A Course in Calculus and Real Analysis

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

• Publisher: Springer
• Year: 2006
• Tongue: English
• Leaves: 442
• Series: Undergraduate Texts in Mathematics
• Edition: 1
• 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......Page 1
Half Title......Page 2
Undergraduate Texts in Mathematics (see end of book)......Page 3
Title......Page 4
Copyright......Page 5
Preface......Page 6
Contents......Page 10
1 Numbers and Functions 1......Page 12
1.1 Properties of Real Numbers 2......Page 13
Completeness Property......Page 16
1.2 Inequalities 10......Page 21
1.3 Functions and Their Geometric Properties 13......Page 24
Definitions and Terminology......Page 25
Basic Examples of Functions......Page 28
Bounded Functions......Page 32
Local Extrema and Points of Inflection......Page 36
Intermediate Value Property......Page 39
Notes and Comments......Page 41
Part A......Page 42
Part B......Page 47
2.1 Convergence of Sequences 43......Page 54
2.2 Subsequences and Cauchy Sequences 55......Page 66
Notes and Comments......Page 70
Part A......Page 71
Part B......Page 73
3.1 Continuity of Functions 67......Page 78
Continuity and Boundedness......Page 83
Continuity and Monotonicity......Page 85
Continuity and Intermediate Value Property......Page 88
Uniform Continuity......Page 90
3.3 Limits of Functions of a Real Variable81......Page 92
Notes and Comments......Page 106
Part A......Page 107
Part B......Page 111
4 Differentiation 103......Page 114
4.1 The Derivative and Its Basic Properties 104......Page 115
2. Implicitly Defined Curves......Page 125
4.2 The Mean Value and Taylor Theorems 117......Page 128
4.3 Monotonicity, Convexity, and Concavity 125......Page 136
4.4 L’Hôpital’s Rule 131......Page 142
Part A......Page 149
Part B......Page 154
5.1 Absolute Minimum and Maximum147......Page 158
5.2 Local Extrema and Points of Inflection 150......Page 161
Points of Inflection......Page 165
5.3 Linear and Quadratic Approximations 157......Page 168
5.4 The Picard and Newton Methods 161......Page 172
Finding a Solution of an Equation......Page 177
Part A......Page 184
Part B......Page 187
6.1 The Riemann Integral 179......Page 190
6.2 Integrable Functions 189......Page 200
Algebraic and Order Properties......Page 205
6.3 The Fundamental Theorem of Calculus 200......Page 211
6.4 Riemann Sums 211......Page 222
Notes and Comments......Page 228
Part A......Page 229
Part B......Page 234
7 Elementary Transcendental Functions 227......Page 238
7.1 Logarithmic and Exponential Functions 228......Page 239
Real Powers of Positive Numbers......Page 244
7.2 Trigonometric Functions240......Page 251
Sine and Cosine Functions......Page 256
7.3 Sine of the Reciprocal 253......Page 264
7.4 Polar Coordinates 260......Page 271
Notion of an Angle......Page 275
7.5 Transcendence 269......Page 280
Notes and Comments......Page 284
Part A......Page 285
Part B......Page 292
Revision Exercises 284......Page 295
8.1 Area of a Region Between Curves291......Page 301
Curves Given by Polar Equations......Page 306
Slicing by Planes Perpendicular to a Fixed Line......Page 308
Slivering by Coaxial Right Circular Cylinders......Page 312
Solids of Revolution......Page 316
8.3 Arc Length of a Curve 311......Page 321
8.4 Area of a Surface of Revolution 318......Page 328
8.5 Centroids 324......Page 334
Curves and Surfaces......Page 336
Planar Regions and Solid Bodies......Page 338
Theorems of Pappus......Page 343
8.6 Quadrature Rules 336......Page 346
Notes and Comments......Page 360
Part A......Page 362
Part B......Page 369
9.1 Convergence of Series 361......Page 371
Tests for Absolute Convergence......Page 377
Tests for Conditional Convergence......Page 383
9.3 Power Series 376......Page 386
9.4 Convergence of Improper Integrals 384......Page 394
Integrals of Derivatives and of Nonnegative Functions......Page 396
9.5 Convergence Tests for Improper Integrals 392......Page 402
Tests for Absolute Convergence of Improper Integrals......Page 403
Tests for Conditional Convergence of Improper Integrals......Page 406
9.6 Related Integrals 398......Page 408
Improper Integrals of the Second Kind......Page 409
Notes and Comments......Page 418
Part A......Page 420
Part B......Page 426
References 419......Page 429
List of Symbols and Abbreviations 423......Page 432
Index 427......Page 436
Undergraduate Texts in Mathematics (continued from p.ii)......Page 442