| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 8 |
| Topic | Taylor series |
| Type | Hyperbolic power functions |
| Difficulty | Challenging +1.2 This is a Further Maths question involving hyperbolic functions and Maclaurin series. Part (a)(i) requires straightforward differentiation using chain rule and hyperbolic identities (cosh² x - sinh² x = 1), which is routine for FM students. Part (a)(ii) extends this pattern. Part (b) requires evaluating derivatives at x=0 using cosh(0)=1, sinh(0)=0, which is mechanical once the derivatives are found. The question is structured with clear guidance ('show that', 'hence') and tests standard FM techniques without requiring novel insight, placing it moderately above average A-level difficulty but well within expected FM scope. |
| Spec | 4.07d Differentiate/integrate: hyperbolic functions4.08a Maclaurin series: find series for function |
$y = \cosh^n x$ \quad $n \geq 5$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that
$$\frac{d^2y}{dx^2} = n^2\cosh^n x - n(n-1)\cosh^{n-2}x$$ [4]
\item Determine an expression for $\frac{d^4y}{dx^4}$ [2]
\end{enumerate}
\item Hence, or otherwise, determine the first three non-zero terms of the Maclaurin series for $y$, simplifying each coefficient and justifying your answer. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2024 Q15 [8]}}