Calculus Simplified

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. Cover......Page 1 Contents......Page 8 Preface......Page 12 To the Student......Page 18 To the Instructor......Page 20 Before You Begin . . .......Page 22 Calculus as a Way of Thinking......Page 26 What Does ?Infinitesimal Change? Mean?......Page 27 1.2 Limits: The Foundation of Calculus......Page 28 1.3 The Three Difficult Problems That Led to the Invention of Calculus......Page 30 2.1 One-Sided Limits: A Graphical Approach......Page 33 2.2 Existence of One-Sided Limits......Page 36 2.3 Two-Sided Limits......Page 38 2.4 Continuity at a Point......Page 40 2.5 Continuity on an Interval......Page 42 2.6 The Limit Laws......Page 46 2.7 Calculating Limits?Algebraic Techniques......Page 50 2.8 Limits Approaching Infinity......Page 55 2.9 Limits Yielding Infinity......Page 58 Chapter 2 Exercises......Page 62 3.1 Solving the Instantaneous Speed Problem......Page 68 3.2 Solving the Tangent Line Problem?The Derivative at a Point......Page 72 3.3 The Instantaneous Rate of Change Interpretation of the Derivative......Page 75 3.4 Differentiability: When Derivatives Do (and Don?t) Exist......Page 76 3.5 The Derivative, a Graphical Approach......Page 78 3.6 The Derivative, an Algebraic Approach......Page 80 Leibniz Notation......Page 84 3.7 Differentiation Shortcuts: The Basic Rules......Page 85 3.8 Differentiation Shortcuts: The Power Rule......Page 86 3.9 Differentiation Shortcuts: The Product Rule......Page 89 3.10 Differentiation Shortcuts: The Chain Rule......Page 90 3.11 Differentiation Shortcuts: The Quotient Rule......Page 93 3.12 (Optional) Derivatives of Transcendental Functions......Page 94 3.13 Higher-Order Derivatives......Page 99 3.14 Parting Thoughts......Page 100 Chapter 3 Exercises......Page 101 4.1 Related Rates......Page 107 4.2 Linearization......Page 114 4.3 The Increasing/Decreasing Test......Page 118 4.4 Optimization Theory: Local Extrema......Page 123 4.5 Optimization Theory: Absolute Extrema......Page 126 4.6 Applications of Optimization......Page 131 4.7 What the Second Derivative Tells Us About the Function......Page 137 4.8 Parting Thoughts......Page 142 Chapter 4 Exercises......Page 143 5.1 Distance as Area......Page 150 5.2 Leibniz?s Notation for the Integral......Page 153 5.3 The Fundamental Theorem of Calculus......Page 155 5.4 Antiderivatives and the Evaluation Theorem......Page 158 5.5 Indefinite Integrals......Page 160 5.6 Properties of Integrals......Page 163 5.7 Net Signed Area......Page 164 5.8 (Optional) Integrating Transcendental Functions......Page 166 5.9 The Substitution Rule......Page 168 5.10 Applications of Integration......Page 173 5.11 Parting Thoughts......Page 177 Chapter 5 Exercises......Page 178 Epilogue......Page 184 Acknowledgments......Page 186 Appendix A: Review of Algebra and Geometry......Page 188 Appendix B: Review of Functions......Page 202 Appendix C: Additional Applied Examples......Page 240 Answers to Appendix and Chapter Exercises......Page 252 Bibliography......Page 264 Index of Applications......Page 266 Index of Subjects......Page 268

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. InCalculus 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 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