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Preparation for Calculus | |

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Graphs and Models | |

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Linear Models and Rates of Change | |

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Functions and Their Graphs | |

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Fitting Models to Data | |

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Inverse Functions | |

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Exponential and Logarithmic Functions | |

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Limits and their Properties | |

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A Preview of Calculus | |

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Finding Limits Graphically and Numerically | |

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Evaluating Limits Analytically | |

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Continuity and One-Sided Limits | |

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Infinite Limits | |

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Section Project: Graphs and Limits of Trigonometric Functions | |

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Differentiation | |

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The Derivative and the Tangent Line Problem | |

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Basic Differentiation Rules and Rates of Change | |

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Product and Quotient Rules and Higher-Order Derivatives | |

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The Chain Rule | |

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Implicit Differentiation | |

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Section Project: Optical Illusions | |

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Derivatives of Inverse Functions | |

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Related Rates | |

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Newton's Method | |

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Applications of Differentiation | |

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Extrema on an Interval | |

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Rolle's Theorem and the Mean Value Theorem | |

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Increasing and Decreasing Functions and the First Derivative Test | |

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Section Project: Rainbows | |

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Concavity and the Second Derivative Test | |

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Limits at Infinity | |

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A Summary of Curve Sketching | |

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Optimization Problems | |

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Section Project: Connecticut River | |

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Differentials | |

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Integration | |

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Antiderivatives and Indefinite Integration | |

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Area | |

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Riemann Sums and Definite Integrals | |

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The Fundamental Theorem of Calculus | |

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Section Project: Demonstrating the Fundamental Theorem | |

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Integration by Substitution | |

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Numerical Integration | |

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The Natural Logarithmic Function: Integration | |

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Inverse Trigonometric Functions: Integration | |

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Hyperbolic Functions | |

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Section Project: St. Louis Arch | |

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Differential Equations | |

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Slope Fields and Euler's Method | |

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Differential Equations: Growth and Decay | |

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Differential Equations: Separation of Variables | |

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The Logistic Equation | |

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First-Order Linear Differential Equations | |

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Section Project: Weight Loss | |

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Predator-Prey Differential Equations | |

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Applications of Integration | |

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Area of a Region Between Two Curves | |

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Volume: The Disk Method | |

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Volume: The Shell Method | |

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Section Project: Saturn | |

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Arc Length and Surfaces of Revolution | |

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Work | |

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Section Project: Tidal Energy | |

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Moments, Centers of Mass, and Centroids | |

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Fluid Pressure and Fluid Force | |

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Integration Techniques, L'H?pital's Rule, and Improper Integrals | |

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Basic Integration Rules | |

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Integration by Parts | |

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Trigonometric Integrals | |

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Section Project: Power Lines | |

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Trigonometric Substitution | |

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Partial Fractions | |

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Integration by Tables and Other Integration Techniques | |

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Indeterminate Forms and L'H?pital's Rule | |

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Improper Integrals | |

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Infinite Series | |

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Sequences | |

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Series and Convergence | |

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Section Project: Cantor's Disappearing Table | |

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The Integral Test and p-Series | |

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Section Project: The Harmonic Series | |

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Comparisons of Series | |

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Section Project: Solera Method | |

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Alternating Series | |

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The Ratio and Root Tests | |

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Taylor Polynomials and Approximations | |

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Power Series | |

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Representation of Functions by Power Series | |

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Taylor and Maclaurin Series | |

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Conics, Parametric Equations, and Polar Coordinates | |

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Conics and Calculus | |

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Plane Curves and Parametric Equations | |

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Section Projects: Cycloids | |

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Parametric Equations and Calculus | |

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Polar Coordinates and Polar Graphs | |

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Section Project: Anamorphic Art | |

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Area and Arc Length in Polar Coordinates | |

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Polar Equations of Conics and Kepler's Laws | |

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Vectors and the Geometry of Space | |

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Vectors in the Plane | |

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Space Coordinates and Vectors in Space | |

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The Dot Product of Two Vectors | |

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The Cross Product of Two Vectors in Space | |

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Lines and Planes in Space | |

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Section Project: Distances in Space | |

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Surfaces in Space | |

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Cylindrical and Spherical Coordinates | |

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Vector-Valued Functions | |

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Vector-Valued Functions | |

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Section Project: Witch of Agnesi | |

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Differentiation and Integration of Vector-Valued Functions | |

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Velocity and Acceleration | |

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Tangent Vectors and Normal Vectors | |

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Arc Length and Curvature | |

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Functions of Several Variables | |

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Introduction to Functions of Several Variables | |

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Limits and Continuity | |

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Partial Derivatives | |

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Section Project: Moire Fringes | |

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Differentials | |

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Chain Rules for Functions of Several Variables | |

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Directional Derivatives and Gradients | |

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Tangent Planes and Normal Lines | |

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Section Project: Wildflowers | |

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Extrema of Functions of Two Variables | |

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Applications of Extrema of Functions of Two Variables | |

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Section Project: Building a Pipeline | |

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Lagrange Multipliers | |

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Multiple Integration | |

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Iterated Integrals and Area in the Plane | |

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Double Integrals and Volume | |

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Change of Variables: Polar Coordinates | |

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Center of Mass and Moments of Inertia | |

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Section Project: Center of Pressure on a Sail | |

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Surface Area | |

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Section Project: Capillary Action | |

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Triple Integrals and Applications | |

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Triple Integrals in Cylindrical and Spherical Coordinates | |

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Section Project: Wrinkled and Bumpy Spheres | |

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Change of Variables: Jacobians | |

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Vector Analysis | |

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Vector Fields | |

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Line Integrals | |

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Conservative Vector Fields and Independence of Path | |

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Green's Theorem | |

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Section Project: Hyperbolic and Trigonometric Functions | |

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Parametric Surfaces | |

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Surface Integrals | |

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Section Project: Hyperboloid of One Sheet | |

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Divergence Theorem | |

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Stoke's Theorem | |