| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795 (Pre-U Further Mathematics) |
| Session | Specimen |
| Topic | Taylor series |
| Type | Differential equation given |
| Difficulty | Challenging +1.2 This is a structured Taylor series question with clear scaffolding through parts (i) and (ii). Part (i) involves straightforward chain rule differentiation and algebraic manipulation. Part (ii) requires differentiating the given differential equation. Part (iii) applies standard Maclaurin series technique using successive derivatives at x=0. While it requires careful algebraic manipulation and understanding of the connection between differential equations and Taylor series, the question provides significant guidance and uses routine Further Maths techniques without requiring novel insight. |
| Spec | 4.08a Maclaurin series: find series for function4.08h Integration: inverse trig/hyperbolic substitutions |
6 (i) Given that $y = \cos ( \ln ( 1 + x ) )$, prove that
\begin{enumerate}[label=(\alph*)]
\item $\quad ( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = - \sin ( \ln ( 1 + x ) )$,
\item $( 1 + x ) ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } + y = 0$.\\
(ii) Obtain an equation relating $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(iii) Hence find the Maclaurin series for $y$, up to and including the term in $x ^ { 3 }$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q6}}