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I eventually got my Applied Math PhD and became the sort of Math
SnobConnoisseur who insists that it's not real calculus until you at least throw away that Riemann crap and use Lebesgue integrals ... but I have to admit there's a ton of students who will do science or engineering or medicine where they would greatly benefit from solidly understanding "superficial" Calculus. If you never quite grokked delta-epsilon proofs, but you understood numerical integration well enough that you could have properly reviewed the discovery of "Tai's Method", that's a better understanding of calculus than at least that medical journal (and that author, and some of her collaborators) had at that time.For either acceleration or enrichment, though, it needs to be periodically reinforced to be worthwhile, and that can be the tricky part. I took an Algebraic Topology class for fun as a college MechE-but-advanced-at-math, was amused by simplicial complexes and exact sequences and so forth but couldn't see what any of it was really useful for, promptly forgot it all because I never used it for anything for a couple years ... and then ended up in a math PhD program where I had to relearn a chunk of it just to understand some of the best visiting lecturers. I assume Mary Tai was the same way: nobody ends up in medical research without taking at least Calc 1, and if she was smart enough to reinvent the trapezoidal rule then she was surely smart enough to understand it as it was taught in Calc 1, but she probably never used it again for years and so had completely forgotten it when she needed it. Being able to rederive ideas you forget is IMHO one of the nicest aspects of math, but it is better to have a fuller toolbox of things you don't have to reinvent, and the more "enrichment" you get, the more connections you can make between ideas, and the easier it is to remember long-unused ideas via their more-obscure connections to more recently used ideas. With narrow acceleration in one subject, you might get so far ahead there that you don't get the same reinforcement schedule that other kids get via the usual connections to other subjects.
Learning on your own makes it a little easier to get some of that reinforcement from "standard" curriculum material, though. A standard high school Physics class won't be based on calculus, because most of the kids who want to take it won't have learned calculus yet, but if you know you're not most kids and you've got basic Calc 1 under your belt then you can just study calculus-based Physics instead, getting in more science and reinforcing math skills (and getting what I'm told is an impressive AP credit) at the same time, and learning something that's still on the critical path for a lot more science+engineering career tracks than e.g. group theory would have been.
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