SEVENTH EDITION Finite Mathematics Enhanced Edition Howard L. Rolf Baylor University Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Finite Mathematics, Seventh Edition © 2011, 2009 Brooks/Cole, Cengage Learning Howard L. Rolf ALL RIGHTS RESERVED. 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For your course and learning solutions, visit academic.cengage.com Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States of America 1 2 3 4 5 6 7 15 14 13 12 11 10 TO A SPECIAL PERSON , Anita Ward Rolf This page intentionally left blank Contents 1 FUNCTIONS AND LINES 1 1.1 Functions 2 Using Your TI-83/84 11 Using EXCEL 12 1.2 Graphs and Lines 14 Using Your TI-83/84: Graphs of Lines and Evaluating Points on a Line 34 Using EXCEL 36 1.3 Mathematical Models and Applications of Linear Functions 38 Using Your TI-83/84: Intersection of Lines 57 Using EXCEL 58 Important Terms in Chapter 1 60 Review Exercises for Chapter 1 61 2 LINEAR SYSTEMS 65 2.1 Systems of Two Equations 66 Using EXCEL 81 2.2 Systems with Three Variables:An Introduction to a Matrix Representation of a Linear System of Equations 82 Using Your TI-83/84: Solving a System of Equations Using Row Operations 103 UsingEXCEL:SolvingSystemsofEquationsUsingRow Operations 105 2.3 Gauss-Jordan Method for General Systems of Equations 107 Using Your TI-83/84: Obtaining the Reduced Echelon Form of a Matrix 126 Using EXCEL: General Procedure for Pivoting 128 2.4 Matrix Operations 129 Using Your TI-83/84: Matrix Operations 140 Using EXCEL: Matrix Addition and Scalar Multiplication 140 v vi Contents 2.5 Multiplication of Matrices 141 Using Your TI-83/84: Matrix Multiplication 158 Using EXCEL 159 2.6 The Inverse of a Matrix 159 Using Your TI-83/84: The Inverse of a Matrix 174 Using EXCEL 175 2.7 Leontief Input–Output Model in Economics 176 Using Your TI-83/84 188 Using EXCEL 189 2.8 Linear Regression 190 Using Your TI-83/84: Regression Lines 199 Using EXCEL: To Draw and Find a Linear Regression Line 200 Important Terms in Chapter 2 202 Review Exercises for Chapter 2 202 3 LINEAR PROGRAMMING 205 3.1 Linear Inequalities in Two Variables 206 3.2 Solutions of Systems of Inequalities: A Geometric Picture 213 Using Your TI-83/84: Finding Corner Points of a Feasible Region 221 Using EXCEL 223 3.3 Linear Programming:A Geometric Approach 225 Using Your TI-83/84: Evaluate the Objective Function 248 Using EXCEL: Evaluate an Objective Function 250 3.4 Applications 250 Important Terms in Chapter 3 258 Review Exercises for Chapter 3 258 4 LINEAR PROGRAMMING: THE SIMPLEX METHOD 261 4.1 Setting Up the Simplex Method 262 4.2 The Simplex Method 273 Using Your TI-83/84: A Program for the Simplex Method 293 Using EXCEL 295 4.3 The Standard Minimum Problem:Duality 299 4.4 Mixed Constraints 307 Using EXCEL 329 Contents vii 4.5 Multiple Solutions,Unbounded Solutions, and No Solutions 330 4.6 What’s Happening in the Simplex Method? (Optional) 339 4.7 Sensitivity Analysis 347 Important Terms in Chapter 4 358 Review Exercises for Chapter 4 358 5 MATHEMATICS OF FINANCE 363 5.1 Simple Interest 364 5.2 Compound Interest 372 Using Your TI-83/84: Making a Table 386 Using EXCEL 387 5.3 Annuities and Sinking Funds 388 Using Your TI-83/84: Growth of Annuities 397 Using EXCEL 398 5.4 Present Value of an Annuity and Amortization 399 Using Your TI-83/84: Monthly Loan Payments 418 Using EXCEL 419 Important Terms in Chapter 5 419 Summary of Formulas in Chapter 5 420 Review Exercises for Chapter 5 420 6 SETS AND COUNTING 423 6.1 Sets 424 6.2 Counting Elements in a Subset Using a Venn Diagram 432 6.3 Basic Counting Principles 440 6.4 Permutations 452 Using Your TI-83/84: Permutations 462 Using EXCEL: Permutations 462 6.5 Combinations 462 Using Your TI-83/84: Combinations 477 Using EXCEL: Combinations 478 6.6 A Mixture of Counting Problems 478 6.7 Partitions (Optional) 484 Important Terms in Chapter 6 491 Review Exercises for Chapter 6 491 viii Contents 7 PROBABILITY 495 7.1 Introduction to Probability 496 7.2 Equally Likely Events 509 7.3 Compound Events:Union,Intersection, and Complement 519 7.4 Conditional Probability 532 7.5 Independent Events 547 7.6 Bayes’Rule 564 7.7 Markov Chains 579 Important Terms in Chapter 7 593 Review Exercises for Chapter 7 593 8 STATISTICS 597 8.1 Frequency Distributions 598 Using Your TI-83/84: Histograms 613 Using EXCEL: Histograms and Pie Charts 615 8.2 Measures of Central Tendency 617 Using Your TI-83/84: Calculating the Mean and Median 628 Using EXCEL: Calculate the Mean and Median 629 8.3 Measures of Dispersion:Range,Variance, and Standard Deviation 630 Using Your TI-83/84 645 Using EXCEL: Measures of Dispersion 647 8.4 Random Variables and Probability Distributions of Discrete Random Variables 648 8.5 Expected Value of a Random Variable 658 8.6 Bernoulli Experiments and Binomial Distribution 668 Using Your TI-83/84: Binomial Probability Distribution 681 8.7 Normal Distribution 683 Using Your TI-83/84: Area Under the Normal Curve 712 Using EXCEL 713 8.8 Estimating Bounds on a Proportion 713 Important Terms in Chapter 8 727 Review Exercises for Chapter 8 728 Contents ix 9 GAME THEORY 731 9.1 Two-Person Games 732 9.2 Mixed-Strategy Games 740 Important Terms in Chapter 9 750 Review Exercises for Chapter 9 750 10 LOGIC 753 10.1 Statements 754 10.2 Conditional Statements 760 10.3 Equivalent Statements 766 10.4 Valid Arguments 768 Important Terms in Chapter 10 775 Review Exercises for Chapter 10 775 A REVIEW TOPICS 777 A.1 Properties of Real Numbers 777 A.2 Solving Linear Equations 780 A.3 Coordinate Systems 784 A.4 Linear Inequalities and Interval Notation 787 Important Terms in Appendix A 793 B USING A TI-83/84 GRAPHING CALCULATOR 795 Notation 795 Arithmetic Operations 795 Graphing 796 Evaluating a Function 797 Finding the Intersection of Two Graphs 797 Constructing a Table 798 Matrices 799 Statistics 801