CALCULUS
About the Authors
Preface
Acknowledgments
Applications Index
Introduction
1.1 Limits of Functions Using Numerical and Graphical Techniques
1.1 Assess Your Understanding
1.2 Limits of Functions Using Properties of Limits
1.2 Assess Your Understanding
1.3 Continuity
1.3 Assess Your Understanding
1.4 Limits and Continuity of Trigonometric, Exponential, and Logarithmic Functions
1.4 Assess Your Understanding
1.5 Infinite Limits; Limits at Infinity; Asymptotes
1.5 Assess Your Understanding
1.6 The ε-δ Definition of a Limit
1.6 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 1 PROJECT
Introduction
2.1 Rates of Change and the Derivative
2.1 Assess Your Understanding
2.2 The Derivative as a Function
2.2 Assess Your Understanding
2.3 The Derivative of a Polynomial Function; The Derivative of y=ex
2.3 Assess Your Understanding
2.4 Differentiating the Product and the Quotient of Two Functions; Higher-Order Derivatives
2.4 Assess Your Understanding
2.5 The Derivative of the Trigonometric Functions
2.5 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 2 Project
Introduction
3.1 The Chain Rule
3.1 Assess Your Understanding
3.2 Implicit Differentiation; Derivatives of the Inverse Trigonometric Functions
3.2 Assess Your Understanding
3.3 Derivatives of Logarithmic Functions
3.3 Assess Your Understanding
3.4 Differentials; Linear Approximations; Newton’s Method
3.4 Assess Your Understanding
3.5 Taylor Polynomials
3.5 Assess Your Understanding
3.6 Hyperbolic Functions
3.6 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 3 PROJECT
Introduction
4.1 Related Rates
4.1 Assess Your Understanding
4.2 Maximum and Minimum Values; Critical Numbers
4.2 Assess Your Understanding
4.3 The Mean Value Theorem
4.3 Assess Your Understanding
4.4 Local Extrema and Concavity
4.4 Assess Your Understanding
4.5 Indeterminate Forms and L’Hôpital’s Rule
4.5 Assess Your Understanding
4.6 Using Calculus to Graph Functions
4.6 Assess Your Understanding
4.7 Optimization
4.7 Assess Your Understanding
4.8 Antiderivatives; Differential Equations
4.8 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 4 PROJECT
Introduction
5.1 Area
5.1 Assess Your Understanding
5.2 The Definite Integral
5.2 Assess Your Understanding
5.3 The Fundamental Theorem of Calculus
5.3 Assess Your Understanding
5.4 Properties of the Definite Integral
5.4 Assess Your Understanding
5.5 The Indefinite Integral; Growth and Decay Models
5.5 Assess Your Understanding
5.6 Method of Substitution; Newton's Law of Cooling
5.6 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
Chapter 5 Project
Introduction
6.1 Area Between Graphs
6.1 Assess Your Understanding
6.2 Volume of a Solid of Revolution: Disks and Washers
6.2 Assess Your Understanding
6.3 Volume of a Solid of Revolution: Cylindrical Shells
6.3 Assess Your Understanding
6.4 Volume of a Solid: Slicing Method
6.4 Assess Your Understanding
6.5 Arc Length
6.5 Assess Your Understanding
6.6 Work
6.6 Assess Your Understanding
6.7 Hydrostatic Pressure and Force
6.7 Assess Your Understanding
6.8 Center of Mass; Centroid; the Pappus Theorem
6.8 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 6 PROJECT
Introduction
7.1 Integration by Parts
7.1 Assess Your Understanding
7.2 Integrals Containing Trigonometric Functions
7.2 Assess Your Understanding
7.3 Integration Using Trigonometric Substitution: Integrands Containing √(a2-x2), √(x2+a2), or √(x2-a2), a>0
7.3 Assess Your Understanding
7.4 Substitution: Integrands Containing ax2+bx+c
7.4 Assess Your Understanding
7.5 Integration of Rational Functions Using Partial Fractions
7.5 Assess Your Understanding
7.6 Integration Using Numerical Techniques
7.6 Assess Your Understanding
7.7 Integration Using Tables and Computer Algebra Systems
7.7 Assess Your Understanding
7.8 Improper Integrals
7.8 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 7 PROJECT
Introduction
8.1 Sequences
8.1 Assess Your Understanding
8.2 Infinite Series
8.2 Assess Your Understanding
8.3 Properties of Series; the Integral Test
8.3 Assess Your Understanding
8.4 Comparison Tests
8.4 Assess Your Understanding
8.5 Alternating Series; Absolute Convergence
8.5 Assess Your Understanding
8.6 Ratio Test; Root Test
8.6 Assess Your Understanding
8.7 Summary of Tests
8.7 Assess Your Understanding
8.8 Power Series
8.8 Assess Your Understanding
8.9 Taylor Series; Maclaurin Series
8.9 Assess Your Understanding
8.10 Approximations Using Taylor/Maclaurin Expansions
8.10 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 8 PROJECT
Introduction
9.1 Parametric Equations
9.1 Assess Your Understanding
9.2 Tangent Lines; Arc Length
9.2 Assess Your Understanding
9.3 Surface Area of a Solid of Revolution
9.3 Assess Your Understanding
9.4 Polar Coordinates
9.4 Assess Your Understanding
9.5 Polar Equations; Parametric Equations of Polar Equations; Arc Length of Polar Equations
9.5 Assess Your Understanding
9.6 Area in Polar Coordinates
9.6 Assess Your Understanding
9.7 The Polar Equation of a Conic
9.7 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 9 PROJECT
Introduction
10.1 Rectangular Coordinates in Space
10.1 Assess Your Understanding
10.2 Introduction to Vectors
10.2 Assess Your Understanding
10.3 Vectors in the Plane and in Space
10.3 Assess Your Understanding
10.4 The Dot Product
10.4 Assess Your Understanding
10.5 The Cross Product
10.5 Assess Your Understanding
10.6 Equations of Lines and Planes in Space
10.6 Assess Your Understanding
10.7 Quadric Surfaces
10.7 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 10 PROJECT
Introduction
11.1 Vector Functions and Their Derivatives
11.1 Assess Your Understanding
11.2 Unit Tangent and Principal Unit Normal Vectors; Arc Length
11.2 Assess Your Understanding
11.3 Arc Length as Parameter; Curvature
11.3 Assess Your Understanding
11.4 Motion Along a Curve
11.4 Assess Your Understanding
11.5 Integrals of Vector Functions; Projectile Motion
11.5 Assess Your Understanding
11.6 Application: Kepler's Laws of Planetary Motion
11.6 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 11 PROJECT
Introduction
12.1 Functions of Two or More Variables and Their Graphs
12.1 Assess Your Understanding
12.2 Limits and Continuity
12.2 Assess Your Understanding
12.3 Partial Derivatives
12.3 Assess Your Understanding
12.4 Differentiability and the Differential
12.4 Assess Your Understanding
12.5 Chain Rules
12.5 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 12 PROJECT
Introduction
13.1 Directional Derivatives; Gradients
13.1 Assess Your Understanding
13.2 Tangent Planes
13.2 Assess Your Understanding
13.3 Extrema of Functions of Two Variables
13.3 Assess Your Understanding
13.4 Lagrange Multipliers
13.4 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 13 PROJECT
Introduction
14.1 The Double Integral over a Rectangular Region
14.1 Assess Your Understanding
14.2 The Double Integral over Nonrectangular Regions
14.2 Assess Your Understanding
14.3 Double Integrals Using Polar Coordinates
14.3 Assess Your Understanding
14.4 Center of Mass; Moment of Inertia
14.4 Assess Your Understanding
14.5 Surface Area
14.5 Assess Your Understanding
14.6 The Triple Integral
14.6 Assess Your Understanding
14.7 Triple Integrals Using Cylindrical Coordinates
14.7 Assess Your Understanding
14.8 Triple Integrals Using Spherical Coordinates
14.8 Assess Your Understanding
14.9 Change of Variables Using Jacobians
14.9 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 14 PROJECT
Introduction
15.1 Vector Fields
15.1 Assess Your Understanding
15.2 Line Integrals
15.2 Assess Your Understanding
15.3 Fundamental Theorem of Line Integrals
15.3 Assess Your Understanding
15.4 An Application of Line Integrals: Work
15.4 Assess Your Understanding
15.5 Green’s Theorem
15.5 Assess Your Understanding
15.6 Parametric Surfaces
15.6 Assess Your Understanding
15.7 Surface and Flux Integrals
15.7 Assess Your Understanding
15.8 The Divergence Theorem
15.8 Assess Your Understanding
15.9 Stokes' Theorem
15.9 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 15 PROJECT
Introduction
16.1 Classification of Ordinary Differential Equations
16.1 Assess Your Understanding
16.2 Separation of Variables in First-Order Differential Equations
16.2 Assess Your Understanding
16.3 Exact Differential Equations
16.3 Assess Your Understanding
16.4 First-Order Linear Differential Equations; Bernoulli Differential Equations
16.4 Assess Your Understanding
16.5 Power Series Methods
16.5 Assess Your Understanding
Chapter Review
REVIEW EXERCISES
CHAPTER 16 PROJECT
Introduction
P.1 Functions and Their Graphs
P.1 Assess Your Understanding
P.2 Library of Functions; Mathematical Modeling
P.2 Assess Your Understanding
P.3 Operations on Functions; Graphing Techniques
P.3 Assess Your Understanding
P.4 Inverse Functions
P.4 Assess Your Understanding
P.5 Exponential and Logarithmic Functions
P.5 Assess Your Understanding
P.6 Trigonometric Functions
P.6 Assess Your Understanding
P.7 Inverse Trigonometric Functions
P.7 Assess Your Understanding
P.8 Technology Used in Calculus
Chapter P
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Introduction
A.1 Algebra Used in Calculus
A.2 Geometry Used in Calculus
A.3 Analytic Geometry Used in Calculus
A.4 Trigonometry Used in Calculus
A.5 Sequences; Summation Notation; the Binomial Theorem
Introduction
B.1 Limit Theorems and Proofs
B.2 Theorems and Proofs Involving Inverse Functions
B.3 Derivative Theorems and Proofs
B.4 Integral Theorems and Proofs
B.5 A Bounded Monotonic Sequence Converges
B.6 Taylor's Formula with Remainder
Photo Credits
Index
Calculus—What Is It?
Table of Derivatives
Table of Integrals