THÔNG TIN TÀI LIỆU
GRAPHS OF PARENT FUNCTIONS Linear Function Absolute Value Function x, x Ն f ͑x͒ ϭ ԽxԽ ϭ ΆϪx, f ͑x͒ ϭ mx ϩ b y Square Root Function f ͑x͒ ϭ Ίx x < y y f(x) = ⏐x⏐ x −2 (− mb , 0( (− mb , 0( f(x) = mx + b, m>0 (0, b) 2 −1 f(x) = mx + b, m0 x −1 −1 Domain: ͑Ϫ ϱ, ϱ͒ Range: ͑Ϫ ϱ, ϱ͒ x-intercept: ͑Ϫb͞m, 0͒ y-intercept: ͑0, b͒ Increasing when m > Decreasing when m < y x x (0, 0) −1 f(x) = f(x) = ax , a < (0, 0) −3 −2 −1 −2 −2 −3 −3 Domain: ͑Ϫ ϱ, ϱ͒ Range ͑a > 0͒: ͓0, ϱ͒ Range ͑a < 0͒ : ͑Ϫ ϱ, 0͔ Intercept: ͑0, 0͒ Decreasing on ͑Ϫ ϱ, 0͒ for a > Increasing on ͑0, ϱ͒ for a > Increasing on ͑Ϫ ϱ, 0͒ for a < Decreasing on ͑0, ϱ͒ for a < Even function y-axis symmetry Relative minimum ͑a > 0͒, relative maximum ͑a < 0͒, or vertex: ͑0, 0͒ x f(x) = x Domain: ͑Ϫ ϱ, ϱ͒ Range: ͑Ϫ ϱ, ϱ͒ Intercept: ͑0, 0͒ Increasing on ͑Ϫ ϱ, ϱ͒ Odd function Origin symmetry Rational (Reciprocal) Function Exponential Function Logarithmic Function f ͑x͒ ϭ x f ͑x͒ ϭ ax, a > 0, a f ͑x͒ ϭ loga x, a > 0, a y y y f(x) = x f(x) = a −x (0, 1) f(x) = a x x −1 f(x) = loga x 1 (1, 0) x x −1 Domain: ͑Ϫ ϱ, 0͒ ʜ ͑0, ϱ) Range: ͑Ϫ ϱ, 0͒ ʜ ͑0, ϱ) No intercepts Decreasing on ͑Ϫ ϱ, 0͒ and ͑0, ϱ͒ Odd function Origin symmetry Vertical asymptote: y-axis Horizontal asymptote: x-axis Domain: ͑Ϫ ϱ, ϱ͒ Range: ͑0, ϱ͒ Intercept: ͑0, 1͒ Increasing on ͑Ϫ ϱ, ϱ͒ for f ͑x͒ ϭ ax Decreasing on ͑Ϫ ϱ, ϱ͒ for f ͑x͒ ϭ aϪx Horizontal asymptote: x-axis Continuous Domain: ͑0, ϱ͒ Range: ͑Ϫ ϱ, ϱ͒ Intercept: ͑1, 0͒ Increasing on ͑0, ϱ͒ Vertical asymptote: y-axis Continuous Reflection of graph of f ͑x͒ ϭ ax in the line y ϭ x Sine Function f ͑x͒ ϭ sin x Cosine Function f ͑x͒ ϭ cos x Tangent Function f ͑x͒ ϭ tan x y y y 3 f(x) = sin x 2 f(x) = cos x 1 x −π f(x) = tan x π π 2π x −π − π π −2 −2 −3 −3 Domain: ͑Ϫ ϱ, ϱ͒ Range: ͓Ϫ1, 1͔ Period: 2 x-intercepts: ͑n, 0͒ y-intercept: ͑0, 0͒ Odd function Origin symmetry π 2π Domain: ͑Ϫ ϱ, ϱ͒ Range: ͓Ϫ1, 1͔ Period: 2 x-intercepts: ϩ n , y-intercept: ͑0, 1͒ Even function y-axis symmetry x − π π 3π ϩ n Range: ͑Ϫ ϱ, ϱ͒ Period: x-intercepts: ͑n, 0͒ y-intercept: ͑0, 0͒ Vertical asymptotes: x ϭ ϩ n Odd function Origin symmetry Domain: all x π Cosecant Function f ͑x͒ ϭ csc x Secant Function f ͑x͒ ϭ sec x f(x) = csc x = y sin x y Cotangent Function f ͑x͒ ϭ cot x f(x) = sec x = cos x f(x) = cot x = y 3 2 1 tan x x x −π π π 2π −π − π π π 3π 2π x −π − π π π 2π −2 −3 Domain: all x n Range: ͑Ϫ ϱ, Ϫ1͔ ʜ ͓1, ϱ͒ Period: 2 No intercepts Vertical asymptotes: x ϭ n Odd function Origin symmetry Domain: all x ϩ n Range: ͑Ϫ ϱ, Ϫ1͔ ʜ ͓1, ϱ͒ Period: 2 y-intercept: ͑0, 1͒ Vertical asymptotes: x ϭ ϩ n Even function y-axis symmetry Domain: all x n Range: ͑Ϫ ϱ, ϱ͒ Period: ϩ n , x-intercepts: Vertical asymptotes: x ϭ n Odd function Origin symmetry Inverse Sine Function f ͑x͒ ϭ arcsin x Inverse Cosine Function f ͑x͒ ϭ arccos x Inverse Tangent Function f ͑x͒ ϭ arctan x y y π y π π f(x) = arccos x x −1 −2 x −1 f(x) = arcsin x −π Domain: ͓Ϫ1, 1͔ Range: Ϫ , 2 Intercept: ͑0, 0͒ Odd function Origin symmetry ΄ ΅ f(x) = arctan x −π x −1 Domain: ͓Ϫ1, 1͔ Range: ͓0, ͔ y-intercept: 0, Domain: ͑Ϫ ϱ, ϱ͒ Range: Ϫ , 2 Intercept: ͑0, 0͒ Horizontal asymptotes: yϭ± Odd function Origin symmetry Precalculus with Limits Second Edition Ron Larson The Pennsylvania State University The Behrend College With the assistance of David C Falvo The Pennsylvania State University The Behrend College Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Precalculus with Limits, Second Edition Ron Larson Publisher: Charlie VanWagner Acquiring Sponsoring Editor: Gary Whalen Development Editor: Stacy Green Assistant Editor: Cynthia Ashton Editorial Assistant: Guanglei Zhang © 2011, 2007 Brooks/Cole, Cengage Learning 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 Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher Associate Media Editor: Lynh Pham Marketing Manager: Myriah FitzGibbon Marketing Coordinator: Angela Kim Marketing Communications Manager: Katy Malatesta Content Project Manager: Susan Miscio Senior Art Director: Jill Ort For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be emailed to permissionrequest@cengage.com Senior Print Buyer: Diane Gibbons Production Editor: Carol Merrigan Text Designer: Walter Kopek Rights Acquiring Account Manager, Photos: Don Schlotman Photo Researcher: Prepress PMG Library of Congress Control Number: 2009930251 Student Edition: ISBN-13: 978-1-4390-4909-9 ISBN-10: 1-4390-4909-2 Cover Designer: Harold Burch Cover Image: Richard Edelman/Woodstock Graphics Studio Compositor: Larson Texts, Inc Brooks/Cole 10 Davis Drive Belmont, CA 94002-3098 USA 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 www.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 13 12 11 10 09 Contents A Word from the Author (Preface) vii chapter Functions and Their Graphs 1.1 Rectangular Coordinates 1.2 Graphs of Equations 13 1.3 Linear Equations in Two Variables 24 1.4 Functions 39 1.5 Analyzing Graphs of Functions 54 1.6 A Library of Parent Functions 66 1.7 Transformations of Functions 73 1.8 Combinations of Functions: Composite Functions 83 1.9 Inverse Functions 92 1.10 Mathematical Modeling and Variation 102 Chapter Summary 114 Review Exercises 116 Chapter Test 121 Proofs in Mathematics 122 Problem Solving 123 chapter Polynomial and Rational Functions 125 2.1 Quadratic Functions and Models 126 2.2 Polynomial Functions of Higher Degree 136 2.3 Polynomial and Synthetic Division 150 2.4 Complex Numbers 159 2.5 Zeros of Polynomial Functions 166 2.6 Rational Functions 181 2.7 Nonlinear Inequalities 194 Chapter Summary 204 Review Exercises 206 Chapter Test 210 Proofs in Mathematics 211 Problem Solving 213 chapter Exponential and Logarithmic Functions 215 3.1 Exponential Functions and Their Graphs 216 3.2 Logarithmic Functions and Their Graphs 227 3.3 Properties of Logarithms 237 3.4 Exponential and Logarithmic Equations 244 3.5 Exponential and Logarithmic Models 255 Chapter Summary 268 Review Exercises 270 Chapter Test 273 Cumulative Test for Chapters 1–3 274 Proofs in Mathematics 276 Problem Solving 277 iii iv Contents chapter Trigonometry 279 4.1 Radian and Degree Measure 280 4.2 Trigonometric Functions: The Unit Circle 292 4.3 Right Triangle Trigonometry 299 4.4 Trigonometric Functions of Any Angle 310 4.5 Graphs of Sine and Cosine Functions 319 4.6 Graphs of Other Trigonometric Functions 330 4.7 Inverse Trigonometric Functions 341 4.8 Applications and Models 351 Chapter Summary 362 Review Exercises 364 Chapter Test 367 Proofs in Mathematics 368 Problem Solving 369 chapter Analytic Trigonometry 371 5.1 Using Fundamental Identities 372 5.2 Verifying Trigonometric Identities 380 5.3 Solving Trigonometric Equations 387 5.4 Sum and Difference Formulas 398 5.5 Multiple-Angle and Product-to-Sum Formulas 405 Chapter Summary 416 Review Exercises 418 Chapter Test 421 Proofs in Mathematics 422 Problem Solving 425 chapter Additional Topics in Trigonometry 427 6.1 Law of Sines 428 6.2 Law of Cosines 437 6.3 Vectors in the Plane 445 6.4 Vectors and Dot Products 458 6.5 Trigonometric Form of a Complex Number 468 Chapter Summary 478 Review Exercises 480 Chapter Test 484 Cumulative Test for Chapters 4–6 485 Proofs in Mathematics 487 Problem Solving 491 chapter Systems of Equations and Inequalities 7.1 7.2 7.3 7.4 7.5 493 Linear and Nonlinear Systems of Equations 494 Two-Variable Linear Systems 505 Multivariable Linear Systems 517 Partial Fractions 530 Systems of Inequalities 538 Contents 7.6 Linear Programming 549 Chapter Summary 558 Chapter Test 565 Problem Solving 567 chapter Matrices and Determinants Review Exercises 560 Proofs in Mathematics 566 569 8.1 Matrices and Systems of Equations 570 8.2 Operations with Matrices 584 8.3 The Inverse of a Square Matrix 599 8.4 The Determinant of a Square Matrix 608 8.5 Applications of Matrices and Determinants 616 Chapter Summary 628 Review Exercises 630 Chapter Test 635 Proofs in Mathematics 636 Problem Solving 637 chapter Sequences, Series, and Probability 639 9.1 Sequences and Series 640 9.2 Arithmetic Sequences and Partial Sums 651 9.3 Geometric Sequences and Series 661 9.4 Mathematical Induction 671 9.5 The Binomial Theorem 681 9.6 Counting Principles 689 9.7 Probability 699 Chapter Summary 712 Review Exercises 714 Chapter Test 717 Cumulative Test for Chapters 7–9 718 Proofs in Mathematics 720 Problem Solving 723 chapter 10 Topics in Analytic Geometry 725 10.1 Lines 726 10.2 Introduction to Conics: Parabolas 733 10.3 Ellipses 742 10.4 Hyperbolas 751 10.5 Rotation of Conics 761 10.6 Parametric Equations 769 10.7 Polar Coordinates 777 10.8 Graphs of Polar Equations 783 10.9 Polar Equations of Conics 791 Chapter Summary 798 Review Exercises 800 Chapter Test 803 Proofs in Mathematics 804 Problem Solving 807 v vi Contents chapter 11 Analytic Geometry in Three Dimensions 809 11.1 The Three-Dimensional Coordinate System 810 11.2 Vectors in Space 817 11.3 The Cross Product of Two Vectors 824 11.4 Lines and Planes in Space 831 Chapter Summary 840 Review Exercises 842 Chapter Test 844 Proofs in Mathematics 845 Problem Solving 847 chapter 12 Limits and an Introduction to Calculus 849 12.1 Introduction to Limits 850 12.2 Techniques for Evaluating Limits 861 12.3 The Tangent Line Problem 871 12.4 Limits at Infinity and Limits of Sequences 881 12.5 The Area Problem 890 Chapter Summary 898 Review Exercises 900 Chapter Test 903 Cumulative Test for Chapters 10–12 904 Proofs in Mathematics 906 Problem Solving 907 Appendix A Review of Fundamental Concepts of Algebra A.1 A.2 A.3 A.4 A.5 A.6 A.7 Real Numbers and Their Properties A1 Exponents and Radicals A14 Polynomials and Factoring A27 Rational Expressions A39 Solving Equations A49 Linear Inequalities in One Variable A63 Errors and the Algebra of Calculus A73 Answers to Odd-Numbered Exercises and Tests Index A211 Index of Applications (web) Appendix B Concepts in Statistics (web) B.1 B.2 B.3 Representing Data Measures of Central Tendency and Dispersion Least Squares Regression A81 A1 Index Conversions between degrees and radians, 284 Convex limaỗon, 787 Coordinate(s), polar, 777 Coordinate axes, reflection in, 75 Coordinate conversion, 778 polar to rectangular, 778 rectangular to polar, 778 Coordinate planes, 810 xy-plane, 810 xz-plane, 810 yz-plane, 810 Coordinate system polar, 777 rectangular, three-dimensional, 810 Correlation coefficient, 103 Correspondence, one-to-one, A2 Cosecant function, 293, 299 of any angle, 310 graph of, 333, 336 Cosine curve, amplitude of, 321 Cosine function, 293, 299 of any angle, 310 common angles, 313 domain of, 295 graph of, 323, 336 inverse, 343 period of, 322 range of, 295 special angles, 301 Cotangent function, 293, 299 of any angle, 310 graph of, 332, 336 Coterminal angles, 280 Counting Principle, Fundamental, 690 Cramer’s Rule, 616, 617 Cross multiplying, A51 Cross product algebraic properties of, 825, 845 determinant form of, 824 geometric properties of, 826, 846 of two vectors in space, 824 Cryptogram, 622 Cube of a binomial, A29 Cube root, A18 Cubic function, 68 Curtate cycloid, 776 Curve bell-shaped, 259 butterfly, 808 logistic, 260 orientation of, 770 plane, 769 rose, 786, 787 sigmoidal, 260 sine, 319 Cycle of a sine curve, 319 Cycloid, 773 curtate, 776 D Damping factor, 335 Decomposition of N͑x͒͞D ͑x͒ into partial fractions, 530 Decreasing function, 57 Defined, 47 Degenerate conic, 733 line, 733 point, 733 two intersecting lines, 733 Degree conversion to radians, 284 fractional part of, 284 measure of angles, 283 of a polynomial, A27 of a term, A27 DeMoivre’s Theorem, 472 Denominator, A8 rationalizing, 382, A20, A21 Dependent system of linear equations, 508 Dependent variable, 41, 47 Depreciation linear, 31 straight-line, 31 Derivative, 876 Descartes’s Rule of Signs, 173 Determinant area of a triangle using, 619 of a square matrix, 608, 611 of a ϫ matrix, 603, 608 Determinant form of the cross product, 824 Diagonal matrix, 598, 615 Diagonal of a polygon, 698 Difference common, of an arithmetic sequence, 651 of complex numbers, 160 of functions, 83 limit of, 855 quotient, 46, 686, 873, A45 limit of, 867 of two cubes, A31 of two squares, A31 of vectors, 447 Differences first, 678 second, 678 Diminishing returns, point of, 148 Dimpled limaỗon, 787 Direct substitution to evaluate a limit, 855 Direct variation, 104 as an nth power, 105 Directed line segment, 445 initial point of, 445 length of, 445 A213 magnitude of, 445 terminal point of, 445 Direction angle of a vector, 451 Direction numbers, 831 Direction vector, 831 Directly proportional, 104 to the nth power, 105 Directrix of a parabola, 734 Discrete mathematics, 40 Discriminant, 765 classification of conics by, 765 Distance between a point and a line, 728, 804, 848 between a point and a plane, 837 between two points in the plane, on the real number line, A6 Distance Formula, in space, 811 Distinguishable permutations, 693 Distributive Property for complex numbers, 161 for matrices, 587, 591 for real numbers, A8 Diverge, 885 Dividing out errors involving, A74 technique for evaluating a limit, 861 Division of complex numbers, 470 of fractions, A10 long, of polynomials, 150 of real numbers, A8 synthetic, 153 Division Algorithm, 151 Divisors, A10 Domain of an algebraic expression, A39 of the cosine function, 295 of a function, 39, 47 implied, 44, 47 of a rational function, 181 of the sine function, 295 Dot product, 458 properties of, 458, 490 of vectors in space, 817 Double inequality, A3, A66 Double subscript notation, 570 Double-angle formulas, 405, 423 Doyle Log Rule, 503 Drag, 492 E e, the number, 220 Eccentricity of a conic, 791 of an ellipse, 746, 791 A214 Index of a hyperbola, 755, 791 of a parabola, 791 Effective yield, 254 Elementary row operations, 572 Eliminating the parameter, 771 Elimination Gaussian, 518, 519 with back-substitution, 576 Gauss-Jordan, 577 method of, 505, 506 Ellipse, 742, 791 center of, 742 classifying by discriminant, 765 by general equation, 757 eccentricity of, 746, 791 foci of, 742 latus rectum of, 750 major axis of, 742 minor axis of, 742 standard form of the equation of, 743 vertices of, 742 Ellipsis points, A1 Endpoints of an interval, A4 Entry of a matrix, 570 main diagonal, 570 Epicycloid, 776 Equal matrices, 584 vectors, 446 in space, 817 Equality of complex numbers, 159 properties of, A9 Equating the coefficients, 533 Equation(s), 13, A49 basic, of a partial fraction decomposition, 531 circle, standard form, 19 conditional, 380, A49 of conics, polar, 791, 806 ellipse, standard form, 743 equivalent, A50 generating, A50 exponential, solving, 244 graph of, 13 hyperbola, standard form, 751 identity, A49 of a line, 24 general form, 32 graph of, 24 intercept form, 34 point-slope form, 28, 32 slope-intercept form, 24, 32 summary of, 32 two-point form, 28, 32, 621 linear, 15 in one variable, A49 in two variables, 24 logarithmic, solving, 244 parabola, standard form, 734, 805 parametric, 769 of a line in space, 831 of a plane general form, 833 standard form, 833 polar, graph of, 783 polynomial, solution of, 140 position, 523 quadratic, 15, A52 quadratic type, 389 second-degree polynomial, A52 solution of, 13, A49 solution point, 13 solving, A49 sphere, standard, 812 symmetric, of a line in space, 831 system of, 494 trigonometric, solving, 387 in two variables, 13 Equilibrium point, 512, 543 Equivalent algebraic expressions, A39 equations, A50 generating, A50 fractions, A10 generate, A10 inequalities, A64 systems, 507, 518 operations that produce, 518 Errors involving dividing out, A74 involving exponents, A74 involving fractions, A73 involving parentheses, A73 involving radicals, A74 Euler’s Formula, 475 Evaluate an algebraic expression, A7 Evaluating trigonometric functions of any angle, 313 Even function, 60 trigonometric, 296 Even/odd identities, 372 Event(s), 699 certain, 700 complement of, 706 probability of, 706 impossible, 700 independent, 705 probability of, 705 mutually exclusive, 703 probability of, 700 the union of two, 703 Existence of a limit, 865 Existence theorems, 166 Expanding a binomial, 684 by cofactors, 611 logarithmic expressions, 239 Expected value, 724 Experiment, 699 outcomes of, 699 sample space of, 699 Exponent(s), A14 errors involving, A74 negative, writing with, A75 properties of, A14 rational, A22 Exponential decay model, 255 Exponential equations, solving, 244 Exponential form, A14 Exponential function, 216 f with base a, 216 graph of, 217 natural, 220 one-to-one property, 218 Exponential growth model, 255 Exponential notation, A14 Exponentiating, 247 Expression algebraic, A7 fractional, A39 rational, A39 Extended Principle of Mathematical Induction, 673 Extracting square roots, A52 Extraneous solution, A51, A57 Extrapolation, linear, 32 F Factor(s) damping, 335 of an integer, A10 of a polynomial, 140, 170, 212 prime, 171 quadratic, 171 repeated linear, 532 quadratic, 534 scaling, 321 and terms, inserting, A76 Factor Theorem, 154, 211 Factorial, 642 Factoring, A30 completely, A30 by grouping, A34 polynomials, guidelines for, A34 solving a quadratic equation by, A52 special polynomial forms, A31 unusual, A75 Family of functions, 74 Far point, 214 Feasible solutions, 549, 550 Fibonacci sequence, 642 Finding a formula for the nth term of a sequence, 676 Index intercepts, 16 an inverse function, 96 an inverse matrix, 601 nth roots of a complex number, 474 test intervals for a polynomial, 194 vertical and horizontal asymptotes of a rational function, 183 Finite sequence, 640 Finite series, 645 First differences, 678 Fixed cost, 30 Fixed point, 395 Focal chord latus rectum, 736 of a parabola, 736 Focus (foci) of an ellipse, 742 of a hyperbola, 751 of a parabola, 734 FOIL Method, A28 Formula(s) change-of-base, 237 common, for area, perimeter, circumference, and volume, A59 for compound interest, 222 Distance, in space, 811 double-angle, 405, 423 Euler’s, 475 half-angle, 408 Heron’s Area, 440, 489 Midpoint, 5, 122 in space, 811 for the nth term of a sequence, 676 power-reducing, 407, 423 product-to-sum, 409 Quadratic, A52 radian measure, 285 recursion, 653 reduction, 400 sum and difference, 398, 422 summation, 890 sum-to-product, 410, 424 Four ways to represent a function, 40 Fractal, 723 Fraction(s) addition of with like denominators, A10 with unlike denominators, A10 complex, A43 division of, A10 equivalent, A10 generate, A10 errors involving, A73 multiplication of, A10 operations of, A10 partial, 530 decomposition, 530 properties of, A10 rules of signs for, A10 subtraction of with like denominators, A10 with unlike denominators, A10 writing as a sum, A75 Fractional expression, A39 Fractional parts of degrees minute, 284 second, 284 Frequency, 354 Function(s), 39, 47 algebraic, 216 arccosine, 343 arcsine, 341, 343 arctangent, 343 arithmetic combinations of, 83 characteristics of, 39 combinations of, 83 common logarithmic, 228 composition of, 85 constant, 40, 57, 67 continuous, 136, 769 cosecant, 293, 299, 310 cosine, 293, 299, 310 cotangent, 293, 299, 310 cubic, 68 decreasing, 57 defined, 47 derivative of, 876 difference of, 83 domain of, 39, 47 even, 60 exponential, 216 family of, 74 four ways to represent, 40 graph of, 54 greatest integer, 69 of half-angles, 405 Heaviside, 124 identity, 67 implied domain of, 44, 47 increasing, 57 inverse, 92, 93 cosine, 343 finding, 96 sine, 341, 343 tangent, 343 trigonometric, 343 limits of, 855, 857, 906 linear, 66 logarithmic, 227 of multiple angles, 405 name of, 41, 47 natural exponential, 220 natural logarithmic, 231 notation, 41, 47 objective, 549 odd, 60 one-to-one, 95 A215 parent, 70 period of, 295 periodic, 295 piecewise-defined, 42 polynomial, 126 power, 137 product of, 83 quadratic, 126 quotient of, 83 range of, 39, 47 rational, 181 limits at infinity, 884 reciprocal, 68 representation, 40 secant, 293, 299, 310 sine, 293, 299, 310 square root, 68 squaring, 67 step, 69 sum of, 83 summary of terminology, 47 tangent, 293, 299, 310 transcendental, 216 transformations of, 73 nonrigid, 77 rigid, 77 trigonometric, 293, 299, 310 undefined, 47 value of, 41, 47 Vertical Line Test, 55 zeros of, 56 Fundamental Counting Principle, 690 Fundamental Theorem of Algebra, 166 of Arithmetic, A10 Fundamental trigonometric identities, 302, 372 G Gaussian elimination, 518, 519 with back-substitution, 576 Gaussian model, 255 Gauss-Jordan elimination, 577 General form of the equation of a line, 32 of the equation of a plane, 833 of a quadratic equation, A53 Generalizations about nth roots of real numbers, A19 Generate equivalent fractions, A10 Generating equivalent equations, A50 Geometric properties of the cross product, 826, 846 Geometric property of the triple scalar product, 828 Geometric sequence, 661 common ratio of, 661 nth term of, 662 A216 Index sum of a finite, 664, 721 Geometric series, 665 sum of an infinite, 665 Geometry, solid analytic, 810 Graph, 13 of cosecant function, 333, 336 of cosine function, 323, 336 of cotangent function, 332, 336 of an equation, 13 of an exponential function, 217 of a function, 54 of an inequality, 538, A63 in two variables, 538 intercepts of, 16 of inverse cosine function, 343 of an inverse function, 94 of inverse sine function, 343 of inverse tangent function, 343 of a line, 24 of a logarithmic function, 229 point-plotting method, 13 of a polar equation, 783 of a polynomial function, x-intercept of, 140 of a rational function, 184 guidelines for analyzing, 184 reflecting, 75 of secant function, 333, 336 shifting, 73 of sine function, 323, 336 slope of, 873 special polar, 787 symmetry of a, 17 of tangent function, 330, 336 Graphical interpretations of solutions, 508 Graphical method, for solving a system of equations, 498 Graphical tests for symmetry, 17 Greatest integer function, 69 Guidelines for analyzing graphs of rational functions, 184 for factoring polynomials, A34 for solving the basic equation of a partial fraction decomposition, 535 for verifying trigonometric identities, 380 H Half-angle formulas, 408 Half-angles, functions of, 405 Half-life, 223 Harmonic motion, simple, 354, 355 Heaviside function, 124 Heron’s Area Formula, 440, 489 Hole, in the graph of a rational function, 186 Hooke’s Law, 111 Horizontal asymptote, 182 of a rational function, 183 Horizontal component of v, 450 Horizontal line, 32 Horizontal Line Test, 95 Horizontal shifts, 73 Horizontal shrink, 77 of a trigonometric function, 322 Horizontal stretch, 77 of a trigonometric function, 322 Horizontal translation of a trigonometric function, 323 Human memory model, 233 Hyperbola, 182, 751, 791 asymptotes of, 753 branches of, 751 center of, 751 classifying by discriminant, 765 by general equation, 757 conjugate axis of, 753 eccentricity of, 755, 791 foci of, 751 standard form of the equation of, 751 transverse axis of, 751 vertices of, 751 Hypocycloid, 808 Hypotenuse of a right triangle, 299 Hypothesis, 676 I i, imaginary unit, 159 Idempotent square matrix, 637 Identities cofunction, 372 even/odd, 372 Pythagorean, 302, 372 quotient, 302, 372 reciprocal, 302, 372 trigonometric fundamental, 302, 372 guidelines for verifying, 380 Identity, 380, A49 function, 67 limit of, 855 matrix of order n ϫ n, 591 Imaginary axis of the complex plane, 468 Imaginary number, 159 pure, 159 Imaginary part of a complex number, 159 Imaginary unit i, 159 Implied domain, 44, 47 Impossible event, 700 Improper rational expression, 151 Inclination of a line, 726 and slope, 726, 804 Inclusive or, A10 Inconsistent system of linear equations, 508, 576 Increasing annuity, 666 Increasing function, 57 Independent events, 705 probability of, 705 Independent system of linear equations, 508 Independent variable, 41, 47 Indeterminate form, 861 Index of a radical, A18 of summation, 644 Indirect proof, 566 Induction, mathematical, 671 Inductive, 611 Inequality (inequalities), A3 absolute value, A67 solution of, A67 double, A3, A66 equivalent, A64 graph of, 538, A63 linear, 539, A65 nonlinear, 194 polynomial, 194 properties of, A64 rational, 198 satisfy, A63 solution of, 538, A63 solution set of, A63 solution of a system of, 540 solution set, 540 solving, A63 symbol, A3 Infinite geometric series, 665 sum of, 665 Infinite sequence, 640 Infinite series, 645 Infinite wedge, 542 Infinitely many solutions, A50 Infinity limit at, 881, 882 for rational functions, 884 negative, A4 positive, A4 Initial point, 445 Initial side of an angle, 280 Inserting factors and terms, A76 Instantaneous rate of change, 876 Integer(s), A1 divisors of, A10 factors of, A10 irreducible over, A30 sums of powers of, 677 Intercept form of the equation of a line, 34 Intercepts, 16 finding, 16 Interest compound, formulas for, 222 compounded n times per year, 221 Index continuously compounded, 221 Intermediate Value Theorem, 143 Interpolation, linear, 32 Intersection, points of, 498 Interval(s), A4 bounded, A4 endpoints of, A4 using inequalities to represent, A4 on the real number line, A4 unbounded, A4 Invariant under rotation, 765 Inverse additive, A8 multiplicative, A8 of a matrix, 599 Inverse function, 92, 93 cosine, 343 finding, 96 graph of, 94 Horizontal Line Test, 95 sine, 341, 343 tangent, 343 Inverse of a matrix, 599 finding, 601 Inverse properties of logarithms, 228 of natural logarithms, 232 of trigonometric functions, 345 Inverse trigonometric functions, 343 Inverse variation, 106 Inversely proportional, 106 Invertible matrix, 600 Involute of a circle, 807 Irrational number, A1 Irreducible over the integers, A30 over the rationals, 171 over the reals, 171 J Joint variation, 107 Jointly proportional, 107 K Kepler’s Laws, 794 Key numbers of a polynomial inequality, 194 of a rational inequality, 198 Key points of the graph of a trigonometric function, 320 intercepts, 320 maximum points, 320 minimum points, 320 L Latus rectum of an ellipse, 750 of a parabola, 736 Law of Cosines, 437, 488 alternative form, 437, 488 standard form, 437, 488 Law of Sines, 428, 487, 848 Law of Tangents, 487 Law of Trichotomy, A6 Leading coefficient of a polynomial, A27 Leading Coefficient Test, 138 Leading 1, 574 Least squares regression line, 103, 515 parabola, 528 Left-handed orientation, 810 Lemniscate, 787 Length of a circular arc, 285 of a directed line segment, 445 of a vector, 446 in space, 817 Lift, 492 Like radicals, A21 Like terms of a polynomial, A28 Limaỗon, 784, 787 convex, 787 dimpled, 787 with inner loop, 787 Limit(s), 851 basic, 855 of a constant function, 855 of a difference, 855 of a difference quotient, 867 evaluating direct substitution, 855 dividing out technique, 861 rationalizing technique, 863 existence of, 865 of the identity function, 855 indeterminate form, 861 at infinity, 881, 882 for rational functions, 884 from the left, 865 nonexistence of, 854 one-sided, 865 of a polynomial function, 857, 906 of a power, 855 of a power function, 855, 906 of a product, 855 properties of, 855 of a quotient, 855 of a radical function, 855 of a rational function, 857 from the right, 865 of a scalar multiple, 855 of a sequence, 885 of a sum, 855 of summations, 890 of a trigonometric function, 855 Limit of summation A217 lower, 644 upper, 644 Line(s) in the plane angle between two, 727, 728 general form of the equation of, 32 graph of, 24 horizontal, 32 inclination of, 726 intercept form of the equation of, 34 least squares regression, 103, 515 normal, 907 parallel, 29 perpendicular, 29 point-slope form of the equation of, 28, 32 secant, 59, 873 segment, directed, 445 slope of, 24, 26 slope-intercept form of the equation of, 24, 32 summary of equations, 32 tangent, 871, 881 to a parabola, 736 two-point form of the equation of, 28, 32, 621 vertical, 25, 32 Line in space, 831 parametric equations of, 831 symmetric equations of, 831 Linear combination of vectors, 450 Linear depreciation, 31 Linear equation, 15 general form, 32 graph of, 24 intercept form, 34 in one variable, A49 point-slope form, 28, 32 slope-intercept form, 24, 32 summary of, 32 in two variables, 24 two-point form, 28, 32, 621 Linear extrapolation, 32 Linear factor, repeated, 532 Linear Factorization Theorem, 166, 212 Linear function, 66 Linear inequality, 539, A65 Linear interpolation, 32 Linear programming, 549 problem optimal solution, 549 solving, 550 Linear speed, 285 Linear system consistent, 508 dependent, 508 inconsistent, 508, 576 independent, 508 nonsquare, 522 number of solutions, 520 A218 Index row operations, 518 row-echelon form, 517 square, 522 Local maximum, 58 Local minimum, 58 Locus, 733 Logarithm(s) change-of-base formula, 237 natural, properties of, 232, 238, 276 inverse, 232 one-to-one, 232 power, 238, 276 product, 238, 276 quotient, 238, 276 properties of, 228, 238, 276 inverse, 228 one-to-one, 228 power, 238, 276 product, 238, 276 quotient, 238, 276 Logarithmic equations, solving, 244 Logarithmic expressions condensing, 239 expanding, 239 Logarithmic function, 227 with base a, 227 common, 228 graph of, 229 natural, 231 Logarithmic model, 255 Logistic curve, 260 growth model, 111, 255 Long division of polynomials, 150 Lower bound, 174 Lower limit of summation, 644 Lower triangular matrix, 615 M Magnitude, A5 of a directed line segment, 445 of a vector, 446 in space, 817 Main diagonal entries of a square matrix, 570 Major axis of an ellipse, 742 Marginal cost, 30 Mathematical induction, 671 Extended Principle of, 673 Principle of, 672 Matrix (matrices), 570 addition, 585 properties of, 587 additive identity, 588 adjoining, 601 augmented, 571 coded row, 622 coefficient, 571, 592 cofactor of, 610 column, 570 constant, 592 determinant of, 603, 608, 611 diagonal, 598, 615 elementary row operations, 572 entry of a, 570 equal, 584 idempotent, 637 identity, 591 inverse of, 599 finding, 601 invertible, 600 lower triangular, 615 main diagonal entries of a, 570 minor of, 610 multiplication, 589 properties of, 591 nonsingular, 600 order of a, 570 reduced row-echelon form, 574 representation of, 584 row, 570 row-echelon form, 574 row-equivalent, 572 scalar identity, 587 scalar multiplication, 585 properties of, 587 singular, 600 square, 570 stochastic, 597 subtraction, 586 transpose of, 638 triangular, 615 uncoded row, 622 upper triangular, 615 zero, 588 Maximum local, 58 relative, 58 value of a quadratic function, 131 Mean, arithmetic, 650 Measure of an angle, 281 degree, 283 radian, 281 Method of elimination, 505, 506 of substitution, 494 Midpoint Formula, 5, 122 in space, 811 Midpoint of a line segment, Minimum local, 58 relative, 58 value of a quadratic function, 131 Minor axis of an ellipse, 742 Minor of a matrix, 610 Minors and cofactors of a square matrix, 610 Minute, fractional part of a degree, 284 Modulus of a complex number, 469 Moment, 847 Monomial, A27 Multiple angles, functions of, 405 Multiplication of complex numbers, 470 of fractions, A10 of matrices, 589 properties of, 591 scalar of matrices, 585 of vectors, 447, 817 Multiplicative identity of a real number, A8 Multiplicative inverse, A8 of a matrix, 599 of a real number, A8 Multiplicity, 140 Multiplier effect, 669 Mutually exclusive events, 703 N n factorial, 642 Name of a function, 41, 47 Natural base, 220 Natural exponential function, 220 Natural logarithm properties of, 232, 238, 276 inverse, 232 one-to-one, 232 power, 238, 276 product, 238, 276 quotient, 238, 276 Natural logarithmic function, 231 Natural numbers, A1 Near point, 214 Negation, properties of, A9 Negative angle, 280 exponents, writing with, A75 infinity, A4 number, principal square root of, 163 of a vector, 447 Newton’s Law of Cooling, 111, 266 Newton’s Law of Universal Gravitation, 111 No solution, A50 Nonexistence of a limit, 854 Nonlinear inequalities, 194 Nonnegative number, A2 Nonrigid transformations, 77 Nonsingular matrix, 600 Nonsquare system of linear equations, 522 Normal line, 907 vector, 833 Normally distributed, 259 Notation Index double subscript, 570 exponential, A14 function, 41, 47 scientific, A16 sigma, 644 standard unit vector, 817 summation, 644 nth partial sum, 645, 655 of an arithmetic sequence, 655 nth root(s) of a, A18 of a complex number, 473, 474 generalizations about, A19 principal, A18 of unity, 475 nth term of an arithmetic sequence, 652 of a geometric sequence, 662 of a sequence, finding a formula for, 676 Number(s) complex, 159 composite, A10 direction, 831 imaginary, 159 pure, 159 irrational, A1 key, 194, 198 natural, A1 negative, principal square root of, 163 nonnegative, A2 of outcomes, 700 prime, A10 rational, A1 real, A1 whole, A1 Number of permutations of n elements, 691 taken r at a time, 692 Number of solutions of a linear system, 520 Numerator, A8 rationalizing, A22 O Objective function, 549 Oblique asymptote, 187 Oblique triangle, 428 area of, 432 Obtuse angle, 281 Octant, 810 Odd/even identities, 372 Odd function, 60 trigonometric, 296 One cycle of a sine curve, 319 One-sided limit, 865 One-to-one correspondence, A2 One-to-one function, 95 One-to-one property of exponential functions, 218 of logarithms, 228 of natural logarithms, 232 Operations of fractions, A10 that produce equivalent systems, 518 Opposite side of a right triangle, 299 Optimal solution of a linear programming problem, 549 Optimization, 549 Order of a matrix, 570 on the real number line, A3 Ordered pair, Ordered triple, 517 Orientation of a curve, 770 Origin, 2, A2 of polar coordinate system, 777 of the real number line, A2 of the rectangular coordinate system, symmetric with respect to, 17 Orthogonal vectors, 460 in space, 818 Outcomes, 699 number of, 700 P Parabola, 126, 734, 791 axis of, 127, 734 classifying by discriminant, 765 by general equation, 757 directrix of, 734 eccentricity of, 791 focal chord of, 736 focus of, 734 latus rectum of, 736 least squares regression, 528 reflective property of, 736 standard form of the equation of, 734, 805 tangent line to, 736 vertex of, 127, 734 Parallel lines, 29 planes, 834 vectors in space, 819 Parallelogram law for vector addition, 447 Parameter, 769 eliminating, 771 Parametric equations, 769 of a line in space, 831 Parent functions, 70 Parentheses, errors involving, A73 Partial fraction, 530 decomposition, 530 Partial sum, nth, 645, 655 Pascal’s Triangle, 683 Perfect cube, A19 A219 square, A19 square trinomial, A31, A32 Perigee, 746 Perihelion distance, 749, 796 Perimeter, common formulas for, A59 Period of a function, 295 of sine and cosine functions, 322 Periodic function, 295 Permutation(s), 691 distinguishable, 693 of n elements, 691 taken r at a time, 692 Perpendicular lines, 29 planes, 834 vectors, 460 Phase shift, 323 Piecewise-defined function, 42 Plane(s) angle between two, 834 coordinate, 810 general form of the equation of, 833 parallel, 834 perpendicular, 834 in space, 833 sketching, 836 standard form of the equation of, 833 trace of, 836 Plane curve, 769 orientation of, 770 Plane region, area of, 894 Plotting, on the real number line, A2 Point(s) break-even, 499 collinear, 12, 620, 820 test for, 620 of diminishing returns, 148 equilibrium, 512, 543 fixed, 395 initial, 445 of intersection, 498 solution, 13 terminal, 445 Point-plotting method, 13 Point-slope form of the equation of a line, 28, 32 Polar axis, 777 Polar coordinate system, 777 pole (origin) of, 777 Polar coordinates, 777 conversion to rectangular, 778 tests for symmetry in, 784, 785 Polar equation, graph of, 783 Polar equations of conics, 791, 806 Polar form of a complex number, 469 Pole, 777 Polygon, diagonal of, 698 Polynomial(s), A27 A220 Index coefficient of, A27 completely factored, A30 constant term, A27 degree of, A27 equation second-degree, A52 solution of, 140 factoring special forms, A31 factors of, 140, 170, 212 finding test intervals for, 194 guidelines for factoring, A34 inequality, 194 irreducible, A30 leading coefficient of, A27 like terms, A28 long division of, 150 operations with, A28 prime, A30 prime quadratic factor, 171 standard form of, A27 synthetic division, 153 test intervals for, 141 Polynomial function, 126 Leading Coefficient Test, 138 limit of, 857, 906 real zeros of, 140 standard form, 139 of x with degree n, 126 x-intercept of the graph of, 140 zeros of, 139 Position equation, 523 Positive angle, 280 infinity, A4 Power, A14 of a complex number, 472 limit of, 855 Power function, 137 limit of, 855, 906 Power property of logarithms, 238, 276 of natural logarithms, 238, 276 Power-reducing formulas, 407, 423 Powers of integers, sums of, 677 Prime factor of a polynomial, 171 factorization, A10 number, A10 polynomial, A30 quadratic factor, 171 Principal nth root of a, A18 of a number, A18 Principal square root of a negative number, 163 Principle of Mathematical Induction, 672 Extended, 673 Probability of a complement, 706 of an event, 700 of independent events, 705 of the union of two events, 703 Producer surplus, 543 Product of functions, 83 limit of, 855 of trigonometric functions, 405 triple scalar, 828 of two complex numbers, 470 Product property of logarithms, 238, 276 of natural logarithms, 238, 276 Product-to-sum formulas, 409 Projection of a vector, 462 Proof, 122 by contradiction, 566 indirect, 566 without words, 636 Proper rational expression, 151 Properties of absolute value, A6 of the cross product algebraic, 825, 845 geometric, 826, 846 of the dot product, 458, 490 of equality, A9 of exponents, A14 of fractions, A10 geometric, of the triple scalar product, 828 of inequalities, A64 inverse, of trigonometric functions, 345 of limits, 855 of logarithms, 228, 238, 276 inverse, 228 one-to-one, 228 power, 238, 276 product, 238, 276 quotient, 238, 276 of matrix addition and scalar multiplication, 587 of matrix multiplication, 591 of natural logarithms, 232, 238, 276 inverse, 232 one-to-one, 232 power, 238, 276 product, 238, 276 quotient, 238, 276 of negation, A9 one-to-one, exponential functions, 218 of radicals, A19 reflective, of a parabola, 736 summation, 890 of sums, 645, 720 of vector addition and scalar multiplication, 449 of zero, A10 Proportional directly, 104 to the nth power, 105 inversely, 106 jointly, 107 Proportionality, constant of, 104 Pure imaginary number, 159 Pythagorean identities, 302, 372 Pythagorean Theorem, 4, 368 Q Quadrants, Quadratic equation, 15, A52 complex solutions of, 163 general form of, A53 solving by completing the square, A52 by extracting square roots, A52 by factoring, A52 using the Quadratic Formula, A52 using the Square Root Principle, A52 Quadratic factor prime, 171 repeated, 534 Quadratic Formula, A52 Quadratic function, 126 maximum value, 131 minimum value, 131 standard form of, 129 Quadratic type equations, 389 Quick tests for symmetry in polar coordinates, 785 Quotient difference, 46, 686, 873, A45 limit of, 867 of functions, 83 limit of, 855 of two complex numbers, 470 Quotient identities, 302, 372 Quotient property of logarithms, 238, 276 of natural logarithms, 238, 276 R Radian, 281 conversion to degrees, 284 Radian measure formula, 285 Radical(s) errors involving, A74 index of, A18 like, A21 properties of, A19 simplest form, A20 symbol, A18 Radical function, limit of, 855 Radicand, A18 Radius of a circle, 19 Random selection with replacement, 689 Index without replacement, 689 Range of the cosine function, 295 of a function, 39, 47 of the sine function, 295 Rate, 30 Rate of change, 30 average, 59, 876 instantaneous, 876 Ratio, 30 Rational exponent, A22 Rational expression(s), A39 improper, 151 proper, 151 Rational function, 181 asymptotes of, 183 domain of, 181 graph of, guidelines for analyzing, 184 hole in the graph, 186 limit of, 857 limit at infinity, 884 test intervals for, 184 Rational inequality, 198 test intervals, 198 Rational number, A1 Rational Zero Test, 167 Rationalizing a denominator, 382, A20, A21 a numerator, A22 Rationalizing technique for evaluating a limit, 863 Real axis of the complex plane, 468 Real number(s), A1 absolute value of, A5 classifying, A1 division of, A8 subset of, A1 subtraction of, A8 Real number line, A2 bounded intervals on, A4 distance between two points, A6 interval on, A4 order on, A3 origin of, A2 plotting on, A2 unbounded intervals on, A4 Real part of a complex number, 159 Real zeros of a polynomial function, 140 Reciprocal function, 68 Reciprocal identities, 302, 372 Rectangular coordinate system, Rectangular coordinates, conversion to polar, 778 Recursion formula, 653 Recursive sequence, 642 Reduced row-echelon form of a matrix, 574 Reducible over the reals, 171 Reduction formulas, 400 Reference angle, 312 Reflection, 75 of a trigonometric function, 322 Reflective property of a parabola, 736 Region, plane, area of, 894 Regression, least squares line, 103, 515 parabola, 528 Relation, 39 Relative maximum, 58 Relative minimum, 58 Remainder, uses in synthetic division, 155 Remainder Theorem, 154, 211 Repeated linear factor, 532 Repeated quadratic factor, 534 Repeated zero, 140 Representation of functions, 40 of matrices, 584 Resultant of vector addition, 447 Right triangle adjacent side of, 299 definitions of trigonometric functions, 299 hypotenuse of, 299 opposite side of, 299 solving, 304 Right-handed orientation, 810 system, 826 Rigid transformations, 77 Root(s) of a complex number, 473, 474 cube, A18 nth, A18 principal nth, A18 square, A18 Rose curve, 786, 787 Rotation of axes, 761 to eliminate an xy-term, 761 invariants, 765 Row matrix, 570 coded, 622 uncoded, 622 Row operations, 518 elementary, 572 Row-echelon form, 517 of a matrix, 574 reduced, 574 Row-equivalent matrices, 572 Rules of signs for fractions, A10 S Sample space, 699 Satisfy the inequality, A63 Scalar, 447, 585 multiple, 585 A221 limit of, 855 of a vector in space, 817 Scalar Identity Property for matrices, 587 Scalar multiplication of matrices, 585 properties of, 587 of a vector, 447 properties of, 449 Scaling factor, 321 Scatter plot, Scientific notation, A16 Scribner Log Rule, 503 Secant function, 293, 299 of any angle, 310 graph of, 333, 336 Secant line, 59, 873 Second differences, 678 Second, fractional part of a degree, 284 Second-degree polynomial equation, A52 Sector of a circle, 287 area of, 287 Sequence, 640 arithmetic, 651 convergence of, 885 divergence of, 885 Fibonacci, 642 finite, 640 first differences of, 678 geometric, 661 infinite, 640 limit of, 885 nth partial sum of, 645 recursive, 642 second differences of, 678 terms of, 640 Series, 645 finite, 645 geometric, 665 infinite, 645 geometric, 665 Shifting graphs, 73 Shrink horizontal, 77 vertical, 77 Sierpinski Triangle, 723 Sigma notation, 644 Sigmoidal curve, 260 Simple harmonic motion, 354, 355 frequency, 354 Simplest form, of an expression involving radicals, A20 Sine curve, 319 amplitude of, 321 one cycle of, 319 Sine function, 293, 299 of any angle, 310 common angles, 313 domain of, 295 graph of, 323, 336 A222 Index inverse, 341, 343 period of, 322 range of, 295 special angles, 301 Sines, cosines, and tangents of special angles, 301 Singular matrix, 600 Sketching the graph of an equation by point plotting, 13 of an inequality in two variables, 538 Sketching planes in space, 836 Slant asymptote, 187 Slope, 831, 873 of a graph, 873 and inclination, 726, 804 of a line, 24, 26 Slope-intercept form of the equation of a line, 24, 32 Solid analytic geometry, 810 Solution(s), 13 of an absolute value inequality, A67 of an equation, 13, A49 extraneous, A51, A57 feasible, 549, 550 of an inequality, 538, A63 infinitely many, A50 of a linear programming problem, optimal, 549 of a linear system, number of, 520 no, A50 of a polynomial equation, 140 of a quadratic equation, complex, 163 of a system of equations, 494 graphical interpretations, 508 of a system of inequalities, 540 solution set, 540 Solution point, 13 Solution set of an inequality, A63 of a system of inequalities, 540 Solving an absolute value inequality, A67 the basic equation of a partial fraction decomposition, 535 an equation, A49 exponential and logarithmic equations, 244 an inequality, A63 a linear programming problem, 550 a polynomial inequality, 195 a quadratic equation by completing the square, A52 by extracting square roots, A52 by factoring, A52 using the Quadratic Formula, A52 using the Square Root Principle, A52 a rational inequality, 198 right triangles, 304 a system of equations, 494 Cramer’s Rule, 616, 617 Gaussian elimination, 518, 519 with back-substitution, 576 Gauss-Jordan elimination, 577 graphical method, 498 method of elimination, 505, 506 method of substitution, 494 a trigonometric equation, 387 Space Distance Formula in, 811 lines in, 831 Midpoint Formula in, 811 planes in, 833 sketching, 836 surface in, 813 vector in, 817 Special angles cosines of, 301 sines of, 301 tangents of, 301 Special products, A29 Speed angular, 285 linear, 285 Sphere, 812 standard equation of, 812 Square of a binomial, A29 of trigonometric functions, 405 Square matrix, 570 determinant of, 608, 611 diagonal, 615 idempotent, 637 lower triangular, 615 main diagonal entries of, 570 minors and cofactors of, 610 triangular, 615 upper triangular, 615 Square root(s), A18 extracting, A52 function, 68 of a negative number, 163 principal, of a negative number, 163 Square Root Principle, A52 Square system of linear equations, 522 Squaring function, 67 Standard equation of a sphere, 812 Standard form of a complex number, 159 of the equation of a circle, 19 of an ellipse, 743 of a hyperbola, 751 of a parabola, 734, 805 of a plane, 833 of Law of Cosines, 437, 488 of a polynomial, A27 of a polynomial function, 139 of a quadratic function, 129 Standard position of an angle, 280 of a vector, 446 Standard unit vector, 450 notation in space, 817 Step function, 69 Stochastic matrix, 597 Straight-line depreciation, 31 Strategies for solving exponential and logarithmic equations, 244 Stretch horizontal, 77 vertical, 77 Strophoid, 808 Subsets, A1 Substitution direct, to evaluate a limit, 855 method of, 494 Substitution Principle, A7 Subtraction of complex numbers, 160 of fractions with like denominators, A10 with unlike denominators, A10 of matrices, 586 of real numbers, A8 Sum(s) of complex numbers, 160 of a finite arithmetic sequence, 654, 721 of a finite geometric sequence, 664, 721 of functions, 83 of an infinite geometric series, 665 limit of, 855 nth partial, 645, 655 of powers of integers, 677 properties of, 645, 720 of square differences, 103 of two cubes, A31 of vectors, 447 in space, 817 Sum and difference formulas, 398, 422 Sum and difference of same terms, A29 Summary of equations of lines, 32 of function terminology, 47 Summation formulas and properties, 890 index of, 644 limit of, 890 lower limit of, 644 notation, 644 upper limit of, 644 Sum-to-product formulas, 410, 424 Supplementary angles, 283 Surface in space, 813 trace of, 814 Surplus consumer, 543 producer, 543 Index Symbol “approximately equal to”, A1 inequality, A3 radical, A18 union, A67 Symmetric equations of a line in space, 831 Symmetry, 17 algebraic tests for, 17 axis of, of a parabola, 127 graphical tests for, 17 in polar coordinates, tests for, 784, 785 with respect to the origin, 17 with respect to the x-axis, 17 with respect to the y-axis, 17 Synthetic division, 153 uses of the remainder in, 155 System of equations, 494 equivalent, 507, 518 solution of, 494 solving, 494 with a unique solution, 604 System of inequalities, solution of, 540 solution set, 540 System of linear equations consistent, 508 dependent, 508 inconsistent, 508, 576 independent, 508 nonsquare, 522 number of solutions, 520 row operations, 518 row-echelon form, 517 square, 522 T Tangent function, 293, 299 of any angle, 310 common angles, 313 graph of, 330, 336 inverse, 343 special angles, 301 Tangent line, 871, 881 to a parabola, 736 Term of an algebraic expression, A7 constant, A7, A27 degree of, A27 of a sequence, 640 variable, A7 Terminal point, 445 Terminal side of an angle, 280 Terms, inserting factors and, A76 Test(s) for collinear points, 620 Horizontal Line, 95 Leading Coefficient, 138 Rational Zero, 167 for symmetry algebraic, 17 graphical, 17 in polar coordinates, 784, 785 Vertical Line, 55 Test intervals polynomial, 141 polynomial inequality, 194 rational function, 184 rational inequality, 198 Theorem of Algebra, Fundamental, 166 of Arithmetic, Fundamental, A10 Binomial, 681, 722 DeMoivre’s, 472 Descartes’s Rule of Signs, 173 existence, 166 Factor, 154, 211 Intermediate Value, 143 Linear Factorization, 166, 212 Pythagorean, 4, 368 Remainder, 154, 211 Three-dimensional coordinate system, 810 left-handed orientation, 810 octant, 810 right-handed orientation, 810 Thrust, 492 Torque, 847 Trace of a plane, 836 of a surface, 814 Transcendental function, 216 Transformations of functions, 73 nonrigid, 77 rigid, 77 Transpose of a matrix, 638 Transverse axis of a hyperbola, 751 Triangle area of using a determinant, 619 Heron’s Area Formula, 440, 489 oblique, 428 area of, 432 Triangular matrix, 615 Trigonometric equations, solving, 387 Trigonometric form of a complex number, 469 argument of, 469 modulus of, 469 Trigonometric functions, 293, 299, 310 of any angle, 310 evaluating, 313 cosecant, 293, 299, 310 cosine, 293, 299, 310 cotangent, 293, 299, 310 even, 296 horizontal shrink of, 322 horizontal stretch of, 322 horizontal translation of, 323 inverse, 343 A223 inverse properties of, 345 key points, 320 intercepts, 320 maximum points, 320 minimum points, 320 limit of, 855 odd, 296 product of, 405 reflection of, 322 right triangle definitions of, 299 secant, 293, 299, 310 sine, 293, 299, 310 square of, 405 tangent, 293, 299, 310 unit circle definitions of, 293 vertical shrink of, 321 vertical stretch of, 321 vertical translation of, 324 Trigonometric identities cofunction, 372 even/odd, 372 fundamental, 302, 372 guidelines for verifying, 380 Pythagorean, 302, 372 quotient, 302, 372 reciprocal, 302, 372 Trigonometric values of common angles, 313 Trigonometry, 280 Trinomial, A27 perfect square, A31, A32 Triple scalar product, 828 geometric property of, 828 Two-point form of the equation of a line, 28, 32, 621 U Unbounded intervals, A4 Uncoded row matrices, 622 Undefined, 47 Union symbol, A67 Union of two events, probability of, 703 Unit circle, 292 definitions of trigonometric functions, 293 Unit vector, 446 in the direction of v, 449 form, 817 notation, standard, 817 in space, in the direction of v, 817 standard, 450 Unity, nth roots of, 475 Unusual factoring, A75 Upper bound, 174 Upper limit of summation, 644 Upper and Lower Bound Rules, 174 Upper triangular matrix, 615 Uses of the remainder in synthetic division, 155 A224 Index V Value of a function, 41, 47 Variable, A7 dependent, 41, 47 independent, 41, 47 term, A7 Variation combined, 106 constant of, 104 direct, 104 as an nth power, 105 inverse, 106 joint, 107 in sign, 173 Vary directly, 104 as nth power, 105 Vary inversely, 106 Vary jointly, 107 Vector(s), 445, 817, 831 addition of, 447, 817 properties of, 449 resultant of, 447 analysis, 845 angle between two, 459, 490, 818 component form of, 446, 817 components of, 446, 461, 462 cross product of, 824 difference of, 447 directed line segment representation, 445 direction, 831 direction angle of, 451 dot product of, 458, 817 properties of, 458, 490 equal, 446, 817 horizontal component of, 450 length of, 446, 817 linear combination of, 450 magnitude of, 446, 817 negative of, 447 normal, 833 orthogonal, 460, 818 parallel, 819 parallelogram law, 447 perpendicular, 460 in the plane, 445 projection of, 462 resultant of, 447 scalar multiplication of, 447, 817 properties of, 449 in space, 817 addition of, 817 angle between two, 818 component form of, 817 cross product of, 824 direction, 831 dot product of, 817 equal, 817 length of, 817 magnitude of, 817 normal, 833 orthogonal, 818 parallel, 819 scalar multiple of, 817 standard unit, 817 sum of, 817 triple scalar product, 828 unit, in the direction of v, 817 unit vector form, 817 zero, 817 standard position of, 446 sum of, 447, 817 triple scalar product, 828 unit, 446 in the direction of v, 449, 817 form, 817 standard, 450, 817 v in the plane, 445 vertical component of, 450 zero, 446, 817 Vertex (vertices) of an angle, 280 of an ellipse, 742 of a hyperbola, 751 of a parabola, 127, 734 Vertical asymptote, 182 of a rational function, 183 Vertical component of v, 450 Vertical line, 25, 32 Vertical Line Test, 55 Vertical shifts, 73 Vertical shrink, 77 of a trigonometric function, 321 Vertical stretch, 77 of a trigonometric function, 321 Vertical translation of a trigonometric function, 324 Volume, common formulas for, A59 W Wedge, infinite, 542 Whole numbers, A1 With replacement, 689 Without replacement, 689 Work, 464 Writing a fraction as a sum, A75 with negative exponents, A75 X x, change in, 873 x-axis, symmetric with respect to, 17 x-coordinate, x-intercepts finding, 16 of the graph of a polynomial function, 140 xy-plane, 810 xz-plane, 810 Y y, change in, 873 y-axis, symmetric with respect to, 17 y-coordinate, y-intercepts, finding, 16 yz-plane, 810 Z Zero(s) of a function, 56 matrix, 588 multiplicity of, 140 of a polynomial function, 139, 140 bounds for, 174 real, 140 properties of, A10 repeated, 140 vector, 446 in space, 817 Zero polynomial, A27 Zero-Factor Property, A10 y Definition of the Six Trigonometric Functions Right triangle definitions, where < < ͞2 Opposite se nu ote p Hy θ Adjacent opp hyp adj cos ϭ hyp opp tan ϭ adj sin ϭ (− 12 , 23 ) π (− 22 , 22 ) 3π 23π 120° 135° (− 23 , 12) 56π 150° hyp opp hyp sec ϭ adj adj cot ϭ opp csc ϭ Circular function definitions, where is any angle y r y csc ϭ sin ϭ 2 r y r= x +y (x , y ) x r sec ϭ cos ϭ r r x θ y y x x cot ϭ tan ϭ x x y ( ( 12 , 23 ) π , 22 ) π ( 60° 45° π ( , ) 2 30° (0, 1) 90° 0° x 360° 2π (1, 0) 330°11π 315° , − 12 300° 74π (−1, 0) π 180° 7π 210° 225° − 2, −2 5π 240° ) (− , − 4π ) 2 − 12 , ( − 270° ) 3π 5π (0, −1) ( ) ( 22 , − 22 ) ( 12 , − 23 ) Double-Angle Formulas Reciprocal Identities csc u csc u ϭ sin u sin u ϭ sec u sec u ϭ cos u cos u ϭ cot u cot u ϭ tan u tan u ϭ sin u cos u cot u ϭ cos u sin u Pythagorean Identities sin2 u ϩ cos2 u ϭ 1 ϩ tan2 u ϭ sec2 u 2 Ϫ u ϭ cos u cos Ϫ u ϭ sin u tan Ϫ u ϭ cot u 2 Ϫ u ϭ tan u sec Ϫ u ϭ csc u csc Ϫ u ϭ sec u cot sin u ϩ sin v ϭ sin u ϩ2 v cosu Ϫ2 v sin u Ϫ sin v ϭ cos u ϩ2 v sinu Ϫ2 v cos u ϩ cos v ϭ cos u ϩ2 v cosu Ϫ2 v cos u Ϫ cos v ϭ Ϫ2 sin u ϩ2 v sinu Ϫ2 v Product-to-Sum Formulas Even/Odd Identities sin͑Ϫu͒ ϭ Ϫsin u cos͑Ϫu͒ ϭ cos u tan͑Ϫu͒ ϭ Ϫtan u Ϫ cos 2u ϩ cos 2u cos u ϭ Ϫ cos 2u tan2 u ϭ ϩ cos 2u Sum-to-Product Formulas ϩ cot2 u ϭ csc2 u Cofunction Identities sin Power-Reducing Formulas sin2 u ϭ Quotient Identities tan u ϭ sin 2u ϭ sin u cos u cos 2u ϭ cos2 u Ϫ sin2 u ϭ cos2 u Ϫ ϭ Ϫ sin2 u tan u tan 2u ϭ Ϫ tan2 u cot͑Ϫu͒ ϭ Ϫcot u sec͑Ϫu͒ ϭ sec u csc͑Ϫu͒ ϭ Ϫcsc u Sum and Difference Formulas sin͑u ± v͒ ϭ sin u cos v ± cos u sin v cos͑u ± v͒ ϭ cos u cos v ϯ sin u sin v tan u ± tan v tan͑u ± v͒ ϭ ϯ tan u tan v sin u sin v ϭ ͓cos͑u Ϫ v͒ Ϫ cos͑u ϩ v͔͒ cos u cos v ϭ ͓cos͑u Ϫ v͒ ϩ cos͑u ϩ v͔͒ sin u cos v ϭ ͓sin͑u ϩ v͒ ϩ sin͑u Ϫ v͔͒ cos u sin v ϭ ͓sin͑u ϩ v͒ Ϫ sin͑u Ϫ v͔͒ FORMULAS FROM GEOMETRY Triangle: Sector of Circular Ring: c a h h ϭ a sin θ b Area ϭ bh c ϭ a ϩ b Ϫ 2ab cos (Law of Cosines) Right Triangle: a Area ϭ Ί3s s Volume ϭ h Ah h A ϭ area of base A s Parallelogram: Right Circular Cone: Area ϭ bh r 2h Lateral Surface Area ϭ rΊr ϩ h h Trapezoid: h Area ϭ ͑a ϩ b͒ h Volume ϭ b a Frustum of Right Circular Cone: h ͑r ϩ rR ϩ R 2͒h Volume ϭ Lateral Surface Area ϭ s͑R ϩ r͒ b a r r s h Circle: Right Circular Cylinder: r Area ϭ Circumference ϭ 2 r Volume ϭ Lateral Surface Area ϭ 2 rh Volume ϭ r 3 s θ r Surface Area ϭ r 4 r Wedge: Circular Ring: Area ϭ ͑ Ϫ ͒ ϭ 2 pw p ϭ average radius, w ϭ width of ring R2 h Sphere: r s ϭ r in radians r r2h r Sector of Circle: R h b Area ϭ Ί a2 ϩ b2 Cone: s a Circumference Ϸ 2 Equilateral Triangle: hϭ w b Area ϭ ab b Ί3s θ Ellipse: c Pythagorean Theorem c2 ϭ a2 ϩ b2 p Area ϭ pw p ϭ average radius, w ϭ width of ring, in radians r2 r p R w A ϭ B sec A ϭ area of upper face, B ϭ area of base A θ B ALGEBRA Factors and Zeros of Polynomials: Given the polynomial p͑x͒ ϭ an x n ϩ anϪ1 x nϪ1 ϩ ϩ a x ϩ a If p͑b͒ ϭ 0, then b is a zero of the polynomial and a solution of the equation p͑x͒ ϭ Furthermore, ͑x Ϫ b͒ is a factor of the polynomial Fundamental Theorem of Algebra: An nth degree polynomial has n (not necessarily distinct) zeros Quadratic Formula: If p͑x͒ ϭ ax ϩ bx ϩ c, a x ϭ ͑Ϫb ± Ίb2 Ϫ 4ac ͒͞2a and b Ϫ 4ac Ն 0, then the real zeros of p are Special Factors: Examples x Ϫ a ϭ ͑x Ϫ a͒͑x ϩ a͒ x Ϫ a ϭ ͑x Ϫ a͒͑x ϩ ax ϩ a 2͒ x ϩ a ϭ ͑x ϩ a͒͑x Ϫ ax ϩ a 2͒ x Ϫ ϭ ͑x Ϫ 3͒͑x ϩ 3͒ x Ϫ ϭ ͑x Ϫ 2͒͑x ϩ 2x ϩ 4͒ x2 Ϫ Ί 4x ϩ Ί 16 x ϩ ϭ ͑x ϩ Ί ͒͑ ͒ x Ϫ a ϭ ͑x Ϫ a͒͑x ϩ a͒͑x ϩ a 2͒ x ϩ a ϭ ͑x ϩ Ί2 ax ϩ a 2͒͑x Ϫ Ί2 ax ϩ a 2͒ x n Ϫ a n ϭ ͑x Ϫ a͒͑x nϪ1 ϩ axnϪ2 ϩ ϩ a nϪ1͒, for n odd x n ϩ a n ϭ ͑x ϩ a͒͑x nϪ1 Ϫ ax nϪ2 ϩ ϩ a nϪ1͒, for n odd x 2n Ϫ a 2n ϭ ͑x n Ϫ a n͒͑x n ϩ a n͒ x Ϫ ϭ ͑x Ϫ Ί2 ͒͑x ϩ Ί2 ͒͑x ϩ 2͒ x ϩ ϭ ͑x ϩ 2x ϩ 2͒͑x Ϫ 2x ϩ 2͒ x Ϫ ϭ ͑x Ϫ 1͒͑x ϩ x ϩ x ϩ x ϩ 1͒ x ϩ ϭ ͑x ϩ 1͒͑x Ϫ x ϩ x Ϫ x ϩ x Ϫ x ϩ 1͒ x Ϫ ϭ ͑x Ϫ 1͒͑x ϩ 1͒ Binomial Theorem: Examples ͑x ϩ a͒2 ϭ x ϩ 2ax ϩ a ͑x Ϫ a͒2 ϭ x Ϫ 2ax ϩ a ͑x ϩ a͒3 ϭ x ϩ 3ax ϩ 3a 2x ϩ a ͑x Ϫ a͒3 ϭ x Ϫ 3ax ϩ 3a 2x Ϫ a ͑x ϩ a͒4 ϭ x ϩ 4ax ϩ 6a 2x ϩ 4a ϩ a ͑x Ϫ a͒4 ϭ x Ϫ 4ax ϩ 6a 2x Ϫ 4a 3x ϩ a n͑n Ϫ 1͒ nϪ2 ͑x ϩ a͒n ϭ xn ϩ naxnϪ1 ϩ a x ϩ ϩ nanϪ1x ϩ a n 2! n͑n Ϫ 1͒ nϪ2 ͑x Ϫ a͒n ϭ x n Ϫ nax nϪ1 ϩ a x Ϫ ± na nϪ1x ϯ a n 2! ͑x ϩ 3͒2 ϭ x ϩ 6x ϩ ͑x Ϫ 5͒2 ϭ x Ϫ 10x ϩ 25 ͑x ϩ 2͒3 ϭ x ϩ 6x ϩ 12x ϩ ͑x Ϫ 1͒3 ϭ x Ϫ 3x ϩ 3x Ϫ ͑x ϩ Ί2 ͒4 ϭ x ϩ 4Ί2 x ϩ 12x ϩ 8Ί2 x ϩ ͑x Ϫ 4͒4 ϭ x Ϫ 16x ϩ 96x Ϫ 256x ϩ 256 ͑x ϩ 1͒5 ϭ x ϩ 5x ϩ 10x ϩ 10x ϩ 5x ϩ ͑x Ϫ 1͒6 ϭ x Ϫ 6x5 ϩ 15x Ϫ 20x ϩ 15x Ϫ 6x ϩ Rational Zero Test: If p͑x͒ ϭ an x n ϩ anϪ1x nϪ1 ϩ ϩ a1 x ϩ a0 has integer coefficients, then every rational zero of p͑x͒ ϭ is of the form x ϭ r͞s, where r is a factor of a0 and s is a factor of an Exponents and Radicals: a0 ϭ 1, a aϪx ϭ ax a xa y ϭ a xϩy ax ϭ a xϪy ay ab ͑a x͒ y ϭ a xy Ίa ϭ a1͞2 n n aΊ n b Ί ab ϭ Ί ͑ab͒ x ϭ a xb x n a ϭ a1͞n Ί Ίab ϭ x ϭ ax bx ͑ n n a Ί am ϭ am͞n ϭ Ί n ͒m n a Ί n b Ί Conversion Table: centimeter Ϸ meter Ϸ Ϸ kilometer Ϸ liter Ϸ newton Ϸ 0.394 inch 39.370 inches 3.281 feet 0.621 mile 0.264 gallon 0.225 pound joule Ϸ 0.738 foot-pound gram Ϸ 0.035 ounce kilogram Ϸ 2.205 pounds inch Ϸ 2.540 centimeters foot Ϸ 30.480 centimeters Ϸ 0.305 meter mile Ϸ 1.609 kilometers gallon Ϸ 3.785 liters pound Ϸ 4.448 newtons foot-lb Ϸ 1.356 joules ounce Ϸ 28.350 grams pound Ϸ 0.454 kilogram ... Range: Ϫ , 2 Intercept: ͑0, 0͒ Horizontal asymptotes: yϭ± Odd function Origin symmetry Precalculus with Limits Second Edition Ron Larson The Pennsylvania State University The Behrend... Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Precalculus with Limits, Second Edition Ron Larson Publisher: Charlie VanWagner Acquiring Sponsoring... Dispersion Least Squares Regression A81 A1 A Word from the Author Welcome to the Second Edition of Precalculus with Limits! We are proud to offer you a new and revised version of our textbook With
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