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

1. the most prominent and
2. 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.

1) Convergence of the Flint Hills series

$$\sum_{n=1}^\infty \frac{1}{n^3 \sin^2 n}$$

is unknown. One can also ask the same question with different exponents - see this paper for more details.

2) Closely related (although $\liminf$ is typically not covered in first year calculus courses, it's too not much of a stretch): whether or not

$$\liminf_{n \to \infty} |n \sin n| = 0$$

While it's certainly got a number-theoretic aspect to it, I'd consider Euler's constant $\gamma = \lim\limits_{n\to\infty}\left(\sum_{k=1}^n\frac1k - \ln n\right)$ to be on-topic for a first-year calculus course (since it features limits, logarithms, and even a fairly simple series), and one of the most fundamental questions one can ask about it — "is this number rational or not?" — is still completely open.

One result that may surprise most calculus students is that there is no algorithm for testing equality of real elementary expressions. This then implies undecidability of other problems, e.g. integration. These are classical results of Daniel Richardson. See below for precise formulations.

$\qquad\qquad$

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 $\zeta(3)$.