AQA FP3 2009 June — Question 6

Exam BoardAQA
ModuleFP3 (Further Pure Mathematics 3)
Year2009
SessionJune
TopicTaylor series
TypeMaclaurin series for composite exponential/root functions

6 The function f is defined by $$\mathrm { f } ( x ) = ( 9 + \tan x ) ^ { \frac { 1 } { 2 } }$$
    1. Find \(f ^ { \prime \prime } ( x )\).
    2. By using Maclaurin's theorem, show that, for small values of \(x\), $$( 9 + \tan x ) ^ { \frac { 1 } { 2 } } \approx 3 + \frac { x } { 6 } - \frac { x ^ { 2 } } { 216 }$$
  1. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { f ( x ) - 3 } { \sin 3 x } \right]$$