Calculus

Front Cover
Title Page
Copyright Page
Dedication
Contents
Preface
1 Preparations for Calculus
	1.1 Real Numbers and Sets
	1.2 Inequalities and Absolute Values
	1.3 Coordinates
	1.4 Lines and Slopes
	1.5 Circles and Parabolas
	1.6 The Ellipse and Hyperbola
	1.7 Polar Coordinates
	1.8 Language
2 Functions and Limits
	2.1 Examples of Functions
	2.2 Graphs of Functions
	2.3 Composite Functions
	2.4 Bounds and Limits
	2.5 Limits of Functions
	2.6 Limit Theorems
	2.7 Limits at Infinity
	2.8 pi and the Circle
	2.9 Trigonometric Functions
3 Continuity and the Derivative
	3.1 Continuous Functions
	3.2 Definition of Derivative
	3.3 Product and Quotient Rules
	3.4 Derivatives of Trigonometric Functions
	3.5 The Chain Rule
	3.6 Derivatives of Implicit Functions
4 Applications of the Derivative
	4.1 Extrema of Functions
	4.2 The Mean Value Theorem
	4.3 Applications of the Mean Value Theorem
	4.4 Indeterminate Forms
	4.5 Maximum and Minimum Function Values
	4.6 More Maximum and Minimum Problems
	4.7 The Second Derivative
	4.8 Concavity
	4.9 Velocity and Acceleration
	4.10 Motion in the Plane
	4.11 Related Rates
5 The Definite Integral
	5.1 Summation Notation
	5.2 Lower and Upper Sums
	5.3 Definition of the Definite Integral
	5.4 Proof of the Existence of the Definite Integral
	5.5 Riemann Sums
	5.6 Proof of the Fundamental Theorem of Calculus
	5.7 Computation of Areas
	5.8 Indefinite Integrals
	5.9 Integration by Substitution
	5.10 Areas in Polar Coordinates
6 Computations Using the Definite Integral
	6.1 Volumes of Certain Solids
	6.2 Work
	6.3 Arc Length
	6.4 Surface Area of Solids of Revolution
	6.5 Moments and Center of Mass
7 Transcendental Functions
	7.1 Inverse Functions
	7.2 The Natural Logarithm Function
	7.3 Derivatives and Integrals Involving ln(x)
	7.4 The Exponential Function
	7.5 The Derivative of the Exponential Function
	7.6 Applications of the Exponential Function
	7.7 Indeterminate Forms
	7.8 Inverse Trigonometric Functions
	7.9 Derivatives of the Inverse Trigonometric Functions
	7.10 Improper Integrals
8 Methods of Integration
	8.1 Integral Formulas and Integral Tables
	8.2 Integration by Parts
	8.3 Reduction Formulas
	8.4 Partial Fraction Decomposition of Rational Functions
	8.5 Integration of Rational Functions
	8.6 Integrals of Algebraic Functions
	8.7 Trigonometric and Other Substitutions
9 Taylor Polynomials and Sequences
	9.1 Taylor Polynomials
	9.2 Sequences
	9.3 Limits of Sequences
10 Power Series
	10.1 Taylor Series
	10.2 Convergence of Infinite Series
	10.3 The Interval of Convergence
	10.4 Differentiation and Integration of Series
	10.5 Computation of Taylor Series
	10.6 Applications of Power Series
	10.7 Additional Convergence Tests
	10.8 Alternating Series and Conditional Convergence
	10.9 The Hyperbolic and Binomial Series
11 Numerical Computations
	11.1 Solution of Equations
	11.2 Numerical Integration
	11.3 Simpson's Rule
12 Vectors in Two and Three Dimensions
	12.1 Coordinates in Three Dimensions
	12.2 Vectors
	12.3 Coordinates for Vectors in Three Dimensions
	12.4 The Dot Product
	12.5 The Cross Product
	12.6 Equations of Lines
	12.7 Equations of Planes
13 Vector Functions
	13.1 Vector Functions
	13.2 Integral of a Vector Function
	13.3 Curves in Parametric and Vector Form
	13.4 Tangents and Normals to Curves
	13.5 Arc Length in Two and Three Dimensions
	13.6 Curvature
	13.7 Curves in Three Dimensions
	13.8 Motion Along a Curve
	13.9 Planetary Motion
14 Partial Derivatives
	14.1 Surfaces in Three Dimensions
	14.2 Quadric Surfaces
	14.3 Functions of Several Variables
	14.4 Partial Derivatives
	14.5 Limits and Continuity
	14.6 The Chain Rule
	14.7 The Gradient
	14.8 Directional Derivatives
	14.9 Extrema of Functions of Several Variables
	14.10 Extrema With Constraints
	14.11 Properties of Continuous Functions
15 Integration in Higher Dimensions
	15.1 Double Integrals
	15.2 Iterated Integrals
	15.3 More Volumes Using Double Integrals
	15.4 Center of Mass and Moments of Inertia
	15.5 Surface Area
	15.6 Triple Integrals
	15.7 Line Integrals
	15.8 Path-Independent Line Integrals
	15.9 Green's Theorem
	15.10 Change of Variables
	15.11 Triple Integrals by Spherical Coordinates
16 Two Theorems in Vector Calculus
	16.1 Oriented Surfaces and Stokes' Theorem
	16.2 The Curl and Divergence
	16.3 The Divergence Theorem
Appendix A: Answers to Selected Exercises
	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
Index
Back Cover