OCR A Level: Core 3 - Iteration [June 2007 Q6]

Calculus Level 3

$(\text{i})$ Given $\displaystyle \int _{ 0 }^{ a }{ 6{ e }^{ 2x }+x } \, dx=42$ , show that $a=\dfrac{1}{2}\ln\left (15-\dfrac{a^2}{6}\right ).$

$(\text{ii})$ Use an iterative formula to find $a$ , correct to 3 decimal places . Use a starting value of 1 and show the result of each iteration.

Input $1000 \times$ your answer to part $(\text{ii})$ .

There are 5 marks available for part (i) and 4 marks for part (ii).

In total, this question is worth 12.5% of all available marks in the paper.

This is part of the set OCR A Level Problems .

The answer is 1344.

1 solution

Michael Fuller
Feb 29, 2016

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

$\color{#3D99F6}{\boxed{\text{M1}}}$ Obtain integral of form $k_1 e^{2x} + k_2 x^2$ .

$\color{#20A900}{\boxed{\text{A1}}}$ Obtain correct $3 e^{2x} + \dfrac{1}{2} x^2$ .

$\color{#20A900}{\boxed{\text{A1}}}$ Obtain $3 e^{2a} + \dfrac{1}{2} a^2 - 3$ .

$\color{#3D99F6}{\boxed{\text{M1}}}$ Equate definite integral to 42 and attempt rearrangement.

$\color{#20A900}{\boxed{\text{A1}}}$ Confirm $a=\dfrac{1}{2}\ln\left (15-\dfrac{a^2}{6}\right )$ .

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

$\color{#69047E}{\boxed{\text{B1}}}$ Obtain correct first iterate 1.3848...

$\color{#3D99F6}{\boxed{\text{M1}}}$ Attempt correct process to find at least 2 iterates.

$\color{#20A900}{\boxed{\text{A1}}}$ Obtain at least 3 correct iterates.

$\color{#20A900}{\boxed{\text{A1}}}$ Obtain 1.344.

$[1 \rightarrow 1.34844 \rightarrow 1.34382 \rightarrow 1.34389 ]$