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Before Calculus | |

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Functions | |

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New Functions from Old | |

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Families of Functions | |

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Inverse Functions | |

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Limits and Continuity | |

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Limits | |

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Computing Limits | |

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Limits at Infinity; End Behavior of a Function | |

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Limits | |

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Continuity | |

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Continuity of Trigonometric Functions | |

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The Derivative | |

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Tangent Lines and Rates of Change | |

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The Derivative Function | |

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Introduction to Techniques of Differentiation | |

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The Product and Quotient Rules | |

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Derivatives of Trigonometric Functions | |

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The Chain Rule | |

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Implicit Differentiation | |

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Related Rates | |

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Local Linear Approximation; Differentials | |

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The Derivative in Graphing and Applications | |

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Analysis of Functions I: Increase, Decrease, and Concavity | |

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Analysis of Functions II: Relative Extrema; Graphing Polynomials | |

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Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents | |

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Absolute Maxima and Minima | |

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Applied Maximum and Minimum Problems | |

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Rectilinear Motion | |

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Newton's Method | |

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Rolle's Theorem; Mean-Value Theorem | |

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Integration | |

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An Overview of the Area Problem | |

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The Indefinite Integral | |

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Integration by Substitution | |

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The Definition of Area as a Limit; Sigma Notation | |

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The Definite Integral | |

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The Fundamental Theorem of Calculus | |

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Rectilinear Motion Revisited: Using Integration | |

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Average Value of a Function and Its Applications | |

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Evaluating Definite Integrals by Substitution | |

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Applications of the Definite Integral in Geometry, Science and Engineering | |

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Area Between Two Curves | |

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Volumes by Slicing; Disks and Washers | |

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Volumes by Cylindrical Shells | |

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Length of a Plane Curve | |

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Area of a Surface Revolution | |

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Work | |

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Moments, Centers of Gravity, and Centroids | |

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Fluid Pressure and Force | |

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Exponential, Logarithmic, and Inverse Trigonometric Functions | |

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Exponential and Logarithmic Functions | |

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Derivatives and Integrals Involving Logarithmic Functions | |

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Derivatives of Inverse Functions; Derivatives and Integrals Involving Exponential Functions | |

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Graphs and Applications Involving Logarithmic and Exponential Functions | |

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L'Hï¿½opital's Rule; Indeterminate Forms | |

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Logarithmic and Other Functions Defined by Integrals | |

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Derivatives and Integrals Involving Inverse Trigonometric Functions | |

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Hyperbolic Functions and Hanging Cubes | |

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Principles of Integral Evaluation | |

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An Overview of Integration Methods | |

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Integration by Parts | |

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Integrating Trigonometric Functions | |

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Trigonometric Substitutions | |

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Integrating Rational Functions by Partial Fractions | |

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Using Computer Algebra Systems and Tables of Integrals | |

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Numerical Integration; Simpson's Rule | |

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Improper Integrals | |

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Mathematical Modeling with Differential Equations | |

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Modeling with Differential Equations | |

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Separation of Variables | |

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Slope Fields; Euler's Method | |

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First-Order Differential Equations and Applications | |

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Infinite Series | |

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Sequences | |

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Monotone Sequences | |

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Infinite Series | |

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Convergence Tests | |

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The Comparison, Ratio, and Root Tests | |

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Alternating Series; Absolute and Conditional Convergence | |

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Maclaurin and Taylor Polynomials | |

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Maclaurin and Taylor Series; Power Series | |

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Convergence of Taylor Series | |

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Differentiating and Integrating Power Series; Modeling with Taylor Series | |

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Parametric and Polar Curves; Conic Sections | |

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Parametric Equations; Tangent Lines and Arc Length for Parametric Curves | |

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Polar Coordinates | |

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Tangent Lines, Arc Length, and Area for Polar Curves | |

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Conic Sections | |

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Rotation of Axes; Second-Degree Equations | |

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Conic Sections in Polar Coordinates | |

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Three-Dimensional Space; Vectors | |

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Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces | |

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Vectors | |

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Dot Product; Projections | |

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Cross Product | |

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Parametric Equations of Lines | |

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Planes in 3-Space | |

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Quadric Surfaces | |

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Cylindrical and Spherical Coordinates | |

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Vector-Valued Functions | |

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Introduction to Vector-Valued Functions | |

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Calculus of Vector-Valued Functions | |

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Change of Parameter; Arc Length | |

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Unit Tangent, Normal, and Binormal Vectors | |

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Curvature | |

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Motion Along a Curve | |

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Kepler's Laws of Planetary Motion | |

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Partial Derivatives | |

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Functions of Two or More Variables | |

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Limits and Continuity | |

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Partial Derivatives | |

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Differentiability, Differentials, and Local Linearity | |

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The Chain Rule | |

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Directional Derivatives and Gradients | |

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Tangent Planes and Normal Vectors | |

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Maxima and Minima of Functions of Two Variables | |

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Lagrange Multipliers | |

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Multiple Integrals | |

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Double Integrals | |

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Double Integrals over Nonrectangular Regions | |

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Double Integrals in Polar Coordinates | |

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Surface Area; Parametric Surfaces} | |

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Triple Integrals | |

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Triple Integrals in Cylindrical and Spherical Coordinates | |

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Change of Variable in Multiple Integrals; Jacobians | |

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Centers of Gravity Using Multiple Integrals | |

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Topics in Vector Calculus | |

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Vector Fields | |

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Line Integrals | |

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Independence of Path; Conservative Vector Fields | |

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Green's Theorem | |

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Surface Integrals | |

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Applications of Surface Integrals; Flux | |

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The Divergence Theorem | |

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Stokes' Theorem | |

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Appendix [order of sections TBD] | |

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Graphing Functions Using Calculators and Computer Algebra Systems | |

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Trigonometry Review | |

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Solving Polynomial Equations | |

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Mathematical Models | |

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Selected Proofs | |

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Web Appendices | |

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Real Numbers, Intervals, and Inequalities | |

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Absolute Value | |

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Coordinate Planes, Lines, and Linear Functions | |

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Distance, Circles, and Quadratic Functions | |

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Second-Order Linear Homogeneous Differential Equations; The Vibrating String | |

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The Discriminant | |

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Answers | |

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Photocredits | |

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Index | |