OCR A Level: Further Pure 2 - Iteration [June 2008 Q4]

Algebra Level pending

\((\text{i})\) Sketch, on the same diagram, \(y=\text{sech} x\) and \(y=x^2\).

\((\text{ii})\) Use the definition of \(\text{sech} x\) in terms of \(e^x\) and \(e^{-x}\) to show the \(x\)-coordinates of the points of intersection are solutions to the equation

\[x^2= \dfrac{2e^x}{e^{2x}+1}.\]

\((\text{iii})\) The iteration \({ x }_{ n+1 }^{ 2 }= \dfrac{2e^{x_n}}{e^{2 x_n }+1}\) can be used to find the positive root. With initial value \(x_1=1\), we obtain \(x_2 = 0.8050\), \(x_3 = 0.8633\), \(x_4 = 0.8463\) and \(x_5 = 0.8513\), correct to 4 decimal places. State, with a reason, whether this iteration produces a "staircase" or a "cobweb" diagram.

Input 1 if the iteration forms a "staircase", or 0 if it forms a "cobweb".

There are 3 marks available for part (i), 3 marks for part (ii) and 2 marks for part (iii).
In total, this question is worth 11.1% of all available marks in the paper.

This is part of the set OCR A Level Problems.

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