Question 0808b

Maclaurin Series

2008 Paper 2 Question 1 Variant

Question

Let

{f(x) = \mathrm{e}^x \sin 3 x.}

(i)

Sketch the graph of

{y=f(x)}

for

{-3 \leq x \leq 3.}

[2]

(ii)

Find the series expansion of

{f(x)}

in ascending powers of

{x,}

up to and including the term in

{x^3.}

[3]

Denote the answer to part (ii) by

{g(x).}

(iii)

On the same diagram as in part (i), sketch the graph of

{y=g(x).}

Label the two graphs clearly.

[1]

(iv)

Find, for

{-3 \leq x \leq 3,}

the set of values of

{x}

for which the value of

{g(x)}

is within

{\pm 0.4}

of the value of

{f(x).}

[3]