Hello, I'm about to be a high school senior next year, and I want to study Calculus on my own. I have a strong base in mathematics, and I'm self motivated. I am currently learning the basics of calculus from a book called "The Complete Idiot's Guide to Calculus", and I'll be done with it in a few days. Now, I want to actually study the hardcore calculus that freshmen-sophomore college courses offer. What do I need to know before I start studying that? Do I need to know the stuff they teach in Calculus AB and BC? or does college freshmen-sophomore level calculus cover the same stuff? All in all, I want to study everything from Basic Calculus (which is hopefully everything covered in the book that i mentioned before) upto Multivariable Calculus. Do you recommend any good books that are easy to understand, explains concepts really well, and has a lot of practice problems? btw, I want to study this before school starts, so please dont tell me to take a calculus course in high school... I would take one in a local community college, but I already have summer school for another advance course. Help would be greatly appreciated.
Hubbard/Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms" is a fun text with a lot of exercises that emphasises both theory and real-world applications. You can also make your knowledge of single-variable calculus more solid by going through at least one of the masterpieces of the genre: Spivak's "Calculus", Courant's "Differential and Integral Calculus", or Apostol's "Calculus". Each one of these books will leave you wanting for nothing more in regards to rigor and proof. No statement is left to analogy or hand-waving.