I'm interested in self-studying the following books over the next year or so:
- Spivak's Calculus (I'm already in Ch. 5 and it is very slow going)
- The Cauchy-Schwarz Master Class by J. Michael Steele
- Analytic Inequalities by Nicholas Kazarinoff
My goal in studying these books is to gain a deeper understanding of calculus, basic real analysis, and manipulations of the standard inequalities, with the ultimate goal of understanding derivations, approximations, and inequalities in probability and statistics (Stirling's approximation, Wallis product, Gamma Function, Normal Distribution, Limit Theorems etc). One of the things I realized when I first started studying Spivak's Calculus is that I have had very little experience in solving challenging problems. I have never had any issues with doing 'Exercises' in the standard engineering style calculus text books, but I am often at a loss of ideas when I do problems in Spivak.
My questions are the following:
Before progressing through Spivak, should I go through a book like Art and Craft of Problem Solving by Paul Zeitz? I guess the point of doing this would be to beef up my problem-solving skills. I should note that I am not very excited about working through the Art and Craft of Problem Solving because a lot of it seems geared toward solving Olympiad geometry problems. I've never had a solid geometry course, so at this point I feel like it might just be a waste of time trying to learn plane geometry.
Should I relearn high school mathematics? To be perfectly honest, I feel robbed by my entire education and I'm very disappointed by my lack of foresight up to this point. I've always used easy textbooks(not my choice) in my college calculus, Linear algebra, and ODE and PDE classes and believed 'good grades' were enough.
Or, should I just keep a copy of Polya's Heuristics on hand while I patiently work through Spivak?
I'm just looking for a bit of advise on the wisest way to proceed.