Understanding Calculus

This is a complete, free, online Calculus textbook/course.

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

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

If you've gone through my online curriculum through Precalculus,

then you should be able to jump right into Chapter 3.

The first two chapters*review* essential material and introduce new readers to *math language* concepts.

then you should be able to jump right into Chapter 3.

The first two chapters

TABLE OF CONTENTS (i–ii)

PREFACE (iii–v)

ACKNOWLEDGEMENTS (vi)

TABLE OF SYMBOLS (viii–ix)

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)

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): also needed for Calculus!

Average Rate of Change (great practice with function notation)

3.1 Limits—The Idea (108–119)Average Rate of Change (great practice with function notation)

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)

4.1 Tangent Lines (182–192)

4.2 The Derivative (193–203)

4.3 Some Very Basic Differentiation Formulas (194–219) 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)Mixed Differentiation Practice

Mixed Differentiation Practice (strict parentheses version)

4.8 Implicit Differentiation (optional) (257–265)

4.9 The Mean Value Theorem (266–272)

SAMPLE TEST, Chapter 4 (273–275)

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)

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)

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) 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)

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)

STUDENT'S COMPLETE SOLUTION MANUAL:

solutions (showing all steps) to ALL in-section and end-of-section exercises, sample tests

solutions (showing all steps) to ALL in-section and end-of-section exercises, sample tests

TABLE OF CONTENTS (i–ii)

SECTION SOLUTIONS:

Chapter 1: | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | ||||

Chapter 2: | 2.1 | 2.2 | 2.3 | 2.4 | |||||

Sample Test, Chapters 1 and 2 (scanned pages, hand-written solutions) |
Zoom to enlarge (as needed): page 40 page 41 page 42 page 43 |
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Chapter 3: | 3.1 | 3.2 | 3.3 | 3.4 | 3.5 | 3.6 | 3.7 | ||

Sample Test, Chapter 3 (scanned pages, hand-written solutions) |
Zoom to enlarge (as needed): page 74 page 75 page 76 |
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Chapter 4: | 4.1 | 4.2 | 4.3 | 4.4 | 4.5 | 4.6 | 4.7 | 4.8 | 4.9 |

Sample Test, Chapter 4 (scanned pages, hand-written solutions) |
Zoom to enlarge (as needed): page 123 page 124 page 125 |
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Chapter 5: | 5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.6 | |||

Sample Test, Chapter 5 (scanned pages, hand-written solutions) |
Zoom to enlarge (as needed): page 158 page 159 page 160 |
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Chapter 6: | 6.1 | 6.2 | 6.3 | 6.4 | 6.5 | 6.6 | |||

Sample Test, Chapter 6 (scanned pages, hand-written solutions) |
Zoom to enlarge (as needed): page 185 page 186 page 187 |
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Chapter 7: | 7.1 | 7.2 | 7.3 | 7.4 | 7.5 | 7.6 | 7.7 | ||

Sample Test, Chapter 7 (scanned pages, hand-written solutions) |
Zoom to enlarge (as needed): page 207 page 208 page 209 |