2021 May TZ1 P2 Q9

Consider the function \(f\) defined by \(f(x) = 90\mathrm{e}^{-0.5x}\) for \(x \in \mathbb{R}^+\).
The graph of \(f\) and the line \(y = x\) intersect at point \(\mathrm{P}\).

1. Find the \(x\)-coordinate of \(\mathrm{P}\).

   [2]

The line \(L\) has a gradient of \(-1\) and is a tangent to the graph of \(f\) at the point \(\mathrm{Q}\).

2. Find the exact coordinates of \(\mathrm{Q}\).

   [4]

3. Show that the equation of \(L\) is \(y = -x + 2\ln{45} + 2\).

   [2]

The shaded region \(A\) is enclosed by the graph of \(f\) and the lines \(y = x\) and \(L\).

4. 1. Find the \(x\)-coordinate of the point where \(L\) intersects the line \(y = x\).

   2. Hence, find the area of \(A\).

      [5]

The line \(L\) is tangent to the graphs of both \(f\) and the inverse function \(f^{ -1}\).

5. Find the shaded area enclosed by the graphs of \(f\) and \(f^{-1}\) and the line \(L\).

   [2]