College mathematics for business ecohomics life sciences and social sciences 13e

www.freebookslides.com www.freebookslides.com College Mathematics For BusIness, econoMIcs, lIFe scIences, And socIAl scIences thirteenth 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 A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM www.freebookslides.com Contents Preface Diagnostic Prerequisite Test 19 Part Chapter A Library of Elementary Functions Linear Equations and Graphs 22 1.1 Linear Equations and Inequalities 23 1.2 Graphs and Lines 32 1.3 Linear Regression 46 Chapter Summary and Review 58 Review Exercises 59 Chapter 2 Functions and Graphs 62 Functions 63 Elementary Functions: Graphs and Transformations 77 Quadratic Functions 89 Polynomial and Rational Functions 104 Exponential Functions 115 Logarithmic Functions 126 Chapter Summary and Review 137 Review Exercises 140 2.1 2.2 2.3 2.4 2.5 2.6 Part Finite Mathematics Chapter 3 Mathematics of Finance 146 Simple Interest 147 Compound and Continuous Compound Interest 154 Future Value of an Annuity; Sinking Funds 167 Present Value of an Annuity; Amortization 175 Chapter Summary and Review 187 Review Exercises 189 3.1 3.2 3.3 3.4 Chapter 4 Systems of Linear Equations; Matrices 193 Review: Systems of Linear Equations in Two Variables 194 Systems of Linear Equations and Augmented Matrices 207 Gauss–Jordan Elimination 216 Matrices: Basic Operations 230 Inverse of a Square Matrix 242 Matrix Equations and Systems of Linear Equations 254 Leontief Input–Output Analysis 262 Chapter Summary and Review 270 Review Exercises 271 4.1 4.2 4.3 4.4 4.5 4.6 4.7 A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM www.freebookslides.com Contents Chapter 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 Summary and Review 302 Review Exercises 303 Chapter 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 Summary and Review 361 Review Exercises 362 Chapter Logic, Sets, and Counting 365 Logic 366 Sets 374 Basic Counting Principles 381 Permutations and Combinations 389 Chapter Summary and Review 400 Review Exercises 402 7.1 7.2 7.3 7.4 Chapter Probability 405 Sample Spaces, Events, and Probability 406 Union, Intersection, and Complement of Events; Odds 419 Conditional Probability, Intersection, and Independence 431 Bayes’ Formula 445 Random Variable, Probability Distribution, and Expected Value 452 Chapter Summary and Review 461 Review Exercises 463 8.1 8.2 8.3 8.4 8.5 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 Summary and Review 503 Review Exercises 504 A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM www.freebookslides.com Part Chapter 10 Contents Calculus Limits and the Derivative 508 Introduction to Limits 509 Infinite Limits and Limits at Infinity 523 Continuity 535 The Derivative 546 Basic Differentiation Properties 561 Differentials 570 Marginal Analysis in Business and Economics 577 Chapter 10 Summary and Review 588 Review Exercises 589 10.1 10.2 10.3 10.4 10.5 10.6 10.7 Chapter 11 Additional Derivative Topics 594 The Constant e and Continuous Compound Interest 595 Derivatives of Exponential and Logarithmic Functions 601 Derivatives of Products and Quotients 610 The Chain Rule 618 Implicit Differentiation 628 Related Rates 634 Elasticity of Demand 640 Chapter 11 Summary and Review 647 Review Exercises 649 11.1 11.2 11.3 11.4 11.5 11.6 11.7 Chapter 12 Graphing and Optimization 651 First Derivative and Graphs 652 Second Derivative and Graphs 668 L’Hôpital’s Rule 685 Curve-Sketching Techniques 694 Absolute Maxima and Minima 707 Optimization 715 Chapter 12 Summary and Review 728 Review Exercises 729 12.1 12.2 12.3 12.4 12.5 12.6 Chapter 13 Integration 733 Antiderivatives and Indefinite Integrals 734 Integration by Substitution 745 Differential Equations; Growth and Decay 756 The Definite Integral 767 The Fundamental Theorem of Calculus 777 Chapter 13 Summary and Review 789 Review Exercises 791 13.1 13.2 13.3 13.4 13.5 A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM www.freebookslides.com Contents Chapter 14 Additional Integration Topics 795 Area Between Curves 796 Applications in Business and Economics 805 Integration by Parts 817 Other Integration Methods 823 Chapter 14 Summary and Review 834 Review Exercises 835 14.1 14.2 14.3 14.4 Chapter 15 Multivariable Calculus 838 Functions of Several Variables 839 Partial Derivatives 848 Maxima and Minima 857 Maxima and Minima Using Lagrange Multipliers 865 Method of Least Squares 874 Double Integrals over Rectangular Regions 884 Double Integrals over More General Regions 894 Chapter 15 Summary and Review 902 Review Exercises 905 15.1 15.2 15.3 15.4 15.5 15.6 15.7 Appendix A Basic Algebra Review 908 A.1 A.2 A.3 A.4 A.5 A.6 A.7 Real Numbers 908 Operations on Polynomials 914 Factoring Polynomials 920 Operations on Rational Expressions 926 Integer Exponents and Scientific Notation 932 Rational Exponents and Radicals 936 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 A01_BARN7668_13_GE_FM.indd Differential Equations 1.1 Basic Concepts 1.2 Separation of Variables 1.3 First-Order Linear Differential Equations Chapter Review Review Exercises 7/18/14 7:09 PM www.freebookslides.com Chapter Taylor Polynomials and Infinite Series Chapter Probability and Calculus Contents 2.1 Taylor Polynomials 2.2 Taylor Series 2.3 Operations on Taylor Series 2.4 Approximations Using Taylor Series Chapter Review Review Exercises 3.1 Improper Integrals 3.2 Continuous Random Variables 3.3 Expected Value, Standard Deviation, and Median 3.4 Special Probability Distributions Chapter 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 Appendix D Special Calculus Topic Table III Area Under the Standard Normal Curve D.1 Interpolating Polynomials and Divided Differences Answers Solutions to Odd-Numbered Exercises Index Applications Index A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM www.freebookslides.com 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 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 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 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 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 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 A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM www.freebookslides.com Preface Finally, the choice and independence of topics make the text readily adaptable to a ­ ariety of courses (see the chapter dependencies chart on page 13) This text is one of v 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 deficiencies 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 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 d­ esign to A01_BARN7668_13_GE_FM.indd 7/18/14 7:09 PM 10 www.freebookslides.com 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 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 24) 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 Solve for x to four decimal places: (A) 10x = 2     (B) ex = 3     (C) 3x = Solution   ( A) 10x = log 10x = log x = log = 0.3010  x (B) e = ln ex = ln x = ln = 1.0986  x (C) = x log = 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 Solve for x to four decimal places: (A)  10x = (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 A01_BARN7668_13_GE_FM.indd 10 7/18/14 7:09 PM Chain rule, 618–625 composite functions and, 618–619 definition of, 623 general derivative rules and, 624–625 general power rule and, 619–622 partial derivatives using, 849–850 reversing, 745–747 using, 623–624 Chains See Markov chains Change-of-base formulas, 133 Change-of-variable method, 749 Circle, 967 Closed intervals, 26 absolute maxima and minima on, finding, 709 continuous on, 538–539 Cobb–Douglas production function, 841–842, 868 Coefficient matrix, 208–209 Coefficients concept of, 915 leading, 105 numerical, 915 of objective functions, 334 of problem constraints, 334 Column matrix, 208 Combinations, 392–395 of n distinct objects taken r at a time, 392–394 Combined factoring polynomials, techniques for, 924–925 Combined matrix, 254 Combined polynomials, 918 Combining like terms, 915–916 Commission schedule, 150–151 Common differences, in arithmetic sequence, 957 Common logarithms, 131 Common ratio, in geometric sequence, 957 Commutative properties, 910 Complement of event, 423–425 of sets, 377 Completely factored numbers, 920 Completing the square, 92–93, 945 Compliment of events, 423 Composite functions, 618–619 Compound events, 406–407 Compound fractions, 929–930 Compound growth rate, 122 Compounding quarterly interest, 154–155 Compound interest, 121–123, 154–157 annual percentage yield, 161–164 continuous, 157–158, 595–598, 759 daily, 158 definition of, 122, 154 graphing, 159 growth of, time and, 158–161 Compound propositions, 368 truth table for, 366–367, 369, 371–372 Concavity definition of, 669 downward, 669–670 graphing, 670–671 for second derivative, graphing of, 668–671 upward, 669–670 www.freebookslides.com Conceptual Insight, 27, 37, 70, 79, 90, 93, 109, 117, 128, 149, 151, 156, 157, 158, 160, 164, 169, 176, 180, 236, 238, 257, 264, 278, 284, 296, 298, 322, 329, 338–339, 367, 375, 383, 395, 408, 421, 439, 449, 457, 470, 484, 496, 514, 517, 526, 530, 537, 551, 562, 570, 579, 595, 597, 604, 608, 611, 619, 622, 631, 635, 641, 642, 654, 656, 670, 673–674, 677, 693, 712, 718, 720, 722, 734, 758, 770, 777, 779, 796, 806–807, 819, 827, 845, 853, 858, 872, 878, 891, 894–895, 896 Conditional probability, 431–434, 441 concept of, 431 definition of, 431–432 events for, 434–435, 437–440 probability trees for, 435–437 product rule for, 434–435 summary of, 441 Conditional propositions, 367, 372 Cone, 968 Conjunction, 367 Connectives, 366–369 Consistent systems of linear equations, 196 Constant e, 595 See also Continuous compound interest Constant functions, 67 continuity properties of, 539 rules of, 561–562 Constant of integration, 735 Constant matrix, 208–209 D Constant multiple property, 564 Constant-profit line, 290 Constant rate of change, 47 Consumer Price Index (CPI), 30 Consumers’ surplus, 811–814 Contingency, 370 Continuity, 535–546 definition of, 536 of functions, 536–538 inequalities and, 540–542 properties of, 539–542 Continuous on closed intervals, 538–539 Continuous compound interest, 595–598 computing, 597 differential equations and, 759 double time and, 598 formula for, 596 graphing, 597 growth time and, 598 models for, 607 Continuous functions, 536–538, 655 average value of, 784 Continuous on half-closed intervals, 539 Continuous income stream, 807–809 future value of, 809–811 Continuous on the left functions, 538 Continuous on the right functions, 538 Continuous stream, 808 Contradiction, 370 Contrapositive propositions, 368–369, 372 Converse propositions, 368–369 Coordinate axis, 33 Coordinates, 33, 307 Coordinate systems rectangular, 842 three-dimensional, 842–845 Corner points, 284, 294, 314 Cost functions, 580, 741, 840 Costs, 40–41, 70–71, 97 average, 701–702 marginal, 577–579, 778–779 marginal average, 582 Counting principles, 381–388 addition principle, 381–383 multiplication principle, 383–385 Counting technique, 381 Critical numbers, 654–656 in domain, 654 local extrema and, 658 Critical points, 858, 866 local extrema and, 859–860 Cross sections, graphing of, 844–845 Cube root, 936 Cumulative matrix, 230, 236 Curve fitting, 48 Curves, area between two, 796–805 Curve-sketching techniques, 694–707 See also Graphing strategy average cost and, 701–702 definition of, 675 for second derivative, graphing of, 675–678 Cylinder, 968 Daily compound interest, 158 Debt, amortizing, 178–179 Decision variables, 290, 293 Decoding matrix, 250 Decreasing functions, 652–656 Definite integral, 767–777 approximation of areas by left and right sums and, 767–770 average value and, 783–785 definition of, 772 evaluating, 779–783 as limit of sums, 770–772 properties of, 772–773 recognizing, 783–785 substitution and, 780–781 Degree of polynomials, 915 Demand, elasticity of, 640–644 Denominators, 912 least common, 24 rationalizing, 940 Dependent events, 438–439 Dependent systems of linear equations, 196 Dependent variables, 66, 839 Derivatives, 546–561 chain rule for, 618–625 concept of, 546 constant e and, 595 continuous compound interest and, 595–598 definition of, 552 differentials and, 570–577 differentiation properties and, 561–570 elasticity of demand and, 640–644 of ex, 601–602 1028 Z06_BARN7668_13_GE_SIDX.indd 1028 09/07/14 8:03 PM of exponential functions, 605 first, 652–667 four-step process for finding, 552–555 general rules for, 624–625 implicit differentiation and, 628–632 interpretations of, 552 of logarithmic functions, 605, 640 marginal analysis and, 577–587 nonexistence of, 556–557 notation for, 561 partial, 848–856 of products, 610–612 of quotients, 612–615 rate of change and, 546–549 related rates and, 634–637 second, 668–685 slope of tangent line and, 549–552 Diameter of a tree at breast height (Dbh), 51–52 E Difference quotients, 70, 518, 547 Differential equations, 756–766 archaeology and, 761–762 continuous compound interest and, 759 definition of, 756 exponential growth law and, 759–760 exponential growth phenomena, comparison of, 762–763 general solution of, 758 learning rate of improvement and, 762 particular solution of, 758 population growth and, 760–761 slope fields and, 757–758 Differentials, 570–577 approximations using, 573–575 definition of, 572, 748 increments and, 570–572 Differentiation See also Differentiation properties of functions, 552, 564 implicit, 628–632 of products, 610–611 of quotients, 613–614 residual, 875 Differentiation properties, 561–570 constant function rule and, 561–562 constant multiple property and, 564 power rule and, 562–563 sum and difference properties and, 565–567 Discontinuous functions, 106, 536–537 Discriminate, 946–948 Disjoint events, 420 Disjoint sets, 381 Disjunction, 367 Distributive properties, 909–910, 915 Division of rational expressions, 927–928 Domains, 65–66 of composite functions, 618 critical numbers in, 654 of elementary functions, 79 of exponential functions, 116–117 finding, 68–69 of functions, 839 of logarithmic functions, 126–128 of polynomial functions, 105 of rational functions, 107 www.freebookslides.com Double inequalities, 26, 28 Double integrals definition of, 887–889 evaluating, 898–899 of exponential function, 888–889 over rectangular regions, 884–894 over regular regions, 894–902 volume and, 890–891, 900–901 Double subscript notation, 208 Double time, 598 Downward concavity, 669–670 Dual problems, 333–345 definition of, 333 formation of, 333–335 fundamental principle of, 335 problem constraints of, 338–339 Effective rate, 162 See also Annual percentage yield (APY) Elastic demand, 643 See also Elasticity of demand Elasticity of demand, 640–644 definition of, 642 at price, 642–643 relative rate of change and, 640–641 revenue and, 644 Elementary functions, 77–89 beginning library of, 78–79 domain of, 79 evaluating, 78 graph of, 79 horizontal shifts, 79–81 range of, 79 reflections, 81–85 shrinks, 81–85 stretches, 81–85 vertical shifts, 79–81 Elements in events, 421 identity, 910 of matrix, 207–208 pivot, 321 of sets, 374, 377–378, 381, 383–384 Elimination by addition, 198–200 Empirical probability, 412, 414 applications to, 426–428 approximate, 412 on graphing calculator, 414 simulation and, 414 Empty sets, 374, 381 Encoding matrix, 250 End behavior, 528–529 Endpoints of intervals, 26–27 Endpoint solution, 722 Entering variables, 320 Equality, absolute, 800 Equality matrix, 254 Equally likely assumption, 413–415 Equal matrix, 230 Equal sets, 375 Equations See also specific types of continuity of functions by, 538 cost, 40–41 differential, 756–766 equivalent, 23, 24–25 exponential, 132 first-order, 757 functions specified by, 63–68 graph of, 33 linear, 23–24, 329 of lines, 38–40 logarithmic, 130–131 matrix, 231, 254–255 price–demand, 42, 579–580 price–supply, 41–42 quadratic, 942–950 second-order, 757 solution of, 23 Equilibrium point, 42, 948 Equilibrium price, 813 Equilibrium quantity, 42, 201, 813 Equity, 181 Equivalent equations, 23–25 Equivalent formulas, 24–25 Equivalent inequalities, 26 Equivalent systems of linear equations, 199 Error in approximation, 768 Error bound, 768–769 e-system, 307 Events, 407 arbitrary, 413–414 compliment of, 423 compound, 406–407 dependent, 438–439 independent, 438–441 intersection of, 419, 420–423 mutually exclusive, 420 odds for, 425–426 probability of, 409–412, 420–423, 445–447 sample spaces and, 406–409 simple, 406–407, 410 union of, 419–423 ex, derivative of, 601–602 Exiting variables, 320 Expanded coordinates, 307 Expected value, 455–456 decision making and, 458 of games, 456 insurance and, 457 of probability distributions, 457 of random variables, 454–457 Experiments, 406–407 Explicit rule for evaluating functions, 628 Explore and Discuss, 24, 25, 27, 33, 38, 47, 50, 64, 65, 80, 82, 83, 90, 92, 111, 119, 122, 127, 133, 148, 161–162, 163, 170–171, 173, 179, 182, 196, 198, 211, 213, 222, 225, 236, 243, 248, 249, 259, 266, 293, 296, 306, 311, 313, 326, 334, 336–337, 357, 370, 372, 376, 378, 383, 395, 415, 423, 425, 436, 440, 447, 455, 468, 474, 480, 490, 491, 514, 519, 523, 536, 554, 556, 562, 572, 583, 598, 601, 606, 610, 612, 619, 623, 629, 635, 640, 641, 652, 668, 692, 699, 737, 758, 768, 784, 802, 818, 821, 843, 858, 870, 878, 879, 890, 897, 900 Exponential decay, 760 Exponential equations, 132 1029 Z06_BARN7668_13_GE_SIDX.indd 1029 09/07/14 8:03 PM Exponential functions, 115–126 See also Logarithmic functions base of, 116 base e, 118–119 bx with base b as, 601 compound interest, 121–123 defined, 115–116 derivatives of, 605 double integrals of, 888–889 domain of, 116–117 ex as, 601–602 formula for, 601, 605 four-step process for, 601–602 graphs of, 116–117, 597 inverse of, 603 (See also Logarithmic functions) limits involving, 689 logarithmic functions, conversion to, 128–129 models of, 606–608 natural, 604 other, 604–606 power rule for, 602 properties of, 118 range of, 116 slope of, 597 Exponential growth, 763 law of, 759–760 phenomena of, comparison of, 762–763 Exponential growth rate, 119–120 Exponential regression, 121, 608 Exponents first property of, 912 integer, 932–933 natural number, 912 properties of, 932 radicals, properties of, 939–940 rational, 936–942 scientific notation and, 933–934 simplifying, 932–933 Extrapolation, 50 Extrema See also Absolute extrema; Local extrema second derivative and, graphing of, 710–713 Extreme value theorem, 708 F Factorability, theorem of, 947 Factored polynomials, 920 Factored form of numbers, 920 Factorials, 388–389, 394, 964 Factoring, 947–948 quadratic equations, solution by, 943–944 quadratic formula and, 947–948 Factoring polynomials, 920–925 combined, techniques for, 924–925 common factors, 920–921 by grouping, 921 second-degree polynomials, factoring, 922–923 special formulas for, 923–924 Fair games, 425, 457 False negative results, 449 False positive results, 449 Feasible region, 284, 293, 296, 308 corner points of, 314 for linear programming, 329 Feasible solution, 307, 317–318 www.freebookslides.com Finance, 146–187 annuities, 167–187 compound interest, 154–167 mathematics of, 146–187 simple interest, 147–154 Finite arithmetic series, 959–960 Finite geometric series, 169, 176 Finite sample space, 454 Finite sequence, 952 Finite series, 953 Finite sets, 375 First derivative, graphing of, 652–667 increasing and decreasing functions and, 652–656 local extrema and, 656–660 First-derivative test, 658–660 First-order equations, 757 First-order partial derivatives, 851 First-state matrix, 469 Fixed costs, 41, 70 Formulas for binomials, 963–964 for continuous compound interest, 596 for exponential functions, 601, 605 geometric, 967–968 for indefinite integrals, 736–737, 747, 749 for integration by parts, 817–819 for logarithmic functions, 601, 605 quadratic, 946 reduction, 829 for slope of line, 777 Formulas See also specific types of for amortization, 179, 181–182, 184 Bayes’, 445–452 change-of-base, 133 equivalent, 24–25 quadratic, 90 of simple interest, 147–149 Fractional expressions, 926 Fractions canceling in, 927 compound, 929–930 definition of, 912 fundamental property of, 926 raising to highest terms, 926 with real numbers, 912 reducing to lowest terms, 926–927 simple, 929, 933 Frequency, 412 relative, 412 Functions, 62–137 See also specific types of absolute value, 511 applications, 70–73 average value of, 785 bounded, 111 Cobb–Douglas production, 841–842, 868 composite, 618–619 constant, 67, 539, 561–562 continuous, 536–538, 655, 784 cost, 97, 580, 741, 840 decreasing, 652–656 definition of, 64–65 differentiation of, 552, 564 discontinuous, 106, 536–537 domain of, 839 elementary, 77–89 end behavior of, 528 equations and, 63–68 evaluation of, 69–70 exponential, 115–126 general notion of, 63 graph/graphing of, 65 increasing, 652–656 of independent variables, 839–840, 865–872 inverse of, 127 limits and, 509–510 linear, 67, 84–85 logarithmic, 126–137 of multiple variables, 839–848 with no absolute extrema, 710 nondifferentiable, 556–557 notation for, 68–70 objective, 290, 293, 317, 334 one-to-one, 127 polynomial, 104–106, 515, 527–528, 539, 660 price–demand, 70, 96 probability, 409, 419 probability density, 805–807 profit, 99, 581, 840 quadratic, 89–104 range of, 839 rational, 104, 107–111, 515, 524–526, 529–530, 539 revenue, 96–97, 580, 840 root of, 90, 106 second-degree, 89 sharp-corner, 106 special notation for, 628 values of, 509–510 vertical-line test for, 67 zero of, 90, 106 Fundamental theorem of calculus, 777–789 definite integrals and, 779–785 Future value (FV), 122, 147–149, 169, 809–811 See also Amount of annuities, 167–175 f 1x2 notation definition of, 68–69 graph of, 93–94, 117 maximum value of, 93 G Games fair, 425, 457 Gauss-Jordan elimination, 216–229, 257, 265 definition of, 214 linear systems of equations, solving by, 218–223 for reduced matrix, 216–218 using graphing calculator, 221 General derivative rules, 624–625 General power rule, 618–622 General problem-solving strategy, 183–184 General regions, double integrals over more, 894–902 General terms of sequence, 951–953 Geometric formulas, 967–968 Geometric sequence, 957–963 definition of, 957 nth-term formulas, 958–959 1030 Z06_BARN7668_13_GE_SIDX.indd 1030 09/07/14 8:03 PM www.freebookslides.com Geometric series sketching on, 63, 111 finite, 169, 176, 960 slope of, 551 infinite, 961 of systems of linear equalities, 277–278, sum formulas for, 960–961 284–285 Gini index, 800–801 of systems of linear equations, 194–197 Graphing See also Graphs of systems of linear inequalities, 278–280 of absolute maxima and minima, 707–715 transformation of, 79–83 of concavity, 670–671 vertical shrink of, 82–83 of continuity of functions, 536–538 vertical stretch of, 82–83 of continuous compound interest, 597 vertical translation of, 80–81, 83 of cross sections, 844–845 Grouping, factoring polynomials by, 921 curve-sketching techniques for, 694–707 Growth rate of exponential functions, 597 compound, 122 of first derivative, 652–667 computing, 160 of inflection points, 674–675 exponential, 119–120 of investment growth, 597 relative, 119 L’Hôpital’s rule for, 685–694 Growth time, 158–161, 598 of limits, 509–514 of local extrema, 657 H Half-closed intervals, continuous on, 539 of optimization problems, 715–727 Half-life, 120 of second derivative, 668–685 Half-planes, 276–277 of triplet of numbers, 845 Histograms Graphing calculator, 35 for probability distributions, 453 for displaying truth table, 370 Horizontal asymptotes empirical probability on, 414 limits at infinity and, 527, 529–531 exponential regression on, 608 of rational functions, 529–531 factorials on, 394 Horizontal asymptotes of rational functions, Gauss-Jordan elimination using, 221 108–111, 117 half-planes on, 277 Horizontal axis, 32–33 identity element on, 242 Hypothesis p and conclusion q, conditional integration on, 782–783 propositions with, 367 linear regression on, 50 I Identity elements, 242, 910 logarithmic regression on, 608 for Markov chains, 484, 497–498 Implicit differentiation, 628–632 matrix on, 236, 249, 473, 498 definition of, 628–629 matrix inverses on, 248 special function notation and, 628 row operations on, 211 Implicit rule for evaluating functions, 628 systems of linear equations, for solving, 197 Increasing functions, 652–656 Graphing strategy Increments, 570–572 asymptotes and, 694–695 Indefinite integrals, 735–740 modifying, 694–695 cost function and, 741 procedure for, 675–678 curves in, 740 using, 695–701 definition of, 735, 737 Graphs See also Graphing formula for, 736–737, 747, 749 compound interest, 159 of products, 740 of elementary functions, 79 properties of, 737–740 of equations, 33 Independent events, 437–440 of exponential equations, 132 Independent systems of linear equations, 196 of exponential functions, 116–117 Independent variables, 66 of functions, 65 definition of, 839 of f(x), 93–94, 117 functions of, 839–840, 865–872 horizontal translation of, 81, 83 Indeterminate form line, 27, 34 0/0, 687–690 linear, 597 infinity/infinity, 691–693 of linear equalities, 276–280 limits of, 517–518 of linear equations, 34 Index of radicals, 937 of linear programming problem, 290 Indicators, 320 of piecewise linear functions, 84–85 Individual retirement account (IRA), 172–173 of polynomial functions, 105–106 Inelastic demand, 643 of price–demand equations, 42 Inequalities of price–supply equations, 42 absolute, 800 of quadratic functions, 90, 92–96 continuity and, 540–542 of rational functions, 108–110, 117 double, 26, 28 reflections of, 82 equivalent, 26 linear, 23, 25–28, 329 properties of, 26 sense of, 26 Infinite geometric series, 961–962 Infinite limits, 523–526 definition of, 523 vertical asymptotes and, 524–526 Infinite sequence, 952 Infinite series, 953 Infinite sets, 375 Infinity, limits at, 526–531, 690–691 Inflection points, 671–674 definition of, 671 graphing, 674–675 locating, 672–673 Initial condition, 762 Initial simplex tableau, 318–319, 347 Initial-state distribution matrix, 469 Initial-state probability matrix, 469 Initial system, 317–318 Input, 66 Input–output analysis, 262, 266–267 Instantaneous rate of change, 546–547 Instantaneous velocity, 546, 549 Integer exponents, 932–933 Integers See numbers Integrals table of, 823, 827–829 definite, 767–777, 779–785 double, 884–902 evaluating, 886–887 formulas for, 968–970 indefinite, 735–740 iterated, 887 Integral sign, 735 Integrand, 735, 772, 887 Integration antiderivatives and, 734–735 area, for computing, 799–800 area between curves and, 796–805 in business and economic applications, 805–816 constant of, 735 in consumers’ and producers’ surplus, 811–814 in continuous income stream, 807–811 definite integral and, 767–777 differential equations and, 756–766 formulas, 968–970 fundamental theorem of calculus and, 777–789 on graphing calculator, 782–783 indefinite integrals and, 735–740 table of integrals and, 827–829 lower limit of, 772 other methods of, 823–833 by parts, 817–823 in probability density functions, 805–807 reduction formulas and, 829 region of, 887 reversing order of, 899–900 Simpson’s rule and, 825–827 by substitution, 745–756 trapezoidal rule and, 824–825 upper limit of, 772 using table of integrals, 827–828 1031 Z06_BARN7668_13_GE_SIDX.indd 1031 09/07/14 8:03 PM Intercepts of polynomial functions, 660 Interchanging rows, 322 Interest See also Compound interest on amortization, 179 compound, 595–598, 759 compounding quarterly, 154–155 definition of, 121, 147 on investments, 149–151 simple, 147–154, 596 Interest rate, 121 annuities, approximating future value of, 173 definition of, 147 on investments, 149–150 on a note, 149 true, 162 Interpolation, 50 Intersection of events, 419, 420–423, 434–435 of sets, 376–377, 381 union and, 419–423 Intervals closed, 26, 538–539, 709 endpoints of, 26–27 half-closed, 539 open, 26, 712–713 sign properties of, 540 Inventory control problem, 722–724 Inverse of exponential functions, 603 See also Logarithmic functions Inverses additive, 243, 910 of functions, 126–127 of M, 244, 248 matrix, 248–249 multiplicative, 243–244, 910 of square matrix, 242–253 Investment growth, graphing, 597 Investments analysis of, 257–259 annual percentage yield of, 162–163 growth time of, 160–161 interest on, 149–151 interest rate earned on, 149–150 present value of, 148 i-system, 307 Iterated integrals, 887 L Lagrange multipliers, 865–874 definition of, 866 for functions of three independent variables, 870–872 for functions of two independent variables, 865–870 method of, 865 Large numbers, law of, 426 Larger problems, 355–357 Law of averages, 426 Law of large numbers, 426 Leading coefficients, 105 Leading term, 528 Learning rate of improvement, 762 Least common denominator (LCD), 24, 928 Least square approximation, 874–879 www.freebookslides.com Least squares line, 875 Left end behavior, 528 Left half-planes, 276 Left-hand limits, 512, 526 Leftmost variables, 222 Left rectangle, 767–768 Left and right sums approximation of areas by, 767–770 limits of, 779 Left sum, 767 See also Left and right sums Leontief input–output analysis, 262–270 three-industry model for, 266–268 two-industry model for, 263–265 L’Hôpital’s rule, 685–694 See also Limits definition of, 685 indeterminate form 0/0 and, 687–690 indeterminate form infinity/infinity and, 691–693 limits at infinity and, 690–691 one-sided limits and, 690–691 Like terms, 915–916 Limited growth, 763 Limiting matrix, 481, 492–497 Limits algebraic approach to, 514–518 analysis of, 510–511 basics of, 509–523 concept of, 509–510 constant e and, 595 continuity and, 535–546 definition of, 511 of difference quotients, 518 evaluating, 515–517 existence of, 512, 526 functions and, 509–510 graphing, 510–514 of indeterminate form, 517–518 infinite, 523–526 at infinity, 526–531, 690–691 involving exponential functions, 689 involving logarithmic functions, 689 involving powers of x, 685 involving powers of x – c, 686 from the left, 512 left-hand, 512, 526 of left and right sums, 779 one-sided, 511–512, 690–691 of polynomial functions, 515 of powers, 685 properties of, 514–515 of quotients, 517 of rational functions, 515 from the right, 512 right-hand, 512, 526 of sums, 770–772 two-sided, 512, 526 unrestricted, 512 Limits at infinity, 526–531 horizontal asymptotes and, 527, 529–531 of polynomial functions, 527–528 of power functions, 527 of rational functions, 529–530 Linear equalities systems of, 277–278, 283–289 in variables, 276–283 Linear equations, 23–25, 33–36, 329 Linear functions, 67, 84 Linear graphs, 597 Linear inequalities, 23, 25–28, 329 Linearly related variables, 47 Linear programming, 290–302, 305–361 See also Problems components of, 334 definition of, 290 feasible region for, 329 fundamental theorem of, 294, 309, 318 general description for, 293–294 geometrically interpreting, 324 introduction to, 306–316 for maximization problem, 317–332 table method for, 306–316 for minimization problem, 333–345 simplex method for, 305–361 slack variables and, 307 summary of, 324–327 variables, basic and nonbasic, 312–314 Linear programming problem definition of, 290, 293 graphically solving, 290 mathematical model for, 290, 293 Linear regression, 46–58, 874 on graphing calculator, 50 slope as rate of change and, 47–48 with spreadsheet, 52 Linear system See Systems of linear equations Line graph, 27 Lines constant-profit, 290 equation of, 38–40 graphing intercepts for, 34 horizontal, 34–36, 40 least squares, 875 real number, 909 regression, 49–50, 875 secant, 550 slope of, 36 tangent, 549–552, 611 vertical, 34–36, 40 ln x, 602–604 See also Logarithmic functions four-step process for finding, 603–604 logb x and, relationship between, 605 Loans, 148, 181–182 Local extrema, 656–658, 858–860 critical numbers and, 658 critical points and, 859–860 first-derivative test for, 658–660 graphing, 657 partial derivatives and, 858 of polynomial functions, 660 second derivative and, 711 second-derivative test for, 711–712, 858 Local maximum, 656–657, 843, 857 Local minimum, 857 Logarithmic equations, 130–131 Logarithmic functions, 126–137 with base, 22, 127–128 definition of, 603, 126, 128 derivatives of, 605, 640 domains of, 126–128 1032 Z06_BARN7668_13_GE_SIDX.indd 1032 09/07/14 8:03 PM exponential function, conversion to, 128–129 formula for, 601, 605 inverse functions and, 126–127 limits involving, 689 ln x as, 602–604 logb x with base b as, 601 models of, 606–608 natural, 604 other, 604–606 properties of, 129–131 range of, 126–128 Logarithmic regression, 134, 608 Logarithms, 131–133, 604 Log to the base b of x, 128 Logb x with base b, 601 Logic, 366–374 connectives and, 366–369 logical equivalences/implications, 370–372 propositions, 366–369 truth tables, 369–370 Logical equivalence, 371–372 Logical implication, 371 Logical reasoning, 366 Logistic growth, 763 Lorenz curve, 800–801 Lower half-planes, 276 Lower limit of integration, 772 Lowest terms, reducing to, 926–927 M M, inverses of, 244, 248 See also Singular matrix Marginal analysis, 577–587 See also Rate of change of marginal cost, 577–579 of marginal profit, 577–579 of marginal revenue, 577–579 Marginal cost, 577–579 area vs, 778–779 average, 582 definition of, 582 exact cost vs, 578–579 Marginal productivity, 850, 869 Marginal profit, 577–579, 581–582 Marginal revenue, 577–582 Markov chains, 467–503 absorbing, 489–503 definition of, 467, 470 introduction to, 468–470 matrices of, 471–472, 480–484 properties of, 468–479 regular, 479–489 state matrices and, 470–472 transitions matrices and, 470–473 Mathematical modeling, 71, 290, 293 Mathematics of finance, 146–187 See also Finance Matrix (matrices) addition of, 230–231, 254 associative, 230 augmented, 208–211 coefficient, 208–209 column, 208 combined, 254 www.freebookslides.com commutative, 230, 236 constant, 208–209 decoding, 250 definition of, 207 dimensions of, 208 elements of, 207–208 encoding, 250 equal, 230 equality, 254 first-state, 469 on graphing calculator/calculating, 498 initial-state distribution, 469 initial-state probability, 469 Leontief input–output analysis, 262–270 limiting, 481 of Markov chains, 471–472, 480–481, 484 multiplication of, 231–238, 242, 254 m * n, 207–208 negative of, 230 notation for, 208 operations of, 230–242 principal diagonal, 208 product, 231–238 properties of, 254 reduced, 216–218 row, 208 singular, 244, 249 solving systems of linear equations, methods for, 207 square, 208, 242–243 state, 470–472, 474, 489–490 subtraction of, 230–231 technology, 264 transitioning, 470–474, 490 transposition of, 333 N zero, 230–231, 236, 496 Matrix equations, 231, 254–255 systems of linear equations and, 254–262 Matrix games See Games Matrix product, 233 Maxima and minima absolute, 707–715 local extrema and, 858–860 multivariable calculus and, 857–865 using Lagrange multipliers, 865–874 Maximization problems, 290, 317–332 initial system for, 317–318 with mixed problem constraints, 346–361 pivot operation for, 319–324 problem constraints in, 339 simplex method for, 336–337 simplex tableau for, 318–319 in standard form, 306 Maximum area, 865 Maximum rate of change, 679 Maximum value of f 1x2, 93 Mean arithmetic, 955 Member of sets, 374 Method of Lagrange multipliers, 865 Method of least squares, 874–884 least square approximation and, 874–879 Midpoint sums, 771, 825 Minima See Maxima and minima Minimization problems, 333–345 big M method for, 353–355 with mixed problem constraints, 346–361 solution of, 335–339 Mixed problem constraints, 346–361 See also Big M method Model/modeling, mathematical, 46, 71, 290, 293 Modified problem, 350–351 Monomials, 915 Motion, related rates and, 634–636 Multiple optimal solutions, 295 Multiplication for linear equations, 329 for linear inequalities, 329 of matrix, 231–238, 242, 254 of polynomials, 917–918 principles of, 383–385 of problem constraints, 338–339 of rational expressions, 927–928 of real numbers, 236, 909–910 square matrix, identity of, 242–243 Multiplicative inverse, 910 Multivariable calculus double integrals over rectangular regions and, 884–894 double integrals over regular regions and, 894–902 functions of multiple variables, 839–848 maxima and minima and, 857–874 method of least squares and, 874–884 partial derivatives and, 848–856 Mutually exclusive events, 420 m * n matrix, 207–208 Natural exponential functions, 604 Natural logarithmic functions, 604 Natural logarithms, 131 Natural number exponents, 912 n distinct objects taken r at a time, permutations of, 390–391 Negation, definition of, 366 Negative of matrix, 230 Negative real numbers, 909–910 n factorials, 389, 964 Nondifferentiable functions, 556–557 Nonnegative constraints, 291, 293, 317, 334, 346–347 Normal curves, 807 Notation for derivative, 561 double subscript, 208 for functions, 68–70 f 1x2, 68–69, 93–94, 117 inequality, 27 interval, 26–27 for matrix, 208 scientific, 933–934 for second derivative, graphing of, 669 for sets, 374–376 special function, 628 summation, 770–771, 953–955 Not defined matrix product, 233 Not defined product matrix, 233 1033 Z06_BARN7668_13_GE_SIDX.indd 1033 09/07/14 8:03 PM Note, interest rate earned on, 149 Not factorable polynomials, 923, 947 nth root, 936–937 nth-term formulas, 958–959 nth terms of sequence, 951–952, 958 Null sets, 374 Numbers See also Real numbers completely factored, 920 critical, 654–656, 658 of elements of sets, 377–378 factored form of, 940 natural, 912 partition, 541, 654–656, 673–674 prime, 920 test, 541 triplet of, 842, 845 Numerator, 912, 940 Numerical coefficient, 915 O Objective functions, 290, 293, 317, 334 Odds, 425–426 One-to-one functions, 127 One-sided limits, 511–512, 690–691 Open intervals, 26, 712–713 Operations canceling, 927 of matrices, 230–242 order of, 918 pivot, 319–324, 348 on polynomials, 914–920 row, 211 Optimal solution, 292–296 Optimal values, 290, 293 Optimization problems for area and perimeter, 715–718 graphing of, 715–727 inventory control and, 722–724 for maximizing revenue and profit, 718–722 strategies for solving, 716 Ordered pair, 33 Ordinary annuities, 167, 169–170 present value of, 177 Ordinate, 33 Origin, 33, 909 Outcomes compound, 406 of experiments, 407 (See also Events) simple, 406–407 Output, 66 P Parabolas, 90, 93 Paraboloid, 843 Parallelogram, 967 Parameter, 200 Parentheses, 70, 916 Partial antiderivatives, 886 Partial antidifferentiation, 885 Partial derivatives, 848–856 definition of, 848 first-order, 851 local extrema and, 858 of f with respect to x, 849 of f with respect to y, 849 www.freebookslides.com second-order, 851–853 using chain rule, 849–850 Partition numbers, 541, 654–656, 673–674 Parts, integration by, 817–823 formula for, 817–819 Payment, 171–172, 179, 181–182 Percentage rate of change, 640–641 Percentages, 912 Perfect squares, 951 Perimeter, 715–718 Permutations, 389–392 definition of, 389–390 of n distinct objects taken r at a time, 390–391 of sets, 390 Piecewise linear functions, 84–85 Pivot element, 321 Pivoting, 322 See also Pivot operation Pivot operation, 319–324 Pivot row, 321 Plot/plotting, 49, 63–64 Point-by-point plotting, 63–64 Points break-even, 581 critical, 858, 859–860, 866 of diminishing returns, 678–679 inflection, 671–674 turning, 657 Point-slope form, 39–40 Polynomial functions continuity properties of, 539 intercepts of, 660 limits of, 515 limits at infinity of, 527–528 local extrema of, 660 Polynomials adding, 916–917 classifying, 915 combined, 918 combining like terms in, 915–916 definition of, 914 degree of, 915 end behavior of, 528–529 equations for, 918 factoring, 920–925, 947 functions for, 104–106 multiplying, 917–918 natural number exponents and, 914 operations on, 914–920 regression of, 106–107 subtracting, 916–917 in variables, 914 Population growth, 760–761 Positive real numbers, 909, 910 Power functions, 527, 562–563 See also Functions Power rule for exponential functions, 602 general, 618–622 Powers, 685–686 Powers of transition matrix, 472–473 Predictions, 50 Preliminary simplex tableau, 347 Present value See also Amortization; Principle amortization and, 178–183 of annuities, 175–187, 181, 183–184 of investments, 148 of ordinary annuities, 177 Price average, 785 elasticity of demand at, 642–643 equilibrium, 813 Price–demand equation, 42, 579–580 Price–demand functions, 70, 96 Price–demand model, 606–607 Prices diamond, 48–49 equilibrium, 42, 201 purchase, 28–29 Price–supply equations, 41–42 Prime numbers, 920 Principal diagonal matrix, 208 Principle, 122, 147–149 See also Present value of addition, 381–383 counting, 381–388 finding, 159 Probabilities, 405–461 Bayes’ formula, 445–452 conditional, 431–434, 441 empirical, 412, 414 equally likely assumption and, 413–415 of events, 409–412, 420–423, 445–447 of simple events, 410 Probability density functions, 805–807 Probability distributions expected value of, 457 histogram for, 453 random variables and, 452–454 of random variables, 453–454 Probability functions, 409, 419 Probability trees, 435–437, 447 Problem constraints, 291, 293, 317 coefficients of, 334 of dual problems, 338–339 in maximization problems, 339 multiplication of, 338–339 Problems dual, 333, 335, 338–339 linear programming, 290, 293 maximization, 290, 306, 336–337, 339 modified, 347, 350–351 word, 28 Producers’ surplus, 811–814 definition of, 812 Product matrix, 232–238 definition of, 233 not defined, 233 of a number k and a matrix M, 231–232 Product rule, 434–435, 441 Products, 610–612 differentiating, 610–611 indefinite integrals of, 740 rules for, 610 tangent lines and, 610–611 Profit changes in, 781–782 marginal, 577–579, 581–582 marginal average, 582 maximizing, 719–721 Profit, 70–71 1034 Z06_BARN7668_13_GE_SIDX.indd 1034 09/07/14 8:03 PM Profit functions, 99, 581, 840 Profit–loss analysis, 70 equality, 23 of exponential functions, 118 of inequalities, 26 of logarithmic functions, 129–131 of Markov chains, 468–479, 481 of matrix, 254 of quadratic functions, 92–96 of real numbers, 37 of sets, 374–376 Properties associative, 910 commutative, 910 constant multiple, 564 of continuity, 539–542 of definite integral, 772–773 differentiation, 561–570 distributive, 909–910, 915 of exponents, 932 of fractions, 926 of indefinite integrals, 737–740 of limits, 514–515 of logarithms, 604 of radicals, 939–940 sign, 540 sum and difference, 565–567 zero, 911 Propositions, 366–369 compound, 368 conditional, 367, 372 contrapositive, 368–369, 372 converse, 368–369 definition of, 366 truth table for, 366–372, 370–372 types of, 370 Pythagorean Theorem, 967 Q Quadrants, 33 Quadratic equations, 942–950 definition of, 942 factoring, solution by, 943–944 polynomial and, 948 quadratic formula for, 944–948 solving, 948 square root, solution by, 942–943 Quadratic formula, 90, 944–948 Quadratic functions, 89–104 analyzing, 95–96 definition of, 89 graph of, 90, 94–96 parabolas of, 90 properties of, 92–96 vertex form of, 92–96 Quotients, 612–615 difference, 518, 547 differentiating, 613–614 limits of, 517 rules for, 612–613 R Radicals, 937 properties of, 939–940 rational exponents and, 936–942 Radicand, 937 Radioactive decay, 761 www.freebookslides.com Raising fractions to higher terms, 926 Random experiments, 406 Random variables definition of, 452–453 expective value of, 454–457 probability distribution of, 452–454 Ranges, 65 of elementary functions, 79 of exponential functions, 116 of functions, 839 of logarithmic functions, 126–128 Rate of change, 546–549 See also Marginal analysis average, 546–547 instantaneous, 546–547 maximum, 679 percentage, 640–641 relative, 640–641 Rate of flow, 808 Rates, 122, 634–637 See also Rate of change of change, 41, 47–48 of descent, 47–48 effective, 162 per compounding period, 155 Rational exponents radicals and, 936–942 working with, 938–939 Rational expressions definition of, 926 operations on, 926–931 Rational functions continuity properties of, 539 horizontal asymptotes of, 529–531 limits of, 515 limits at infinity of, 529–530 vertical asymptotes of, 524–526, 531 Rational functions, 107–111 asymptotes of, 108–111, 117 definition of, 104, 107 domain of, 107 graphs of, 108–110, 117 Rationalizing denominator, 940 Rationalizing numerator, 940 Real nth root, 937 Real number line, 909 Real numbers, 908–914 addition of, 919–910 associative properties of, 910 commutative properties of, 910 definition of, 966 distributive properties of, 909–910, 915 division of, 911 fractions with, 912 identity elements of, 910 multiplication of, 236, 909–910 negative, 909–911 nth root of, 936–937 positive, 909, 910 properties of, 37, 909–911 real number line for, 909 real roots of, 937 sets of, 908–910 subtraction of, 911 zero, 911 Real roots, 937 Reasonable probability assignment, 410, 412 Rectangle, 767–768, 967 Rectangular coordinate system, 32–33, 842 See also Cartesian (rectangular) ­ coordinate system Rectangular regions average value over, 889–890 double integrals over, 884–894 Recursive Markov chains, 472 Reduced matrix, 217 Reduced row echelon form (reduced form), 217 See also Reduced matrix Reduced system, 218, 222 See also ­Gauss-Jordan elimination Reducing fractions to lowest terms, 926–927 Reduction formulas, 829 Reflections, of graphs, 82 Region of integration, 887 Regression analysis of, 48, 874 exponential, 121, 608 linear, 48–52, 874 logarithmic, 134, 608 of polynomials, 106–107 Regression line, 875 Regular Markov chains, 479–489 definition of, 480–481 properties of, 481 stationary matrix and, 479–480 Regular x region, 894–897 Regular y region, 894, 897 Related rates, 634–637 business and, 637 motion and, 634–636 suggestions for solving, 635 Related-rates problem, 634 Relative frequency, 412 Relative growth rate, 119 Relative rate of change, 640–641 Removing parentheses, 916 Representing sets, 375 Residual differences, 875 Restriction of variables, 926 Revenue, 70–71, 96–97 analysis of, 547 definition of, 580 elasticity of demand and, 644 marginal, 577–581 marginal average, 582 maximizing, 718–719, 721–722 Revenue function, 580, 840 Reversing chain rule, 745–747 Reversing order of integration, 899–900 Riemann sum, 771 Right end behavior, 528 Right half-planes, 276 Right-hand limits, 512, 526 Right rectangle, 768 Right sum, 768 See also Left and right sums Rise, 36 Roots cube, 936 of functions, 90, 106 1035 Z06_BARN7668_13_GE_SIDX.indd 1035 09/07/14 8:03 PM Roots (continued) nth, 936–937 real, 937 Row equivalent augmented matrix, 209–211 Row matrix, 208 Rows interchanging, 322 operations, 211 pivot, 321 Rules chain, 618–625 general derivative, 624–625 general power, 618–622 L’Hôpital’s, 685–694 power, 602, 619–622 for products, 610 for quotients, 612–613 Simpson’s, 823, 825–827 trapezoidal, 770, 823–825 Run, 36 S Saddle point, 844, 857 Sample space definition of, 407 events and, 406–409 finite, 454 Schedules of amortization, 179–183 commission, 150–151 Scientific notation, 933–934 Secant line, 550–551 Second-degree functions, 89 See also Quadratic functions Second-degree polynomials, 922–923 Second derivative, graphing of, 668–685 absolute maxima and minima and, 710–713 analysis of, 674–675 concavity for, 668–671 curve sketching technique for, 675–678 extrema and, 710–713 inflection points and, 671–674 local extrema and, 711 notation for, 669 point of diminishing returns and, 678–679 Second-derivative test, 711–712, 858 Second-order equations, 757 Second-order partial derivatives, 851–853 Sequence arithmetic, 957–963 definition of, 951 finite, 952 geometric, 957–958 infinite, 952 terms of, 951–953 Series arithmetic, 959 definition of, 953 finite, 953 geometric, 960–961 infinite, 953 Sets, 374–380 complement of, 377 definition of, 374 www.freebookslides.com disjoint, 381 element of, 374, 377–378, 381–384 empty, 374, 381 equal, 375 finite, 375 infinite, 375 intersection of, 376–377, 381 member of, 374 notations for, 374–376 null, 374 permutation of, 390 properties of, 374–376 of real numbers, 908–910 representing, 375 solutions to, 23, 194 subset of, 375 union of, 376, 381 universal, 376 Venn diagrams and, 376–378 Sharp-corner functions, 106 Sign chart, 541–542 Sign properties of intervals, 540 Simple events, 406–407, 410 Simple fractions, 929, 933 Simple interest, 147–154, 596 definition of, 147 formula of, 147–149 investments and, 149–151 Simplex method See also Linear programming algorithm for, 324 defined, 305–306 for maximization problems, 336–337 variables for, 319 Simplex tableau initial, 318–319, 347 for maximization problems, 318–319 preliminary, 347 procedure for, 319 Simpson’s rule, 823, 825–827 Simulation, empirical probability and, 414 Singular matrix, 244, 249 Sinking funds, 171–173 Slack variables, 307, 338, 349 Slope, 37–38 of exponential functions, 597 formula for, 777 geometric interpretation of, 37 graphing, 551 of line, 36–38 as rate of change, 47–48 of secant line, 550–551 of tangent line, 549–554 Slope fields, 757–758 Slope-intercept form, 38–40 Solution region for system of linear inequalities, 284, 286 Solutions basic, 308–309 big M method, summary of, 355 of equations, 23 feasible, 307, 309, 318 to initial system, 317–318 to i-system, 307 of linear equations, 33 for linear programming, 307–312 of minimization problem, 335–339 optimal, 292–296 to problems, summary of, 342 to sets, 23, 194 by square root, 942–943 to systems of linear equations, 196, 200 unique, 196 Solving of augmented matrix, 209–214 of double inequalities, 28 geometric method for, 294–296 larger problems, 355–357 logarithmic equations, 130–131 quadratic equations, 948 systems of linear equations, 194–207 (See also Substitution in solving system of linear equations) Special function notation, 628 Speed, 47–48 Sphere, 968 Spreadsheets input–output analysis in, 267 inverses matrix in, 248 linear regression with, 52 multiplication of matrix in, 238 notation for matrix in, 208 singular matrix in, 249 Square matrix, 208, 242–243 inverse of, 242–253 multiplication, identity of, 242–243 Square root, solution by, 942–943 Squares, perfect, 951 Standard maximization problem in standard form, 306 State matrix, 470–472, 474, 489–491 Stationary Markov chains, 480–481, 484 Stationary matrix, 479–480 Stochastic process, 436, 468 Subset of set, 375 Substitution additional techniques for, 750–753 definite integral and, 780–781 table of integrals and, 828–829 integration by, 747–750 method of, 748–752 reversing chain rule and, 745–747 Substitution in solving system of linear ­equations, 194–207 addition, by elimination of, 198–200 graphing calculator for, 197 methods for, 194, 207 solution set for, 194 Subtraction of matrices, 230–231 of polynomials, 916–917 of rational expressions, 928–929 of real numbers, 911 Sum See also Addition formulas for, 176, 959–962 of two matrices of the same size, 230 Sum and difference properties, 565–567 Summation, 953–955 Summation notation, 770–771 1036 Z06_BARN7668_13_GE_SIDX.indd 1036 09/07/14 8:03 PM Summing index, 953 Sums left and right, 767–770, 779 limit of, 770–772 midpoint, 771, 825 Riemann, 771 trapezoidal, 824 Supply and demand, 41–43, 948–949 Surface, 843, 890–891 Surplus variables, 346, 349 Systems of linear equalities, 277–278, 284–285 Systems of linear equations augmented matrices and, 207–214 consistent, 196 defined, 194 dependent, 196 equivalent, 199 independent, 196 matrix equations and, 254–262 solutions to, 196, 200 substitution, 194–207 in variables, 194–207 Systems of linear inequalities, 278–280 T Table method for linear programming, 306–316, 310–312 definition of, 306, 308 procedure for, 308, 313–314 U solutions, 307–312 Table of integrals, 823, 827–828 integration using, 827–828 substitution and, 828–829 Tables, 967–970 corner point, 294 truth, 366–372 Tangent, 655 Tangent lines, 611 See also Secant line definition of, 551 slope of, 549–554 Tautology, 370, 371 Technology matrix, 264 V Terms of sequence, 951–953 Test/testing ac, 922 for independent events, 439–440 vertical-line, 67 Test number, 541 Three-dimensional coordinate systems, 842–845 Three-industry model, input–output analysis, 266–268 Time doubling, 133–134 growth and, 158–161 Total income, 808 www.freebookslides.com Transition diagram, 468 Transition matrix, 470–474, 490 graphing calculator for, 473 of Markov chains, 470–473, 496 powers of, 472–473 probability of, 468 Transposition of matrix, 333 Trapezoid, 967 Trapezoidal rule, 770, 823–825 Trapezoidal sum, 824 Tree diagram, 920 Triangle, 967 Triplet of numbers, 842, 845 True interest rate, 162 See also Annual ­percentage yield (APY) Truth table, 369–370 for compound propositions, 366–367, 369, 371–372 constructing, 369–370 definition of, 369 graphing calculator for displaying, 370 for propositions, 366–372 Turning points, 657 Two-industry model, input–output analysis, 263–265 Two-sided limits, 512, 526 Unbounded solution regions, 286 Union, 537 of events, 419–423 intersection and, 419–423 of sets, 376, 381 Unique solution to system of linear equations, 196 Unit elasticity, 643 Universal set, 376 Unlimited growth, 763 Unrestricted limits, 512 Upper half-planes, 276 Upper limit of integration, 772 Upward concavity, 669–670 Vacuously true conditional propositions, 367 Values absolute, 79, 511 average, 783–785, 889–890 expected, 455–457 of functions, 509–510 future, 122, 147–149, 167–175, 809–811 maximum, of f(x), 93 optimal, 290, 293 Variable costs, 71 Variables artificial, 346–347, 349 decision, 290, 293 dependent, 66, 839 entering, 320 exiting, 320 independent, 66, 839–840, 865–872 leftmost, 222 linear equalities in, 276–283 linearly related, 47 multiple, 839–848 polynomials in, 914 random, 452–455 restriction of, 926 for simplex method, 319 slack, 307, 338, 349 surplus, 346, 349 in systems of linear equations, 194–207 three-dimensional coordinate systems and, 842–845 two decision, 310–312 Velocity average, 546–549 instantaneous, 546, 549 Venn diagram definition of, 376 elements of sets, for determining, 383 set operations and, 376–378 Vertex, of parabolas, 93 of quadratic functions, 92–96 Vertical asymptotes infinite limits and, 524–526 of rational functions, 524–526, 531 Vertical asymptotes of rational functions, 108–111 Vertical axis, 32–33 Vertical-line test, 67 Vertical shrink of graphs, 82–83 Vertical stretch of graphs, 82–83 Vertical tangent, 655 Vertical translation of graphs, 80–81, 83 Volume double integrals and, 890–891, 900–901 under surface, 890–891 X x axis, 797 See also Horizontal axis x axis, reflection in, 82 x coordinates, 33 Y y coordinates, 33 Z Zero factorials, 389, 964 Zero of a function, 90, 106 Zero matrix, 230–231, 236, 496 Zero properties of real numbers, 911 1037 Z06_BARN7668_13_GE_SIDX.indd 1037 09/07/14 8:03 PM www.freebookslides.com Index of Applications Business & Economics Advertising, 124, 332, 359, 477, 569, 576, 639, 683–684, 732, 741–742, 765 Point of diminishing returns and, 732 And sales, 847, 855 Agriculture, 137, 144, 269–270, 327–329, 725, 874 Exports and imports of, 661 Amortization, 179 Annuities, 174 Future values of, 169–170 Guaranteed rate of, 184–185 Interest rate of, 186 Present value of, 177 Automation-labor mix for minimum cost, 864 Average cost, 533, 576, 667, 701–702, 731, 732, 787 (See also Minimizing average cost) Average and marginal costs, 705 Average price, 833 Average revenue, 630 Blending—food processing, 364 Bonus incentives, 444 Break-even analysis, 29–30, 32, 97–99, 102–103, 143, 205, 273, 587, 593 Budgeting For least cost, 873 For maximum production, 873 Business closings, 399 Buying and selling commissions schedule, 167 Cable television, 56 Revenue from, 883 Subscribers to, 607 Capital expansion, 301 Car rental, 725 Coal, 269 Cobb-Douglas production function, 893 Coffee blends, 206 Committee selection, 380, 400, 418 Communications, 388 Compound interest, 630 Computers, 300 Control systems for, 440 Testing on, 384–385 Construction, 143–144, 731 Costs of, 705, 726, 732 Consumer financing rate, 185–186 Consumer Price Index (CPI), 30, 61 Consumer’s surplus, 811–812, 813–814, 823, 832, 837 Consumer survey/testing, 418, 434–435 Continuous compound interest, 599, 600, 607, 609, 650, 765 Continuous income stream, 808, 810–811, 822, 832, 837 Cost, revenue, and profit rates, 639 Costs, 787, 793, 832 Analysis of, 44, 240, 583, 584–585, 592, 682 Average, 113–114 Equation for, 40–41 Function of, 626–627, 744, 755, 847 Hospital, 87 Material, 273 Minimum average, 114 Credit cards, 151, 183 Annual interest rate on, 154 Minimum payment due on, 153 Decision analysis, 460, 466 Delivery charges, 205–206 Demand equation, 650, 855 Demand function, 788 Depreciation, 44–45, 60, 121 Distribution of wealth, 804–805 Doubling rate, 600 Doubling time, 133, 136, 600, 650 Economy, 270, 962, 963 Electricity Consumption of, 560 Natural gas, 270 Oil, 270 Rates for, 87, 143 Electronics, 206, 304, 404 Employees Benefits for, 381–382 Evaluation of, 483 Layoffs for, 399 Rating of, 451 Screening of, 451 Training of, 111, 501–502, 506 Employee training, 534, 593, 705, 776, 787 Energy, 53–54 Consumption of, 881 Costs of, 533 Equilibrium point, 136, 144 Equilibrium price, 813–814 Equipment rental, 31, 545 Eraser, 774 Exit polling, 378 Finance/financing, 102, 124, 182 Furniture, 288–289, 300 Future value, 847 Of continuous income stream, 837 Gross receipts, 920 Growth Compound, 124, 174, 192 Exponential, 119–120 in individual retirement account, 172–173 Internet, 125 Money, 124, 143 Growth time, 600 Home Building/construction of, 332 Equity in, 181, 186–187, 192 Insurance for, 477–478 Values of, 31 Housing trends, 478–479 Ice cream, 345 Incentive plan, 262 Income, 61, 545 Distribution of, 801, 804, 822, 832, 837 Operating, 55 State tax on, 88 Taxable, 228 Individual retirement account (IRA), 31, 172–173, 175, 192 Inflation, 682 Input–output analysis, 274 Insurance, 430–431, 457, 460, 466, 471–472, 482–483, 487 Interest Compound, 124, 136, 166–167, 963 on loans, 185 Interest rate, 174, 190–191, 950 Annual, 153–154, 185, 191–192 of annuities, 186 Approximating, 173 Earned on Note, 149, 153 Graphical approximation techniques for, 175, 187 Internet, 125, 506 Inventory, 788, 793 Control over, 726, 731 Inventory value, 240–241 Investment, 60, 136, 301, 331–332, 364, 919 Analysis on, 257–259, 274 Annual percentage yield (APY) on, 162–163, 175 Interest rate earned on, 149–150 Present value of, 148 Strategy for, 360 Labor costs, 745 Learning and, 788 Labor force, 486 Costs, 233, 237–238, 240, 273, 282 Relations with, 444 Leases Airplane, 227–228 Business, 273 Tank car, 227 Loans, 148, 501 Distribution of, 360 Refund anticipation, 154 Repayment of, 963 Maintenance costs, 787 Management, 387 Manufacturing, 359, 364, 725 Bicycle, 332 Resource allocation, 331 Sail, 304 1038 Z07_BARN7668_13_GE_AIDX.indd 1038 09/07/14 8:46 PM Marginal analysis, 650, 731 Market/marketing, 502, 506, 755, 794, 833 Analysis of, 465 Research on, 382–383, 387–388, 403–404, 426–428, 430, 465 Shares in, 487 Markups, 44, 240 Mattresses, 283 Maximizing profit, 864, 883, 905 Maximum profit, 724–725, 731 Maximum revenue, 646, 724–725, 731 Maximum shipping volume, 865 Maximum volume, 865, 873–874 Mineral consumption, 560 Minimizing average costs, 706 Minimizing cost, 864 Minimizing material, 906 Minimum material, 864 Minimum wage, 388 Mining, 345 Multiplier principle, 893 Natural gas Consumption of, 593 Rates for, 545, 592 Natural gas rates, 85 Oil production, 755–756, 788, 804 Oil refining, 360 Operational costs, 726 Package design, 841, 847, 862–863, 870–872 Packaging, 76, 725–726 Parking meter coins, 31 Parking receipts, 261 Pensions, 502 Personnel Selection of, 400, 418, 444 Petroleum blending, 355–357 Politics, 577, 727, 732, 788, 823, 833, 837 Postal rates, 544–545 Present value, 599–600 Price analysis, 666–667, 731 Price-demand, 61, 71–73, 75, 87, 576, 639, 753–754, 765, 833 Equation for, 569, 617, 627, 755 Model for, 606 Prices/pricing, 60–61 of car, 31 of diamonds, 48–49 of gasoline, 914 Purchase, 28–29 Supply, 87 of tickets, 274 Price-supply, 765, 793 Equation for, 617, 627, 755 Producer’s surplus, 812–814, 823, 830, 832, 837 Product/production, 822 of automobiles, 102 of boats, 227 Costs of, 744 Defects in, 437, 449, 451–452 Demand for, 856 Mix of, 383–384 Point of diminishing returns, 683 www.freebookslides.com Scheduling for, 227, 261, 300 Strategy for, 579–582 Switching of, 506 Testing of, 431 Warranty on, 836 Productivity, 842, 847, 850–851, 855, 873, 906 Product mix for maximum profit, 864 Profit, 75–76, 560, 683, 705, 822, 833, 850, 861, 906 Analysis of, 585, 666 Functions of, 793, 855 And production, 793 Profit–loss analysis, 103, 143 Public debt, 935–936 Purchasing, 224–225 Quality control, 399, 431, 444, 466 Rapid transit, 488 Rate of change Of cost, 646 Of revenue, 650 Real estate development, 492–494 Rental income, 725, 731 Replacement time, 114, 705 Resale value, 609 Resource allocation, 273 Retail sales, 31 Retirement planning, 177–178 Revenue, 75, 102, 559, 683, 705, 744, 833 Analysis of, 585, 662, 666 And elasticity, 646, 650 Functions of, 755, 837, 847 Maximum, 96–97 Modeling of, 71–73 And profit, 576, 855 Revenue, cost, and profit, 585–587, 847 Salaries, 125 Sales, 280–281 Commission, 31–32, 232 Net, 55 Tax on, 912, 914 Ticket, 31 Sales analysis, 555–556, 560, 568–569, 593, 614–615, 616–617, 744, 822–823 Salvage value, 609, 787 Sausage, 206 Scheduling, 477 Service contracts, 478 Shipping schedules, 364 Steel, 269 Supply and demand, 41–43, 45–46, 57–58, 136, 202–203, 205, 948–949, 950 Supply function, 788 Telephone Calls on, duration of, 806 Rates for, 521–522, 545 Telephone expenditures, 57 Textiles, 283 Tire mileage, 101–102 Traffic flow, 229, 274 Trail mix, 360 Training, 478 Transportation, 301, 360, 387, 403, 486 Manufacturing of, 269 Problems with, 340–342 Useful life, 782, 787, 793, 804 Vehicles, customized, 283 Volume discount, 522 Water skies, 288–289, 300 Zero coupon bonds, 167, 191 Life Sciences Agriculture, 137, 144, 269–270, 327–329 Air pollution, 560 Animals Diet for, 206 Feed/feeding, 228, 301–302, 304 Nutrition for, 332 Animal supply, 545 Archaeology, 137, 761, 766, 794 Atmospheric pressure, 50 Bacterial control, 727, 732 Bacterial growth, 609 Biochemistry, 534 Biology, 627, 756, 765, 788, 805, 883–884 Biophysics, 633 Bird flights, 727 Birthday problem, 424–425 Blood flow, 847, 856 Blood pressure, 609, 765 And age, 627 Blood types, 380 Body surface area (BSA), 47 Boiling point, 45 Botany, 727 Cancer screening, 452 Carbon-14 dating, 137 Cereal, 241 Decibels, 136–137 Dental insurance, 478 Diet, 114, 200–201, 262 And minimum cost, 874 Drugs Assimilation of, 837 Concentration of, 533, 627, 650, 727, 766 Sensitivity to, 576, 617, 684 Earthquakes, 206 Ecology, 569, 950 Exponential decay, 120 Family planning, 388 Fertilizer, 206, 282–283, 301 Fish weight, 106–107 Flight conditions/navigation, 45 Forestry, 51, 56, 61 Gene mutation, 488 Genetics, 444, 460, 466, 487, 506 Global warming, 884 Gravity, 634 1039 Z07_BARN7668_13_GE_AIDX.indd 1039 09/07/14 8:46 PM Health plans, 478 Heredity, 241 Herpetology, 88 Insecticides, 766 Life expectancy, 125–126, 228–229, 907 Marine biology, 125, 144, 847, 906 Measurement, 576 Medical research, 400, 404, 418, 452 Medicare, 144 Medication/medicine, 103, 144, 296–297, 388, 400, 418, 431, 502, 560, 569, 576, 593, 617, 667, 706, 756, 776, 788, 823, 837, 856 Cardiogram test, 466 Sports, 60 Muscle contraction, 77 Natural resource depletion, 805 Nuclear accident, 766 Nutrition, 228, 286–287, 289, 301, 920 for animal, 332 Human, 345, 359, 360 Organic farming, 884 Outboard motors, 99–100, 103–104 Physical anthropology, 848, 856 Physics, 46, 206, 534–535, 593 Physiology, 706 Plants Food for, 228, 289, 301, 345, 359 Pollution, 301, 522, 533–534, 593, 639, 705, 727, 756, 794, 823, 833, 893, 906, 936 Population growth: bacteria, 684 Psychology, 32, 206–207, 289, 302, 445, 502–503 Pulse rate, 576 Radioactive decay, 600 Rate of descent, 53 Renewable energy, 744 Resource depletion, 793 Seed costs, 282 Simple epidemic, 766 Smoking, 45, 54, 506 www.freebookslides.com Sound Intensity of, 136–137 Speed of, 53, 633, 634 Waves of, 206 Speed of sound, 633, 634 Marriage, 115 Mixing feed, 360 Temperature, 32, 55–56, 61, 788 Tuberculosis screening, 448, 452 Perception, 766 Physiology, 114–115 Police science, 452 Politics, 77, 241, 388, 400, 418, 431, 452, 461 Advertising of, 283 Filibuster in, 488 Population/population growth, 794 Composition of, 788–789 Density of, 906, 936 Distribution of, 893 Growth of, 125, 144, 228, 601, 647, 760–761 Shifts in, 488–489 U.S., 601 World, 137, 600–601 Pregnancy testing, 452 Psychology, 837, 848, 893–894 Learning, 650 Retention, 706 Stimulus/response, 609 Training, 609 Underwater pressure, 53 U.S Food and Drug Administration, 444 Water pollution, 627 Weight Human, 88 Ideal, 53 Weight-height, 745 Wildlife management, 32 Wound healing, 650, 745 Social Sciences Code words, 385 College enrollment, 569, 756 Concert tickets, 261 Crime, 144, 647, 936 Rates of, 883 Cryptography, 250–251, 253, 274 Demographics, 45 Divorce, 115 Education, 262, 906 Enrollments, 54, 56–57, 380, 496–497 High school dropout rates, 61 Resource allocation for, 345 Student retention, 473–474 Tests/testing, 241–242 Exam scores, 879–881 Home ownership, 134, 487 Learning, 88–89, 115, 124–125, 545–546, 569–570, 576–577, 593, 639, 650, 684, 727, 745, 756, 762, 766, 776, 794, 805, 823, 833 Licensed drivers, 54–55 Olympic games, 57, 883 Opinion poll/survey, 332 Rumor propagation, 766 Safety research, 88, 433–434, 848, 856, 893, 950 Small-group analysis, 766 Sociology, 229, 302, 906 Urban growth, 745 Voters/voting Coalition for, 380 Patterns of, 445 Preferences of, 466 Voter turnout, 522 1040 Z07_BARN7668_13_GE_AIDX.indd 1040 09/07/14 8:46 PM www.freebookslides.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 Z07_BARN7668_13_GE_AIDX.indd 1041 x 09/07/14 8:46 PM www.freebookslides.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 f(x) ϭ xϪ3 xϪ2 x Ϫ5 Ϫ5 f(x) ϭ x2 Ϫ 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 h Ϫ5 Vertical shift g(x) ϭ f(x) ϩ h(x) ϭ f(x) Ϫ Z07_BARN7668_13_GE_AIDX.indd 1042 x Ϫ5 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) 09/07/14 8:46 PM ... 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. .. 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...www.freebookslides.com College Mathematics For BusIness, econoMIcs, lIFe? ?scIences, And socIAl scIences thirteenth edition Raymond A Barnett Michael R Ziegler Karl E Byleen Merritt College Marquette