| |
| |
| Preface | |
| |
| |
| |
| Functions | |
| |
| |
| |
| Functions and Their Graphs | |
| |
| |
| |
| Combining Functions; Shifting and Scaling Graphs | |
| |
| |
| |
| Trigonometric Functions | |
| |
| |
| |
| Exponential Functions | |
| |
| |
| |
| Inverse Functions and Logarithms | |
| |
| |
| |
| Graphing with Calculators and Computers | |
| |
| |
| |
| Limits and Continuity | |
| |
| |
| |
| Rates of Change and Tangents to Curves | |
| |
| |
| |
| Limit of a Function and Limit Laws | |
| |
| |
| |
| The Precise Definition of a Limit | |
| |
| |
| |
| One-Sided Limits and Limits at Infinity | |
| |
| |
| |
| Infinite Limits and Vertical Asymptotes | |
| |
| |
| |
| Continuity | |
| |
| |
| |
| Tangents and Derivatives at a Point | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Differentiation | |
| |
| |
| |
| The Derivative as a Function | |
| |
| |
| |
| Differentiation Rules for Polynomials, Exponentials, Products, and Quotients | |
| |
| |
| |
| The Derivative as a Rate of Change | |
| |
| |
| |
| Derivatives of Trigonometric Functions | |
| |
| |
| |
| The Chain Rule and Parametric Equations | |
| |
| |
| |
| Implicit Differentiation | |
| |
| |
| |
| Derivatives of Inverse Functions and Logarithms | |
| |
| |
| |
| Inverse Trigonometric Functions | |
| |
| |
| |
| Related Rates | |
| |
| |
| |
| Linearization and Differentials | |
| |
| |
| |
| Hyperbolic Functions | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Applications of Derivatives | |
| |
| |
| |
| Extreme Values of Functions | |
| |
| |
| |
| The Mean Value Theorem | |
| |
| |
| |
| Monotonic Functions and the First Derivative Test | |
| |
| |
| |
| Concavity and Curve Sketching | |
| |
| |
| |
| Applied Optimization | |
| |
| |
| |
| Indeterminate Forms and L'Hopital's Rule | |
| |
| |
| |
| Newton's Method | |
| |
| |
| |
| Antiderivatives | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Integration | |
| |
| |
| |
| Estimating with Finite Sums | |
| |
| |
| |
| Sigma Notation and Limits of Finite Sums | |
| |
| |
| |
| The Definite Integral | |
| |
| |
| |
| The Fundamental Theorem of Calculus | |
| |
| |
| |
| Indefinite Integrals and the Substitution Rule | |
| |
| |
| |
| Substitution and Area Between Curves | |
| |
| |
| |
| The Logarithm Defined as an Integral | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Applications of Definite Integrals | |
| |
| |
| |
| Volumes by Slicing and Rotation About an Axis | |
| |
| |
| |
| Volumes by Cylindrical Shells | |
| |
| |
| |
| Lengths of Plane Curves | |
| |
| |
| |
| Areas of Surfaces of Revolution | |
| |
| |
| |
| Exponential Change and Separable Differential Equations | |
| |
| |
| |
| Work | |
| |
| |
| |
| Moments and Centers of Mass | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Techniques of Integration | |
| |
| |
| |
| Integration by Parts | |
| |
| |
| |
| Trigonometric Integrals | |
| |
| |
| |
| Trigonometric Substitutions | |
| |
| |
| |
| Integration of Rational Functions by Partial Fractions | |
| |
| |
| |
| Integral Tables and Computer Algebra Systems | |
| |
| |
| |
| Numerical Integration | |
| |
| |
| |
| Improper Integrals | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Infinite Sequences and Series | |
| |
| |
| |
| Sequences | |
| |
| |
| |
| Infinite Series | |
| |
| |
| |
| The Integral Test | |
| |
| |
| |
| Comparison Tests | |
| |
| |
| |
| The Ratio and Root Tests | |
| |
| |
| |
| Alternating Series, Absolute and Conditional Convergence | |
| |
| |
| |
| Power Series | |
| |
| |
| |
| Taylor and Maclaurin Series | |
| |
| |
| |
| Convergence of Taylor Series | |
| |
| |
| |
| The Binomial Series | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Polar Coordinates and Conics | |
| |
| |
| |
| Polar Coordinates | |
| |
| |
| |
| Graphing in Polar Coordinates | |
| |
| |
| |
| Areas and Lengths in Polar Coordinates | |
| |
| |
| |
| Conic Sections | |
| |
| |
| |
| Conics in Polar Coordinates | |
| |
| |
| |
| Conics and Parametric Equations; The Cycloid | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Vectors and the Geometry of Space | |
| |
| |
| |
| Three-Dimensional Coordinate Systems | |
| |
| |
| |
| Vectors | |
| |
| |
| |
| The Dot Product | |
| |
| |
| |
| The Cross Product | |
| |
| |
| |
| Lines and Planes in Space | |
| |
| |
| |
| Cylinders and Quadric Surfaces | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Vector-Valued Functions and Motion in Space | |
| |
| |
| |
| Vector Functions and Their Derivatives | |
| |
| |
| |
| Integrals of Vector Functions | |
| |
| |
| |
| Arc Length in Space | |
| |
| |
| |
| Curvature of a Curve | |
| |
| |
| |
| Tangential and Normal Components of Acceleration | |
| |
| |
| |
| Velocity and Acceleration in Polar Coordinates | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Partial Derivatives | |
| |
| |
| |
| Functions of Several Variables | |
| |
| |
| |
| Limits and Continuity in Higher Dimensions | |
| |
| |
| |
| Partial Derivatives | |
| |
| |
| |
| The Chain Rule | |
| |
| |
| |
| Directional Derivatives and Gradient Vectors | |
| |
| |
| |
| Tangent Planes and Differentials | |
| |
| |
| |
| Extreme Values and Saddle Points | |
| |
| |
| |
| Lagrange Multipliers | |
| |
| |
| |
| Taylor's Formula for Two Variables | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Multiple Integrals | |
| |
| |
| |
| Double and Iterated Integrals over Rectangles | |
| |
| |
| |
| Double Integrals over General Regions | |
| |
| |
| |
| Area by Double Integration | |
| |
| |
| |
| Double Integrals in Polar Form | |
| |
| |
| |
| Triple Integrals in Rectangular Coordinates | |
| |
| |
| |
| Moments and Centers of Mass | |
| |
| |
| |
| Triple Integrals in Cylindrical and Spherical Coordinates | |
| |
| |
| |
| Substitutions in Multiple Integrals | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| Integration in Vector Fields | |
| |
| |
| |
| Line Integrals | |
| |
| |
| |
| Vector Fields, Work, Circulation, and Flux | |
| |
| |
| |
| Path Independence, Potential Functions, and Conservative Fields | |
| |
| |
| |
| Green's Theorem in the Plane | |
| |
| |
| |
| Surfaces and Area | |
| |
| |
| |
| Surface Integrals and Flux | |
| |
| |
| |
| Stokes' Theorem | |
| |
| |
| |
| The Divergence Theorem and a Unified Theory | |
| |
| |
| Questions to Guide Your Review | |
| |
| |
| Practice Exercises | |
| |
| |
| Additional and Advanced Exercises | |
| |
| |
| |
| First-Order Differential Equations (online) | |
| |
| |
| |
| Solutions, Slope Fields, and Picard's Theorem | |
| |
| |
| |
| First-order Linear Equations | |
| |
| |
| |
| Applications | |
| |
| |
| |
| Euler's Method | |
| |
| |
| |
| Graphical Solutions of Autonomous Equations | |
| |
| |
| |
| Systems of Equations and Phase Planes | |
| |
| |
| |
| Second-Order Differential Equations (online) | |
| |
| |
| |
| Second-Order Linear Equations | |
| |
| |
| |
| Nonhomogeneous Linear Equations | |
| |
| |
| |
| Applications | |
| |
| |
| |
| Euler Equations | |
| |
| |
| |
| Power Series Solutions | |
| |
| |
| Appendices | |
| |
| |
| |
| Real Numbers and the Real Line | |
| |
| |
| |
| Mathematical Induction | |
| |
| |
| |
| Lines, Circles, and Parabolas | |
| |
| |
| |
| Trigonometry Formulas | |
| |
| |
| |
| Proofs of Limit Theorems | |
| |
| |
| |
| Commonly Occurring Limits | |
| |
| |
| |
| Theory of the Real Numbers | |
| |
| |
| |
| The Distributive Law for Vector Cross Products | |
| |
| |
| |
| The Mixed Derivative Theorem and the Increment Theorem | |
| |
| |
| Answers | |
| |
| |
| Index | |
| |
| |
| A Brief Table of Integrals | |
| |
| |
| Credits | |