## Table of Contents

(Practice Exercises, Additional Exercises, and Questions to Guide Your Review appear at the end of each chapter.)

**Preliminaries **

Real Numbers and the Real Line

Lines, Circles, and Parabolas

Functions and Their Graphs

Identifying Functions; Mathematical Models

Combining Functions; Shifting and Scaling Graphs

Trigonometric Functions

Graphing with Calculators and Computers

**2. Limits and Derivatives **

Rates of Change and Limits

Calculating Limits Using the Limit Laws

Precise Definition of a Limit

One-Sided Limits and Limits at Infinity

Infinite Limits and Vertical Asymptotes

Continuity

Tangents and Derivatives

**3. Differentiation **

The Derivative as a Function

Differentiation Rules

The Derivative as a Rate of Change

Derivatives of Trigonometric Functions

The Chain Rule and Parametric Equations

Implicit Differentiation

Related Rates

Linearization and Differentials

**4. Applications of Derivatives **

Extreme Values of Functions

The Mean Value Theorem

Monotonic Functions and the First Derivative Test

Concavity and Curve Sketching

Applied Optimization Problems

Indeterminate Forms and L'Hopital's Rule

Newton's Method

Antiderivatives

**5. 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

**6. Applications of Definite Integrals **

Volumes by Slicing and Rotation About an Axis

Volumes by Cylindrical Shells

Lengths of Plane Curves

Moments and Centers of Mass

Areas of Surfaces of Revolution and The Theorems of Pappus

Work

Fluid Pressures and Forces

**7. Transcendental Functions **

Inverse Functions and their Derivatives

Natural Logarithms

The Exponential Function

ax and loga x

Exponential Growth and Decay

Relative Rates of Growth

Inverse Trigonometric Functions

Hyperbolic Functions

**8. Techniques of Integration **

Basic Integration Formulas

Integration by Parts

Integration of Rational Functions by Partial Fractions

Trigonometric Integrals

Trigonometric Substitutions

Integral Tables and Computer Algebra Systems

Numerical Integration

Improper Integrals

**9. Further Applications of Integration **

Slope Fields and Separable Differential Equations

First-Order Linear Differential Equations

Euler's Method

Graphical Solutions of Autonomous Equations

Applications of First-Order Differential Equations

**10. Conic Sections and Polar Coordinates **

Conic Sections and Quadratic Equations

Classifying Conic Sections by Eccentricity

Quadratic Equations and Rotations

Conics and Parametric Equations; The Cycloid

Polar Coordinates

Graphing in Polar Coordinates

Area and Lengths in Polar Coordinates

Conic Sections in Polar Coordinates

**11. 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; Error Estimates

Applications of Power Series

Fourier Series

**12. 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

**13. Vector-Valued Functions and Motion in Space **

Vector Functions

Modeling Projectile Motion

Arc Length and the Unit Tangent Vector T

Curvature and the Unit Normal Vector N

Torsion and the Unit Binormal Vector B

Planetary Motion and Satellites

**14. 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

*Partial Derivatives with Constrained Variables

Taylor's Formula for Two Variables

**15. Multiple Integrals **

Double Integrals

Areas, Moments and Centers of Mass*

Double Integrals in Polar Form

Triple Integrals in Rectangular Coordinates

Masses and Moments in Three Dimensions

Triple Integrals in Cylindrical and Spherical Coordinates

Substitutions in Multiple Integrals

**16. 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

Surface Area and Surface Integrals

Parametrized Surfaces

Stokes' Theorem

The Divergence Theorem and a Unified Theory

**Appendices **

Mathematical Induction

Proofs of Limit Theorems

Commonly Occurring Limits

Theory of the Real Numbers

Complex Numbers

The Distributive Law for Vector Cross Products

Determinants and Cramer's Rule

The Mixed Derivative Theorem and the Increment Theorem

The Area of a Parallelogram's Projection on a Plane

**17. Second-order Differential Equations**

Second-Order Linear Equations

Nonhomogeneous Linear Equations

Applications

Euler Equations

Power-Series Solutions