In calculus, which questions can the naive ask that the learned cannot answer? Number theory is known to be a field in which many questions that can be understood by secondary-school pupils have defied the most formidable mathematicians' attempts to answer them.
Calculus is not known to be such a field, as far as I know. (For now, let's just assume this means the basic topics included in the staid and stagnant conventional first-year calculus course.)
What are


*

*the most prominent and 

*the most readily comprehensible


questions that can be understood by those who know the concepts taught in first-year calculus and whose solutions are unknown?
I'm not looking for problems that people who know only first-year calculus can solve, but only for questions that they can understand.  It would be acceptable to include questions that can be understood only in a somewhat less than logically rigorous way by students at that level.

A common calculus problem is to prove that the series $1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots$ converges.  More difficult is to find the sum:
$$1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6}.$$
Much more mysterious is the series:
$$1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots = 1.2020569\ldots$$
After a few hundred years of searching, it seems unlikely that there is a simple closed form.  This number is known as Apéry's constant or just $\zeta(3)$.
