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About the Author | |

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Preface to the Instructor | |

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Acknowledgments | |

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Preface to the Student | |

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The Real Numbers | |

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The Real Line | |

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Construction of the Real Line | |

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Is Every Real Number Rational? | |

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Problems | |

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Algebra of the Real Numbers | |

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Commutativity and Associativity | |

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The Order of Algebraic Operations | |

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The Distributive Property | |

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Additive Inverses and Subtraction | |

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Multiplicative Inverses and the Algebra of Fractions | |

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Symbolic Calculators | |

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Exercises, Problems, and Worked-out Solutions | |

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Inequalities | |

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Positive and Negative Numbers | |

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Lesser and Greater | |

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Intervals | |

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Absolute Value | |

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Exercises, Problems, and Worked-out Solutions | |

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Chapter Summary and Chapter Review Questions | |

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Combining Algebra and Geometry | |

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The Coordinate Plane | |

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Coordinates | |

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Graphs of Equations | |

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Distance Between Two Points | |

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Length, Perimeter, and Circumference | |

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Exercises, Problems, and Worked-out Solutions | |

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Lines | |

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Slope | |

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The Equation of a Line | |

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Parallel Lines | |

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Perpendicular Lines | |

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Midpoints | |

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Exercises, Problems, and Worked-out Solutions | |

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Quadratic Expressions and Conic Sections | |

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Completing the Square | |

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The Quadratic Formula | |

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Circles | |

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Ellipses | |

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Parabolas | |

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Hyperbolas | |

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Exercises, Problems, and Worked-out Solutions | |

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Area | |

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Squares, Rectangles, and Parallelograms | |

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Triangles and Trapezoids | |

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Stretching | |

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Circles and Ellipses | |

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Exercises, Problems, and Worked-out Solutions | |

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Chapter Summary and Chapter Review Questions | |

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Functions and Their Graphs | |

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Functions | |

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Definition and Examples | |

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The Graph of a Function | |

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The Domain of a Function | |

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The Range of a Function | |

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Functions via Tables | |

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Exercises, Problems, and Worked-out Solutions | |

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Function Transformations and Graphs | |

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Vertical Transformations: Shifting, Stretching, and Flipping | |

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Horizontal Transformations: Shifting, Stretching, Flipping | |

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Combinations of Vertical Function Transformations | |

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Even Functions | |

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Odd Functions | |

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Exercises, Problems, and Worked-out Solutions | |

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Composition of Functions | |

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Combining Two Functions | |

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Definition of Composition | |

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Order Matters in Composition | |

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Decomposing Functions | |

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Composing More than Two Functions | |

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Function Transformations as Compositions | |

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Exercises, Problems, and Worked-out Solutions | |

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Inverse Functions | |

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The Inverse Problem | |

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One-to-one Functions | |

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The Definition of an Inverse Function | |

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The Domain and Range of an Inverse Function | |

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The Composition of a Function and Its Inverse | |

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Comments about Notation | |

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Exercises, Problems, and Worked-out Solutions | |

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A Graphical Approach to Inverse Functions | |

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The Graph of an Inverse Function | |

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Graphical Interpretation of One-to-One | |

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Increasing and Decreasing Functions | |

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Inverse Functions via Tables | |

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Exercises, Problems, and Worked-out Solutions | |

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Chapter Summary and Chapter Review Questions | |

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Polynomial and Rational Functions | |

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Integer Exponents | |

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Positive Integer Exponents | |

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Properties of Exponents | |

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Defining x0 | |

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Negative Integer Exponents | |

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Manipulations with Exponents | |

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Exercises, Problems, and Worked-out Solutions | |

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Polynomials | |

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The Degree of a Polynomial | |

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The Algebra of Polynomials | |

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Zeros and Factorization of Polynomials | |

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The Behavior of a Polynomial Near -1 | |

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Graphs of Polynomials | |

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Exercises, Problems, and Worked-out Solutions | |

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Rational Functions | |

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Ratios of Polynomials | |

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The Algebra of Rational Functions | |

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Division of Polynomials | |

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The Behavior of a Rational Function Near -1 | |

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Graphs of Rational Functions | |

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Exercises, Problems, and Worked-out Solutions | |

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Complex Numbers | |

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The Complex Number System | |

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Arithmetic with Complex Numbers | |

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Complex Conjugates and Division of Complex Numbers | |

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Zeros and Factorization of Polynomials, Revisited | |

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Exercises, Problems, and Worked-out Solutions | |

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Chapter Summary and Chapter Review Questions | |

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Exponents and Logarithms | |

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Exponents and Exponential Functions | |

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Roots | |

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Rational Exponents | |

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Real Exponents | |

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Exponential Functions | |

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Exercises, Problems, and Worked-out Solutions | |

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Logarithms as Inverses of Exponential Functions | |

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Logarithms Base 2 | |

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Logarithms with Any Base | |

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Common Logarithms and the Number of Digits | |

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Logarithm of a Power | |

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Radioactive Decay and Half-Life | |

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Exercises, Problems, and Worked-out Solutions | |

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Applications of Logarithms | |

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Logarithm of a Product | |

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Logarithm of a Quotient | |

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Earthquakes and the Richter Scale | |

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Sound Intensity and Decibels | |

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Star Brightness and Apparent Magnitude | |

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Change of Base | |

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Exercises, Problems, and Worked-out Solutions | |

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Exponential Growth | |

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Functions with Exponential Growth | |

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Population Growth | |

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Compound Interest | |

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Exercises, Problems, and Worked-out Solutions | |

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Chapter Summary and Chapter Review Questions | |

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e and the Natural Logarithm | |

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Defining e and ln | |

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Estimating Area Using Rectangles | |

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Defining e | |

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Defining the Natural Logarithm | |

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Properties of the Exponential Function and ln | |

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Exercises, Problems, and Worked-out Solutions | |

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Approximations with e and ln | |

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Approximation of the Natural Logarithm | |

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Inequalities with the Natural Logarithm | |

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Approximations with the Exponential Function | |

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An Area Formula | |

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Exercises, Problems, and Worked-out Solutions | |

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Exponential Growth Revisited | |

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Continuously Compounded Interest | |

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Continuous Growth Rates | |

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Doubling Your Money | |

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Exercises, Problems, and Worked-out Solutions | |

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Chapter Summary and Chapter Review Questions | |

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Trigonometric Functions | |

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The Unit Circle | |

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The Equation of the Unit Circle | |

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Angles in the Unit Circle | |

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Negative Angles | |

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Angles Greater Than 360- | |

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Length of a Circular Arc | |

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Special Points on the Unit Circle | |

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Exercises, Problems, and Worked-out Solutions | |

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Radians | |

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A Natural Unit of Measurement for Angles | |

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Negative Angles | |

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Angles Greater Than 2- | |

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Length of a Circular Arc | |

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Area of a Slice | |

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Special Points on the Unit Circle | |

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Exercises, Problems, and Worked-out Solutions | |

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Cosine and Sine | |

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Definition of Cosine and Sine | |

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Cosine and Sine of Special Angles | |

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The Signs of Cosine and Sine | |

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The Key Equation Connecting Cosine and Sine | |

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The Graphs of Cosine and Sine | |

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Exercises, Problems, and Worked-out Solutions | |

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More Trigonometric Functions | |

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Definition of Tangent | |

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Tangent of Special Angles | |

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The Sign of Tangent | |

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Connections between Cosine, Sine, and Tangent | |

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The Graph of Tangent | |

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Three More Trigonometric Functions | |

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Exercises, Problems, and Worked-out Solutions | |

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Trigonometry in Right Triangles | |

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Trigonometric Functions via Right Triangles | |

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Two Sides of a Right Triangle | |

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One Side and One Angle of a Right Triangle | |

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Exercises, Problems, and Worked-out Solutions | |

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Trigonometric Identities | |

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The Relationship Between Cosine and Sine | |

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Trigonometric Identities for the Negative of an Angle | |

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Trigonometric Identities with | |

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Trigonometric Identities Involving a Multiple of | |

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Exercises, Problems, and Worked-out Solutions | |

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Chapter Summary and Chapter Review Questions | |

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Trigonometric Algebra and Geometry | |

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Inverse Trigonometric Functions | |

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The Arccosine Function | |

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The Arcsine Function | |

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The Arctangent Function | |

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Exercises, Problems, and Worked-out Solutions | |

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Inverse Trigonometric Identities | |

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The Arccosine, Arcsine, and Arctangent of | |

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t: Graphical | |

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Approach | |

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The Arccosine, Arcsine, and Arctangent of | |

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t: Algebraic | |

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Approach | |

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Arccosine Plus Arcsine | |

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The Arctangent of 1t | |

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Composition of Trigonometric Functions and Their Inverses | |

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More Compositions with Inverse Trigonometric Functions | |

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Exercises, Problems, and Worked-out Solutions | |

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Using Trigonometry to Compute Area | |

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The Area of a Triangle via Trigonometry | |

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Ambiguous Angles | |

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The Area of a Parallelogram via Trigonometry | |

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The Area of a Polygon | |

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Exercises, Problems, and Worked-out Solutions | |

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The Law of Sines and the Law of Cosines | |

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The Law of Sines | |

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Using the Law of Sines | |

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The Law of Cosines | |

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Using the Law of Cosines | |

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When to Use Which Law | |

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Exercises, Problems, and Worked-out Solutions | |

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Double-Angle and Half-Angle Formulas | |

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The Cosine of 2- | |

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The Sine of 2- | |

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The Tangent of 2- | |

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The Cosine and Sine of -2 | |

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The Tangent of -2 | |

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Exercises, Problems, and Worked-out Solutions | |

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Addition and Subtraction Formulas | |

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The Cosine of a Sum and Difference | |

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The Sine of a Sum and Difference | |

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The Tangent of a Sum and Difference | |

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Exercises, Problems, and Worked-out Solutions | |

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Chapter Summary and Chapter Review Questions | |

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Applications of Trigonometry | |

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Parametric Curves | |

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Curves in the Coordinate Plane | |

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Graphing Inverse Functions as Parametric Curves | |

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Shifting, Stretching, or Flipping a Parametric Curve | |

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Exercises, Problems, and Worked-out Solutions | |

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Transformations of Trigonometric Functions | |

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Amplitude | |

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Period | |

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Phase Shift | |

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Exercises, Problems, and Worked-out Solutions | |

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Polar Coordinates | |

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Defining Polar Coordinates | |

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Converting from Polar to Rectangular Coordinates | |

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Converting from Rectangular to Polar Coordinates | |

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Graphs of Polar Equations | |

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Exercises, Problems, and Worked-out Solutions | |

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Vectors | |

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An Algebraic and Geometric Introduction to Vectors | |

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Vector Addition | |

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Vector Subtraction | |

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The Dot Product | |

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Exercises, Problems, and Worked-out Solutions | |

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The Complex Plane | |

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Complex Numbers as Points in the Plane | |

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Geometric Interpretation of Complex Multiplication and Division | |

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De Moivre's Theorem | |

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Finding Complex Roots | |

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Exercises, Problems, and Worked-out Solutions | |

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Chapter Summary and Chapter Review Questions | |

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Systems of Equations and Inequalities | |

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Equations and Systems of Equations | |

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Solving an Equation | |

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Solving a System of Equations | |

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Systems of Linear Equations | |

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Matrices | |

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Exercises, Problems, and Worked-out Solutions | |

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Solving Systems of Linear Equations | |

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Gaussian Elimination | |

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Gaussian Elimination with Matrices | |

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Special Cases-No Solutions | |

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Special Cases-Infinitely Many Solutions | |

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Exercises, Problems, and Worked-out Solutions | |

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Matrix Algebra | |

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Adding and Subtracting Matrices | |

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Multiplying Matrices | |

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The Inverse of a Matrix | |

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Exercises, Problems, and Worked-out Solutions | |

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Chapter Summary and Chapter Review Questions | |

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Sequences, Series, and Limits | |

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Sequences | |

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Introduction to Sequences | |

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Arithmetic Sequences | |

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Geometric Sequences | |

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Recursively-Defined Sequences | |

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Exercises, Problems, and Worked-out Solutions | |

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Series | |

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Sums of Sequences | |

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Arithmetic Series | |

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Geometric Series | |

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Summation Notation | |

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The Binomial Theorem | |

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Exercises, Problems, and Worked-out Solutions | |

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Limits | |

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Introduction to Limits | |

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Infinite Series | |

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Decimals as Infinite Series | |

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Special Infinite Series | |

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Exercises, Problems, and Worked-out Solutions | |

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Chapter Summary and Chapter Review Questions | |