All but the most straightforward proofs are worked out in detail before being presented formally in this book. Thus most of the ideas are expressed in two different ways: the first encourages and develops the intuition and the second gives a feeling for what constitutes a proof. In this way, intuition and rigour appear as partners rather than competitors. The informal discussions, the examples and the exercises may assume some familiarity with calculus, but the definitions, theorems and formal proofs are presented in the correct logical order and assume no prior knowledge of calculus. Thus some basic knowledge of calculus is blended into the presentation rather than being completely excluded.
Author(s): Neil A. Watson
Series: Advanced Series on Ocean Engineering
Publisher: World Scientific
Year: 1993
Language: English
Pages: x+179
Preface vii
Chapter 1 THE REAL NUMBERS
1.1. Motivation 1
1.2. The real number system 4
1.3. Upper and lower bounds 6
1.4. Exercises 10
Chapter 2 SEQUENCES AND SERIES
2.1. Sequences 12
2.2. Algebraic operations on limits 15
2.3. Monotone sequences 19
2.4. Infinite series 21
2.5. Basic properties of infinite series 27
2.6. Exercises 29
Chapter 3 CONTINUOUS FUNCTIONS
3.1. Functions, limits, and continuity 32
3.2. Elementary properties of continuous functions 36
3.3. The intermediate value property for continuous functions 38
3.4. Boundedness of continuous functions 39
3.5. Uniform continuity 43
3.6. Increasing functions 47
3.7. Exercises 50
Chapter 4 DIFFERENTIABLE FUNCTIONS
4.1. Differentiation52
4.2. Repeated differentiation58
4.3. Mean value theorems 61
4.4. The intermediate value property for derivatives 66
4.5. Local maxima and minima 68
4.6. Taylor's theorem70
4.7. Indeterminate forms77
4.8. Exercises81
Chapter 5 FURTHER RESULTS ON INFINITE SERIES
5.1. Tests for convergence 85
5.2. Absolute and conditional convergence 92
5.3. Series of complex terms 95
5.4. Power series 97
5.5. Multiplication of series 100
5.6. Exercises 104
Chapter 6 SPECIAL FUNCTIONS
6.1. The exponential function 108
6.2. The logarithm 115
6.3. Powers 117
6.4. Trigonometric functions 119
6.5. Inverse trigonometric functions 124
6.6. Exercises 127
Chapter 7 THE RIEMANN INTEGRAL
7.1. Definition of the integral 130
7.2. Integrability 137
7.3. Properties of integrable functions 142
7.4. Integration and differentiation 151
7.5. Integral forms of the mean value theorems 157
7.6. Integration over unbounded intervals 158
7.7. Integration of unbounded functions 161
7.8. Exercises 162
Chapter 8 THE NUMBER $\pi$
8.1. The geometric significance of $\pi$ 166
8.2. Calculation of $\pi$ 167
8.3. Irrationality of $\pi$ 172
8.4. Exercise 175
Index 176
