Calculus simplified

✍ Scribed by Fernandez, Oscar Edward

• Publisher: Princeton University Press
• Year: 2019
• Tongue: English
• Leaves: 270
• Category: Library

✦ Synopsis

Cover; Contents; Preface; To the Student; To the Instructor; Before You Begin . . .; 1. The Fast Track Introduction to Calculus; 1.1 What Is Calculus?; Calculus as a Way of Thinking; What Does "Infinitesimal Change" Mean?; 1.2 Limits: The Foundation of Calculus; 1.3 The Three Difficult Problems That Led to the Invention of Calculus; 2. Limits: How to Approach Indefinitely (and Thus Never Arrive); 2.1 One-Sided Limits: A Graphical Approach; 2.2 Existence of One-Sided Limits; 2.3 Two-Sided Limits; 2.4 Continuity at a Point; 2.5 Continuity on an Interval; 2.6 The Limit Laws;An accessible, streamlined, and user-friendly approach to calculusCalculus 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.

✦ Table of Contents

Cover......Page 1
Contents......Page 8
Preface......Page 11
To the Student......Page 17
To the Instructor......Page 19
Before You Begin . . .......Page 21
Calculus as a Way of Thinking......Page 25
What Does “Infinitesimal Change” Mean?......Page 26
1.2 Limits: The Foundation of Calculus......Page 27
1.3 The Three Difficult Problems That Led to the Invention of Calculus......Page 29
2.1 One-Sided Limits: A Graphical Approach......Page 32
2.2 Existence of One-Sided Limits......Page 35
2.3 Two-Sided Limits......Page 37
2.4 Continuity at a Point......Page 39
2.5 Continuity on an Interval......Page 41
2.6 The Limit Laws......Page 45
2.7 Calculating Limits—Algebraic Techniques......Page 49
2.8 Limits Approaching Infinity......Page 54
2.9 Limits Yielding Infinity......Page 57
Chapter 2 Exercises......Page 61
3.1 Solving the Instantaneous Speed Problem......Page 67
3.2 Solving the Tangent Line Problem—The Derivative at a Point......Page 71
3.3 The Instantaneous Rate of Change Interpretation of the Derivative......Page 74
3.4 Differentiability: When Derivatives Do (and Don’t) Exist......Page 75
3.5 The Derivative, a Graphical Approach......Page 77
3.6 The Derivative, an Algebraic Approach......Page 79
Leibniz Notation......Page 83
3.7 Differentiation Shortcuts: The Basic Rules......Page 84
3.8 Differentiation Shortcuts: The Power Rule......Page 85
3.9 Differentiation Shortcuts: The Product Rule......Page 88
3.10 Differentiation Shortcuts: The Chain Rule......Page 89
3.11 Differentiation Shortcuts: The Quotient Rule......Page 92
3.12 (Optional) Derivatives of Transcendental Functions......Page 93
3.13 Higher-Order Derivatives......Page 98
3.14 Parting Thoughts......Page 99
Chapter 3 Exercises......Page 100
4.1 Related Rates......Page 106
4.2 Linearization......Page 113
4.3 The Increasing/Decreasing Test......Page 117
4.4 Optimization Theory: Local Extrema......Page 122
4.5 Optimization Theory: Absolute Extrema......Page 125
4.6 Applications of Optimization......Page 130
4.7 What the Second Derivative Tells Us About the Function......Page 136
4.8 Parting Thoughts......Page 141
Chapter 4 Exercises......Page 142
5.1 Distance as Area......Page 149
5.2 Leibniz’s Notation for the Integral......Page 152
5.3 The Fundamental Theorem of Calculus......Page 154
5.4 Antiderivatives and the Evaluation Theorem......Page 157
5.5 Indefinite Integrals......Page 159
5.6 Properties of Integrals......Page 162
5.7 Net Signed Area......Page 163
5.8 (Optional) Integrating Transcendental Functions......Page 165
5.9 The Substitution Rule......Page 167
5.10 Applications of Integration......Page 172
5.11 Parting Thoughts......Page 176
Chapter 5 Exercises......Page 177
Epilogue......Page 183
Acknowledgments......Page 185
Appendix A: Review of Algebra and Geometry......Page 187
Appendix B: Review of Functions......Page 201
Appendix C: Additional Applied Examples......Page 239
Answers to Appendix and Chapter Exercises......Page 251
Bibliography......Page 263
Index of Applications......Page 265
Index of Subjects......Page 267

✦ Subjects

Calculus;MATHEMATICS--Calculus;MATHEMATICS--Mathematical Analysis;Electronic books;MATHEMATICS -- Calculus;MATHEMATICS -- Mathematical Analysis