AQA FP3 2013 June — Question 6

Exam BoardAQA
ModuleFP3 (Further Pure Mathematics 3)
Year2013
SessionJune
TopicTaylor series
TypeMaclaurin series for composite exponential/root functions

6 It is given that \(y = ( 4 + \sin x ) ^ { \frac { 1 } { 2 } }\).
  1. Express \(y \frac { \mathrm {~d} y } { \mathrm {~d} x }\) in terms of \(\cos x\).
  2. Find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) when \(x = 0\).
  3. Hence, by using Maclaurin's theorem, find the first four terms in the expansion, in ascending powers of \(x\), of \(( 4 + \sin x ) ^ { \frac { 1 } { 2 } }\).
    (2 marks)