A Course in Calculus And Real Analysis

by Ghorpade, Sudhir R.; Limaye, Balmohan Vishnu

• ISBN13: 9780387305301
• ISBN10: 0387305300
• Format: Hardcover
• Copyright: 2006-07-30
• Publisher: Springer Nature

Summary

This book provides a self-contained and rigorous introduction to calculus of functions of one variable. The presentation and sequencing of topics emphasizes the structural development of calculus. At the same time, due importance is given to computational techniques and applications. The authors have strived to make a distinction between the intrinsic definition of a geometric notion and its analytic characterization. Throughout the book, the authors highlight the fact that calculus provides a firm foundation to several concepts and results that are generally encountered in high school and accepted on faith. For example, one can find here a proof of the classical result that the ratio of the circumference of a circle to its diameter is the same for all circles. Also, this book helps students get a clear understanding of the concept of an angle and the definitions of the logarithmic, exponential and trigonometric functions together with a proof of the fact that these are not algebraic functions. A number of topics that may have been inadequately covered in calculus courses and glossed over in real analysis courses are treated here in considerable detail. As such, this book provides a unified exposition of calculus and real analysis. The only prerequisites for reading this book are topics that are normally covered in high school; however, the reader is expected to possess some mathematical maturity and an ability to understand and appreciate proofs. This book can be used as a textbook for a serious undergraduate course in calculus, while parts of the book can be used for advanced undergraduate and graduate courses in real analysis. Each chapter contains several examples and a large selection of exercises, as well as "Notes and Comments" describing salient features of the exposition, related developments and references to relevant literature.

Table of Contents

1 Numbers and Functions	1	(42)
1.1 Properties of Real Numbers	2	(8)
1.2 Inequalities	10	(3)
1.3 Functions and Their Geometric Properties	13	(18)
Exercises	31	(12)
2 Sequences	43	(24)
2.1 Convergence of Sequences	43	(12)
2.2 Subsequences and Cauchy Sequences	55	(5)
Exercises	60	(7)
3 Continuity and Limits	67	(36)
3.1 Continuity of Functions	67	(5)
3.2 Basic Properties of Continuous Functions	72	(9)
3.3 Limits of Functions of a Real Variable	81	(15)
Exercises	96	(7)
4 Differentiation	103	(44)
4.1 The Derivative and Its Basic Properties	104	(13)
4.2 The Mean Value and Taylor Theorems	117	(8)
4.3 Monotonicity, Convexity, and Concavity	125	(6)
4.4 L'Hôpital's Rule	131	(7)
Exercises	138	(9)
5 Applications of Differentiation	147	(32)
5.1 Absolute Minimum and Maximum	147	(3)
5.2 Local Extrema and Points of Inflection	150	(7)
5.3 Linear and Quadratic Approximations	157	(4)
5.4 The Picard and Newton Methods	161	(12)
Exercises	173	(6)
6 Integration	179	(48)
6.1 The Riemann Integral	179	(10)
6.2 Integrable Functions	189	(11)
6.3 The Fundamental Theorem of Calculus	200	(11)
6.4 Riemann Sums	211	(7)
Exercises	218	(9)
7 Elementary Transcendental Functions	227	(64)
7.1 Logarithmic and Exponential Functions	228	(12)
7.2 Trigonometric Functions	240	(13)
7.3 Sine of the Reciprocal	253	(7)
7.4 Polar Coordinates	260	(9)
7.5 Transcendence	269	(5)
Exercises	274	(10)
Revision Exercises	284	(7)
8 Applications and Approximations of Riemann Integrals	291	(70)
8.1 Area of a Region Between Curves	291	(7)
8.2 Volume of a Solid	298	(13)
8.3 Arc Length of a Curve	311	(7)
8.4 Area of a Surface of Revolution	318	(6)
8.5 Centroids	324	(12)
8.6 Quadrature Rules	336	(16)
Exercises	352	(9)
9 Infinite Series and Improper Integrals	361	(58)
9.1 Convergence of Series	361	(6)
9.2 Convergence Tests for Series	367	(9)
9.3 Power Series	376	(8)
9.4 Convergence of Improper Integrals	384	(8)
9.5 Convergence Tests for Improper Integrals	392	(6)
9.6 Related Integrals	398	(12)
Exercises	410	(9)
References	419	(4)
List of Symbols and Abbreviations	423	(4)
Index	427