Thomas' Calculus

Cover
Contents
Preface
1 - Functions
1.1 - Functions and Their Graphs
1.2 - Combining Functions; Shifting and Scaling Graphs
1.3 - Trigonometric Functions
1.4 - Graphing with Calculators and Computers
Review
2 - Limits and Continuity
2.1 - Rates of Change and Tangents to Curves
2.2 - Limit of a Function and Limit Laws
2.3 - The Precise Definition of a Limit
2.4 - One-Sided Limits
2.5 - Continuity
2.6 - Limits Involving Infinity; Asymptotes of Graphs
Review
3 - Differentiation
3.1 - Tangents 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 - Related Rates
3.9 - Linearization and Differentials
Review
4 - Applications of Derivatives
4.1 - Extreme Values of Functions
4.2 - The Mean Value Theorem
4.3 - Monotonic Functions and the First Derivative Test
4.4 - Concavity and Curve Sketching
4.5 - Applied Optimization
4.6 - Newton's Method
4.7 - Antiderivatives
Review
5 - Integration
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 - Substitution and Area Between Curves
Review
6 - Applications of Definite Integrals
6.1 - Volumes Using Cross-Sections
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
Review
7 - Transcendental Functions
7.1 - Inverse Functions and Their Derivatives
7.2 - Natural Logarithms
7.3 - Exponential Functions
7.4 - Exponential Change and Seperable Differential Equations
7.5 - Indeterminate Forms and L'Hopital's Rule
7.6 - Inverse Trigonometric Functions
7.7 - Hyperbolic Functions
7.8 - Relative Rates of Growth
Review
8 - Techniques of Integration
8.1 - Integration by Parts
8.2 - Trigonometric Integrals
8.3 - Trigonometric Substitutions
8.4 - Integration of Rational Functions by Partial Fractions
8.5 - Integral Tables and Computer Algebra Systems
8.6 - NumericaL Integration
8.7 - Improper Integrals
Review
9 - First-Order Differential Equations
9.1 - Solutions, Slope Fields, and Euler's Method
9.2 - First-Order Linear Equations
9.3 - Applications
9.4 - Graphical Solutions of Autonomous Equations
9.5 - Systems of Equations and Phase Planes
Review
10 - Infinite Sequences and Series
10.1 - Sequences
10.2 - Infinite Series
10.3 - The Integral Test
10.4 - Comparison Tests
10.5 - The Ratio and Root Tests
10.6 - Alternating Series, Absolute and Conditional Convergence
10.7 - Power Series
10.8 - Taylor and Maclaurin Series
10.9 - Convergence of Taylor Series
10.10 - The Binomial Series and Applications of Taylor Series
Review
11 - Parametric Equations and Polar Coordinates
11.1 - Parametrizations of Plrane Curves
11.2 - Calculus with Parametric Curves
11.3 - Polar Coordinates
11.4 - Graphing in Polarr Coordinates
11.5 - Areas and Lengths in Polar Coordinates
11.6 - Conic Sections
11.7 - Conics in Polar Coordinates
Review
12 - Vectors and the Geometry of Space
12.1 - Three-Dimensional 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
Review
13 - Vector-Valued 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
Review
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
Review
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 - Moments and Centers of Mass
15.7 - Triple Integrals in Cylindrical and Spherical Coordinates
15.8 - Substitutions in Multiple Integrals
Review
16 - Integration in Vector Fields
16.1 - Line Integrals
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 - Stoke's Theorem
16.8 - The Divergence Theorem and a Unified Theory
Review
17 - Second-Order Differential Equations
17.1 - Second-Order Linear Equations
17.2 - Nonhomogeneous Linear Equations
17.3 - Applications
17.4 - Euler Equations
17.5 - Power-Series Solutions
Appendices
A.1 - Real Numbers and the Real Line
A.2 - Mathematical Induction
A.3 - Lines, Circles, and Parabolas
A.4 - Proofs of Limit Theorems
A.5 - Commonly Occurring Limits
A.6 - Theory of the Real Numbers
A.7 - Complex Numbers
A.8 - The Distributive Law for Vector Cross Products
A.9 - The Mixed Derivative Theorem and the Increment Theorem
Answers to Odd-Numbered 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
Appendices
Chapter 17
Index
A Brief Table of Integrals
Credits
Formulas
Basic Algebra Formulas
Geometry Formulas
Limits
Differentiation Rules
Integration Rules
Trigonometry Formulas
Series
Vector Operator Formulas (Cartesian Form)