Question 0811

Maclaurin Series

2011 Paper 1 Question 4 Variant

Question

(i)

Use the first two non-zero terms of the Maclaurin series for

{\sin x}

to find the Maclaurin series for

{g(x),}

where

{g(x) = (1 + \sin^2 x)^7,}

up to and including the term in

{x^4.}

[3]

(iia)

Use your answer to part (i) to give an approximation for

\int_0^a g(x) \; \mathrm{d}x,

and evaluate this approximation in the case where

{a=\frac{1}{3}\pi.}

[3]

(iib)

Use your calculator to find an accurate value for

\int_0^{\frac{1}{3}\pi} g(x) \; \mathrm{d}x.

Why is the approximation in part (ii)(a) not very good?

[2]