10 Let \(\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )\).
- Determine \(\mathrm { f } ^ { \prime \prime } ( x )\).
- Determine the first two non-zero terms of the Maclaurin expansion for \(\mathrm { f } ( x )\).
- 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.