| |
| |
| About the Author | |
| |
| |
| Preface to the Instructor | |
| |
| |
| Acknowledgments | |
| |
| |
| Preface to the Student | |
| |
| |
| |
| The Real Numbers | |
| |
| |
| |
| The Real Line | |
| |
| |
| Construction of the Real Line | |
| |
| |
| Is Every Real Number Rational? | |
| |
| |
| Problems | |
| |
| |
| |
| Algebra of the Real Numbers | |
| |
| |
| Commutativity and Associativity | |
| |
| |
| The Order of Algebraic Operations | |
| |
| |
| The Distributive Property | |
| |
| |
| Additive Inverses and Subtraction | |
| |
| |
| Multiplicative Inverses and the Algebra of Fractions | |
| |
| |
| Symbolic Calculators | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Inequalities | |
| |
| |
| Positive and Negative Numbers | |
| |
| |
| Lesser and Greater | |
| |
| |
| Intervals | |
| |
| |
| Absolute Value | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| Chapter Summary and Chapter Review Questions | |
| |
| |
| |
| Combining Algebra and Geometry | |
| |
| |
| |
| The Coordinate Plane | |
| |
| |
| Coordinates | |
| |
| |
| Graphs of Equations | |
| |
| |
| Distance Between Two Points | |
| |
| |
| Length, Perimeter, and Circumference | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Lines | |
| |
| |
| Slope | |
| |
| |
| The Equation of a Line | |
| |
| |
| Parallel Lines | |
| |
| |
| Perpendicular Lines | |
| |
| |
| Midpoints | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Quadratic Expressions and Conic Sections | |
| |
| |
| Completing the Square | |
| |
| |
| The Quadratic Formula | |
| |
| |
| Circles | |
| |
| |
| Ellipses | |
| |
| |
| Parabolas | |
| |
| |
| Hyperbolas | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Area | |
| |
| |
| Squares, Rectangles, and Parallelograms | |
| |
| |
| Triangles and Trapezoids | |
| |
| |
| Stretching | |
| |
| |
| Circles and Ellipses | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| Chapter Summary and Chapter Review Questions | |
| |
| |
| |
| Functions and Their Graphs | |
| |
| |
| |
| Functions | |
| |
| |
| Definition and Examples | |
| |
| |
| The Graph of a Function | |
| |
| |
| The Domain of a Function | |
| |
| |
| The Range of a Function | |
| |
| |
| Functions via Tables | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Function Transformations and Graphs | |
| |
| |
| Vertical Transformations: Shifting, Stretching, and Flipping | |
| |
| |
| Horizontal Transformations: Shifting, Stretching, Flipping | |
| |
| |
| Combinations of Vertical Function Transformations | |
| |
| |
| Even Functions | |
| |
| |
| Odd Functions | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Composition of Functions | |
| |
| |
| Combining Two Functions | |
| |
| |
| Definition of Composition | |
| |
| |
| Order Matters in Composition | |
| |
| |
| Decomposing Functions | |
| |
| |
| Composing More than Two Functions | |
| |
| |
| Function Transformations as Compositions | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Inverse Functions | |
| |
| |
| The Inverse Problem | |
| |
| |
| One-to-one Functions | |
| |
| |
| The Definition of an Inverse Function | |
| |
| |
| The Domain and Range of an Inverse Function | |
| |
| |
| The Composition of a Function and Its Inverse | |
| |
| |
| Comments about Notation | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| A Graphical Approach to Inverse Functions | |
| |
| |
| The Graph of an Inverse Function | |
| |
| |
| Graphical Interpretation of One-to-One | |
| |
| |
| Increasing and Decreasing Functions | |
| |
| |
| Inverse Functions via Tables | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| Chapter Summary and Chapter Review Questions | |
| |
| |
| |
| Polynomial and Rational Functions | |
| |
| |
| |
| Integer Exponents | |
| |
| |
| Positive Integer Exponents | |
| |
| |
| Properties of Exponents | |
| |
| |
| Defining x0 | |
| |
| |
| Negative Integer Exponents | |
| |
| |
| Manipulations with Exponents | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Polynomials | |
| |
| |
| The Degree of a Polynomial | |
| |
| |
| The Algebra of Polynomials | |
| |
| |
| Zeros and Factorization of Polynomials | |
| |
| |
| The Behavior of a Polynomial Near -1 | |
| |
| |
| Graphs of Polynomials | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Rational Functions | |
| |
| |
| Ratios of Polynomials | |
| |
| |
| The Algebra of Rational Functions | |
| |
| |
| Division of Polynomials | |
| |
| |
| The Behavior of a Rational Function Near -1 | |
| |
| |
| Graphs of Rational Functions | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Complex Numbers | |
| |
| |
| The Complex Number System | |
| |
| |
| Arithmetic with Complex Numbers | |
| |
| |
| Complex Conjugates and Division of Complex Numbers | |
| |
| |
| Zeros and Factorization of Polynomials, Revisited | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| Chapter Summary and Chapter Review Questions | |
| |
| |
| |
| Exponents and Logarithms | |
| |
| |
| |
| Exponents and Exponential Functions | |
| |
| |
| Roots | |
| |
| |
| Rational Exponents | |
| |
| |
| Real Exponents | |
| |
| |
| Exponential Functions | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Logarithms as Inverses of Exponential Functions | |
| |
| |
| Logarithms Base 2 | |
| |
| |
| Logarithms with Any Base | |
| |
| |
| Common Logarithms and the Number of Digits | |
| |
| |
| Logarithm of a Power | |
| |
| |
| Radioactive Decay and Half-Life | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Applications of Logarithms | |
| |
| |
| Logarithm of a Product | |
| |
| |
| Logarithm of a Quotient | |
| |
| |
| Earthquakes and the Richter Scale | |
| |
| |
| Sound Intensity and Decibels | |
| |
| |
| Star Brightness and Apparent Magnitude | |
| |
| |
| Change of Base | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Exponential Growth | |
| |
| |
| Functions with Exponential Growth | |
| |
| |
| Population Growth | |
| |
| |
| Compound Interest | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| Chapter Summary and Chapter Review Questions | |
| |
| |
| |
| e and the Natural Logarithm | |
| |
| |
| |
| Defining e and ln | |
| |
| |
| Estimating Area Using Rectangles | |
| |
| |
| Defining e | |
| |
| |
| Defining the Natural Logarithm | |
| |
| |
| Properties of the Exponential Function and ln | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Approximations with e and ln | |
| |
| |
| Approximation of the Natural Logarithm | |
| |
| |
| Inequalities with the Natural Logarithm | |
| |
| |
| Approximations with the Exponential Function | |
| |
| |
| An Area Formula | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Exponential Growth Revisited | |
| |
| |
| Continuously Compounded Interest | |
| |
| |
| Continuous Growth Rates | |
| |
| |
| Doubling Your Money | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| Chapter Summary and Chapter Review Questions | |
| |
| |
| |
| Trigonometric Functions | |
| |
| |
| |
| The Unit Circle | |
| |
| |
| The Equation of the Unit Circle | |
| |
| |
| Angles in the Unit Circle | |
| |
| |
| Negative Angles | |
| |
| |
| Angles Greater Than 360- | |
| |
| |
| Length of a Circular Arc | |
| |
| |
| Special Points on the Unit Circle | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Radians | |
| |
| |
| A Natural Unit of Measurement for Angles | |
| |
| |
| Negative Angles | |
| |
| |
| Angles Greater Than 2- | |
| |
| |
| Length of a Circular Arc | |
| |
| |
| Area of a Slice | |
| |
| |
| Special Points on the Unit Circle | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Cosine and Sine | |
| |
| |
| Definition of Cosine and Sine | |
| |
| |
| Cosine and Sine of Special Angles | |
| |
| |
| The Signs of Cosine and Sine | |
| |
| |
| The Key Equation Connecting Cosine and Sine | |
| |
| |
| The Graphs of Cosine and Sine | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| More Trigonometric Functions | |
| |
| |
| Definition of Tangent | |
| |
| |
| Tangent of Special Angles | |
| |
| |
| The Sign of Tangent | |
| |
| |
| Connections between Cosine, Sine, and Tangent | |
| |
| |
| The Graph of Tangent | |
| |
| |
| Three More Trigonometric Functions | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Trigonometry in Right Triangles | |
| |
| |
| Trigonometric Functions via Right Triangles | |
| |
| |
| Two Sides of a Right Triangle | |
| |
| |
| One Side and One Angle of a Right Triangle | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Trigonometric Identities | |
| |
| |
| The Relationship Between Cosine and Sine | |
| |
| |
| Trigonometric Identities for the Negative of an Angle | |
| |
| |
| Trigonometric Identities with | |
| |
| |
| Trigonometric Identities Involving a Multiple of | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| Chapter Summary and Chapter Review Questions | |
| |
| |
| |
| Trigonometric Algebra and Geometry | |
| |
| |
| |
| Inverse Trigonometric Functions | |
| |
| |
| The Arccosine Function | |
| |
| |
| The Arcsine Function | |
| |
| |
| The Arctangent Function | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Inverse Trigonometric Identities | |
| |
| |
| The Arccosine, Arcsine, and Arctangent of | |
| |
| |
| t: Graphical | |
| |
| |
| Approach | |
| |
| |
| The Arccosine, Arcsine, and Arctangent of | |
| |
| |
| t: Algebraic | |
| |
| |
| Approach | |
| |
| |
| Arccosine Plus Arcsine | |
| |
| |
| The Arctangent of 1t | |
| |
| |
| Composition of Trigonometric Functions and Their Inverses | |
| |
| |
| More Compositions with Inverse Trigonometric Functions | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Using Trigonometry to Compute Area | |
| |
| |
| The Area of a Triangle via Trigonometry | |
| |
| |
| Ambiguous Angles | |
| |
| |
| The Area of a Parallelogram via Trigonometry | |
| |
| |
| The Area of a Polygon | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| The Law of Sines and the Law of Cosines | |
| |
| |
| The Law of Sines | |
| |
| |
| Using the Law of Sines | |
| |
| |
| The Law of Cosines | |
| |
| |
| Using the Law of Cosines | |
| |
| |
| When to Use Which Law | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Double-Angle and Half-Angle Formulas | |
| |
| |
| The Cosine of 2- | |
| |
| |
| The Sine of 2- | |
| |
| |
| The Tangent of 2- | |
| |
| |
| The Cosine and Sine of -2 | |
| |
| |
| The Tangent of -2 | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Addition and Subtraction Formulas | |
| |
| |
| The Cosine of a Sum and Difference | |
| |
| |
| The Sine of a Sum and Difference | |
| |
| |
| The Tangent of a Sum and Difference | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| Chapter Summary and Chapter Review Questions | |
| |
| |
| |
| Applications of Trigonometry | |
| |
| |
| |
| Parametric Curves | |
| |
| |
| Curves in the Coordinate Plane | |
| |
| |
| Graphing Inverse Functions as Parametric Curves | |
| |
| |
| Shifting, Stretching, or Flipping a Parametric Curve | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Transformations of Trigonometric Functions | |
| |
| |
| Amplitude | |
| |
| |
| Period | |
| |
| |
| Phase Shift | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Polar Coordinates | |
| |
| |
| Defining Polar Coordinates | |
| |
| |
| Converting from Polar to Rectangular Coordinates | |
| |
| |
| Converting from Rectangular to Polar Coordinates | |
| |
| |
| Graphs of Polar Equations | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Vectors | |
| |
| |
| An Algebraic and Geometric Introduction to Vectors | |
| |
| |
| Vector Addition | |
| |
| |
| Vector Subtraction | |
| |
| |
| The Dot Product | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| The Complex Plane | |
| |
| |
| Complex Numbers as Points in the Plane | |
| |
| |
| Geometric Interpretation of Complex Multiplication and Division | |
| |
| |
| De Moivre's Theorem | |
| |
| |
| Finding Complex Roots | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| Chapter Summary and Chapter Review Questions | |
| |
| |
| |
| Systems of Equations and Inequalities | |
| |
| |
| |
| Equations and Systems of Equations | |
| |
| |
| Solving an Equation | |
| |
| |
| Solving a System of Equations | |
| |
| |
| Systems of Linear Equations | |
| |
| |
| Matrices | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Solving Systems of Linear Equations | |
| |
| |
| Gaussian Elimination | |
| |
| |
| Gaussian Elimination with Matrices | |
| |
| |
| Special Cases-No Solutions | |
| |
| |
| Special Cases-Infinitely Many Solutions | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Matrix Algebra | |
| |
| |
| Adding and Subtracting Matrices | |
| |
| |
| Multiplying Matrices | |
| |
| |
| The Inverse of a Matrix | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| Chapter Summary and Chapter Review Questions | |
| |
| |
| |
| Sequences, Series, and Limits | |
| |
| |
| |
| Sequences | |
| |
| |
| Introduction to Sequences | |
| |
| |
| Arithmetic Sequences | |
| |
| |
| Geometric Sequences | |
| |
| |
| Recursively-Defined Sequences | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Series | |
| |
| |
| Sums of Sequences | |
| |
| |
| Arithmetic Series | |
| |
| |
| Geometric Series | |
| |
| |
| Summation Notation | |
| |
| |
| The Binomial Theorem | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| |
| Limits | |
| |
| |
| Introduction to Limits | |
| |
| |
| Infinite Series | |
| |
| |
| Decimals as Infinite Series | |
| |
| |
| Special Infinite Series | |
| |
| |
| Exercises, Problems, and Worked-out Solutions | |
| |
| |
| Chapter Summary and Chapter Review Questions | |