Calculus Simplified

by Oscar E. Fernandez

Overview

An accessible, streamlined, and user-friendly approach to calculus

Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a “Goldilocks approach” to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure.

Fernandez begins by offering an intuitive introduction to the three key ideas in calculus—limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources—including video tutorials and interactive graphs—are available on the book’s website.

Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions—these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics—chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus.

Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence.

• An accessible, intuitive introduction to first-semester calculus
• Nearly 200 solved problems and more than 300 exercises (all with answers)
• No prior knowledge of exponential, logarithmic, or trigonometric functions required
• Additional online resources—video tutorials and supplementary exercises—provided

Product Details

ISBN-13:	9780691175393
Publisher:	Princeton University Press
Publication date:	06/11/2019
Pages:	272
Sales rank:	582,959
Product dimensions:	7.90(w) x 9.50(h) x 0.90(d)
Age Range:	18 Years

About the Author

Oscar E. Fernandez is associate professor of mathematics at Wellesley College. He is the author of Everyday Calculus and The Calculus of Happiness (both Princeton).

Table of Contents

Preface xi

To the Student xvii

To the Instructor xix

Before You Begin… xxi

1 The Fast Track Introduction to Calculus 1

1.1 What Is Calculus? 1

Calculus as a Way of Thinking 1

What Does "Infinitesimal Change" Mean? 2

1.2 Limits: The Foundation of Calculus 3

1.3 The Three Difficult Problems That Led to the Invention of Calculus 5

2 Limits: How to Approach Indefinitely (and Thus Never Arrive) 8

2.1 One-Sided Limits: A Graphical Approach 8

2.2 Existence of One-Sided Limits 11

2.3 Two-Sided Limits 13

2.4 Continuity at a Point 15

2.5 Continuity on an Interval 17

2.6 The Limit Laws 21

2.7 Calculating Limits-Algebraic Techniques 25

2.8 Limits Approaching Infinity 30

2.9 Limits Yielding Infinity 33

2.10 Parting Thoughts 37

Chapter 2 Exercises 37

3 Derivatives; Change, Quantified 43

3.1 Solving the Instantaneous Speed Problem 43

3.2 Solving the Tangent Line Problem-The Derivative at a Point 47

3.3 The Instantaneous Rate of Change Interpretation of the Derivative 50

3.4 Differentiability: When Derivatives Do (and Don't) Exist 51

3.5 The Derivative, a Graphical Approach 53

3.6 The Derivative, an Algebraic Approach 55

Leibniz Notation 59

3.7 Differentiation Shortcuts: The Basic Rules 60

3.8 Differentiation Shortcuts: The Power Rule 61

3.9 Differentiation Shortcuts: The Product Rule 64

3.10 Differentiation Shortcuts: The Chain Rule 65

3.11 Differentiation Shortcuts: The Quotient Rule 68

3.12 (Optional) Derivatives of Transcendental Functions 69

3.13 Higher-Order Derivatives 74

3.14 Parting Thoughts 75

Chapter 3 Exercises 76

4 Applications of Differentiation 82

4.1 Related Rates 82

4.2 Linearization 89

4.3 The Increasing/Decreasing Test 93

4.4 Optimization Theory: Local Extrema 98

4.5 Optimization Theory: Absolute Extrema 101

4.6 Applications of Optimization 106

4.7 What the Second Derivative Tells Us About the Function 112

4.8 Parting Thoughts 117

Chapter 4 Exercises 118

5 Integration: Adding Up Change 125

5.1 Distance as Area 125

5.2 Leibniz's Notation for the Integral 128

5.3 The Fundamental Theorem of Calculus 130

5.4 Antiderivatives and the Evaluation Theorem 133

5.5 Indefinite Integrals 135

5.6 Properties of Integrals 138

5.7 Net Signed Area 139

5.8 (Optional) Integrating Transcendental Functions 141

5.9 The Substitution Rule 143

5.10 Applications of Integration 148

5.11 Parting Thoughts 152

Chapter 5 Exercises 153

Epilogue 159

Acknowledgments 161

Appendix A Review of Algebra and Geometry 163

Appendix B Review of Functions 177

Appendix C Additional Applied Examples 215

Answers to Appendix and Chapter Exercises 227

Bibliography 239

Index of Applications 241

Index of Subjects 243

What People are Saying About This

From the Publisher

"This reader-friendly book is specifically designed for first-year calculus students. Fernandez writes in a welcoming style and the text is neither repetitive nor jargon-laden. With a strong presentation, well-chosen examples, and useful exercises, he does a marvelous job distilling the subject down to the essentials."—David R. Dorman, Middlebury College

"A brilliant, mind-easing, mind-straightening, and mind-empowering companion guide for all those currently working—and struggling—to discover and understand the meaning and story of calculus."—James Tanton, Mathematical Association of America and Global Math Project

"Calculus Simplified contains just the right amount of content and depth of insight for students who need neither the rigor of a calculus class for mathematicians nor the detail of a calculus class for engineers and physicists. This book balances intuition with theory to present the appropriate level for readers working outside of those subject areas."—John A. Adam, author of A Mathematical Nature Walk

"Calculus Simplified provides an ideal mix of conceptual development of the big ideas and formal treatment of the process. It answers, 'What is calculus?' and 'How do I use it?' without getting bogged down in the muck that often inhibits the joy and productivity for so many students using other materials. This excellent new resource will fill an important niche."—Brent A. Ferguson, Lawrenceville School

Editorial Reviews

"A well-produced and well-written introduction to calculus which nicely balances the technical 'how do you do it' and motivational 'why are we doing this' aspects of the subject."—-Nick Lord, Mathematical Gazette

From the Publisher