Note: Each chapter concludes with Review Exercises and P.S. Problem Solving.
P. Preparation for Calculus
Eruptions of Old Faithful
P.1 Graphs and Models
P.2 Linear Models and Rates of Changes
P.3 Functions and Their Graphs
P.4 Fitting Models to Data
P.5 Inverse Functions
P.6 Exponential and Logarithmic Functions
1. Limits and Their Properties
Swimming Speed: Taking it to the Limit
1.1 A Preview of Calculus
1.2 Finding Limits Graphically and Numerically
1.3 Evaluating Limits Analytically
1.4 Continuity and One-Sided Limits
1.5 Infinite Limits
Section Project: Graphs and Limits of Trigonometric Functions
Review Exercises
P.S. Problem Solving
2. Differentiation
Gravity: Finding it Experimentally
2.1 The Derivative and the Tangent Line Problem
2.2 Basic Differentiation Rules and Rates of Change
2.3 The Product and Quotient Rules and Higher Order Derivatives
2.4 The Chain Rule
2.5 Implicit Differentiation
Section Project: Optical Illusions
2.6 Derivatives of Inverse Functions
2.7 Related Rates
2.8 Newton's Method
3. Applications of Differentiation
Packaging: The Optimal Form
3.1 Extrema on an Interval
3.2 Rolle's Theorem and the Mean Value Theorem
3.3 Increasing and Decreasing Functions and the First Derivative Test
Section Project: Rainbows
3.4 Concavity and the Second Derivative Test
3.5 Limits at Infinity
3.6 A Summary of Curve Sketching
3.7 Optimization Problems
Section Project: Connecticut River
3.8 Differentials
4. Integration
The Wankel Rotary Engine and Area
4.1 Antiderivatives and Indefinite Integration
4.2 Area
4.3 Riemann Sums and Definite Integrals
4.4 The Fundamental Theorem of Calculus
Section Project: Demonstrating the Fundamental Theorem
4.5 Integration by Substitution
4.6 Numerical Integration
4.7 The Natural Logarithmic Functions: Integration
4.8 Inverse Trigonometric Functions: Integration
4.9 Hyperbolic Functions
Section Project:St. Louis Arch
5. Differential Equations
Plastics and Cooling
5.1 Differential Equations: Growth and Decay
5.2 Differential Equations: Separation of Variables
5.3 First-Order Linear Differential Equations
Section Project: Weight Loss
6. Applications of Integration
Constructing an Arch Dam
6.1 Area of a Region Between Two Curves
6.2 Volume: The Disk Method
6.3 Volume: The Shell Method
Section Project: Saturn
6.4 Arc Length and Surfaces of Revolution
6.5 Work
Section Project: Tidal Energy
6.6 Moments, Centers of Mass, and Centroids
6.7 Fluid Pressure and Fluid Force
7. Integration Techniques, LHopital;s Rule, and Improper
Integrals
Making a Mercator Map
7.1 Basic Integration Rules
7.2 Integration by Parts
7.3 Trigonometric Integrals
Section Project: Power Lines
7.4 Trigonometric Substitution
7.5 Partial Fractions
7.6 Integration by Tables and Other Integration Techniques
7.7 Indeterminate Forms and LHopitals Rule
7.8 Improper Integrals 8. Infinite Series
The Koch Snowflake: Infinite Perimeter?
8.1 Sequences
8.2 Series and Convergence
Section Project: Cantor's Disappearing Table
8.3 The Integral Test and p-Series
Section Project: The Harmonic Series
8.4 Comparisons of Series
Section Project: Solera Method
8.5 Alternating Series
8.6 The Ratio and Root Tests
8.7 Taylor Polynomials and Approximations
8.8 Power Series
8.9 Representation of Functions by Power Series
8.10 Taylor and Maclaurin Series
9. Conics, Parametric Equations, and Polar Coordinates
Exploring New Planets
9.1 Conics and Calculus
9.2 Plane Curves and Parametric Equations
Section Project: Cycloids
9.3 Parametric Equations and Calculus
9.4 Polar Coordinates and Polar Graphs
Section Project: Anamorphic Art
9.5 Area and Arc Length in Polar Coordinates
9.6 Polar Equations of Conics and Kepler's Laws
10. Vectors and the Geometry of Space
Suspension Bridges
10.1 Vectors in the Plane
10.2 Space Coordinates and Vectors in Space
10.3 The Dot Product of Two Vectors
10.4 The Cross Product of Two Vectors in Space
10.5 Lines and Planes in Space
Section Project: Distances in Space
10.6 Surfaces in Space
10.7 Cylindrical and Spherical Coordinates
11. Vector-Valued Functions
Race Car Cornering
11.1 Vector-Valued Functions
Section Project: Witch of Agnesi
11.2 Differentiation and Integration of Vector-Valued Functions
11.3 Velocity and Acceleration
11.4 Tangent Vectors and Normal Vectors
11.5 Arc Length and Curvature
12. Functions of Several Variables
Satellite Receiving Dish
12.1 Introduction to Functions of Several Variables
12.2 Limits and Continuity
12.3 Partial Derivatives
Section Project: Moire Fringes
12.4 Differentials
12.5 Chain Rules for Functions of Several Variables
12.6 Directional Derivatives and Gradients
12.7 Tangent Planes and Normal Lines
Section Project: Wildflowers
12.8 Extrema of Functions of Two Variables
12.9 Applications of Extrema of Functions of Two Variables
Section Project: Building a Pipeline
12.10 Lagrange Multipliers
13. Multiple Integration
Hyperthermia Treatments for Tumors
13.1 Iterated Integrals and Area in the Plane
13.2 Double Integrals and Volume
13.3 Change of Variables: Polar Coordinates
13.4 Center of Mass and Moments of Inertia
Section Project: Center of Pressure on a Sail
13.5 Surface Area
Section Project: Capillary Action
13.6 Triple Integrals and Applications
13.7 Triple Integrals in Cylindrical and Spherical Coordinates
Section Project: Wrinkled and Bumpy Spheres
13.8 Change of Variables: Jacobians
14. Vector Analysis
Mathematical Sculpture
14.1 Vector Fields
14.2 Line Integrals
14.3 Conservative Vector Fields and Independence of Path
14.4 Green's Theorem
Section Project: Hyperbolic and Trigonometric Functions
14.5 Parametric Surfaces
14.6 Surface Integrals
Section Project: Hyperboloid of One Sheet
14.7 Divergence Theorem
14.8 Stoke's Theorem
Section Project: The Planimeter
Appendices
A. Business and Economic Applications
B. Proofs of Selected Theorems
C. Integration Tables
D. Precalculus Review
E. Rotation and the General Second-Degree Equation
F. Complex
