Analysis: A Gateway To Understanding Mathematics

by Sean Dineen

Overview

This book shows that it is possible to provide a fully rigorous treatment of calculus for those planning a career in an area that uses mathematics regularly (e.g., statistics, mathematics, economics, finance, engineering, etc.). It reveals to students on the ways to approach and understand mathematics. It covers efficiently and rigorously the differential and integral calculus, and its foundations in mathematical analysis. It also aims at a comprehensive, efficient, and rigorous treatment by introducing all the concepts succinctly. Experience has shown that this approach, which treats understanding on par with technical ability, has long term benefits for students.

Product Details

ISBN-13:	9789814401388
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	07/03/2012
Pages:	320
Product dimensions:	6.10(w) x 9.00(h) x 0.90(d)

Table of Contents

Preface xi

1 Quadratic Equations 1

1.1 Quadratic Equations 1

1.2 Further Consequences 14

1.3 Exercises 19

2 Diagrams 21

2.1 Graphs 21

2.2 Exercises 37

3 Functions 39

3.1 Functions 39

3.2 Generating New Functions 53

3.3 Exercises 57

4 Inverse Functions 59

4.1 Injective and Surjective Functions 60

4.2 Bijective Functions and Inverses 63

4.3 Increasing, Decreasing, Convex and Indicator Functions 69

4.4 Exercises 72

5 Numbers 77

5.1 The Positive Integers 77

5.2 The Integers and Rational Numbers 85

5.3 Arithmetic and Geometry 93

5.4 Exercises 101

6 Sets of Real Numbers 103

6.1 Countable Sets 103

6.2 Uncountable Sets 108

6.3 Sets Which Are Bounded Above 110

6.4 Intervals 114

6.5 Exercises 115

7 Series with Positive Entries 117

7.1 Increasing and Decreasing Sequences 117

7.2 Series with Positive Entries 124

7.3 The Exponential Function 128

7.4 Multiplication of Series 130

7.5 The Log Function 131

7.6 Exercises 135

8 Review and Transition 137

8.1 Review and Preview 137

8.2 Convergent Sequences 139

8.3 Convergent Series 142

8.4 Power Series 148

8.5 Exercises 151

9 Continuous Functions 155

9.1 Continuous Functions 155

9.2 Intermediate Value Theorem 159

9.3 Fundamental Existence Theorem for Maxima and Minima 162

9.4 Exercises 166

10 Differentiable Functions 169

10.1 Differentiable Functions 169

10.2 Basic Functions and Rules for Differentiation 174

10.3 Mean Value Theorem 177

10.4 Exercises 185

11 Applications of Differentiation 189

11.1 Sequences 189

11.2 Limits 192

11.3 Curve Sketching 195

11.4 Applications of the Mean Value Theorem 201

11.5 In Retrospect 211

11.6 Exercises 213

12 The Real Numbers 219

12.1 Upper Bounds 219

12.2 Dyadic Sections and the Bisection Principle 222

12.3 Foundational Developments 229

12.4 Infinitesimals 234

12.5 Exercises 235

13 Integration Theory 237

13.1 The Riemann Integral 237

13.2 Fundamental Theorem of Calculus 245

13.3 Methods of Integration 247

13.4 Exercises 251

14 Applications of Integration Theory 253

14.1 Integral Inequalities 254

14.2 Integrals over Infinite Intervals 255

14.3 Taylor's Theorem 262

14.4 Exercises 265

Solutions 269

Bibliography 291

Index 295