GIáo trình finite mathematics for business economics life sciences, and social sciences 13th global edit ion by barnett

GlobAl edITIon Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences THIRTeenTH edITIon Raymond A Barnett • Michael R Ziegler • Karl E Byleen www.ebookslides.com FInIte M AtheM AtIcs For BusIness, econoMIcs, LIFe scIences, And socIAL scIences thirteenth edition Global edition rAyMond A BArnett MIchAeL r ZIeGLer KArL e ByLeen Merritt college Marquette university Marquette universit y Boston columbus Indianapolis new york san Francisco upper saddle river Amsterdam cape town dubai London Madrid Milan Munich Paris Montréal toronto delhi Mexico city são Paulo sydney hong Kong seoul singapore taipei tokyo www.ebookslides.com Editor in Chief: Deirdre Lynch Head of Learning Asset Acquisition, Global Edition: Laura Dent Executive Editor: Jennifer Crum Project Manager: Kerri Consalvo Assistant Acquisitions Editor, Global Edition: Murchana Borthakur Associate Project Editor, Global Edition: Uttaran Das Gupta Editorial Assistant: Joanne Wendelken Senior Managing Editor: 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as trademarks Where those designations appear in this book, and Pearson was aware of a trademark claim, the designations have been printed in initial caps or all caps Pearson Education Limited Edinburgh Gate Harlow Essex CM 20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2015 The rights of Raymond A Barnett, Michael R Ziegler, and Karl E Byleen to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences, 13th edition, ISBN 978-0-321-94552-5, by Raymond A Barnett, Michael R Ziegler, and Karl E Byleen, published by Pearson Education © 2015 All rights reserved 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 or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners ISBN 10: 1-292-06229-0 ISBN 13: 978-1-292-06229-7 (Print) ISBN 13: 978-1-292-06645-5 (PDF) British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 14 13 12 11 Typeset by Intergra in Times LT Std 11 pt Printed and bound by Courier Kendallville in The United States of America www.ebookslides.com contents Preface Diagnostic Prerequisite Test 16 Part chapter A LibrAry of ELEmEnTAry funcTions Linear Equations and Graphs 18 1.1 Linear equations and Inequalities 19 1.2 Graphs and Lines 28 1.3 Linear regression 42 chapter summary and review 54 review exercises 55 chapter functions and Graphs 58 Functions elementary Functions: Graphs and transformations Quadratic Functions Polynomial and rational Functions exponential Functions Logarithmic Functions chapter summary and review review exercises 2.1 2.2 2.3 2.4 2.5 2.6 Part chapter 59 73 85 100 111 122 133 136 finiTE mAThEmATics mathematics of finance 142 simple Interest compound and continuous compound Interest Future Value of an Annuity; sinking Funds Present Value of an Annuity; Amortization chapter summary and review review exercises 3.1 3.2 3.3 3.4 chapter 143 150 163 171 183 185 systems of Linear Equations; matrices 189 review: systems of Linear equations in two Variables systems of Linear equations and Augmented Matrices Gauss–Jordan elimination Matrices: Basic operations Inverse of a square Matrix Matrix equations and systems of Linear equations Leontief Input–output Analysis chapter summary and review review exercises 4.1 4.2 4.3 4.4 4.5 4.6 4.7 190 203 212 226 238 250 258 266 267 www.ebookslides.com conTEnTs chapter Linear inequalities and Linear Programming 271 5.1 Linear Inequalities in two Variables 272 5.2 systems of Linear Inequalities in two Variables 279 5.3 Linear Programming in two dimensions: A Geometric Approach 286 chapter summary and review 298 review exercises 299 chapter Linear Programming: The simplex method 301 6.1 the table Method: An Introduction to the simplex Method 302 6.2 the simplex Method: Maximization with Problem constraints of the Form … 6.3 the dual Problem: Minimization with Problem constraints of the Form Ú 6.4 Maximization and Minimization with Mixed Problem constraints chapter summary and review review exercises chapter 329 342 357 358 Logic, sets, and counting 361 Logic sets Basic counting Principles Permutations and combinations chapter summary and review review exercises 7.1 7.2 7.3 7.4 chapter 313 362 370 377 385 396 398 Probability 401 sample spaces, events, and Probability union, Intersection, and complement of events; odds conditional Probability, Intersection, and Independence Bayes’ Formula random Variable, Probability distribution, and expected Value chapter summary and review review exercises 8.1 8.2 8.3 8.4 8.5 chapter 402 415 427 441 448 457 459 markov chains 463 9.1 Properties of Markov chains 464 9.2 regular Markov chains 475 9.3 Absorbing Markov chains 485 chapter summary and review 499 review exercises 500 www.ebookslides.com chapter 10 conTEnTs Games and Decisions 503 10.1 strictly determined Games 504 10.2 Mixed-strategy Games 510 10.3 Linear Programming and * Games: A Geometric Approach 10.4 Linear Programming and m * n Games: simplex Method and the dual Problem chapter 10 summary and review review exercises chapter 11 Graphing data Measures of central tendency Measures of dispersion Bernoulli trials and Binomial distributions normal distributions chapter 11 summary and review review exercises 537 548 558 564 574 584 585 real numbers operations on Polynomials Factoring Polynomials operations on rational expressions Integer exponents and scientific notation rational exponents and radicals Quadratic equations 588 594 600 606 612 616 622 special Topics 631 B.1 B.2 B.3 Appendix c basic Algebra review 588 A.1 A.2 A.3 A.4 A.5 A.6 A.7 Appendix b 527 532 534 Data Description and Probability Distributions 536 11.1 11.2 11.3 11.4 11.5 Appendix A 521 sequences, series, and summation notation 631 Arithmetic and Geometric sequences 637 Binomial theorem 643 Tables 647 table I table II Area under the standard normal curve 647 Basic Geometric Formulas 648 Answers A-1 index i-1 index of Applications i-10 www.ebookslides.com PreFAce The thirteenth edition of Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences is designed for a one-term course in finite mathematics for students who have had one to two years of high school algebra or the equivalent The book’s overall approach, refined by the authors’ experience with large sections of college freshmen, addresses the challenges of teaching and learning when prerequisite knowledge varies greatly from student to student The authors had three main goals when writing this text: ▶ To write a text that students can easily comprehend ▶ To make connections between what students are learning and how they may apply that knowledge ▶ To give flexibility to instructors to tailor a course to the needs of their students Many elements play a role in determining a book’s effectiveness for students Not only is it critical that the text be accurate and readable, but also, in order for a book to be effective, aspects such as the page design, the interactive nature of the presentation, and the ability to support and challenge all students have an incredible impact on how easily students comprehend the material Here are some of the ways this text addresses the needs of students at all levels: ▶ Page layout is clean and free of potentially distracting elements ▶ Matched Problems that accompany each of the completely worked examples help students gain solid knowledge of the basic topics and assess their own level of understanding before moving on ▶ Review material (Appendix A and Chapters and 2) can be used judiciously to help remedy gaps in prerequisite knowledge ▶ A Diagnostic Prerequisite Test prior to Chapter helps students assess their skills, while the Basic Algebra Review in Appendix A provides students with the content they need to remediate those skills ▶ Explore and Discuss problems lead the discussion into new concepts or build upon a current topic They help students of all levels gain better insight into the mathematical concepts through thought-provoking questions that are effective in both small and large classroom settings ▶ Instructors are able to easily craft homework assignments that best meet the needs of their students by taking advantage of the variety of types and difficulty levels of the exercises Exercise sets at the end of each section consist of a Skills Warm-up (four to eight problems that review prerequisite knowledge specific to that section) followed by problems divided into categories A, B, and C by level of difficulty, with level-C exercises being the most challenging ▶ The MyMathLab course for this text is designed to help students help themselves and provide instructors with actionable information about their progress The immediate feedback students receive when doing homework and practice in MyMathLab is invaluable, and the easily accessible e-book enhances student learning in a way that the printed page sometimes cannot Most important, all students get substantial experience in modeling and solving real-world problems through application examples and exercises chosen from business and economics, life sciences, and social sciences Great care has been taken to write a book that is mathematically correct, with its emphasis on computational skills, ideas, and problem solving rather than mathematical theory www.ebookslides.com PrEfAcE Finally, the choice and independence of topics make the text readily adaptable to a variety of courses (see the chapter dependencies chart on page 11) This text is one of three books in the authors’ college mathematics series The others are Calculus for Business, Economics, Life Sciences, and Social Sciences, and College Mathematics for Business, Economics, Life Sciences, and Social Sciences; the latter contains selected content from the other two books Additional Calculus Topics, a supplement written to accompany the Barnett/Ziegler/Byleen series, can be used in conjunction with any of these books new to This Edition Fundamental to a book’s effectiveness is classroom use and feedback Now in its thirteenth edition, Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences has had the benefit of a substantial amount of both Improvements in this edition evolved out of the generous response from a large number of users of the last and previous editions as well as survey results from instructors, mathematics departments, course outlines, and college catalogs In this edition, ▶ The Diagnostic Prerequisite Test has been revised to identify the specific deficiencies in prerequisite knowledge that cause students the most difficulty with finite mathematics ▶ Most exercise sets now begin with a Skills Warm-up—four to eight problems that review prerequisite knowledge specific to that section in a just-in-time approach References to review material are given for the benefit of students who struggle with the warm-up problems and need a refresher ▶ Section 6.1 has been rewritten to better motivate and introduce the simplex method and associated terminology ▶ Examples and exercises have been given up-to-date contexts and data ▶ Exposition has been simplified and clarified throughout the book ▶ An Annotated Instructor’s Edition is now available, providing answers to exercises directly on the page (whenever possible) Teaching Tips provide less-experienced instructors with insight on common student pitfalls, suggestions for how to approach a topic, or reminders of which prerequisite skills students will need Lastly, the difficulty level of exercises is indicated only in the AIE so as not to discourage students from attempting the most challenging “C” level exercises ▶ MyMathLab for this text has been enhanced greatly in this revision Most notably, a “Getting Ready for Chapter X” has been added to each chapter as an optional resource for instructors and students as a way to address the prerequisite skills that students need, and are often missing, for each chapter Many more improvements have been made See the detailed description on pages 14 and 15 for more information Trusted features emphasis and style As was stated earlier, this text is written for student comprehension To that end, the focus has been on making the book both mathematically correct and accessible to students Most derivations and proofs are omitted, except where their inclusion adds significant insight into a particular concept as the emphasis is on computational skills, ideas, and problem solving rather than mathematical theory General concepts and results are typically presented only after particular cases have been discussed design One of the hallmark features of this text is the clean, straightforward design of its pages Navigation is made simple with an obvious hierarchy of key topics and a judicious use of call-outs and pedagogical features We made the decision to maintain a two-color design to help students stay focused on the mathematics and applications Whether students start in www.ebookslides.com PrEfAcE the chapter opener or in the exercise sets, they can easily reference the content, examples, and Conceptual Insights they need to understand the topic at hand Finally, a functional use of color improves the clarity of many illustrations, graphs, and explanations, and guides students through critical steps (see pages 77, 124, and 418) examples and Matched Problems More than 300 completely worked examples are used to introduce concepts and to demonstrate problem-solving techniques Many examples have multiple parts, significantly increasing the total number of worked examples The examples are annotated using blue text to the right of each step, and the problem-solving steps are clearly identified To give students extra help in working through examples, dashed boxes are used to enclose steps that are usually performed mentally and rarely mentioned in other books (see Example on page 20) Though some students may not need these additional steps, many will appreciate the fact that the authors not assume too much in the way of prior knowledge ExamplE solving exponential equations (A) 10x = (B) ex = Solve for x to four decimal places: (C) 3x = Solution 10x = log 10x = log x = log = 0.3010 x (B) e = ln ex = ln x = ln = 1.0986 x (C) = (A) log 3x = log x log = log Take common logarithms of both sides Property Use a calculator To four decimal places Take natural logarithms of both sides Property Use a calculator To four decimal places Take either natural or common logarithms of both sides (We choose common logarithms.) Property Solve for x log Use a calculator log = 1.2619 To four decimal places x = Matched Problem x (A) 10 = Solve for x to four decimal places: (B) ex = (C) 4x = Each example is followed by a similar Matched Problem for the student to work while reading the material This actively involves the student in the learning process The answers to these matched problems are included at the end of each section for easy reference explore and discuss Most every section contains Explore and Discuss problems at appropriate places to encourage students to think about a relationship or process before a result is stated or to investigate additional consequences of a development in the text This serves to foster critical thinking and communication skills The Explore and Discuss material can be used for in-class discussions or out-of-class group activities and is effective in both small and large class settings www.ebookslides.com PrEfAcE Explore and Discuss How many x intercepts can the graph of a quadratic function have? How many y intercepts? Explain your reasoning New to this edition, annotations in the instructor’s edition provide tips for lessexperienced instructors on how to engage students in these Explore and Discuss activities, expand on the topic, or simply guide student responses exercise sets The book contains over 4,200 carefully selected and graded exercises Many problems have multiple parts, significantly increasing the total number of exercises Exercises are paired so that consecutive odd- and even-numbered exercises are of the same type and difficulty level Each exercise set is designed to allow instructors to craft just the right assignment for students Exercise sets are categorized as Skills Warm-up (review of prerequisite knowledge), and within the Annotated Instructor’s Edition only, as A (routine easy mechanics), B (more difficult mechanics), and C (difficult mechanics and some theory) to make it easy for instructors to create assignments that are appropriate for their classes The writing exercises, indicated by the icon , provide students with an opportunity to express their understanding of the topic in writing Answers to all odd-numbered problems are in the back of the book Answers to application problems in linear programming include both the mathematical model and the numeric answer Applications A major objective of this book is to give the student substantial experience in modeling and solving real-world problems Enough applications are included to convince even the most skeptical student that mathematics is really useful (see the Index of Applications at the back of the book) Almost every exercise set contains application problems, including applications from business and economics, life sciences, and social sciences An instructor with students from all three disciplines can let them choose applications from their own field of interest; if most students are from one of the three areas, then special emphasis can be placed there Most of the applications are simplified versions of actual real-world problems inspired by professional journals and books No specialized experience is required to solve any of the application problems Additional Pedagogical features The following features, while helpful to any student, are particularly helpful to students enrolled in a large classroom setting where access to the instructor is more challenging or just less frequent These features provide much-needed guidance for students as they tackle difficult concepts ▶ Call-out boxes highlight important definitions, results, and step-by-step processes (see pages 106, 112–113) ▶ Caution statements appear throughout the text where student errors often occur (see pages 154, 159, and 192) ! Caution Note that in Example 11 we let x = represent 1900 If we let x = represent 1940, for example, we would obtain a different logarithmic regression equation, but the prediction for 2015 would be the same We would not let x = represent 1950 (the first year in Table 1) or any later year, because logarithmic func▲ tions are undefined at www.ebookslides.com This page is intentionally left blank www.ebookslides.com Index A Abscissa, 29 Absolute values, 43–44, 75 Absorbing Markov chains, 485–499 absorbing states, 485–487 definition of, 486 graphing calculator approximations for, 493–494 limiting matrix and, 488–493 recognizing, 487 standard forms for, 487–488 transition matrix for, 492 Absorbing states, 470, 485–487 Acceptable probability assignment, 406, 408 ac test, 602 Addition elimination by, 194–196 of matrix, 226–227, 250 of polynomials, 596–597 principle of, 377–379 of rational expressions, 608–609 of real numbers, 589–590 Additive inverse, 590 Algebra, 588–630 integer exponents, 612–613 polynomials, 594–600, 600–605 quadratic equations, 622–630 radicals, 619–620 rational exponents, 616–622 rational expressions, 606–611 real numbers, 588–594 scientific notation, 613–614 Algebraic expressions, 594 Amortization, 174–175 of debt, 174–175 formula for, 175, 177–178, 180 interest on, 175 schedules of, 175–179 Amount, 118, 143–145 See also future value (FV) Analysis break-even, 25–26, 66 input–output, 258, 262–263 of investments, 253–255, 528–531 profit–loss, 66 regression, 44 Analytic geometry, fundamental theorem of, 29 Annual nominal rates, 151, 159 See also interest rate Annual percentage yield (APY), 157–159 Annuities defined, 163 future value of, 163–171 (See also sinking funds) ordinary, 163, 165–166, 173 present value of, 171–183 (See also amortization) Applications See Index of Applications Arbitrary event, 409–410 Arithmetic average, 548 See also mean Arithmetic mean, 635 Arithmetic sequences, 637–643 common differences in, 637 nth-term formulas and, 638–639 sum formulas and, 639–642 Arithmetic series, 639 Artificial variables, 342–343, 345 Associative matrix, 226 Associative properties, 590 Asymptotes of rational functions finding, 106–107 horizontal, 104–107, 113 vertical, 104–107 Augmented matrix, 203–205 row equivalent, 205–207 solving, 205–210 summary of, 210 Average daily balance method, 147 Averages, 453, 548, 550 See also mean law of, 422, 560 Axis coordinate, 29 horizontal, 28–29 of parabolas, 89 vertical, 28–29 x, reflection in, 78 B Balance sheet, 166 Bar graphs, 537–539 Base of exponential functions, 112 Base e, 114–115 Base logarithmic functions, 123–124 Basic algebra See algebra Bayes’ formula, 441–448 Bernoulli trials, 565–567 Best fit, 45 Big M method See also modified problem definition of, 301, 342, 345 introduction to, 342–345 for minimization problem, 349–351 modified problem, for solving, 346–347 procedure for, 345 slack variables and, 345 solutions to, 351 summary of, 345–349 Binomial distribution, 565, 568–570 approximating, 578–582 mean of, 570 standard deviation of, 570 Binomials, 595 expansion of, 568 formulas for, 567–568, 643–644 theorem for, 643–646 Body surface area (BSA), 43 Boundary line of half-planes, 272–273 Bounded functions, 107 Bounded solution region, 282 Break-even analysis, 25–26, 66 Broken-line graphs, 538–539, 539 C Canceling, in fractions, 607 Cartesian (rectangular) coordinate system, 28–29 Central tendency See measures of central tendency Certificates of deposit (CDs), 157 Chains See Markov chains Change-of-base formulas, 129 Circle, 648 Class frequency, 540–541 Class intervals, 540 Closed intervals, 22 Coefficient matrix, 204–205 Coefficients concept of, 595 leading, 101 numerical, 595 of objective functions, 330 of problem constraints, 330 Column matrix, 204 Columns dominant, 516 players in, 504 recessive, 516–518 Combinations, 388–391 of n distinct objects taken r at a time, 388–390 Combined factoring polynomials, techniques for, 604–605 Combined matrix, 250 Combined polynomials, 598 Combining like terms, 595–596 Commission schedule, 146–147 Common differences, in arithmetic sequence, 637 Common logarithms, 127 Common ratio, in geometric sequence, 637 Commutative properties, 590 Complement of event, 419–421 of sets, 373 Completely factored numbers, 600 Completing the square, 88–89, 625 Complement of events, 419 Compound events, 402–403 Compound fractions, 609–610 Compound growth rate, 118 Compounding quarterly interest, 150–151 Compound interest, 117–119, 150–153 annual percentage yield, 157–160 I-1 I-2 www.ebookslides.com Index Compound interest (continued) continuous, 153–154 daily, 154 definition of, 118, 150 graphing, 155 growth of, time and, 154–157 Compound propositions, 364 truth table for, 362–363, 365, 367–368 Conditional probability, 427–430, 437 concept of, 427 definition of, 427–428 events for, 430–431, 433–436 probability trees for, 431–433 product rule for, 430–431 summary of, 437 Conditional propositions, 363, 368 Cone, 649 Conjunction, 363 Connectives, 362–365 Consistent systems of linear equations, 192 Constant functions, 63 Constant matrix, 204–205 Constant-profit line, 286 Constant rate of change, 43 Consumer Price Index (CPI), 26 Contingency, 366 Continuous random variables, 575 Contradiction, 366 Contrapositive propositions, 364–365, 368 Converse propositions, 364–365 Coordinate axis, 29 Coordinates, 29, 303 Corner points, 280, 290, 310 Costs, 36–37, 66–67, 93 Counting principles, 377–384 addition principle, 377–379 multiplication principle, 379–381 Counting technique, 377 Cube root, 616 Cumulative matrix, 226, 232 Curve fitting, 44 Curves, 543, 574, 576, 647 Cylinder, 649 least common, 20 rationalizing, 620 Dependent events, 434–435 Dependent systems of linear equations, 192 Dependent variables, 62 Deviation, 558 See also standard deviation Diameter of a tree at breast height (Dbh), 47–48 Difference quotient, 66 Discontinuous functions, 102 Discrete random variables, 575 Discriminate, 626–628 Disjoint events, 416 Disjoint sets, 377 Disjunction, 363 Dispersion See measures of dispersion Distributions, 574–584 See also specific types of binomial, 565, 568–570, 578–582 of curves, 575–578 frequency, 540–542, 558, 564 hypothetical, 564 normal, 574–584 probabilities for, 541, 564, 576–577 theoretical, 564 Distributive properties, 589–590, 595 Divided bar graph, 537–538 Division of rational expressions, 607–608 Domains, 61–62 of elementary functions, 75 of exponential functions, 112–113 finding, 64–65 of logarithmic functions, 122–124 of polynomial functions, 101 of rational functions, 103 Dominant column, 516 Double bar graph, 537–538 Double inequalities, 22, 24 Double subscript notation, 204 Dual problems, 329–341 definition of, 329 formation of, 329–331 fundamental principle of, 331 problem constraints of, 334–335 for * games, 527–528 D E Daily compound interest, 154 Data, 536–584 See also probability distributions graphing, 537–548 grouped, 550–553 measures of central tendency of, 548–557 measures of dispersion of, 558–564 quantitative, 540 ranges of, 540 standard deviation of, 559 ungrouped, 558 variations of, 558–559 Debt, amortizing, 174–175 Decision variables, 286, 289 Decoding matrix, 246 Degree of polynomial, 595 Denominators, 592 Effective rate, 158 See also annual percentage yield (APY) Elementary functions, 73–85 beginning library of, 74–75 domain of, 75 evaluating, 74 graph of, 75 horizontal shifts, 75–77 range of, 75 reflections, 77–81 shrinks, 77–81 stretches, 77–81 vertical shifts, 75–77 Elements in events, 417 identity, 238, 590 of matrix, 203–204 pivot, 317 of sets, 370, 373–374, 377, 379–380 Elimination by addition, 194–196 Empirical distribution of frequency, 564 Empirical probability, 408, 410 applications to, 422–424 approximate, 408 distributions for, 564 on graphing calculator, 410 simulation and, 410 Empty sets, 370, 377 Encoding matrix, 246 Endpoints of intervals, 22–23 Entering variables, 316 Equality matrix, 250 Equality of saddle values, 505 Equally likely assumption, 409–411 Equal matrix, 226 Equal sets, 371 Equations See also specific types of cost, 36–37 for curves, 574 equivalent, 19, 20–21 exponential, 128 functions specified by, 59–64 graph of, 29 linear, 19–20, 325 of lines, 34–36 logarithmic, 126–127 matrix, 227, 250–251 price–demand, 38 price–supply, 37–38 quadratic, 622–630 solution of, 19 Equilibrium point, 38, 628 Equilibrium quantity, 38, 197 Equity, 177 Equivalent equations, 19–21 Equivalent formulas, 20–21 Equivalent inequalities, 22 Equivalent systems of linear equations, 195 e-system, 303 Events, 403 arbitrary, 409–410 complement of, 419 compound, 402–403 dependent, 434–435 independent, 434–437 intersection of, 415, 416–419 mutually exclusive, 416 odds for, 421–422 probability of, 405–408, 416–419, 441–443 sample spaces and, 402–405 simple, 402–403, 406 union of, 415–419 Exiting variables, 316 Expanded coordinates, 303 Expansion of binomials, 568 Expected value, 451–452 of binomial distribution, 570 decision making and, 454 of games, 452, 511–513 www.ebookslides.com insurance and, 453 of mixed-strategy games, 511–513 for optimal strategies, 514 of probability distributions, 453 of random variables, 450–453 of two-finger Morra games, 511–513 Experiments, 402–403 See also Bernoulli trials; events Exponential equations, 128 Exponential functions, 111–122 See also logarithmic functions base e, 114–115 base of, 112 compound interest, 117–119 defined, 111–112 domain of, 112–113 graphs of, 112–113 logarithmic functions, conversion to, 124–125 properties of, 114 range of, 112 Exponential growth rate, 115–116 Exponential regression, 117 Exponents first property of, 592 integer, 612–613 natural number, 592 properties of, 612 radicals, properties of, 619–620 rational, 616–622 scientific notation and, 613–614 simplifying, 612–613 Extrapolation, 46 F Factorability, theorem of, 627 Factored polynomials, 600 Factored form of numbers, 600 Factorials, 384–385, 390, 644 Factoring, 627–628 quadratic equations, solution by, 623–624 quadratic formula and, 627–628 Factoring polynomials, 600–605 combined, techniques for, 604–605 common factors, 600–601 by grouping, 601 second-degree polynomials, factoring, 602–603 special formulas for, 603–604 Fair games, 421, 453 False negative results, 445 False positive results, 445 Feasible region, 280, 289, 292, 304 corner points of, 310 for linear programming, 325 Feasible solution, 303, 313–314 Finance, 142–183 annuities, 163–183 compound interest, 150–163 mathematics of, 142–183 simple interest, 143–150 Finite arithmetic series, 639–640 Finite geometric series, 165, 172 Finite sample space, 450 Finite sequence, 632 Finite series, 633 Finite sets, 371 First-state matrix, 465 Fixed costs, 37, 66 Formulas See also specific types of for amortization, 175, 177–178, 180 Bayes’, 441–448 for binomials, 567–568, 643–644 change-of-base, 129 equivalent, 20–21 geometric, 648–649 quadratic, 86, 626 of simple interest, 143–145 Fractional expressions, 606 Fractions canceling in, 607 compound, 609–610 definition of, 592 fundamental property of, 606 raising to highest terms, 606 with real numbers, 592 reducing to lowest terms, 606–607 simple, 609, 613 Frequency, 408 class, 540–541 curve for, 543 distribution of, 540–542, 558, 564 relative, 408, 541 Frequency polygons, 543–544 Functions, 58–133 applications, 66–69 bounded, 107 constant, 63 cost, 93 definition of, 60–61, 61 discontinuous, 102 elementary, 73–85 equations and, 59–64 evaluation of, 65–66 exponential, 111–122 general notion of, 59 graph/graphing of, 61 inverse of, 123 linear, 63, 80–81, 544 logarithmic, 122–133 notation for, 64–66 objective, 286, 289, 313, 330 one-to-one, 123 polynomial, 100–102 price–demand, 66, 92 probability, 405, 415 profit, 95 quadratic, 85–100 rational, 100, 103–107 revenue, 92–93 root of, 86, 102 second-degree, 85 sharp-corner, 102 vertical-line test for, 63 zero of, 86, 102 Index I-3 Future value (FV), 118, 143–145, 165 See also amount of annuities, 163–171 f (x) notation definition of, 64–65 graph of, 89–90, 113 maximum value of, 89 G Games expected value of, 511–513 fair, 421, 453 mixed-strategy, 510–521 m * n, 527–532 nonstrictly determined, 507–508, 521 solutions to, 514 solving, 514 strategies for, 514 strictly determined, 504–510, 505–506, 508 two-finger Morra, 510–513 two-person zero-sum, 504, 510 * 3, 527–528 * 2, 521–527 value of, 514 Game theory, fundamental principle of, 505, 513 Gauss–Jordan elimination, 212–225, 253, 261 definition of, 210 linear systems of equations, solving by, 214–219 for reduced matrix, 212–214 using graphing calculator, 217 General problem-solving strategy, 179–180 General terms of sequence, 631–633 Geometric formulas, 648–649 Geometric sequence, 637–643 See also sum definition of, 637 nth-term formulas, 638–639 Geometric series finite, 165, 172, 640 infinite, 641 sum formulas for, 640–641 Graphing calculator, 31 for binomial distribution, 569 for displaying truth table, 366 empirical probability on, 410 factorials on, 390 Gauss–Jordan elimination using, 217 half-planes on, 273 identity element on, 238 linear regression on, 46 for Markov chains, 480, 493–494 matrix inverses on, 244 matrix on, 232, 245, 469, 494 row operations on, 207 for standard deviation, 560 systems of linear equations, for solving, 193 Graphs/graphing bar, 537–539 broken-line, 538–539 compound interest, 155 of data, 537–548 of elementary functions, 75 I-4 Index Graphs/graphing (continued) of equations, 29 of exponential equations, 128 of exponential functions, 112–113 of functions, 61 of f (x), 89–90, 113 horizontal translation of, 77, 79 line, 23, 30 of linear equalities, 272–276 of linear equations, 30 of linear programming problem, 286 pie, 539 of piecewise linear functions, 80–81 of polynomial functions, 101–102 of price–demand equations, 38 of price–supply equations, 38 of quadratic functions, 86, 88–92 of rational functions, 104–106, 113 reflections of, 78 sketching on, 59, 107 of systems of linear equalities, 273–274, 280–281 of systems of linear equations, 190–193 of systems of linear inequalities, 274–276 transformation of, 75–79 vertical shrink of, 78–79 vertical stretch of, 78–79 vertical translation of, 76–77, 79 Grouped data, 550–553 Grouping, factoring polynomials by, 601 Growth rate compound, 118 computing, 156 exponential, 115–116 relative, 115 Growth time, 154–157 H Half-life, 116 Half-planes, 272–273 Histograms, 541–543 for binomial distribution, 569 constructing, graphing calculator for, 542–543 definition of, 541–542 for frequency distributions, 542 for mean, 550 for median, 552–553 for mode, 554 for probability distributions, 449 Horizontal asymptotes of rational functions, 104–107, 113 Horizontal axis, 28–29 Horizontal bar graphs, 537 Hypothesis p and conclusion q, conditional propositions with, 363 Hypothetical distribution, 564 I Identity elements, 238, 590 Independent Bernoulli trials, 565 Independent events, 433–436 Independent systems of linear equations, 192 www.ebookslides.com Independent variables, 62 Index of radicals, 617 Indicators, 316 Individual retirement account (IRA), 168–169 Inequalities double, 22, 24 equivalent, 22 linear, 19, 21–24, 325 properties of, 22 sense of, 22 Inferential statistics, 541 Infinite geometric series, 641–642 Infinite sequence, 632 Infinite series, 633 Infinite sets, 371 Initial simplex tableau, 314–315, 343 Initial-state distribution matrix, 465 Initial-state probability matrix, 465 Initial system, 313–314 Input, 62 Input–output analysis, 258, 262–263 Integer exponents, 612–613 Integers See numbers Interchanging rows, 318 Interest See also compound interest on amortization, 175 compounding quarterly, 150–151 definition of, 117, 143 on investments, 145–147 simple, 143–150 Interest rate, 117 annuities, approximating future value of, 169 definition of, 143 on investments, 145–146 on a note, 145 true, 158 Interpolation, 46 Intersection of events, 415, 416–419, 430–431 of sets, 372–373, 377 union and, 415–419 Intervals class, 540 closed, 22 endpoints of, 22–23 open, 22 Invariant optimal strategies, 521–524 Inverses additive, 239, 590 of functions, 122–123 of M, 240, 244 matrix, 244–245 multiplicative, 239–240, 590 of square matrix, 238–249 Investments analysis of, 253–255, 528–531 annual percentage yield of, 158–159 growth time of, 156–157 interest on, 145–147 interest rate earned on, 145–146 present value of, 144 i-system, 303 L Large numbers, law of, 422, 560 Larger problems, 351–353 Law of averages, 422, 560 Law of large numbers, 422, 560 Leading coefficients, 101 Least common denominator (LCD), 20, 608 Left half-planes, 272 Leftmost variables, 218 Leontief input–output analysis, 258–266 three-industry model for, 262–264 two-industry model for, 259–261 Like terms, 595–596 Limiting matrix, 477, 488–493 Linear equalities systems of, 273–274, 279–285 in variables, 272–279 Linear equations, 19–21, 29–32, 325 Linear functions, 63, 80, 544 Linear inequalities, 19, 21–24, 325 Linearly related variables, 43 Linear programming, 286–298, 301–357 See also problems components of, 330 definition of, 286 feasible region for, 325 fundamental theorem of, 290, 305, 314 general description for, 289–290 geometrically interpreting, 320 introduction to, 302–312 for maximization problem, 313–328 for minimization problem, 329–341 and m * n games, simplex method and dual problem for, 527–532 simplex method for, 301–357 slack variables and, 303 summary of, 320–323 table method for, 302–312 and * games, 527–528 and * games, geometric approach to, 521–527 variables, basic and nonbasic, 308–310 Linear programming problem definition of, 286, 289 graphically solving, 286 mathematical model for, 286, 289 Linear regression, 42–54 on graphing calculator, 46 slope as rate of change and, 43–44 with spreadsheet, 48 Linear system See systems of linear equations Line graph, 23 Lines constant-profit, 286 equation of, 34–36 graphing intercepts for, 30 horizontal, 30–32, 36 real number, 589 regression, 45–46 slope of, 32 vertical, 30–32, 36 www.ebookslides.com Loans, 144, 177–178 Logarithmic equations, 126–127 Logarithmic functions, 122–133 with base 2, 123–124 definition of, 122, 124 domains of, 122–124 exponential function, conversion to, 124–125 inverse functions and, 122–123 properties of, 125–127 range of, 122–124 Logarithmic regression, 130 Logarithms, 127–129 Logic, 362–370 connectives and, 362–365 logical equivalences/implications, 366–368 propositions, 362–365 truth tables, 365–366 Logical equivalence, 367–368 Logical implication, 367 Logical reasoning, 362 Log to the base b of x, 124 Lower half-planes, 272 Lowest terms, reducing to, 606–607 M M, inverses of, 240, 244 See also singular matrix Markov chains, 463–499 absorbing, 485–499 definition of, 463, 466 introduction to, 464–466 matrices of, 467–468, 476–480 properties of, 464–475 regular, 475–485 state matrices and, 466–468 transitions matrices and, 466–469 Mathematical modeling, 67, 286, 289 Mathematics of finance, 142–183 See also finance Matrix (matrices) addition of, 226–227, 250 associative, 226 augmented, 204–207 coefficient, 204–205 column, 204 combined, 250 constant, 204–205 commutative, 226, 232 decoding, 246 definition of, 203 dimensions of, 204 elements of, 203–204 encoding, 246 equal, 226 equality, 250 first-state, 465 game, 504 on graphing calculator/calculating, 494 initial-state distribution, 465 initial-state probability, 465 Leontief input–output analysis, 258–266 limiting, 477 of Markov chains, 467–468, 476–477, 480 multiplication of, 227–234, 238, 250 m * n, 203–204, 504 negative of, 226 notation for, 204 operations of, 226–238 payoff, 504 principal diagonal, 204 product, 227–234 properties of, 250 reduced, 212–214 row, 204 singular, 240, 245 solving systems of linear equations, methods for, 203 square, 204, 238–239 state, 466–468, 470, 485–486 subtraction of, 226–227 technology, 260 transitioning, 466–470, 486 transposition of, 329 zero, 226–227, 232, 492 Matrix equations, 227, 250–251 systems of linear equations and, 250–258 Matrix games See games Matrix product, 229 Maximization problems, 286, 313–328 initial system for, 313–314 with mixed problem constraints, 342–357 pivot operation for, 315–320 problem constraints in, 335 simplex method for, 332–333 simplex tableau for, 314–315 in standard form, 302 Maximum value of f (x), 89 Mean, 570 See also averages arithmetic, 635 of binomial distribution, 570 definition of, 548–549 deviation from, 558 finding, 549, 554 of grouped data, 550–551 histogram for, 550 measures of central tendency and, 548–551 of ungrouped data, 549, 551 Measures of central tendency, 548–557 definition of, 548 mean and, 548–551 median and, 551–553 mode and, 553–555 Measures of dispersion, 548, 558–564 See also standard deviation range and, 558 Median definition of, 551 finding, 552, 554 for grouped data, 552–553 histogram for finding, 552–553 measures of central tendency and, 551–553 for ungrouped data, 552 Index I-5 Member of sets, 370 Minimization problems, 329–341 big M method for, 349–351 with mixed problem constraints, 342–357 solution of, 331–335 Mixed problem constraints, 342–357 See also big M method Mixed strategies, 511 See also mixed-strategy games Mixed-strategy games, 510–521 expected value of, 511–513 game theory, fundamental theorem of, 514 nonstrictly determined, 510–511 pure and, 511 recessive rows/columns of, 516–518 * 2, solution to, 514–516 Mode, 553–555 Model/modeling, mathematical, 42, 67, 286, 289 Modified problem, 346–347 Monomials, 595 Multiple optimal solutions, 291 Multiplication for linear equations, 325 for linear inequalities, 325 of matrix, 227–234, 238, 250 of polynomials, 597, 597–598 principles of, 379–381 of problem constraints, 334–335 of rational expressions, 607–608 of real numbers, 232, 589–590 square matrix, identity of, 238–239 Multiplicative inverse, 590 Mutually exclusive events, 416 m * n games, 527–532 m * n matrix, 203–204, 504 N Natural logarithms, 127 Natural number exponents, 592 n distinct objects taken r at a time, permutations of, 386–387 Negation, definition of, 362 Negative of matrix, 226 Negative real numbers, 589–590 n factorials, 385, 644 Nonnegative constraints, 287, 289, 313, 330, 343–342 Nonstrictly determined games, 507–508 mixed-strategy games, 510–511 positive payoffs of, 521 two-person zero-sum, 510 Normal curves See curves Notation double subscript, 204 for functions, 64–66 f (x), 64–65, 89–90, 113 inequality, 23 interval, 22–23 for matrix, 204 scientific, 613–614 for sets, 370–372 summation, 633–635 I-6 Index Not defined matrix product, 229 Not defined product matrix, 229 Note, interest rate earned on, 145 Not factorable polynomials, 603, 627 nth root, 616–617 nth-term formulas, 638–639 nth terms of sequence, 631–632, 638 Null sets, 370 Numbers See also real numbers completely factored, 600 of elements of sets, 373–374 factored form of, 600 natural, 592 prime, 600 Numerator, 592, 620 Numerical coefficient, 595 O Objective functions, 286, 289, 313, 330 Odds, 421–422 Ogive, 544 One-to-one functions, 123 Open intervals, 22 Operations canceling, 607 of matrices, 226–238 order of, 598 pivot, 315–320, 344 on polynomials, 594–600 row, 207 Optimal solution, 288–292 Optimal strategies, 505–506 definition of, 514 expected value for, 514 finding, 506–507 invariant, 521–524 Optimal values, 286, 289 Ordered pair, 29 Ordinary annuities, 163, 165–166 present value of, 173 Ordinate, 29 Origin, 29, 589 Outcomes compound, 402 of experiments, 403 (See also events) simple, 402–403 Output, 62 P Parabolas, 86, 89 Parallelogram, 648 Parameter, 196 Parentheses, 66, 596 Payment, 167–168, 175, 177–178 Payoff matrix, 504 Payoff values, 504 Percentages, 592 Perfect squares, 631 Permutations, 385–388 definition of, 385–386 of n distinct objects taken r at a time, 386–387 of sets, 386 www.ebookslides.com Piecewise linear functions, 80–81, 544 Pie graphs, 539 Pivot element, 317 Pivoting, 318 See also pivot operation Pivot operation, 315–320 Pivot row, 317 Player, column/row, 504 Plot/plotting, 45, 59–60 Point-by-point plotting, 59–60 Point-slope form, 35–36 Polygons, 543–544 Polynomials adding, 596–597 classifying, 595 combined, 598 combining like terms in, 595–596 definition of, 594 degree of, 595 equations for, 628 factoring, 600–605, 627 functions for, 100–102 multiplying, 597–598 natural number exponents and, 594 operations on, 594–600 regression of, 102–103 subtracting, 596–597 in variables, 594 Population, 559–561 Positive real numbers, 589, 590 Powers of transition matrix, 468–469 Predictions, 46 Preliminary simplex tableau, 343 Present value See also amortization; principle amortization and, 174–179 of annuities, 171–183, 177, 179–180 of investments, 144 of ordinary annuities, 173 Price–demand equations, 38 Price–demand functions, 66, 92 Prices diamond, 44–45 equilibrium, 38, 197 purchase, 24–25 Price–supply equations, 37–38 Prime numbers, 600 Principal diagonal matrix, 204 Principle, 118, 143–145 See also present value of addition, 377–379 counting, 377–384 finding, 155 Probabilities, 401–457 Bayes’ formula, 441–448 Bernoulli trials, of failure/success in, 565–567 conditional, 427–430, 437 for distributions, 541, 564, 576–577 empirical, 408, 410 equally likely assumption and, 409–411 of events, 405–408, 416–419, 441–443 of simple events, 406 Probability distributions, 536–584, 541, 564 See also data Bernoulli trials and, 564–573 binomial distributions and, 564–573 empirical, 564 expected value of, 453 histogram for, 449 measures of central tendency and, 548–557 measures of dispersion and, 558–564 normal, 574–584 of random variables, 449–450 random variables and, 448–450 Probability functions, 405, 415 Probability trees, 431–433, 443 Problem constraints, 287, 289, 313 coefficients of, 330 of dual problems, 334–335 in maximization problems, 335 multiplication of, 334–335 Problems dual, 329, 331, 334–335 linear programming, 286, 289 maximization, 286, 302, 332–333, 335 modified, 343, 346–347 word, 24 Product matrix, 228–234 definition of, 229 not defined, 229 of a number k and a matrix M, 227–228 Product rule, 430–431, 437 Profit, 66–67 Profit functions, 95 Profit–loss analysis, 66 Properties associative, 590 commutative, 590 distributive, 589–590, 595 equality, 19 of exponential functions, 114 of exponents, 612 of fractions, 606 of inequalities, 22 of logarithmic functions, 125–127 of Markov chains, 464–475, 477 of matrix, 250 of quadratic functions, 88–92 of radicals, 619–620 of real numbers, 33, 589–591 of sets, 370–372 zero, 591 Propositions, 362–365 compound, 364 conditional, 363, 368 contrapositive, 364–365, 368 converse, 364–365 definition of, 362 truth table for, 362–368, 366–368 types of, 366 Pure games, 511 Pure strategies, 511 Pythagorean Theorem, 648 www.ebookslides.com Q Quadrants, 29 Quadratic equations, 622–630 definition of, 622 factoring, solution by, 623–624 polynomial and, 628 quadratic formula for, 624–628 solving, 628 square root, solution by, 622–623 Quadratic formula, 86, 624–628 Quadratic functions, 85–100 analyzing, 91–92 definition of, 85 graph of, 86, 90–92 parabolas of, 86 properties of, 88–92 vertex form of, 88–92 Quantitative data, 540 R Radicals, 617 properties of, 619–620 rational exponents and, 616–622 Radicand, 617 Raising fractions to higher terms, 606 Random experiments, 402 Random variables continuous, 575 definition of, 448–449 discrete, 575 expective value of, 450–453 probability distribution of, 448–450 Ranges, 61, 558 of data, 540 for distribution of frequency, 558 of elementary functions, 75 of exponential functions, 112 of logarithmic functions, 122–124 for ungrouped data, 558 Rates, 118 of change, 37, 43–44 of descent, 43–44 effective, 158 per compounding period, 151 Rational exponents radicals and, 616–622 working with, 618–619 Rational expressions definition of, 606 operations on, 606–611 Rational functions, 103–107 asymptotes of, 104–107, 113 definition of, 100, 103 domain of, 103 graphs of, 104–106, 113 Rationalizing denominator, 620 Rationalizing numerator, 620 Real nth root, 617 Real number line, 589 Real numbers, 588–594 addition of, 589–590 associative properties of, 590 commutative properties of, 590 definition of, 588 distributive properties of, 589–590, 595 division of, 591 fractions with, 592 identity elements of, 590 multiplication of, 232, 589–590 negative, 589–591 nth root of, 616–617 positive, 589, 590 properties of, 33, 589–591 real number line for, 589 real roots of, 617 sets of, 588–590 subtraction of, 591 zero, 591 Real roots, 617 Reasonable probability assignment, 406, 408 Recessive column, 516–518 Recessive rows/columns, 516–518 Rectangle, 648 Rectangular coordinate system, 28–29 Recursive Markov chains, 468 Reduced matrix, 213 Reduced row echelon form (reduced form), 213 See also reduced matrix Reduced system, 214, 218 See also Gauss–Jordan elimination Reducing fractions to lowest terms, 606–607 Reflections, of graphs, 78 Regression analysis of, 44 exponential, 117 linear, 44–48 logarithmic, 130 of polynomials, 102–103 Regular Markov chains, 475–485 definition of, 476–477 properties of, 477 stationary matrix and, 475–476 Relative frequency, 408, 541 Relative growth rate, 115 Removing parentheses, 596 Representing sets, 371 Restriction of variables, 606 Revenues, 66–67, 92–93 Right half-planes, 272 Rise, 32 Roots cube, 616 of functions, 86, 102 nth, 616–617 real, 617 Row equivalent augmented matrix, 205–207 Row matrix, 204 Rows dominant, 516 interchanging, 318 operations, 207 Index pivot, 317 player, 504 recessive, 516–518 Run, 32 S Saddle values, 505–507 Sample space definition of, 403 events and, 402–405 finite, 450 Schedules of amortization, 175–179 commission, 146–147 Scientific notation, 613–614 Second-degree functions, 85 See also quadratic functions Second-degree polynomials, 602–603 Security level, 505 Sequence arithmetic, 637–643 of Bernoulli trials, 565 definition of, 631 finite, 632 geometric, 637–638 infinite, 632 terms of, 631–633 Series arithmetic, 639 definition of, 633 finite, 633 geometric, 640–641 infinite, 633 Sets, 370–376 complement of, 373 definition of, 370 disjoint, 377 element of, 370, 373–374, 377–380 empty, 370, 377 equal, 371 finite, 371 infinite, 371 intersection of, 372–373, 377 member of, 370 notations for, 370–372 null, 370 permutation of, 386 properties of, 370–372 of real numbers, 588–590 representing, 371 solutions to, 19, 190 subset of, 371 union of, 372, 377 universal, 372 Venn diagrams and, 372–374 Sharp-corner functions, 102 Simple events, 402–403, 406 Simple fractions, 609, 613 Simple interest, 143–150 definition of, 143 formula of, 143–145 investments and, 145–147 I-7 I-8 Index Simplex method See also linear programming algorithm for, 320 defined, 301–302 for maximization problems, 332–333 for * games, 527–528 variables for, 315 Simplex tableau initial, 314–315, 343 for maximization problems, 314–315 preliminary, 343 procedure for, 315 Simulation, empirical probability and, 410 Singular matrix, 240, 245 Sinking funds, 167–169 Slack variables, 303, 334, 345 Slope, 33–34 geometric interpretation of, 33 of line, 32–34 as rate of change, 43–44 Slope-intercept form, 34–36 Solution region for system of linear inequalities, 280, 282 Solutions basic, 304–305 big M method, summary of, 351 of equations, 19 feasible, 303, 305, 314 to games, 514 to initial system, 313–314 to i-system, 303 of linear equations, 29 for linear programming, 303–308 of minimization problem, 331–335 optimal, 288–292 to problems, summary of, 338 to sets, 19, 190 by square root, 622–623 to systems of linear equations, 192, 196 unique, 192 Solving of augmented matrix, 205–210 of double inequalities, 24 geometric method for, 290–292 larger problems, 351–353 logarithmic equations, 126–127 quadratic equations, 628 systems of linear equations, 190–203 (See also substitution in solving system of linear equations) Speed, 43–44 Sphere, 649 Spreadsheets for bar graphs, 539 for broken-line graphs, 539 input–output analysis in, 263 inverses matrix in, 244 linear regression with, 48 multiplication of matrix in, 234 www.ebookslides.com notation for matrix in, 204 for pie graphs, 539 singular matrix in, 245 Square matrix, 204, 238–239 inverse of, 238–249 multiplication, identity of, 238–239 Square root, solution by, 622–623 Squares, perfect, 631 Standard deviation, 570, 574 of binomial distribution, 570 concept of, 558–559 of data, 559 finding, 560 of grouped data, 561–562 population, 559–561 sample, 559, 561 significance of, 562 of ungrouped data, 558–561 Standard maximization problem in standard form, 302 State matrix, 466–468, 470, 485–487 Stationary Markov chains, 476–477, 480 Stationary matrix, 475–476 Statistics, 541 Stochastic process, 432, 464 Strategies, 505 definition of, 511 for games, 514 mixed, 511 optimal, 505–507, 514, 521–524 pure, 511 Strictly determined games, 504–510 two-person zero-sum games, 510 Subset of set, 371 Substitution in solving system of linear equations, 190–203 addition, by elimination of, 194–196 graphing calculator for, 193 methods for, 190, 203 solution set for, 190 Subtraction of matrices, 226–227 of polynomials, 596–597, 597 of rational expressions, 608–609 of real numbers, 591 Sum See also addition formulas for, 172, 639–642 of two matrices of the same size, 226 Summation, 633–635 Summing index, 633 Supply and demand, 37–39, 628–629 Surplus variables, 342, 345 Systems of linear equalities, 273–274, 280–281 Systems of linear equations augmented matrices and, 203–210 consistent, 192 defined, 190 dependent, 192 equivalent, 195 independent, 192 matrix equations and, 250–258 solutions to, 192, 196 substitution, 190–203 in variables, 190–203 Systems of linear inequalities, 274–276 T Table method for linear programming, 302–312, 306–308 definition of, 302, 304 procedure for, 304, 309–310 solutions, 303–308 Tables, 647–649 for binomial distribution, 569 corner point, 290 frequency, 540–541, 543 truth, 362–368 Tautology, 366, 367 Technology matrix, 260 Terms of sequence, 631–633 Test/testing ac, 602 for independent events, 435–436 vertical-line, 63 Theoretical distribution, 564 Three-industry model, input–output analysis, 262–264 Time doubling, 129–130 growth and, 154–157 Transition diagram, 464 Transition matrices, 486 Transition matrix, 466–470, 486 graphing calculator for, 469 of Markov chains, 466–469, 492 powers of, 468–469 probability of, 464 Transposition of matrix, 329 Trapezoid, 648 Tree diagram, 600 Triangle, 648 True interest rate, 158 See also annual percentage yield (APY) Truth table, 365–366 for compound propositions, 362–363, 365, 367–368 constructing, 365–366 definition of, 365 graphing calculator for displaying, 366 for propositions, 362–368 Two-finger Morra games, 510–511 expected value of, 511–513 Two-industry model, input–output analysis, 259–261 Two-person zero-sum games, 504, 510 * games, 527–528 * games, 521–527 solution to, 514–516 www.ebookslides.com U Unbounded solution regions, 282 Ungrouped data, 558–560 mean of, 549, 551 median for, 552 Union of events, 415–419 intersection and, 415–419 of sets, 372, 377 Unique solution to system of linear equations, 192 Universal set, 372 Upper half-planes, 272 V Vacuously true conditional propositions, 363 Values absolute, 75 expected, 451–453, 511–513, 570 future, 118, 143–145, 163–171 of games, 514 maximum, of f (x), 89 optimal, 286, 289 payoff, 504 saddle, 505–506 Variable costs, 67 Variables artificial, 342–343, 345 decision, 286, 289 dependent, 62 entering, 316 exiting, 316 independent, 62 leftmost, 218 linear equalities in, 272–279 linearly related, 43 polynomials in, 594 random, 448–451, 575 restriction of, 606 for simplex method, 315 slack, 303, 334, 345 surplus, 342, 345 in systems of linear equations, 190–203 two decision, 306–308 Variance/variation, 558–559 Venn diagram definition of, 372 elements of sets, for determining, 379 set operations and, 372–374 Vertex of parabolas, 89 of quadratic functions, 88–92 Index Vertical asymptotes of rational functions, 104–107 Vertical axis, 28–29 Vertical bar graphs, 537 Vertical-line test, 63 Vertical shrink of graphs, 78–79 Vertical stretch of graphs, 78–79 Vertical translation of graphs, 76–77, 79 W Weighted average, 550 X x axis, reflection in, 78 x coordinates, 29 Y y coordinates, 29 Z Zero factorials, 385, 644 Zero matrix, 226–227, 232, 492 Zero of a function, 86, 102 Zero properties of real numbers, 591 I-9 www.ebookslides.com Index of ApplIcAtIons Business & Economics Advertising, 120, 328, 355, 473, 535 Agriculture, 133, 140, 265–266, 323–325, 534, 583 Amortization, 175 Annuities, 170 Future values of, 165–166 Guaranteed rate of, 180–181 Interest rate of, 182 Present value of, 173 Bank promotion, 520, 527 Bonus incentives, 440 Break-even analysis, 25–26, 28, 56, 93–95, 98–99, 139, 201 Business closings, 395 Buying and selling commissions schedule, 163 Cable television, 52 Capital expansion, 297 Checkout times, 564 Coal, 265 Coffee blends, 202 Committee selection, 376, 396, 414 Communications, 384 Commute times, 546 Computers, 296 Control systems for, 436 Testing on, 380–381 Construction, 139–140 Consumer financing rate, 181–182 Consumer Price Index (CPI), 26, 57 Consumer survey/testing, 414, 430–431 Corporate farming, 520–521, 527 Corporate revenues, 545 Costs Analysis of, 40, 236 Average, 109–110 Equation for, 36–37 Hospital, 83 Material, 269 Minimum average, 110 Credit cards, 147, 179 Annual interest rate on, 150 Minimum payment due on, 149 Decision analysis, 456, 462 Delivery charges, 201–202 Depreciation, 40–41, 117 Double timing, 129, 132 DVD sales, 531–532 Earnings per share, 563 Economy, 266, 642, 643 Election campaigning, 461 Electricity Natural gas, 266 I-10 Oil, 266 Rates for, 83, 139 Electronics, 202, 300, 400 Employees Benefits for, 377–378 Evaluation of, 479 Layoffs for, 395 Rating of, 447 Screening of, 447 Training of, 107, 497–498, 502 Turnover of, 572 Energy, 49–50 Entrance test, 587 Equilibrium point, 132, 140 Equipment rental, 27 Exit polling, 374 Finance/financing, 98, 120, 178 Financial aid, 556 Furniture, 284–285, 296 Gasoline tax, 556 Gross domestic product, 545 Gross receipts, 600 Growth Compound, 120, 170, 188 Exponential, 115–116 in individual retirement account, 168–169 Internet, 121 Money, 120, 139 Guarantees, 572, 583 Home Building/construction of, 328 Equity in, 177, 182–183, 188 Insurance for, 473–474 Values of, 27 Housing trends, 474–475 Ice cream, 341 Incentive plan, 258 Income, 57 Federal, 545 Operating, 51 State tax on, 84 Taxable, 224 Individual retirement account (IRA), 27, 168–169, 171, 188 Input–output analysis, 270 Insurance, 426–427, 453, 456, 462, 467–468, 478–479, 483 Interest Compound, 120, 132, 162–163, 643 on loans, 181 Interest rate, 170, 186–187, 630 Annual, 149–150, 181, 187–188 of annuities, 182 Approximating, 169 Earned on Note, 145, 149 Graphical approximation techniques for, 171, 183 Internet, 121, 502 Inventory value, 236–237 Investment, 56, 132, 297, 327–328, 509, 520, 527, 599 Analysis on, 253–255, 270, 528–531 Annual percentage yield (APY) on, 158–159, 171 Interest rate earned on, 145–146 Present value of, 144 Strategy for, 356 Labor force, 482 Costs, 229, 233–234, 236, 269, 278 Relations with, 440, 583 Leases Airplane, 223–224 Tank car, 223 Loans, 144, 497 Distribution of, 356 Refund anticipation, 150 Repayment of, 643 Management, 383, 572–573 Manufacturing, 355, 360 Bicycle, 328 Resource allocation, 327 Sail, 300 Market/marketing, 498, 502 Analysis of, 461 Claims, 583 Research on, 378–379, 383–384, 399–400, 422–424, 426, 461, 578–580, 587 Shares in, 483 Markups, 40, 236 Mattresses, 279 Minimum wage, 384 Mining, 341 Natural gas rates, 81 Oil refining, 356 Optimization, 269 Packaging, 72 Parking meter coins, 27 Parking receipts, 257 Pensions, 498 Personnel Selection of, 396, 414, 440 Petroleum blending, 351–353 Population migration, 269 Preference survey, 587 Prices/pricing, 56–57 of car, 27 www.ebookslides.com Demand, 57, 67–69, 71, 83 of diamonds, 44–45 Earnings ratios for, 556 of gasoline, 545, 546, 594 Purchase, 24–25 Supply, 83 of tickets, 270 War of, 509 Product/production of automobiles, 98 of boats, 223 Defects in, 433, 445, 447–448 of gold, 545 Mix of, 379–380 Scheduling for, 223, 257, 296 Switching of, 502 Testing of, 427 Profit, 71–72 Profit–loss analysis, 99, 139 Public debt, 615–616 Purchasing, 220–221 Quality control, 395, 427, 440, 462, 564, 572, 580–582, 583 Railroad freight, 545 Rapid transit, 484 Real estate development, 488–490 Replacement time, 110 Resource allocation, 269 Retail sales, 27, 587 Retirement planning, 173–174 Revenue, 71, 98 Maximum, 92–93 Modeling of, 67–69 Salaries, 121, 546 Sales, 276–277, 583 Commission, 27–28, 228 Net, 51 Tax on, 592, 594 Ticket, 27 Sausage, 202 Scheduling, 473 Service contracts, 474 Shipping schedules, 360 Steel, 265 Stocks, 546, 564 Store location, 509–510 Supply and demand, 37–39, 41–42, 53–54, 132, 198–199, 201, 628–629, 630 Telephone expenditures, 53 Textiles, 279 Tire mileage, 97–98 Tour agency, 532 Tourism, 556 Traffic flow, 225 Trail mix, 356 Training, 474 Transportation, 297, 356, 383, 399, 482 Manufacturing of, 265 Problems with, 336–338 Vehicles, customized, 279 Viewer ratings, 520, 527 Water skies, 284–285, 296 Zero coupon bonds, 163, 187 Life Sciences Index of Applications Medication/medicine, 99, 140, 292–293, 384, 396, 414, 427, 498, 564, 583 Cardiogram test, 462 Sports, 56 Muscle contraction, 73 Nutrition, 224, 282–283, 285, 297, 547, 600 for animal, 328, 564 Human, 341, 355, 356 Outboard motors, 95–96, 99–100 Agriculture, 133, 140, 265–266, 323–325, 360, 534, 583 AIDS epidemic, 546–547 Animals Diet for, 202 Feed/feeding, 224, 297–298 Nutrition for, 328, 564 Archaeology, 133 Atmospheric pressure, 46 Patient recovery, 570–571 Physics, 42, 202 Plants Food for, 224, 285, 297, 341, 355 Safety for, 587 Pollution, 297, 616 Poultry, 300 Psychology, 28, 202–203, 285, 298, 441, 498–499, 584 Birthday problem, 420–421 Blood cholesterol levels, 557 Blood types, 376 Body surface area (BSA), 43 Boiling point, 41 Rate of descent, 49 Cancer screening, 448 Carbon-14 dating, 133 Cereal, 237 Decibels, 132–133 Dental insurance, 474 Diet, 110, 196–197, 258 Drugs, side effects of, 573 Earthquakes, 202 Ecology, 630 Epidemics, 573 Exponential decay, 116 Family planning, 384 Fertilizer, 202, 278–279, 297 Fish weight, 102–103 Flight conditions/navigation, 41 Food delivery, 587 Forestry, 47, 52, 57 Gene mutation, 484, 573 Genetics, 440, 456, 483, 502, 573, 584 Greenhouse gases, 547 Health, 56 Health plans, 474 Heredity, 237 Herpetology, 84 Life expectancy, 121–122, 224–225 Marine biology, 121, 140 Medical diagnosis, 573 Medical research, 396, 414, 448 Medicare, 140 I-11 Seed costs, 278 Smoking, 41, 50, 502 Social research, 400 Sound Intensity of, 132–133 Speed of, 49 Waves of, 202 Temperature, 28, 51–52, 57 Tuberculosis screening, 444, 448 Underwater pressure, 49 U.S Food and Drug Administration, 440 Weight Human, 84 Ideal, 49 Mouse, 546, 557 Wildlife management, 28 Social Sciences Code words, 381 Concert tickets, 257 Crime, 140, 616 Cryptography, 246–247, 249, 270 Demographics, 41 Divorce, 111 Education, 258, 547 Enrollments, 50, 52–53, 376, 492–493 Entrance examination scores, 557 Grade-point averages, 547–548, 557, 564 Grading on a curve, 584 High school dropout rates, 57 Resource allocation for, 341 Student retention, 469–470 Studying abroad, 547 Tests/testing, 237–238, 573, 584 I-12 Index of Applications Home ownership, 130, 483 Immigration, 462, 557 Learning, 84–85, 111, 120–121 Licensed drivers, 50–51 Lightbulb lifetime, 556 Marriage, 111 Median age, 547 Mixing feed, 356 Olympic games, 53 Opinion poll/survey, 328, 573 www.ebookslides.com Psychology, 56 Physiology, 110–111 Police science, 448 Politics, 73, 237, 384, 396, 414, 427, 448, 457, 584 Advertising of, 279 Filibuster in, 484 Population Density of, 616 Growths in, 121, 140, 224, 546 Shifts in, 484–485 World, 133 Pregnancy testing, 448 Presidents, 557 Reading scores, 564 Safety research, 84, 429–430, 630 Sociology, 225, 298, 573 State prisoners, 547 Voters/voting Coalition for, 376 Patterns of, 441 Preferences of, 462 www.ebookslides.com A Library of Elementary Functions BASIC FUNCTIONS f(x) g(x) h(x) 5 5 Ϫ5 x Ϫ5 x Ϫ5 Ϫ5 Ϫ5 Ϫ5 Identity function f(x) ϭ x Absolute value function g(x) ϭ ͉x͉ Square function h(x) ϭ x2 m(x) n(x) p(x) 5 5 Ϫ5 x Ϫ5 Ϫ5 x Ϫ5 Ϫ5 Cube function m(x) ϭ x3 x x Ϫ5 Square root function n(x) ϭ ͙x Cube root function p(x) ϭ ͙x L I N E A R A N D C O N S TA N T F U N C T I O N S f(x) f(x) f(x) b b b x mϾ0 Rising x x mϭ0 Horizontal mϽ0 Falling Linear function f(x) ϭ mx ϩ b Linear function f(x) ϭ mx ϩ b Constant function f(x) ϭ b Q U A D R AT I C F U N C T I O N S f(x) f(x) aϽ0 Opens downward k k x h h aϾ0 Opens upward f(x) ϭ ax2 ϩ bx ϩ c ϭ a(x Ϫ h)2 ϩ k x www.ebookslides.com EXPONENTIAL AND LOGARITHMIC FUNCTIONS f(x) f(x) f(x) x x x 0ϽbϽ1 bϾ1 Exponential function f(x) ϭ bx bϾ1 Exponential function f(x) ϭ bx Logarithmic function f(x) ϭ logb x R E P R E S E N TAT I V E P O L Y N O M I A L F U N C T I O N S ( D E G R E E Ͼ ) f(x) f(x) 40 f(x) 40 x Ϫ5 40 Ϫ5 Ϫ40 x Ϫ40 Third-degree polynomial f(x) ϭ x3 Ϫ x2 Ϫ 14x ϩ 11 x Ϫ5 Ϫ40 Fourth-degree polynomial f(x) ϭ x4 Ϫ 3x3 Ϫ 9x2 ϩ 23x ϩ Fifth-degree polynomial f(x) ϭ Ϫx5 Ϫ x4 ϩ 14x3 ϩ 6x2 Ϫ 45x Ϫ R E P R E S E N TAT I V E R AT I O N A L F U N C T I O N S f(x) f(x) f(x) 5 x Ϫ5 Ϫ5 Ϫ5 x Ϫ5 xϪ3 f(x) ϭ xϪ2 x Ϫ5 Ϫ5 f(x) ϭ x Ϫ4 f(x) ϭ x ϩ x G R A P H T R A N S F O R M AT I O N S y y y g f g f x Ϫ5 Ϫ5 x Ϫ5 h Ϫ5 Vertical shift g(x) ϭ f(x) ϩ h(x) ϭ f(x) Ϫ x h g f Ϫ5 h Horizontal shift g(x) ϭ f(x ϩ 3) h(x) ϭ f(x Ϫ 2) Ϫ5 Stretch, shrink and reflection g(x) ϭ 2f(x) h(x) ϭ Ϫ0.5f(x) ... college mathematics series The others are Calculus for Business, Economics, Life Sciences, and Social Sciences, and College Mathematics for Business, Economics, Life Sciences, and Social Sciences; ... Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences, 13th edition, ISBN... PreFAce The thirteenth edition of Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences is designed for a one-term course in finite mathematics for students who have had