Precalculus
Precalculus
A Prelude to Calculus
Axler, Sheldon
John Wiley & Sons Inc
08/2017
576
Loose-leaf
Inglês
9781119443339
1053
Descrição não disponível.
About the Author v Preface to the Instructor xv Acknowledgments xxi Preface to the Student xxiii Chapter 0 The Real Numbers 1 0.1 The Real Line 2 Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems 5 0.2 Algebra of the Real Numbers 6 Commutativity and Associativity 6 The Order of Algebraic Operations 7 The Distributive Property 8 Additive Inverses and Subtraction 9 Multiplicative Inverses and the Algebra of Fractions 10 Symbolic Calculators 13 Exercises, Problems, and Worked-out Solutions 15 0.3 Inequalities, Intervals, and Absolute Value 20 Positive and Negative Numbers 20 Inequalities 21 Intervals 23 Absolute Value 25 Exercises, Problems, and Worked-out Solutions 29 Chapter Summary and Chapter Review Questions 35 Chapter 1 Functions and Their Graphs 37 1.1 Functions 38 Definition and Examples 38 The Domain of a Function 41 The Range of a Function 42 Functions via Tables 44 Exercises, Problems, and Worked-out Solutions 45 1.2 The Coordinate Plane and Graphs 50 The Coordinate Plane 50 The Graph of a Function 52 Determining the Domain and Range from a Graph 54 Which Sets are Graphs of Functions? 56 Exercises, Problems, and Worked-out Solutions 56 1.3 Function Transformations and Graphs 63 Vertical Transformations: Shifting, Stretching, and Flipping 63 Horizontal Transformations: Shifting, Stretching, Flipping 66 Combinations of Vertical Function Transformations 68 Even Functions 71 Odd Functions 72 Exercises, Problems, and Worked-out Solutions 73 1.4 Composition of Functions 81 Combining Two Functions 81 Definition of Composition 82 Decomposing Functions 85 Composing More than Two Functions 85 Function Transformations as Compositions 86 Exercises, Problems, and Worked-out Solutions 88 1.5 Inverse Functions 93 The Inverse Problem 93 One-to-one Functions 94 The Definition of an Inverse Function 95 The Domain and Range of an Inverse Function 97 The Composition of a Function and Its Inverse 98 Comments About Notation 99 Exercises, Problems, and Worked-out Solutions 101 1.6 A Graphical Approach to Inverse Functions 106 The Graph of an Inverse Function 106 Graphical Interpretation of One-to-One 107 Increasing and Decreasing Functions 108 Inverse Functions via Tables 110 Exercises, Problems, and Worked-out Solutions 111 Chapter Summary and Chapter Review Questions 115 Chapter 2 Linear, Quadratic, Polynomial, and Rational Functions 119 2.1 Lines and Linear Functions 120 Slope 120 The Equation of a Line 121 Parallel Lines 125 Perpendicular Lines 126 Exercises, Problems, and Worked-out Solutions 128 2.2 Quadratic Functions and Conics 135 Completing the Square and the Quadratic Formula 135 Parabolas and Quadratic Functions 138 Circles 140 Ellipses 142 Hyperbolas 144 Exercises, Problems, and Worked-out Solutions 146 2.3 Exponents 157 Positive Integer Exponents 157 Defining x0 159 Negative Integer Exponents 160 Roots 161 Rational Exponents 164 Properties of Exponents 165 Exercises, Problems, and Worked-out Solutions 166 2.4 Polynomials 174 The Degree of a Polynomial 174 The Algebra of Polynomials 175 Zeros and Factorization of Polynomials 177 The Behavior of a Polynomial Near +/- 179 Graphs of Polynomials 181 Exercises, Problems, and Worked-out Solutions 182 2.5 Rational Functions 187 The Algebra of Rational Functions 187 Division of Polynomials 188 The Behavior of a Rational Function Near +/- 191 Graphs of Rational Functions 194 Exercises, Problems, and Worked-out Solutions 195 Chapter Summary and Chapter Review Questions 201 Chapter 3 Exponential Functions, Logarithms, and e 203 3.1 Logarithms as Inverses of Exponential Functions 204 Exponential Functions 204 Logarithms Base 2 206 Logarithms with Any Base 207 Common Logarithms and the Number of Digits 208 Exercises, Problems, and Worked-out Solutions 209 3.2 The Power Rule for Logarithms 214 Logarithm of a Power 214 Radioactive Decay and Half-Life 215 Change of Base 217 Exercises, Problems, and Worked-out Solutions 219 3.3 The Product and Quotient Rules for Logarithms 223 Logarithm of a Product 223 Logarithm of a Quotient 224 Earthquakes and the Richter Scale 225 Sound Intensity and Decibels 226 Star Brightness and Apparent Magnitude 227 Exercises, Problems, and Worked-out Solutions 228 3.4 Exponential Growth 235 Functions with Exponential Growth 236 Population Growth 239 Compound Interest 241 Exercises, Problems, and Worked-out Solutions 245 3.5 e and the Natural Logarithm 250 Estimating Area Using Rectangles 250 Defining e 252 Defining the Natural Logarithm 254 Properties of the Exponential Function and Natural Logarithm 255 Exercises, Problems, and Worked-out Solutions 256 3.6 Approximations and Area with e and ln 262 Approximation of the Natural Logarithm 262 Approximations with the Exponential Function 263 An Area Formula 265 Exercises, Problems, and Worked-out Solutions 267 3.7 Exponential Growth Revisited 270 Continuously Compounded Interest 270 Continuous Growth Rates 271 Doubling Your Money 272 Exercises, Problems, and Worked-out Solutions 274 Chapter Summary and Chapter Review Questions 278 Chapter 4 Trigonometric Functions 281 4.1 The Unit Circle 282 The Equation of the Unit Circle 282 Angles in the Unit Circle 283 Negative Angles 284 Angles Greater than 360? 286 Length of a Circular Arc 287 Special Points on the Unit Circle 287 Exercises, Problems, and Worked-out Solutions 289 4.2 Radians 295 A Natural Unit of Measurement for Angles 295 The Radius Corresponding to an Angle 298 Length of a Circular Arc 300 Area of a Slice 301 Special Points on the Unit Circle 301 Exercises, Problems, and Worked-out Solutions 302 4.3 Cosine and Sine 307 Definition of Cosine and Sine 307 The Signs of Cosine and Sine 309 The Key Equation Connecting Cosine and Sine 310 The Graphs of Cosine and Sine 311 Exercises, Problems, and Worked-out Solutions 313 4.4 More Trigonometric Functions 317 Definition of Tangent 317 The Sign of Tangent 318 Connections Among Cosine, Sine, and Tangent 319 The Graph of Tangent 320 Three More Trigonometric Functions 321 Exercises, Problems, and Worked-out Solutions 322 4.5 Trigonometry in Right Triangles 327 Trigonometric Functions via Right Triangles 327 Two Sides of a Right Triangle 328 One Side and One Angle of a Right Triangle 329 Exercises, Problems, and Worked-out Solutions 331 4.6 Trigonometric Identities 336 The Relationship Among Cosine, Sine, and Tangent 336 Trigonometric Identities for the Negative of an Angle 338 Trigonometric Identities with /2 339 Trigonometric Identities Involving a Multiple of 341 Exercises, Problems, and Worked-out Solutions 343 Chapter Summary and Chapter Review Questions 348 Chapter 5 Trigonometric Algebra and Geometry 351 5.1 Inverse Trigonometric Functions 352 The Arccosine Function 352 The Arcsine Function 354 The Arctangent Function 357 Exercises, Problems, and Worked-out Solutions 359 5.2 Inverse Trigonometric Identities 365 Composition of Trigonometric Functions and Their Inverses 365 More Inverse Functions 366 More Compositions with Inverse Trigonometric Functions 367 The Arccosine, Arcsine, and Arctangent of t 369 Arccosine Plus Arcsine 370 Exercises, Problems, and Worked-out Solutions 371 5.3 Using Trigonometry to Compute Area 375 The Area of a Triangle via Trigonometry 375 Ambiguous Angles 376 The Area of a Parallelogram via Trigonometry 377 The Area of a Polygon 378 Trigonometric Approximations 380 Exercises, Problems, and Worked-out Solutions 383 5.4 The Law of Sines and the Law of Cosines 388 The Law of Sines 388 The Law of Cosines 390 When to Use Which Law 393 Exercises, Problems, and Worked-out Solutions 395 5.5 Double-Angle and Half-Angle Formulas 402 The Cosine of 2 402 The Sine of 2 403 The Tangent of 2 404 The Cosine and Sine of /2 404 The Tangent of /2 406 Exercises, Problems, and Worked-out Solutions 407 5.6 Addition and Subtraction Formulas 414 The Cosine of a Sum and Difference 414 The Sine of a Sum and Difference 416 The Tangent of a Sum and Difference 417 Products of Trigonometric Functions 418 Exercises, Problems, and Worked-out Solutions 418 5.7 Transformations of Trigonometric Functions 423 Amplitude 423 Period 425 Phase Shift 426 Fitting Transformations of Trigonometric Functions to Data 429 Exercises, Problems, and Worked-out Solutions 430 Chapter Summary and Chapter Review Questions 437 Chapter 6 Sequences, Series, and Limits 439 6.1 Sequences 440 Introduction to Sequences 440 Arithmetic Sequences 442 Geometric Sequences 443 Recursively Defined Sequences 445 Exercises, Problems, and Worked-out Solutions 448 6.2 Series 453 Sums of Sequences 453 Arithmetic Series 453 Geometric Series 455 Summation Notation 457 Pascal's Triangle 459 The Binomial Theorem 462 Exercises, Problems, and Worked-out Solutions 465 6.3 Limits 470 Introduction to Limits 470 Infinite Series 473 Decimals as Infinite Series 476 Special Infinite Series 477 Exercises, Problems, and Worked-out Solutions 479 Chapter Summary and Chapter Review Questions 482 Chapter 7 Polar Coordinates, Vectors, and Complex Numbers 483 7.1 Polar Coordinates 484 Defining Polar Coordinates 484 Converting from Polar to Rectangular Coordinates 485 Converting from Rectangular to Polar Coordinates 485 Graphs of Polar Equations 488 Exercises, Problems, and Worked-out Solutions 491 7.2 Vectors 494 An Algebraic and Geometric Introduction to Vectors 494 Vector Addition 496 Vector Subtraction 498 Scalar Multiplication 500 The Dot Product 500 Exercises, Problems, and Worked-out Solutions 503 7.3 Complex Numbers 506 The Complex Number System 506 Arithmetic with Complex Numbers 507 Complex Conjugates and Division of Complex Numbers 508 Zeros and Factorization of Polynomials, Revisited 511 Exercises, Problems, and Worked-out Solutions 514 7.4 The Complex Plane 518 Complex Numbers as Points in the Plane 518 Geometric Interpretation of Complex Multiplication and Division 519 De Moivre's Theorem 522 Finding Complex Roots 523 Exercises, Problems, and Worked-out Solutions 524 Chapter Summary and Chapter Review Questions 526 Appendix: Area 527 Circumference 527 Squares, Rectangles, and Parallelograms 528 Triangles and Trapezoids 529 Stretching 531 Circles and Ellipses 531 Exercises, Problems, and Worked-out Solutions 534 Photo Credits 543 Index 545 Colophon: Notes on Typesetting 551
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sheldon axler's precalculus; algebra prerequisites; topics; students; trigonometry; calculus; need; courses; prelude; functions; logarithms; half-life; growth; area; function; logarithm
About the Author v Preface to the Instructor xv Acknowledgments xxi Preface to the Student xxiii Chapter 0 The Real Numbers 1 0.1 The Real Line 2 Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems 5 0.2 Algebra of the Real Numbers 6 Commutativity and Associativity 6 The Order of Algebraic Operations 7 The Distributive Property 8 Additive Inverses and Subtraction 9 Multiplicative Inverses and the Algebra of Fractions 10 Symbolic Calculators 13 Exercises, Problems, and Worked-out Solutions 15 0.3 Inequalities, Intervals, and Absolute Value 20 Positive and Negative Numbers 20 Inequalities 21 Intervals 23 Absolute Value 25 Exercises, Problems, and Worked-out Solutions 29 Chapter Summary and Chapter Review Questions 35 Chapter 1 Functions and Their Graphs 37 1.1 Functions 38 Definition and Examples 38 The Domain of a Function 41 The Range of a Function 42 Functions via Tables 44 Exercises, Problems, and Worked-out Solutions 45 1.2 The Coordinate Plane and Graphs 50 The Coordinate Plane 50 The Graph of a Function 52 Determining the Domain and Range from a Graph 54 Which Sets are Graphs of Functions? 56 Exercises, Problems, and Worked-out Solutions 56 1.3 Function Transformations and Graphs 63 Vertical Transformations: Shifting, Stretching, and Flipping 63 Horizontal Transformations: Shifting, Stretching, Flipping 66 Combinations of Vertical Function Transformations 68 Even Functions 71 Odd Functions 72 Exercises, Problems, and Worked-out Solutions 73 1.4 Composition of Functions 81 Combining Two Functions 81 Definition of Composition 82 Decomposing Functions 85 Composing More than Two Functions 85 Function Transformations as Compositions 86 Exercises, Problems, and Worked-out Solutions 88 1.5 Inverse Functions 93 The Inverse Problem 93 One-to-one Functions 94 The Definition of an Inverse Function 95 The Domain and Range of an Inverse Function 97 The Composition of a Function and Its Inverse 98 Comments About Notation 99 Exercises, Problems, and Worked-out Solutions 101 1.6 A Graphical Approach to Inverse Functions 106 The Graph of an Inverse Function 106 Graphical Interpretation of One-to-One 107 Increasing and Decreasing Functions 108 Inverse Functions via Tables 110 Exercises, Problems, and Worked-out Solutions 111 Chapter Summary and Chapter Review Questions 115 Chapter 2 Linear, Quadratic, Polynomial, and Rational Functions 119 2.1 Lines and Linear Functions 120 Slope 120 The Equation of a Line 121 Parallel Lines 125 Perpendicular Lines 126 Exercises, Problems, and Worked-out Solutions 128 2.2 Quadratic Functions and Conics 135 Completing the Square and the Quadratic Formula 135 Parabolas and Quadratic Functions 138 Circles 140 Ellipses 142 Hyperbolas 144 Exercises, Problems, and Worked-out Solutions 146 2.3 Exponents 157 Positive Integer Exponents 157 Defining x0 159 Negative Integer Exponents 160 Roots 161 Rational Exponents 164 Properties of Exponents 165 Exercises, Problems, and Worked-out Solutions 166 2.4 Polynomials 174 The Degree of a Polynomial 174 The Algebra of Polynomials 175 Zeros and Factorization of Polynomials 177 The Behavior of a Polynomial Near +/- 179 Graphs of Polynomials 181 Exercises, Problems, and Worked-out Solutions 182 2.5 Rational Functions 187 The Algebra of Rational Functions 187 Division of Polynomials 188 The Behavior of a Rational Function Near +/- 191 Graphs of Rational Functions 194 Exercises, Problems, and Worked-out Solutions 195 Chapter Summary and Chapter Review Questions 201 Chapter 3 Exponential Functions, Logarithms, and e 203 3.1 Logarithms as Inverses of Exponential Functions 204 Exponential Functions 204 Logarithms Base 2 206 Logarithms with Any Base 207 Common Logarithms and the Number of Digits 208 Exercises, Problems, and Worked-out Solutions 209 3.2 The Power Rule for Logarithms 214 Logarithm of a Power 214 Radioactive Decay and Half-Life 215 Change of Base 217 Exercises, Problems, and Worked-out Solutions 219 3.3 The Product and Quotient Rules for Logarithms 223 Logarithm of a Product 223 Logarithm of a Quotient 224 Earthquakes and the Richter Scale 225 Sound Intensity and Decibels 226 Star Brightness and Apparent Magnitude 227 Exercises, Problems, and Worked-out Solutions 228 3.4 Exponential Growth 235 Functions with Exponential Growth 236 Population Growth 239 Compound Interest 241 Exercises, Problems, and Worked-out Solutions 245 3.5 e and the Natural Logarithm 250 Estimating Area Using Rectangles 250 Defining e 252 Defining the Natural Logarithm 254 Properties of the Exponential Function and Natural Logarithm 255 Exercises, Problems, and Worked-out Solutions 256 3.6 Approximations and Area with e and ln 262 Approximation of the Natural Logarithm 262 Approximations with the Exponential Function 263 An Area Formula 265 Exercises, Problems, and Worked-out Solutions 267 3.7 Exponential Growth Revisited 270 Continuously Compounded Interest 270 Continuous Growth Rates 271 Doubling Your Money 272 Exercises, Problems, and Worked-out Solutions 274 Chapter Summary and Chapter Review Questions 278 Chapter 4 Trigonometric Functions 281 4.1 The Unit Circle 282 The Equation of the Unit Circle 282 Angles in the Unit Circle 283 Negative Angles 284 Angles Greater than 360? 286 Length of a Circular Arc 287 Special Points on the Unit Circle 287 Exercises, Problems, and Worked-out Solutions 289 4.2 Radians 295 A Natural Unit of Measurement for Angles 295 The Radius Corresponding to an Angle 298 Length of a Circular Arc 300 Area of a Slice 301 Special Points on the Unit Circle 301 Exercises, Problems, and Worked-out Solutions 302 4.3 Cosine and Sine 307 Definition of Cosine and Sine 307 The Signs of Cosine and Sine 309 The Key Equation Connecting Cosine and Sine 310 The Graphs of Cosine and Sine 311 Exercises, Problems, and Worked-out Solutions 313 4.4 More Trigonometric Functions 317 Definition of Tangent 317 The Sign of Tangent 318 Connections Among Cosine, Sine, and Tangent 319 The Graph of Tangent 320 Three More Trigonometric Functions 321 Exercises, Problems, and Worked-out Solutions 322 4.5 Trigonometry in Right Triangles 327 Trigonometric Functions via Right Triangles 327 Two Sides of a Right Triangle 328 One Side and One Angle of a Right Triangle 329 Exercises, Problems, and Worked-out Solutions 331 4.6 Trigonometric Identities 336 The Relationship Among Cosine, Sine, and Tangent 336 Trigonometric Identities for the Negative of an Angle 338 Trigonometric Identities with /2 339 Trigonometric Identities Involving a Multiple of 341 Exercises, Problems, and Worked-out Solutions 343 Chapter Summary and Chapter Review Questions 348 Chapter 5 Trigonometric Algebra and Geometry 351 5.1 Inverse Trigonometric Functions 352 The Arccosine Function 352 The Arcsine Function 354 The Arctangent Function 357 Exercises, Problems, and Worked-out Solutions 359 5.2 Inverse Trigonometric Identities 365 Composition of Trigonometric Functions and Their Inverses 365 More Inverse Functions 366 More Compositions with Inverse Trigonometric Functions 367 The Arccosine, Arcsine, and Arctangent of t 369 Arccosine Plus Arcsine 370 Exercises, Problems, and Worked-out Solutions 371 5.3 Using Trigonometry to Compute Area 375 The Area of a Triangle via Trigonometry 375 Ambiguous Angles 376 The Area of a Parallelogram via Trigonometry 377 The Area of a Polygon 378 Trigonometric Approximations 380 Exercises, Problems, and Worked-out Solutions 383 5.4 The Law of Sines and the Law of Cosines 388 The Law of Sines 388 The Law of Cosines 390 When to Use Which Law 393 Exercises, Problems, and Worked-out Solutions 395 5.5 Double-Angle and Half-Angle Formulas 402 The Cosine of 2 402 The Sine of 2 403 The Tangent of 2 404 The Cosine and Sine of /2 404 The Tangent of /2 406 Exercises, Problems, and Worked-out Solutions 407 5.6 Addition and Subtraction Formulas 414 The Cosine of a Sum and Difference 414 The Sine of a Sum and Difference 416 The Tangent of a Sum and Difference 417 Products of Trigonometric Functions 418 Exercises, Problems, and Worked-out Solutions 418 5.7 Transformations of Trigonometric Functions 423 Amplitude 423 Period 425 Phase Shift 426 Fitting Transformations of Trigonometric Functions to Data 429 Exercises, Problems, and Worked-out Solutions 430 Chapter Summary and Chapter Review Questions 437 Chapter 6 Sequences, Series, and Limits 439 6.1 Sequences 440 Introduction to Sequences 440 Arithmetic Sequences 442 Geometric Sequences 443 Recursively Defined Sequences 445 Exercises, Problems, and Worked-out Solutions 448 6.2 Series 453 Sums of Sequences 453 Arithmetic Series 453 Geometric Series 455 Summation Notation 457 Pascal's Triangle 459 The Binomial Theorem 462 Exercises, Problems, and Worked-out Solutions 465 6.3 Limits 470 Introduction to Limits 470 Infinite Series 473 Decimals as Infinite Series 476 Special Infinite Series 477 Exercises, Problems, and Worked-out Solutions 479 Chapter Summary and Chapter Review Questions 482 Chapter 7 Polar Coordinates, Vectors, and Complex Numbers 483 7.1 Polar Coordinates 484 Defining Polar Coordinates 484 Converting from Polar to Rectangular Coordinates 485 Converting from Rectangular to Polar Coordinates 485 Graphs of Polar Equations 488 Exercises, Problems, and Worked-out Solutions 491 7.2 Vectors 494 An Algebraic and Geometric Introduction to Vectors 494 Vector Addition 496 Vector Subtraction 498 Scalar Multiplication 500 The Dot Product 500 Exercises, Problems, and Worked-out Solutions 503 7.3 Complex Numbers 506 The Complex Number System 506 Arithmetic with Complex Numbers 507 Complex Conjugates and Division of Complex Numbers 508 Zeros and Factorization of Polynomials, Revisited 511 Exercises, Problems, and Worked-out Solutions 514 7.4 The Complex Plane 518 Complex Numbers as Points in the Plane 518 Geometric Interpretation of Complex Multiplication and Division 519 De Moivre's Theorem 522 Finding Complex Roots 523 Exercises, Problems, and Worked-out Solutions 524 Chapter Summary and Chapter Review Questions 526 Appendix: Area 527 Circumference 527 Squares, Rectangles, and Parallelograms 528 Triangles and Trapezoids 529 Stretching 531 Circles and Ellipses 531 Exercises, Problems, and Worked-out Solutions 534 Photo Credits 543 Index 545 Colophon: Notes on Typesetting 551
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