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Question id: 9. This question is similar to one that appeared in an IB Standard paper in 2010. The use of a calculator is allowed.

The diagram shows part of the graph of \(f(x)=Ae^{kx}+2\).

The y-intercept is at \((0,10)\).

(a) Show that \(A=8\).

(b) Given that \(f(8)=3.62\) (correct to 3 significant figures), find the value of \(k\).

(c) (i) Using your value of \(k\), find \(f'(x)\).

(ii) Hence, explain why \(f\) is a decreasing function.

(iii) Find the equation of the horizontal asymptote of the graph \(f\).

Let \(g(x)=-x^2+7x+5\)

(d) Find the area enclosed by the graphs of \(f\) and \(g\).

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