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.