I started writing this book as a junior in high school in my second semester of AP Calculus BC with the goal of writing a textbook from the perspective of a student so that better analogies could be made to increase understanding, since I remember what it was like not long ago to not understand.

Calculus is the mathematical study of change. In this text, we outline the basics of the three main subjects of calculus: Differential Calculus (Calculus I), Integral & Series Calculus (Calculus II), and Multivariable & Vector Calculus (Calculus III). We attempt to do this in a fluid and intuitive manner that is easy for those of various mathematical backgrounds to understand. This book is intended as an introductory text to calculus and is best used when supplemented with lectures on the main topics of each chapter.

At the end of each chapter, following the chapter summary, there will be exercises split up by which section of the chapter they should be assigned with. Solutions to odd-numbered exercises will be provided at the end of the book. Also at the end of the book will be some of the more involved derivations of different derivative and integral rules, like the chain rule, derivatives of inverse trig functions, and more.

This textbook is very much a work in progress. In its current stage, no formatting has been finalized (in fact, in most areas, it will be changed). Exercises will be added in future versions. In this version, chapter 5 on some of the applications of integrals has been added. Small refinements have been made to other sections, particularly to the section on limits, though there are many more to come in the near future.

In the next version, I hope to add more examples and a few exercises for the currently existing sections. I would also like to formalize my write-up of chapter 3 and roughly complete that section here. Moreover, I plan on making major modifications to Chapter 1 on limits for the reason that, while most of the explanations are okay for an intuitive understanding, they lie on very shaky ground mathematically. I would like to be slightly more precise (while still avoiding epsilons and deltas). I will also finish the main content of chapters 2 (derivatives) and 5 (applications of integrals) by finishing the sections on differentiability, implicit differentiation, and initial value problems, along with adding numerous more examples. My formatting largely varies by chapter as I am trying to discover which formatting I like the best. It should again be noted that no formatting is finalized, and that much of the formatting is currently in a very prototypical state.

A rough draft of chapter 3 is complete, and chapter 6's rough draft is about halfway done. Suggestions and corrections are welcome and encouraged, both of current content and suggestions for future content (examples, exercises, formatting tips, etc.). You can contact me here.

Calculus (v2) (3/09/2026)
Calculus (v1) (1/29/2026)