\\
(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 Q7}}