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
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