Differentiation - Tangents and Normals at a Point

Let \(f(x) = \mathrm{e}^{2x}\cos{x}\), \(-1 \leq x \leq 2\).

1. Show that \(f'(x) = \mathrm{e}^{2x}(2x \cos{x} - \sin{x})\).[3]

Let the line \(L\) be the normal to the curve of \(f\) at \(x =0\).

2. Find the equation of \(L\).[5]

The graph of \(f\) and the line \(L\) intersect at the point \( (0,1)\) and at a second point \(\mathrm{P}\).

3. 1. Find the \(x\)-coordinate of \(\mathrm{P}\).
   2. Find the area of the region enclosed by the graph of \(f\) and the line \(L\).[6]