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

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 $c$ .

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

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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.

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This is part of the set OCR A Level Problems .