Question 7 - A-Level Maths

Exam Board	AQA
Module	FP3 (Further Pure Mathematics 3)
Year	2014
Session	June
Marks	4
Topic	Taylor series
Type	Maclaurin series for ln(trigonometric expressions)

It is given that \(y = \ln ( \cos x + \sin x )\).
1. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 2 } { 1 + \sin 2 x }\).
2. Find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
1. Hence use Maclaurin's theorem to show that the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln ( \cos x + \sin x )\) are \(x - x ^ { 2 } + \frac { 2 } { 3 } x ^ { 3 }\).
2. Write down the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln ( \cos x - \sin x )\).
Hence find the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \left( \frac { \cos 2 x } { \mathrm { e } ^ { 3 x - 1 } } \right)\).
[0pt] [4 marks]
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