Interesting calculus problems of medium difficulty?

I would like to know sources, and examples of good "challenge" problems for students who have studied pre-calculus and some calculus. (differentiation and the very basics of integration.) Topics could be related to things such as:

Taylor Series.
Product Rule, Quotient Rule, Chain Rule.
Simple limits.
Delta Epsilon Proofs.
Induction proofs for the sum of the first n, integer, squares, etc.
Integration by substitution.
Other topics...

What I have found so far are too many problems that are just a bit too difficult. The problem can have a "trick" but it needs to be something a freshman could do.

Here is one problem that I thought was just at the right level:

If $f(x) = \frac{x}{x+\frac{x}{x+ \frac{x}{x+ \vdots}}}$, find $f'(x)$*

*To be honest this problem makes me a little nervous. Still, I like it.

Solution 1:

If $C_0 + C_1/2 + \ldots + C_n/(n+1) = 0$, where each $C_i \in \mathbb{R}$, then prove that the equation $C_0 + C_1x + \ldots + C_nx^n = 0$ has at least one real root between 0 and 1. (Rudin, Ch 5 Exercise 4)

If $|f(x)| \leq |x|^2$ for all $x \in \mathbb{R}$, then prove that $f$ is differentiable at $x = 0$. (Spivak?) (OK, so this one is easier than "medium" perhaps, but I like it anyway. It illustrates nicely that growth conditions can have an impact on smoothness.)