Ask HN: Can calculus be taught without differentiating or integrating by hand?

I wholeheartedly agree with this.

No. Completely wrong. Engineers and scientists are given the very same rigorous fundamentals of mathematics. We (EE/CS) also had to take the very same weeder courses as mathematicians including abstract mathematics and concrete mathematics, in addition to modern physics for physicists and engineers, followed by the entire computer science and electrical engineering series. We did learn the Monte Carlo method for problems that may not be directly computable in reasonable time, but these are approximations that can be achieved with a qualified amount of error. In fact, I helped port a FORTRAN-based nuclear reactor simulator for UNIX to Windows that was, at its core, a quadruple integral Monte Carlo approximation considering configurations of voxels of matter and voids over time with their various properties.

We started with the fundamental in high school on paper with area under the curve and learning mathematical transformations of integrals, and derivatives as rate of change. The Erable CAS found in the HP 48 helped provide transformations and reductions sometimes, but not every time for basic calculus, vector calculus, and ordinary diff eq. (We didn't have MAPLE, Mathematica, or GIAC which are far more powerful.)

Finally, in college, we ended up with discrete FFTs and Laplace transforms in EE and the Schrödinger eave eq for hydrogen in physics. This isn't as far as physicists or mathematicians go in grad school such as groups, fields, and such.

So, I think you need to reevaluate your assumptions about curricula that seems wholly inadequate.

I think, while your mapping from solution methods to professions makes sense, the act of solving should be taught as an afterthought.

The goal should be to build a robust mental model. Personally i find playing around with computer visualisations helps a lot with that.

The same is even more important for differential equations, imo. I wish i would have spent less time studying algorithmic DE solving techniques in my school days. The beauty and power of expression of these has nothing to do with the way you solve them, if you ask me.

At first order the engineer need to be able to provide some sort of solution to any problem and get something done. The numerical toolbox is kind of an easy reliable Swiss knife that will get you through anything related to derivatives and integrals. It

Ten years down the line when your nuclear core is melting and you need to pull the right lever will you be confident enough to trust your rusty manual computations tricks and not miss some minus sign somewhere in the transcription of your derivative.

The derivative and integral game, is kind of an interesting mathematical artefact about the study of functions from before the time we had computers. As far as I know it's not yet a solved game. It's more about heuristics and picking rules than using a specific formula.

Rule based Integration https://rulebasedintegration.org/ is a kind of collection of all the tips and tricks to play this game, but even then it's not foolproof.

Closed formula are nice to have but hard to get by in real problems.

The thing which makes the difference between something that is nicely behaved or not, is whether or not the operations are closed ("closure" concept). When it's the case you can build some nice algebra with nice properties. The computations becomes deterministic and formulaic. When it's not you end-up playing a NP-complete game to find the solution.

For example in the case of the study of infinitesimal rotations, the natural extension of calculus for physics, we have the Lie Algebra. Even there your numerical toolbox will be more useful than your pages of Feynman Diagrams.