Table of Contents

1 Preparation for Calculus
 1.1 Graphs and Models
 1.2 Linear Models and Rates of Change
 1.3 Functions and Their Graphs
 1.4 Fitting Models to Data
 1.5 Inverse Functions
 1.6 Exponential and Logarithmic Functions

2 Limits and Their Properties
 2.1 A Preview of Calculus
 2.2 Finding Limits Graphically and Numerically
 2.3 Evaluating Limits Analytically
 2.4 Continuity and OneSided Limits
 2.5 Infinite Limits

3 Differentiation
 3.1 The Derivative and the Tangent Line Problem
 3.2 Basic Differentiation Rules and Rates of Change
 3.3 Product and Quotient Rules and HigherOrder Derivatives
 3.4 The Chain Rule
 3.5 Implicit Differentiation
 3.6 Derivatives of Inverse Functions
 3.7 Related Rates
 3.8 Newton’s Method

4 Applications of Differentiation
 4.1 Extrema on an Interval
 4.2 Rolle’s Theorem and the Mean Value Theorem
 4.3 Increasing and Decreasing Functions and the First Derivative Test
 4.4 Concavity and the Second Derivative Test
 4.5 Limits at Infinity
 4.6 A Summary of Curve Sketching
 4.7 Optimization Problems 256
 4.8 Differentials

5 Integration
 5.1 Antiderivatives and Indefinite Integration

5.2 Area
 5.3 Riemann Sums and Definite Integrals
 5.4 The Fundamental Theorem of Calculus
 5.5 Integration by Substitution
 5.6 Numerical Integration
 5.7 The Natural Logarithmic Function: Integration
 5.8 Inverse Trigonometric Functions: Integration
 5.9 Hyperbolic Functions

6 Differential Equations

6.1 Slope Fields and Euler’s Method
 6.2 Differential Equations: Growth and Decay
 6.3 Differential Equations: Separation of Variables
 6.4 The Logistic Equation
 6.5 FirstOrder Linear Differential Equations
 6.6 PredatorPrey Differential Equations

6.1 Slope Fields and Euler’s Method

7 Applications of Integration
 7.1 Area of a Region between Two Curves
 7.2 Volume: The Disk Method
 7.3 Volume: The Shell Method
 7.4 Arc Length and Surfaces of Revolution
 7.5 Work
 7.6 Moments, Centers of Mass, and Centroids
 7.7 Fluid Pressure and Fluid Force

8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals
 8.1 Basic Integration Rules
 8.2 Integration by Parts
 8.3 Trigonometric Integrals
 8.4 Trigonometric Substitution
 8.5 Partial Fractions
 8.6 Integration by Tables and Other Integration Techniques
 8.7 Indeterminate Forms and L’Hôpital’s Rule
 8.8 Improper Integrals

9 Infinite Series
 9.1 Sequences
 9.2 Series and Convergence
 9.3 The Integral Test and 𝑝Series
 9.4 Comparisons of Series
 9.5 Alternating Series
 9.6 The Ratio and Root Tests

9.7 Taylor Polynomials and Approximations
 9.8 Power Series
 9.9 Representation of Functions by Power Series
 9.10 Taylor and Maclaurin Series

10 Conics, Parametric Equations, and Polar Coordinates
 10.1 Conics and Calculus
 10.2 Plane Curves and Parametric Equations

10.3 Parametric Equations and Calculus
 10.4 Polar Coordinates and Polar Graphs
 10.5 Area and Arc Length in Polar Coordinates
 10.6 Polar Equations of Conics and Kepler’s Laws

11 Vectors and the Geometry of Space
 11.1 Vectors in the Plane
 11.2 Space Coordinates and Vectors in Space
 11.3 The Dot Product of Two Vectors
 11.4 The Cross Product of Two Vectors in Space
 11.5 Lines and Planes in Space

11.6 Surfaces in Space
 11.7 Cylindrical and Spherical Coordinates

12 VectorValued Functions
 12.1 VectorValued Functions
 12.2 Differentiation and Integration of VectorValued Functions

12.3 Velocity and Acceleration

12.4 Tangent Vectors and Normal Vectors
 12.5 Arc Length and Curvature

13 Functions of Several Variables
 13.1 Introduction to Functions of Several Variables
 13.2 Limits and Continuity
 13.3 Partial Derivatives
 13.4 Differentials
 13.5 Chain Rules for Functions of Several Variables
 13.6 Directional Derivatives and Gradients
 13.7 Tangent Planes and Normal Lines
 13.8 Extrema of Functions of Two Variables
 13.9 Applications of Extrema

13.10 Lagrange Multipliers

14 Multiple Integration
 14.1 Iterated Integrals and Area in the Plane
 14.2 Double Integrals and Volume

14.3 Change of Variables: Polar Coordinates

14.4 Center of Mass and Moments of Inertia

14.5 Surface Area
 14.6 Triple Integrals and Applications

14.7 Triple Integrals in Other Coordinates
 14.8 Change of Variables: Jacobians

15 Vector Analysis
 15.1 Vector Fields
 15.2 Line Integrals
 15.3 Conservative Vector Fields and Independence of Path
 15.4 Green’s Theorem
 15.5 Parametric Surfaces

15.6 Surface Integrals

15.7 Divergence Theorem

15.8 Stokes’s Theorem

Appendices
 Appendix A Proofs of Selected Theorems
 Appendix B Integration Tables

Appendix C Precalculus Review
 C.1 Real Numbers and the Real Number Line
 C.2 The Cartesian Plane
 C.3 Review of Trigonometric Functions
 Appendix D Rotation and the General SecondDegree Equation
 Appendix E Complex Numbers
 Appendix F Business and Economic Applications