CAIE A-Level Further Math A2 2.3.1 Hyperbolic Functions Questions

Practise differentiating hyperbolic and inverse hyperbolic functions, including parametric forms, with Further Mathematics Paper 2 questions and mark schemes.

Exam points

Differentiate hyperbolic and inverse hyperbolic functions, including parametric forms.

[Maximum number: 4]

Figure for Question CAIE A-Level Further Math A2 2.3.1 Hyperbolic Functions Questions question 1 — CAIE A-Level Further Math A2

The diagram shows part of the curve $y=x \operatorname{sech}^{2} x$ and its maximum point M.

Show that, at M,

2 x \tanh x-1=0

and verify that this equation has a root between 0.7 and 0.8 .

$\begin{aligned} \frac{\mathrm{d} y}{\mathrm{~d} x}=-2 x \operatorname{sech}^{2} x \tanh x+\operatorname{sech}^{2} x {\left[=1-\tanh ^{2} x-2 x \tanh x+2 x \tanh ^{3} x\right]} \end{aligned}$
Differentiating using product rule
and chain rule for M1.

$\operatorname{sech}^{2} x \neq 0 \Rightarrow 2 x \tanh x-1=0$
AG.

$\begin{aligned} 2(0.7) \tanh (0.7)-1=-0.154<0 2(0.8) \tanh (0.8)-1=0.062>0 \end{aligned}$
Shows sign change. Must write
down values correct to at least 1dp
for B 1 .

4

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CAIE A-Level Further Math A2 2.3.1 Hyperbolic Functions Questions question 1