Finite Mathematics E-Book

Carla C. Morris

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 682

Veröffentlichungsjahr: 2015

Beschreibung

Features step-by-step examples based on actual data and connects fundamental mathematical modeling skills and decision making concepts to everyday applicability

Featuring key linear programming, matrix, and probability concepts, Finite Mathematics: Models and Applications emphasizes cross-disciplinary applications that relate mathematics to everyday life. The book provides a unique combination of practical mathematical applications to illustrate the wide use of mathematics in fields ranging from business, economics, finance, management, operations research, and the life and social sciences.

In order to emphasize the main concepts of each chapter, Finite Mathematics: Models and Applications features plentiful pedagogical elements throughout such as special exercises, end notes, hints, select solutions, biographies of key mathematicians, boxed key principles, a glossary of important terms and topics, and an overview of use of technology. The book encourages the modeling of linear programs and their solutions and uses common computer software programs such as LINDO. In addition to extensive chapters on probability and statistics, principles and applications of matrices are included as well as topics for enrichment such as the Monte Carlo method, game theory, kinship matrices, and dynamic programming.

Supplemented with online instructional support materials, the book features coverage including:

• Algebra Skills
• Mathematics of Finance
• Matrix Algebra
• Geometric Solutions
• Simplex Methods
• Application Models
• Set and Probability Relationships
• Random Variables and Probability Distributions
• Markov Chains
• Mathematical Statistics
• Enrichment in Finite Mathematics

An ideal textbook, Finite Mathematics: Models and Applications is intended for students in fields from entrepreneurial and economic to environmental and social science, including many in the arts and humanities.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Preface

About the Authors

Chapter Content

Algebra Skills

Basic Finance

Matrices

Linear Programming

Probability and Statistics

Enrichment

Supplements

Suggestions

Acknowledgements

About the Companion Website

Chapter 1: Linear Equations and Mathematical Concepts

1.1 Solving Linear Equations

1.2 Equations of Lines and Their Graphs

1.3 Solving Systems of Linear Equations

1.4 The Numbers π and

e

1.5 Exponential and Logarithmic Functions

1.6 Variation

1.7 Unit Conversions and Dimensional Analysis

Historical Notes and Comments

Chapter 2: Mathematics of Finance

2.1 Simple and Compound Interest

2.2 Ordinary Annuity

2.3 Amortization

2.4 Arithmetic and Geometric Sequences

Historical Notes

Chapter 3: Matrix Algebra

3.1 Matrices

3.2 Matrix Notation, Arithmetic, and Augmented Matrices

3.3 Gauss–Jordan Elimination

3.4 Matrix Inversion and Input–Output Analysis

Historical Notes

Chapter 4: Linear Programming – Geometric Solutions

Introduction

4.1 Graphing Linear Inequalities

4.2 Graphing Systems of Linear Inequalities

4.3 Geometric Solutions to Linear Programs

Historical Notes

Chapter 5: Linear Programming – Simplex Method

5.1 The Standard Maximization Problem (SMP)

5.2 Tableaus and Pivot Operations

5.3 Optimal Solutions and The Simplex Method

5.4 Dual Programs

5.5 Non-SMP Linear Programs

Historical Notes

Chapter 6: Linear Programming – Application Models

Product Mix and Feed Mix Problems

Fluid Blending Problem

Assignment Problems

Transportation Problem

The Knapsack Problem

The Trim Problem

A Caterer Problem

Another Planning Horizon

Historical Notes

Chapter 7: Set and Probability Relationships

7.1 Sets

7.2 Venn Diagrams

7.3 Tree Diagrams

7.4 Combinatorics

7.5 Mathematical Probability

7.6 Bayes' Rule and Decision Trees

Historical Notes

Further Reading

Chapter 8: Random Variables and Probability Distributions

8.1 Random Variables

8.2 Bernoulli Trials and the Binomial Distribution

8.3 The Hypergeometric Distribution

8.4 The Poisson Distribution

Historical Notes

Chapter 9: Markov Chains

9.1 Transition Matrices and Diagrams

9.2 Transitions

9.3 Regular Markov Chains

9.4 Absorbing Markov Chains

Historical Notes

Chapter 10: Mathematical Statistics

10.1 Graphical Descriptions of Data

10.2 Measures of Central Tendency and Dispersion

10.3 The Uniform Distribution

10.4 The Normal Distribution

10.5 Normal Distribution Applications

10.6 Developing and Conducting a Survey

Historical Notes

Chapter 11: Enrichment in Finite Mathematics

11.1 Game Theory

Historical Notes

11.2 Applications in Finance and Economics

11.3 Applications in Social and Life Sciences

11.4 Monte Carlo Method

Historical Notes

11.5 Dynamic Programming

Historical Notes

Answers to Odd Numbered Exercises

Using Technology

Matrices

Matrix Operations

Linear Programming

Using Lindo

Using Excel for Factorial, Combinations, and Permutations

To Create a Frequency Histogram

Using Excel to Determine Summary Statistics

Glossary

Index

End User License Agreement

Pages

ix

x

xi

xiii

xiv

xv

xvi

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

Guide

Cover

Table of Contents

Preface

Begin Reading

Finite Mathematics

Models and Applications

Carla C. Morris

University of Delaware

Robert M. Stark

University of Delaware

Copyright © 2016 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, [phone], fax [phone], or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, [phone], fax [phone], or online at http://www.wiley.com/go/permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at [phone], outside the United States at [phone] or fax [phone].

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

Morris, Carla C.

Finite Mathematics : Models and Applications / Carla C. Morris and Robert M. Stark.

pages cm

Includes bibliographical references and index.

ISBN 978-1-119-01550-5 (cloth)

1. Mathematics–Textbooks. I. Stark, Robert M., 1930- II. Title.

QA39.3.M68 2015

511′.1–dc23

2014042187

Preface

Mathematical development has recently grown at an exponential rate. The mathematics of computers, statistics, matrices, linear programming, and game theory, to name a few, has changed and expanded the landscape of mathematical influence. Many view films such as A Beautiful Mind, and Pi, plays such as Proof, and TV shows such as Numb3rs to witness a wider recognition of mathematics and enhance its awareness in daily lives.

Prior to World War II, basic college and university mathematics consisted mainly of college algebra and calculus. The enormous success of those Newtonian era mathematical discoveries in the development of physical sciences and engineering continues today.

War often brings scientific and technological advances, and World War II was no exception. Developments in the immediate post-World War II years and remarkable advances in automated computation led to new mathematics and to new uses for older mathematics.

Among the many post-World War II developments: three stand out that have become a basis for finite mathematics courses:

Linear Programming

Linear programming, the mathematics for the optimal allocation of resources, has found wide usage in the cutting and planting of forests, automobile production lines, oil refining, hospital menus, and much more. A Google search actually returns millions of references.

Matrix Algebra

The algebra of this mathematical subject is now vital to the operation of computers that store, move, manipulate, and analyze huge arrays of data and variables. It has wide usage in engineering, social, business, and health contexts. Leontief's Nobel Prize winning input – output economic models of firms, governments, and nations are well known.

Chance and Probability

With origins predating the 17th century, mathematical probability matured in the 20th century and with related topics has found applications in virtually every branch of science, technology, and commerce. Examples include the proliferation of medical tests with false negatives and positives, risk assessments, artificial intelligence usages of Bayes' Rule, tracking demographics and behavior with Markov chains, and the development of mathematical statistics.

Recognition of the fuller import of these three primary topics led to their aggregation into a new course, “Finite Mathematics”, in the 1960's. The course and its topics have these distinctions:

Topics are mostly independent of one another.

Topics often differ from pre World War II curricula.

Topics deal with discrete or finite events from which the course and text derive their title. This contrasts with Calculus, which is based upon continuous variation.

Finite mathematics is more a course title than a branch of mathematics marked by an intrinsic coherence. In a sense, it is a hybrid branch of mathematics.

While calculus remains fundamental to physical science and technology, its impact upon the post-World War II social, behavioral, life, health, business, management, and economic sciences has been markedly less. However, it is finite mathematics topics that have become more important to post-World War II business and social sciences applications. One expects finite mathematics topics to find a niche in college curricula.

While a finite mathematics course is often required at many colleges and universities, students in our classes range from freshmen to seniors. Although business, economics, finance, and management students are often the primary enrollment, other students often enroll and can easily benefit as the text and exercises span many disciplines in the life, natural, and social sciences.

The course typically features matrix algebra, linear programming, probability, and related topics. For a one-semester course, this text has ample material for teachers to enrich or offer a two-semester course, perhaps for honors students.

Many institutions have shown an interest in developing General Education guidelines to encourage student development in oral, written communication, and quantitative reasoning skills. Several of the exercises in this text were included with the idea that they would be well suited for instructors looking to aid students in developing skills that suit General Education guidelines. The exercises in this text may be completed with or without the use of technological aids. That decision is for the teacher.

About the Authors

Carla C. Morris has taught courses ranging from college algebra to calculus and statistics since 1987 at the Dover Campus of the University of Delaware. Her B.S. (Mathematics) and M.S. (Operations Research & Statistics) are from Rensselaer Polytechnic Institute, and her Ph.D. (Operations Research) is from the University of Delaware.

Robert M. Stark is Professor Emeritus of Mathematical Sciences at the University of Delaware. His undergraduate and doctoral degrees were awarded by, respectively, Johns Hopkins University and the University of Delaware in physics, mathematics, and operations research. Among his publications is the 2004 Dover Edition of Mathematical Foundations for Design with R.L. Nicholls.

Chapter Content

Algebra Skills

Chapter 1, Linear Equations and Mathematical Concepts is provided for students not fully prepared for a finite mathematics course. The chapter reviews algebra skills and emphasizes solving linear equations and basic graphing skills. In addition, exponential and logarithmic functions, π and e, variation and dimensional analysis are briefly considered.

Basic Finance

Chapter 2, Mathematics of Finance addresses the importance of complex finance considerations of modern life. Here the basics underlying consumer financial products and more advanced topics are included. This chapter, requiring only elementary mathematics, appears early in the text for the benefit of some curricula and can be omitted.

Matrices

Chapter 3, Matrix Algebra is one of the pillars of finite mathematics courses. Matrices have gained a vital role in the operations of large computer databases, manipulating large systems of equations, as in linear programs, and in the increasing quantification of economics, finance, and social sciences. The text covers the encoding of data into matrices, their characteristics, and appropriate algebraic manipulations as addition, multiplication, and inversion. The application to solving systems of equations using Gauss-Jordan elimination schemes is an entire section.

Still, teachers find higher quality examples of actual applications of matrices to be elusive. Examples from cryptology (a major government and industrial security usage) and Input-Output analysis (a fundamental application for economics) are included. A novel section in Chapter 11 has examples of sociological usage in studies of kinship structures through generations and of occupational mobility among populations.

Linear Programming

Linear optimization, or linear programming as it is better known, is another pillar of finite mathematics courses. It is the subject of the second portion of the text in Chapters 4, 5, and 6 and in game theory and finance in Chapter 11. This text mirrors others and has added features and enrichment. In Chapter 4, for Geometric Solutions to linear optimization problems, graphing and solving simple systems of linear equations, some of the introductory skills of Chapter 1 are utilized. However, graphical solutions are limited to problems of few variables.

The Simplex Method appears in Chapter 5. Linear programming is likely the most widely used mathematics to allocate scarce resources in industry and government. Here the matrix algebra skills of Chapter 3 aid in solutions. However, the use of LINDO™ the widely available popular software, enables students to solve larger problems if an instructor wishes to take a more technological approach to optimization. The basic Standard Maximization Problem (SMP) with “less than” constraints is in Sections 5.1 – 5.3. A flow chart aids students to master the simplex iterations.

Section 5.4 on duality and Section 5.5 on non – SMP programs have an extended treatment of linear programming. Two solution methods are given for minimization objectives and excess variable situations. One uses artificial variables and the other algebraic manipulation. A unique flow chart guides students.

Chapter 6, Application Models enhances the valuable skill of formulating linear programs. The use of linear programming for diverse industrial and governmental purposes is widespread. Google has well over a million references! The many prototype models presented in Chapter 6 such as the knapsack, trim waste, and caterer problems are classics of linear programming literature. Virtually all applications require automated computation and computer software is widely available.

Two additional linear programming features are its mathematical history and application to game theory in Chapter 11. Game theory is important to contemporary economics and industrial competition. In addition to the elementary methods usual in finite texts, this text places emphasis on the connection between game theory and linear programming. This aids applications that require automated computation as well as understanding of mathematical theory.

Probability and Statistics

Mathematical probability and statistical topics usually form another pillar of a finite mathematics course. It comprises the third major part of the text. Chapter 7, Set and Probability Relationships; Chapter 8, Random Variables and Probability Distributions; Chapter 9, Markov Chains; Chapter 10, Mathematical Statistics; and a section on Monte Carlo Method, unique to this text, in Chapter 11.

Chapter 7, Set and Probability Relationships has ample examples to illustrate the basics of sets and the relations to probabilities. One section emphasizes tree diagrams both as an aid to teaching probability and as an introduction to their widespread use in managing large projects from building construction to varied components of space ship design. A unique feature of combinatorics in this text is a table aid to help with these challenging exercises. Also included is a fourth combinatoric possibility, usually omitted, that samples with replacement but without regard to order. Newer applications of combinatorics in computer program speed estimates are included.

Bayes' Rule has earned a niche in much of everyday life in applications from spam filters to medical tests and oil exploration. The text provides a robust treatment of the Total Probability Theorem and Bayes' Rule. Both algebraic and tree diagrams are used to aid students' skill and appreciation.

While Chapter 8, Random Variables and Probability Distributions has more material than can usually be included in a semester course, it can be used for enrichment or a second semester. Aside from the customary basic topics of the distributions, the hypergeometric and Poisson distributions are included. The hypergeometric distribution, easily taught, has a role in industrial quality control and estimation of sizes of animal populations that our students find interesting. The Poisson's Law's ubiquitousness in everyday situations is briefly explored.

Chapter 9, Markov Chains is a staple of many finite mathematical courses. This text uniquely pairs matrix and diagram solutions to each example. Students are aided to grasp matrix methods with a diagrammatic solution alongside. Students learn to form and execute transition matrices in regular and absorbing Markov chains.

Chapter 10, Mathematical Statistics an occasional topic, has become fundamental in many social science and business contexts. Some feel that statistics should be a part of a collegiate education. Included here are topics in statistical graphs, and surveys, basic definitions and interpretations of mean and variance. The normal distribution receives robust attention in view of its basic theoretical and practical importance.

Enrichment

Chapter 11Enrichment in Finite Mathematics may be unique among finite mathematics texts. Each section of this chapter deals with a nontraditional and useful topic related to finite mathematics. One section has the elements of game theory and its connection to linear programming. Other sections, somewhat novel to finite mathematics texts, are both interesting and useful in seeking fresh topics.

There is a section on financial and economic applications and another that uses matrix applications to the social and life sciences. The last two sections involve the Monte Carlo Method and Dynamic Programming. Monte Carlo method, often called simulation, is a staple of planning, research, and operations in a huge variety of contexts. A popular topic with students it is easily mastered. Dynamic Programming, a relatively new optimization format, relies on mathematical recursion. A useful topic in its own right, additionally, it alerts students to the limits of automated computation even for seemingly “small” problems.

Supplements

A modestly priced companion “Student Solutions Manual” has detailed solutions to the odd numbered exercises and is recommended.

For teachers, a complete solutions manual, PowerPoint® slides by chapter, and a test bank are available from John Wiley & Sons.

Suggestions

Suggestions for improvements are welcome.

Acknowledgements

We have benefitted from advice and discussions with Professors Louise Amick, Washington College; Nancy Hall, University of Delaware Associate in Arts Program-Georgetown; Richard Schnackenberg, Florida Gulf Coast University; Robert H. Mayer, Jr, US Naval Academy, Dr Wiseley Wong, University of Maryland; and Carolyn Krause, Delaware Technical and Community College-Terry Campus.

We acknowledge the University of Delaware's Morris Library for use of its resources during the preparation of this text.

About the Companion Website

This book is accompanied by a companion website:

http://www.wiley.com/go/morris/finitemathematics

The website includes:

Instructors' Solutions Manual

PowerPoint® slides by chapter

Test banks by chapter

Teacher Commentary

Chapter 1Linear Equations and Mathematical Concepts

1.1 Solving Linear Equations

Example 1.1.1 Solving a Linear Equation

Example 1.1.2 Solving for y

Example 1.1.3 Simple Interest

Example 1.1.4 Investment

Example 1.1.5 Gasoline Prices

Example 1.1.6 Breaking a Habit

1.2 Equations of Lines and Their Graphs

Example 1.2.1 Ordered Pair Solutions

Example 1.2.2 Intercepts and Graph of a Line

Example 1.2.3 Slope-Intercept Form

Example 1.2.4 Point-Slope Form

Example 1.2.5 Temperature Conversion

Example 1.2.6 Salvage Value

Example 1.2.7 Parallel or Perpendicular Lines

1.3 Solving Systems of Linear Equations

Example 1.3.1 Graphical Solutions to a System of Linear Equations

Example 1.3.2 Substitution – Consistent System

Example 1.3.3 Substitution–Inconsistent System

Example 1.3.4 Elimination Method – Consistent System

Example 1.3.5 Market Equilibrium

1.4 The Numbers π and

e

The Number

π

Example 1.4.1 A Biblical π

The Number

e

1.5 Exponential and Logarithmic Functions

Exponential Functions

Example 1.5.1 Using Laws of Exponents

Example 1.5.2 Simplifying Exponential Equations

Example 1.5.3 Simplifying Exponential Exponents

Logarithmic Functions

Example 1.5.4 Exponential and Logarithmic Forms

Example 1.5.5 Simplifying Logarithms

Example 1.5.6 Simplifying Logarithmic Expressions

Example 1.5.7 Base 10 Exponents

Example 1.5.8 Base e Exponents

1.6 Variation

Example 1.6.1 Summer Wages

Example 1.6.2 Hooke's Law

Example 1.6.3 Geometric Similarity

Example 1.6.4 Volume of a Sphere

Example 1.6.5 Seawater Density

Example 1.6.6 Illumination

Example 1.6.7 Beam Strength

1.7 Unit Conversions and Dimensional Analysis

Example 1.7.1 Dimensional Correctness

Example 1.7.2 Speed Conversion

Example 1.7.3 Time Conversion

Example 1.7.4 Atoms of Nitrogen

Example 1.7.5 Atoms of Hydrogen

Historical Notes and Comments

1.1 Solving Linear Equations

Mathematical descriptions, often as algebraic expressions, usually consist of alphanumeric characters and special symbols.

Physicists describe the distance, s, that an object falls under gravity in time, t, by . Here, the letters s and t are variables since their values may change, while, g, the acceleration of gravity is considered constant. While any letters can represent variables, typically the later letters of the alphabet are customary. The use of x and y is generic. Sometimes, it is convenient to use a letter that is descriptive of the variable, as t for time.

Earlier letters of the alphabet are customary for fixed values or constants. However, exceptions are widespread. The equal sign, a special symbol, is used to form an . An equation equates algebraic expressions. Numerical values for variables that preserve equality are called to the equations.