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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'hopital'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 | |