Derivatives of Exponential, Logarithmic and Trigonometric Functions

Let \(f(x) = A\mathrm{e}^{kx} + 3\). Part of the graph of \(f\) is shown below.

The \(y\)-intercept is at \( (0,13)\).

1. Show that \(A = 10\).[2]
2. Given that \(f(15) = 3.49\) (correct to \(3\) significant figures), find the value of \(k\).[3]
3. 1. Using your value of \(k\), find \(f'(x)\).
   2. Hence, explain why \(f\) is a decreasing function.
   3. Write down the equation of the horizontal asymptote of the graph \(f\).[5]

Let \(g(x) = -x^2 +12x -24\).

4. Find the area enclosed by the graph of \(f\) and \(g\).[5]