Question 0819

Maclaurin Series

2019 Paper 2 Question 4 Variant

Question

(i)

Given that

{f(x)=\sec 3 x,}

find

{f'(x)}

and

{f''(x).}

Hence, or otherwise, find the Maclaurin series for ${f(x),}$ up to and including the term in ${x^2.}$

[5]

(ii)

Use your series from part (i) to estimate

\int_0^{0.03} \sec 3 x \; \mathrm{d}x,

correct to 5 decimal places.

[2]

(iii)

Use your calculator to find

\int_0^{0.03} \sec 3 x \; \mathrm{d}x,

correct to 5 decimal places.

[1]

(iv)

Comparing your answers to parts (ii) and (iii), and with reference to the value of

{x,}

comment on the accuracy of your approximations.

[2]

(v)

Explain why a Maclaurin series for

{g(x)=\cosec 3 x}

cannot be found.

[1]