# OCR A Level: Further Pure 2 - Hyperbolics [June 2008 Q7]

Calculus Level 3

It is given that $$f(x) = \tanh ^{ -1 }{ \left( \cfrac { 1-x }{ 2+x } \right) }$$, for $$x> -\dfrac{1}{2}$$

$$\text{(i)}$$ Show that $$f'(x) = -\dfrac{1}{1+2x}$$, and find $$f''(x)$$.

$$\text{(ii)}$$ Show that the first three terms of the Maclaurin series for $$f(x)$$ can be written as $$\ln a + bx + cx^2$$, for some constants $$a$$, $$b$$ and $$c$$.

Input $$a^2+b^2+c^2$$ as your answer.

×