**AMS/MAA Textbooks**

Volume: 2;
1997;
345 pp;
Softcover

MSC: Primary 00;
Secondary 58

**Print ISBN: 978-0-88385-707-6
Product Code: TEXT/2**

List Price: $51.00

AMS Member Price: $38.25

MAA Member Price: $38.25

**Electronic ISBN: 978-1-61444-601-9
Product Code: TEXT/2.E**

List Price: $51.00

AMS Member Price: $38.25

MAA Member Price: $38.25

# Elementary Mathematical Models: Order Aplenty and a Glimpse of Chaos

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*Dan Kalman*

MAA Press: An Imprint of the American Mathematical Society

The language of mathematics has proven over centuries of application to be an indispensable tool for the expression and analysis of real problems. With numerical, graphical, and theoretical methods, this book examines the relevance of mathematical models to phenomena ranging from population growth and economics to medicine and the physical sciences. In a book written for the intelligent and literate non-mathematician, Kalman aims at an understanding of the power and utility of quantitative methods rather than at technical mastery of mathematical operations. He shows first that mathematical models can serve a critical function in understanding the world, and he concludes with a discussion of the problems encountered by traditional algebraic assumptions in chaos theory. Though models can often approximate future events based on existing data and quantitative relationships, Kalman shows that the appearance of regularity and order can often be misleading. By beginning with quantitative models and ending with an introduction to chaos, Kalman offers a broad treatment of both the power and limitations of quantitatively-based predictions.

#### Reviews & Endorsements

Kalman uses basic growth models...not only to convey the power of mathematics in solving real-world problems, but also to motivate the study of the elementary functions usually encountered in college algebra courses. There is a natural evolution from a simple hypotheses to difference equations, to their solutions, to the study of the elementary functions associated with the solutions. There is an emphasis on the "why" of algebra and on manipulation associated with applications rather than for its own sake.Numerical, graphical, and symbolic approaches are used throughout, and the numerous exercises include reading comprehension exercises and group selected exercises...Aimed at students at the college algebra or liberals arts mathematics level, the slow careful development should be clear even to those with a weak algebraic background. Highly recommended.

-- Choice

An innovative alternative to introductory college mathematics intended for any student not headed for calculus. Uses discrete and continuous models of growth to introduce increasingly sophisticated algebraic patterns...An effective blend of narrative, motivation, calculation, and graphical representation that introduces algebraic thinking with a minimum of algebraic formalisms.

-- The American Mathematical Monthly

The book is well balanced and succeeds in introducing the use of discrete models to students who might view a mathematics class with a weary eye...the author does a superb job of addressing a difficult audience. This is especially true of the problem sets. An excellent mix of reading, simple/short answer, and word problems of varying difficulties are given. Furthermore, complete answers to some of the problems are given in the same chapter, rather than in an appendix.

-- Kelly Black, University of New Hampshire

This book can be described as the protocol of the ultimate (and apparently successful) teaching experiment, namely to lead students with hardly any mathematical background at all, to a respectable level in the fundamentals of mathematics in such a way, that they will always have positive thoughts about it. ...The expert reader may use this book as a rich source of growth problems.

-- Springer-Verlag, Zentallblatt für Mathematik

I found the book a refreshing alternative to college algebra textbooks and would recommend it to instructors who are seeking changes.

-- The Mathematics Teacher

# Table of Contents

## Elementary Mathematical Models: Order Aplenty and a Glimpse of Chaos

- cover cover11
- copyright page ii3
- title page iii4
- Preface v6
- Contents xiii14
- 1 Overview 118
- 2 Sequences and Difference Equations 1128
- 3 Arithmetic Growth 3754
- 4 Linear Graphs, Functions, and Equations 5572
- 5 Quadratic Growth Models 7996
- A New School Population Example 7996
- Quadratic Growth Models 8097
- Quadratic Growth Difference Equations 8299
- Applications of Quadratic Growth 91108
- Properties of Quadratic Growth Models 101118
- Whole Number Sums and sum Notation 104121
- Summary 107124
- Exercises 108125
- Solutions to Selected Exercises 111128

- 6 Quadratic Graphs, Functions, andEquations 117134
- 7 Polynomial and Rational Functions 149166
- 8 Fitting a Line to Data 165182
- 9 Geometric Growth 181198
- 10 Exponential Functions 213230
- 11 More On Logarithms 233250
- 12 Geometric Sums and Mixed Models 255272
- 13 Logistic Growth 287304
- 14 Chaos in Logistic Models 319336
- Index 341358