I am visiting my in-laws for the holidays. My sister in law is a statistician. She asked me to take a stab at a calculus problem which was coming up in her research. The Lambert $W_0$ function is defined as the inverse of the function $f(x) = xe^x$ for $x \geq -1$. It is clear that $g(x) = W_0(f(x)) = x$ when $x \geq -1$. Her problem was to bound the derivative of $g$ when $x < -1$. I figured it out and found it to be a very fun calculus problem! The solution didn't use anything which would be inaccessible to a calc 1 student, but there were several creative steps which were needed. I think that, with a bit of hand-holding, this could make a nice project in a calc 1 class. It would have the added bonus of being motivated by real research, not just being a contrived exercise.

I am hoping for a big list of interesting calculus problems, solvable using calc 1 techniques, which have come up in actual research. Links to papers are especially welcome!

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    $\begingroup$ I don’t know how this can be useful: calculus problems appear e.g. in many places in engineering and physics, and you would end up with a heterogeneous list that can go on forever. Moreover, since calculus problems are usually of relatively straightforward solution, one usually skips the details when publishing actual research. And finally, many of these problems would be understandable only with some specific knowledge about the topic, and you wouldn’t be able to use them in a calc 1 class. $\endgroup$ Commented Dec 26, 2022 at 12:47
  • $\begingroup$ @MassimoOrtolano The unusual feature of the problem I mention as an example in the OP is that it is actually an interesting calculus problem (try it yourself!), which can be posed independently of the background statistical question. $\endgroup$ Commented Dec 26, 2022 at 13:18
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    $\begingroup$ "It would have the added bonus of being motivated by real research, not just being a contrived exercise." Not to be a Negative Nelly, but are you sure your students will perceive it that way? It's a real-world problem to you because your statistician SIL asked you about it IRL (in your real life). I'm not sure whether this RL aspect can effectively be conveyed to your students. To your students, a real-world problem is probably something where they can grasp why it arises in the mythical "real world", rather than a problem where you tell them that your statistican SIL asked you about it. $\endgroup$ Commented Dec 26, 2022 at 20:28
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    $\begingroup$ I agree with the previous comment. The main problem I see here is that the students (i) will not have heard of Lambert's $W$-function before, so they will not find it compelling even if it is arising in a real problem and (ii) bounding the derivative doesn't sound like an interesting task for its own sake for someone who doesn't already care about math. It can sound close to "this function you never heard of before, with a technical definition, can be used to do something mathematical", which I suspect isn't going to make the target audience interested. $\endgroup$
    – KCd
    Commented Dec 27, 2022 at 3:23
  • $\begingroup$ Sure, I can agree that taking an abstract problem and slapping an "applied" label on it because it comes from a research paper is not going to be the best motivation for students. I really appreciate your suggestions @KCd! The problems you pose are natural, well motivated, and accessible. $\endgroup$ Commented Dec 27, 2022 at 11:40

6 Answers 6


Modeling Basketball Free Throws

by Joerg M. Gablonsky and Andrew S. I. D. Lang, SIAM Review vol. 47, no. 4, pp. 775-798, 2005, https://doi.org/10.1137/S0036144598339555


This paper presents a mathematical model for basketball free throws. It is intended to be a supplement to an existing calculus course and could easily be used as a basis for a calculus project. Students will learn how to apply calculus to model an interesting real-world problem, from problem identification all the way through to interpretation and verification. Along the way we will introduce topics such as optimization (univariate and multiobjective), numerical methods, and differential equations.


Consider Betz's law, which was worked out around $100$ years ago by three different scientists independently (in Germany, the UK, and Russia). It determines the maximum power that can be extracted from a wind turbine. Without getting into the physical background here, the problem eventually turns into the task of finding the point $x$ in $[0,1]$ where the polynomial $P(x) = (1/2)(1+x-x^2-x^3)$ is maximized.

The derivative of $P(x)$ is $(1/2)(1 - 2x - 3x^2) = (1/2)(1-3x)(1+x)$, whose only zero in $[0,1]$ is at $x = 1/3$. Since $P(1/3) = 16/27 \approx .592$, while the boundary values are $P(0) = 1/2$ and $P(1) = 0$, the maximum occurs at $x = 1/3$. That its value there is around $.592$ has the real-world meaning that a wind turbine can convert at most around $59\%$ of a wind's power into mechanical power.


Consider trying to find a curve that will let you link up two straight tracks smoothly, such as two straight parts of a roller coaster track. By "smoothly" meeting, we want the derivatives to match. This can be done with cubic splines. Degree $2$ is not enough because there are four conditions imposes by equality of function values and their derivatives at a point, so the three coefficients of a quadratic polynomial are inadequate but the four coefficients of a cubic polynomial are a good model. As a simple example of this task you can present in a calculus course, find the cubic polynomial on $[0,1]$ that will let you smoothly transition from $y = -.5x$ for $x \leq 0$ and $y = .6(x-1)$ for $x \geq 1$.

Splines show up all over the place: computer animation (movies, video games), automotive and aircraft design (where two pieces of metal should meet up in a smooth way), and scalable fonts used on the screen on which you are reading this.

For transition curves used for railroad tracks or highways exits and entrances, the related math is Fresnel integrals.


Consider the reflective property of a parabola: rays coming into a parabola parallel to its axis of symmetric will all bounce off the parabola (angle of incidence equals angle of reflection) and meet at the focus. Verifying this property is a calculus problem related to normal directions at a point on a parabola, which is perpendicular to the tangent direction at that point.

This property also holds for a paraboloid of revolution and is why radar dishes are shaped like parts of a paraboloid, with the receiver at the focus. Going in the other direction, this is why the casing around some front-facing (not rear-facing!) car headlights are shaped like paraboloids with the bulb at the focus.

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    $\begingroup$ A wonderful related problem is the find the free surface of a body of fluid undergoing solid-body rotation. That is how you make really big telescopes. $\endgroup$
    – Philip Roe
    Commented Dec 27, 2022 at 22:08

There should be a lot of interesting examples in ecology. The first one that comes to mind for me is population dynamics: Wikipedia Link

The simplest model is exponential growth:

$$ dN/dt = rN $$

Where $t$ is time, $r$ is the population growth rate, and $N$ is the population size.

This can be relevant for things like predicting bacteria growth. But in general, many factors prevent this type of growth for most species, so there are many ways to extend this model for practical applications. The most common example is probably the logistic growth model:

$$ dN/dt = rN(1 - N/K) $$

This incorporates $K$ as carrying capacity to limit the maximum population size. It may be feasible to have calc 1 students derive this themselves from the exponential growth equation.

Other variations might be interesting as well. For example, the Explosion-Extinction model:

$$ dN/dt = rN(N/M - 1) $$

Here, $M$ represents a threshold population size. If $N > M$, the population will grow. If $N < M$, the population will shrink. This type of equation is useful for modeling invasive species (which can have ecological and economical consequences), which usually have to meet some threshold $M$ to become established.


The economic spacing of horizontal wells in a "shale" is a practical max problem. [First semester calculus] You want to get the most profit. Closer than about 2000 feet and the wells start to interfere (cannibalize or steal from each other). However, each new well bore does break (literally, that is what hydraulic fracturing is) new rock. So it is not 100% cannibalization. But they cost money. Based on the price of oil/gas, and the cost of drilling wells, there is an optimal density. You will always "leave some behind" and always "have some cannibalization". In addition, the cost function does not scale exactly. Each new well bore (drilling) is essentially a scaled cost. But the costs of the "pad" (road, permit, etc.) are essentially constant (thus divided over more wells). Also, fracturing and even drilling have some cost advantages when done in a campaign (less equipment moves).

After solving a base case, you can do some interesting questions. Like "is it possible to drill 6.5 wells (no) in a unit. So then they have to pick 6 or 7 and see which is better. Or you can discuss unitizing sections (combining two units, with the right leases/permits) to allow drilling "13 in a double wide". Plus you can do useful variances from the base case (what is the optimal density if oil price is $150/bb? How about $50?)

Personally, I think the easy thing is to just write out reasonable functions yourself, than do the analysis. But here are some papers. (Not sure they are the best, so many junky little papers!)




Maybe there is a little local connect for your kids, since you have the Utica in Ohio...although it's on the other side and further south.

P.s. In general, I'm not a fan of this sort of thing. I think research oriented teachers are too interested in the research and also don't realize that the classical examples are new/interesting to their students, even the techniques are new to them. So, it ends up being too much "push" from the teacher and a diversion. You do seem very organized in teaching your kids and getting results, so maybe you have the slack to spend time on this sort of thing. But, still..."Danger, Will Robinson!"

P.s.s. In practice companies aren't really doing calculus. They just do a spreadsheet and look at 1-20 wells per section and pick the max profit. Where added oil and diminishing returns counterbalance. I'm not sure this is even cutting edge "research" really, since companies can do it without the literature. It's just interesting FP&A. But it's definitely calculus driving the bus in terms of the math that actually affects the economics. So an econ grad student would think of it that way, write about it that way. In a sense, there are some beautiful equations and functions and calculus concepts lurking behind the matrix of the Excel. And I think it helps as an engineer/executive/consultant to have some functional insight as to what is going on, not just the black box of the Excel tables.

(Another P.S.) I understand your joy/pride in the collaboration with your sister, but I think it would be better form to lead with the question, rather than your anecdote. Looking, now at the commentary, I see a lot of discussion of the mathematics details of the problem itself. Almost as if this was more the interest than the "math ed" part at the end. So that it feels to me, more discussion of the math itself, rather than education. As a rule-breaker, of course, I applaud your (mis)use of the forum, to accomplish any objective, rather than the stated objective. All that said, I would just be very wary of this sort of "I like this hard math" that does not translate well to kids learning things (just the chain rule is new to them).

(Yet one more) If you don't like the shale industry (I love it, see capitalism in action), you can search on environmental cost benefit analysis. The Google Scholar returns lots of academic lit (if that's important to you). The regular Google search returns some case studies and the like, which may be more quickly relevant. Here is one review:


(not immediately usable for a class, but gets you into the lit, since it's a review article. Really I think a book chapter or simple case study is better than nitty gritty cutting edge research.)


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