CAIE A-Level Further Math A2 2.1.4 Hyperbolic Functions Questions

Practise deriving inverse hyperbolic logarithmic forms and related derivatives with Further Mathematics Paper 2 questions.

Exam points

Derive and use logarithmic forms of inverse hyperbolic functions, including inverse tanh and sech derivatives.

[Maximum number: 3]

Starting from the definition of tanh in terms of exponentials, prove that $\tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$ . [3]

$x=\frac{\mathrm{e}^{u}-\mathrm{e}^{-u}}{\mathrm{e}^{u}+\mathrm{e}^{-u}}$

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М1

Write in exponential form

$\mathrm{e}^{2 u}(1-x)=1+x$

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Rearrange

$u=\tanh ^{-1} x=\frac{1}{2} \ln \left(\frac{1+x}{1-x}\right)$

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AG CAO

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CAIE A-Level Further Math A2 2.1.4 Hyperbolic Functions Questions question 1