### 基本信息

- 原书名：A First Course Mathematical Modeling(Third Edition)
- 原出版社： Thomson

### 内容简介

数学书籍

数学建模这门课程在数学及其在各个领域的应用之间架起一座桥梁。本书介绍了速个建模过程的原理，通过本书的学习，学生将有机会在以下建模活动中亲身实践，增强解决问题的能力；设计创意模型和经验模型、模型分析以及模型研究。

本书特点

论证了离散动态系统、离散优化和仿真等技术如何促进现代应用数学的发展

强调通过模型设计提高学生的创造性，展现模型构建的艺术特性，包括经验建模和仿真建模的思想

将数学建模方法与多样化建模和置信度建立等更具创造性的方面结合起来

在设计创意模型和经验模型、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随书光盘包含软件、附加的建模场景和项目以及过去数学建模竞赛的题目

### 目录

Introduction 1

Example 1: Testing for Proportionality 2

1.1 Modeling Change with Difference Equations 4

Example I: A Savings Certificate 5

Example 2: Mortgaging a Home 6

1.2 Approximating Change with Difference Equations

Example 1: Growth of a Yeast Culture 9

Example 2: Growth of a Yeast Culture Revisited 10

Example 3: Spread of a Contagious Disease 12

Example 4: Decay of Digoxin in the Bloodstream 13

Example 5: Heating of a Cooled Object 14

1.3 Solutions to Dynamical Systems 18

Example 1: A Savings Certificate Revisited 18

Example 2: Sewage Treatment 21

Example 3: Prescription for Digoxin 25

### 前言

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 preparing the CD and 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. The student will have occasions 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 as- sumptions 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 9, 10, and 11 is a basic understanding of single-variable differential and integral calculus. Although some unfamiliar mathematical ideas are taught as part of the modeling process, the em-

phasis is on using mathematics already known by the students after completing high school. This emphasis is especially tree 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 eight 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 which are then revisited in the next two chapters. In Chapter 3 the student is presented with three criteria for fitting a specific curve-type to a collected data set, with emphasis on the least-squares cfiteflon. 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 oneterm models approximating collected data sets and 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 probabihsfic 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 findingthe 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 8, which is devoted to dimensional analysis, a topic of great importance in the physical sciences and engineering.a combination of graphing calculators and computers to be advantageous throughout the course. The use of a spreadsheet is beneficial in Chapters 1, 5, and 7, and the capability for graphical displays of data is enormously useful, even essential,whenever data is provided. Students will find computers useful, too, in transforming data, least-squares curve fitting, divided difference tables and cubic splines,programming simulation models, linear programming and numerical search methods, and numerical solutions to differential equations. The CD accompanying this text provides some basic technology tools that students and instructors can use as a foundation for modeling with technology. Several FORTRAN executable programs are provided to execute the methodologies presented in Chapter 4. Also included is a tutorial on the computer algebra system MAPLE and its use for this text.

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