# How much rigor?

I am working on teaching myself calculus and have a few texts that employ different teaching methods. One covers a short chapter on limits and then goes right into calculating derivatives. Another covers logic, set theory, and rigorous definitions of continuity and limits before introducing derivatives. I started on the first text and am still working through the exercises, but have also been reading the front matter of the other text.

Is it more important to take my time and slowly learn the rigorous definitions of everything at this stage, or is it more important to just learn how to do the calculations and learn the proofs in detail at a later stage?

Learn things properly the first time round. Trust me it helps a lot. But the definition of properly depends on why you are learning the material and what you plan to use it for.

Definately understand what you are doing before going to the next topic.

My HS calc teacher explained limits as being "very close to the number, but not the number", which has caused some problems. I don't think I ever even saw the formal definition for a limit until this year when we started doing differentiables...

Anyway, in the case of limits, it wasn't so bad, but you want to make sure you understand things before you move on. If you spend more time that it required, all that will happens is you will remember it better.

PL

Tom Mattson
Staff Emeritus
Gold Member
Do both!

Seriously, synthesize the two approaches, and move forward with the two in step. It's important for your long-term growth to be rigorous, and it's important for your short-term growth to be able to successfully solve exercises.

By the way, you'll find that the logic will help you solve some of the exercises too. I can't tell you how many calculus students I've had who have made dumb mistakes in exercises because they didn't properly understand conditionals and their inverses, converses, and contrapositives and the logical equivalences between them. I always do a short suicide crash course on symbolic logic as early as possible.

Thanks for the advice. I read this chapter on logic http://faculty.swosu.edu/michael.dougherty/book/chapter01.pdf" [Broken] and found it quite helpful. I felt kind of dumb for being a college graduate and not knowing any of this stuff. I will continue to work hard at correctly solving calculus exercises, but also read analysis at the same time.

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