Find a nonnegative function 
whose ordinate set on an interval has area proportional to the difference of the function values at the endpoints of the interval.
With reference to the previous exercise (Section 8.24, Exercise #17) we can write the area 
as
![\[ A(x) = \int_a^x f(t) \, dt = C (f(x) - f(a)) \quad \implies \quad A'(x) = f(x) = C f'(x). \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-51840e067b253f5e90108cc3d74a9d28_l3.png "Rendered by QuickLaTeX.com")
Therefore,
![\[ \frac{f'(x)}{f(x)} = \frac{1}{C} \quad \implies \quad \log |f(x)| = \frac{x}{C} + A. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-93049d79ea462e949e7e4f44e375812a_l3.png "Rendered by QuickLaTeX.com")
Thus,
![\[ f(x) = Ae^{\frac{x}{c}}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-6619cd514a669ebc120af13f3774e4b0_l3.png "Rendered by QuickLaTeX.com")