Differentiation - Tangents and Normals at a Point

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

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

Part of the graph of \(y = f(x)\), for \(-6 \leq x \leq 2\), is shown below. The coordinates of the local minimum and maximum points are \(r\) and \(s\) respectively.

2. Write down the equation of the horizontal asymptote.[1]
3. Write down the value of \(r\) and of \(s\).[4]
4. Let \(L\) be the normal to the curve of \(f\) at \(\mathrm{P}(0,1)\). Show that \(L\) has equation \(x +y = 1\).[4]
5. Let \(R\) be the region enclosed by the curve \(y = f(x)\) and the line \(L\).[5]
   1. Find an expression for the area of \(R\).
   2. Calculate the area of \(r\).