Preparation for Calculus | |

Graphs and Models | |

Linear Models and Rates of Change | |

Functions and Their Graphs | |

Fitting Models to Data | |

Inverse Functions | |

Exponential and Logarithmic Functions | |

Limits and their Properties | |

A Preview of Calculus | |

Finding Limits Graphically and Numerically | |

Evaluating Limits Analytically | |

Continuity and One-Sided Limits | |

Infinite Limits | |

Section Project: Graphs and Limits of Trigonometric Functions | |

Differentiation | |

The Derivative and the Tangent Line Problem | |

Basic Differentiation Rules and Rates of Change | |

Product and Quotient Rules and Higher-Order Derivatives | |

The Chain Rule | |

Implicit Differentiation | |

Section Project: Optical Illusions | |

Derivatives of Inverse Functions | |

Related Rates | |

Newton's Method | |

Applications of Differentiation | |

Extrema on an Interval | |

Rolle's Theorem and the Mean Value Theorem | |

Increasing and Decreasing Functions and the First Derivative Test | |

Section Project: Rainbows | |

Concavity and the Second Derivative Test | |

Limits at Infinity | |

A Summary of Curve Sketching | |

Optimization Problems | |

Section Project: Connecticut River | |

Differentials | |

Integration | |

Antiderivatives and Indefinite Integration | |

Area | |

Riemann Sums and Definite Integrals | |

The Fundamental Theorem of Calculus | |

Section Project: Demonstrating the Fundamental Theorem | |

Integration by Substitution | |

Numerical Integration | |

The Natural Logarithmic Function: Integration | |

Inverse Trigonometric Functions: Integration | |

Hyperbolic Functions | |

Section Project: St. Louis Arch | |

Differential Equations | |

Slope Fields and Euler's Method | |

Differential Equations: Growth and Decay | |

Differential Equations: Separation of Variables | |

The Logistic Equation | |

First-Order Linear Differential Equations | |

Section Project: Weight Loss | |

Predator-Prey Differential Equations | |

Applications of Integration | |

Area of a Region Between Two Curves | |

Volume: The Disk Method | |

Volume: The Shell Method | |

Section Project: Saturn | |

Arc Length and Surfaces of Revolution | |

Work | |

Section Project: Tidal Energy | |

Moments, Centers of Mass, and Centroids | |

Fluid Pressure and Fluid Force | |

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

Basic Integration Rules | |

Integration by Parts | |

Trigonometric Integrals | |

Section Project: Power Lines | |

Trigonometric Substitution | |

Partial Fractions | |

Integration by Tables and Other Integration Techniques | |

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

Improper Integrals | |

Infinite Series | |

Sequences | |

Series and Convergence | |

Section Project: Cantor's Disappearing Table | |

The Integral Test and p-Series | |

Section Project: The Harmonic Series | |

Comparisons of Series | |

Section Project: Solera Method | |

Alternating Series | |

The Ratio and Root Tests | |

Taylor Polynomials and Approximations | |

Power Series | |

Representation of Functions by Power Series | |

Taylor and Maclaurin Series | |

Conics, Parametric Equations, and Polar Coordinates | |

Conics and Calculus | |

Plane Curves and Parametric Equations | |

Section Projects: Cycloids | |

Parametric Equations and Calculus | |

Polar Coordinates and Polar Graphs | |

Section Project: Anamorphic Art | |

Area and Arc Length in Polar Coordinates | |

Polar Equations of Conics and Kepler's Laws | |

Vectors and the Geometry of Space | |

Vectors in the Plane | |

Space Coordinates and Vectors in Space | |

The Dot Product of Two Vectors | |

The Cross Product of Two Vectors in Space | |

Lines and Planes in Space | |

Section Project: Distances in Space | |

Surfaces in Space | |

Cylindrical and Spherical Coordinates | |

Vector-Valued Functions | |

Vector-Valued Functions | |

Section Project: Witch of Agnesi | |

Differentiation and Integration of Vector-Valued Functions | |

Velocity and Acceleration | |

Tangent Vectors and Normal Vectors | |

Arc Length and Curvature | |

Functions of Several Variables | |

Introduction to Functions of Several Variables | |

Limits and Continuity | |

Partial Derivatives | |

Section Project: Moire Fringes | |

Differentials | |

Chain Rules for Functions of Several Variables | |

Directional Derivatives and Gradients | |

Tangent Planes and Normal Lines | |

Section Project: Wildflowers | |

Extrema of Functions of Two Variables | |

Applications of Extrema of Functions of Two Variables | |

Section Project: Building a Pipeline | |

Lagrange Multipliers | |

Multiple Integration | |

Iterated Integrals and Area in the Plane | |

Double Integrals and Volume | |

Change of Variables: Polar Coordinates | |

Center of Mass and Moments of Inertia | |

Section Project: Center of Pressure on a Sail | |

Surface Area | |

Section Project: Capillary Action | |

Triple Integrals and Applications | |

Triple Integrals in Cylindrical and Spherical Coordinates | |

Section Project: Wrinkled and Bumpy Spheres | |

Change of Variables: Jacobians | |

Vector Analysis | |

Vector Fields | |

Line Integrals | |

Conservative Vector Fields and Independence of Path | |

Green's Theorem | |

Section Project: Hyperbolic and Trigonometric Functions | |

Parametric Surfaces | |

Surface Integrals | |

Section Project: Hyperboloid of One Sheet | |

Divergence Theorem | |

Stoke's Theorem | |

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