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The Cartesian Plane and Functions | |
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Real Numbers and the Real Line | |
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The Cartesian Plane | |
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Graphs of Equations | |
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Lines in the Plane | |
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Functions | |
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Limits and Their Properties | |
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An Introduction to Limits | |
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Techniques for Evaluating Limits | |
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Continuity | |
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Infinite Limits | |
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e-d Definition of Limits | |
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Differentiation | |
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The Derivative and the Tangent Line Problem | |
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Velocity, Acceleration, and Other Rates of Change | |
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Differentiation Rules for Powers, Constant Multiples, and Sums | |
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Differentiation Rules for Products and Quotients | |
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The Chain Rule | |
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Implicit Differentiation | |
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Related Rates | |
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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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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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Newton's Method | |
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Differentials | |
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Business and Economics Applications | |
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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 the Definite Integral | |
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The Fundamental Theorem of Calculus | |
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Integration by Substitution | |
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Numerical Integration | |
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Applications of Integration | |
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Area of a Region Between Two Curves | |
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Volume: The Disc Method | |
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Volume: The Shell Method | |
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Arc Length and Surfaces of Revolution | |
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Work | |
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Fluid Pressure and Fluid Force | |
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Moments, Centers of Mass, and Centroids | |
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Exponential and Logarithmic Functions | |
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Exponential Functions | |
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Differentiation and Integration of Exponential Functions | |
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Inverse Functions | |
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Logarithmic Functions | |
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Logarithmic Functions and Differentiation | |
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Logarithmic Functions and Integration | |
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Growth and Decay | |
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Indeterminate Forms and L'Hopital's Rule | |
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Trigonometric Functions and Inverse Trigonometric Functions | |
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Review of Trigonometric Functions | |
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Graphs and Limits of Trigonometric Functions | |
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Derivatives of Trigonometric Functions | |
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Integrals of Trigonometric Functions | |
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Inverse Trigonometric Functions and Differentiation | |
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Inverse Trigonometric Functions: Integration and Completing the Square | |
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Hyperbolic Functions | |
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Integration Techniques and Improper Integrals | |
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Basic Integration Formulas | |
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Integration by Parts | |
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Trigonometric Integrals | |
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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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Improper Integrals | |
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Infinite Series | |
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Sequences | |
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Series and Convergence | |
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The Integral Test and p-Series | |
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Comparisons of Series | |
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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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Conic Sections | |
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Parabolas | |
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Ellipses | |
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Hyperbolas | |
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Rotation and the General Second-Degree Equation | |
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Plane Curves, Parametric Equations, and Polar Coordinates | |
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Plane Curves and Parametric Equations | |
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Parametric Equations and Calculus | |
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Polar Coordinates and Polar Graphs | |
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Tangent Lines and Curve Sketching in Polar Coordinates | |
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Area and Arc Length in Polar Coordinates | |
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Polar Equations for Conics and Kepler's Laws | |
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Vectors and Curves in the Plane | |
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Vectors in the Plane | |
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The Dot Product of Two Vectors | |
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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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Solid Analytic Geometry and Vectors in Space | |
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Space Coordinates and Vectors in Space | |
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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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Surfaces in Space | |
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Curves and Vector-Valued Functions in Space | |
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Tangent Vectors, Normal Vectors, and Curvature in Space | |
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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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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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Extrema of Functions of Two Variables | |
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Applications of Extrema of Functions of Two Variables | |
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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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Surface Area | |
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Triple Integrals and Applications | |
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Cylindrical and Spherical Coordinates | |
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Triple Integrals in Cylindrical and Spherical Coordinates | |
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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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Parametric Surfaces | |
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Surface Integrals | |
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Divergence Theorem | |
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Stokes's Theorem | |
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Differential Equations | |
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Definitions and Basic Concepts | |
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Separation of Variables in First-Order Equations | |
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Exact First-Order Equations | |
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First-Order Linear Differential Equations | |
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Second-Order Homogeneous Linear Equations | |
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Second-Order Nonhomogeneous Linear Equations | |
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Series Solutions of Differential Equations | |
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Proofs of Selected Theorems | |
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Basic Differentiation Rules for Elementary Functions | |
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Integration Tables Answers to Odd-Numbered Exercises | |