Precalculus

Cover......Page 1
Title Page......Page 8
Copyright Page......Page 9
Acknowledgments......Page 25
Contents......Page 10
Three Distinct Series......Page 21
The Contemporary Series......Page 22
Preface to the Instructor......Page 23
Resources for Success......Page 27
 C......Page 29
 D......Page 30
 F......Page 31
 L......Page 32
 P......Page 33
 V......Page 34
 W......Page 35
1 Graphs......Page 36
 1.1 The Distance and Midpoint Formulas......Page 37
  Use the Distance Formula......Page 38
  Use the Midpoint Formula......Page 40
  Graph Equations by Plotting Points......Page 44
  Find Intercepts from a Graph......Page 46
  Test an Equation for Symmetry with Respect to the x-Axis, the y-Axis, and the Origin......Page 47
  Know How to Graph Key Equations......Page 50
  Calculate and Interpret the Slope of a Line......Page 54
  Find the Equation of a Vertical Line......Page 57
  Use the Point–Slope Form of a Line; Identify Horizontal Lines......Page 58
  Write the Equation of a Line in Slope–Intercept Form......Page 59
  Identify the Slope and y-Intercept of a Line from Its Equation......Page 60
  Graph Lines Written in General Form Using Intercepts......Page 61
  Find Equations of Parallel Lines......Page 62
  Find Equations of Perpendicular Lines......Page 63
  Write the Standard Form of the Equation of a Circle......Page 69
  Graph a Circle......Page 70
  Work with the General Form of the Equation of a Circle......Page 71
 Chapter Review......Page 75
 Chapter Test......Page 77
 Chapter Project......Page 78
2 Functions and Their Graphs......Page 79
  Determine Whether a Relation Represents a Function......Page 80
  Find the Value of a Function......Page 83
  Find the Difference Quotient of a Function......Page 86
  Find the Domain of a Function Defined by an Equation......Page 87
  Form the Sum, Difference, Product, and Quotient of Two functions......Page 89
  Identify the Graph of a Function......Page 95
  Obtain Information from or about the Graph of a Function......Page 96
  Determine Even and Odd Functions from a Graph......Page 104
  Use a Graph to Determine Where a Function Is Increasing, Decreasing, or Constant......Page 106
  Use a Graph to Locate Local Maxima and Local Minima......Page 107
  Use a Graph to Locate the Absolute Maximum and the Absolute Minimum......Page 108
  Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine Where a Function Is Increasing or Decreasing......Page 110
  Find the Average Rate of Change of a Function......Page 111
  Graph the Functions Listed in the Library of Functions......Page 118
  Graph Piecewise-defined Functions......Page 123
 2.5 Graphing Techniques: Transformations......Page 128
  Graph Functions Using Vertical and Horizontal Shifts......Page 129
  Graph Functions Using Compressions and Stretches......Page 132
  Graph Functions Using Reflections about the x-Axis and the y-Axis......Page 134
  Build and Analyze Functions......Page 141
 Chapter Review......Page 147
 Chapter Test......Page 151
 Chapter Projects......Page 152
3 Linear and Quadratic Functions......Page 154
  Use Average Rate of Change to Identify Linear Functions......Page 155
  Determine Whether a Linear Function Is Increasing, Decreasing, or Constant......Page 158
  Build Linear Models from Verbal Descriptions......Page 159
  Draw and Interpret Scatter Diagrams......Page 165
  Distinguish between Linear and Nonlinear Relations......Page 166
  Use a Graphing Utility to Find the Line of Best Fit......Page 167
 3.3 Quadratic Functions and Their Properties......Page 171
  Graph a Quadratic Function Using Transformations......Page 173
  Graph a Quadratic Function Using Its Vertex, Axis, and Intercepts......Page 175
  Find a Quadratic Function Given Its Vertex and One Other Point......Page 178
  Find the Maximum or Minimum Value of a Quadratic Function......Page 179
  Build Quadratic Models from Verbal Descriptions......Page 183
  Build Quadratic Models from Data......Page 187
  Solve Inequalities Involving a Quadratic Function......Page 193
 Chapter Review......Page 196
 Chapter Test......Page 199
 Cumulative Review......Page 200
 Chapter Projects......Page 201
4 Polynomial and Rational Functions......Page 202
  Identify Polynomial Functions and Their Degree......Page 203
  Graph Polynomial Functions Using Transformations......Page 207
  Know Properties of the Graph of a Polynomial Function......Page 208
  Analyze the Graph of a Polynomial Function......Page 215
  Build Cubic Models from Data......Page 217
  Find the Domain of a Rational Function......Page 224
  Find the Vertical Asymptotes of a Rational Function......Page 227
  Find the Horizontal or Oblique Asymptote of a Rational Function......Page 229
  Analyze the Graph of a Rational Function......Page 234
  Solve Applied Problems Involving Rational Functions......Page 245
  Solve Polynomial Inequalities......Page 249
  Solve Rational Inequalities......Page 251
  Use the Remainder and Factor Theorems......Page 256
  Use Descartes’ Rule of Signs to Determine the Number of Positive and the Number of Negative Real Zeros of a Polynomial Function......Page 259
  Use the Rational Zeros Theorem to List the Potential Rational Zeros of a Polynomial Function......Page 260
  Find the Real Zeros of a Polynomial Function......Page 261
  Solve Polynomial Equations......Page 263
  Use the Theorem for Bounds on Zeros......Page 264
  Use the Intermediate Value Theorem......Page 265
 4.6 Complex Zeros; Fundamental Theorem of Algebra......Page 271
  Use the Conjugate Pairs Theorem......Page 272
  Find a Polynomial Function with Specified Zeros......Page 273
  Find the Complex Zeros of a Polynomial Function......Page 274
 Chapter Review......Page 277
 Cumulative Review......Page 280
 Chapter Projects......Page 281
5 Exponential and Logarithmic Functions......Page 283
  Form a Composite Function......Page 284
  Find the Domain of a Composite Function......Page 285
  Determine Whether a Function Is One-to-One......Page 292
  Determine the Inverse of a Function Defined by a Map or a Set of Ordered Pairs......Page 294
  Obtain the Graph of the Inverse Function from the Graph of the Function......Page 297
  Find the Inverse of a Function Defined by an Equation......Page 298
  Evaluate Exponential Functions......Page 304
  Graph Exponential Functions......Page 308
  Define the Number e......Page 311
  Solve Exponential Equations......Page 313
  Change Exponential Statements to Logarithmic Statements and Logarithmic Statements to Exponential Statements......Page 321
  Determine the Domain of a Logarithmic Function......Page 322
  Graph Logarithmic Functions......Page 323
  Solve Logarithmic Equations......Page 327
  Work with the Properties of Logarithms......Page 333
  Write a Logarithmic Expression as a Sum or Difference of Logarithms......Page 335
  Write a Logarithmic Expression as a Single Logarithm......Page 336
  Evaluate Logarithms Whose Base Is Neither 10 Nor e......Page 338
  Solve Logarithmic Equations......Page 342
  Solve Exponential Equations......Page 344
  Solve Logarithmic and Exponential Equations Using a Graphing Utility......Page 345
  Determine the Future Value of a Lump Sum of Money......Page 349
  Calculate Effective Rates of Return......Page 352
  Determine the Present Value of a Lump Sum of Money......Page 353
  Determine the Rate of Interest or the Time Required to Double a Lump Sum of Money......Page 354
  Find Equations of Populations That Obey the Law of Uninhibited Growth......Page 359
  Find Equations of Populations That Obey the Law of Decay......Page 361
  Use Newton’s Law of Cooling......Page 362
  Use Logistic Models......Page 364
  Build an Exponential Model from Data......Page 370
  Build a Logistic Model from Data......Page 372
 Chapter Review......Page 378
 Cumulative Review......Page 383
 Chapter Projects......Page 384
6 Trigonometric Functions......Page 386
 6.1 Angles and Their Measure......Page 387
  Convert between Decimals and Degrees, Minutes, Seconds Measures for Angles......Page 389
  Find the Length of an Arc of a Circle......Page 390
  Convert from Degrees to Radians and from Radians to Degrees......Page 391
  Find the Area of a Sector of a Circle......Page 394
  Find the Linear Speed of an Object Traveling in Circular Motion......Page 395
 6.2 Trigonometric Functions: Unit Circle Approach......Page 401
  Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle......Page 403
  Find the Exact Values of the Trigonometric Functions of Quadrantal Angles......Page 404
  Find the Exact Values of the Trigonometric Functions of Π/4= 45º......Page 406
  Find the Exact Values of the Trigonometric Functions of Π/6 = 30º and Π/3 = 60º......Page 407
  Find the Exact Values of the Trigonometric Functions for Integer Multiples of Π/6 = 30º, Π/4 = 45º, and Π/3 = 60º......Page 409
  Use a Calculator to Approximate the Value of a Trigonometric Function......Page 411
  Use a Circle of Radius r to Evaluate the Trigonometric Functions......Page 412
  Determine the Domain and the Range of the Trigonometric Functions......Page 418
  Determine the Period of the Trigonometric Functions......Page 420
  Determine the Signs of the Trigonometric Functions in a Given Quadrant......Page 422
  Find the Values of the Trigonometric Functions Using Fundamental Identities......Page 423
  Find the Exact Values of the Trigonometric Functions of an Angle Given one of the Functions and the Quadrant of the Angle......Page 425
  Use Even–Odd Properties to find the Exact Values of the Trigonometric Functions......Page 428
 6.4 Graphs of the Sine and Cosine Functions......Page 432
  Graph Functions of the form y = A sin (ωx) Using Transformations......Page 433
  Graph Functions of the Form y = A cos (ωx) Using Transformations......Page 435
  Determine the Amplitude and Period of Sinusoidal Functions......Page 436
  Graph Sinusoidal Functions Using Key Points......Page 438
  Find an Equation for a Sinusoidal Graph......Page 441
 6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions......Page 447
  Graph functions of the form y = A tan (ωx) + B and y = A cot (ωx) + B......Page 449
  Graph Functions of the Form y = A csc (ωx) + B and y = A sec (ωx) + B......Page 451
  Graph Sinusoidal functions of the form y = A sin (ωx – φ) + B......Page 455
  Build Sinusoidal Models from Data......Page 458
 Chapter Review......Page 465
 Chapter Test......Page 471
 Cumulative Review......Page 472
 Chapter Projects......Page 473
7 Analytic Trigonometry......Page 475
 7.1 The Inverse Sine, Cosine, and Tangent Functions......Page 476
  Find the Exact Value of an Inverse Sine Function......Page 477
  Find an Approximate Value of an Inverse Sine Function......Page 478
  Use Properties of Inverse Functions to Find Exact Values of Certain Composite Functions......Page 479
  Find the Inverse Function of a Trigonometric Function......Page 484
  Solve Equations Involving Inverse Trigonometric Functions......Page 485
  Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent Functions......Page 489
  Define the Inverse Secant, Cosecant, and Cotangent Functions......Page 490
  Use a Calculator to Evaluate sec[sup(–1)] x, csc[sup(–1)] x, and cot[sup(–1)]x......Page 491
  Write a Trigonometric Expression as an Algebraic Expression......Page 492
  Solve Equations Involving a Single Trigonometric Function......Page 495
  Solve Trigonometric Equations Quadratic in Form......Page 498
  Solve Trigonometric Equations Using Fundamental Identities......Page 499
  Solve Trigonometric Equations Using a Graphing Utility......Page 500
 7.4 Trigonometric Identities......Page 505
  Use Algebra to Simplify Trigonometric Expressions......Page 506
  Establish Identities......Page 507
 7.5 Sum and Difference Formulas......Page 513
  Use Sum and Difference Formulas to Find Exact Values......Page 514
  Use Sum and Difference Formulas to Establish Identities......Page 517
  Use Sum and Difference Formulas Involving Inverse Trigonometric Functions......Page 519
  Solve Trigonometric Equations Linear in Sine and Cosine......Page 520
 7.6 Double-angle and Half-angle Formulas......Page 525
  Use Double-angle Formulas to Establish Identities......Page 526
  Use Half-angle Formulas to Find Exact Values......Page 529
  Express Products as Sums......Page 535
  Express Sums as Products......Page 536
 Chapter Review......Page 539
 Chapter Test......Page 542
 Cumulative Review......Page 543
 Chapter Projects......Page 544
8 Applications of Trigonometric Functions......Page 545
  Find the Value of Trigonometric Functions of Acute Angles Using Right Triangles......Page 546
  Solve Right Triangles......Page 548
  Solve Applied Problems......Page 549
 8.2 The Law of Sines......Page 559
  Solve SAA or ASA Triangles......Page 560
  Solve SSA Triangles......Page 561
  Solve Applied Problems......Page 563
 8.3 The Law of Cosines......Page 569
  Solve SAS Triangles......Page 570
  Solve Applied Problems......Page 571
  Find the Area of SAS Triangles......Page 576
  Find the Area of SSS Triangles......Page 577
  Build a Model for an Object in Simple Harmonic Motion......Page 582
  Analyze Simple Harmonic Motion......Page 584
  Analyze an Object in Damped Motion......Page 585
  Graph the Sum of Two Functions......Page 586
 Chapter Review......Page 591
 Chapter Test......Page 594
 Cumulative Review......Page 595
 Chapter Projects......Page 596
9 Polar Coordinates; Vectors......Page 598
  Plot Points Using Polar Coordinates......Page 599
  Convert from Polar Coordinates to Rectangular Coordinates......Page 601
  Convert from Rectangular Coordinates to Polar Coordinates......Page 603
  Transform Equations between Polar and Rectangular Forms......Page 605
 9.2 Polar Equations and Graphs......Page 608
  Identify and Graph Polar Equations by Converting to Rectangular Equations......Page 609
  Test Polar Equations for Symmetry......Page 612
  Graph Polar Equations by Plotting Points......Page 613
  Plot Points in the Complex Plane......Page 623
  Convert a Complex Number between Rectangular Form and Polar Form......Page 624
  Find Products and Quotients of Complex Numbers in Polar Form......Page 625
  Use De Moivre’s Theorem......Page 626
  Find Complex Roots......Page 627
 9.4 Vectors......Page 631
  Find a Position Vector......Page 633
  Add and Subtract Vectors Algebraically......Page 635
  Find a Unit Vector......Page 636
  Find a Vector from Its Direction and Magnitude......Page 637
  Model with Vectors......Page 638
  Find the Dot Product of Two Vectors......Page 645
  Find the Angle between Two Vectors......Page 646
  Decompose a Vector into Two Orthogonal Vectors......Page 647
  Compute Work......Page 649
 9.6 Vectors in Space......Page 652
  Find the Distance between Two Points in Space......Page 653
  Perform Operations on Vectors......Page 654
  Find the Angle between Two Vectors......Page 656
  Find the Direction Angles of a Vector......Page 657
  Find the Cross Product of Two Vectors......Page 662
  Know Algebraic Properties of the Cross Product......Page 663
  Know Geometric Properties of the Cross Product......Page 664
  Find the Area of a Parallelogram......Page 665
 Chapter Review......Page 668
 Chapter Test......Page 671
 Chapter Projects......Page 672
10 Analytic Geometry......Page 673
  Know the Names of the Conics......Page 674
  Analyze Parabolas with Vertex at the Origin......Page 675
  Analyze Parabolas with Vertex at (h, k)......Page 678
  Solve Applied Problems Involving Parabolas......Page 680
  Analyze Ellipses with Center at the Origin......Page 684
  Analyze Ellipses with Center at (h, k)......Page 688
  Solve Applied Problems Involving Ellipses......Page 690
  Analyze Hyperbolas with Center at the Origin......Page 694
  Find the Asymptotes of a Hyperbola......Page 699
  Analyze Hyperbolas with Center at (h, k)......Page 701
  Solve Applied Problems Involving Hyperbolas......Page 702
  Identify a Conic......Page 707
  Use a Rotation of Axes to Transform Equations......Page 708
  Analyze an Equation Using a Rotation of Axes......Page 710
  Identify Conics without a Rotation of Axes......Page 712
  Analyze and Graph Polar Equations of Conics......Page 715
  Convert the Polar Equation of a Conic to a Rectangular Equation......Page 719
  Graph Parametric Equations......Page 721
  Find a Rectangular Equation for a Curve Defined Parametrically......Page 722
  Use Time as a Parameter in Parametric Equations......Page 724
  Find Parametric Equations for Curves Defined by Rectangular Equations......Page 727
 Chapter Review......Page 733
 Cumulative Review......Page 736
 Chapter Projects......Page 737
11 Systems of Equations and Inequalities......Page 738
 11.1 Systems of Linear Equations: Substitution and Elimination......Page 739
  Solve Systems of Equations by Substitution......Page 741
  Solve Systems of Equations by Elimination......Page 742
  Express the Solution of a System of Dependent Equations Containing Two Variables......Page 744
  Solve Systems of Three Equations Containing Three Variables......Page 745
  Identify Inconsistent Systems of Equations Containing Three Variables......Page 747
  Express the Solution of a System of Dependent Equations Containing Three Variables......Page 748
  Write the Augmented Matrix of a System of Linear Equations......Page 754
  Write the System of Equations from the Augmented Matrix......Page 755
  Perform Row Operations on a Matrix......Page 756
  Solve a System of Linear Equations Using Matrices......Page 757
  Evaluate 2 by 2 Determinants......Page 769
  Use Cramer’s Rule to Solve a System of Two Equations Containing Two Variables......Page 770
  Evaluate 3 by 3 Determinants......Page 772
  Use Cramer’s Rule to Solve a System of Three Equations Containing Three Variables......Page 774
  Know Properties of Determinants......Page 775
 11.4 Matrix Algebra......Page 779
  Find the Sum and Difference of Two Matrices......Page 780
  Find Scalar Multiples of a Matrix......Page 782
  Find the Product of Two Matrices......Page 783
  Find the Inverse of a Matrix Solve a System of Linear Equations Using an Inverse Matrix......Page 787
 11.5 Partial Fraction Decomposition......Page 796
  Decompose P/Q Where Q Has only Nonrepeated Linear Factors......Page 797
  Decompose P/Q Where Q Has Repeated Linear Factors......Page 799
  Decompose P/Q Where Q Has a Nonrepeated Irreducible Quadratic Factor......Page 801
  Decompose P/Q Where Q Has a Repeated Irreducible Quadratic Factor......Page 802
  Solve a System of Nonlinear Equations Using Substitution......Page 804
  Solve a System of Nonlinear Equations Using Elimination......Page 805
  Graph an Inequality......Page 813
  Graph a System of Inequalities......Page 815
 11.8 Linear Programming......Page 820
  Solve a Linear Programming Problem......Page 821
 Chapter Review......Page 827
 Chapter Test......Page 831
 Chapter Projects......Page 832
12 Sequences; Induction; the Binomial Theorem......Page 834
  Write the First Several Terms of a Sequence......Page 835
  Write the Terms of a Sequence Defined by a Recursive Formula......Page 838
  Use Summation Notation......Page 839
  Find the Sum of a Sequence......Page 840
  Determine Whether a Sequence Is Arithmetic......Page 845
  Find a Formula for an Arithmetic Sequence......Page 846
  Find the Sum of an Arithmetic Sequence......Page 847
  Determine Whether a Sequence Is Geometric......Page 851
  Find a Formula for a Geometric Sequence......Page 852
  Find the Sum of a Geometric Sequence......Page 853
  Determine Whether a Geometric Series Converges or Diverges......Page 854
  Solve Annuity Problems......Page 857
  Prove Statements Using Mathematical Induction......Page 862
  Evaluate(nj)......Page 866
  Use the Binomial Theorem......Page 868
 Chapter Review......Page 872
 Cumulative Review......Page 875
 Chapter Projects......Page 876
13 Counting and Probability......Page 877
  Count the Number of Elements in a Set......Page 878
  Solve Counting Problems Using the Multiplication Principle......Page 880
  Solve Counting Problems Using Permutations Involving n Distinct Objects......Page 883
  Solve Counting Problems Using Combinations......Page 886
  Solve Counting Problems Using Permutations Involving n Nondistinct Objects......Page 888
  Construct Probability Models......Page 892
  Compute Probabilities of Equally Likely Outcomes......Page 894
  Find Probabilities of the Union of Two Events......Page 896
  Use the Complement Rule to Find Probabilities......Page 897
 Chapter Review......Page 902
 Chapter Test......Page 904
 Chapter Projects......Page 905
14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function......Page 907
  Find a Limit Using a Table......Page 908
  Find a Limit Using a Graph......Page 910
 14.2 Algebra Techniques for Finding Limits......Page 913
  Find the Limit of a Sum, a Difference, and a Product......Page 914
  Find the Limit of a Polynomial......Page 915
  Find the Limit of a Power or a Root......Page 916
  Find the Limit of a Quotient......Page 917
  Find the Limit of an Average Rate of Change......Page 918
  Find the One-sided Limits of a Function......Page 920
  Determine Whether a Function Is Continuous......Page 922
  Find an Equation of the Tangent Line to the Graph of a Function......Page 927
  Find the Derivative of a Function......Page 929
  Find the Instantaneous Speed of a Particle......Page 930
 14.5 The Area Problem; The Integral......Page 934
  Approximate the Area under the Graph of a Function......Page 935
  Approximate Integrals Using a Graphing Utility......Page 939
 Chapter Review......Page 940
 Chapter Test......Page 943
 Chapter Projects......Page 944
  Work with Sets......Page 946
  Find Distance on the Real Number Line......Page 950
  Evaluate Algebraic Expressions......Page 951
  Determine the Domain of a Variable......Page 952
  Use the Laws of Exponents......Page 953
  Evaluate Square Roots......Page 954
  Use a Calculator to Evaluate Exponents......Page 955
  Use the Pythagorean Theorem and Its Converse......Page 959
  Know Geometry Formulas......Page 960
  Understand Congruent Triangles and Similar Triangles......Page 961
  Recognize Polynomials......Page 967
  Know Formulas for Special Products......Page 969
  Divide Polynomials Using Long Division......Page 970
  Factor Polynomials......Page 972
  Complete the Square......Page 973
  Divide Polynomials Using Synthetic Division......Page 976
  Reduce a Rational Expression to Lowest Terms......Page 980
  Multiply and Divide Rational Expressions......Page 981
  Add and Subtract Rational Expressions......Page 982
  Use the Least Common Multiple Method......Page 983
  Simplify Complex Rational Expressions......Page 984
 A.6 Solving Equations......Page 988
  Solve Equations Involving Absolute Value......Page 990
  Solve a Quadratic Equation by Factoring......Page 991
  Solve a Quadratic Equation by Completing the Square......Page 992
  Solve a Quadratic Equation Using the Quadratic Formula......Page 993
 A.7 Complex Numbers; Quadratic Equations in the Complex Number System......Page 998
  Add, Subtract, Multiply, and Divide Complex Numbers......Page 999
  Solve Quadratic Equations in the Complex Number System......Page 1003
  Translate Verbal Descriptions into Mathematical Expressions......Page 1007
  Solve Interest Problems......Page 1008
  Solve Mixture Problems......Page 1009
  Solve Uniform Motion Problems......Page 1010
  Solve Constant Rate Job Problems......Page 1012
  Use Interval Notation......Page 1017
  Use Properties of Inequalities......Page 1018
  Solve Inequalities......Page 1020
  Solve Combined Inequalities......Page 1021
  Solve Inequalities Involving Absolute Value......Page 1022
  Work with nth Roots......Page 1027
  Simplify Radicals......Page 1028
  Rationalize Denominators......Page 1029
  Simplify Expressions with Rational Exponents......Page 1030
 B.1 The Viewing Rectangle......Page 1036
 B.2 Using a Graphing Utility to Graph Equations......Page 1038
 B.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry......Page 1040
 B.4 Using a Graphing Utility to Solve Equations......Page 1041
 B.5 Square Screens......Page 1043
 B.7 Using a Graphing Utility to Solve Systems of Linear Equations......Page 1044
 B.9 Using a Graphing Utility to Graph Parametric Equations......Page 1046
Answers......Page 1048
Credits......Page 1153
 A......Page 1154
 C......Page 1155
 D......Page 1157
 E......Page 1158
 G......Page 1159
 H......Page 1160
 I......Page 1161
 L......Page 1162
 O......Page 1163
 P......Page 1164
 R......Page 1165
 S......Page 1166
 T......Page 1168
 W......Page 1169
 Z......Page 1170