Precalculus

About the Author v Preface to the Instructor xv Acknowledgments xx Preface to the Student xxii 0 The Real Numbers 1 0.1 The Real Line 2 Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems 6 0.2 Algebra of the Real Numbers 7 Commutativity and Associativity 7 The Order of Algebraic Operations 8 The Distributive Property 10 Additive Inverses and Subtraction 11 Multiplicative Inverses and Division 12 Exercises, Problems, and Worked-out Solutions 14 0.3 Inequalities 18 Positive and Negative Numbers 18 Lesser and Greater 19 Intervals 21 Absolute Value 24 Exercises, Problems, and Worked-out Solutions 26 Chapter Summary and Chapter Review Questions 32 1 Functions and Their Graphs 33 1.1 Functions 34 Examples of Functions 34 Equality of Functions 35 The Domain of a Function 37 Functions via Tables 38 The Range of a Function 38 Exercises, Problems, and Worked-out Solutions 40 1.2 The Coordinate Plane and Graphs 47 The Coordinate Plane 47 The Graph of a Function 49 Determining a Function from Its Graph 50 Which Sets Are Graphs? 52 Determining the Range of a Function from Its Graph 53 Exercises, Problems, and Worked-out Solutions 54 1.3 Function Transformations and Graphs 62 Shifting a Graph Up or Down 62 Shifting a Graph Right or Left 63 Stretching a Graph Vertically or Horizontally 65 Reflecting a Graph Vertically or Horizontally 67 Even and Odd Functions 68 Exercises, Problems, and Worked-out Solutions 70 1.4 Composition of Functions 80 Definition of Composition 80 Order Matters in Composition 81 The Identity Function 82 Decomposing Functions 82 Exercises, Problems, and Worked-out Solutions 83 1.5 Inverse Functions 88 Examples of Inverse Functions 88 One-to-one Functions 89 The Definition of an Inverse Function 90 Finding a Formula for an Inverse Function 92 The Domain and Range of an Inverse Function 92 The Composition of a Function and Its Inverse 93 Comments about Notation 95 Exercises, Problems, and Worked-out Solutions 96 1.6 A Graphical Approach to Inverse Functions 102 The Graph of an Inverse Function 102 Inverse Functions via Tables 104 Graphical Interpretation of One-to-One 104 Increasing and Decreasing Functions 105 Exercises, Problems, and Worked-out Solutions 108 Chapter Summary and Chapter Review Questions 113 2 Linear, Quadratic, Polynomial, and Rational Functions 115 2.1 Linear Functions and Lines 116 Slope 116 The Equation of a Line 117 Parallel Lines 120 Perpendicular Lines 122 Exercises, Problems, and Worked-out Solutions 125 2.2 Quadratic Functions and Parabolas 133 The Vertex of a Parabola 133 Completing the Square 135 The Quadratic Formula 138 Exercises, Problems, and Worked-out Solutions 140 2.3 Integer Exponents 146 Exponentiation by Positive Integers 146 Properties of Exponentiation 147 Defining x0 148 Exponentiation by Negative Integers 149 Manipulations with Powers 150 Exercises, Problems, and Worked-out Solutions 152 2.4 Polynomials 158 The Degree of a Polynomial 158 The Algebra of Polynomials 160 Zeros and Factorization of Polynomials 161 The Behavior of a Polynomial Near + 163 Graphs of Polynomials 166 Exercises, Problems, and Worked-out Solutions 168 2.5 Rational Functions 173 Ratios of Polynomials 173 The Algebra of Rational Functions 174 Division of Polynomials 175 The Behavior of a Rational Function Near + 177 Graphs of Rational Functions 180 Exercises, Problems, and Worked-out Solutions 181 2.6 Complex Numbers 188 The Complex Number System 188 Arithmetic with Complex Numbers 189 Complex Conjugates and Division of Complex Numbers 190 Zeros and Factorization of Polynomials, Revisited 193 Exercises, Problems, and Worked-out Solutions 196 2.7 Systems of Equations and Matrices* 202 Solving a System of Equations 202 Systems of Linear Equations 204 Matrices and Linear Equations 208 Exercises, Problems, and Worked-out Solutions 215 Chapter Summary and Chapter Review Questions 221 3 Exponents and Logarithms 223 3.1 Rational and Real Exponents 224 Roots 224 Rational Exponents 227 Real Exponents 229 Exercises, Problems, and Worked-out Solutions 231 3.2 Logarithms as Inverses of Exponentiation 237 Logarithms Base 2 237 Logarithms with Arbitrary Base 238 Change of Base 240 Exercises, Problems, and Worked-out Solutions 242 3.3 Algebraic Properties of Logarithms 247 Logarithm of a Product 247 Logarithm of a Quotient 248 Common Logarithms and the Number of Digits 249 Logarithm of a Power 250 Exercises, Problems, and Worked-out Solutions 251 3.4 Exponential Growth 258 Functions with Exponential Growth 259 Population Growth 261 Compound Interest 263 Exercises, Problems, and Worked-out Solutions 268 3.5 Additional Applications of Exponents and Logarithms 274 Radioactive Decay and Half-Life 274 Earthquakes and the Richter Scale 276 Sound Intensity and Decibels 278 Star Brightness and Apparent Magnitude 279 Exercises, Problems, and Worked-out Solutions 281 Chapter Summary and Chapter Review Questions 287 4 Area, e, and the Natural Logarithm 289 4.1 Distance, Length, and Circles 290 Distance between Two Points 290 Midpoints 291 Distance between a Point and a Line 292 Circles 293 Length 295 Exercises, Problems, and Worked-out Solutions 297 4.2 Areas of Simple Regions 303 Squares 303 Rectangles 304 Parallelograms 304 Triangles 304 Trapezoids 305 Stretching 306 Circles 307 Ellipses 310 Exercises, Problems, and Worked-out Solutions 312 4.3 e and the Natural Logarithm 320 Estimating Area Using Rectangles 320 Defining e 322 Defining the Natural Logarithm 325 Properties of the Exponential Function and ln 326 Exercises, Problems, and Worked-out Solutions 328 4.4 Approximations with e and ln 335 Approximations of the Natural Logarithm 335 Inequalities with the Natural Logarithm 336 Approximations with the Exponential Function 337 An Area Formula 338 Exercises, Problems, and Worked-out Solutions 341 4.5 Exponential Growth Revisited 345 Continuously Compounded Interest 345 Continuous Growth Rates 346 Doubling Your Money 347 Exercises, Problems, and Worked-out Solutions 349 Chapter Summary and Chapter Review Questions 354 5 Trigonometric Functions 356 5.1 The Unit Circle 357 The Equation of the Unit Circle 357 Angles in the Unit Circle 358 Negative Angles 360 Angles Greater Than 360 361 Length of a Circular Arc 362 Special Points on the Unit Circle 363 Exercises, Problems, and Worked-out Solutions 364 5.2 Radians 370 A Natural Unit of Measurement for Angles 370 Negative Angles 373 Angles Greater Than 2pi 374 Length of a Circular Arc 375 Area of a Slice 375 Special Points on the Unit Circle 376 Exercises, Problems, and Worked-out Solutions 377 5.3 Cosine and Sine 382 Definition of Cosine and Sine 382 Cosine and Sine of Special Angles 384 The Signs of Cosine and Sine 385 The Key Equation Connecting Cosine and Sine 387 The Graphs of Cosine and Sine 388 Exercises, Problems, and Worked-out Solutions 390 5.4 More Trigonometric Functions 395 Definition of Tangent 395 Tangent of Special Angles 396 The Sign of Tangent 397 Connections between Cosine, Sine, and Tangent 398 The Graph of Tangent 398 Three More Trigonometric Functions 400 Exercises, Problems, and Worked-out Solutions 401 5.5 Trigonometry in Right Triangles 407 Trigonometric Functions via Right Triangles 407 Two Sides of a Right Triangle 409 One Side and One Angle of a Right Triangle 410 Exercises, Problems, and Worked-out Solutions 410 5.6 Trigonometric Identities 417 The Relationship Between Cosine and Sine 417 Trigonometric Identities for the Negative of an Angle 419 Trigonometric Identities with pi2 420 Trigonometric Identities Involving a Multiple of pi 422 Exercises, Problems, and Worked-out Solutions 426 5.7 Inverse Trigonometric Functions 432 The Arccosine Function 432 The Arcsine Function 435 The Arctangent Function 437 Exercises, Problems, and Worked-out Solutions 440 5.8 Inverse Trigonometric Identities 443 The Arccosine, Arcsine, and Arctangent of -t: Graphical Approach 443 The Arccosine, Arcsine, and Arctangent of -t: Algebraic Approach 445 Arccosine Plus Arcsine 446 The Arctangent of 1t 446 Composition of Trigonometric Functions and Their Inverses 447 More Compositions with Inverse Trigonometric Functions 448 Exercises, Problems, and Worked-out Solutions 451 Chapter Summary and Chapter Review Questions 455 6 Applications of Trigonometry 457 6.1 Using Trigonometry to Compute Area 458 The Area of a Triangle via Trigonometry 458 Ambiguous Angles 459 The Area of a Parallelogram via Trigonometry 461 The Area of a Polygon 462 Exercises, Problems, and Worked-out Solutions 463 6.2 The Law of Sines and the Law of Cosines 469 The Law of Sines 469 Using the Law of Sines 470 The Law of Cosines 472 Using the Law of Cosines 473 When to Use Which Law 475 Exercises, Problems, and Worked-out Solutions 476 6.3 Double-Angle and Half-Angle Formulas 483 The Cosine of 2theta 483 The Sine of 2theta 484 The Tangent of 2theta 485 The Cosine and Sine of theta2 485 The Tangent of theta2 488 Exercises, Problems, and Worked-out Solutions 489 6.4 Addition and Subtraction Formulas 497 The Cosine of a Sum and Difference 497 The Sine of a Sum and Difference 499 The Tangent of a Sum and Difference 500 Exercises, Problems, and Worked-out Solutions 501 6.5 Transformations of Trigonometric Functions 507 Amplitude 507 Period 509 Phase Shift 512 Exercises, Problems, and Worked-out Solutions 514 6.6 Polar Coordinates* 523 Defining Polar Coordinates 523 Converting from Polar to Rectangular Coordinates 524 Converting from Rectangular to Polar Coordinates 525 Graphs of Polar Equations 529 Exercises, Problems, and Worked-out Solutions 531 6.7 Vectors and the Complex Plane* 534 An Algebraic and Geometric Introduction to Vectors 534 The Dot Product 540 The Complex Plane 542 De Moivre's Theorem 546 Exercises, Problems, and Worked-out Solutions 547 Chapter Summary and Chapter Review Questions 551 7 Sequences, Series, and Limits 553 7.1 Sequences 554 Introduction to Sequences 554 Arithmetic Sequences 556 Geometric Sequences 557 Recursive Sequences 559 Exercises, Problems, and Worked-out Solutions 562 7.2 Series 568 Sums of Sequences 568 Arithmetic Series 568 Geometric Series 570 Summation Notation 572 Exercises, Problems, and Worked-out Solutions 573 7.3 Limits 578 Introduction to Limits 578 Infinite Series 582 Decimals as Infinite Series 584 Special Infinite Series 586 Exercises, Problems, and Worked-out Solutions 588 Chapter Summary and Chapter Review Questions 591 Index 592