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.

There are 6 marks available for part (i) and 4 marks for part (ii).
In total, this question is worth 13.9% of all available marks in the paper.

This is part of the set OCR A Level Problems.

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