Question 1 - A-Level Maths

Exam Board	OCR MEI
Module	FP2 (Further Pure Mathematics 2)
Year	2009
Session	January
Topic	Taylor series
Type	Series for reciprocal functions

1. By considering the derivatives of $\cos x$, show that the Maclaurin expansion of $\cos x$ begins $$1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 4 }$$
2. The Maclaurin expansion of $\sec x$ begins $$1 + a x ^ { 2 } + b x ^ { 4 }$$ where $a$ and $b$ are constants. Explain why, for sufficiently small $x$, $$\left( 1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 4 } \right) \left( 1 + a x ^ { 2 } + b x ^ { 4 } \right) \approx 1$$ Hence find the values of $a$ and $b$.
1. Given that $y = \arctan \left( \frac { x } { a } \right)$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a } { a ^ { 2 } + x ^ { 2 } }$.
2. Find the exact values of the following integrals. $$\begin{aligned} & \text { (A) } \int _ { - 2 } ^ { 2 } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x
  & \text { (B) } \int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 4 } { 1 + 4 x ^ { 2 } } \mathrm {~d} x \end{aligned}$$