Why Teach Calculus in the Age of AI

https://dibeos.net/wp-content/uploads/2025/08/what_happens_t...

For the [current] layperson, each of those things I mentioned I might as well be speaking in Martian.

You can’t “brush up” on something you never learned

You'll (probably) never apply the ability out the kinetic energy vs. time of a ball rolling down a hill, but these exercises build understanding of the tools. Derivatives are everywhere in a fundamental electric circuits course, you need to have an intuitive understanding of basic calculus. The relationship between current through and voltage across ideal inductors and capacitors are directly described in the language of calculus, even if you're not "using" the calculus substitutions you learned each time you analyse a circuit.

And good luck getting through a couple weeks of an introductory quantum mechanics course without using calculus as a fundamental building block. You can solve many of these problems with computers, but it's not going to build intuition on how to approach future problems. (I don't mean this as a joke or picking an arbitrary complicated-sounding topic; this is a core course in some engineering programs.)

Many engineering problems have nice closed-form equations (at least to get approximations). Obtaining those equations often involves calculus, and someone has to do that in the first place.

(I'm giving examples from the lens of my education, but each field of science, engineering, and mathematics will have their own context, and will vary from little-to-no calculus to being all-calculus.)

But when things break and you have to understand why, or you have to fix it, or even describe the problem in some detail that someone else is able to guide you to fix it, it sure helps to have an idea of how it works.

Sounds like a job for AI.

https://en.wikipedia.org/wiki/Tai%27s_model

Calculus is an elective course in most schools.

For something more traditional, take a look at textbooks by Piskunov, Courant, or Apostol. Spivak's Calculus has excellent problems if you are looking for something more abstract and rigorous (probably better after a first course). https://archive.org/details/n.-piskunov-differential-and-int... ; https://archive.org/details/ost-math-courant-differentialint... ; https://archive.org/details/calculus-tom-m.-apostol-calculus... ; https://archive.org/details/introductory-calculus-book-colle...

Finally, if you want a strategy for those tricky integrals, per se, take a look at Schoenfeld's "Integration: Getting it All Together", https://files.eric.ed.gov/fulltext/ED214787.pdf ; some results of teaching the solution of integrals by this method were presented in https://www.jstor.org/stable/2320344

Not only will you be even more capable of picking up solving things numerically, but you'll also have the prerequisites for studying physics or probability or machine learning or Knuth's Concrete Mathematics. It opens doors to new intellectual vistas.

Solving things analytically (when possible) also can reveal more about the nature of the problem than doing so numerically, and can give the same satisfaction as finding an elegant solution in code.

You can definitely go an entire programming career without ever using it, but if you ever do run into a problem it solves, having this tool available to you is only a benefit.

There will always be some brilliant people that breeze through a hard course, and effortlessly acquire the intuition, and then wrongly attribute their understanding to the course. That doesn't help the rest of us, who are much more likely to build intuition by guessing about the outcome of a program and then checking if we are right.