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PRECALCULUS Mathematics for Calculus FIFTH EDITION This page intentionally left blank PRECALCULUS Mathematics for Calculus FIFTH EDITION James Stewart McMaster University Lothar Redlin The Pennsylvania State University Saleem Watson California State University, Long Beach Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Precalculus: Mathematics for Calculus, Fifth Edition, Enhanced WebAssign Edition James Stewart, Lothar Redlin, Saleem Watson Acquisitions Editor: Gary Whalen Assistant Editor: Natasha Coats ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Sec- Editorial Assistant: Rebecca Dashiell Technology Project Manager: Sam Subity ten permission of the publisher Marketing Manager: Joe Rogove For product information and technology assistance, contact us at Marketing Assistant: Ashley Pickering Marketing Communications Manager: Darlene Amidon-Brent For permission to use material from this text or product, submit all requests online at cengage.com/permissions Further permissions questions can be emailed to permissionrequest@cengage.com Project Manager, Editorial Production: Jennifer Risden Creative Director: Rob Hugel Art Director: Vernon Boes Print Buyer: Judy Inouye Student Edition: Permissions Editor: Bob Kauser Production Service: Martha Emry Text Designer: John Edeen Art Editor: Martha Emry Brooks/Cole Cengage Learning Photo Researcher: Stephen Forsling Copy Editor: Luana Richards USA Illustrator: Jade Myers, Matrix Cover Designer: Roy E Neuhaus Cover Image: Bill Ralph Compositor: Newgen–India Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan Locate your local office at international.cengage.com/region Cengage Learning products are represented in Canada by Nelson Education, Ltd For your course and learning solutions, visit academic.cengage.com Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States of America 12 11 10 09 08 07 To our students, from whom we have learned so much About the Cover The art on the cover was created by Bill Ralph, a mathematician who uses modern mathematics to produce visual representations of “dynamical systems.” Examples of dynamical systems in nature include the weather, blood pressure, the motions of the planets, and other phenomena that involve continual change Such systems, which tend to be unpredictable and even chaotic at times, are modeled mathematically using the concepts of composition and iteration of functions (see Section 2.7 and the Discovery Project on pages 223–224) The basic idea is to start with a particular function and evaluate it at some point in its domain, yielding a new number The function is then evaluated at the new number Repeating this process produces a sequence of numbers called iterates of the function The original domain is “painted” by assigning a color to each starting point; the color is determined by certain properties of its sequence of iterates and the mathematical concept of “dimension.” The result is a picture that reveals the complex patterns of the dynamical system In a sense, these pictures allow us to look, through the lens of mathematics, at exotic little universes that have never been seen before Professor Ralph teaches at Brock University in Canada He can be contacted by e-mail at bralph@spartan.ac.brocku.ca About the Authors James Stewart was educated at the University of Toronto and Stanford University, did research at the University of London, and now teaches at McMaster University His research field is harmonic analysis He is the author of a best-selling calculus textbook series published by Brooks/Cole, including Calculus, 5th Ed., Calculus: Early Transcendentals, 5th Ed., and Calculus: Concepts and Contexts, 3rd Ed., as well as a series of high-school mathematics textbooks Lothar Redlin grew up on Vancouver Island, received a Bachelor of Science degree from the University of Victoria, and a Ph.D from McMaster University in 1978 He subsequently did research and taught at the University of Washington, the University of Waterloo, and California State University, Long Beach He is currently Professor of Mathematics at The Pennsylvania State University, Abington College His research field is topology Saleem Watson received his Bachelor of Science degree from Andrews University in Michigan He did graduate studies at Dalhousie University and McMaster University, where he received his Ph.D in 1978 He subsequently did research at the Mathematics Institute of the University of Warsaw in Poland He also taught at The Pennsylvania State University He is currently Professor of Mathematics at California State University, Long Beach His research field is functional analysis The authors have also published College Algebra, Fourth Edition (Brooks/Cole, 2004), Algebra and Trigonometry, Second Edition (Brooks/Cole, 2007), and Trigonometry (Brooks/Cole, 2003) Contents Preface xiii To the Student xxi Calculators and Calculations Fundamentals ■ 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 2.1 2.2 Chapter Overview Real Numbers Exponents and Radicals 12 Algebraic Expressions 24 ● DISCOVERY PROJECT Visualizing a Formula 34 Rational Expressions 35 Equations 44 Modeling with Equations 58 ● DISCOVERY PROJECT Equations through the Ages 75 Inequalities 76 Coordinate Geometry 87 Graphing Calculators; Solving Equations and Inequalities Graphically 101 Lines 111 Modeling Variation 123 Chapter Review 130 Chapter Test 135 ■ FOCUS ON PROBLEM SOLVING General Principles 138 Functions ■ xxii 146 Chapter Overview 147 What is a Function? 148 Graphs of Functions 158 ● DISCOVERY PROJECT Relations and Functions 171 vii viii Contents 2.3 2.4 2.5 2.6 2.7 2.8 Polynomial and Rational Functions ■ 3.1 3.2 3.3 3.4 3.5 3.6 Increasing and Decreasing Functions; Average Rate of Change 173 Transformations of Functions 182 Quadratic Functions; Maxima and Minima 193 Modeling with Functions 203 Combining Functions 214 ● DISCOVERY PROJECT Iteration and Chaos 223 One-to-One Functions and Their Inverses 225 Chapter Review 233 Chapter Test 237 ■ FOCUS ON MODELING Fitting Lines to Data 239 248 Chapter Overview 249 Polynomial Functions and Their Graphs 250 Dividing Polynomials 265 Real Zeros of Polynomials 272 ● DISCOVERY PROJECT Zeroing in on a Zero 283 Complex Numbers 285 Complex Zeros and the Fundamental Theorem of Algebra 291 Rational Functions 299 Chapter Review 316 Chapter Test 319 ■ FOCUS ON MODELING Fitting Polynomial Curves to Data 320 Exponential and Logarithmic 326 Functions ■ 4.1 4.2 4.3 4.4 4.5 Chapter Overview 327 Exponential Functions 328 ● DISCOVERY PROJECT Exponential Explosion 341 Logarithmic Functions 342 Laws of Logarithms 352 Exponential and Logarithmic Equations 358 Modeling with Exponential and Logarithmic Functions 369 Chapter Review 382 Chapter Test 385 ■ FOCUS ON MODELING Fitting Exponential and Power Curves to Data 386 Contents Trigonometric Functions of Real 398 Numbers ■ 5.1 5.2 5.3 5.4 5.5 Trigonometric Functions of Angles ■ 6.1 6.2 6.3 6.4 6.5 Chapter Overview 399 The Unit Circle 400 Trigonometric Functions of Real Numbers 408 Trigonometric Graphs 418 ● DISCOVERY PROJECT Predator/Prey Models 432 More Trigonometric Graphs 434 Modeling Harmonic Motion 442 Chapter Review 454 Chapter Test 458 ■ FOCUS ON MODELING Fitting Sinusoidal Curves to Data 459 Chapter Overview 467 Angle Measure 468 Trigonometry of Right Triangles 478 Trigonometric Functions of Angles 488 ● DISCOVERY PROJECT Similarity 499 The Law of Sines 501 The Law of Cosines 508 Chapter Review 516 Chapter Test 520 ■ FOCUS ON MODELING Surveying 522 Analytic Trigonometry ■ 7.1 7.2 7.3 7.4 7.5 466 526 Chapter Overview 527 Trigonometric Identities 528 Addition and Subtraction Formulas 535 Double-Angle, Half-Angle, and Sum-Product Formulas 541 Inverse Trigonometric Functions 550 ● DISCOVERY PROJECT Where to Sit at the Movies Trigonometric Equations 561 Chapter Review 571 Chapter Test 574 ■ FOCUS ON MODELING Traveling and Standing Waves 575 560 ix Index Abel, Niels Henrik, 282 Absolute value, 8–10 of complex numbers, 597–598 equations, 54, 91 properties of, Absolute value function, 162, 166 Absolute value inequalities, 81–82 Addition of complex numbers, 286 graphical, 215 of inequalities, 76 of matrices, 676–678 of polynomials, 25 of rational expressions, 37–38 of vectors, 608, 610, 611 Addition and subtraction formulas, 535–541 Additive identity, Adleman, Leonard, 308 Agnesi, Maria Gaetana, 802 Ahmes (Rhind papyrus scribe), 716 Airplane design, 245 Algebraic errors, avoiding, 40–41 Algebraic expressions, 24–33, 35 AM (amplitude modulation) radio, 428 Ambiguous case, of solving triangles, 503–505, 508 Amortization schedule, 854 Amplitude, 421 decaying, 428 harmonic motion and, 443 period and, 423–425 variable, 427–428 Amplitude modulation (AM) radio, 428 Analogy, used in problem solving, 138–139 Ancillary circle of ellipse, 760 Angle measure, 468–478 Angles see also Trigonometric functions, of angles angle of depression, 482 angle of elevation, 482 angle of incidence, 570 angle of inclination, 482 angle of refraction, 570 defined, 468 quadrantal, 490 reference, 491–492 standard position of, 470–471 supplement of, 504 Angular speed, 473–474 Annual percentage yield, 366, 367 Annuities calculating amount of, 848–850 in perpetuity, 853–854 present value of, 850–851 Aphelion, 760, 801 Apolune, 761 Approval voting, 682 Arccosine function, 553 Archimedes, 69, 414, 748–749, 902 Architecture, conics in, 771–775 Arcsine function, 551 Arctangent function, 555 Area of circular sector, 472–473 of a triangle, 494–495, 512–513, 711–712, 714–715 Area problem, calculus, 916–925 approximating area with calculator, 925 under a curve, 922–924 defined, 920–922 estimating using rectangles, 917–918 under graphs, 929–931 limit of approximating sums, 919–920 Areas, formulas for, inside front cover Argument of complex number, 598 Aristarchus of Samos, 480 Aristotle, 54 Arithmetic mean, 838 Arithmetic sequences, 833–838 defined, 833 partial sums, 834–836 Arrow, Kenneth, 682 Arrow diagram, of functions, 150 Assets, division of, 834–835 Associative Property, Astroid, 808 Asymptotes, 300–302 defined, 301 horizontal, 301, 304–306, 307, 308, 910–911 of hyperbolas, 763–764, 767 of rational functions, 303–306, 307, 308 slant, 309–310 vertical, 301, 303–308, 434–436, 886–887 Atmospheric pressure formula, 368 Augmented matrix, 635, 662–663 Automotive design, 256 Average rate of change, 174–178, 904 Avogadro’s number, 23 Axes see also Rotation of axes of a conic, 797 of ellipses, 754, 755 of hyperbolas, 762 of parabolas, 745–748 polar axis, 582 real and imaginary, 596 Axis of symmetry, parabolas, 744 Back-substitution, solving linear equations, 652, 653 Base, change of, 355–356 Base 10 logarithm, 346–347 Bearing, 511 Beer-Lambert Law, 364, 394 Bell, E.T., 678 Bernoulli, Johann, 805 I1 I2 Index Best fit exact fit vs., 660–661 finding, 240–242, 320–323 measuring, 242–243 polynomials of, 320–323 Bhaskara, 75, 144 Binomial coefficients, 863–865 Binomial expansion, 861–863, 865–867 Binomials, 24, 860 Binomial Theorem, 865–868 Bits, changing words/sounds/pictures to, 30 Bounded regions, of planes, 725 Boyle’s Law, 125, 128 Brahe, Tycho, 780 Brams, 834 CAD (computer-aided design), 256 Calculators evaluating trigonometric functions, 413, 436 graphing calculators, 102–104, 884–885, 890, 925 as graphing device, 425 Calculus addition and subtraction formulas in, 537 preview of see Limits Cardano, Gerolamo, 282, 296 Cardioid, 590 Carrier signals, radio, 428 Carrying capacity, 393 Cartesian plane, 87–88, 112 see also Coordinate plane CAT (Computer Aided Tomography) scan, 746 Catenary, 331 Cayley, Arthur, 692 Center of ellipse, 754 of hyperbola, 762 Central box, of hyperbolas, 763, 764 Chang Ch’iu-Chien, 75 Change of Base Formula, 356 Chaos, 224 Chevalier, Auguste, 273 Chu Shikie, 862 Circadian rhythms, 453, 464 Circles, 92–95 ancillary, of ellipse, 760 area of, 154 equations of, 93, 94–95 graphing, 93, 103–104 involute of a, 810 as polar graph, 594 Circular arc, length of, 471–472 Circular function see Trigonometric functions Circular motion, 473–474 Circular sector, area of, 472–473 Closed curves, 806 Codes, unbreakable, 308–309 Coefficient matrix, 694 Coefficients binomial, 863–865 correlation, 242 of polynomials, 250, 253 Cofunction identities, 528, 537 Collinear points, 123, 715 Column transformations, of determinants, 707–708 Comets, paths of, 766 Common (base 10) logarithms, 346–347 Common difference of sequence, 833 Commutative Property, Complete Factorization Theorem, 291–293 Completing the square, 48–49 Complex conjugates, 287, 289 Conjugate Zeros Theorem, 296, 299 Complex numbers, 285–290 arithmetic operations on, 286–287 complex roots of quadratic equations, 288–289, 290 defined, 285 DeMoivre’s Theorem, 600–601 fractals and iterating functions of, 605–607 graphing, 596–598 multiplication and division of, 599–600 polar (trigonometric) form of, 598–600 roots of, 601–603 square roots of negative numbers, 287–288 Complex plane, 596 Complex roots, of quadratic equations, 288–289, 290 Complex zeros, 291–299 Composite function, 216–219 Compound fractions, 38–39 Compound interest, 334–336, 340, 367 annuities and, 849–850 continuously compounded, 336 formula for, 335 using logarithmic equations for, 365–366, 367 Computer-aided design (CAD), 256 Computer Aided Tomography (CAT) scan, 746 Computer graphics applying matrices to generation of, 683–684, 700–703 rotating an image, 792–794 Computers applications of, 178 as graphing device, 425 Confocal conics family of, 783 hyperbolas, 769–770 parabolas, 782 Conics see also by type in architecture, 771–775 basic shapes of, 743–744 confocal, 769–770, 782, 783 degenerate, 780–781 equivalent description of, 795 graphing rotated, 787–789 identifying by discriminant, 789–790 polar equations of, 795–801 shifted, 775–783 simplifying general equation for, 785–787 Conjecture, mathematical induction and, 854–855 Conjugate hyperbolas, 769 Conjugate Zeros Theorem, 296, 299 Constant(s) growth, 223 of proportionality, 124 spring, 127, 452, 931 Constant coefficient, 250 Constant function, 158, 159 Constant rate of change, 178 Constant term, 250 Constraints, 725, 736, 737 Continuous functions, 893 Continuously compounded interest, 336 Contradiction, proof by, 133 Cooling, Newton’s Law of, 375–376, 381, 878 Coordinate geometry, 87–101 circles, 92–95 coordinate plane, 87–88 graphing equations, 90–91 intercepts, 92 symmetry, 95–96 Coordinate line (real number line), 7, 9–10 Coordinate plane, 1, 87–88 Coordinates see Homogenous coordinates; Polar coordinates; Rectangular coordinates Correlation, 242–243 Correlation coefficient, 242 Index Cosecant function, 408 cosecant curves, 439–440 formula for, 488 graphing, 435–436, 439–440 inverse, 556 special values of, 410 trigonometric ratios, 478 Cosine function, 408 addition and subtraction formulas for, 535–536 cosine curves, 422, 427–428, 459–461 double-angle formula for, 542, 786 formula for, 488 graphing, 418–420 graphing transformations of, 420–425 half-angle formula for, 544 inverse cosine, 553–554 Law of Cosines, 508–516, 561 periodic properties of, 419 product-sum formula for, 546 shifted curves, 423–425 special values of, 410 sum of sines and cosines, 538–539 sum-to-product formula for, 547 trigonometric ratios, 478 Cost function, 163 Cotangent function, 408 cotangent curves, 437, 438–439 formula for, 488 graphing, 435, 436–439 inverse cotangent, 556, 557 special values of, 410 trigonometric ratios, 478 Coterminal angles, 470–471 Cramer’s Rule, 708–711 Cubic formula, 282 Cubic splines, 249, 252 Cumulative voting, 682–683 Curtate cycloid (trochoid), 808 Curve area under, 922–924 slope of a, 900 Cycles, of vibration, 443 Cycloid curtate (trochoid), 808 parametric equations, 804–805 prolate, 808 Cylindrical projection, 630–631, 632 Damped harmonic motion, 449–451, 569 Damping constant, 449 Data, linearizing, 389–390 Data matrices, 701 Daylight, modeling hours of, 447–448 Decibel scale, 378 Degenerate conics, 780–781 Degrees as angle measure, 468 compared with radians, 469 Demand equation, 247 Demand function, 232 DeMoivre’s Theorem, 600–601 Denominators, of partial fractions, 716–719 rationalizing, 20–21, 40 Dependent systems, linear equations, 644, 645–646, 654, 655–656, 668–672 Dependent variables, 150 Depressed cubic, 282 Depression, angle of, 482 Derivatives, 902–904 defined, 902 estimating from graphs, 907 finding at a point, 903 Descartes, René, 87, 112, 140, 275 Descartes’ Rule of Signs, 275, 297 Determinants, 691, 704–715 areas of triangles, 711–712, 714–715 collinear points and, 715 invertibility criterion, 707 row and column transformations, 707–708 zero, matrices with, 715 Difference of cubes, 29 of functions, 214 of matrices, 767 of squares, 29 Difference quotients, 151, 177 Digital images, 683–684, 687 Digital numbers, 30 Diophantus, 20, 75 Directed quantities see Vectors Directrix, 744, 795, 796 Direct substitution, finding limits using, 893–894 Direct variation, 123–125 Discriminant identifying conics by, 789–790 invariant under rotation, 789, 791 of quadratic formula, 50–51 Distance, between points on the real line, 9–10 Distance formula, 88–89, 587 Distributive Property combining algebraic expressions, 25 factoring with, 27–28 real numbers and, 1, 3–4 I3 Dividends, 266 Division of complex numbers, 287, 599–600 long, 265–267, 720 overview of, of polynomials, 265–272 of rational expressions, 36–37 synthetic, 267–268 Division Algorithm, 266 Divisors, 5, 266 Domains of algebraic expression, 35 of combined functions, 214–215 finding, from graphs, 161 of functions, 150, 153 of an inverse function, 227 of rational function, 300 of relation, 171 of trigonometric functions, 411 Doppler effect, 315, 454 Dot product, 617–625 calculating work, 623–624 component of u along v, 620–622 defined, 618 projection of u onto v, 622–623 properties of, 618 of vectors, 617–620 Dot Product Theorem, 618–619 Double-angle formulas, 541–543, 550, 786 Earthquakes, magnitude of, 377–378 Ebbinghaus, Hermann, 354–355, 357, 395 Eccentricity of a conic, 795, 796 of an ellipse, 757–758 of planetary orbits, 758 Ecology, mathematical study of, 696–697 Economics, use of mathematics in, 850 Einstein, Albert, 104, 141, 710, 816 Elementary row operations, 663–664 Elements, of sets, Elevation, angle of, 482 Elimination method, 638–640 Ellipses, 476, 743, 753–761 ancillary circle of, 760 with center at origin, 754, 755 constructing, 775 eccentricity of, 757–758 equation of, 757, 758–759 foci of, 758 geometric definition of, 753 graphing shifted, 776–777 latus rectum of, 761 orbits of planets as, 758 I4 Index Ellipses (continued) rotating, 799 sketching, 755–756 vertices of, 754, 755 Elongation, 487, 508 Empty set л, Encryption, 308–309 End behavior of polynomials, 252–254, 255 of rational functions, 309–311 e (number), 332–333, 347–348 Envelope of lines, parabola as, 703 Epicycloid, 808 Equality of matrices, 767 properties of, 44 of vectors, 608, 610 Equations, 1, 44–58 see also Systems of equations; Systems of linear equations absolute value, 54, 91 of circles, 93, 94–95 demand, 247 equivalent, 44 exponential, 358–361 false, 654 family of, 57 of functions, 164–165 graphic solutions for, 104–108 graph of, 90–91 of horizontal lines, 115 of a hyperbola, 762 involving fractional expressions, 52–53 involving fractional powers, 54 involving radicals, 53 linear, 45–46, 115–116, 118–120 of lines, 113–116 logarithmic, 361–364 matrix, 694–697 modeling with see Mathematical models nonlinear, 45 of a parabola, 194 polynomial, 277–279 Properties of Equality and, 44 quadratic, 46–52 of quadratic type, 53–54 roots of, 254 of a shifted conic, 780–781 solving for unknown functions, 222, 233 solving using analogy strategy, 138–139 two-intercept form of, 121 in two variables, 90–91 of vertical lines, 115 Equations, trigonometric, 527, 561–570 with functions of multiple angles, 566–567 solving, 561–565, 567–568 Equivalent equations, 44 Equivalent inequalities, 76 Equivalent systems, 652 Eratosthenes, 476, 825 Error-correcting codes, 38–39 Euclid, 532 Eudoxus, 902 Euler, Leonhard, 138, 288, 332, 708 Even function, 188–189, 193, 222 Even-odd identities, 528 Even-odd properties, 413–414 Everest, Sir George, 505 Existence theorem, 283 Exponential data, linearizing, 389–390 Exponential equations, 358–361 Exponential form, 342–343 Exponential function, 327, 328–341 compared with power function, 332 compound interest, 334–336 family of, 330 graphs of, 329–332 natural, 332–334 transformations of, 331, 333 Exponential growth, 341 Exponential modeling, 369–376, 386–387, 390–392 Exponential notation, 13, 16–17 Exponents fractional, 19, 31, 54 integer, 12–16 integer, exponential notation, 13 integer, zero and negative exponents, 13, 16 Laws of, 14–16, 19, 328 rational, 19–20 Extraneous solutions, 53 Extreme values, 193–203 of quadratic functions, 194–198 using graphing devices for, 198–200 Factoring common factors, 27–28 complex solutions and, 295 differences and sums of cubes, 29–30 differences of squares, 29 expressions with fractional exponents, 31 Finding limit by canceling common factors, 894 by grouping, 31 inequalities, 79–81 polynomials, 291–293, 294 quadratics, 28 solving trigonometric equations by, 563–565 by trial and error, 28 Factoring formulas, 29 Factor Theorem, 269–270, 272 Falling objects, instantaneous velocity of, 905 False equations, 654 Family of equations, 57 of exponential functions, 330 of lines, graphing, 118 of logarithmic functions, 344 of polynomials, 261 of power functions, 160 Fechner, Gustav, 347 Fermat, Pierre de, 20, 87, 288 Ferrari, 282 Fibonacci, Leonardo, 825 Fibonacci numbers, 678, 825–826, 829, 832 Fitt’s Law, 352 FM (frequency modulation) radio, 428 Focal diameter, of parabolas, 748, 749 Focal length, 752 Focus of a conic, 795 of an ellipse, 753, 755, 756–757 of a hyperbola, 762, 766–767 of a parabola, 744, 752 prime, 752 FOIL method, 26 Force, modeling, 614–615 Forgetting, Law of (Forgetting Curve), 355, 357, 395 Fourier, Jean Baptiste Joseph, 427, 536 Fourier analysis, 30 Four-leaved rose, 591, 594 Fractal image compression, 600 Fractals, 600, 605–607 Fractional exponents, 19, 31, 54 Fractional expressions, 35 see also Rational expressions compound fractions, 38–39 solving equations involving, 52–53 Fractions compound, 38–39 LCD and adding, 5–6 partial, 715–721 properties of, Frequency, harmonic motion and, 443 Index Frequency modulation (FM) radio, 428 Functions, 146–247 algebra of, 214–215 average rate of change and, 174–178 combining, 214–222 common examples of, 148–149 composition of, 216–219 defined, 149–150 demand, 232 domain of, 153 equations of, 164–165 evaluating, 151–152 even, 188–189, 193, 222 extreme values, 193–203 graphing, 158–170, 306–312, 315, 329–332 greatest integer, 162 identity, 233 increasing/decreasing, 173–174 inverse, 226–230 iterates of, 223–224 limits of, 882–890 logistic, 223 methods for representing, 153–154 modeling with, 203–213 modeling with, guidelines for, 205 objective, 736, 737, 738 odd, 188–189, 193, 222 one-to-one, 225–226, 228–230 relations and, 171–172 transformations of, 182–193 trigonometric see Trigonometric functions Fundamental identities, 414–415, 493, 528 Fundamental Principle of Analytic Geometry, 90, 93 Fundamental Theorem of Algebra, 291 Galilei, Galileo, 816, 817 Galois, Evariste, 273, 282 Galton, Sir Francis, 247 Gateway Arch, 331 Gaudi, Antoni, 771 Gauss, Carl Friedrich, 294, 665, 834–835 Gaussian elimination, 652–653, 664–667 Gauss-Jordan elimination, 667–668 Gear ratio, 517 General conic equation, simplifying, 785–787 Geometric mean, 845 Geometric sequences, 838–846 Geometry, analytic, 742–819 see also Conics; Ellipses; Hyperbolas; Parabolas; Parametric equations GIMPS (Great Internet Mersenne Prime Search), 824 Global Positioning System (GPS), 643, 656–657 Golden ratio, 829 Googol, 352 Googolplex, 352 Grads, measuring angles with, 478 Graphical addition, 216 Graphical solutions, 104–108 compared with algebraic method, 104, 105–106 for equations, 104–108 for inequalities, 108 for systems of equations, 640–641 using graphing calculator, 102–104 Graphing calculators approximating area with, 925 choosing viewing rectangle, 426–427 for extreme values of functions, 198–200 pitfalls of, 890 for trigonometric graphs, 425–428 using, 102–104 zoom and trace features of, 884–885 Graphing devices see Graphing calculators Graphing functions, 158–170 exponential functions, 329–332 rational functions, 306–312, 315 Graphs of complex numbers, 596–598 of equations of two variables, 90–91 of nonlinear inequalities, 721–723 of polar equations, 587–596 of polynomials, 251–260 reflecting, 185–186 shifted, 776–780 shifts, horizontal, 183–185 shifts, vertical, 182–183, 184–185 stretching and shrinking, 186–188 of systems of inequalities, 723–728 Gravity, Newton’s Law of, 46, 126, 388 Greater than (Ͼ), Greatest integer function, 162–163, 166 Great Internet Mersenne Prime Search (GIMPS), 824 Great Trigonometric Survey of India, 505, 525 Grouping, factoring by, 31 Growth constant, 223 Half-angle formulas, 541, 543–546 Half-life of radioactive elements, 373–374 I5 Halley, Edmund, 894 Hamming, Richard, 39 Hardy, G.H., 840 Harmonic mean, 837 Harmonic motion, 417, 442–454 damped, 449–451, 569 modeling periodic behavior, 443–448, 459–462 simple, 443, 575 Harmonic sequences, 837 Heating degree-hour, 932–933 Heaviside, Oliver, 885 Heaviside function, 885 Herons’ Formula, 512–513 Hilbert, David, 103, 708 Hilbert’s tenth problem, 678 Hipparchus, 479 Homogenous coordinates, 794 Hooke’s Law, 127, 134, 931 Horizontal asymptotes, 301, 304–306, 307, 308, 910–911 Horizontal lines, 115, 225, 226 Horizontal line test, 225, 226 Horizontal shifts, of graphs, 183–185 Horizontal stretching and shrinking, of graphs, 187–188 Huygens, Christian, 805 Hyperbolas, 743, 762–770 with center at origin, 763–764 confocal, 769–770 conjugate, 769 constructing, 774 degenerate, 781 equation of, 766–767 finding tangent line to, 901 geometric definition of, 762 rotating, 784–785 shifted, 778–780 sketching, 764–767 with transverse axis, 764–766 Hyperbolic cosine function, 337 Hyperbolic sine function, 337 Hypocycloid, 808 Hypothesis, induction, 856 Identities addition and subtraction formulas for, 537 Pythagorean, 414, 493, 528 reciprocal, 413, 414, 493, 528 trigonometric, 413, 414–415, 492–494, 527, 528–534, 563–564 Identity function, 233 Identity matrices, 689–690 I6 Index Image of x under f , 150 Imaginary axis, 596 Imaginary part, of complex numbers, 285 Incidence, angle of, 570 Inclination, angle of, 482 Inconsistent systems, linear equations, 644–645, 654, 668–670 Independent variables, 150 Index of refraction, 570 Index of summation, 828 Induction, mathematical, 854–860 conjecture and proof, 854–855 induction step, 855–856 principle of, 856–858 sums of powers and, 858–859 Induction hypothesis, 856 Inequalities, 76–87 see also Systems of inequalities, graphing absolute value, 81–82 equivalent, 76 graphic solutions for, 108 graphing, 721–723 linear, 77, 724 modeling with, 82–84 nonlinear, 77–81 proving by induction, 858–859 rules for, 76 with three factors, 80–81 Infinite geometric series, 843–844 Infinite series, 841–844 Infinity limits at, 908–913 symbol, Initial point, vectors, 607 Initial side, of angles, 468 Inner product, of matrices, 678–679 Installment buying, 851–852 Instantaneous rate of change, 177, 881–882, 904–905 defined, 904 estimating, 905 instantaneous velocity of falling objects, 905 Integer exponents, 12–16 Integers, as real number type, Intensity levels of sound, 347, 378–379 Intercepts, 92 Interest, on investment, 59–60 Intermediate Value Theorem, 255, 283 Intersect command, in calculators, 106 Intersections finding intersection points, 562–563 of intervals, of sets, Intervals, 7–8 graphing, open and closed, test values for, 78 unions and intersections, Invariants under rotation, 789, 791 Invariant Theory, 710 Inverse cosecant, 556 Inverse cosine, 553–554 Inverse cotangent, 556, 557 Inverse functions, 226–230 defined, 227 finding, 227–230 linear functions becoming, 232 properties of, 227 Inverse numbers, Inverse of matrices, 689–693, 695 Inverse secant, 556 Inverse sine, 551–552 Inverse square law for sound, 382 Inverse tangent, 554–556 Inverse trigonometric functions, 527–528, 550–559 solving trigonometric equations using, 567–568 Inverse variation, 125–126 Invertibility criterion, 707 Involute of a circle, 810 Irrational numbers, Irreducible quadratic factor, 297–298 Iterates of functions, 223–224 Mandelbrot set and bounded, 605–607 Joint variation, 126 Jordan, Camille, 273 Kantorovick, Leonid, 735 Karmarkar, Narendra, 737 Kepler, Johannes, 388, 389, 580, 758 Kepler’s Third Law, 23, 129 Kirchhoff’s Laws, 659 Knuth, Donald, 165 Koopmans, T.C., 735 Kovalevsky, Sonya, 188 Laminar flow, law of, 156 Latus rectum, 748, 761 Law enforcement, use of mathematics for, 344–345 Law of Cooling, Newton’s, 375–376, 381 Law of Cosines, 508–516, 561 Law of Forgetting (Forgetting Curve), 355, 357, 395 Law of Gravity, 46, 126, 388 Law of laminar flow, 156 Law of Sines, 501–508 Law of the Lever, 69, 748 Law of the pendulum, 127 Laws of Exponents, 14–16, 328 for rational exponents, 19 Laws of Logarithms, 352–358 LCD see Least common denominator (LCD) Leading coefficients, 250, 253 Leading entry in row-echelon form, 665 Leading terms, 250 end behavior of polynomial and, 252–254 Leading variable, 668 Learning curve, 368 Least common denominator (LCD) adding fractions, 5–6 using with rational expressions, 37–38 Least squares line, 240–242, 650–651 Left-hand limits, 887–888, 895–897 Lemniscates, as polar graph, 594 Length, vectors, 608, 610, 611 Lens equation, 56 Leontief, Wassily, 850 Less than (Ͻ), Lever, Law of the, 69, 748 Limaỗon, 592, 593, 594 Limit Laws, 890893 nding limits using, 894–895 limits at infinity and, 911 Limits, 880–933 derivative problems, 902–904 finding by direct substitution, 893–894 finding by using algebra and Limit Laws, 894–895 of a function, 882–890 instantaneous rates of change, 881–882, 904–905 left- and right-hand limits, 895–897 Newton on, 902 special, 892–893 tangent line problems, 898–902 Limits, area problems, 881, 916–925 area defined, 920–922 area under a curve, 922–924 area under a graph, 929–931 estimating area using rectangles, 917–918 limit of approximating sums, 919–920 modeling with, 929–931 Limits at infinity, 908–913 defined, 909 finding, 912 Index functions with no limit at infinity, 913 at negative infinity, 910, 912 Limits of sequences, 913–915 defined, 913 finding, 914–915 limits of recursive sequences, 916 Linear and Quadratic Factors Theorem, 297–298 Linear depreciation, 122 Linear equations, 115 see also Systems of linear equations applying to rate of change, 118–120 graph of, 115–116 solving, 45–46 two-intercept form of, 121 Linear factors, 297–298 Linear fractional transformations, 302–303 Linear functions composing, 222 defined, 158 graphs of, 166 as mathematical models, 239–242 Linear inequalities, 77, 724 graphing systems of, 724–726 Linearizing exponential data, 389–390 power data, 390 Linear programming, 735–741 guidelines for, 737 Karmakar’s technique, 737 Linear speed, 473–474 Line of sight, 482 Lines, 111–123 of best fit, 240–242 family of, graphing, 118 general equation of, 115 parallel, 116–117 perpendicular, 117–118 point-slope form of equation of, 113–114 slope-intercept form of equation of, 114 slope of, 111–113 slope as rate of change, 118–120 vertical and horizontal, 115 Lissajous figure, 806 Lithotripsy, reflection property used in, 759 LnReg command, in calculator, 397 Local extrema, of polynomials, 260–261, 265 Local maximum, 199, 260 Local minimum, 199, 260 loga, 342 Logarithmic equations, 361–364 Logarithmic form, 342–343 Logarithmic functions, 327, 342–352 applications of, 365–366, 376–379 common (base 10) logarithms, 346–347 family of, 344 graphs of, 343–346 natural logarithms, 347–349 properties of, 343 Logarithmic model, 397 Logarithmic scales, 376–379 Logarithms, Laws of, 352–358 Logistic command, in calculator, 392, 397 Logistic curves (or logistic growth model), 334, 339, 392–393, 397 Logistic function, 223 Logistic population growth, 878–879 Longbow curve, 808 Long division partial fractions and, 720 of polynomials, 265–267 LORAN (LOng RAnge Navigation), 768 Lorenz Contraction Formula, 898 Lotka, Alfred J., 696–697 Lower bounds, 276, 278 Machine, function as, 150 Magnetic resonance imaging (MRI), 746 Magnitude of an earthquake, 377–378 of a star, 358 of vectors, 608, 610 Main diagonal, of matrices, 689 Major axes, of ellipses, 754, 755 Majority voting, 682 Mandelbrot, Benoit, 600, 605 Mandelbrot set, 605–607 Manning Equation, 23–24 Mathematical models, 58–75 constructing, 59–67 defined, 239 finding line of best fit, 240–242 functions as, 203–213 guidelines for, 58–59 guidelines for modeling functions, 205 linear functions as, 239–242 logarithmic model, 397 measuring fit, 242–243 using inequalities, 82–84 variation, 123–129 Matijasevicˇ, Yuri, 678 Matrices, algebra of, 675–687 see also Determinants applied to computer graphics, 683–684, 700–703 I7 Equality of matrices, 767 identity matrices, 689–690 inverse of matrices, 689–693, 695 matrix equations, 681–682, 694–697 multiplication, 678–683, 700 no Zero-Product Property for, 699 rotating images in plane, 792–794 rotating points in plane, 792 rotation of axes formulas, 791 singular matrix, 693 square roots of matrix, 687 stochastic matrices, 683 sum, difference, and scalar product, 767–778 transition matrix, 688–689, 697 Matrices, solving linear equations, 662–675 augmented matrix, 635, 662–663 elementary row operations, 663–664 Gaussian elimination, 664–667 matrix defined, 662 reduced row-echelon form, 665, 667–668 row-echelon form, 665–667 Matrix equations, 681–682, 694–697 linear programming for, 735–741 local, 199, 260 modeling with functions to find, 207–208 Maximum command, in calculators, 200 Maximum value(s), maxima, 195–198, 203 Mean arithmetic, 838 geometric, 845 harmonic, 837 Mersenne numbers, 824 Midpoint formula, 90 Mill, John Stuart, 112 Minimum command, in calculators, 200 Minimum value(s), minima, 195–198, 203 local, 199, 260 modeling with functions to find, 209–210 Minor axes, of ellipses, 754, 755 Modeling see also Mathematical models with area, 929–931 cylindrical projection, 630–631, 632 defined, 203 with equations, 58–75 exponential, 369–376, 386–387, 390–392 force and velocity, 612–615 harmonic motion, 442–454 I8 Index Modeling (continued ) with linear systems, 646–648, 656–657, 672–673 logarithmic, 376–379 with logistic functions, 392–393 mapping world, 630–633 path of a projectile, 816–818 with polynomial functions, 320–323 population growth, 327, 369–373, 386–387, 392–393 with power functions, 388–392 prey/predator models, 432–433, 464, 696–697 with recursive sequences, 874–876 standing waves, 576–577 stereographic projection, 631, 632, 633 surveying, 522–525 traveling waves, 575–576 using linear programming, 735–741 using matrix equations, 696–697 Modulus of complex numbers, 597–598 Monomials, 24, 250, 251–252 Mortgage payments, 852 amortizing a mortgage, 854 MRI (magnetic resonance imaging), 746 Multiple angles, trigonometric functions of, 566–567 Multiplication of algebraic expressions, 26 of complex numbers, 286, 599–600 of functions, 214 of inequalities, 76 of matrices, 678–683, 700 of polynomials, 25–26 of rational expressions, 36 of vectors by scalars, 608, 611 Multiplicative identity, Multiplicities, zeros and, 259, 293–295 Napier, John, 346 Nash, John, 850 Natural exponential functions, 332–334 Natural logarithms, 347–349 Natural numbers, Nautical mile, 476 Navigation bearings, 511 LORAN, 768 Negative exponents, 13, 16 Negative numbers, square roots of, 287–288 Negative of image, 687 Newton, Sir Isaac, 758, 766, 816, 894–895, 902 Newton’s Law of Cooling, 375–376, 381, 878 Newton’s Law of Gravitation, 46, 126, 388 n-leaved rose, 591, 594 n! (n factorial), 863 Nodes, standing wave, 576–577 Noether, Emmy, 710 Nonlinear equations, 45 Nonlinear inequalities, 77–81 graphing, 721–723 guidelines for solving, 79 Notation exponential, 13, 16–17 scientific, 16–17 set-builder, sigma, 828–830 summation, 828 use in problem solving, 138 Nowak, Martin, 824 nth root, 18–19 of complex number, 601–602 Numbers complex see Complex numbers converting sound, pictures, and text into, 30 imaginary, 285–286 inverse, irrational, negative, ordered pair of, 87 polygonal numbers, 847–848 prime, 824, 825 rational, 2–3 real see Real numbers Reference, 404–406, 411–412 representing functions with, 154 square, 847 using geometric shapes to represent, 847 Numerators, rationalizing, 40, 895 Numerical method finding values of functions with, 412 for finding zeros, 283–284 to find trigonometric ratios, 480 Objective function, 736, 737, 738 Oblique asymptotes, 310 Oblique triangles, 501 Odd functions, 188–189, 193, 222 One-sided limits, 887–888, 895–897 One-to-one function, 225–226 finding inverse of, 228–230 Orbits see Planetary orbits Ordered pair, of numbers, 87 relation as collection of, 171 Origin (O), 6, 87, 582 hyperbola with center at, 763–764 symmetry with respect to, 95 p, value of, 414 Parabolas, 640, 721, 743, 744–752 confocal, 782 constructing, 772–774 family of, 749 focal diameter of, 748, 749 focal point of, 750 geometric definition of, 744 graph of, 91 graph of shifted, 777–778 with horizontal axis, 747–748 latus rectum of, 748 as quadratic function, 194 sketching, 748–749 with vertical axis, 745–746 Parallax, 487 Parallel lines, 116–117 Parameters, 57, 656, 801, 803 Parametric curve, graphing, 805–806 Parametric equations, 801–810 for cycloid, 804–805 eliminating parameter, 803 graphing parametric curves, 805–806 for path of projectile, 816–818 plane curves and, 801–802 polar equations in parametric form, 806 Pareto, Vilfredo, 357 Pareto’s Principle, 357 Partial fraction decomposition, 716–720 Partial fractions, 715–721 Partial sums, of sequences, 827–828, 834–836, 840–841 Pascal, Blaise, 805, 858 Pascal’s triangle, 861–863, 864 Pattern recognition, 138, 847–848 Paulos, John Allen, 242 Pendulum, law of the, 127 Pentagonal numbers, 847 Perfect square, 30, 48 Perihelion, 760, 801 Perilune, 761 Period amplitude and, 423–425 harmonic motion and, 443 Periodic behavior, modeling, 443–448, 459–462 Periodic functions, 419, 427, 431 Periodic properties, 434 Periodic rent, 849 Index Perpendicular lines, 117–118 Phase shift, of sine and cosine curves, 423–425 pH scale, 376–377 Pi (p), value of, 414 Piecewise defined function, 151, 888 graphing, 161–162 limit of, 896–897 Plane(s) bounded and unbounded regions, 725 complex, 596 coordinate, 1, 87–88 as graph of linear equation in three variables, 654 Plane curves, 801–802 Planetary orbits eccentricities of, 758 Kepler’s description of, 23, 129, 580 perihelion and aphelion, 760, 801 power model for planetary periods, 388–389 Plurality voting, 682 Point-slope form of equation of lines, 113–114 Polar axis, 582 Polar coordinates, 581, 582–587 graphing polar equations, 587–596 relationship between rectangular coordinates and, 584–585 Polar equations, 585–586 of conics, 795–801 family of, 593 graphs of, 587–596 in parametric form, 806 Polar form of complex numbers, 598–600 Polya, George, 138 Polygonal numbers, 847–848 Polynomial function, 249, 250, 320–323 Polynomials, 24 adding and subtracting, 25 of best fit, 320–323 defined, 250 degrees of, 24–26 dividing, 265–272 end behavior of, 252–254, 255 family of, 261 graphs of, 251–260 guidelines for graphing, 255 local extrema of, 260–261 nested form, 272 product of, 25–26 quadratic, 660–661 real zeros of, 254, 272–284 Tchebycheff, 549 zeros of, 254–260, 269 Population growth, 327, 369–373, 386–387, 392–393 carrying capacity and, 393 logistic, 878–879 Power data, linearizing, 390 Power functions compared with exponential functions, 332 graphs of, 160, 166 modeling with, 388–392 Powers finding, using DeMoivre’s Theorem, 601 formulas for lowering, 544 Predator/prey models, 432–433, 464, 696–697 Preference voting, 682 Present value, 340 of an annuity (Ap), 850–851 Prime focus, 752 Prime numbers, 824, 825 Principal, compound interest and, 334 Principal nth root, 18 Principal square root, 17 of complex numbers, 288 Principle of Mathematical Induction, 856–858 Principle of Substitution, 26 Problem solving, principles, 138–141 Products see also Multiplication of functions, 214 inner, 678–679 of polynomials, 25–26 positive/negative, 77 scalar, 676, 677, 678 sign of, 78 Product-sum formulas, 541, 546–547 Projectile modeling path of, 51–52, 816–818 range of, 569 Projection cylindrical, 630–631, 632 stereographic, 631, 632, 633 Projection laws, 514 Projection of vectors, 622–623 Prolate cycloid, 808 Proof by contradiction, 133 mathematical induction and, 854–855 Proportionality, 123–129 constant of, 124 direct, 123–125 inverse, 125–126 I9 joint, 126 Pure imaginary number, 285–286 Pythagoras, 54 Pythagorean identities, 414, 493, 528 Pythagorean Theorem, 54, 144 Quadrantal angles, 490 Quadrants, of coordinate plane, 87 Quadratic equations, 46–52 complex roots of, 288–289, 290 factoring, 28 form of, 47 fourth-degree equation of quadratic type, 53–54 path of projectile modeled by, 51–52 solving by completing the square, 48–49 solving by factoring, 47 solving simple, 47–48 trigonometric identities and, 563–564 Quadratic factors, 297–298 Quadratic formula, 49–50 complex solutions and, 295 discriminant of, 50–51 using Rational Zeros Theorem and, 274–275 Quadratic function, 194–198 extreme values of, 195–198 graphing, 194 maximum/minimum value of, 195–198 standard form of, 194–195 Quadratic inequalities, 78–79 Quadratic polynomial of best fit vs exact fit, 660–661 QuadReg command, in calculator, 661 Quotients, 266 difference quotients, 151 in division, of functions, 214 inequalities and, 79–80 positive/negative, 78 Radian measure, of angles, 468–469, 472 Radicals, 17–19 combining, 19 equations for, 53 nth root and, 18–19 using, with rational exponents, 20 Radio, AM and FM, 428 Radioactive decay model, 374–375 Radioactive elements, half-lives of, 373–374 Radiocarbon dating, 351, 360 Ramanujan, Srinivasa, 840 I10 Index Range finding from graphs, 161 of functions, 150 of an inverse function, 227 of a projectile, 569 of a relation, 171 Rate of change average, 174–178, 904 concavity and changing, 181 constant, 178 instantaneous, 177, 881–882, 904–905 slope as, 118–120, 175 Rational exponents, 19–20 Rational expressions, 35–44 adding and subtracting, 37–38 avoiding common errors, 40–41 compound fractions, 38–39 multiplying and dividing, 36–37 rationalizing denominator or numerator, 40 simplifying, 36 Rational functions, 299–316 graphing, 306–312, 315 simple, 300–302 slant asymptotes and end behavior, 309–311 transformations, 302–303, 315–316 Rationalizing the denominator or numerator, 20–21, 40, 895 Rational numbers, 2–3 Rational zeros see Real zeros, of polynomials Rational Zeros Theorem, 272–275, 295 Real axis, 596 Real number line, 6, 9–10 Real numbers, 1, 2–12 absolute values and distance, 8–10 Law of Exponents and, 328 natural numbers as, order of (less than, greater than), properties of, 3–6 real lines and, sets and intervals, 6–8 Real part, of complex numbers, 285 Real zeros, of polynomials, 254, 272–284 Reciprocal functions, 166 Reciprocal identities, 413, 414, 493, 528 Reciprocal relations, 480 Reciprocals of inequalities, direction of inequality and, 76 Rectangles, using to estimate area, 917–918 Rectangular coordinates, 581, 584–586 Recursive sequences, 824–825 limits of, 916 as models, 874–876 Reduced row-echelon form of a matrix, 665, 667–668 Reduction formulas, 418, 442 Ref command, in calculator, 667 Reference angle, 491–492 Reference numbers, 404–406 finding value of trigonometric function with, 411–412 Reflecting graphs, 185–186, 343, 345 Reflection, total internal, 570 Reflection property of ellipses, 759 of hyperbolas, 767 of parabolas, 750 Refraction, angle of, 570 Refraction, Index of, 570 Regression line, 240–242, 650–651 Relations, 171–172 reciprocal, 480 Relativity, Theory of, 157, 710, 816 Remainders, 266 Remainder Theorem, 268–269 Repeating decimal, Resistance, electrical, 43, 312 Resultant force, 614–615 Rhind papyrus, 75, 716 Richter, Charles, 377 Richter scale, 377–378 Right angles, 478–483 Right-hand limits, 887–888, 895–897 Right triangle trigonometry, 467–468, 478–487 applications, 481–483 Rise, vs run in slope, 111 Rivest, Ron, 308 Robinson, Julia, 678 Romanus, Adrianus, 414 Root functions, 166 Root-mean-square (rms) method, 448 Roots complex, 288–289, 290 of complex numbers, 601–603 of equations, 44 of polynomial equations, 254 of unity, 299 Roses (polar curve), 591, 594 Rotation of axes, 783–794 eliminating xy-term, 786–787 equations of, 784 graphing rotated conics, 787–789 matrix form of formulas, 791 rotating hyperbolas, 784–785 Row-echelon form of a matrix, 665–667 reduced, 665, 667–668 solving linear equations, 666–667, 669 Row transformations, of determinants, 707–708 Rref command, in calculators, 668, 673 RSA code, 308–309 Rule of Signs (Descartes), 275, 297 Rules, for inequalities, 76 Run, vs rise in slope, 111 Scalar product, of matrices, 676, 677, 678 Scalars, 607, 608 Scatter plots, 239–242, 320–323, 386–387 Scientific notation, 16–17 Secant formula for, 488 inverse, 556 trigonometric ratios, 479 Secant function, 408 graphing, 435, 436, 440 secant curves, 439, 440 special values of, 410 Secant line, average rate of change as slope of, 175 Sectors, circular, 472–473 Semiperimeter, 512 Seq mode, calculators, 823–824 Sequences, 821–828 arithmetic, 833–838 defined, 822 Fibonacci, 678, 825–826, 829, 832 finding terms of, 823–824, 840 geometric, 838–846 harmonic, 837 infinite series, 841–844 partial sums of, 827–828, 834–836, 840–841 polygonal numbers, 847–848 properties of sums of, 830 recursive, 824–825, 874–876, 916 sigma notation of, 828–830 Sequences, limits of, 913–915 Series, infinite, 841–844 Set-builder notation, Sets as collection of objects, unions and intersections, Shamir, Adi, 308 Shanks, William, 414 Shifted conics, 775–783 Sieve of Eratosthenes, 825 Sight, line of, 482 Index Sigma notation, 828–830 Signs, of trigonometric functions, 411, 490 Similarity and similarity ratio, in trigonometry, 499–501 Simple harmonic motion, 443, 575 Sine addition and subtraction formulas for, 535, 536, 541 curves, 422, 428, 461–462 double-angle formula for, 542, 786 formula for, 488 half-angle formula for, 544 inverse, 551–552 Law of, 501–508 product-sum formula for, 546 sum of sines and cosines, 538–539 sum-to-product formula for, 547 trigonometric ratios, 479 Sine function, 408 applications, 432–433 graphing, 418–420 graphing transformations of, 420–425 periodic properties of, 419 shifted curves, 423–425 special values of, 410 SinReg command, in calculator, 461, 462 Sinusoidal curves, 422, 431 Slant asymptotes, 309–310 Slope indicating rate of change, 118–120, 175 of lines, 111–113 Slope-intercept form of equation of a line, 114 Slope of the line tangent to a curve, 899–900 Snell’s Law, 570 Solutions see Roots Sound see also Harmonic motion intensity levels of, 347, 378–379 inverse square law for, 382 Special Product Formulas, 26–27, 34 Special Theory of Relativity, 816 Species, study of survival of, 688–689 Species-Area relationship, 357–358 Sphere, area of, 156 Splines, polynomial curves, 249, 252, 256 Spring constant, 127, 452, 931 Square matrix, 704–708 Square numbers, 847 Square roots, 17–19 of negative numbers, 287–288 nth root and, 18–19 Squaring function, 150 Standard form, of equation of a circle, 93 Standard position, of angles, 470–471 Standing waves, 576–577 Stars, modeling brightness of, 446 Step functions, 163, 170 Stereographic projection, 631, 632, 633 Stochastic matrices, 683 Substitution, Principle of, 26 Substitution, trigonometric, 532 Substitution method for solving linear systems, 637–638 using direct substitution for finding limits, 893–894 Subtraction of complex numbers, 286 of inequalities, 76 overview of, of polynomials, 25 of rational expressions, 37–38 of vectors, 608 Subtraction and addition formulas, 535–541 Summation notation, 828 Summation variable, 828 Sums of cubes, 30 of functions, 214 of infinite geometric series, 843–844 limits of approximating, 919–920 of matrices, 676–678 partial sums of sequences, 827–828, 834–836, 840–841 of powers, 858–859 of sequences, properties of, 830 of sines and cosines, 538–539 Sum-to-product formulas, 547 Supplement of angle, 504 Surveying, 522–525 using triangulation for, 504–505 Symmetry, 95–96 tests for, 591–592 Synthetic division, 267–268 Systems of equations, 635, 636–644 elimination method for solving, 638–640 graphical method for solving, 640–641 modeling with, 646–648 substitution method for solving, 637–638 Systems of inequalities, graphing, 723–728 see also Inequalities Systems of linear equations dependent and inconsistent, 644–646, 654–656 graph of, 654 I11 modeling with, 646–648, 656–657, 672–673 several variables, 651–661 two variables, 644–651 using Cramer’s rule for solving, 708–711 writing as matrix equations, 681–682 Table command, in calculators, 824 Tables, finding limits using, 884–885 Taking cases, 139 Tangent, 488, 547 addition and subtraction formulas for, 535, 541, 560 double-angle formula for, 542 half-angle formula for, 544 inverse, 554–556 to parabola, 773, 774 trigonometric ratios, 479 Tangent function, 408 graphing, 434–439 special values of, 410 tangent curves, 437–438 Tangent line, 898–902 to a hyperbola, finding, 901 Taussky-Todd, Olga, 672 Taylor, Brook, 436, 834 Tchebycheff, P.L., 549 Tchebycheff polynomials, 549 Terminal points reference numbers and, 404–406 on unit circle, 401–404 of vectors, 607 Terminal side, of angles, 468 Terminal velocity, 338 Terms combining like, 25 of polynomial, 24 Terms, of sequences defined, 822 finding, 823–824, 834, 840 for recursive sequences, 825 Test points, graphing, 255, 256, 257, 722 Test values for intervals, 78 Thales of Miletus, 482 Theodolite, 504 Theory of Relativity, 157, 710, 816 Tide, modeling height of, 459–462 Torricelli’s Law, 156, 232, 325 Total internal reflection, 570 Trace command, in calculators, 106, 199, 725, 884–885 Transformations of exponential functions, 331, 333 I12 Index Transformations (continued) of functions, 182–193 by matrix multiplication, 700 of monomials, 251–252 of rational functions, 302–303, 315–316 of sine and cosine functions, 420–425 Transition matrix, 688–689, 697 Translation of image, 794 Transverse axes, of hyperbolas, 762, 764–766 Traveling waves, 575–576 Triangles ambiguous case, 503–505, 508 area of, 494–495, 512–513, 711–712, 714–715 Pascal’s triangle, 861–863, 864 right triangle trigonometry, 467–468, 478–487 solving height problems, 62–63 solving oblique, 501 special, 479–481 Triangular form, of linear systems, 651–652 Triangular numbers, 847 Triangulation, for surveying, 504–505 Trigonometric equations, 527, 561–570 Trigonometric functions, inverse, 527–528, 550–559, 567–568 Trigonometric functions, of angles, 466–525 defined, 488 reference angle and, 491–492 relationship to trigonometric functions of real numbers, 489 signs of, 490 Trigonometric functions, of real numbers, 398–465 of angles, 409 defined, 408 domains of, 411 even-odd properties, 413–414 relationship to trigonometric functions of angles, 489 signs of, 411 trigonometric identities, 413, 414–415 unit circle, 400–408 values of, 411–414, 436 Trigonometric graphs of cosecant and secant functions, 439–440 graphing devices used for, 425–428 of sine and cosine functions, 418–420 of tangent and cotangent functions, 434–439 Trigonometric identities, 527, 528–534 of angles, 492–494 basic types of, 528 proving, 529–532 quadratic equations and, 563–564 of real numbers, 413, 414–415 simplifying, 528–529 Trigonometric ratios, 467, 478–479, 480, 481, 488 Trigonometric substitution, 532 Trinomials, 24 Triple-angle formula, 543 Trochoid, 808 Tsu Ch’ung-chih, 414 Turing, Alan, 103, 178 Two-intercept form of linear equation, 121 Two-sided limits, 895 Unbounded regions, of planes, 725 Unbreakable codes, 308–309 Unions of intervals, of sets, Unit circle, 400–408 points on, 400 reference numbers, 404–406, 411–412 terminal points, 401–404 Unit vector, 611 Universal machine, 178 Upper and Lower Bounds Theorem, 276–277, 278 Upper bounds, 276, 278 Value of f at x, 150 Variables correlation of, 242–243 defined, 24 dependent and independent, 150 leading, 668 in linear systems, 644–661 summation, 828 Variation, modeling direct, 123–125 inverse, 125–126 joint, 126 Variation in sign, 255 Vectors, 581–582 algebraic operations on, 610–611 analytic description of, 609–612 angle between, 619 calculating components of, 621–622 direction of, 608, 609, 612, 620–622 dot product of, 617–620 expressing in terms of i and j, 611–612 geometric description of, 607–608 horizontal and vertical components, 609, 612 modeling velocity and force, 612–615 orthogonal, 619–620 perpendicularity, checking for, 620 properties of, 611 use of, 607 wind as, tacking against, 626 zero, 608, 611 Velocity estimating, 907 instantaneous, 904–905 modeling, 612–614 terminal, 338 of traveling waves, 575–576 Vertical asymptotes, 301, 303–308, 434–436, 886–887 Vertical axes, of parabolas, 745–746 Vertical lines, 115 Vertical line test, 163–164 Vertical shifts, graphs, 182–183, 184–185 Vertical stretching and shrinking, graphs, 186–187 Vertices of ellipses, 754, 755 of feasible region, 737, 739 of hyperbolas, 762, 766–767 of parabolas, 744 of systems of inequalities, 723, 724 Viốte, Franỗois, 49, 498 Viewing rectangle, of graphing calculator, 102 Voltage, measuring, 448 Volterra, Vito, 696–697 Von Neumann, John, 178 Voting, fair methods, 682–683 Wankel, Felix, 809 Wavelet theory, 30 Waves standing, 576–577 traveling, 575–576 Weather prediction, 562 Weber-Fechner Law, 378 Whispering galleries, reflection property used in, 759 Witch of Maria Agnesi (curve), 809 Words, representing functions with, 153, 154 Work calculating with dot product, 623–624 modeled by area, 929–931 Index x-axis, 87, 95 x-coordinate, 87 x-intercepts, 92 graphing rational functions and, 306–312 y-axis, 87, 95 y-coordinate, 87 y-intercepts, 92 graphing rational functions and, 306–312 Zero(s) additive identity, complex, 291–299 Factor Theorem and, 269–270 multiplicities and, 259, 293–295 numerical method of finding, 283–284 of polynomials, 254–260, 269 Rational Zeros Theorem, 272–275, 295 real, 254, 272–284 Zero exponents, 13 Zero-Product Property, 47, 563 Zeros Theorem, 293 Zero vector, 608, 611 Zoom feature, in calculators, 884 ZSquare command, 104 I13 This page intentionally left blank Photo Credits This page constitutes an extension of the copyright page We have made every effort to trace the ownership of all copyrighted material and to secure permission from copyright holders In the event of any question arising as to 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Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Precalculus: Mathematics for Calculus, Fifth Edition, Enhanced WebAssign Edition James Stewart,... mathematics as a problem-solving endeavor In this book we have included all these methods of teaching precalculus as enhancements to a central core of fundamental skills These methods are tools to be