やさしく学べる大学第２学期の積分と無限級数 Calculus 2 Simplified : Integration and Infinite Series

Fernandez, Oscar E.

Princeton University Press（2025/04発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 256 p.
• 言語 ENG
• 商品コード 9780691263755
• DDC分類 515

Full Description

From the author of Calculus Simplified, an accessible, personalized approach to Calculus 2

Second-semester calculus is rich with insights into the nature of infinity and the very foundations of geometry, but students can become overwhelmed as they struggle to synthesize the range of material covered in class. Oscar Fernandez provides a "Goldilocks approach" to learning the mathematics of integration, infinite sequences and series, and their applications—the right depth of insights, the right level of detail, and the freedom to customize your student experience. Learning calculus should be an empowering voyage, not a daunting task. Calculus 2 Simplified gives you the flexibility to choose your calculus adventure, and the right support to help you master the subject.

Provides an accessible, user-friendly introduction to second-semester college calculus
The unique customizable approach enables students to begin first with integration (traditional) or with sequences and series (easier)
Chapters are organized into mini lessons that focus first on developing the intuition behind calculus, then on conceptual and computational mastery
Features more than 170 solved examples that guide learning and more than 400 exercises, with answers, that help assess understanding
Includes optional chapter appendixes
Comes with supporting materials online, including video tutorials and interactive graphs

Contents

Preface
Before You Begin
To the Student
To the Instructor

1 The Fast-Track Introduction to Calculus 2

1.1 First Things First: What Is Calculus?
1.2 Limits: (Still) The Foundation of Calculus
1.3 The Three Difficult Questions That Drove the Development of Calculus 2

2 Integration Techniques and Approximations

2.1 Integrating, Leibniz's Way
2.2 Approximating Integrals, Riemann's Way
2.3 The Trapezoidal Rule
2.4 How to Approximate Integrals to Any Desired Accuracy
2.5 Integration by Parts
2.6 Trigonometric Integrals
2.7 Trigonometric Substitution
2.8 Partial Fraction Decomposition
2.9 Parting Thoughts

Chapter 2 Exercises

Chapter 2 Appendix
A2.1 Evaluating Riemann Sums Using Summation Formulas
A2.2 Additional Error Theorems for Riemann Sums and the Trapezoidal Rule
A2.3 The Tabular Method for Integration by Parts
A2.4 Integrands That Are Products of Powers of Sine and Cosine
A2.5 Integrands That Are Sine-Cosine Products with Different Arguments
A2.6 A Brief Review of Long Division and Its Uses in Partial Fraction Decomposition

3 Applications of Integration

3.1 A Quick Preview of What's to Come
3.2 Area between Curves
3.3 Volumes by Cross Sections
3.4 Volumes of Revolution: The Disk Method
3.5 Volumes of Revolution: The Washer Method
3.6 Volumes of Revolution: The Shell Method
3.7 Calculating the Length of a Curve
3.8 Calculating the (Lateral) Area of a Surface
3.9 Parting Thoughts

Chapter 3 Exercises

Chapter 3 Appendix
A3.1 Area between Two Curves: The Riemann Sums Approach
A3.2 Riemann Sums Approach to Volumes by Cross Sections
A3.3 Riemann Sums Approach to the Disk Method
A3.4 Riemann Sums Approach to the Washer Method
A3.5 Riemann Sums Approach to the Shell Method
A3.6 Volumes of Revolution: Noncoordinate Axes of Revolution

4 Sequences and Series

4.1 Introduction to Sequences
4.2 Convergence of Sequences
4.3 Infinite Series
4.4 Special Series and the Series Laws
4.5 The Limit and Direct Comparison Tests
4.6 Alternating Series
4.7 The Ratio Test
4.8 Approximating the Sum of an Alternating Series
4.9 Taylor Polynomials
4.10 Taylor's Theorem
4.11 Power Series and Their Convergence
4.12 Power Series as Functions
4.13 Taylor Series
4.14 Convergence of Taylor Series
4.15 Applications of Taylor Series
4.16 Parting Thoughts

Chapter 4 Exercises

5 Connections between Integration and Series

5.1 Partial Fractions and Telescoping Series
5.2 Power/Taylor Series and Integration
5.3 Improper Integrals
5.4 The Integral Test for Series
5.5 Parting Thoughts

Chapter 5 Exercises

Epilogue
Acknowledgments

Appendixes A--D: Precalculus and Calculus Review

A Algebra and Geometry Review
B Precalculus (Functions) Review
C Calculus Review I—Through Differentiation
D Calculus Review II—Integration

Appendix E: Integration Basics

E.1 The Fundamental Theorem of Calculus
E.2 Antiderivatives and the Evaluation Theorem
E.3 Properties of Integrals
E.4Net Signed Area
E.5 Integrating Transcendental Functions
E.6 The Substitution Rule

Appendix F: L'Hôpital's Rule

Appendix F Exercises

Answers to Exercises
Bibliography
Index of Applications
Subject Index