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Prerequisites | |

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Real Numbers | |

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Representing Real Numbers | |

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Order and Interval Notation | |

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Basic Properties of Algebra | |

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Integer Exponents | |

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Scientific Notation | |

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Cartesian Coordinate System | |

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Cartesian Plane | |

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Absolute Value of a Real Number | |

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Distance Formulas | |

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Midpoint Formulas | |

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Equations of Circles | |

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Applications | |

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Linear Equations and Inequalities | |

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Equations | |

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Solving Equations | |

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Linear Equations in One Variable | |

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Linear Inequalities in One Variable | |

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Lines in the Plane | |

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Slope of a Line | |

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Point-Slope Form Equation of a Line | |

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Slope-Intercept Form Equation of a Line | |

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Graphing Linear Equations in Two Variables | |

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Parallel and Perpendicular Lines | |

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Applying Linear Equations in Two Variables | |

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Solving Equations Graphically, Numerically, and Algebraically | |

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Solving Equations Graphically | |

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Solving Quadratic Equations | |

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Approximating Solutions of Equations Graphically | |

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Approximating Solutions of Equations Numerically with Tables | |

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Solving Equations by Finding Intersections | |

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Complex Numbers | |

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Complex Numbers | |

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Operations with Complex Numbers | |

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Complex Conjugates and Division | |

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Complex Solutions of Quadratic Equations | |

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Solving Inequalities Algebraically and Graphically | |

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Solving Absolute Value Inequalities | |

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Solving Quadratic Inequalities | |

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Approximating Solutions to Inequalities | |

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Projectile Motion | |

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Key Ideas | |

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Review Exercises | |

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Functions and Graphs | |

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Modeling and Equation Solving | |

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Numerical Models | |

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Algebraic Models | |

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Graphical Models | |

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The Zero Factor Property | |

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Problem Solving | |

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Grapher Failure and Hidden Behavior | |

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A Word About Proof | |

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Functions and Their Properties | |

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Function Definition and Notation | |

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Domain and Range | |

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Continuity | |

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Increasing and Decreasing Functions | |

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Boundedness | |

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Local and Absolute Extrema | |

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Symmetry | |

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Asymptotes | |

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End Behavior | |

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Twelve Basic Functions | |

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What Graphs Can Tell Us | |

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Twelve Basic Functions | |

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Analyzing Functions Graphically | |

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Building Functions from Functions | |

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Combining Functions Algebraically | |

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Composition of Functions | |

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Relations and Implicitly Defined Functions | |

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Parametric Relations and Inverses | |

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Relations Defined Parametrically | |

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Inverse Relations and Inverse Functions | |

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Graphical Transformations | |

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Transformations | |

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Vertical and Horizontal Translations | |

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Reflections Across Axes | |

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Vertical and Horizontal Stretches and Shrinks | |

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Combining Transformations | |

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Modeling With Functions | |

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Functions from Formulas | |

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Functions from Graphs | |

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Functions from Verbal Descriptions | |

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Functions from Data | |

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Math at Work | |

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Key Ideas | |

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Review Exercises | |

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Chapter Project | |

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Polynomial, Power, and Rational Functions | |

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Linear and Quadratic Functions and Modeling | |

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Polynomial Functions | |

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Linear Functions and Their Graphs | |

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Average Rate of Change | |

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Linear Correlation and Modeling | |

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Quadratic Functions and Their Graphs | |

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Applications of Quadratic Functions | |

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Power Functions with Modeling | |

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Power Functions and Variation | |

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Monomial Functions and Their Graphs | |

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Graphs of Power Functions | |

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Modeling with Power Functions | |

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Polynomial Functions of Higher Degree with Modeling | |

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Graphs of Polynomial Functions | |

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End Behavior of Polynomial Functions | |

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Zeros of Polynomial Functions | |

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Intermediate Value Theorem | |

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Modeling | |

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Real Zeros of Polynomial Functions | |

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Long Division and the Division Algorithm | |

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Remainder and Factor Theorems | |

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Synthetic Division | |

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Rational Zeros Theorem | |

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Upper and Lower Bounds | |

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Complex Zeros and the Fundamental Theorem of Algebra | |

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Two Major Theorems | |

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Complex Conjugate Zeros | |

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Factoring with Real Number Coefficients | |

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Graphs of Rational Functions | |

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Rational Functions | |

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Transformations of the Reciprocal Function | |

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Limits and Asymptotes | |

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Analyzing Graphs of Rational Functions | |

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Exploring Relative Humidity | |

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Solving Equations in One Variable | |

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Solving Rational Equations | |

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Extraneous Solutions | |

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Applications | |

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Solving Inequalities in One Variable | |

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Polynomial Inequalities | |

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Rational Inequalities | |

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Other Inequalities | |

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Applications | |

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Math at Work | |

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Key Ideas | |

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Review Exercises | |

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Chapter Project | |

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Exponential, Logistic, and Logarithmic Functions | |

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Exponential and Logistic Functions | |

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Exponential Functions and Their Graphs | |

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The Natural Base e | |

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Logistic Functions and Their Graphs | |

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Population Models | |

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Exponential and Logistic Modeling | |

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Constant Percentage Rate and Exponential Functions | |

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Exponential Growth and Decay Models | |

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Using Regression to Model Population | |

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Other Logistic Models | |

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Logarithmic Functions and Their Graphs | |

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Inverses of Exponential Functions | |

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Common Logarithms-Base 10 | |

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Natural Logarithms-Base e | |

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Graphs of Logarithmic Functions | |

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Measuring Sound Using Decibels | |

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Properties of Logarithmic Functions | |

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Properties of Logarithms | |

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Change of Base | |

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Graphs of Logarithmic Functions with Base b | |

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Re-expressing Data | |

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Equation Solving and Modeling | |

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Solving Exponential Equations | |

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Solving Logarithmic Equations | |

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Orders of Magnitude and Logarithmic Models | |

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Newton's Law of Cooling | |

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Logarithmic Re-expression | |

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Mathematics of Finance | |

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Interest Compounded Annually | |

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Interest Compounded k Times per Year | |

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Interest Compounded Continuously | |

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Annual Percentage Yield | |

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Annuities-Future Value | |

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Loans and Mortgages-Present Value | |

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Key Ideas | |

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Review Exercises | |

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Chapter Project | |

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Trigonometric Functions | |

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Angles and Their Measures | |

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The Problem of Angular Measure | |

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Degrees and Radians | |

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Circular Arc Length | |

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Angular and Linear Motion | |

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Trigonometric Functions of Acute Angles | |

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Right Triangle Trigonometry | |

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Two Famous Triangles | |

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Evaluating Trigonometric Functions with a Calculator | |

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Common Calculator Errors When Evaluating Trig Functions | |

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Applications of Right Triangle Trigonometry | |

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Trigonometry Extended: The Circular Functions | |

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Trigonometric Functions of Any Angle | |

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Trigonometric Functions of Real Numbers | |

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Periodic Functions | |

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The 16-Point Unit Cricle | |

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Graphs of Sine and Cosine: Sinusoids | |

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The Basic Waves Revisited | |

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Sinusoids and Transformations | |

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Modeling Periodic Behavior with Sinusoids | |

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Graphs of Tangent, Cotangent, Secant, and Cosecant | |

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The Tangent Function | |

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The Cotangent Function | |

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The Secant Function | |

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The Cosecant Function | |

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Graphs of Composite Trigonometric Functions | |

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Combining Trigonometric and Algebraic Functions | |

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Sums and Differences of Sinusoids | |

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Damped Oscillation | |

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Inverse Trigonometric Functions | |

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Inverse Sine Function | |

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Inverse Cosine and Tangent Functions | |

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Composing Trigonometric and Inverse Trigonometric Functions | |

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Applications of Inverse Trigonometric Functions | |

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Solving Problems with Trigonometry | |

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More Right Triangle Problems | |

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Simple Harmonic Motion | |

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Key Ideas | |

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Review Exercises | |

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Chapter Project | |

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Analytic Trigonometry | |

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Fundamental Identities | |

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Identities | |

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Basic Trigonometric Identities | |

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Pythagorean Identities | |

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Cofunction Identities | |

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Odd-Even Identities | |

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Simplifying Trigonometric Expressions | |

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Solving Trigonometric Equations | |

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Proving Trigonometric Identities | |

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A Proof Strategy | |

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Proving Identities | |

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Disproving Non-Identities | |

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Identities in Calculus | |

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Sum and Difference Identities | |

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Cosine of a Difference | |

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Cosine of a Sum | |

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Sine of a Difference or Sum | |

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Tangent of a Difference or Sum | |

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Verifying a Sinusoid Algebraically | |

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Multiple-Angle Identities | |

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Double-Angle Identities | |

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Power-Reducing Identities | |

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Half-Angle Identities | |

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Solving Trigonometric Equations | |

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The Law of Sines | |

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Deriving the Law of Sines | |

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Solving Triangles (AAS, ASA) | |

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The Ambiguous Case (SSA) | |

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Applications | |

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The Law of Cosines | |

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Deriving the Law of Cosines | |

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Solving Triangles (SAS, SSS) | |

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Triangle Area and Heron's Formula | |

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Applications | |

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Math at Work | |

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Key Ideas | |

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Review Exercises | |

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Chapter Project | |

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Applications of Trigonometry | |

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Vectors in the Plane | |

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Two-Dimensional Vectors | |

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Vector Operations | |

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Unit Vectors | |

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Direction Angles | |

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Applications of Vectors | |

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Dot Product of Vectors | |

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The Dot Product | |

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Angle Between Vectors | |

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Projecting One Vector onto Another | |

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Work | |

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Parametric Equations and Motion | |

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Parametric Equations | |

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Parametric Curves | |

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Eliminating the Parameter | |

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Lines and Line Segments | |

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Simulating Motion with a Grapher | |

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Polar Coordinates | |

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Polar Coordinate System | |

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Coordinate Conversion | |

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Equation Conversion | |

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Finding Distance Using Polar Coordinates | |

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Graphs of Polar Equations | |

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Polar Curves and Parametric Curves | |

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Symmetry | |

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Analyzing Polar Graphs | |

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Rose Curves | |

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Limacon Curves | |

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Other Polar Curves | |

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De Moivre's Theorem and nth Roots | |

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The Complex Plane | |

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Trigonometric Form of Complex Numbers | |

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Multiplication and Division of Complex Numbers | |

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Powers of Complex Numbers | |

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Roots of Complex Numbers | |

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Key Ideas | |

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Review Exercises | |

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Chapter Project | |

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Systems and Matrices | |

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Solving Systems of Two Equations | |

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The Method of Substitution | |

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Solving Systems Graphically | |

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The Method of Elimination | |

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Applications | |

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Matrix Algebra | |

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Matrices | |

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Matrix Addition and Subtraction | |

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Matrix Multiplication | |

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Identity and Inverse Matrices | |

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Determinant of a Square Matrix | |

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Applications | |

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Multivariate Linear Systems and Row Operations | |

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Triangular Form for linear Systems | |

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Gaussian Elimination | |

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Elementary Row Operations and Row Echelon Form | |

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Reduced Row Echelon Form | |

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Solving Systems with Inverse Matrices | |

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Applications | |

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Partial Fractions | |

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Partial Fraction Decomposition | |

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Denominators with Linear Factors | |

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Denominators with Irreducible Quadratic Factors | |

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Applications | |

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Systems of Inequalities in Two Variables | |

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Graph of an Inequality | |

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Systems of Inequalities | |

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Linear Programming | |

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Math at Work | |

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Key Ideas | |

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Review Exercises | |

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Chapter Project | |

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Analytic Geometry in Two and Three Dimensions | |

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Conic Sections and Parabolas | |

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Conic Sections | |

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Geometry of a Parabola | |

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Translations of Parabolas | |

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Reflective Property of a Parabola | |

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Ellipses | |

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Geometry of an Ellipse | |

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Translations of Ellipses | |

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Orbits and Eccentricity | |

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Reflective Property of an Ellipse | |

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Hyperbolas | |

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Geometry of a Hyperbola | |

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Translations of Hyperbolas | |

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Eccentricity and Orbits | |

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Reflective Property of a Hyperbola | |

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Long-Range Navigation | |

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Translation and Rotation of Axes | |

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Second-Degree Equations in Two Variables | |

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Translating Axes versus Translating Graphs | |

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Rotation of Axes | |

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Discriminant Test | |

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Polar Equations of Conics | |

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Eccentricity Revisited | |

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Writing Polar Equations for Conics | |

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Analyzing Polar Equations of Conics | |

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Orbits Revisited | |

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Three-Dimensional Cartesian Coordinate System | |

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Three-Dimensional Cartesian Coordinates | |

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Distance and Midpoint Formulas | |

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Equation of a Sphere | |

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Planes and Other Surfaces | |

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Vectors in Space | |

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Lines in Space | |

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Key Ideas | |

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Review Exercises | |

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Chapter Project | |

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Discrete Mathematics | |

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Basic Combinatorics | |

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Discrete Versus Continuous | |

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The Importance of Counting | |

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The Multiplication Principle of Counting | |

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Permutations | |

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Combinations | |

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Subsets of an n-Set | |

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The Binomial Theorem | |

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Powers of Binomials | |

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Pascal's Triangle | |

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The Binomial Theorem | |

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Factorial Identities | |

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Probability | |

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Sample Spaces and Probability Functions | |

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Determining Probabilities | |

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Venn Diagrams and Tree Diagrams | |

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| |

Conditional Probability | |

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Binomial Distributions | |

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Sequences | |

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Infinite Sequences | |

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Limits of Infinite Sequences | |

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Arithmetic and Geometric Sequences | |

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Sequences and Graphing Calculators | |

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Series | |

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Summation Notation | |

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Sums of Arithmetic and Geometric Sequences | |

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Infinite Series | |

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Convergence of Geometric Series | |

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| |

Mathematical Induction | |

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The Tower of Hanoi Problem | |

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Principle of Mathematical Induction | |

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Induction and Deduction | |

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Statistics and Data (Graphical) | |

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Statistics | |

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Displaying Categorical Data | |

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Stemplots | |

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Frequency Tables | |

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Histograms | |

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Time Plots | |

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Statistics and Data (Algebraic) | |

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Parameters and Statistics | |

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Mean, Median, and Mode | |

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The Five-Number Summary | |

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Boxplots | |

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Variance and Standard Deviation | |

| |

| |

Normal Distributions | |

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| |

Math at Work | |

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Key Ideas | |

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Review Exercises | |

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| |

Chapter Project | |

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An Introduction to Calculus: Limits, Derivatives, and Integrals | |

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| |

Limits and Motion: The Tangent Problem | |

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Average Velocity | |

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Instantaneous Velocity | |

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Limits Revisited | |

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The Connection to Tangent Lines | |

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The Derivative | |

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Limits and Motion: The Area Problem | |

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Distance from a Constant Velocity | |

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Distance from a Changing Velocity | |

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Limits at Infinity | |

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The Connection to Areas | |

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| |

The Definite Integral | |

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| |

More on Limits | |

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| |

A Little History | |

| |

| |

Defining a Limit Informally | |

| |

| |

Properties of Limits | |

| |

| |

Limits of Continuous Functions | |

| |

| |

One-sided and Two-sided Limits | |

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| |

Limits Involving Infinity | |

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| |

Numerical Derivatives and Integrals | |

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| |

Derivatives on a Calculator | |

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| |

Definite Integrals on a Calculator | |

| |

| |

Computing a Derivative from Data | |

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Computing a Definite Integral from Data | |

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| |

Key Ideas | |

| |

| |

Review Exercises | |

| |

| |

Chapter Project | |

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| |

| |

Algebra Review | |

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| |

Radicals and Rational Exponents | |

| |

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Radicals | |

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| |

Simplifying Radical Expressions | |

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Rationalizing the Denominator | |

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Rational Exponents | |

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Polynomials and Factoring | |

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Adding, Subtracting, and Multiplying Polynomials | |

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Special Products | |

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Factoring Polynomials Using Special Products | |

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Factoring Trinomials | |

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Factoring by Grouping | |

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Fractional Expressions | |

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Domain of an Algebraic Expression | |

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Reducing Rational Expressions | |

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Operations with Rational Expressions | |

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Compound Rational Expressions | |

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Key Formulas | |

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Formulas from Algebra | |

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Exponents | |

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Radicals and Rational Expressions | |

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Special Products | |

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Factoring Polynomials | |

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Inequalities | |

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Quadratic Formula | |

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Logarithms | |

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Determinants | |

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Arithmetic Sequences and Series | |

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Geometric Sequences and Series | |

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Factorial | |

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Binomial Coefficient | |

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Binomial Theorem | |

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Formulas from Geometry | |

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Triangle | |

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Trapezoid | |

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Circle | |

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Sector of Circle | |

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Right Circular Cone | |

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Right Circular Cylinder | |

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Right Triangle | |

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Parallelogram | |

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Circular Ring | |

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Ellipse | |

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Cone | |

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Sphere | |

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Formulas from Trigonometry | |

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Angular Measure | |

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Reciprocal Identities | |

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Quotient Identities | |

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Pythagorean Identities | |

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Odd-Even Identities | |

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Sum and Difference Identities | |

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Cofunction Identities | |

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Double-Angle Identities | |

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Power-Reducing Identities | |

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Half-Angle Identities | |

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Triangles | |

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Trigonometric Form of a Complex Number | |

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De Moivre's Theorem | |

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Formulas from Analytic Geometry | |

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Basic Formulas | |

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Equations of a Line | |

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Equation of a Circle | |

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Parabolas with Vertex (h, k) | |

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Ellipses with Center (h, k) and a > b > 0 | |

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Hyperbolas with Center (h, k) | |

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Gallery of Basic Functions | |

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Logic | |

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Logic: An Introduction | |

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Statements | |

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Compound Statements | |

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Conditionals and Biconditionals | |

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Forms of Statements | |

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Valid Reasoning | |

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Glossary | |

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Selected Answers | |

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Applications Index | |

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Index | |