One Mathematical Cat, Please! Understanding Calculus

Calculus book main cover

Curious about Calculus? Want honest content, but a bit ‘slower and kinder’ than traditional Calculus books?

Like the way I explain things in Algebra I, Geometry, Algebra II, and Precalculus? Then, this book is for you.

It's a complete Calculus course. You can read it online or buy a printed version.

I call it the ‘warrior cat’ book because of the print-version cover. (You can read about these four cats in my Mathematical Cat Gallery: numbers 23, 24, 25 and 26.)

Read the ‘Warrior Cat’ Book For Free Online

The entire book is available below. Or, you can get a Complete Calculus Course bundle at my Teachers Pay Teachers Store that includes pdfs of the complete book, solution manual, Calculus index cards, prerequisites, and end-of-year review.

Gone through my online curriculum through Precalculus? Then, jump right into Chapter 3.

Haven't been using my lessons? No problem. The first two chapters review essential material and introduce you to all the math language concepts you'll need.

Print Version of the ‘Warrior Cat’ Book (and Complete Solution Manual)

Calculus book main cover
Understanding Calculus
(the main text)

The PRINT version of the Warrior Cat book is for those of us who like to highlight, write in margins, feel the paper between our fingers.

Both the online and print versions include selected solutions: odd-numbered end-of-section exercises and Quick Quiz Problems.

Calculus solution manual
Complete Solution Manual

For ALL the solutions, I also offer a Complete Solution Manual. It includes completely-written-out solutions to:

(A pdf version of the Complete Solution Manual is included in my Complete Calculus Course bundle at my Teachers Pay Teachers store.)

Calculus Index Cards

I also offer a set of Calculus Index Cards. Hold the entire course in the palm of your hand!

Calculus index cards

(These index cards are included in my Complete Calculus Course bundle at my Teachers Pay Teachers store.)

Classic Calculus Textbooks

These are the Calculus textbooks I learned and/or taught from in my early days (before using my own materials). You can see the well-worn bindings. I even put tabs on Stewart to help me quickly turn to various sections.

my Calculus books

Of course, you can't (easily, at least) get the ones that I have anymore. I can't comment directly on any of the current editions (below), but all the ones I have are excellent. If you want a more traditional text, likely any of these would be a great choice.

READ THE ‘WARRIOR CAT’ BOOK: UNDERSTANDING CALCULUS

The materials in green give unlimited online practice and/or supplement the text.

TABLE OF CONTENTS (i–ii)
PREFACE (iii–v)
ACKNOWLEDGEMENTS (vi)
STUDY STRATEGIES for Students of Mathematics (vii)
TABLE OF SYMBOLS (viii–ix)
CHAPTER 1. ESSENTIAL PRELIMINARIES
1.1 The Language of Mathematics—Expressions versus Sentences (1–11)
1.2 The Role of Variables (12–21)
1.3 Sets and Set Notation (22–28)
1.4 Mathematical Equivalence (29–38)
1.5 Graphs (39–53)
CHAPTER 2. FUNCTIONS
2.1 Functions and Function Notation (54–68)
2.2 Graphs of Functions (69–81)
2.3 Composite Functions (82–91)
2.4 One-to-One Functions and Inverse Functions (92–103)
SAMPLE TEST, Chapters 1 and 2 (104–107)
Prerequisites for Precalculus (with solutions): These prerequisites are also needed for Calculus!
Average Rate of Change (great practice with function notation)
CHAPTER 3. LIMITS AND CONTINUITY
3.1 Limits—The Idea (108–119)
Introduction to Limits (online practice)
3.2 Limits—Making it Precise (120–132)
3.3 Properties of Limits (133–144)
3.4 Continuity (145–153)
3.5 Indeterminate Forms (154–159)
3.6 The Intermediate Value Theorem (160–170)
3.7 The Max-Min Theorem (171–178)
SAMPLE TEST, Chapter 3 (179–181)
CHAPTER 4. THE DERIVATIVE
4.1 Tangent Lines (182–192)
4.2 The Derivative (193–203)
4.3 Some Very Basic Differentiation Formulas (194–219)
Basic Differentiation Shortcuts (differentiating constants and linear functions; the Simple Power Rule)
4.4 Instantaneous Rates of Change (220–227)
4.5 The Chain Rule (Differentiating Composite Functions) (228–238)
4.6 Differentiating Products and Quotients (239–248)
Differentiation Formula Practice
Mixed Differentiation Practice
Mixed Differentiation Practice (strict parentheses version)
4.7 Higher Order Derivatives (249–256)
4.8 Implicit Differentiation (optional) (257–265)
4.9 The Mean Value Theorem (266–272)
SAMPLE TEST, Chapter 4 (273–275)
CHAPTER 5. USING THE INFORMATION GIVEN BY THE DERIVATIVE
5.1 Increasing and Decreasing Functions (276–286)
5.2 Local Maxima and Minima—Critical Points (287–298)
5.3 The Second Derivative—Inflection Points (299–308)
5.4 Graphing Functions—Some Basic Techniques (309–319)
5.5 More Graphing Techniques (320–329)
5.6 Asymptotes—Checking Behavior at Infinity (330–338)
SAMPLE TEST, Chapter 5 (339–341)
CHAPTER 6. ANTIDIFFERENTIATION
6.1 Antiderivatives (342–353)
6.2 Some Basic Antidifferentiation Formulas (354–361)
6.3 Analyzing a Falling Object (optional) (362–375)
6.4 The Substitution Technique for Antidifferentiation (376–384)
6.5 More on Substitution (385–390)
6.6 Integration By Parts (391–397)
SAMPLE TEST, Chapter 6 (398–400)
CHAPTER 7. THE DEFINITE INTEGRAL
7.1 Using Antiderivatives to Find Area (401–407)
7.2 The Definite Integral (408–417)
7.3 The Definite Integral as the Limit of Riemann Sums (418–422)
7.4 The Substitution Technique applied to Definite Integrals (423–427)
Mixed Integration Practice
7.5 The Area Between Two Curves (428–435)
7.6 Finding the Volume of a Solid of Revolution—Disks (436–443)
7.7 Finding the Volume of a Solid of Revolution—Shells (444–449)
SAMPLE TEST, Chapter 7 (450–452)
ABBREVIATED SOLUTIONS: Odd-Numbered End-of-Section Exercises and Quick Quiz Problems
Chapter 1 (453–457)
Chapter 2 (458–462)
Chapter 3 (463–469)
Chapter 4 (470–476)
Chapter 5 (477–479)
Chapter 6 (480–482)
Chapter 7 (483–486)
INDEX (487–497)