Calculus

This volume provides notes on calculus. Because calculus doesn't have clear boundaries — some topics are more algebraic-flavored, others more geometric, and yet others more discrete — the notes aren't necessarily grouped by traditional course headings (i.e., "Calculus I" and "Calculus II"). That said, readers can find this volume mapping to the typical U.S. calculus sequence: Precalculus, Calculus I, Calculus II, and Calculus III. Some of the materials later in the sequence touch on Real Analysis, but I'm slowly sifting through these notes to place them in a separate volume.

That said, here are a few personal comments on the topic of calculus. In my experience, the hardest part of calculus (at least early on) isn't the introduction of new theorems and ideas, but the necessity of quick and careful algebraic manipulation. Derivatives and integrals can be nasty constructions, and the tiniest mistake — forgetting that puny subscript, misplacing a parenthesis, or neglecting the subtle minus sign — can plunge even the most skilled algebraists down to the bottom margins. I've personally seen professionals, some of them Fields Medalists, come perilously close to these outcomes (thank goodness for large blackboards and attentive audiences).

All this is to say that algebra is hard. If I could do it over again, I would spend much more time solidifying algebra and trigonometry before stepping into calculus. If, dear reader, it's not entirely clear why ${\sin^2(x) + \cos^2(x) = 1,}$ I encourage revisiting algebra and trigonometry before proceeding.