Maclaurin series for inverse trigonometric functions

Questions asking to find Maclaurin series by differentiation for inverse trigonometric functions such as arcsin(x), arctan(x), or arccos(x).

3 questions

Given that $\mathrm { f } ( x ) = \arccos x$,
1. sketch the graph of $y = \mathrm { f } ( x )$,
2. show that $\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$,
3. obtain the Maclaurin series for $\mathrm { f } ( x )$ as far as the term in $x ^ { 3 }$.
A curve has polar equation $r = \theta + \sin \theta , \theta \geqslant 0$.
1. By considering $\frac { \mathrm { d } r } { \mathrm {~d} \theta }$ show that $r$ increases as $\theta$ increases. Sketch the curve for $0 \leqslant \theta \leqslant 4 \pi$.
2. You are given that $\sin \theta \approx \theta$ for small $\theta$. Find in terms of $\alpha$ the approximate area bounded by the curve and the lines $\theta = 0$ and $\theta = \alpha$, where $\alpha$ is small.

1. Determine $\mathrm { f } ^ { \prime \prime } ( x )$.
2. Determine the first two non-zero terms of the Maclaurin expansion for $\mathrm { f } ( x )$.
3. By considering the first two non-zero terms of the Maclaurin expansion for $\mathrm { f } ( x )$, find an approximation to $\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer correct to 6 decimal places.
By writing $\mathrm { f } ( x )$ as $\sin ^ { - 1 } ( x ) \times 1$, determine the value of $\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer in exact form.

Determine the first two non-zero terms, in ascending powers of $x$, of the Maclaurin series for $\mathrm { f } ( x )$, giving each coefficient in its simplest form.
Substitute $x = \frac { 1 } { 2 }$ into the answer to part (a) and hence find an approximate value for $\pi$ Give your answer in the form $\frac { p } { q }$ where $p$ and $q$ are integers to be determined.