Sullivan precalculus enhanced with graphing utilities 6th txtbk

www.elsolucionario.net http://www.elsolucionario.net LIBROS UNIVERISTARIOS Y SOLUCIONARIOS DE MUCHOS DE ESTOS LIBROS LOS SOLUCIONARIOS CONTIENEN TODOS LOS EJERCICIOS DEL LIBRO RESUELTOS Y EXPLICADOS DE FORMA CLARA VISITANOS PARA DESARGALOS GRATIS www.elsolucionario.net Innovative Technology to Help You Succeed MyMathLab can improve any learning environment—whether you are taking a lab-based, hybrid, fully online, or a traditional lecture-style course INTERACTIVE FIGURES Math comes alive with new Interactive Figures in MyMathLab! Your instructor may choose to assign assessment RVFTUJPOTUIBUBSFXSJUUFOUPBDDPNQBOZFBDImHVSF5IJTJOUFSBDUJPOXJMMMFBEZPVUPGVMMZVOEFSTUBOELFZ mathematical concepts in a hands-on, engaging way A HISTORY OF SUCCESS 3FTVMUTTIPXUIBUZPVDBOJNQSPWFZPVSHSBEFCZVTJOHUIFWJEFPT BOJNBUJPOT JOUFSBDUJWFmHVSFT TUFQCZTUFQ FYBNQMFT BOEQFSTPOBMJ[FEGFFECBDLJO.Z.BUI-BC5PTFFUIFHSPXJOHMJTUPGDBTFTUVEJFTGPSZPVSTFMG  visit www.mymathlab.com/success-stories www.mymathlab.com www.elsolucionario.net Prepare for Class “Read the Book” Feature Description Benefit Page Every chapter begins with… Chapter Opening Article & Project Each chapter begins with a current article and ends with a related project The Article describes a real situation The Project lets you apply what you learned to solve a related problem 250, 353 NEW! Internet-based Projects The projects allow for the integration of spreadsheet technology that students will need to be a productive member of the workforce The projects allow the opportunity for students to collaborate and use mathematics to deal with issues that come up in their lives 250, 353 Every section begins with… Learning Objectives Each section begins with a list of objectives Objectives also appear in the text where the objective is covered These focus your studying by emphasizing what’s most important and where to find it 271 Most sections contain… Most sections begin with a list of key concepts to review with page numbers Ever forget what you’ve learned? This feature highlights previously learned material to be used in this section Review it, and you’ll always be prepared to move forward 271 Now Work the ‘Are You Prepared?’ Problems Problems that assess whether you have the prerequisite knowledge for the upcoming section Not sure you need the Preparing for This Section review? Work the ‘Are You Prepared?’ problems If you get one wrong, you’ll know exactly what you need to review and where to review it! 271, 282 “Now Work ” These follow most examples and direct you to a related exercise We learn best by doing You’ll solidify your understanding of examples if you try a similar problem right away, to be sure you understand what you’ve just read 280 PROBLEMS WARNING Warnings are provided in the text These point out common mistakes and help you to avoid them 305 Explorations and Seeing the Concept These represent graphing utility activities to foreshadow a concept or solidify a concept just presented You will obtain a deeper and more intuitive understanding of theorems and definitions 104, 277 These provide alternative descriptions of select definitions and theorems Does math ever look foreign to you? This feature translates math into plain English 273 Calculus Icon These appear next to information essential for the study of calculus Pay attention if you spend extra time now, you’ll better later! 217 Showcase EXAMPLES These examples provide “how-to” instruction by offering a guided, stepby-step approach to solving a problem With each step presented on the left and the mathematics displayed on the right, students can immediately see how each step is employed 189–190 Model It! Marked with These are examples and problems that require you to build a mathematical model from either a verbal description or data The homework Model It! problems are marked by purple numbers It is rare for a problem to come in the form, “Solve the following equation” Rather, the equation must be developed based on an explanation of the problem These problems require you to develop models that will allow you to describe the problem mathematically and suggest a solution to the problem 296, 325 PREPARING FOR THIS SECTION In Words Examples and Problems www.elsolucionario.net Practice “Work the Problems” Feature Description Benefit Page “Assess Your Understanding” contains a variety of problems at the end of each section ‘Are You Prepared?’ Problems These assess your retention of the prerequisite material you’ll need Answers are given at the end of the section exercises This feature is related to the Preparing for This Section feature Do you always remember what you’ve learned? Working these problems is the best way to find out If you get one wrong, you’ll know exactly what you need to review and where to review it! 271, 282 Concepts and Vocabulary These short-answer questions, mainly Fill-in-the-Blank and True/False items, assess your understanding of key definitions and concepts in the current section It is difficult to learn math without knowing the language of mathematics These problems test your understanding of the formulas and vocabulary 283 Skill Building Correlated to section examples, these problems provide straightforward practice It’s important to dig in and develop your skills These problems provide you with ample practice to so 283–285 Mixed Practice These problems offer comprehensive assessment of the skills learned in the section by asking problems that relate to more than one concept or objective These problems may also require you to utilize skills learned in previous sections Learning mathematics is a building process Many concepts are interrelated These problems help you see how mathematics builds on itself and also see how the concepts tie together 285 Applications and Extensions These problems allow you to apply your skills to real-world problems These problems also allow you to extend concepts leamed in the section You will see that the material learned within the section has many uses in everyday life 285–287 Explaining Concepts: Discussion and Writing “Discussion and Writing” problems are colored red These support class discussion, verbalization of mathematical ideas, and writing and research projects To verbalize an idea, or to describe it clearly in writing, shows real understanding These problems nurture that understanding Many are challenging but you’ll get out what you put in 288 NEW! Interactive Exercises In selected exercise sets, applets are provided to give a “hands-on” experience “Now Work ” Many examples refer you to a related homework problem These related problems are marked by a pencil and yellow numbers If you get stuck while working problems, look for the closest Now Work problem and refer back to the related example to see if it helps 281 Every chapter concludes with a comprehensive list of exercises to pratice Use the list of objectives to determine the objective and examples that correspond to the problems Work these problems to verify you understand all the skills and concepts of the chapter Think of it as a comprehensive review of the chapter 348–351 PROBLEMS Chapter Review Problems The applets allow students to interact with mathematics in an active learning environment By exploring a variety of scenarios, the student is able to visualize the mathematics and develop a deeper conceptual understanding of the material www.elsolucionario.net 197–198 Review “Study for Quizzes and Tests” Feature Description Benefit Page Chapter Reviews at the end of each chapter contain… “Things to Know” A detailed list of important theorems, formulas, and definitions from the chapter Review these and you’ll know the most important material in the chapter! 346–347 “You should be able to…” Contains a complete list of objectives by section, examples that illustrate the objective, and practice exercises that test your understanding of the objective Do the recommended exercises and you’ll have mastery over the key material If you get something wrong, review the suggested examples and page numbers and try again 347–348 Review Exercises These provide comprehensive review and practice of key skills, matched to the Learning Objectives for each section Practice makes perfect These problems combine exercises from all sections, giving you a comprehensive review in one place 348–351 CHAPTER TEST About 15–20 problems that can be taken as a Chapter Test Be sure to take the Chapter Test under test conditions—no notes! Be prepared Take the sample practice test under test conditions This will get you ready for your instructor’s test If you get a problem wrong, watch the Chapter Test Prep video 351–352 CUMULATIVE REVIEW These problem sets appear at the end of each chapter, beginning with Chapter They combine problems from previous chapters, providing an ongoing cumulative review These are really important They will ensure that you are not forgetting anything as you go These will go a long way toward keeping you constantly primed for the final exam 352 CHAPTER PROJECTS The Chapter Project applies what you’ve learned in the chapter Additional projects are available on the Instructor’s Resource Center (IRC) The Project gives you an opportunity to apply what you’ve learned in the chapter to solve a problem related to the opening article If your instructor allows, these make excellent opportunities to work in a group, which is often the best way of learning math 353 NEW! Internet-based Projects In selected chapters, a web-based project is given The projects allow the opportunity for students to collaborate and use mathematics to deal with issues that come up in their lives 353 www.elsolucionario.net This page intentionally left blank www.elsolucionario.net PRECALCULUS Enhanced with Graphing Utilities Sixth Edition www.elsolucionario.net This page intentionally left blank www.elsolucionario.net PRECALCULUS Enhanced with Graphing Utilities Sixth Edition Michael Sullivan Chicago State University Michael Sullivan, III Joliet Junior College Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo www.elsolucionario.net I2 INDEX Bessel, Friedrich, 517 Best fit cubic function of, 192–193 line of, 142–143 Beta of a stock, 129 Bezout, Etienne, 772 Binomial(s), A22 cubing, A24 squares of (perfect squares), A24 Binomial coefficient, 833, 834 Binomial Theorem, 831–837 n to evaluate j , 831–833 expanding a binomial, 833–834 historical feature on, 836 proof of, 835 using, 833–835 Bisection Method, 208, 210–211 Blood alcohol concentration (BAC), 296 Bode, Johann, 811 Bode’s Law, 811 Bonds, zero-coupon, 326 Book value, 134–135 Boole, George, 864 Bounded graphs, 782 Bounds on zeros, 205–207 Box, volume and surface area of, A15 Brachistochrone, 693–694 Brancazio, Peter, 78 Branches of hyperbola, 657 Break-even point, 138 Brewster’s Law, 468 Briggs, Henry, 308 Bürgi, Joost, 308 Calculator(s), A10 See also Graphing utility(ies) approximating roots on, A83 converting between decimals and degrees, minutes, seconds on, 357 converting from polar coordinates to rectangular coordinates, 563, 564 to evaluate powers of 2, 272 functions on, 63–64 inverse sine on, 443 kinds of, A10 logarithms on, 306 trigonometric equations solved using, 462 Calculus “approaches infinity” concept, 217 approximating e x, 811 area under curve, 452 area under graph, 91 complicated functions in, 67 composite functions in, 255 derivative, 226 difference quotient in, 63, 92, 287, 309, 487 double-angle formulas in, 490 e in, 279, 811 end behavior of graphs, 187 exponential equations in, 314 functions and exponential, 272, 811 increasing, decreasing, or constant, 83, 334 local maxima and local minima in, 84–85 graph of polynomial functions in, 180 independent variable in, 398 inflection points, 79 integral, 492, 903 area under graph, 898–902 graphing utility to approximate, 903 Intermediate Value Theorem, 207 limit of a sequence in, B1 limits, 187, 217 logarithms and, 304, 314 partial fraction decomposition and, 759 polar equations and, 582 projectile motion, 688–690 radians in, 382 secant line and, 88, 92 simplifying expressions with rational exponents in, A87 Simpson’s Rule, 166 Snell’s Law and, 467 tangent line and, 542 trigonometric functions and equations in, 464, 466, 490, 498 trigonometric identities useful in, 492 turning points and, 185 values of function in, 62 of variations, 694 Cancellation Property, A35 Carbon dating, 330 Cardano, Girolamo, 208, 591, 863 Cardioid, 575–576, 581 Carlson, Tor, 341 Carrying capacity, 332–333 Cartesian (rectangular) coordinates, 2–3 converted to polar coordinates, 564–565 polar coordinates converted to, 562–563 polar coordinates vs., 560 polar equations graphed by converting to, 569–570 in space, 614 Cartesian (rectangular) form of complex number, 586–587 Cassegrain telescope, 669 Catenary, 162n, 645 Cayley, Arthur, 702, 755 Ceilometer, 513 Cell division, 327, 332 Cellular telephones, 57 Center of circle, 45 of hyperbolas, 657 of sphere, 622 Central angle, 358 Change-of-Base Formula, 306–307 Chu Shih-chieh, 836 Circle(s), 45–52, 635 arc length of, 359 area of, A15 area of sector of, 362–363 center of, 45 central angle of, 358 circumference of, A15 defined, 45 general equation of, 49 general form of equation of, 48–49 graphing, 46–47, 581 inscribed, 543 intercepts of, 47 polar equation of, 569–570, 572–574 radius of, 45 standard form of equation of, 45–46 unit, 46, 368–371 Circular functions, 370 Circular motion, 363–364 simple harmonic motion and, 544, 545 Circumference, A15 Clark, William, 507, 557 Clock, Huygens’s, 694 Closed interval, A73, A74 Coefficient, A22, A23 binomial, 833, 834 correlation, 143 damping, 547 leading, 211, A23 Coefficient matrix, 718 Cofactors, 736 Cofunctions, 510 names of, 380 Coincident lines, 705 Column index, 718, 742 Column vector, 746 www.elsolucionario.net Combinations, 851–853 defined, 851 listing, 851–852 of n distinct objects taken r at a time, 852 Combinatorics, 843 Common difference, 812 Common logarithms (log), 293, 305–306, 308 Common ratio, 818, 819 Commutative property of dot products, 607, 618 of matrix addition, 744, 749 of vector addition, 594 Complementary angles, 510 Complementary Angle Theorem, 510 Complement of event, 862 Complement of set, A2 Complement Rule, 862–863 Complete graph, Completing the square, A28–A29, A48–A49 identifying conics without, 671 Complex fraction, A39n Complex number(s), 602, 611 argument of, 586, 587 conjugates of, 586 De Moivre’s Theorem and, 588–589 geometric interpretation of, 585 magnitude (modulus) of, 586 in polar form converting from rectangular form to, 586–587 converting to rectangular form, 586–587 products and quotients of, 587–588 product of, 587–588 quotient of, 587–588 Complex numbers, A56–A59 addition, subtraction, and multiplication of, A56–A59 conjugates of, A58–A59 definition of, A56 equality of, A57 imaginary part of, A56 real part of, A56 in standard form, A56 power of, A59 reciprocal of, A58 Complex number system, A56 quadratic equations in, A60–A61 Complex plane, 585–588 defined, 585 imaginary axis of, 585 plotting points in, 585–586 real axis of, 585 Complex polynomial function, 211 Complex rational expressions, A39–A40 Complex roots, 589–591 Complex variable, 211 Complex zeros of polynomials, 211–216 Conjugate Pairs Theorem, 212–213 defined, 211 finding, 214–215 polynomial function with specified zeros, 213–214 Components of vectors, 596, 597 in space, 615 Composite functions, 251–258 calculus application of, 255 components of, 255 defined, 251 domain of, 252–255 equal, 254–255 evaluating, 252 finding, 252–255 forming, 251–252 Compound interest, 318–323 computing, 318–319 continuous, 320–321 defined, 318 doubling or tripling time for money, 323 INDEX effective rates of return, 321 formula, 319 future value of lump sum of money, 317–321 present value of lump sum of money, 322 Compound probabilities, 860 Compressions, 106–108, 110 Computer software graphing packages See Graphing utility(ies) Conditional equation, 469 Cone axis of, 635 generators of, 635 right circular, 635 vertex of, 635 Congruent triangles, A16–A18 Conics defined, 678 degenerate, 635 directrix of, 678 eccentricity of, 678 ellipse, 635, 645–656 with center at (h, k), 650–651 with center at the origin, 646–650 with center not at origin, 651–652 center of, 646 defined, 646, 678 eccentricity of, 656, 678, 680 foci of, 646 graphing of, 648–650 length of major axis, 646 major axis of, 646, 678 minor axis of, 646 solving applied problems involving, 653 vertices of, 646 focus of, 678 general form of, 670–671 hyperbolas, 634, 635, 656–669 asymptotes of, 662–663 branches of, 657 with center at (h, k), 663–665 with center at the origin, 657–661 with center not at the origin, 664 center of, 657 conjugate, 669 conjugate axis of, 657 defined, 657, 678 eccentricity of, 669, 680 equilateral, 669 foci of, 657 graphing equation of, 658–659 solving applied problems involving, 665–666 transverse axis of, 657, 678 vertices of, 657 identifying, 670–671 without a rotation of axes, 675–676 names of, 635 parabola, 9, 148–150, 635, 636–645 axis of symmetry of, 148, 636 defined, 636, 678 directrix of, 636 eccentricity of, 680 focus of, 636 graphing equation of, 637 solving applied problems involving, 641–642 with vertex at (h, k), 640–641 with vertex at the origin, 636–640 vertex of, 148, 636 paraboloids of revolution, 634, 641–642 parametric equations, 684–697 applications to mechanics, 693–694 for curves defined by rectangular equations, 691–694 cycloid, 693 defined, 684 describing, 687–688 graphing by hand, 684–685 graphing using graphing utility, 685–686 rectangular equation for curve defined parametrically, 686–688 time as parameter in, 688–691 polar equations of, 678–684 analyzing and graphing, 678–682 converting to rectangular equation, 682 focus at pole; eccentricity e, 679–681 rotation of axes to transform equations of, 671–673 analyzing equation using, 673–675 formulas for, 672 Conjugate of complex number, A58–A59 of conjugate of complex number, A59 of product of two complex numbers, A59 of real number, A59 of sum of two complex numbers, A59 Conjugate axis, 657 Conjugate golden ratio, 811 Conjugate hyperbola, 669 Conjugate of complex numbers, 586 Conjugate Pairs Theorem, 212–213 Connected mode, 98 Consistent systems of equations, 704, 705, 709 Constant(s), A6, A21–A22 Constant, limit of, 877 Constant functions, 83–84, 85, 95 Constant linear functions, 133–134 Constant rate job problems, A68–A70 Constraints, 785 Consumer Price Index (CPI), 326 Continued fractions, A41 Continuous compounding, 320–321 Continuous function, 97, 207, 886–888 Continuous graph, 180 Convergent geometric series, 821–824 Convergent infinite series, B5 Convergent sequence, B2–B3 Cooling, Newton’s Law of, 331–332 Coordinates, See also Rectangular (Cartesian) coordinates on graphing utility screen, of ordered triple, 614 of point on number line, A4 Copernicus, 364 Corner points, 782 Correlation coefficient, 143 Correspondence between two sets, 58 Cosecant defined, 508 graph of, 417–418 periodic properties of, 387 Cosecant function, 370 continuous, 887, 888 domain of, 385, 386 inverse, 455 approximate value of, 456 calculator to evaluate, 455–456 definition of, 455 exact value of, 455 range of, 386 Cosine(s) defined, 508 direction, 619–620 exact value of, 477 Law of, 531–537 in applied problems, 533–534 defined, 531 historical feature on, 534 proof of, 532 Pythagorean Theorem as special case of, 532 SAS triangles solved using, 532–533 SSS triangles solved using, 533 periodic properties of, 387 Sum and Difference Formula for, 476–477 trigonometric equations linear in, 483–484 www.elsolucionario.net I3 Cosine function, 369 continuous, 888 domain of, 385, 386, 401 graphs of, 398–413 amplitude and period, 402–404 equation for, 407–408 key points for, 404–407 hyperbolic, 287 inverse, 445–447 defined, 445 exact value of, 446–447 exact value of expressions involving, 453–454 implicit form of, 445 properties of, 401 range of, 386, 401 Cost(s) fixed, 42 marginal, 158 variable, 42 Cotangent defined, 508 periodic properties of, 387 Cotangent function, 370 continuous, 887, 888 domain of, 385, 386 graph of, 416–417 inverse, 455 approximating the value of, 456 calculator to evaluate, 455–456 definition of, 455 range of, 386 Counting, 843–848 addition principle of, 844–845 combinations, 851–853 defined, 851 listing, 851–852 of n distinct objects taken r at a time, 852 formula, 844 multiplication principle of, 845–846 number of possible meals, 845–846 permutations, 848–851 computing, 851 defined, 848 distinct objects without repetition, 849–851 distinct objects with repetition, 849 involving n nondistinct objects, 853–854 Counting numbers (natural numbers), 829, A3 Cramer, Gabriel, 702 Cramer’s Rule, 702, 732 inconsistent or dependent systems, 738 for three equations containing three variables, 737–738 for two equations containing two variables, 733–735 Cross (vector) product, 602, 623–628 defined, 623 determinants to find, 623–624 to find the area of a parallelogram, 626 to find vector orthogonal to two given vectors, 625–626 properties of, 624–625 algebraic, 624–625 geometric, 625 of two vectors in space, 623–624 Cube(s) of binomials (perfect cubes), A24 difference of two, A24 sum of two, A24 Cube function, 62, 96 Cube root, 93–94, 96, A83 complex, 589–591 Cubic function of best fit, 192–193 Cubic models from data, 192–193 Curve(s) defined by rectangular equations, 691–694 defined parametrically, 686–688 of quickest descent, 693–694 sawtooth, 551 I4 INDEX Curve fitting, 713 sinusoidal, 424–428 hours of daylight, 427–428 sine function of best fit, 428 temperature data, 424–426 Curvilinear motion, 688 Cycle of sinusoidal graph, 398, 403 Cycloid, 693 inverted, 693 Damped motion, 546–548 Damping factor (damping coefficient), 547 Data arrangement in matrix, 742 cubic models from, 192–193 fitting exponential functions to, 339–340 linear models from, 140–146 quadratic models from, 163–164 sinusoidal model from, 424–428 Day length, 178, 354 “Deal or No Deal” (TV show), 842 Decay, Law of, 329–331 See also Exponential growth and decay Decimals, A3 approximate, A3 converting between degrees, minutes, seconds and, 357–358 repeating, 822–823, A3 Declination of the Sun, 451 Decomposition, 609 Decreasing functions, 83–84, 85, 86–87 Decreasing linear functions, 133–134 Deflection, force of, 559 Degenerate conics, 635 Degree of monomial, A22 Degree of polynomial, 179–183, A23, A27 odd, 205, 213 Degree of power function, 180 Degrees, 356–358 converting between decimals and, 357–358 converting between radians and, 359–362 historical note on, 356 Demand equation, 159 De Moivre, Abraham, 588 De Moivre’s Theorem, 588–589 Denominator, A35 rationalizing the, A85 Dependent systems of equations, 705 containing three variables, 712–713 containing two variables, 708–709 Cramer’s Rule with, 738 matrices to solve, 724–726 Dependent variable, 62 Depreciation, 250 Depressed equation, 203 Depression, angle of, 512–513 Derivative, 226, 893–894 Descartes, René, 1, 2, 57 Determinants, 623–624, 702, 732–741 cofactors, 736 Cramer’s Rule to solve a system of three equations containing three variables, 737–738 Cramer’s Rule to solve a system of two equations containing two variables, 733–735 expanding across a row or column, 736 minors of, 735–736 properties of, 738–739 by 3, 735–737 by 2, 732, 738–739 Diagonal entries, 749 Difference(s) See also Subtraction common, 812 of complex numbers, A57 first, 420 limits of, 878 of logarithms, 304 of two cubes, A24 of two functions, 66–67 of two matrices, 743–744 of two squares, A24, A27 of vectors, 594 Difference quotient, 63, 92, 287, 309, 487 Diophantus, 824 Directed line segment, 593 Direction angle, 599 Direction angles of vector, 618–621 Direction (bearing), 515 Direction cosines, 619–620 Direction of vectors, 593, 598–600 Directrix, 678 of parabola, 636 Dirichlet, Lejeune, 57 Discontinuity, 97–98 Discontinuous function, 886 Discriminant, A50, A61 Disjoint sets, A3 Distance, mean, 656, 701 Distance formula, 4–6 proof of, in space, 615 using, 4–6 Distributive Property of dot products, 607, 618 of matrix multiplication, 749 of real numbers, A4 Divergent geometric series, 821–824 Divergent infinite series, B5 Divergent sequence, B2–B3 Dividend, 199, A25 Division, A10 See also Quotient(s) of complex numbers, A58 of polynomials, A24–A26 algorithm for, 199 synthetic, A31–A34 of rational expressions, A35–A36 of two integers, A25 Divisor, 199, A25 Domain, 59, 64–66 of absolute value function, 97 of composite function, 252–255 of constant function, 95 of cosecant function, 385, 386 of cosine function, 385, 386, 401 of cotangent function, 385, 386 of cube function, 96 of cube root function, 96 defined by an equation, 64–66 of difference function, 66 of greatest integer function, 97 of identity function, 95 of inverse function, 262 of logarithmic function, 290–291 of logistic models, 333 of one-to-one function, 259 of product function, 66 of quotient function, 66–67 of rational function, 216–220 of reciprocal function, 97 of secant function, 385, 386 of sine function, 385, 386, 399, 400 of square function, 96 of square root function, 96 of sum function, 66 of tangent function, 385, 386, 415 of the trigonometric functions, 385 unspecified, 68 of variable, A7 Domain-restricted function, 266–267 Doppler, Christian, 235 Doppler effect, 235 Dot mode, 98 Dot product, 602, 606–613 angle between vectors using, 607–608 to compute work, 610–611 www.elsolucionario.net defined, 606 finding, 607–608 historical feature on, 611 orthogonal vectors and, 608–610 parallel vectors and, 609 properties of, 607, 618 of two vectors, 606–607 in space, 616, 617–618 Double-angle Formulas, 488–492 to establish identities, 489–492 to find exact values, 488–489 Double root (root of multiplicity 2), A46 Drag, 632 Dry adiabatic lapse rate, 817 e, 279–280, 287 defined, 279 Earthquakes, magnitude of, 301 Eccentricity, 678 of ellipse, 656, 678, 680 of hyperbola, 669, 680 of parabola, 680 Eddin, Nasir, 364, 534 Effective rates of return, 321 Egyptians, ancient, 824 Elements (Euclid), 534, 824 Elements of sets, 843–845, A1 Elevation, angle of, 512–513 Elimination, Gauss-Jordan, 724 Elimination method, 702, 706–708, 709, 710 systems of nonlinear equations solved using, 767–772 Ellipse, 635, 645–656 with center at (h, k), 650–651 with center at the origin, 646–650 major axis along x-axis, 647 major axis along y-axis, 649 with center not at origin, 651–652 center of, 646 defined, 646, 678 eccentricity of, 656, 678, 680 foci of, 646 graphing of, 648–650 major axis of, 646, 678 length of, 646 minor axis of, 646 solving applied problems involving, 653 vertices of, 646 Ellipsis, A3 Elliptical orbits, 634 Elongation angle, 530 Empty (null) sets, 843, A1 End behavior, 187–188 of rational function, 219–220 Engels, Friedrich, 871 Entries of matrix, 718, 742 diagonal, 749 Equality of complex numbers, A57 of sets, 843, A2 of vectors, 593, 597 in space, 616 Equally likely outcomes, 859–860 Equation(s) algebraic and graphic solutions of, 28 conditional, 469 demand, 159 depressed, 203 domain of a function defined by, 64–66 equivalent, 10, A42–A43 even and odd functions identified from, 82–83 exponential, 280–282, 295, 312–315 quadratic in form, 314 in the form y = {expression in x}, 10 as function, 61 intercepts from, 18–19 inverse function defined by, 264–267 INDEX linear See Linear equation(s) polar See Polar equations quadratic See Quadratic equation(s) quadratic in form, A51–A52 satisfying the, 7–8, A42 second-degree See Quadratic equation(s) sides of, 7, A42 solution set of, A42 solving, A42–A55 algebraically, A42–A43 by factoring, A46–A47, A52–A53 involving absolute value, A52 systems of See Systems of equations in two variables, graphs of, 7–11 intercepts from, 12–13 by plotting points, 7–9 symmetry test using, 19–21 x = y 2, 22 y = , x, 22–23 y = x 3, 21 Equilateral hyperbola, 669 Equilateral triangle, 16 Equilibrium, static, 601–602 Equilibrium price, 135–136 Equilibrium quantity, 135–136 Equilibrium (rest) position, 544 Equivalent equations, 10, A42–A43 Equivalent systems of equations, 707 Error triangle, 17 Euclid, 534, 824, 836 Euler, Leonhard, 57, 364, 864 Even functions, 403 determining from graph, 81–82 identifying from equation, 82–83 Evenness ratio, 299 Even-Odd identity, 469 Even-Odd Properties, 394 Events, 859 complement of, 862–863 mutually exclusive, 861–862 probabilities of union of two, 861–862 Explicit form of function, 64 Exponent(s), A8–A9 Laws of, 272, 281, A7–A9 logarithms related to, 288–289 Exponential equations, 280–282 defined, 280 solving, 280–282, 295 equations quadratic in form, 314 using graphing utility, 314–315 solving graphically and algebraically, 312–314 Exponential expressions, changing between logarithmic expressions and, 289 Exponential functions, 271–288, 888 continuous, 888 defined, 273 e, 279–280, 287 evaluating, 271–275 fitting to data, 339–340 graph of, 275–279 using transformations, 278–279, 280 identifying, 273–275 power function vs., 273 properties of, 276–277, 278, 282 ratio of consecutive outputs of, 273–275 Exponential growth and decay, 272–273, 327–338 law of decay, 329–331 logistic models, 332–335 defined, 332 domain and range of, 333 graph of, 332–333 properties of, 333 uninhibited growth, 327–329 Exponential law, 327 Exponents, calculator to evaluate, A10 Extended Principle of Mathematical Induction, 830 Extraneous solutions, A85–A86 Extreme values of functions, 85 Extreme Value Theorem, 86 Factored completely, A27 Factorial symbol, 801–802 Factoring defined, A27 equations solved by, A46–A47, A52–A53 of expression containing rational exponents, A88 over the integers, A27 polynomials, A27–A28 by grouping, A28 Factors, A27 linear, 759–762 nonrepeated, 759–760 repeated, 761–762 quadratic, 204–205, 763 synthetic division to verify, A33 Factor Theorem, 200–201 Family of lines, 43 of parabolas, 114 Feasible point, 786, 787 Fermat, Pierre de, 1, 287, 863 Ferrari, Lodovico, 208 Ferris, George W., 51, 466 Fertility rate, 798 Fibonacci, 824 Fibonacci numbers, 803 Fibonacci sequences, 803, 810–811 Financial models, 317–327 compound interest, 318–323 doubling time for investment, 323 effective rates of return, 321 future value of a lump sum of money, 317–321 present value of a lump sum of money, 319, 322 tripling time for investment, 323 Finck, Thomas, 364, 380 Finite sets, 843 First-degree equation See Linear equation(s) First differences, 420 Fixed costs, 42 Focus/foci, 678 of ellipse, 646 of hyperbola, 657 of parabola, 636 FOIL method, A23–A24 Foot-pounds, 610 Force(s), 544 aerodynamic, 632 of deflection, 559 resultant, 600 Force vector, 599 Formulas, geometry, A15–A16 Fractions complex, A39n continued, A41 partial, 759 Frequency, 412 in simple harmonic motion, 545 Frobenius, Georg, 755 Function(s), 57–128 See also Composite functions; Exponential functions; Inverse functions; Linear functions; Polynomial functions; Trigonometric functions absolute value, 94–95, 97 area under graph of, 899–902 argument of, 62 average cost, 75 average rate of change of, 87–89 finding, 87–89 secant line and, 88–89 building and analyzing, 116–121 on calculators, 63–64 circular, 370 constant, 83–84, 85, 95 continuous, 97, 207, 886–888 www.elsolucionario.net cube, 62, 96 cube root, 93–94, 96 decreasing, 83–84, 85, 86–87 defined, 59 derivative of, 893–894 difference of two, 66 difference quotient of, 63 discontinuous, 886 domain of, 59, 64–66 unspecified, 68 domain-restricted, 266–267 equation as, 61 even and odd, 403 determining from graph, 81–82 identifying from equation, 82–83 explicit form of, 64 graph of, 71–80, 104–116 combining procedures, 106, 110–112 determining odd and even functions from, 81–82 determining properties from, 83–84 identifying, 72–73 information from or about, 73–75 using compressions and stretches, 106–108, 110 using reflections about the x-axis or y-axis, 108–110 using vertical and horizontal shifts, 104–106, 110 greatest integer, 97 identically equal, 469 identity, 95–96 implicit form of, 64 important facts about, 64 increasing, 83–84, 85, 86–87 library of, 93–98 local maxima and local minima of, 84–85 with no limit at 0, 875 nonlinear, 130 objective, 785–789 one-to-one, 258–261 periodic, 386–387 piecewise-defined, 98–100, 887–888 power, 180–182 graph of, 181 of odd degree, 181–182 properties of, 182 product of two, 66 quotient of two, 66–67 range of, 59 reciprocal, 96–97, 417 See also Cosecant function; Secant function relation as, 58–61 square, 96 square root, 93, 96 step, 98 sum of two, 66 graph of, 548–549 value (image) of, 59, 61–64 zeros of, Bisection Method for approximating, 208, 210–211 Function keys, A10 Function notation, 68 Fundamental identities of trigonometric functions, 389–391 quotient, 389 reciprocal, 389 Fundamental period, 387 Fundamental Theorem of Algebra, 211–212 Conjugate Pairs Theorem and, 212–213 proof of, 212 Future value, 317–321 Galois, Evariste, 208 Gauss, Karl Friedrich, 211, 591, 702 Gauss-Jordan method, 724 General addition principle of counting, 845 I5 I6 INDEX General form of conics, 670–671 of equation of circle, 48–49 linear equation in, 36–37 General (nth) term, 800, B4 Generators of cone, 635 Geometric mean, A82 Geometric progression See Geometric sequences Geometric sequences, 818–821 common ratio of, 818, 819 defined, 818 determining, 818–819 formula for, 819–820 nth term of, 819–820 sum of, 820–821 Geometric series, 821-824, B5–B6 infinite, 821–822 sum of, B6 Geometric vectors, 593–594 Geometry essentials, A13–A21 congruent and similar triangles, A16–A18 formulas, A15–A16 Pythagorean Theorem and its converse, A13–A15 Geometry problems, algebra to solve, Gibbs, Josiah, 602 Golden ratio, 811 conjugate, 811 Grade, 44 Graph(s)/graphing area under, 899–902 bounded, 782 of circles, 46–47, 581 complete, of cosecant function, 417–418 using transformations, 418 of cosine function, 400–402 of cotangent function, 416–417 of ellipse, 648–650 equation of hyperbola, 658–661, 663–665 of equations in two variables, 7–11 intercepts from, 12–13 by plotting points, 7–9 symmetry test using, 19–21 x = y 2, 22 y = , x, 22–23 y = x 3, 21 of exponential functions, 275–279 using transformations, 278–279, 280 of function, 71–80, 104–116 combining procedures, 106, 110–112 determining odd and even functions from 81–82 determining properties from, 83–84 identifying, 72–73 information from or about, 73–75 in library of functions, 93–98 using compressions and stretches, 106–108, 110 using reflections about the x-axis or y-axis, 108–110 using vertical and horizontal shifts, 104–106, 110 of inequalities, 776–779, A4–A5 linear inequalities, 777–778 steps for, 777 of inverse functions, 263–264 limit using, 873–875 of lines given a point and the slope, 32 using intercepts, 36–37 to locate absolute maximum and absolute minimum of function, 85–86 of logarithmic functions, 291–294 base not 10 or e, 307 inverse, 292–294 of logistic models, 332–333 of one-to-one function, 264 of parabola, 637 of parametric equations, 684–686 of piecewise-defined functions, 98–100 of polar equations, 569–585 cardioid, 575–576, 581 circles, 581 of conics, 679–681 by converting to rectangular coordinates, 569570 defined, 569 lemniscate, 579580, 581 limaỗon with inner loop, 577578, 581 limaỗon without inner loop, 576577, 581 by plotting points, 575–580 polar grids for, 569 rose, 578–579, 581 sketching, 581–582 spiral, 580 using graphing utility, 571–574 of polynomial functions, 182–193 analyzing, 189–192 end behavior of, 187–188 smooth and continuous, 180 turning points of, 185–186 using bounds on zeros, 206–207 using transformations, 182–183 using x-intercepts, 185 of polynomial inequalities, 237–238 of quadratic functions by hand, 151–152 properties of, 150 steps for, 151–152, 155 using its vertex, axis, and intercepts, 150–153 using transformations, 148–150 of rational functions, 227–236 analyzing, 227–233 constructing rational function from, 232–233 end behavior of, 219–220 using transformations, 218 of rational inequalities, 239 of secant function, 417–418 using transformations, 418 of sequences, 799, 800 of sine and cosine functions, 398–413, 424, 549 amplitude and period, 402–404 equation for, 407–408 key points for, 404–407 of sum of two functions, 548–549 of systems of nonlinear inequalities, 779–782 of vectors, 595 of y = , 217–218 x Graphing calculator(s), A10 composite functions on, 252 Graphing calculators See Graphing utility(ies) Graphing software See Graphing utility(ies) Graphing utility(ies), 3–4 to approximate intercepts, 13 circle on, 47, 48–49 connected mode, 98 derivative of function using, 893 dot mode, 98 ellipses on, 648, 652 equations solved using, 26–29 equations using, 10–11 eVALUEate feature, 200 even and odd functions identified using, 82–83 exponents on, A10 to find limit, 875 to find sum of arithmetic sequence, 814–816 to fit exponential function to data, 339–340 to fit logarithmic function to data, 340–341 to fit logistic function to data, 340–341 functions on, 86–87 geometric sequences using, 820, 821 hyperbola using, 659 identity established with, 471 inequalities using, 778–779 system of linear inequalities, 779–780 www.elsolucionario.net INTERSECT feature, 27 limit of sequence using, B1 line of best fit from, 142–143 logarithmic and exponential equations solved using, 314–315 matrix operations on, 743–744, 745 reduced row echelon form, 724 MAXIMUM and MINIMUM features, 86, 162 parabolas using, 637, 641 PARametric mode, 690 polar equations using, 571–574 polynomial function graph analyzed with, 190–192 REF command, 723 REGression options, 339 RREF command, 724 screen of, square, 31 sine function of best fit on, 428 sum of a sequence using, 804–806 system of nonlinear equations solved using, 766, 767–771 systems of equations on, 705 TABLE feature, 207–208, 800 tables using, 12 TRACE feature, 800 trigonometric equations on, 464 turning points in, 185–186 ZERO (or ROOT) feature, 26–27, 162 Grassmann, Hermann, 602, 611 Greatest integer function, 97–98 Greek letters, to denote angles, 355 Greeks, ancient, 364 Grouping, factoring by, A28 Growth, uninhibited, 327–329 Growth factor, 273 Hale-Bopp comet, orbit of, 634, 701 Half-angle Formulas, 493–494 to find exact values, 493–494 for tangent, 494 Half-life, 330 Half-line (ray), 355 Half-open/half-closed intervals, A73, A74 Half-planes, 777 Hamilton, William Rowan, 602 Harmonic mean, A82 Heron of Alexandria, 539, 540, 824 Heron’s Formula, 539 historical feature on, 540 proof of, 539–540 Home, valuing a, 1, 55–56 Horizontal asymptote, 219 Horizontal component of vector, 597 Horizontal compression or stretches, 107 Horizontal lines, 33–34, 570, 580 Horizontal-line test, 260–261 Horizontal shifts, 104–106, 110 Huygens, Christiaan, 694, 863 Huygens’s clock, 694 Hyperbolas, 634, 635, 656–669 asymptotes of, 662–663 branches of, 657 with center at (h, k), 663–665 with center at the origin, 657–661 transverse axis along x-axis, 658–660, 663, 664 transverse axis along y-axis, 660–661, 664 with center not at the origin, 664 center of, 657 conjugate, 669 conjugate axis of, 657 defined, 657, 678 eccentricity of, 669, 680 equilateral, 669 foci of, 657 graphing equation of, 658–661 solving applied problems involving, 665–666 INDEX transverse axis of, 657, 678 vertices of, 657 Hyperbolic cosine function, 287 Hyperbolic sine function, 287 Hyperboloid, 669 Hypocycloid, 697 Hypotenuse, 508, A13–A14 i, A59 Identically equal functions, 469 Identity(ies), A42 definition of, 469 polarization, 613 Pythagorean, 390, 469 trigonometric, 468–475 basic, 469 establishing, 470–473, 478–481, 489–492 Even-Odd, 469 Pythagorean, 469 Quotient, 469 Reciprocal, 389, 469 trigonometric equations solved using, 463–464 Identity function, 95–96 Identity matrix, 749–750 Identity Properties, 750 Image (value) of function, 59, 61–64 Imaginary axis of complex plane, 585 Imaginary unit, A56 Implicit form of function, 64 Improper rational expression, 759 Improper rational function, 221 Incidence, angle of, 467 Inclination, 384 Inconsistent systems of equations, 704, 705, 709, 711 containing three variables, 711 containing two variables, 708 Cramer’s Rule with, 738 matrices to solve, 726 Increasing functions, 83–84, 86–87 Increasing linear functions, 133–134 Independent systems of equations, 705 Independent variable, 62 in calculus, 398 Index/indices of radical, A83 of refraction, 467 row and column, 718, 742 of sum, 803 Induction, mathematical, 827–830 Extended Principle of, 830 principle of, 827–828, 830 proving statements using, 827–829 Inequality(ies) combined, A77–A78 graphing, 776–779 linear inequalities, 777–778 steps for, 777 interval notation to write, A74 involving absolute value, A78–A79 involving quadratic functions, 168–172 nonstrict, A5 in one variable, A76 polynomial, 237–238 algebraically and graphically solving, 237–238 steps for solving, 238 properties of, A75–A76, A78 rational, 238–240 algebraically and graphically solving, 238–240 steps for solving, 239–240 satisfying, 776 sides of, A5 solving, A76–A77 strict, A5 systems of, 776–784 graphing, 779–782 in two variables, 776 Inequality symbols, A4 Inertia moment of, 501 product of, 497 Infinite geometric series, 821–822 Infinite limit, 187 Infinite series, B3–B5 convergent and divergent, B5 defined, B4 sum of, B5 terms of, B4 Infinite sets, 843 Infinity, A74 Infinity, limits at, 187, 217 Inflation, 325 Inflection point, 79, 333 Initial point of directed line segment, 593 Initial side of angle, 355 Initial value of exponential function, 273 Input to relation, 58 Inscribed circle, 543 Instantaneous rate of change, 894 Integers, A3 dividing, A25 factoring over the, A27 Integrals, 492, 903 area under graph, 899–902 graphing utility to approximate, 903 Intercept(s) of circle, 47 from an equation, 18–19 from a graph, 12–13 graphing an equation in general form using, 36–37 graphing utility to approximate, 13 graph of lines using, 36–37 of quadratic function, 150–154 Interest compound, 318–323 computing, 318–319 continuous, 320–321 defined, 318 doubling or tripling time for money, 323 effective rates of return, 321 formula, 319 future value of lump sum of money, 317–321 present value of lump sum of money, 322 problems involving, A64 –A65 rate of, 317–318, A64 –A65 effective, 321 simple, 318, A64 –A65 Intermediate Value Theorem, 207–208 Intersection of sets, A2 Intervals, A73–A74 closed, A73, A74 endpoints of, A74 half-open, or half-closed, A73, A74 open, A73, A74 writing, using inequality notation, A74 Invariance, 677 Inverse of matrix, 750–753 finding, 750–753 multiplying matrix by, 750–752 solving system of linear equations using, 753–754 Inverse functions, 261–267, 440–458 See also Logarithmic functions cosine, 445–447 defined, 445 exact value of, 446–447 exact value of expressions involving, 453–454 implicit form of, 445 defined by a map or an ordered pair, 261–263 domain of, 262 of domain-restricted function, 266–267 finding, 261–263, 449 defined by an equation, 264–267 www.elsolucionario.net I7 graph of, 263–264 range of, 262 secant, cosecant, and cotangent, 455 approximating the value of, 456 calculator to evaluate, 455–456 definition of, 455 sine, 440–444 approximate value of, 442–443 defined, 441 exact value of, 441–442 exact value of expressions involving, 453–454, 482 implicit form of, 441 properties of, 443–444 solving equations involving, 450 Sum and Difference Formulas involving, 482 tangent, 447–449 defined, 448 exact value of, 448–449 exact value of expressions involving, 453–454 implicit form of, 448 verifying, 263 written algebraically, 456–457 Inverse trigonometric equations, 450 Inverted cycloid, 693 Irrational numbers, A3, A56 decimal representation of, A3 Irreducible quadratic factor, 204–205, 763 Isosceles triangle, 16 Jı¯ba, 380 Jı¯va, 380 Jordan, Camille, 702 Joules (newton-meters), 610 Khayyám, Omar, 836 Kirchhoff’s Rules, 716, 731 Koukounas, Teddy, 542n Kôwa, Takakazu Seki, 702 Latitude, 178 Latus rectum, 637, 638 Law of Cosines, 531–537 in applied problems, 533–534 defined, 531 historical feature on, 534 proof of, 532 Pythagorean Theorem as special case of, 532 SAS triangles solved using, 532–533 SSS triangles solved using, 533 Law of Decay, 329–331 See also Exponential growth and decay Law of Sines in applied problems, 525–527 defined, 521 historical feature on, 534 proof of, 526–527 SAA or ASA triangles solved using, 521–522 SSA triangles solved using, 522–525 Law of Tangents, 531, 534 Laws of Exponents, 272, 281, A7–A9 Leading coefficient, 211, A23 Least common multiple (LCM) to add rational expressions, A38 Left endpoint of interval, A74 Left limit, 884 Left stochastic transition matrix, 758 Legs of triangle, 508, A13–A14 Leibniz, Gottfried Wilhelm, 57, 702 Lemniscate, 579–580, 581 Length of arc of a circle, 359 Lensmaker’s equation, A41 Lewis, Meriwether, 507, 557 Lift, 559, 632 Light detector, 513 Light projector, 513 Like radicals, A84 I8 INDEX Like terms, A22 Limaỗon with inner loop, 577–578, 581 without inner loop, 576–577, 581 Limits, 187, 217, 872–890 algebra techniques for finding, 877–883 of average rate of change, 882 of constant, 877 of difference, 878 finding, 872–876 by graphing, 873–875 using a table, 872–873 infinite, 187 at infinity, 187 of monomial, 879 one-sided, 884–885 of polynomial, 879–880 of power or root, 880 of product, 878 of quotient, 881 of sequence, B1–B3 defined, B2 finding, B2 of sum, 878 of x, 877 Line(s), 29–45 of best fit, 143–144 coincident, 705 equations of See also Linear equation(s); Systems of linear equations secant, 88 family of, 43 graphing given a point and the slope, 32 using intercepts, 36–37 horizontal, 33–34, 570, 580 point-slope form of, 33–34 polar equation of, 570, 580 slope of, 29–32, 35–36 containing two points, 30 from linear equation, 35–36 tangent, 51 vertical, 29, 571–572, 580 y-intercept of, 35–36 Linear algebra, 742 Linear equation(s) See also Line(s); Systems of linear equations defined, 37 equation that leads to, A44–A45 in general form, 36–37 given two points, 34 for horizontal line, 33–34 in one variable, A42, A43–A44 for parallel line, 37–38 for perpendicular line, 38–40 slope from, 35–36 in slope-intercept form, 34–35 solving, A43–A45 for vertical line, 32–33 Linear factors, 759–762 nonrepeated, 759–760 repeated, 761–762 Linear functions, 130–139 average rate of change of, 130–133 building from data, 140–146 defined, 130 graphing utility to find the line of best fit, 143–144 graph of, 130 identifying, 273–275 increasing, decreasing, or constant, 133–134 nonlinear relations vs., 141–142 scatter diagrams, 140–142 Linear models from data, 140–146 from verbal descriptions, 134–136 Linear programming problems, 702, 785–791 maximum, 788–789 minimum, 787–788 setting up, 785 solution to, 787 location of, 787 solving, 786–789 in two variables, 786 Linear speed, 363–364 Linear trigonometric equation, 460–461 Line segment, 593 length of, 5–6 midpoint of, Local maxima and local minima of functions, 84–85 Logarithmic equations, 310–312 defined, 295 solving, 295–296 graphically and algebraically, 310–312 Logarithmic functions, 288–301 changing between logarithmic expressions and exponential expressions, 289 continuous, 888 defined, 288 domain of, 290–291 evaluating, 289–290 fitting to data, 340–341 graph of, 291–294 base not 10 or e, 307 properties of, 291, 297 range of, 290 Logarithmic spiral, 580 Logarithms, 301–309 on calculators, 306 common (log), 293, 305–306, 308 evaluating, with bases other than 10 or e, 305–307 historical feature on, 308 logarithmic expression as single, 304–305 logarithmic expression as sum or difference of, 304 natural (ln), 292, 305–306, 308 properties of, 301–309 establishing, 302 proofs of, 302–303 summary of, 307 using, with even exponents, 312 relating to exponents, 288–289 Logistic functions, fitting to data, 340–341 Logistic models, 332–335 defined, 332 domain and range of, 333 graph of, 332–333 properties of, 333 Long division of polynomials, A24–A26 Loudness, 300 Lowest terms rational expressions in, A35 rational function in, 217, 220 Magnitude of earthquake, 301 vector in terms of direction cosines and, 620–621 of vectors, 593, 595, 597, 598, 599–600 in space, 616 Magnitude (modulus), 586, 587 Major axis, 678 Malthus, Thomas Robert, 871, 908 Mandel, Howie, 842 Mandelbrot sets, 592 Mapping, 58 Marginal cost, 158 Marginal propensity to consume, 826 Markov chains, 796 Marx, Karl, 871 Mathematical induction, 827–830 Extended Principle of, 830 principle of, 827–828, 830 proving statements using, 827–829 www.elsolucionario.net Mathematical modeling, A63 Matrix/matrices, 702, 717, 741–758 arranging data in, 742 augmented, 718–720 coefficient, 718 defined, 717, 742 entries of, 718, 742, 749 equal, 743 examples of, 743 graphing utilities for, 743–744, 745 historical feature on, 755 identity, 749–750 inverse of, 750–753 finding, 750–753 multiplying matrix by, 750–752 solving system of linear equations using, 753–754 left stochastic transition, 758 m by n, 742 nonsingular, 750, 752 product of two, 745–750 algebraic and graphing solutions of, 747 defined, 747 identity matrices, 749–750 properties of, 749–750 square matrices, 749 in reduced row echelon form, 724, 726, 727 row and column indices of, 718, 742 in row echelon form, 720–728 row operations on, 719–720 scalar multiples of, 744–745 singular, 750 to solve system of linear equations, 720–728 algebraic and graphing methods, 723–724 square, 742–743 sum and difference of two, 743–744 transition, 758, 796 zero, 744 Maxima of functions absolute, 85–86 local, 84–85 Maximum value of a quadratic function, 154 Mean arithmetic, A82 geometric, A82 harmonic, A82 Mean distance, 656, 701 Mechanics, parametric equations applied to, 693–694 Medians of triangle, 16 Menelaus of Alexandria, 364 Metrica (Heron), 540 Midpoint formula, Mind, mapping of, 439 Mindomo (software), 506 Minima of functions absolute, 85–86 local, 84–85 Minimum value of a quadratic function, 154 Minors, 735–736 Minutes, 357–358 Miranda, Kathleen, 542n Mixture problems, A66 Model(s), A63 linear from data, 140–146 from verbal descriptions, 134–136 sinusoidal, 424–428 best-fit, 428 daylight hours, 427–428 temperature data, 424–427 Modeling process, A63 Modulus (magnitude), 586, 587 Mollweide, Karl, 530 Mollweide’s Formula, 530 Moment of inertia, 501 Monomial(s), A22 common factors, A27 degree of, A22 INDEX examples of, A22 limit of, 879 recognizing, A22 Monter, 44 Motion circular, 363–364, 544, 545 curvilinear, 688 damped, 546–548 Newton’s second law of, 595 projectile, 688–690 simple harmonic, 543–546 uniform, A67–A68 Multiplication, A10 See also Product(s) of complex numbers, A57–A58 of rational expressions, A35–A36 scalar, 744–745 of vectors, by numbers See also Dot product of vectors, by numbers geometrically, 594–595 Multiplication principle of counting, 845–846 Multiplication properties for inequalities, A75, A76 Multiplier, 826 Mutually exclusive events, 861–862 Napier, John, 308 Nappes, 635 Natural logarithms (ln), 292, 305–306, 308 Natural numbers (counting numbers), 829, A3 Nautical miles, 367 Negative angle, 355 Negative numbers real, A4 square root of, A9, A60 Newton-meters (joules), 610 Newton’s Law of Cooling, 331–332, 336 Newton’s Law of Heating, 336 Newton’s Law of universal gravitation, 243 Newton’s Method, 226 Newton’s Second Law of Motion, 544, 595 Niccolo of Brescia (Tartaglia), 208, 591 Nonlinear equations, systems of, 766–775 elimination method for solving, 767–772 historical feature on, 772 substitution method for solving, 766–767 Nonlinear functions, 130 Nonlinear inequalities, systems of, 781 Nonlinear relations, 141–142 Nonnegative property of inequalities, A75 Nonsingular matrix, 750, 752 Nonstrict inequalities, A5 nth (general) term of series, 800, B4 nth roots, A83–A84 rationalizing the denominator, A85 simplifying, A83 simplifying radicals, A83–A84 Null (empty) sets, 843, A1 Numbers Fibonacci, 803 irrational, A3 natural (counting), 829, A3 rational, A3 triangular, 811 Numerator, A35 Objective function, 785–789 Oblique asymptote, 220, 221–224, 229 Oblique triangle, 520–521 Odd functions, 403 determining from graph, 81–82 identifying from equation, 82–83 One-sided limits, 884–885 One-to-one functions, 258–261 defined, 259 graph of, 264 horizontal-line test for, 260–261 Open interval, A73, A74 Opens down, 148 Opens up, 148 Optical (scanning) angle, 497 Optimization, quadratic functions and, 159 Orbits elliptical, 634 planetary, 656 Ordered pair(s), inverse function defined by, 261–263 as relations, 58–59 Ordinary annuity, 806 Ordinary (statute) miles, 367 Ordinate (y-coordinate), Orientation, 685 Origin, 2, 614 distance from point to, 117 ellipses with center at, 646–650 hyperbolas with center at, 657–661 parabolas with vertex at, 636–640 of real number line, A4 symmetry with respect to, 19, 20–21 Orthogonal vectors, 608–610 Outcome of probability, 857 equally likely, 859–860 Output of relation, 58 Parabola, 9, 148–150, 635, 636–645 axis of symmetry of, 148, 636 defined, 636, 678 directrix of, 636 eccentricity of, 680 family of, 114 focus of, 636 graphing equation of, 637 solving applied problems involving, 641–642 with vertex at (h, k), 640–641 with vertex at the origin, 636–640 finding equation of, 639–640 focus at (a, 0), a > 0, 637–638 vertex of, 148, 636 Paraboloids of revolution, 634, 641–642 Parallax, 517–518 Parallelepiped, 628 Parallel lines, 37–38 Parallelogram, area of, 626 Parallel vectors, 609 Parameter, 684 time as, 688–691 Parametric equations, 684–697 for curves defined by rectangular equations, 691–694 applications to mechanics, 693–694 cycloid, 693 defined, 684 describing, 687–688 graphing, 684–686 rectangular equation for curve defined parametrically, 686–688 time as parameter in, 688–691 Partial fraction decomposition, 702, 758–765 defined, 759 where denominator has nonrepeated irreducible quadratic factor, 763 where denominator has only nonrepeated linear factors, 759–760 where denominator has repeated irreducible quadratic factors, 764 where denominator has repeated linear factors, 761–762 Partial fractions, 759 Partial sums, B4 sequence of, B5 Participation rate, 71 Partitioning, 899–902 Pascal, Blaise, 694, 833, 863 Pascal triangle, 833, 836 Payment period, 318 www.elsolucionario.net I9 Peano, Giuseppe, 864 Pendulum with damped motion, 547–548 period of, A90 Perfect cubes, A24 Perfect roots, A83 Perfect squares, A24, A28 Perfect triangle, 542 Perihelion, 656, 684, 701 Perimeter, formulas for, A15 Period fundamental, 387 of simple harmonic motion, 544 of sinusoidal functions, 402–404, 421–424 of trigonometric functions, 386–388 Periodic functions, 387 Period of pendulum, A90 Permutations, 848–851 computing, 851 defined, 848 distinct objects without repetition, 849–851 distinct objects with repetition, 849 involving n nondistinct objects, 853–854 Perpendicular lines, equations of, 38–40 Phase shift, 420–424 to graph y = A sin(vx - f) + B, 420–422 Phones, cellular, 57 Physics, vectors in, 593 Piecewise-defined functions, 98–100 continuous, 887–888 Pitch, 44 Pixels, Plane(s) complex, 585–588 defined, 585 imaginary axis of, 585 plotting points in, 585–586 real axis of, 585 Plane curve, 684 Planets, orbit of, 656 Plotting points, 2, 560–562 graph equations by, 7–9 Point(s) coordinates of on number line, A4 corner, 782 distance between two, 4–5 distance from the origin to, 117 feasible, 786, 787 inflection, 79, 333 initial, 593 plotting, 2, 560–562 graph equations by, 7–9 polar coordinates of, 561–562 of tangency, 51 terminal, 593 turning, 185–186 Point–slope form of equation of line, 33–34 Polar axis, 560 Polar coordinates, 560–568 conversion from rectangular coordinates, 564–565 conversion to rectangular coordinates, 562–563 defined, 560 plotting points using, 560–562 of a point, 561–562 polar axis of, 560 pole of, 560 rectangular coordinates vs., 560 Polar equations calculus and, 582 classification of, 580–581 of conics, 678–684 analyzing and graphing, 679–681 converting to rectangular equation, 682 focus at pole; eccentricity e, 679–681 defined, 569 I10 INDEX Polar equations, (continued) graph of, 569–585 cardioid, 575–576, 581 circles, 581 by converting to rectangular coordinates, 569570 defined, 569 lemniscate, 579580, 581 limaỗon with inner loop, 577578, 581 limaỗon without inner loop, 576577, 581 by plotting points, 575–580 polar grids for, 569 rose, 578–579, 581 sketching, 581–582 spiral, 580 using graphing utility, 571–574 historical feature on, 582 identifying, 569–570 testing for symmetry, 574–575 transforming rectangular form to, 566–567 transforming to rectangular form, 566 Polar form of complex number, 586 Polar grids, 569 Polarization identity, 613 Pole, 560 Polynomial(s), A21–A30 definition of, A23 degree of, 179–183, A23, A27 odd, 205, 213 dividing, 199–201, A24–A26 synthetic division, A31–A34 examples of, A23 factoring, A27–A28 by grouping, A28 limit of, 879–880 prime, A27 recognizing, A22–A23 solving, 204–205 special products formulas, A23–A24 in standard form, A23 terms of, A23 zero, A23 Polynomial functions, 179–198, 886 complex, 211 complex zeros of, 211–216 Conjugate Pairs Theorem, 212–213 defined, 211 finding, 214–215 polynomial function with specified zeros, 213–214 continuous, 886 cubic models from data, 192–193 defined, 179 end behavior of, 187–188 graph of, 182–193 analyzing, 189–192 end behavior of, 187–188 smooth and continuous, 180 turning points of, 185–186 using bounds on zeros, 206–207 using transformations, 182–183 using x-intercepts, 185 historical feature on, 208 identifying, 179–182 multiplicity of, 183–189 real zeros (roots) of, 183–185, 198–211 finding, 202–204 Intermediate Value Theorem, 207–208 number of, 201 Rational Zeros Theorem, 201–202, 214 Remainder Theorem and Factor Theorem, 199–201 repeated, 184 theorem for bounds on, 205–207 solving, 202–204 unbounded in the negative direction, 187 Polynomial inequalities, 237–238 algebraically and graphically solving, 237–238 steps for solving, 238 Population, world, 798 Population increases, 871, 908 Position vector, 596–597 in space, 615–616 Positive angle, 355 Positive real numbers, A4 Power(s), 412 See also Exponent(s) of i, A59 limit of, 880 log of, 303 Power functions, 180–182 exponential function vs., 273 graph of, 181 of odd degree, 181–182 properties of, 182 Present value, 319, 322 Price, equilibrium, 135–136 Prime polynomials, A27 Principal, 317–318, A64 Principal nth root of real number, A83 Principal square root, A9, A60 Probability(ies), 796–797, 857–867 Complement Rule to find, 862–863 compound, 860 constructing models, 857–859 defined, 857 of equally likely outcomes, 859–860 of event, 859 mutually exclusive, 861–862 historical feature on, 863–864 outcome of, 857 sample space, 857 of union of two events, 861–862 Product(s) See also Dot product; Multiplication of complex numbers, A57–A58 in polar form, 587–588 of inertia, 497 limits of, 878–879 log of, 303 special, A23–A24 of two functions, 66–68 of two matrices, 745–750 algebraic and graphing solutions of, 747 defined, 747 identity matrices, 749–750 properties of, 749–750 square matrices, 749 vector (cross), 602 Product function, 66 Product-to-Sum Formulas, 498–499 Projectile motion, 688–690 Projection, vector, 609–610 Projection of P on the x-axis, 544–545 Projection of P on the y-axis, 545 Prolate spheroid, 655 Proper rational expressions, 759 Proper rational function, 221–222 Proper subsets, 843 Propertyshark.com, Ptolemy, 468, 534 Pure imaginary number, A56 Pythagorean Identities, 390, 469 Pythagorean Theorem, 508, A13–A15 applying, A14–A15 converse of, A14 as special case of Law of Cosines, 532 Pythagorean triples, A21 Quadrant, angle lying in, 356 Quadrantal angles, 356 trigonometric functions of, 371–373 Quadrants, Quadratic equation(s), A46–A51 character of the solutions of, A61 in the complex number system, A60–A61 definition of, A46 www.elsolucionario.net factoring, A46–A47 solving completing the square, A48–A49 procedure for, A51 quadratic formula, A49–A51, A60–A61 Square Root Method, A47 in standard form, A46 Quadratic factors, irreducible, 204–205, 763 Quadratic formula, A49–A51, A60–A61 Quadratic functions, 147–158 defined, 147 finding, given its vertex and one other point, 153–154 graph of properties of, 150 steps for, 155 using its vertex, axis, and intercepts, 150–153 using transformations, 148–150 inequalities involving, 168–172 hand solution and graphing utility solution, 169 maximum or minimum value of, 154, 159 optimizations and, 159 vertex and axis of symmetry of, 150–154 Quadratic models, 159–168 from data, 163–164 from verbal descriptions, 159–164 Quantity, equilibrium, 135–136 Quantity demanded, 135–136 Quantity supplied, 135–136 Quaternions, 602 Quotient(s), 199, A25 See also Division of complex numbers in polar form, 587–588 difference, 63, 92, 287, 309, 487 limit of, 881 log of, 303 synthetic division to find, A32–A33 of two functions, 66–67 Quotient identity(ies), 469 of trigonometric functions, 389 Radians, 358 converting between degrees and, 359–362 Radical equations, A85–A86 defined, A85 solving, A85–A86 Radicals, A83 fractional exponents as, A86 index of, A83 like, A84 properties of, A84 rational exponents defined using, A86 simplifying, A83–A84 Radical sign, A9 Radicand, A83 Radioactive decay, 329–330 Radius, 45 of sphere, 622 Range, 59 of absolute value function, 97 of constant function, 95 of cosecant function, 386 of cosine function, 386, 401 of cotangent function, 386 of cube function, 96 of cube root function, 96 of greatest integer function, 97 of identity function, 95 of inverse function, 262 of logarithmic function, 290 of logistic models, 333 of one-to-one function, 259 of projectile, 491–492 of reciprocal function, 97 of secant function, 386 of sine function, 386, 399, 400 of square function, 96 INDEX of square root function, 96 of tangent function, 386, 415 of the trigonometric functions, 385–386 vertical shifts and, 104 Rate of change average, 30, 87–89, 130–133 limit of, 882 of linear and exponential functions, 274–275 instantaneous, 894 Rate of interest, 317–318, A64–A65 Rates of return, effective, 321 Ratio common, 818, 819 golden, 811 conjugate, 811 Rational equations, A45 Rational exponents, A86–A88 Rational expressions, A34–A41 adding and subtracting, A36–A37 least common multiple (LCM) method for, A37–A38 complex, A39–A40 decomposing See Partial fraction decomposition defined, A35 improper, 759 multiplying and dividing, A35–A36 proper, 759 reducing to lowest terms, A35 Rational functions, 216–236 applied problems involving, 233 asymptotes of, 218–220 horizontal, 219, 221–224 vertical, 220–221 continuous, 886–887 defined, 216 domain of, 216–220 graph of, 227–236 analyzing, 227–233 constructing rational function from, 232–233 end behavior of, 219–220 using transformations, 218 with a hole, 231–232 improper, 221 in lowest terms, 217, 220 proper, 221–222 unbounded in positive direction, 217 Rational inequalities, 238–240 algebraically and graphically solving, 238–240 steps for solving, 239–240 Rationalizing the denominator, A85 Rational numbers, 216, A3, A56 Rational Zeros Theorem, 201–202, 214 Rays (half-lines), 355 of central angle, 358 vertex of, 355 Real axis of complex plane, 585 RealestateABC.com, Real number(s), A3–A6, A56 approximate decimals, A3 conjugate of, A59 defined, A3 principal nth root of, A83 principal nth root of, A83 Real number line, A4 Real part of complex numbers, A56 Real zeros (roots) of polynomial functions, 198–211 of even multiplicity, 185 finding, 202–204 Intermediate Value Theorem, 207–208 number of, 201 of odd multiplicity, 185 Rational Zeros Theorem, 201–202, 214 Remainder Theorem and Factor Theorem, 199–201 repeated, 184 theorem for bounds on, 205–207 Reciprocal function, 96–97, 417 See also Cosecant function; Secant function Reciprocal identities, 389, 469 Reciprocal property for inequalities, A76, A78 Rectangle, area and perimeter of, A15 Rectangular (Cartesian) coordinates, 2–3 converted to polar coordinates, 564–565 polar coordinates converted to, 562–563 polar coordinates vs., 560 polar equations graphed by converting to, 569–570 in space, 614 Rectangular (Cartesian) form of complex number, 586–587 Rectangular equations, 580–581 for curve defined parametrically, 686–688 polar equations converted to, 566, 682 transforming to polar equation, 566–567 Rectangular grid, 569 Recursive formula, 802–803 for arithmetic sequences, 814 terms of sequences defined by, 802–803 Reduced row echelon form, 724, 726, 727 Reflections about x-axis or y-axis, 108–110 Refraction, 467 Regiomontanus, 364, 534 Relation(s), 58 See also Function(s) defined, 58 as function, 58–61 input to, 58 nonlinear, 141–142 ordered pairs as, 58–59 Relative maxima and minima of functions, 84–85 Remainder, 199, A25 synthetic division to find, A32–A33 Remainder Theorem, 199–200 Repeated solution, A46 Repeated zeros (solutions), 184 Repeating decimals, 822–823, A3 Reply.com, Repose, angle of, 458 Rest (equilibrium) position, 544 Resultant force, 600 Review, A1–A90 of algebra, A1–A13 distance on the real number line, A5–A6 domain of variable, A7 evaluating algebraic expressions, A6–A7 evaluating exponents, A10 graphing inequalities, A4–A5 Laws of Exponents, A7–A9 sets, A1–A4 square roots, A9–A10 complex numbers, A56–A59 of geometry, A13–A21 congruent and similar triangles, A16–A18 formulas, A15–A16 Pythagorean Theorem and its converse, A13–A15 inequalities combined, A77–A78 properties of, A75–A76 solving, A76–A77 interval notation, A73–A74 of nth roots, A83–A84 rationalizing the denominator, A85 simplifying, A83 simplifying radicals, A83–A84 of polynomials, A21–A30 dividing, A24–A26 factoring, A27–A28 monomials, A22 recognizing, A22–A23 special products formulas, A23–A24 synthetic division of, A31–A34 of rational exponents, A86–A88 www.elsolucionario.net of rational expressions, A34–A41 adding and subtracting, A36–A37 complex, A39–A40 multiplying and dividing, A35–A36 reducing to lowest terms, A35 of solving equations, A42–A55 absolute value equations, A52 algebraically, A42–A43 linear equations, A43–A45 quadratic equations, A46–A51 quadratic in form, A51–A52 rational equations, A45 Revolutions per unit of time, 364 Rhaeticus, 364 Rhind papyrus, 824 Richter scale, 301 Right angle, 356, A13 Right circular cone, 635 Right circular cylinder, volume and surface area of, A15 Right endpoint of interval, A74 Right-hand rule, 614 Right limit, 884–885 Right triangles, 508, 510–515, A13–A14 applications of, 511–515 solving, 510–515 Right triangle trigonometry, 508–520 Complementary Angle Theorem, 510 fundamental identities, 389–391 values of trigonometric functions of acute angles, 508–510 Rise, 29, 30 Root(s), A42 See also Solution(s); Zeros complex, 589–591 limit of, 880 of multiplicity (double root), A46 perfect, A83 Rose, 578–579, 581 Roster method, A1 Rotation of axes, 671–676 analyzing equation using, 673–675 formulas for, 672 identifying conic without, 675–676 Rounding, A10 Round-off errors, 511 Row echelon form, 720–728 reduced, 724, 726, 727 Row index, 718, 742 Row operations, 719–720 Row vector, 746 Ruffini, P., 208 Rumsey, David, 507 Run, 29, 30 Rutherford, Ernest, 669 SAA triangles, 521–522 Sample space, 857 SAS triangles, 521, 532–533, 538–539 Satisfying equations, 7–8, A42 Satisfying inequalities, 776 Sawtooth curve, 551 Scalar, 594, 744 Scalar multiples of matrix, 744–745 Scalar product See Dot product Scale of number line, A4 Scanning (optical) angle, 497 Scatter diagrams, 140–142, 424–425 Schroeder, E., 864 Scientific calculators, A10 Screen of a graphing utility, Secant defined, 508 graph of, 417–418 periodic properties of, 387 Secant function, 370 continuous, 887, 888 domain of, 385, 386 I11 I12 INDEX Secant function, (continued) inverse, 455 approximating the value of, 456 calculator to evaluate, 455–456 definition of, 455 range of, 386 Secant line, 88–89, 92 Second-degree equation See Quadratic equation(s) Seconds, 357–358 Seed, 592 Sequences, 799–812 annuity problems, 806–808 arithmetic, 812–817 common difference in, 812 defined, 812 determining, 812–813 formula for, 813–814 nth term of, 813 recursive formula for, 814 sum of, 814–816 convergent, B2–B3 defined, 799 divergent, B2–B3 factorial symbol, 801–802 Fibonacci, 803, 810–811 geometric, 818–821 common ratio of, 818, 819 defined, 818 determining, 818–819 formula for, 819–820 nth term of, 819–820 sum of, 820–821 graph of, 799, 800 historical feature on, 824 limit of, B1–B3 defined, B2 finding, B2 of partial sums, B5 from a pattern, 801 properties of, 804 summation notation, 803–804 sum of, 804–806 formulas for, 805 terms of, 799–801 alternating, 801 defined by a recursive formula, 802–803 general, 800 Series geometric, B5–B6 infinite, B3–B5 terms of, B4 Set(s), A1–A4 complement of, A2 correspondence between two, 58 disjoint, A3 elements of, 843–845, A1 empty (null), 843, A1 equal, 843, A2 finite, 843 infinite, 843 intersection of, A2 Mandelbrot, 592 of numbers, A1–A4 solution, A42 subsets of, 843 proper, 843 union of, A2, A78n universal, 844, A2 Set-builder notation, A1–A2 Shannon’s diversity index, 299 Shifts, graphing functions using vertical and horizontal, 104–106, 110 Side–angle–side case of congruent triangle, A16 Side–angle–side case of similar triangle, A17 Sides of equation, 7, A42 of inequality, A5 Side–side–side case of congruent triangle, A16 Side–side–side case of similar triangle, A17 Similar triangles, A16–A18 Simple harmonic motion, 543–546 amplitude of, 544 analyzing, 546 circular motion and, 544, 545 defined, 544 equilibrium (rest) position, 544 frequency of object in, 545 model for, 544–546 period of, 544 Simple interest, 318, A64–A65 Simplifying complex rational expressions, A39–A40 expressions with rational exponents, A86–A88 nth roots, A83 radicals, A83–A84 Simpson’s rule, 166 Sine defined, 508 historical feature on, 380 Law of in applied problems, 525–527 defined, 521 historical feature on, 534 proof of, 526–527 SAA or ASA triangles solved using, 521–522 SSA triangles solved using, 522–525 periodic properties of, 387 Sum and Difference Formula for, 478–479 trigonometric equations linear in, 483–484 Sine function, 369 of best fit, 428 continuous, 888 domain of, 385, 386, 399, 400 graphs of, 398–413 amplitude and period, 402–404 equation for, 407 key points for, 404–407 hyperbolic, 287 inverse, 440–444 approximate value of, 442–443 defined, 441 exact value of, 441–442 exact value of expressions involving, 453–454 implicit form of, 441 properties of, 443–444 properties of, 399 range of, 386, 399, 400 Singular matrix, 750 Sinusoidal graphs, 398–413, 549 amplitude and period, 402–404 equation for, 407–408 key points for, 404–407 steps for, 424 Sinusoidal models, 424–428 best-fit, 428 daylight hours, 427–428 temperature data, 424–426 Six trigonometric functions exact values of, 379 of quadrantal angles, 371–373 of t, 369–370 Slope, 29–32, 35–36 containing two points, 30 graphing lines given, 32 from linear equation, 35–36 of secant line, 88 undefined, 29 Slope-intercept form of equation of line, 34–35 Smooth graph, 180 Snell, Willebrord, 467 Snell’s Law of Refraction, 467 Solution(s), A42 See also Zeros approximate, 26–27 extraneous, A85–A86 www.elsolucionario.net of inequality, A76 of linear programming problems, 787 location of, 787 repeated, 184, A46 of systems of equations, 704, 709–710 of trigonometric equations, 459 Solution set of equation, A42 Special products, A23–A24 Speed angular, 363–364 instantaneous, 894–896 linear, 363–364 Sphere, 622 volume and surface area of, A15 Spherical trigonometry, 557 Spheroid, prolate, 655 Spiral, 580 Square(s) of binomials (perfect squares), A24, A28 difference of two, A24, A27 perfect, A24, A28 Square function, 96 Square matrix, 742–743 Square root(s), A9–A10, A83 complex, 589 of negative number, A9, A60 principal, A9, A60 Square root function, 93, 96 Square Root Method, A47 in complex number system, A60–A61 Square screens, 31 SSA triangles, 522–525 SSS triangles, 521, 533, 539–540 Standard deviation, A82 Standard form complex number in, A56 power of, A59 quotient of two, A58 reciprocal of, A58 of equation of circle, 45–46 polynomials in, A23 quadratic equation in, A46 Standard form See General form Standard position, angle in, 355–356 Static equilibrium, 601–602 Statute (ordinary) miles, 367 Step function, 98 Stirling’s formula, 837 Stock valuation, 129 Straight angle, 356 Stretches, graphing functions using, 106–108, 110 Strict inequalities, A5 Subscript, A23n Subscripted letters, 800 Subsets, 843, A2 proper, 843 Substitution method, 702, 705–706 systems of nonlinear equations solved using, 766–767 Subtraction, A10 See also Difference(s) of complex numbers, A57 of rational expressions, A36–A37 least common multiple (LCM) method for, A37–A38 of vectors, 597–598 in space, 616 Sum(s) See also Addition of arithmetic sequences, 814–816 of geometric sequences, 820–821 of geometric series, B6 index of, 803 of infinite geometric series, 822 of infinite series, B5 limits of, 878 of logarithms, 304 partial, B4 sequence of, B5 INDEX of sequences, 804–806 formulas for, 805 of two cubes, A24 of two functions, 66 graph of, 548–549 of two matrices, 743–744 Sum and Difference Formulas, 476–488 for cosines, 476–477 defined, 476 to establish identities, 478–481 to find exact values, 477, 479–480 involving inverse trigonometric function, 482 for sines, 478–479 for tangents, 481 Sum function, 66 Summation notation, 803–804 Sum-to-Product Formulas, 499–500 Sun, declination of, 451 Surface area, formulas for, A15 Sylvester, James J., 755 Symmetry, 19–21 axis of of parabola, 148 of quadratic function, 150–154 axis of, of parabola, 636 of polar equations, 574–575 with respect to origin, 19, 20–21 p with respect to the line u = (y@axis), 574 with respect to the polar axis (x-axis), 574 with respect to the pole (origin), 574 with respect to the x-axis, 19–20 with respect to the y-axis, 19–20 Synthetic division, A31–A34 Systems of equations consistent, 704, 709 dependent, 705 containing three variables, 712–713 containing two variables, 708–709 Cramer’s Rule with, 738 equivalent, 707 graphing, 705 inconsistent, 704, 709, 711 containing three variables, 711 containing two variables, 708 Cramer’s Rule with, 738 independent, 705 solutions of, 704, 709–710 Systems of inequalities, 776–784 graphing, 779–782 bounded and unbounded graphs, 782 with graphing utility, 779–780 by hand, 779, 780–781 vertices or corner points, 782 Systems of linear equations, 703–741 consistent, 705, 709 defined, 705 dependent, 705 containing three variables, 712–713 containing two variables, 708–709 matrices to solve, 724–726 determinants, 732–741 cofactors, 736 Cramer’s Rule to solve a system of three equations containing three variables, 737–738 Cramer’s Rule to solve a system of two equations containing two variables, 733–735 minors of, 735–736 properties of, 738–739 by 3, 735–737 by 2, 732–733, 738–739 elimination method of solving, 706–708, 709, 710 equivalent, 707 examples of, 703–704 graphing, 705 inconsistent, 705, 709, 711 containing three variables, 711 containing two variables, 708 matrices to solve, 726 independent, 705 matrices See Matrix/matrices partial fraction decomposition, 758–765 defined, 759 where denominator has a nonrepeated irreducible quadratic factor, 763 where denominator has only nonrepeated linear factors, 759–760 where denominator has repeated irreducible quadratic factors, 764 where denominator has repeated linear factors, 761–762 solution of, 704, 709–710 solving, 704 substitution method of, 705–706 three equations containing three variables, 709–711 Systems of nonlinear equations, 766–775 elimination method for solving, 767–772 historical feature on, 772 substitution method for solving, 766–767 Systems of nonlinear inequalities, graphing, 781 Tables, graphing utility to create, 12 Tangency, point of, 51 Tangent(s) defined, 508 graph of, 413–416 Half-angle Formulas for, 494 historical feature on, 380 Law of, 531, 534 periodic properties of, 387 Sum and Difference Formulas for, 481 Tangent function, 370 continuous, 887, 888 domain of, 385, 386, 415 inverse, 447–449 defined, 448 exact value of, 448–449 exact value of expressions involving, 453–454 implicit form of, 448 properties of, 415 range of, 386, 415 Tangent line, 51 to the graph of a function, 891–892 Greek method for finding, 51 Tangent problem, 891–892 Tartaglia (Niccolo of Brescia), 208, 591 Tautochrone, 693–694 Telescope, Cassegrain, 669 Terminal point of directed line segment, 593 Terminal side of angle, 355 Terminating decimals, A3 Terms like, A22 nth (general term), B4 of polynomial, A23 of sequences, 799–801 alternating, 801 defined by a recursive formula, 802–803 general, 800 of series, B4 by determinants, 623, 735–737 Thrust, 632 TI-84 Plus, 10 See also Graphing utility(ies) eVALUEate feature of, 13 ZERO feature of, 13 Time, as parameter, 688–691 Transformations, 640, 651, 664 combining, 106, 110–112 compressions and stretches, 106–108, 110 cosecant and secant graphs using, 418 of cosine function, 401–402 www.elsolucionario.net I13 defined, 104 graphs using of exponential functions, 278–279, 280 of polynomial functions, 182–183 of quadratic functions, 148–150 of rational functions, 218 reflections about the x-axis or y-axis, 108–110 of sine function, 399–401 vertical and horizontal shifts, 104–106, 110 Transition matrix, 758, 796 Transverse axis, 657, 678 Tree diagram, 846 Triangle(s) See also Law of Sines area of, 537–543, A15 ASA, 521–522 congruent, A16–A18 equilateral, 16 error, 17 isosceles, 16 legs of, 508, A13–A14 medians of, 16 oblique, 520–521 Pascal, 833, 836 perfect, 542 right, 508, 510–515, A13–A14 applied problems involving, 511–515 solving, 510–515 SAA, 521–522 SAS, 521, 532–533, 538–539 similar, A16–A18 SSA, 522–525 SSS, 521, 533, 539–540 Triangular addition, 835 Triangular number, 811 Trigonometric equations, 459–468 calculator for solving, 462 graphing utility to solve, 464 identities to solve, 463–464 involving single trigonometric function, 459–462 linear, 460–461 linear in sine and cosine, 483–484 quadratic in from, 462–463 solutions of, defined, 459 Trigonometric expressions, written algebraically, 456–457, 482 Trigonometric functions of acute angles, 508–510 applications of, 507–558 damped motion, 546–548 graphing sum of two functions, 548–549 involving right triangles, 511–515 Law of Cosines, 533–534 Law of Sines, 532 Law of Tangents, 531, 534 simple harmonic motion, 543–546 calculator to approximate values of, 378–379 circle of radius r to evaluate, 379 cosecant and secant graphs, 417–418 domain and the range of, 385–386 exact value of p = 45Њ, 373–374, 377–378 of p p of = 30Њ and = 60Њ, 374–378 given one function and the quadrant of the angle, 391–393 using even-odd properties, 394 fundamental identities of, 389–391 quotient, 389 reciprocal, 389 of general angles, signs of, in a given quadrant, 388–389 historical feature on, 364, 380 period of, 386–388 phase shift, 420–424 to graph y = A sin(vx - f) + B, 420–422 properties of, 368–397 Even-Odd, 394 I14 INDEX Trigonometric functions, (continued) of quadrantal angles, 371–373 right triangle trigonometry, 508–520 Complementary Angle Theorem, 510 fundamental identities, 389–391 sine and cosine graphs, 398–413 amplitude and period, 402–404, 421–424 equation for, 407–408 key points for, 404–407 sinusoidal curve fitting, 424–428 of t, 369–370 tangent and cotangent graphs, 413–417 unit circle approach to, 368–384 Trigonometric identities, 468–475 basic, 469 establishing, 470–473 Double-angle Formulas for, 489–492 Sum and Difference Formulas for, 478–481 Even-Odd, 469 Pythagorean, 469 Quotient, 469 Reciprocal, 469 Trinomials, A22–A23 Truncation, A10 Turning points, 185–186 by determinants, 623, 732–733 proof for, 738–739 Umbra versa, 380 Unbounded graphs, 782 Unbounded in positive direction, 217 Unbounded in the negative direction, polynomial functions, 187 Uniform motion, A67–A68 Uninhibited growth, 327–329 Union of sets, A2, A78n of two events, probabilities of, 861–862 Unit circle, 46, 368–371 Unit vector, 595, 598–599 in space, 617 Universal sets, 844, A2 Value (image) of function, 59, 61–64 Variable(s), A6, A21–A22 complex, 211 dependent, 62 domain of, A7 independent, 62 in calculus, 398 Variable costs, 42 Vector(s), 593–606 adding, 594, 597–598 algebraic, 596 angle between, 607–608 column, 746 components of, 596, 597 horizontal and vertical, 597 decomposing, 609 defined, 593 difference of, 594 direction of, 593, 598–600 dot product of two, 606–607 equality of, 593, 597 finding, 599–600 force, 599 geometric, 593–594 graphing, 595 historical feature on, 602 magnitudes of, 593, 595, 598, 599–600 modeling with, 600–602 multiplying by numbers geometrically, 594–595 objects in static equilibrium, 601–602 orthogonal, 608–610 parallel, 609 in physics, 593 position, 596–597 row, 746 scalar multiples of, 594–595, 597, 598, 606 in space, 613–622 angle between two vectors, 618–621 cross product of two, 623–624 direction angles, 618–621 distance formula, 615 dot product, 617–618 operations on, 616–617 orthogonal to two given vectors, 625–626 position vectors, 615–616 rectangular coordinates, 614 subtracting, 597–598 unit, 595, 598–599 velocity, 599–601 writing, 600 zero, 593, 594 Vector product See Cross (vector) product Vector projection, 609–610 Velocity, instantaneous, 894–896 Velocity vector, 599–601 Venn diagrams, A2 Verbal descriptions linear models from, 134–136 quadratic models from, 159–164 Vertex/vertices, 782 of cone, 635 of ellipse, 646 of hyperbola, 657 of parabola, 148, 636 of quadratic function, 150–154 of ray, 355 Vertical asymptote, 219, 220–221 Vertical component of vector, 597 Vertical line, 29, 571–572, 580 Vertical-line test, 72 Vertically compressed or stretched graphs, 106–107 Vertical shifts, 104106, 110 Viốte, Franỗois, 534 Viewing angle, 452 Viewing rectangle, Volume, formulas for, A15 www.elsolucionario.net Wallis, John, 591 Waves See also Simple harmonic motion Waves, traveling speeds of, 467 Weight, 632 Whispering galleries, 653 Window, viewing, Wings, airplane, 559, 632 Work, 622 dot product to compute, 610–611 World population, 798, 871, 908 x-axis, projection of P on the, 544–545 reflections about, 108–110 symmetry with respect to, 19–20 x-coordinate, x-intercept, 12 polynomial graphed using, 185 of quadratic function, 151 xy-plane, 2, 614 xz-plane, 614 Yang Hui, 836 y-axis, projection of P on the, 545 reflections about, 108–110 symmetry with respect to, 19–20 y-coordinate (ordinate), y-intercept, 12, 35–36 from linear equation, 35–36 yz-plane, 614 Zero-coupon bonds, 326 Zero-level earthquake, 301 Zero matrix, 744 Zero polynomial, A23 Zero-Product Property, A4 Zeros bounds on, 205–207 complex, of polynomials, 211–216 Conjugate Pairs Theorem, 212–213 defined, 211 finding, 214–215 polynomial function with specified zeros, 213–214 real, of polynomials, 183–185, 198–211 of even multiplicity, 185 finding, 202–204 Intermediate Value Theorem, 207–208 number of, 201 of odd multiplicity, 185 Rational Zeros Theorem, 201–202, 214 Remainder Theorem and Factor Theorem, 199–201 repeated, 184 theorem for bounds on, 205–207 Zero vector, 593, 594 Zestimate home valuation, 1, 55 Zillow.com, CONICS Parabola D: x = –a y V D: x = a y y F = (–a, 0) V F = (a, 0) x y F = (0, a) V x x x D: y = –a y2 ϭ 4ax Ellipse y2 ϭ Ϫ4ax x2 ϭ Ϫ4ay y V = (0, a ) F = (0, c) (0, b) V2 = (a, 0) x F2 = (c, 0) (0, –b) F1 = (–c, 0) (b, 0) x (–b, 0) F = (0, – c) V = (0, – a) y2 x2 + = 1, a b, c = a - b2 a b2 y2 x2 + = 1, a b, c = a - b2 b2 a y Hyperbola y V = (– a, 0) F = (– c, 0) F = (0, –a) x2 ϭ 4ay y V1 = (–a, 0) D: y = a V F = (0, c) V = (a, 0) F = (c, 0) x V = (0, a) x V = (0, –a ) F = (0, – c ) y2 x2 - = 1, c = a + b2 a b b b Asymptotes: y = x, y = - x a a y2 x2 = 1, c = a + b2 a b2 a a Asymptotes: y = x, y = - x b b - PROPERTIES OF LOGARITHMS BINOMIAL THEOREM log a 1MN2 = log a M + log a N n n 1a + b2 n = a n + a b ba n - + a b b2a n - 2 log a a M b = log a M - log a N N + g+ a log a M = r log a M r log a M = log M ln M = log a ln a a x = e x ln a PERMUTATIONS/COMBINATIONS n b b n - 1a + b n n -1 ARITHMETIC SEQUENCE a + 1a + d2 + 1a + 2d2 + g+ a + 1n - 12d n n = 2a + 1n - 12d = a + a n 2 0! = 1! = n! = n 1n - 12 # p # 132 122 112 n! P1n, r2 = 1n - r2! GEOMETRIC SEQUENCE n n! C 1n, r2 = a b = 1n - r2! r! r If r 1, a + a 1r + a 1r + g = a a 1r k - a + a 1r + a 1r + g + a 1r n - = a # GEOMETRIC SERIES - rn 1-r ϱ k=1 = www.elsolucionario.net a1 1-r LIBRARY OF FUNCTIONS Identity Function f 1x2 = x Square Function f1x2 = x2 Cube Function f1x2 = x3 y y y (1, 1) (1, 1) (0, 0) –3 (–1, –1) (2, 4) (–2, 4) (–1, 1) 3x (1, 1) 4x (0, 0) –4 (0, 0) x –4 (– 1, – 1) –4 Square Root Function Reciprocal Function f1x2 = x f1x2 = 1x Cube Root Function f1x2 = 1x y y y (1, 1) 1– (2, ) (4, 2) x (0, 0) –1 –2 (Ϫ2,Ϫ ) Ϫ3 –2 Exponential Function f1x2 = e x y (2, e (2, 2) (1, 1) y 2) (e, 1) x (0, 0) –3 (–1, 1–e ) x (0, 0) (Ϫ1, Ϫ1) Natural Logarithm Function f 1x2 = ln x y (– 2, 2) (– 1, 1) ( 1–8 , 1–2) Ϫ3 2x (–1, –1) Absolute Value Function f1x2 = x (1, 1) (Ϫ 1–8,Ϫ 1–2) (1, 1) (2, ) (1, 0) x ( 1–e , Ϫ1) (1, e) (0, 1) x Sine Function f 1x2 = sin x Cosine Function f1x2 = cos x y Ϫ␲ – Ϫ␲ Ϫ1 y 3–– ␲ ␲ – ␲ ␲ x 2␲ 5–– Cosecant Function f 1x2 = csc x y Ϫ␲ – 3␲ Ϫ––– Ϫ1 Ϫ␲ Ϫ␲ –Ϫ1 2 ␲ – ␲ 3–– ␲ ␲ 2␲ 5–– y x Secant Function f1x2 = sec x ␲ ␲ Ϫ 3–– ␲ Ϫ –– Ϫ 5–– 2Ϫ1 2 x 3␲ Ϫ ––– ␲ Ϫ –– 2Ϫ1 ␲ – ␲ 3–– 2 5␲ –– Cotangent Function f 1x2 = cot x y 3␲ ––– ␲ – Tangent Function f 1x2 = tan x y ␲ – 3␲ –– x www.elsolucionario.net 3␲ Ϫ ––– ␲ Ϫ␲ – Ϫ1 – 2 3␲ ––– 5␲ ––– x x ... 194 2Precalculus : enhanced with graphing utilities/ Michael Sullivan, Michael Sullivan, III.? ?6th ed p cm Includes index ISBN 978-0-321-79546-5 Functions—Textbooks Graphic calculators—Textbooks I Sullivan, ... Formulas; Graphing Utilities; Introduction to Graphing Equations Quadrant IV x > 0, y < PROBLEM 13 Graphing Utilities All graphing utilities (graphing calculators and computer software graphing. .. Pearson representative NEW! Resources for Sullivan/ Sullivan, Precalculus Enhanced with Graphing Utilities, 6e • Author Solves It videos feature author Michael Sullivan, III, solving MathXL® exercises