| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Topic | Hyperbolic functions |
| Type | Arc length with hyperbolic curves |
| Difficulty | Challenging +1.8 This question requires differentiation of a composite hyperbolic function (non-trivial but systematic), then an arc length integral involving asymptotic analysis of $\sqrt{1 + \operatorname{cosech}^2 x}$ for large $x$, and finally geometric interpretation. The calculus is moderately challenging and requires understanding hyperbolic identities and asymptotic behavior, placing it well above average but not at the extreme difficulty level of multi-stage proof questions. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials4.07d Differentiate/integrate: hyperbolic functions8.06a Reduction formulae: establish, use, and evaluate recursively |
**Question 12** [3+5+2 marks]
**(i)**
$y = \ln\left(\tanh\frac{1}{2}x\right) \Rightarrow \frac{dy}{dx} = \frac{1}{\tanh\frac{1}{2}x}\cdot\frac{1}{2}\mathrm{sech}^2\frac{1}{2}x$ M1A1
$= \mathrm{cosech}\, x$ A1 **(AG)**
**[3]**
**(ii)(a)**
$L_n = \int_n^{2n}\sqrt{1+\mathrm{cosech}^2 x}\,\mathrm{d}x$ M1
$= \int_n^{2n}\coth x\,\mathrm{d}x$ A1
$= \left[\ln(\sinh x)\right]_n^{2n}$ A1 (correct integrn.)
$\ln\left(\frac{\sinh 2n}{\sinh n}\right) = \ln\left(\frac{e^{2n}-e^{-2n}}{e^n - e^{-n}}\right)$ M1
$\approx \ln\left(\frac{e^{2n}}{e^n}\right)$, for large $n$, $= \ln(e^n) = n$ A1 **legitimately**
**[5]**
OR: $\ln\left(\frac{\sinh 2n}{\sinh n}\right) = \ln(2\cosh n) = \ln(e^n+e^{-n})$ **M1**
$\approx \ln(e^n)$ for large $n$, $= n$ **A1** **legitimately**
**(b)**
Method (sketch or statement) to indicate that $C$ asymptotically "merges" with the $x$-axis so that $C$ is approximately a horizontal straight-line from $(n,0)$ to $(2n,0)$ M1
A1
**[2]**12 The curve $C$ has equation $y = \ln \left( \tanh \frac { 1 } { 2 } x \right)$, for $x > 0$.\\
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosech } x$.\\
(ii) For positive integers $n$, the length of the arc of $C$ between $x = n$ and $x = 2 n$ is $L _ { n }$.
\begin{enumerate}[label=(\alph*)]
\item Show by calculus that, when $n$ is large, $L _ { n } \approx n$.
\item Explain how this result corresponds to the shape of $C$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q12 [10]}}