E-Book Overview
Students of analysis are often beset with frustration. They ask "Why did you bound that quantity with that other quantity?" The typical answer, "Because it works out in the proof!" is certainly true, yet wholly unsatisfactory for the student.This book begins with models, real-world problems, that originally motivated the development of analysis. The student easily grasps how, and more importantly why, quantities are bounded. The days of staring at an algebraic form for hours are gone! (Well, mostly.)Instead of the normal calculus-style, simple-to-complex development of the material, Estep introduces concepts in the natural order of the real-world problems. For example, Lipschitz continuity is introduced early to solve obvious extensions to previous problems. The mathematical idea of continuity is progressively extended and provides much of the motivation for the second half of the book.By orienting on the problems solved by analysis, Estep avoids many of the bewildering difficulties encountered in traditional introductory treatments. This is the best introductory analysis book I've seen. I'm very surprised that it hasn't received more attention.
E-Book Content
Practical Analysis in One Variable Donald Estep Springer Contents Preface vii Introduction I 1 Numbers and Functions, Sequences and Limits 5 1 Mathematical Modeling 1.1 The Dinner Soup Model . . . . . . . . . . . . . . . . . . . . 1.2 The Muddy Yard Model . . . . . . . . . . . . . . . . . . . . 1.3 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . 7 8 10 11 2 Natural Numbers Just Aren’t Enough 2.1 The Natural Numbers . . . . . . . . . . . . . . . . 2.2 Infinity or Is There a Largest Natural Number? . . 2.3 A Controversy About the Set of Natural Numbers 2.4 Subtraction and the Integers . . . . . . . . . . . . 2.5 Division and the Rational Numbers . . . . . . . . . 2.6 Distance and the Absolute Value . . . . . . . . . . 2.7 Computer Representation of Integers . . . . . . . . . . . . . . . 15 15 18 19 21 23 24 25 3 Infinity and Mathematical Induction 3.1 The Need for Induction . . . . . . . . . . . . . . . . . . . . 3.2 The Principle of Mathematical Induction . . . . . . . . . . 29 29 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Contents 3.3 3.4 Using Induction . . . . . . . . . . . . . . . . . . . . . . . . . A Model of an Insect Population . . . . . . . . . . . . . . . 32 33 4 Rational Numbers 4.1 Operating with Rational Numbers . . . . 4.2 Decimal Expansions of Rational Numbers 4.3 The Set of Rational Numbers . . . . . . . 4.4 The Verhulst Model of Populations . . . . 4.5 A Model of Chemical Equilibrium . . . . 4.6 The Rational Number Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 38 40 45 45 46 47 5 Functions 5.1 Functions . . . . . . . . . . . . 5.2 Functions and Sets . . . . . . . 5.3 Graphing Functions of Integers 5.4 Graphing Functions of Rational . . . . . . . . . . . . . . . . . . Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 51 53 55 58 6 Polynomials 6.1 Polynomials . . . . . . . . . . . 6.2 The Σ Notation for Sums . . . 6.3 Arithmetic with Polynomials . 6.4 Equality of Polynomials . . . . 6.5 Graphs of Polynomials . . . . . 6.6 Piecewise Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . .