Chapter 0: Preliminaries
0.1 The Real Numbers and the Cartesian Plane
0.2 Lines and Functions
0.3 Graphing Calculators and Computer Algebra Systems
0.4 Trigonometric Functions
0.5 Transformations of Functions
Chapter 1: Limits and Continuity
1.1 A Brief Preview of Calculus: Tangent Lines and the Length of a Curve
1.2 The Concept of Limit
1.3 Computation of Limits
1.4 Continuity and its Consequences
The Method of Bisections
1.5 Limits Involving Infinity
Asysmptotes
1.6 The Formal Definition of the Limit
1.7 Limits and Loss-of-Significance Errors
Computer Representation or Real Numbers
Chaper 2: Differentiation
2.1 Tangent Lines and Velocity
2.2 The Derivative
Alternative Derivative Notations
Numerical Differentiation
2.3 Computation of Derivatives: The Power Rule
Higher Order Derivatives
Acceleration
2.4 The Product and Quotient Rules
2.5 The Chain Rule
2.6 Derivatives of the Trigonometric Functions
2.7 Implicit Differentiation
2.8 The Mean Value Theorem
Chapter 3: Applications of Differentiation
3.1 Linear Approximations and Newton's Method
3.2 Maximum and Minimum Values
3.3 Increasing and Decreasing Functions
3.4 Concavity and the Second Derivative Test
3.5Overview of Curve Sketching
3.6Optimization
3.8Related Rates
3.8Rates of Change in Economics and the Sciences
Chapter 4: Integration
4.1 Antiderivatives
4.2 Sums and Sigma Notation
Principle of Mathematical Induction
4.3 Area under a Curve
4.4 The Definite Integral
Average Value of a Function
4.5 The Fundamental Theorem of Calculus
4.6 Integration by Substitution
4.7 Numerical Integration
Error bounds for Numerical Integration
Chapter 5: Applications of the Definite Integral
5.1 Area Between Curves
5.2 Volume: Slicing, Disks, and Washers
5.3 Volumes by Cylindrical Shells
5.4 Arc Length and Srface Area
5.5 Projectile Motion
5.6 Applications of Integration to Physics and Engineering
Chapter 6: Exponentials, Logarithms and other Transcendental Functions
6.1 The Natural Logarithm
6.2 Inverse Functions
6.3 Exponentials
6.4 The Inverse Trigonometric Functions
6.5 The Calculus of the Inverse Trigonometric Functions
6.6 The Hyperbolic Function
Chapter 7: First-Order Differential Equations
7.1 Modeling with Differential Equations
Growth and Decay Problems
Compound Interest
7.2 Separable Differential Equations
Logistic Growth
7.3 Direction Fields and Euler's Method
7.4 Systems of First-Order Differential Equations
Predator-Prey Systems
7.6 Indeterminate Forms and L'Hopital's Rule
Improper Integrals
A Comparison Test
7.8 Probability
Chapter 8: First-Order Differential Equations
8.1 modeling with Differential Equations
Growth and Decay Problems
Compound Interest
8.2 Separable Differential Equations
Logistic Growth
8.3 Direction Fields and Euler's Method
Systems of First Order Equations
Chapter 9: Infinite Series
9.1 Sequences of Real Numbers
9.2 Infinite Series
9.3 The Integral Test and Comparison Tests
9.4 Alternating Series
Estimating the Sum of an Alternating Series
9.5 Absolute Convergence and the Ratio Test
The Root Test
Summary of Convergence Test
9.6 Power Series
9.7 Taylor Series
Representations of Functions as Series
Proof of Taylor's Theorem
9.8 Applications of Taylor Series
The Binomial Series
9.9 Fourier Series
Chapter 10: Parametric Equations and Polar Coordinates
10.1 Plane Curves and Parametric Equations
10.2 Calculus and Parametric Equations
10.3 Arc Length and Surface Area in Parametric Equations
10.4 Polar Coordinates
10.5 Calculus and Polar Coordinates
10.6 Conic Sections
10.7 Conic Sections in Polar Coordinates
Chapter 11: Vectors and the Geometry of Space
11.1 Vectors in the Plane
11.2 Vectors in Space
11.3 The Dot Product
Components and Projections
11.4 The Cross Product
11.5 Lines and Planes in Space
11.6 Surfaces in Space
Chapter 12: Vector-Valued Functions
12.1 Vector-Valued Functions
12.2 The Calculus Vector-Valued Functions
12.3 Motion in Space
12.4 Curvature
12.5 Tangent and Normal Vectors
Components of Acceleration, Kepler's Laws
11.6 Parametric Surfaces
Chapter 13: Functions of Several Variables and Partial Differentiation
13.1 Functions of Several Variables
13.2 Limits and Continuity
13.3 Partial Derivatives
13.4 Tangent Planes and Linear Approximations
Increments and Differentials
13.5 The Chain Rule
Implicit Differentiation
13.6 The Gradient and Directional Derivatives
13.7 Extrema of Functions of Several Variables
13.8 Constrained Optimization and Lagrange Multipliers
Chapter 14: Multiple Integrals
14.1 Double Integrals
14.2 Area, Volume, and Center of Mass
14.3 Double Integrals in Polar Coordinates
14.4 Surface Area
14.5 Triple Integrals
Mass and Center of Mass
14.6 Cylindrical Coordinates
14.7 Spherical Coordinates
14.8 Change of Variables in Multiple Integrals
Chapter 15: Vector Calculus
15.1 Vector Fields
15.2 Line Integrals
15.3 Independence of Path and Conservative Vector Fields
15.4 Green's Theorem
15.5 Curl and Divergence
15.6 Surface Integrals
15.7 The Divergence Theorem
15.8 Stokes' Theorem
15.9 Applications of Vector Calculus
Chapter 16: Second-Order Differential Equations
16.1 Second-Order Equations with Constant Coefficients
16.2 Nonhomogeneous Equations: Undetermined Coefficients
16.3 Applications of Second-Order Differential Equations
16.4 Power Series Solutions of Differential Equations
Appendix A: Proofs of Selected Theorems
Appendix B: Answers to Odd-Numbered Exercises
