A Course in Calculus and Real Analysis (Undergraduate Texts in Mathematics)

✍ Scribed by Sudhir R. Ghorpade, Balmohan V. Limaye

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 442
• Category: Library

✦ Synopsis

This book provides a self-contained and rigorous introduction to calculus of functions of one variable, in a presentation which emphasizes the structural development of calculus. Throughout, the authors highlight the fact that calculus provides a firm foundation to concepts and results that are generally encountered in high school and accepted on faith; for example, the classical result that the ratio of circumference to diameter is the same for all circles. A number of topics are treated here in considerable detail that may be inadequately covered in calculus courses and glossed over in real analysis courses.

✦ Table of Contents

Cover
Title Page
Copyright Page
Preface
Contents
1. Numbers and Functions
1.1 Properties of Real Numbers
1.2 Inequalities
1.3 Functions and Their Geometric Properties
Exercises
2. Sequences
2.1 Convergence of Sequences
2.2 Subsequences and Cauchy Sequences
Exercises
3. Continuity and Limits
3.1 Continuity of Functions
3.2 Basic Properties of Continuous Functions
3.3 Limits of Functions of a Real Variable
Exercises
4. Differentiation
4.1 The Derivative and Its Basic Properties
4.2 The Mean Value and Taylor Theorems
4.3 Monotonicity, Convexity, and Concavity
4.4 L’Hôpital’s Rule
Exercises
5. Applications of Differentiation
5.1 Absolute Minimum and Maximum
5.2 Local Extrema and Points of Inflection
5.3 Linear and Quadratic Approximations
5.4 The Picard and Newton Methods
Exercises
6. Integration
6.1 The Riemann Integral
6.2 Integrable Functions
6.3 The Fundamental Theorem of Calculus
6.4 Riemann Sums
Exercises
7. Elementary Transcendental Functions
7.1 Logarithmic and Exponential Functions
7.2 Trigonometric Functions
7.3 Sine of the Reciprocal
7.4 Polar Coordinates
7.5 Transcendence
Exercises
8. Applications and Approximations ofRiemann Integrals
8.1 Area of a Region Between Curves
8.2 Volume of a Solid
8.3 Arc Length of a Curve
8.4 Area of a Surface of Revolution
8.5 Centroids
8.6 Quadrature Rules
Exercises
9. Infinite Series and Improper IntegralsIf
9.1 Convergence of Series
9.2 Convergence Tests for Series
9.3 Power Series
9.4 Convergence of Improper Integrals
9.5 Convergence Tests for Improper Integrals
9.6 Related Integrals
Exercises
References
List of Symbols and Abbreviations
Index