I don’t know if this thread is stale but I just saw it. For what it’s worth, I have some observations from almost 30 years of teaching physics and calculus.
1) AP calc is almost worthless. Unless you are the unusually gifted student who self taught everything missing from your AP course you should plan on starting at calc 1 at a decent university. Going from BC to calc 3 virtually guarantees you will spend most of the semester catching up.
2) If you want to learn calculus for physics skip Apostol and Spivak, you won’t learn any physical intuition. Get a copy of Thomas’ Calculus, edition 1, 2, or 3. He died after those and later editions were reworked by other people. The usefulness for physics is beyond compare. I never use a textbook anymore, I just post my lecture notes but if students want a book I recommend Thomas. They are reasonably priced used. All current calculus books suck. Something like 9798% of students who take calculus don’t want to be mathematicians but calculus books are all written by pure mathematicians who don’t seem to understand who their audience is. I can’t think of who does it at the moment but some engineering schools are starting to teach their own math courses because the math dept can’t do it in a useful manner.
3) Don’t become enamored with epsilon-delta proofs. When you are in a physics class your professor will say “assume dt is infinitesimal...” Jerome Keisler has a good book for free on his website that teaches calculus using infinitesimals. That’s the way every physicist and engineer uses calculus and it’s very intuitive. Not only do people not use epsilon-delta proofs in application, they are not intuitively helpful.
4) I don’t know where other people took their ODE courses but you can not pass mine without a solid understanding of calc 2. Even calc 3 is prerequisite but I do let exceptional students take it concurrently.
