In this question you may use expansions from the List of Formulae (MF26). Find the Maclaurin expansi

Question

In this question you may use expansions from the List of Formulae (MF26).

Find the Maclaurin expansion of $\ln \left( 1+\cos 3x \right)$ in ascending powers of $x$, up to and including the term in ${{x}^{4}}$, for $0\le x<\frac{\pi }{3}$.[4]
Use your expansion from part (i) and to find an approximate value for $\int_{0}^{0.5} x \ln (1 + \cos 3x) \mathrm{d}x$, giving your answer to $5$ decimal places.
[2]
Use your calculator to find the value of $\int_{0}^{0.5} x \ln (1 + \cos 3x) \mathrm{d}x$ up to $5$ decimal places.
[1]
With the aid of a suitable diagram, comparing your answers in (ii) and (iii), comment on the accuracy of your approximations.
[2]

Accepted Answer

(i) $\ln 2 - \frac{9}{4}x^2 - \frac{27}{32}x^4 + \dots$ (ii) $0.04929$ ($5$ d.p.) (iii) $0.04900$ ($5$ d.p.) [Accurate] (iv) percentage error $0.5918\%$

2022 SAJC P2 Q3

2022 SAJC P2 Q3

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