# Sourcing and verifying calculus applications

There are many questions on this site about specific (or not-so-specific) applications of calculus to the "real world". However, one issue I've noticed in using textbooks for this purpose is that the applications are not always well-sourced, and hence I suspect are (in some sense) not really used.

The question I have is what strategies educators use (if at all) to actually verify for students that these applications are used in some actual practitioner's toolkit. I'm referring to those not involving conceptual ideas (such as area under a force-distance curve), but actual computation with algebra (such as integrating said area from a specific cubic model to help compute how much better a kangaroo can jump than a human).

Here are the type of answers I am looking for, as long as there is at least anecdotal evidence that it pays off. I happen to think these techniques don't always work well, though you could still use them as answers if your experience differs.

• Doing straight-up web searches for keywords
• Using references in "application calculus" textbooks (because there often aren't references)
• Asking colleagues in discipline X about calculus example in discipline X

As an aside, the story motivating this question is my hunt for a verified source outside a calculus text that medical researchers use $$R(q)=q^2\left(\frac{k}{2}-\frac{q}{3}\right)$$ as a general formula to measure the response $$R$$ to a dose of quantity $$q$$ of some medication. If your method actually finds a source for that, it will get an upvote for sure!

As a second aside, I'm mostly referring to applications outside of the traditional physics-chem package, and especially to mathematical models of real-life data which calculus could be nicely applied to if they were really used, and which are purported to be very similar to actual uses, but which aren't.

• What's the motivation for "verifying" these models for students? Here's an article on using restricted cubic splines to model dose-response relationships. Here's a more readable article on clinical modeling with cubic splines. My sense is that "models" given in well-written calculus textbooks usually have some basis in reality, but they are often highly simplified to the point of being unrecognizable. Oct 2 at 23:00
• @JustinHancock Baldly stated, the motivation is truth. Some of the examples of how various models are used are not, in any way that I've been able to determine, actually used. I remember one particularly egregious example of a random cubic polynomial that was supposed to model occupancy rates at a specific hotel - or another one about a cubic that was supposed to be a cost function for making salsa. That isn't to say that cubics aren't used to model cost functions, but rather that the example was ridiculously worded! Oct 3 at 10:55
• @JustinHancock thanks! Though, interestingly, from their abstracts both of these seem to be more about using splines to connect data, than a single cubic polynomial to model dose-response relationship. This is the sort of misleading thing that causes students to be skeptical about "applications" (though it certainly means they should know more math!). Oct 3 at 10:58
• I agree that if these kinds of problems are framed as "real," then students are being misled. I typically present them as a kind of fiction. A nice-sounding story with perhaps some nuggets of truth. There's a good argument that they're not worth using in either case. Oct 3 at 12:39

I vaguely recall seeing the reference to modeling response in a calculus textbook, probably an edition of Stewart or Thomas. I took part of your question and did a "ChatGPT search." My prompt was

"give me a source for this assertion: medical researchers use $$R(q)=q^2\left(\frac{k}{2}-\frac{q}{3}\right)$$ as a general formula to measure the response $$R$$ to a dose of quantity $$q$$ of some medication."

The return was a reference to a book "Principles of Pharmacology: The Pathophysiologic Basis of Drug Therapy" by Golan et al. I was able to confirm this is a real book, and not a hallucination. The ChatGPT response also described the Hill equation and said it was a type of sigmoid function. So I started using search engines with queries like "Hill equation" and "Hill equation response rate."

There was an abundance of interesting things to read about the Hill equation, which I was not familiar with. You will find that the Hill equation is a real thing used in applications in biochemistry, pharmacology, etc. There is lots of literature. Nothing I found seemed to explicitly use a cubic as in the the expression $$q^2\left(\frac{k}{2}-\frac{q}{3}\right)$$.

But it seems likely that this cubic is a simple but practical representation (approximation) of a sigmoid curve that avoids transcendental functions, which would not come up until later in the course. The maximum provided by the cubic approximation is at $$q=k$$ and the inflection point is halfway, at $$q=k/2$$, which seems like a decent model. In the context, the inflection point would correspond to the "dosage saturation point."

The key information that ChatGPT gave me that I would have had difficulty getting from "straight-up web searches" is Hill equation.

• ChatGPT is an answer I didn't think of to my question, but I think that's a fair answer, so +1! Oct 3 at 14:57
• On the specific example, thanks; Hill functions are one of the examples of this kind of modeling I'm not skeptical of, because I've been able to find good sources for them :-) as you point out, and certainly not just in pharmacology, as you mention. Oct 3 at 14:59

The primary use of calculus is not day to day "application" by working/paid professionals, but it IS a needed tool for undergrad engineers, scientists (and sometimes social scientists) within their majors coursework. It's needed to follow derivations and do homework problems in physics, EE, etc. That a practicing EE will rarely calculate impedance using integrals and derivatives is irrelevant to their need to have those tools when learning their basic courses.

And it's not even really required that they have practiced the specific applications (circuits or fluids or whatever), as they will also be taught in majors courses. Nice sure (good to see things twice, including from different perspectives). But still not really vital. They need the basic algebraic calculus toolkit and at least a bit of eventual practice in some applications (since "word problems are hard", they represent a higher cognitive load training of the basic calculus material.)

P.s. It never ceases to amaze me the naivete of math educators, when asking questions like this. If you want to know how often engineers use calculus, ask them...not your fellow math educators (most of whom lack a strong generalist STEM background...I know...I do have one.). Or, go walk over to the library (or better yet colleagues' offices) and look at the texts for all the basic majors courses in physics, engineering, p-chem, etc. A very simple starting point would be Lindeburg's EIT reference manual (one of the most amazing compilations of most undergrad STEM coursework). https://www.amazon.com/Engineer-Training-Reference-Michael-Lindeburg/dp/0912045566

• I don't think this is answering the question asked, which was not "how is calculus used?", but rather "are any of those alleged modeling problems at all related to reality?" Oct 3 at 0:58
• @SueVanHattum has precisely articulated it. And while there are some alleged more physical applications that I'm doubtful about the reality of (how many Norman windows do we really need to optimize light coming through?), typically the problem isn't whether the calculus is needed in engineering! It's non-physics/chem/engineering areas where this problem often comes up in terms of whether the model is even remotely connected to actual use. Oct 3 at 11:01
• To more directly address what I think you real point is, how can a math educator address student objections about "when am I going to need this" if every single example in the textbook that supposedly relates to reality is just made up by the author of the text or exercises? I have zero problem with integrals for work or derivatives for projectile motion, even if engineer X will not happen to do those in the day job, because your point is 100% accurate in that case. I'm less happy about models in bio or business, even those based on real-life data, that no practitioner has ever used! Oct 3 at 11:06