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Preface | |

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Functions | |

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Functions and Their Graphs | |

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Combining Functions; Shifting and Scaling Graphs | |

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Trigonometric Functions | |

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Exponential Functions | |

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Inverse Functions and Logarithms | |

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Graphing with Calculators and Computers | |

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Limits and Continuity | |

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Rates of Change and Tangents to Curves | |

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Limit of a Function and Limit Laws | |

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The Precise Definition of a Limit | |

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One-Sided Limits and Limits at Infinity | |

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Infinite Limits and Vertical Asymptotes | |

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Continuity | |

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Tangents and Derivatives at a Point | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Differentiation | |

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The Derivative as a Function | |

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Differentiation Rules for Polynomials, Exponentials, Products, and Quotients | |

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The Derivative as a Rate of Change | |

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Derivatives of Trigonometric Functions | |

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The Chain Rule and Parametric Equations | |

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Implicit Differentiation | |

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Derivatives of Inverse Functions and Logarithms | |

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Inverse Trigonometric Functions | |

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Related Rates | |

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Linearization and Differentials | |

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Hyperbolic Functions | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Applications of Derivatives | |

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Extreme Values of Functions | |

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The Mean Value Theorem | |

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Monotonic Functions and the First Derivative Test | |

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Concavity and Curve Sketching | |

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Applied Optimization | |

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Indeterminate Forms and L'Hopital's Rule | |

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Newton's Method | |

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Antiderivatives | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Integration | |

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Estimating with Finite Sums | |

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Sigma Notation and Limits of Finite Sums | |

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The Definite Integral | |

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The Fundamental Theorem of Calculus | |

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Indefinite Integrals and the Substitution Rule | |

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Substitution and Area Between Curves | |

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The Logarithm Defined as an Integral | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Applications of Definite Integrals | |

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Volumes by Slicing and Rotation About an Axis | |

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Volumes by Cylindrical Shells | |

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Lengths of Plane Curves | |

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Areas of Surfaces of Revolution | |

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Exponential Change and Separable Differential Equations | |

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Work | |

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Moments and Centers of Mass | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Techniques of Integration | |

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Integration by Parts | |

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Trigonometric Integrals | |

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Trigonometric Substitutions | |

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Integration of Rational Functions by Partial Fractions | |

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Integral Tables and Computer Algebra Systems | |

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Numerical Integration | |

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Improper Integrals | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Infinite Sequences and Series | |

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Sequences | |

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Infinite Series | |

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The Integral Test | |

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Comparison Tests | |

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The Ratio and Root Tests | |

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Alternating Series, Absolute and Conditional Convergence | |

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Power Series | |

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Taylor and Maclaurin Series | |

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Convergence of Taylor Series | |

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The Binomial Series | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Polar Coordinates and Conics | |

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Polar Coordinates | |

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Graphing in Polar Coordinates | |

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Areas and Lengths in Polar Coordinates | |

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Conic Sections | |

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Conics in Polar Coordinates | |

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Conics and Parametric Equations; The Cycloid | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Vectors and the Geometry of Space | |

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Three-Dimensional Coordinate Systems | |

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Vectors | |

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The Dot Product | |

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The Cross Product | |

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Lines and Planes in Space | |

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Cylinders and Quadric Surfaces | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Vector-Valued Functions and Motion in Space | |

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Vector Functions and Their Derivatives | |

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Integrals of Vector Functions | |

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Arc Length in Space | |

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Curvature of a Curve | |

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Tangential and Normal Components of Acceleration | |

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Velocity and Acceleration in Polar Coordinates | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Partial Derivatives | |

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Functions of Several Variables | |

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Limits and Continuity in Higher Dimensions | |

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Partial Derivatives | |

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The Chain Rule | |

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Directional Derivatives and Gradient Vectors | |

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Tangent Planes and Differentials | |

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Extreme Values and Saddle Points | |

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Lagrange Multipliers | |

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Taylor's Formula for Two Variables | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Multiple Integrals | |

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Double and Iterated Integrals over Rectangles | |

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Double Integrals over General Regions | |

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Area by Double Integration | |

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Double Integrals in Polar Form | |

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Triple Integrals in Rectangular Coordinates | |

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Moments and Centers of Mass | |

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Triple Integrals in Cylindrical and Spherical Coordinates | |

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Substitutions in Multiple Integrals | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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Integration in Vector Fields | |

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Line Integrals | |

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Vector Fields, Work, Circulation, and Flux | |

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Path Independence, Potential Functions, and Conservative Fields | |

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Green's Theorem in the Plane | |

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Surfaces and Area | |

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Surface Integrals and Flux | |

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Stokes' Theorem | |

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The Divergence Theorem and a Unified Theory | |

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Questions to Guide Your Review | |

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Practice Exercises | |

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Additional and Advanced Exercises | |

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First-Order Differential Equations (online) | |

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Solutions, Slope Fields, and Picard's Theorem | |

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First-order Linear Equations | |

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Applications | |

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Euler's Method | |

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Graphical Solutions of Autonomous Equations | |

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Systems of Equations and Phase Planes | |

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Second-Order Differential Equations (online) | |

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Second-Order Linear Equations | |

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Nonhomogeneous Linear Equations | |

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Applications | |

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Euler Equations | |

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Power Series Solutions | |

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Appendices | |

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Real Numbers and the Real Line | |

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Mathematical Induction | |

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Lines, Circles, and Parabolas | |

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Trigonometry Formulas | |

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Proofs of Limit Theorems | |

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Commonly Occurring Limits | |

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Theory of the Real Numbers | |

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The Distributive Law for Vector Cross Products | |

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The Mixed Derivative Theorem and the Increment Theorem | |

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Answers | |

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Index | |

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A Brief Table of Integrals | |

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Credits | |