目次

1. Functions 1.1 Functions and Their Graphs 1.2 Combining Functions;Shifting and Scaling Graphs 1.3 Trigonometric Functions 1.4 Graphing withCalculators and Computers 1.5 Exponential Functions 1.6 Inverse Functionsand Logarithms 2. Limits and Continuity 2.1 Rates of Change and Tangents toCurves 2.2 Limit of a Function and Limit Laws 2.3 The Precise Definition ofa Limit 2.4 One-Sided Limits 2.5 Continuity 2.6 Limits Involving Infinity;Asymptotes of Graphs 3. Differentiation 3.1 Tangents and the Derivative ata Point 3.2 The Derivative as a Function 3.3 Differentiation Rules 3.4 TheDerivative as a Rate of Change 3.5 Derivatives of Trigonometric Functions3.6 The Chain Rule 3.7 Implicit Differentiation 3.8 Derivatives of InverseFunctions and Logarithms 3.9 Inverse Trigonometric Functions 3.10 RelatedRates 3.11 Linearization and Differentials 4. Applications of Derivatives4.1 Extreme Values of Functions 4.2 The Mean Value Theorem 4.3 MonotonicFunctions and the First Derivative Test 4.4 Concavity and Curve Sketching4.5 Indeterminate Forms and L'Hopital's Rule 4.6 Applied Optimization 4.7Newton's Method 4.8 Antiderivatives 5. Integration 5.1 Area and Estimatingwith Finite Sums 5.2 Sigma Notation and Limits of Finite Sums 5.3 TheDefinite Integral 5.4 The Fundamental Theorem of Calculus 5.5 IndefiniteIntegrals and the Substitution Rule 5.6 Substitution and Area Between Curves6. Applications of Definite Integrals 6.1 Volumes Using Cross-Sections 6.2Volumes Using Cylindrical Shells 6.3 Arc Length 6.4 Areas of Surfaces ofRevolution 6.5 Work 6.6 Moments and Centers of Mass 7. Integrals andTranscendental Functions 7.1 The Logarithm Defined as an Integral 7.2Exponential Change and Separable Differential Equations 7.3 HyperbolicFunctions 8. Techniques of Integration 8.1 Integration by Parts 8.2Trigonometric Integrals 8.3 Trigonometric Substitutions 8.4 Integration ofRational Functions by Partial Fractions 8.5 Integral Tables and ComputerAlgebra Systems 8.6 Numerical Integration 8.7 Improper Integrals 9.Infinite Sequences and Series 9.1 Sequences 9.2 Infinite Series 9.3 TheIntegral Test 9.4 Comparison Tests 9.5 The Ratio and Root Tests 9.6Alternating Series, Absolute and Conditional Convergence 9.7 Power Series9.8 Taylor and Maclaurin Series 9.9 Convergence of Taylor Series 9.10 TheBinomial Series and Applications of Taylor Series 10. Parametric Equationsand Polar Coordinates 10.1 Parametrizations of Plane Curves 10.2 Calculuswith Parametric Curves 10.3 Polar Coordinates 10.4 Graphing in PolarCoordinates 10.5 Areas and Lengths in Polar Coordinates 10.6 Conics inPolar Coordinates 11. Vectors and the Geometry of Space 11.1Three-Dimensional Coordinate Systems 11.2 Vectors 11.3 The Dot Product11.4 The Cross Product 11.5 Lines and Planes in Space 11.6 Cylinders andQuadric Surfaces 12. Vector-Valued Functions and Motion in Space 12.1Curves in Space and Their Tangents 12.2 Integrals of Vector Functions;Projectile Motion 12.3 Arc Length in Space 12.4 Curvature and NormalVectors of a Curve 12.5 Tangential and Normal Components of Acceleration12.6 Velocity and Acceleration in Polar Coordinates 13. Partial Derivatives13.1 Functions of Several Variables 13.2 Limits and Continuity in HigherDimensions 13.3 Partial Derivatives 13.4 The Chain Rule 13.5 DirectionalDerivatives and Gradient Vectors 13.6 Tangent Planes and Differentials 13.7Extreme Values and Saddle Points 13.8 Lagrange Multipliers 14. MultipleIntegrals 14.1 Double and Iterated Integrals over Rectangles 14.2 DoubleIntegrals over General Regions 14.3 Area by Double Integration 14.4 DoubleIntegrals in Polar Form 14.5 Triple Integrals in Rectangular Coordinates14.6 Moments and Centers of Mass 14.7 Triple Integrals in Cylindrical andSpherical Coordinates 14.8 Substitutions in Multiple Integrals 15.Integration in Vector Fields 15.1 Line Integrals 15.2 Vector Fields andLine Integrals: Work, Circulation, and Flux 15.3 Path Independence,Conservative Fields, and Potential Functions 15.4 Green's Theorem in thePlane 15.5 Surfaces and Area 15.6 Surface Integrals 15.7 Stokes' Theorem15.8 The Divergence Theorem and a Unified Theory 16. First-OrderDifferential Equations (Online) 16.1 Solutions, Slope Fields, and Euler'sMethod 16.2 First-Order Linear Equations 16.3 Applications 16.4 GraphicalSolutions of Autonomous Equations 16.5 Systems of Equations and Phase Planes17. Second-Order Differential Equations (Online) 17.1 Second-Order LinearEquations 17.2 Nonhomogeneous Linear Equations 17.3 Applications 17.4Euler Equations 17.5 Power Series Solutions Appendices 1. Real Numbers andthe Real Line 2. Mathematical Induction 3. Lines, Circles, and Parabolas4. Conic Sections 5. Proofs of Limit Theorems 6. Commonly Occurring Limits7. Theory of the Real Numbers 8. Complex Numbers 9. The Distributive Lawfor Vector Cross Products 10. The Mixed Derivative Theorem and the IncrementTheorem 11. Taylor's Formula for Two Variables