college MAtheMAtIcs For BusIness, econoMIcs, lIFe scIences, And socIAl scIences thirteenth edition rAyMond A. BArnett Merritt college MIchAel r. ZIegler Marquette university KArl e. Byleen Marquette university 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 A01_BARN7668_13_GE_FM.indd 1 7/18/14 7:09 PM contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Diagnostic Prerequisite Test . . . . . . . . . . . . . . . . . . . . 19 Part 1 A LibrAry of ELEmEnTAry funcTions chapter 1 Linear Equations and Graphs . . . . . . . . . . . . . . . . 22 1.1 linear equations and Inequalities . . . . . . . . . . . . . . . . 23 1.2 graphs and lines. . . . . . . . . . . . . . . . . . . . . . . . 32 1.3 linear regression. . . . . . . . . . . . . . . . . . . . . . . . 46 chapter 1 summary and review . . . . . . . . . . . . . . . . . . . 58 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 59 chapter 2 functions and Graphs . . . . . . . . . . . . . . . . . . . 62 2.1 Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.2 elementary Functions: graphs and transformations . . . . . . . . 77 2.3 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . 89 2.4 Polynomial and rational Functions . . . . . . . . . . . . . . . 104 2.5 exponential Functions. . . . . . . . . . . . . . . . . . . . . 115 2.6 logarithmic Functions . . . . . . . . . . . . . . . . . . . . . 126 chapter 2 summary and review . . . . . . . . . . . . . . . . . . 137 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 140 Part 2 finiTE mAThEmATics chapter 3 mathematics of finance . . . . . . . . . . . . . . . . . . 146 3.1 simple Interest . . . . . . . . . . . . . . . . . . . . . . . . 147 3.2 compound and continuous compound Interest . . . . . . . . . 154 3.3 Future Value of an Annuity; sinking Funds . . . . . . . . . . . 167 3.4 Present Value of an Annuity; Amortization . . . . . . . . . . . 175 chapter 3 summary and review . . . . . . . . . . . . . . . . . . 187 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 189 chapter 4 systems of Linear Equations; matrices . . . . . . . . . . . . 193 4.1 review: systems of linear equations in two Variables . . . . . . 194 4.2 systems of linear equations and Augmented Matrices. . . . . . 207 4.3 gauss–Jordan elimination. . . . . . . . . . . . . . . . . . . 216 4.4 Matrices: Basic operations . . . . . . . . . . . . . . . . . . 230 4.5 Inverse of a square Matrix . . . . . . . . . . . . . . . . . . 242 4.6 Matrix equations and systems of linear equations. . . . . . . . 254 4.7 leontief Input–output Analysis. . . . . . . . . . . . . . . . . 262 chapter 4 summary and review . . . . . . . . . . . . . . . . . . 270 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 271 3 A01_BARN7668_13_GE_FM.indd 3 7/18/14 7:09 PM 4 conTEnTs chapter 5 Linear inequalities and Linear Programming . . . . . . . . . 275 5.1 linear Inequalities in two Variables . . . . . . . . . . . . . . 276 5.2 systems of linear Inequalities in two Variables . . . . . . . . . 283 5.3 linear Programming in two dimensions: A geometric Approach. 290 chapter 5 summary and review . . . . . . . . . . . . . . . . . . 302 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 303 chapter 6 Linear Programming: The simplex method . . . . . . . . . 305 6.1 the table Method: An Introduction to the simplex Method . . . . 306 6.2 the simplex Method: Maximization with Problem constraints of the Form … . . . . . 317 6.3 the dual Problem: Minimization with Problem constraints of the Form Ú . . . . . . 333 6.4 Maximization and Minimization with Mixed Problem constraints . . . . . . . . . . . . . . . . . . 346 chapter 6 summary and review . . . . . . . . . . . . . . . . . . 361 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 362 chapter 7 Logic, sets, and counting . . . . . . . . . . . . . . . . . 365 7.1 logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 7.2 sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 7.3 Basic counting Principles . . . . . . . . . . . . . . . . . . . 381 7.4 Permutations and combinations . . . . . . . . . . . . . . . . 389 chapter 7 summary and review . . . . . . . . . . . . . . . . . . 400 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 402 chapter 8 Probability . . . . . . . . . . . . . . . . . . . . . . . . . 405 8.1 sample spaces, events, and Probability . . . . . . . . . . . . 406 8.2 union, Intersection, and complement of events; odds. . . . . . 419 8.3 conditional Probability, Intersection, and Independence . . . . . 431 8.4 Bayes’ Formula. . . . . . . . . . . . . . . . . . . . . . . . 445 8.5 random Variable, Probability distribution, and expected Value. . 452 chapter 8 summary and review . . . . . . . . . . . . . . . . . . 461 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 463 chapter 9 markov chains . . . . . . . . . . . . . . . . . . . . . . 467 9.1 Properties of Markov chains. . . . . . . . . . . . . . . . . . 468 9.2 regular Markov chains . . . . . . . . . . . . . . . . . . . . 479 9.3 Absorbing Markov chains . . . . . . . . . . . . . . . . . . 489 chapter 9 summary and review . . . . . . . . . . . . . . . . . . 503 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 504 A01_BARN7668_13_GE_FM.indd 4 7/18/14 7:09 PM conTEnTs 5 Part 3 cALcuLus chapter 10 Limits and the Derivative . . . . . . . . . . . . . . . . . . 508 10.1 Introduction to limits. . . . . . . . . . . . . . . . . . . . . 509 10.2 Infinite limits and limits at Infinity. . . . . . . . . . . . . . . 523 10.3 continuity. . . . . . . . . . . . . . . . . . . . . . . . . . 535 10.4 the derivative. . . . . . . . . . . . . . . . . . . . . . . . 546 10.5 Basic differentiation Properties . . . . . . . . . . . . . . . . 561 10.6 differentials. . . . . . . . . . . . . . . . . . . . . . . . . 570 10.7 Marginal Analysis in Business and economics . . . . . . . . . 577 chapter 10 summary and review . . . . . . . . . . . . . . . . . 588 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 589 chapter 11 Additional Derivative Topics . . . . . . . . . . . . . . . . 594 11.1 the constant e and continuous compound Interest . . . . . . 595 11.2 derivatives of exponential and logarithmic Functions. . . . . . 601 11.3 derivatives of Products and Quotients. . . . . . . . . . . . . 610 11.4 the chain rule . . . . . . . . . . . . . . . . . . . . . . . 618 11.5 Implicit differentiation . . . . . . . . . . . . . . . . . . . . 628 11.6 related rates . . . . . . . . . . . . . . . . . . . . . . . . 634 11.7 elasticity of demand. . . . . . . . . . . . . . . . . . . . . 640 chapter 11 summary and review . . . . . . . . . . . . . . . . . 647 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 649 chapter 12 Graphing and optimization . . . . . . . . . . . . . . . . 651 12.1 First derivative and graphs . . . . . . . . . . . . . . . . . 652 12.2 second derivative and graphs. . . . . . . . . . . . . . . . 668 12.3 l’hôpital’s rule . . . . . . . . . . . . . . . . . . . . . . . 685 12.4 curve-sketching techniques . . . . . . . . . . . . . . . . . 694 12.5 Absolute Maxima and Minima . . . . . . . . . . . . . . . . 707 12.6 optimization . . . . . . . . . . . . . . . . . . . . . . . . 715 chapter 12 summary and review . . . . . . . . . . . . . . . . . 728 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 729 chapter 13 integration . . . . . . . . . . . . . . . . . . . . . . . . . 733 13.1 Antiderivatives and Indefinite Integrals . . . . . . . . . . . . 734 13.2 Integration by substitution . . . . . . . . . . . . . . . . . . 745 13.3 differential equations; growth and decay. . . . . . . . . . . 756 13.4 the definite Integral . . . . . . . . . . . . . . . . . . . . . 767 13.5 the Fundamental theorem of calculus . . . . . . . . . . . . 777 chapter 13 summary and review . . . . . . . . . . . . . . . . . 789 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 791 A01_BARN7668_13_GE_FM.indd 5 7/18/14 7:09 PM 6 conTEnTs chapter 14 Additional integration Topics . . . . . . . . . . . . . . . . 795 14.1 Area Between curves . . . . . . . . . . . . . . . . . . . . 796 14.2 Applications in Business and economics . . . . . . . . . . . 805 14.3 Integration by Parts . . . . . . . . . . . . . . . . . . . . . 817 14.4 other Integration Methods . . . . . . . . . . . . . . . . . . 823 chapter 14 summary and review . . . . . . . . . . . . . . . . . 834 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 835 chapter 15 multivariable calculus . . . . . . . . . . . . . . . . . . . 838 15.1 Functions of several Variables . . . . . . . . . . . . . . . . 839 15.2 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . 848 15.3 Maxima and Minima . . . . . . . . . . . . . . . . . . . . 857 15.4 Maxima and Minima using lagrange Multipliers . . . . . . . 865 15.5 Method of least squares. . . . . . . . . . . . . . . . . . . 874 15.6 double Integrals over rectangular regions . . . . . . . . . . 884 15.7 double Integrals over More general regions . . . . . . . . . 894 chapter 15 summary and review . . . . . . . . . . . . . . . . . 902 review exercises . . . . . . . . . . . . . . . . . . . . . . . . . 905 Appendix A basic Algebra review . . . . . . . . . . . . . . . . . . . 908 A.1 real numbers . . . . . . . . . . . . . . . . . . . . . . . . 908 A.2 operations on Polynomials . . . . . . . . . . . . . . . . . . 914 A.3 Factoring Polynomials. . . . . . . . . . . . . . . . . . . . . 920 A.4 operations on rational expressions . . . . . . . . . . . . . . 926 A.5 Integer exponents and scientific notation . . . . . . . . . . . 932 A.6 rational exponents and radicals . . . . . . . . . . . . . . . 936 A.7 Quadratic equations . . . . . . . . . . . . . . . . . . . . . 942 Appendix b special Topics . . . . . . . . . . . . . . . . . . . . . . . 951 B.1 sequences, series, and summation notation . . . . . . . . . . 951 B.2 Arithmetic and geometric sequences . . . . . . . . . . . . . 957 B.3 Binomial theorem . . . . . . . . . . . . . . . . . . . . . . 963 Appendix c Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . 967 Answers . . . . . . . . . . . . . . . . . . . . . . . . . .971 index . . . . . . . . . . . . . . . . . . . . . . . . . . 1027 index of Applications . . . . . . . . . . . . . . . . . . . 1038 Available separately: calculus Topics to Accompany calculus, 13e, and college mathematics, 13e chapter 1 Differential Equations 1.1 Basic concepts 1.2 separation of Variables 1.3 First-order linear differential equations chapter 1 review review exercises A01_BARN7668_13_GE_FM.indd 6 7/18/14 7:09 PM conTEnTs 7 chapter 2 Taylor Polynomials and infinite series 2.1 taylor Polynomials 2.2 taylor series 2.3 operations on taylor series 2.4 Approximations using taylor series chapter 2 review review exercises chapter 3 Probability and calculus 3.1 Improper Integrals 3.2 continuous random Variables 3.3 expected Value, standard deviation, and Median 3.4 special Probability distributions chapter 3 review review exercises Appendixes A and b (refer to back of College Mathematics for Business, Economics, Life Sciences, and Social Sciences, 13e) Appendix c Tables table III Area under the standard normal curve Appendix D special calculus Topic d.1 Interpolating Polynomials and divided differences Answers solutions to odd-numbered Exercises index Applications index A01_BARN7668_13_GE_FM.indd 7 7/18/14 7:09 PM PreFAce The thirteenth edition of College Mathematics for Business, Economics, Life Sciences, and Social Sciences is designed for a two-term (or condensed one-term) course in finite mathematics and calculus for students who have had one to two years of high school al- gebra 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 e ffective, 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 com- prehend 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 under- standing before moving on. ▶ Review material (Appendix A and Chapters 1 and 2) can be used judiciously to help remedy gaps in prerequisite knowledge. ▶ A Diagnostic Prerequisite Test prior to Chapter 1 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 mathemati- cal 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 of varying levels of difficulty. ▶ The MyMathLab course for this text is designed to help students help themselves and provide instructors with actionable information about their progress. The immedi- ate 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 econom- ics, 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. 8 A01_BARN7668_13_GE_FM.indd 8 7/18/14 7:09 PM PrEfAcE 9 Finally, the choice and independence of topics make the text readily adaptable to a variety of courses (see the chapter dependencies chart on page 13). This text is one of three books in the authors’ college mathematics series. The others are Finite Mathematics for Business, Economics, Life Sciences, and Social Sciences, and Calculus for Business, Economics, Life Sciences, and Social Sciences. 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, College 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 deficien- cies in prerequisite knowledge that cause students the most difficulty with finite mathematics and calculus. ▶ 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. ▶ Section 14.4 has been rewritten to cover the trapezoidal rule and Simpson’s rule. ▶ Examples and exercises have been given up-to-date contexts and data. ▶ Exposition has been simplified and clarified throughout the book. ▶ 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 17 and 18 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 pre- sented 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 d esign to A01_BARN7668_13_GE_FM.indd 9 7/18/14 7:09 PM 10 PrEfAcE help students stay focused on the mathematics and applications. Whether students start in 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 81, 128, and 422). examples and Matched Problems More than 490 completely worked examples are used to introduce concepts and to dem- onstrate 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 2 on page 24). Though some students may not need these additional steps, many will appreciate the fact that the authors do not assume too much in the way of prior knowledge. ExamplE 9 solving exponential equations Solve for x to four decimal places: (A) 10x = 2 (B) ex = 3 (C) 3x = 4 Solution (A) 10x = 2 Take common logarithms of both sides. log 10x = log 2 Property 3 x = log 2 Use a calculator. = 0.3010 To four decimal places (B) ex = 3 Take natural logarithms of both sides. ln ex = ln 3 Property 3 x = ln 3 Use a calculator. = 1.0986 To four decimal places (C) 3x = 4 Take either natural or common logarithms of both sides. (We choose common logarithms.) log 3x = log 4 Property 7 x log 3 = log 4 Solve for x. log 4 x = Use a calculator. log 3 = 1.2619 To four decimal places Matched Problem 9 Solve for x to four decimal places: (A) 10x = 7 (B) ex = 6 (C) 4x = 5 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. 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