| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Topic | Hyperbolic functions |
| Type | Find stationary points of hyperbolic curves |
| Difficulty | Challenging +1.3 This is a Further Maths calculus question requiring differentiation of hyperbolic functions and algebraic manipulation. Part (a) involves setting dy/dx = 0 and solving sech²(36x) = 1, which leads to straightforward logarithmic manipulation. Part (b) requires substituting back and simplifying using hyperbolic identities. While it involves multiple steps and hyperbolic functions (a FM topic), the techniques are standard and the path is clear once you know to differentiate and solve. The 'show that' format in part (b) provides the target, reducing difficulty. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07d Differentiate/integrate: hyperbolic functions |
The curve with equation
$$y = -x + \tanh(36x), \quad x \geq 0,$$
has a maximum turning point $A$.
\begin{enumerate}[label=(\alph*)]
\item Find, in exact logarithmic form, the $x$-coordinate of $A$.
[4 marks]
\item Show that the $y$-coordinate of $A$ is
$$\frac{\sqrt{35}}{6} - \frac{1}{36}\ln(6 + \sqrt{35}).$$
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2021 Q7 [7]}}