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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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