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.