Question 1 - A-Level Maths

A curve has polar equation \(r = 2 ( \cos \theta + \sin \theta )\) for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\).
1. Show that a cartesian equation of the curve is \(x ^ { 2 } + y ^ { 2 } = 2 x + 2 y\). Hence or otherwise sketch the curve.
2. Find, by integration, the area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 2 } \pi\). Give your answer in terms of \(\pi\).
1. Given that \(\mathrm { f } ( x ) = \arctan \left( \frac { 1 } { 2 } x \right)\), find \(\mathrm { f } ^ { \prime } ( x )\).
2. Expand \(\mathrm { f } ^ { \prime } ( x )\) in ascending powers of \(x\) as far as the term in \(x ^ { 4 }\). Hence obtain an expression for \(\mathrm { f } ( x )\) in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\).