In calculus, which questions can the naive ask that the learned cannot answer?

Question

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.

Answer 1

Many indefinite integrals fit this, the poster child being $\int \exp(-x^2)\,dx$ Easy to ask, hard to answer (unless you count the error function as an answer, but in this case that sounds like giving a name to the unknowable).

Answer 2

Maybe explain why an elementary function might not have an elementary antiderivative, for example.

Answer 3

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 $\zeta(3)$.

Answer 4

Whether the real numbers and the theory of limits correctly model the physical universe.

In other words, is spacetime infinitely divisible and mathematically complete, ie a continuum? Or is it discrete, in which case calculus is only a continuous approximation but not the literal truth.

This is an ancient problem that nobody knows the answer to. It relates to Zeno's paradoxes.

To be fair, this is a problem of physics, not mathematics. But it's within the spirit of the question.

Answer 5

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.

Answer 6

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

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

Answer 7

With all the recent advances in understanding infinitesimals, we still don't fully understand why Leibniz's definition of $\frac{dy}{dx}$ as literally a ratio works the way it does and seems to explain numerous facts including chain rule.

To respond to the comments below, note that Robinson modified Leibniz's approach as follows. Suppose we have a function $y=f(x)$. Let $\Delta x$ be an infinitesimal hyperreal $x$-increment. Consider the corresponding $y$-increment $\Delta y=f(x+\Delta x)-f(x)$. The ratio of hyperreals $\frac{\Delta y}{\Delta x}$ is not quite the derivative. Rather, we must round off the ratio to the nearest real number (its standard part) and so we set $f'(x)=\text{st}\big(\frac{\Delta y}{\Delta x}\big)$. To be consistent with the traditional Leibnizian notation one then defines new variables $dx=\Delta x$ and $dy=f'(x)dx$ so as to get $f'(x)=\frac{dy}{dx}$ but of course here $dy$ is not the $y$-increment corresponding to the $x$-increment. Thus the Leibnizian notation is not made fully operational.

Leibniz himself handled the problem (of which he was certainly aware, contrary to Bishop Berkeley's allegations) by explaining that he was working with a more general relation of equality "up to" negligible terms, in a suitable sense to be determined. Thus if $y=x^2$ then the equality sign in $\frac{dy}{dx}=2x$ does not mean, to Leibniz, exactly what we think it means.

Another approach to $dy=f'(x)dx$ is smooth infinitesimal analysis where infinitesimals are nilsquare so you get equality on the nose though you can't form the ratio. On the other hand, Leibniz worked with arbitrary nonzero orders of infinitesimals $dx^n$ so this doesn't fully capture the Leibnizian framework either.

In an algebraic context one may be able to assign a precise sense to the Leibnizian generalized equality using global considerations, but I personally don't know how to do that precisely.

Answer 8

What is $\int x^x$ ? Although not an open problem, it is hard to think of the correct response right off the top of your head unless you have seen it before.

Answer 9

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.

$\qquad\qquad$

Answer 10

Is $e+\pi$ an irrational number?

This is a variation of Steven Stadnicki's answer, but without the need to introduce definitions of new constants, which $\gamma$ presumably could be to a first-year calculus student.

Answer 11

Some interesting integrals do not have closed-form solutions.

For example, elliptic integrals can only be approximated using table look-ups. This makes it impossible to exactly calculate the arc-length of part of an ellipse, or the surface area of part of a planet (as represented by an ellipsoid).

Answer 12

For what number $k$ is $\int\left(\sin(x^2) + k \cos(x^2)\right)\ dx$ an elementary function?

Hint: $k$ is rational.

Answer 13

Some discussion of the three-body problem, especially if your students have followed a physics course where Newtonian mechanics have been taught, can be relevant:

• It is very easy to state (it is a simple ODE)
• It is useful (think space travel)
• Very few things are known about it, if we compare against the two-body problem: no closed form of trajectories (series exist though), new periodic solutions have been discovered in 2012, boundedness of solutions is difficult to assess, etc.

As a bonus, this can be used as a motivation for the need of asymptotic expansions.