I agree. This is what I was going for in that paragraph. If you define derivatives & integrals with infinitesimals, then you can actually do things like treating dy/dx as a fraction without partaking in the half-in half-out dance that calc 1 teachers currently have to do.
I don’t think the pedagogical benefit of nonstandard analysis is to replace Analysis I courses, but rather to give a rigorous backing to doing algebra with infinitesimals (“an infinitely small thing plus a real number is the same real number, an infinitely small thing times a real number is zero”). *Improper integrals would make a lot more sense this way, IMO.
Indefinite integrals would make a lot more sense this way, IMO
Why so? I thought they already made sense, they’re “antiderivatives”, so a function such that taking its derivative gives you the original functions. Do you need anything further to define them?
(I know about the definite integral Riemann and Lebesgue definitions, but I thought indefinite integrals were much easier in comparison.
Now that I’m thinking about it, my memory’s fuzzy on how you’d actually calculate them rigorously w/infinitesimals. Will get back to you with an example.
I agree. This is what I was going for in that paragraph. If you define derivatives & integrals with infinitesimals, then you can actually do things like treating dy/dx as a fraction without partaking in the half-in half-out dance that calc 1 teachers currently have to do.
I don’t think the pedagogical benefit of nonstandard analysis is to replace Analysis I courses, but rather to give a rigorous backing to doing algebra with infinitesimals (“an infinitely small thing plus a real number is the same real number, an infinitely small thing times a real number is zero”). *Improper integrals would make a lot more sense this way, IMO.
Thank you, that makes sense!
Why so? I thought they already made sense, they’re “antiderivatives”, so a function such that taking its derivative gives you the original functions. Do you need anything further to define them?
(I know about the definite integral Riemann and Lebesgue definitions, but I thought indefinite integrals were much easier in comparison.
Language mix-up. Meant improper integrals.
Now that I’m thinking about it, my memory’s fuzzy on how you’d actually calculate them rigorously w/infinitesimals. Will get back to you with an example.