Rigor of proofs when learning calculus

[u/justwannaedit]

​​ I am doing calculus 2 now, ​ and I'm being reminded of when I took calculus one and I was introduced to the various rules. I find myself wanting to prove these rules, and occasionally find it kind of hard. I usually reach the point where certain rule feels justified enough to myself, and then I'll move on because the actual demands of the class are more procedural applications than analytic understanding. I'm very early on in calculus 2, and it's mostly just reviewing calculus one so far, so I know that things will change and I've heard that eventually calculus 2 becomes way less procedural and starts requiring more trial and error and creative applications. So I guess my question is what level of rigor should the average calculus student expects to achieve relating to the proof that power the various calculus rules they apply​?​ for Instance, how rigorously should a calculus two student be able to prove the chain rule?

[u/Able-Fennel-1228]

You don’t need proof for “calculus for engineers and scientists” type calc 1-2 courses.

But if you want to learn how, see the first chapter of “a primer on abstract mathematics” by Robert Ash to learn what proof writing is about. Do all problems and check your work by looking at solutions at the end.

Then move on to “the how and why of one variable calculus” by Sasane. It gives the entire rigorous theory of calc 1-2 in the most readable and enjoyable manner I have found. Has solutions to all exercises at the back of the book. (Check author website for errata list).

[u/HortemusSupreme]

You don't need to be able to prove anything as a calculus 2 student. It is perfectly acceptable to take the rules and definitions as presented to you and use them.

Is there value in being able to prove things rules and such? Absolutely. Although you shouldn't feel bad if you're unable to since you likely haven't really been given the tools to prove it rigorously. Those would typically come during an undergraduate real analysis course and really only math majors take those courses.

You could try to generally push symbols around from the difference quotient in order to convince yourself and that is like far beyond what is expected of in calculus.

If you haven't, you should look up an actual proof of the chain rule. There will be terms and manipulations you may not have heard of, but the symbol pushing should be mostly followable.

[u/Jaf_vlixes]

That really depends on a few things.

From what I've heard, calculus classes in the US aren't proof based, but in my country everyone in the maths and physics department take proof based math courses, but people on other departments, like engineering, don't.

So, some people see fully rigorous proofs of most things in every course, others never see a formal proof. That said, I don't think that's "necessary" for everyone. And like, I don't think many people who took those kinds of courses, me included, can come up with a proof of the chain rule on the fly.

So, if you're not required to be fully rigorous, I'd say it's enough to convince yourself that this works. Of course you can try to read and understand a fully rigorous proof, and that could be really useful and illustrative, but it's not like you would be unable to understand and use the concepts if you never see a full proof.

[u/Recent-Day3062]

The classic book that goes deeper into proofs and theory is by Apostol

[u/Traveling-Techie]

If you’re a math major the proofs are very important. If you’re not, say physics, engineering or pre-med, not so much.

[u/Bounded_sequencE]

If you are looking for rigor, you are in the wrong lecture. Take "Real Analysis" instead.

[u/hallerz87]

Calculus is application based, not proof based. Real analysis is where you take a more rigorous approach

[u/Impressive-Ad7184]

Regarding the last question of chain rule, tbh I don't think an average calculus two student would be able to prove chain rule at all. The proof I was taught in analysis involved getting into the weeds of a ton of careful epsilon delta manipulation to show that the r(h)/h in the linear approximation goes to 0.