The function $f$ is defined by $f(x) = {{\text{e}}^{ - x}}\cos x + x - 1$.

By finding a suitable number of derivatives of $f$, determine the first non-zero term in its Maclaurin series.

$f(0) = 0$     A1

$f'(x) =  - {{\text{e}}^{ - x}}\cos x - {{\text{e}}^{ - x}}\sin x + 1$     M1A1

$f'(0) = 0$     (M1)

$f''(x) = 2{{\text{e}}^{ - x}}\sin x$     A1

$f''(0) = 0$

${f^{(3)}}(x) =  - 2{{\text{e}}^{ - x}}\sin x + 2{{\text{e}}^{ - x}}\cos x$     A1

${f^{(3)}}(0) = 2$

the first non-zero term is $\frac{{2{x^3}}}{{3!}}\;\;\;\left( { = \frac{{{x^3}}}{3}} \right)$     A1

Note:     Award no marks for using known series.

[7 marks]

Most students had a good understanding of the techniques involved with this question. A surprising number forgot to show $f(0) = 0$. Some candidates did not simplify the second derivative which created extra work and increased the chance of errors being made.