Good final exam bonus problem for calculus students? I'm making a final exam for a first course in calculus at a university.  I need some suggestions for a good bonus problem.  By "good" I mean an interesting problem that some of the brighter students could solve in about 15 minutes.  

Hopefully it should involve limits, derivatives, or integrals. 

The problem should NOT involve trig functions.

The problem should NOT be theoretical. These students can't do real analysis proofs, for instance.
Here are a few interesting problems:


Let $f^n(x)$ denote the nth iterate of a function (for example, $f^2(x)=(f\circ f)(x)$). If $f$ has a fixed point at $x=x_0$ (meaning that $f(x_0)=x_0$) then what is $\frac{d}{dx}f^n(x_0)$ in terms of $f'(x)$?
Prove that the intersection of the parabola and the rectangle in the picture is $\frac{2}{3}$ the area of the rectangle:





A triangle starts is an equilateral triangle $ABC$ at $t=0$. However, each second, both $BC$ and the length of the altitude to $BC$ increase by $1$ unit. What does the measure of angle $BAC$ approach as $t$ approaches $\infty$?
Browse old problems from the Putnam exam. http://kskedlaya.org/putnam-archive/