Calculus Book

Chapter 1 PRECALCULUS REVIEW
1.1 Real Numbers, Functions, and Graphs
1.2 Linear and Quadratic Functions
1.3 The Basic Classes of Functions
1.4 Trigonometric Functions
1.5 Technology: Calculators and Computers

Chapter 2 LIMITS
2.1 Limits, Rates of Change, and Tangent Lines
2.2 Limits: A Numerical and Graphical Approach
2.3 Basic Limit Laws
2.4 Limits and Continuity
2.5 Evaluating Limits Algebraically
2.6 Trigonometric Limits
2.7 Intermediate Value Theorem
2.8 The Formal Definition of a Limit

Chapter 3 DIFFERENTIATION
3.1 Definition of the Derivative
3.2 The Derivative as a Function
3.3 Product and Quotient Rules
3.4 Rates of Change
3.5 Higher Derivatives
3.6 Trigonometric Functions
3.7 The Chain Rule
3.8 Implicit Differentiation
3.9 Related Rates

Chapter 4 APPLICATIONS OF THE DERIVATIVE
4.1 Linear Approximation and Applications
4.2 Extreme Values
4.3 The Mean Value Theorem and Monotonicity
4.4 The Shape of a Graph
4.5 Graph Sketching and Asymptotes
4.6 Applied Optimization
4.7 Newton's Method
4.8 Antiderivatives

Chapter 5 THE INTEGRAL
5.1 Approximating and Computing Area
5.2 The Definite Integral
5.3 The Fundamental Theorem of Calculus, Part I
5.4 The Fundamental Theorem of Calculus, Part II
5.5 Net or Total Change as the Integral of a Rate
5.6 Substitution Method

Chapter 6 APPLICATIONS OF THE INTEGRAL
6.1 Area Between Two Curves
6.2 Setting Up Integrals: Volume, Density, Average Value
6.3 Volumes of Revolution
6.4 The Method of Cylindrical Shells
6.5 Work and Energy

Chapter 7 EXPONENTIAL FUNCTIONS
7.1 Derivative of f(x)=b^x and the Number e
7.2 Inverse Functions
7.3 Logarithms and their Derivatives
7.4 Exponential Growth and Decay
7.5 Compound Interest and Present Value
7.6 Models Involving y'= k(y-b)
7.7 L'Hoˆpital's Rule
7.8 Inverse Trigonometric Functions
7.9 Hyperbolic Functions

Chapter 8 TECHNIQUES OF INTEGRATION
8.1 Numerical Integration
8.2 Integration by Parts
8.3 Trigonometric Integrals
8.4 Trigonometric Substitution
8.5 The Method of Partial Fractions
8.6 Improper Integrals

Chapter 9 FURTHER APPLICATIONS OF THE INTEGRAGAL TAYLOR POLYNOMIALS
9.1 Arc Length and Surface Area
9.2 Fluid Pressure and Force
9.3 Center of Mass
9.4 Taylor Polynomials

Chapter 10 INTRODUCTION TO DIFFERENTIAL EQUATIONS
10.1 Solving Differential Equations
10.2 Graphical and Numerical Methods
10.3 The Logistic Equation
10.4 First-Order Linear Equations

Chapter 11 INFINITE SERIES
11.1 Sequences
11.2 Summing an Infinite Series
11.3 Convergence of Series with Positive Terms
11.4 Absolute and Conditional Convergence
11.5 The Ratio and Root Tests
11.6 Power Series
11.7 Taylor Series

Chapter 12 PARAMETRIC EQUATIONS, POLAR COORDINATES, AND CONIC SECTIONS
12.1 Parametric Equations
12.2 Arc Length and Speed
12.3 Polar Coordinates
12.4 Area and Arc Length in Polar Coordinates
12.5 Conic Sections

Chapter 13 VECTOR GEOMETRY
13.1 Vectors in the Plane
13.2 Vectors in Three Dimensions
13.3 Dot Product and the Angle Between Two Vectors
13.4 The Cross Product
13.5 Planes in Three-Space
13.6 Survey of Quadric Surfaces
13.7 Cylindrical and Spherical Coordinates

Chapter 14 CALCULUS OF VECTOR-VALUED FUNCTIONS
14.1 Vector-Valued Functions
14.2 Calculus of Vector-Valued Functions
14.3 Arc Length and Speed
14.4 Curvature
14.5 Motion in Three-Space
14.6 Planetary Motion According to Kepler and Newton

Chapter 15 DIFFERENTIATION IN SEVERAL VARIABLES
15.1 Functions of Two or More Variables
15.2 Limits and Continuity in Several Variables
15.3 Partial Derivatives
15.4 Differentiability, Linear Approximation,and Tangent Planes
15.5 The Gradient and Directional Derivatives
15.6 The Chain Rule
15.7 Optimization in Several Variables
15.8 Lagrange Multipliers: Optimizing with a Constraint

Chapter 16 MULTIPLE INTEGRATION
16.1 Integration in Several Variables
16.2 Double Integrals over More General Regions
16.3 Triple Integrals
16.4 Integration in Polar, Cylindrical, and Spherical Coordinates
16.5 Change of Variables

Chapter 17 LINE AND SURFACE INTEGRALS
17.1 Vector Fields
17.2 Line Integrals
17.3 Conservative Vector Fields
17.4 Parametrized Surfaces and Surface Integrals
17.5 Surface Integrals of Vector Fields

Chapter 18 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS
18.1 Green's Theorem
18.2 Stokes' Theorem
18.3 Divergence Theorem

APPENDICES
A. The Language of Mathematics B. Properties of Real Numbers C. Mathematical Induction and the BinomialTheorem D. Additional Proofs of Theorems

ANSWERS TO ODD-NUMBERED EXERCISES