Table of Contents

1 Functions

1.1 Functions and Their Graphs
 Graphing Linear Functions Using Intercepts
 Graphing Linear Functions Using Tables
 Graphing Quadratic Functions in Standard Form
 Domain and Range of a Piecewise Function
 Evaluating Piecewise Functions
 Graphing Piecewise Functions
 Increasing and Decreasing Intervals of a Function
 Even and Odd Functions
 1.2 Combining Functions; Shifting and Scaling Graphs
 1.3 Trigonometric Functions
 1.4 Graphing with Software
 1.5 Exponential Functions
 1.6 Inverse Functions and Logarithms

1.1 Functions and Their Graphs

2 Limits and Continuity
 2.1 Rates of Change and Tangent Lines to Curves
 2.2 Limit of a Function and Limit Laws

2.3 The Precise Definition of a Limit
 2.4 OneSided Limits
 2.5 Continuity
 2.6 Limits Involving Infinity; Asymptotes of Graphs

3 Derivatives
 3.1 Tangent Lines and the Derivative at a Point
 3.2 The Derivative as a Function
 3.3 Differentiation Rules
 3.4 The Derivative as a Rate of Change
 3.5 Derivatives of Trigonometric Functions
 3.6 The Chain Rule
 3.7 Implicit Differentiation
 3.8 Derivatives of Inverse Functions and Logarithms
 3.9 Inverse Trigonometric Functions
 3.10 Related Rates
 3.11 Linearization and Differentials

4 Applications of Derivatives
 4.1 Extreme Values of Functions on Closed Intervals
 4.2 The Mean Value Theorem
 4.3 Monotonic Functions and the First Derivative Test
 4.4 Concavity and Curve Sketching
 4.5 Indeterminate Forms and L’Hôpital’s Rule
 4.6 Applied Optimization
 4.7 Newton’s Method
 4.8 Antiderivatives

5 Integrals
 5.1 Area and Estimating with Finite Sums
 5.2 Sigma Notation and Limits of Finite Sums
 5.3 The Definite Integral
 5.4 The Fundamental Theorem of Calculus
 5.5 Indefinite Integrals and the Substitution Method
 5.6 Definite Integral Substitutions and the Area between Curves

6 Applications of Definite Integrals
 6.1 Volumes Using CrossSections
 6.2 Volumes Using Cylindrical Shells
 6.3 Arc Length
 6.4 Areas of Surfaces of Revolution
 6.5 Work and Fluid Forces
 6.6 Moments and Centers of Mass

7 Integrals and Transcendental Functions
 7.1 The Logarithm Defined as an Integral
 7.2 Exponential Change and Separable Differential Equations
 7.3 Hyperbolic Functions
 7.4 Relative Rates of Growth

8 Techniques of Integration
 8.1 Using Basic Integration Formulas
 8.2 Integration by Parts
 8.3 Trigonometric Integrals
 8.4 Trigonometric Substitutions
 8.5 Integration of Rational Functions by Partial Fractions
 8.6 Integral Tables and Computer Algebra Systems
 8.7 Numerical Integration
 8.8 Improper Integrals
 8.9 Probability

9 FirstOrder Differential Equations

9.1 Solutions, Slope Fields, and Euler’s Method
 9.2 FirstOrder Linear Equations
 9.3 Applications
 9.4 Graphical Solutions of Autonomous Equations
 9.5 Systems of Equations and Phase Planes

9.1 Solutions, Slope Fields, and Euler’s Method

10 Infinite Sequences and Series
 10.1 Sequences
 10.2 Infinite Series
 10.3 The Integral Test
 10.4 Comparison Tests
 10.5 Absolute Convergence; The Ratio and Root Tests
 10.6 Alternating Series and Conditional Convergence
 10.7 Power Series
 10.8 Taylor and Maclaurin Series

10.9 Convergence of Taylor Series
 10.10 Applications of Taylor Series

11 Parametric Equations and Polar Coordinates
 11.1 Parametrizations of Plane Curves
 11.2 Calculus with Parametric Curves
 11.3 Polar Coordinates
 11.4 Graphing Polar Coordinate Equations
 11.5 Areas and Lengths in Polar Coordinates
 11.6 Conic Sections
 11.7 Conics in Polar Coordinates

12 Vectors and the Geometry of Space
 12.1 ThreeDimensional Coordinate Systems
 12.2 Vectors
 12.3 The Dot Product
 12.4 The Cross Product
 12.5 Lines and Planes in Space
 12.6 Cylinders and Quadric Surfaces

13 VectorValued Functions and Motion in Space
 13.1 Curves in Space and Their Tangents
 13.2 Integrals of Vector Functions; Projectile Motion
 13.3 Arc Length in Space
 13.4 Curvature and Normal Vectors of a Curve
 13.5 Tangential and Normal Components of Acceleration
 13.6 Velocity and Acceleration in Polar Coordinates

14 Partial Derivatives
 14.1 Functions of Several Variables
 14.2 Limits and Continuity in Higher Dimensions
 14.3 Partial Derivatives
 14.4 The Chain Rule
 14.5 Directional Derivatives and Gradient Vectors
 14.6 Tangent Planes and Differentials
 14.7 Extreme Values and Saddle Points
 14.8 Lagrange Multipliers
 14.9 Taylor’s Formula for Two Variables
 14.10 Partial Derivatives with Constrained Variables

15 Multiple Integrals
 15.1 Double and Iterated Integrals over Rectangles
 15.2 Double Integrals over General Regions
 15.3 Area by Double Integration
 15.4 Double Integrals in Polar Form
 15.5 Triple Integrals in Rectangular Coordinates
 15.6 Applications
 15.7 Triple Integrals in Cylindrical and Spherical Coordinates
 15.8 Substitutions in Multiple Integrals

16 Integrals and Vector Fields
 16.1 Line Integrals of Scalar Functions
 16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux
 16.3 Path Independence, Conservative Fields, and Potential Functions
 16.4 Green’s Theorem in the Plane
 16.5 Surfaces and Area
 16.6 Surface Integrals
 16.7 Stokes’ Theorem
 16.8 The Divergence Theorem and a Unified Theory

17 SecondOrder Differential Equations
 17.1 SecondOrder Linear Equations
 17.2 Nonhomogeneous Linear Equations
 17.3 Applications
 17.4 Euler Equations
 17.5 PowerSeries Solutions