本书旨在指导学生初步掌握数学建模的思想和方法，共分两大部分：离散建模和连续建模。本书对于用到的数学知识力求深入浅出，涉及的应用领域相当广泛，适合作为高等院校相关专业的数学建模教材和参考书，也可作为参加国内外数学建模竞赛的指导用书。

作者简介
Frank R. Giordano 毕业于美国西点军校，曾任西点军校数学系系主任，现为美国海军研究生院教授，多年来一直是美国大学生数学建模竞赛的主要组织者，也是美国大学生数学建模竞赛组委会主任。
William P. Fox 曾任教于美国西点军校，现为美国海军研究生院教授，是美国中学生数学建模竞赛组委会主任。
Steven B. Horton 美国西点军校教授。
Maurice D. Weir 美国海军研究生院荣誉退休教授，曾任该校副教务长。
数学建模是用数学方法解决各种实际问题的桥梁。本书分离散建模和连续建模两部分介绍了整个建模过程的原理，通过本书的学习，学生将有机会在创造性模型和经验模型的构建、模型分析以及模型研究方面进行实践，增强解决问题的能力。
本书特点
论证了离散动力系统、离散优化等技术对现代应用数学发展的促进作用。
在创造性模型和经验模型的构建、模型分析以及模型研究中融入个人项目和小组项目，并且包含大量的例子和习题。
本版新增了关于图论建模的新的一章，从数学建模的角度介绍图论并鼓励学生对图论进行更深入的学习。
随书光盘中包含大学数学应用教学单元（UMAP），过去的建模竞赛试题，充满活力的跨学科应用研究课题，利用电子表格（Excel）、计算机代数系统（Maple、Mathematica、Matlab）以及图形计算器（TI）等技术的广泛的例子，在实验室环境下为学生设计的例子和习题。

To facilitate an early initiation of the modeling experience, the first edition of this text was designed to be taught concurrently or immediately after an introductory business or
engineering calculus course. In the second edition, we added chapters treating discrete
dynamical systems, linear programming and numerical search methods, and an introduction
to probabilistic modeling. Additionally, we expanded our introduction of simulation. In
the third edition we included solution methods for some simple dynamical systems to
reveal their long-term behavior. We also added basic numerical solution methods to the
chapters covering modeling with differential equations. In this edition, we have added a
new chapter to address modeling using graph theory. Graph theory is an area of burgeoning
interest for modeling contemporary scenarios. Our chapter is intended to introduce graph
theory from a modeling perspective and encourage students to pursue the subject in greater
detail. We have also added two new sections to the chapter on modeling with a differential
equation: discussions of separation of variables and linear equations. Many of our readers
had expressed a desire that analytic solutions to first-order differential equations be included
as part of their modeling course. The text has been reorganized into two parts: Part One,
Discrete Modeling (Chapters 1–9), and Part Two, Continuous Modeling (Chapters 10–13).
　This organizational structure allows for teaching an entire modeling course that is based on Part One and does not require the calculus. Part Two then addresses continuous models based on optimization and differential equations that can be presented concurrently with freshman calculus. The text gives students an opportunity to cover all phases of the mathematical modeling process. The new CD-ROM accompanying the text contains software, additional modeling scenarios and projects, and a link to past problems from the Mathematical Contest in Modeling. We thank Sol Garfunkel and the COMAP staff for their support of modeling activities that we refer to under Resource Materials below.
Goals and Orientation The course continues to be a bridge between the study of mathematics and the applications of mathematics to various fields. The course affords the student an early opportunity to see how the pieces of an applied problem fit together. The student investigates meaningful and practical problems chosen from common experiences encompassing many academic disciplines, including the mathematical sciences, operations research, engineering, and the management and life sciences.
　　This text provides an introduction to the entire modeling process. Students will have
opportunities to practice the following facets of modeling and enhance their problem-solving
capabilities:
　　1. Creative and Empirical Model Construction: Given a real-world scenario, the student
learns to identify a problem, make assumptions and collect data, propose a model, test
the assumptions, refine the model as necessary, fit the model to data if appropriate, and
analyze the underlying mathematical structure of the model to appraise the sensitivity
of the conclusions when the assumptions are not precisely met.
　　2. Model Analysis: Given a model, the student learns to work backward to uncover the
implicit underlying assumptions, assess critically how well those assumptions fit the
scenario at hand, and estimate the sensitivity of the conclusions when the assumptions
are not precisely met.
　　3. Model Research: The student investigates a specific area to gain a deeper understanding of some behavior and learns to use what has already been created or discovered.
　　Student Background and Course Content
Because our desire is to initiate the modeling experience as early as possible in the student’s program, the only prerequisite for Chapters 10, 11, and 12 is a basic understanding of singlevariable differential and integral calculus. Although some unfamiliar mathematical ideas are taught as part of the modeling process, the emphasis is on using mathematics that the students already know after completing high school. This is especially true in Part One.
　　The modeling course will then motivate students to study the more advanced courses such
as linear algebra, differential equations, optimization and linear programming, numerical
analysis, probability, and statistics. The power and utility of these subjects are intimated
throughout the text.
　　Further, the scenarios and problems in the text are not designed for the application
of a particular mathematical technique. Instead, they demand thoughtful ingenuity in using
fundamental concepts to find reasonable solutions to “open-ended” problems. Certain
mathematical techniques (such as Monte Carlo simulation, curve fitting, and dimensional
analysis) are presented because often they are not formally covered at the undergraduate
level. Instructors should find great flexibility in adapting the text to meet the particular
needs of students through the problem assignments and student projects.We have used this
material to teach courses to both undergraduate and graduate students—and even as a basis
for faculty seminars.
　　Organization of the Text
　　The organization of the text is best understood with the aid of Figure 1. The first nine chapters
constitute Part One and require only precalculus mathematics as a prerequisite. We begin
with the idea of modeling change using simple finite difference equations. This approach
is quite intuitive to the student and provides us with several concrete models to support our discussion of the modeling process in Chapter 2. There we classify models, analyze the
modeling process, and construct several proportionality models or submodels that are then
revisited in the next two chapters. In Chapter 3 the student is presented with three criteria
for fitting a specific type of curve to a collected data set, with emphasis on the least-squares
criterion. Chapter 4 addresses the problem of capturing the trend of a collected set of
data. In this empirical construction process, we begin with fitting simple one-term models
approximating collected data sets and then progress to more sophisticated interpolating
models, including polynomial smoothing models and cubic splines. Simulation models are
discussed in Chapter 5. An empirical model is fit to some collected data, and then Monte
Carlo simulation is used to duplicate the behavior being investigated. The presentation
motivates the eventual study of probability and statistics.
　　Chapter 6 provides an introduction to probabilistic modeling. The topics of Markov
processes, reliability, and linear regression are introduced, building on scenarios and analysis
presented previously. Chapter 7 addresses the issue of finding the best-fitting model using the
other two criteria presented in Chapter 3. Linear programming is the method used for finding
the “best” model for one of the criteria, and numerical search techniques can be used for the
other. The chapter concludes with an introduction to numerical search methods, including the
dichotomous and Golden Section methods. Part One ends with Chapter 9, which is devoted
to dimensional analysis, a topic of great importance in the physical sciences and engineering.
　　Part Two is dedicated to the study of continuous models. Chapter 10 treats the construction of continuous graphical models and explores the sensitivity of the models constructed to the assumptions underlying them. In Chapters 11 and 12 we model dynamic (time varying) scenarios. These chapters build on the discrete analysis presented in Chapter 1 by now considering situations where time is varying continuously. Chapter 13 is devoted to the study of continuous optimization. Students get the opportunity to solve continuous optimization problems requiring only the application of elementary calculus and are introduced to constrained optimization problems as well.
　　Student Projects
　　Student projects are an essential part of any modeling course. This text includes projects in creative and empirical model construction, model analysis, and model research. Thus we recommend a course consisting of a mixture of projects in all three facets of modeling.
　　These projects are most instructive if they address scenarios that have no unique solution.
　　Some projects should include real data that students are either given or can readily collect.
　　A combination of individual and group projects can also be valuable. Individual projects
are appropriate in those parts of the course in which the instructor wishes to emphasize
the development of individual modeling skills. However, the inclusion of a group project
early in the course gives students the exhilaration of a “brainstorming” session. A variety
of projects is suggested in the text, such as constructing models for various scenarios,
completing UMAP1 modules, or researching a model presented as an example in the text
1UMAP modules are developed and distributed through COMAP, Inc., 57 Bedford Street, Suite 210, Lexington,
　　MA 02173.
or class. It is valuable for each student to receive a mixture of projects requiring either
model construction, model analysis, or model research for variety and confidence building
throughout the course. Students might also choose to develop a model in a scenario of
particular interest, or analyze a model presented in another course. We recommend five
to eight short projects in a typical modeling course. Detailed suggestions on how student
projects can be assigned and used are included in the Instructor’s Manual that accompany
this text.
　　In terms of the number of scenarios covered throughout the course, as well as the
number of homework problems and projects assigned, we have found it better to pursue a
few that are developed carefully and completely. We have provided many more problems
and projects than can reasonably be assigned to allow for a wide selection covering many
different application areas.
　　Resource Materials
　　We have found material provided by the Consortium for Mathematics and Its Application
(COMAP) to be outstanding and particularly well suited to the course we propose.
　　Individual modules for the undergraduate classroom, UMAP Modules, may be used in
a variety of ways. First, they may be used as instructional material to support several
lessons. In this mode a student completes the self-study module by working through its
exercises (the detailed solutions provided with the module can be conveniently removed
before it is issued). Another option is to put together a block of instruction using one or
more UMAP modules suggested in the projects sections of the text. The modules also
provide excellent sources for “model research,” because they cover a wide variety of applications of mathematics in many fields. In this mode, a student is given an appropriate
module to research and is asked to complete and report on the module. Finally, the
modules are excellent resources for scenarios for which students can practice model construction.
　　In this mode the instructor writes a scenario for a student project based on an
application addressed in a particular module and uses the module as background material,
perhaps having the student complete the module at a later date. The CD accompanying
the text contains most of the UMAPs referenced throughout. Information on the availability
of newly developed interdisciplinary projects can be obtained by writing COMAP
at the address given previously, calling COMAP at 1-800-772-6627, or electronically:
order@comap.com.
　　A great source of student-group projects are the Mathematical Contest in Modeling
(MCM) and the Interdisciplinary Contest in Modeling (ICM). These projects can be taken
from the link provided on the CD and tailored by the instructor to meet specific goals
for their class. These are also good resources to prepare teams to compete in the MCM
and ICM contests. The contest is sponsored by COMAP with funding support from the
National Security Agency, the Society of Industrial and Applied Mathematics, the Institute
for Operations Research and the Management Sciences, and the Mathematical Association
of America. Additional information concerning the contest can be obtained by contacting
COMAP, or visiting their website at www.comap.com.
　　The Role of Technology
　　Technology is an integral part of doing mathematical modeling with this textbook. Technology can be used to support the modeling of solutions in all of the chapters. Rather than incorporating lots of varied technologies into the explanations of the models directly in the text, we decided to include the use of various technology on the enclosed CD. There
the student will find templates in Microsoft Excel , Maple , Mathematica , and Texas
Instruments graphing calculators, including the TI-83 and 84 series.
　　We have chosen to illustrate the use of Maple in our discussion of the following topics
that are well supported by Maple commands and programming procedures: difference
equations, proportionality, model fitting (least squares), empirical models, simulation, linear programming, dimensional analysis, modeling with differential equations, modeling with systems of differential equations, and optimization of continuous models. Mapleworksheets for the illustrative examples appearing in the referenced chapters are provided on the CD.
　　Mathematica was chosen to illustrate difference equations, proportionality, model fitting (least squares), empirical models, simulation, linear programming, graph theory, dimensional analysis, modeling with differential equations, modeling with systems of differential equations, and optimization of continuous models. Mathematica worksheets for illustrative examples in the referenced chapters are provided on the CD.
　　Excel is a spreadsheet that can be used to obtain numerical solutions and conveniently
obtain graphs. Consequently, Excel was chosen to illustrate the iteration process and graphical solutions to difference equations. It was also selected to calculate and graph functions in proportionality, model fitting, empirical modeling (additionally used for divided difference tables and the construction and graphing of cubic splines), Monte Carlo simulation, linear programming (Excel’s Solver is illustrated), modeling with differential equations (numerical approximations with both the Euler and the Runge-Kutta methods), modeling with systems of differential equations (numerical solutions), and Optimization of Discrete and Continuous Models (search techniques in single-variable optimization such as the dichotomous and Golden Section searches). Excel worksheets can be found on the website.
The TI calculator is a powerful tool for technology as well. Much of this textbook
can be covered using the TI calculator. We illustrate the use of TI calculators with difference equations, proportionality, modeling fitting, empirical models (Ladder of Powers and other transformations), simulation, and differential equations (Euler’s method to construct numerical solutions).　
　　Acknowledgments
　　It is always a pleasure to acknowledge individuals who have played a role in the development of a book. We are particularly grateful to Brigadier General (retired) Jack M. Pollin and Dr. Carroll Wilde for stimulating our interest in teaching modeling and for support and guidance in our careers. We’re indebted to many colleagues for reading the first edition manuscript and suggesting modifications and problems: Rickey Kolb, John Kenelly, Robert Schmidt, Stan Leja, Bard Mansager, and especially Steve Maddox and Jim McNulty.
　　We are indebted to a number of individuals who authored or co-authored UMAP materials
that support the text: David Cameron, Brindell Horelick, Michael Jaye, Sinan Koont,
Stan Leja, MichaelWells, and CarrollWilde. In addition, we thank Solomon Garfunkel and
the entire COMAP staff for their cooperation on this project, especially Roland Cheyney
for his help with the production of the CD that accompanies the text. We also thank Tom
O’Neil and his students for their contributions to the CD and for helpful suggestions in
support of modeling activities. We would like to thank Dr. Amy H. Erickson for her many
contributions to the CD and website.
　　Thanks to the following reviewers: Stephen Alessandrini, Rutgers-Camden; John
Cannon, University of Central Florida; Donald Cathcart, Salisbury State University;
Catherine Crawford, Elmhurst College; Hajrudin Fejzic, California State University San
Bernadino; Michael Frantz, University of La Verne; Larry Hill, Lafayette College; Dick
Jardine, Keene State University; Theresa Jeevanjee, Fontbonne University; Andy Keck,
Western State College of Colorado; Aprillya Lanz, Clayton State University; Abdelhamid
Meziani, Florida International University; Ho-Kuen Ng, San Jose State University;William
Paulsen, Arkansas State University;Ken Roblee, Troy State University; Todd Smith, Athens
State College; Alexandros Sopasakis, University of Massachusetts; and Ray Toland,
Clarkson University.
　　The production of any mathematics text is a complex process, and we have been
especially fortunate in having a superb and creative production staff at Brooks/Cole. In
particular, we express our thanks to Craig Barth, our editor for the first edition, Gary
Ostedt, the second edition, and Gary Ostedt and Bob Pirtle, our editors for the third edition.
　　We would like to thank everyone from Brooks/Cole who worked with us on this edition,
especially Charlie VanWagner, our Acquisitions Editor, and Stacy Green, our Development
Editor. Thanks also to Sara Planck and Matrix Productions for production service.
Frank R. Giordano
　　William P. Fox
　　Steven B. Horton
　　Maurice D. Weir

数学

数学建模是用数学方法解决各种实际问题的桥梁。本书分离散建模和连续建模两部分介绍了整个建模过程的原理，通过本书的学习，学生将有机会在创造性模型和经验模型的构建、模型分析以及模型研究方面进行实践，增强解决问题的能力。
? 论证了离散动力系统、离散优化等技术对现代应用数学的发展的促进作用。
? 在创造性模型和经验模型的构建、模型分析以及模型研究中融入个人项目和小组项目，并且包含大量的例子和习题。
? 本版新增了关于图论建模的新的一章，从数学建模的角度介绍图论并鼓励学生对图论进行更深入的学习。
? 随书光盘中包含大学数学应用教学单元，过去的建模竞赛试题，充满活力的跨学科应用研究课题，利用电子表格（Excel）、计算机代数系统（Maple、Mathematica、Matlab）以及图形计算器（TI）等技术的广泛的例子，在实验室环境下为学生设计的例子和习题。

(美）Frank R.Giordano　 William P.Fox　 Steven B. Horton　 Maurice D.Weir著：Frank R. Giordano 毕业于美国西点军校，曾任西点军校数学系系主任，现为美国海军研究生院教授，多年来一直是美国大学生数学建模竞赛的主要组织者，也是美国大学生数学建模竞赛组委会主任。 William P. Fox 曾任教于美国西点军校，现为美国海军研究生院教授，是美国中学生数学建模竞赛组委会主任。 Steven B. Horton 美国西点军校教授。 Maurice D. Weir 美国海军研究生院荣誉退休教授，曾任该校副教务长。

1 Modeling Change 1
Introduction 1
Testing for Proportionality 2
1.1 Modeling Change with Difference Equations 4
A Savings Certificate 5
Mortgaging a Home 6
1.2 Approximating Change with Difference Equations 10
Growth of a Yeast Culture Revisited 10
Spread of a Contagious Disease 13
Decay of Digoxin in the Bloodstream 14
Heating of a Cooled Object 15
1.3 Solutions to Dynamical Systems 18
A Savings Certificate Revisited 19
Sewage Treatment 21
Prescription for Digoxin 25
An Investment Annuity 26
A Checking Account 28
An Investment Annuity Revisited 30
1.4 Systems of Difference Equations 37
A Car Rental Company 37
The Battle of Trafalgar 40
Competitive Hunter Model—Spotted Owls and Hawks 43
Voting Tendencies of the Political Parties 47
2 The Modeling Process, Proportionality,
and Geometric Similarity 53
Introduction 53
2.1 Mathematical Models 55
Vehicular Stopping Distance 60
2.2 Modeling Using Proportionality 65
Kepler's Third Law 67
2.3 Modeling Using Geometric Similarity 75
Raindrops from a Motionless Cloud 77
Modeling a Bass Fishing Derby 78
Modeling the Size of the "Terror Bird” 83
2.4 Automobile Gasoline Mileage 89
2.5 Body Weight and Height, Strength and Agility 92
3 Introduction 98
3.1 Fitting Models to Data Graphically 101
3.2 Analytic Methods of Model Fitting 108
3.3 Applying the Least-Squares Criterion 114
3.4 Choosing a Best Model 121
Vehicular Stopping Distance 123
4 Experimental Modeling 128
Introduction 128
4.1 Harvesting in the Chesapeake Bay and Other
One-Term Models 129
Harvesting Bluefish 131
Harvesting Blue Crabs 133
4.2 High-Order Polynomial Models 138
Elapsed Time of a Tape Recorder 140
Model Fitting 98
4.3 Smoothing: Low-Order Polynomial Models 146
Elapsed Time of a Tape Recorder Revisited 147
Elapsed Time of a Tape Recorder Revisited Again 151
Vehicular Stopping Distance 153
Growth of a Yeast Culture 155
4.4 Cubic Spline Models 158
Vehicular Stopping Distance Revisited 166
5 Simulation Modeling 173
Introduction 173
5.1 Simulating Deterministic Behavior: Area Under a Curve 175
5.2 Generating Random Numbers 179
5.3 Simulating Probabilistic Behavior 183
5.4 Inventory Model: Gasoline and Consumer Demand 190
5.5 Queuing Models 200
A Harbor System 200
Morning Rush Hour 207
6 Discrete Probabilistic Modeling 211
Introduction 211
6.1 Probabilistic Modeling with Discrete Systems 211
Rental Car Company Revisited 211
Voting Tendencies 213
6.2 Modeling Component and System Reliability 217
Series Systems 217
Parallel Systems 218
Series and Parallel Combinations 218
6.3 Linear Regression 220
Ponderosa Pines 222
The Bass Fishing Derby Revisited 226
7 Optimization of Discrete Models 229
Introduction 229
7.1 An Overview of Optimization Modeling 230
Determining a Production Schedule 230
Space Shuttle Cargo 233
Approximation by a Piecewise Linear Function 234
7.2 Linear Programming I: Geometric Solutions 242
The Carpenter's Problem 243
A Data-Fitting Problem 244
7.3 Linear Programming II: Algebraic Solutions 252
Solving the Carpenter's Problem Algebraically 253
7.4 Linear Programming III: The Simplex Method 256
The Carpenter's Problem Revisited 260
Using the Tableau Format 263
7.5 Linear Programming IV: Sensitivity Analysis 266
8 Modeling Using Graph Theory 273
Introduction 273
8.1 Graphs as Models 274
The Seven Bridges of K nigsberg 274
Graph Coloring 275
8.2 Describing Graphs 280
9 Dimensional Analysis and Similitude 283
Introduction 283
Introductory Example: A Simple Pendulum 284
9.1 Dimensions as Products 286
A Simple Pendulum Revisited 288
Wind Force on a Van 291
9.2 The Process of Dimensional Analysis 294
Terminal Velocity of a Raindrop 299
Automobile Gas Mileage Revisited 301
9.3 A Damped Pendulum 303
9.4 Examples Illustrating Dimensional Analysis 309
Explosion Analysis 309
How Long Should You Roast a Turkey 313
9.5 Similitude 319
Drag Force on a Submarine 320
10 Graphs of Functions as Models 325
10.1 An Arms Race 325
Civil Defense 333
Mobile Launching Pads 334
Multiple Warheads 335
MIRVs Revisited: Counting Warheads 336
10.2 Modeling an Arms Race in Stages 338
10.3 Managing Nonrenewable Resources: The Energy Crisis 342
10.4 Effects of Taxation on the Energy Crisis 346
10.5 A Gasoline Shortage and Taxation 350
11 Modeling with a Differential Equation 355
Introduction 355
11.1 Population Growth 358
11.2 Prescribing Drug Dosage 368
11.3 Braking Distance Revisited 376
11.4 Graphical Solutions of Autonomous Differential Equations 380
Drawing a Phase Line and Sketching Solution Curves 382
Cooling Soup 384
Logistic Growth Revisited 385
12 Modeling with Systems
of Differential Equations 389
Introduction 389
12.1 Graphical Solutions of Autonomous Systems of First-Order
Differential Equations 390
A Linear Autonomous System 391
A Nonlinear Autonomous System 392
12.2 A Competitive Hunter Model 395
12.3 A Predator--Prey Model 404
12.4 Two Military Examples 413
Lanchester Combat Models 413
Economic Aspects of an Arms Race 419
13 Optimization of Continuous Models 425
Introduction 425
13.1 An Inventory Problem: Minimizing the Cost
of Delivery and Storage 426
13.2 A Manufacturing Problem: Maximizing Profit
in Producing Competing Products 434
13.3 Constrained Continuous Optimization 440
An Oil Transfer Company 441
A Space Shuttle Water Container 442
13.4 Managing Renewable Resources: The Fishing Industry 446