### 基本信息

- 原书名：First Course in Mathematical Modeling
- 原出版社： Brooks/Cole

- 作者：
**(美)Frank R. Giordano****William P. Fox****Steven B. Horton****Maurice D. Weir** - 丛书名：
**经典原版书库** - 出版社：机械工业出版社
- ISBN：
**9787111282495** - 上架时间：2009-10-20
- 出版日期：2009 年10月
- 开本：16开
- 页码：454
- 版次：4-1
- 所属分类：数学 > 数学实验与数学建模 > 数学建模

教材

### 内容简介

数学书籍

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。...

### 作译者

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。...

### 目录

Introduction

1.1 Modeling Change with Difference Equations

1.2 Approximating Change with DifferenceEquations

1.3 Solutions to Dynamical Systems

1.4 Systems of Difference Equations

2 The Modeling Process, Proportionality,and Geometric Similarity

2.1 Mathematical Models

2.2 Modeling Using Proportionality

2.3 Modeling Using Geometric Similarity

2.4 Automobile Gasoline Mileage

2.5 Body Weight and Height, Strength and Agility

3 Model Fitting

3.1 Fitting Models to Data Graphically

3.2 Analytic Methods of Model Fitting

3.3 Applying the Least-Squares Criterion

3.4 Choosing a Best Model

4 Experimental Modeling

4.1 Harvesting in the Chesapeake Bay and Other One-Term Models

4.2 High-Order Polynomial Models

### 前言

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 single-variable 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 us-ing 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 modeels, 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-fitfing 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. Pan 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 construc-tion 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 vary-ing) 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 opti-mization problems requiring only the application of elementary calculus and are introduced to constrained optimization problems as well. ..

Student Projects

Student projects are an essential pan 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 pans 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 UMAP modules, or researching a model presented as an example in the text 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