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 .

The answer is 0.

1 solution

Michael Fuller
Feb 29, 2016

Marks for part $(\text{i})$ :

$\color{#69047E}{\boxed{\text{B1}}}$ Correct $y=x^2$ .

$\color{#69047E}{\boxed{\text{B1}}}$ Correct shape/asymptote of $y= \text{sech} x$ .

$\color{#69047E}{\boxed{\text{B1}}}$ Label $(0,1)$ intercept.

Marks for part $(\text{ii})$ :

$\color{#69047E}{\boxed{\text{B1}}}$ Define $\text{sech} x = \dfrac{2}{e^x + e^{-x}}$ .

$\color{#3D99F6}{\boxed{\text{M1}}}$ Equate your expression to $x^2$ and attempt to simplify.

$\color{#20A900}{\boxed{\text{A1}}}$ Clearly get answer given.

Marks for part $(\text{iii})$ :

$\color{#69047E}{\boxed{\text{B1}}}$ Cobweb

$\color{#69047E}{\boxed{\text{B1}}}$ The iterates oscillate above (>) and below (<) the root.