CAIE A-Level Further Math 2.1.4 Hyperbolic functions

Practise deriving inverse hyperbolic logarithmic forms and differentiating inverse hyperbolic expressions in constrained domains.

Exam points

derive tanh^-1 x as a logarithmic expression by rearranging exponentials
use implicit differentiation on sech u=t to obtain d/dt(sech^-1 t)
find exact inverse hyperbolic values and give the answer in logarithmic form

[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]

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

1

М1

Write in exponential form

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

1

M1

Rearrange

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

1

A1

AG CAO

3

All question bank results loaded

2.1.4—Hyperbolic functions question 1