Calculus: Early Transcendentals 3rd Edition © 2019

William Briggs has been on the mathematics faculty at the University of Colorado at Denver for twenty-three years. He received his BA in mathematics from the University of Colorado and his MS and PhD in applied mathematics from Harvard Univeristy. He teaches undergraduate and graduate courses throughout the mathematics curriculum with a special interest in mathematical modeling and differential equations ans it applies to problems in the biosciences. He has written a quantitative reasoning textbook, Using and Understanding Mathematics; an undergraduate problem solving book, Ants, Bikes, and Clocks; and two tutorial monographs, The Multigrid Tutorial and The DFT: An Owner’s Manual for the Discrete Fourier Transform. He is the Society for Industrial and Applied Mathematics (SIam) Vice President for Education, a University of Colorado President’s Teaching Scholar, a recipient of the Outstanding Teacher Award of the Rocky Mountain Section of the Mathematical Association of America (MAA), and the recipient of a Fulbright Fellowship to Ireland.

Lyle Cochran is a professor of mathematics at Whitworth University in Spokane, Washington. He holds BS degrees in mathematics and mathematics education from Oregon State University and a MS and PhD in mathematics from Washington State University. He has taught a wide variety of undergraduate mathematics courses at Washington State University, Fresno Pacific University, and, since 1995, at Whitworth University. His expertise is in mathematical analysis, and he has a special interest in the integration of technology and mathematics education. He has written technology materials for leading calculus and linear algebra textbooks including the Instructor’s Mathematica Manual for Linear Algebra and Its Applications by David C. Lay and the Mathematica Technology Resource Manual for Thomas’ Calculus. He is a member of the MAA and a former chair of the Department of Mathematics and Computer Science at Whitworth University.

Bernard Gillett is a Senior Instructor at the University of Colorado at Boulder; his primary focus is undergraduate education. He has taught a wide variety of mathematics courses over a twenty-year career, receiving five teaching awards in that time. Bernard authored a software package for algebra, trigonometry, and precalculus; the Student’s Guide and Solutions Manual and the Instructor’s Guide and Solutions Manual for Using and Understanding Mathematics by Briggs and Bennett; and the Instructor’s Resource Guide and Test Bank for Calculus and Calculus: Early Transcendentals by Briggs, Cochran, and Gillett. Bernard is also an avid rock climber and has published four climbing guides for the mountains in and surrounding Rocky Mountain National Park.

Eric Schulz has been teaching mathematics at Walla Walla Community College since 1989 and began his work with Mathematica in 1992. He has an undergraduate degree in mathematics from Seattle Pacific University and a graduate degree in mathematics from the University of Washington. Eric loves working with students and is passionate about their success. His interest in innovative and effective uses of technology in teaching mathematics has remained strong throughout his career. He is the developer of the Basic Math Assistant, Classroom Assistant, and Writing Assistant palettes that ship in Mathematica worldwide. He is an author on multiple textbooks: Calculus and Calculus: Early Transcendentals with Briggs, Cochran, Gillett, and Precalculus with Sachs, Briggs — where he writes, codes, and creates dynamic eTexts combining narrative, videos, and Interactive Figures using Mathematica and CDF technology.

1. Functions

• 1.1 Review of Functions
• 1.2 Representing Functions
• 1.3 Inverse, Exponential, and Logarithmic Functions
• 1.4 Trigonometric Functions and Their Inverses

Review Exercises

2. Limits

• 2.1 The Idea of Limits
• 2.2 Definitions of Limits
• 2.3 Techniques for Computing Limits
• 2.4 Infinite Limits
• 2.5 Limits at Infinity
• 2.6 Continuity
• 2.7 Precise Definitions of Limits

Review Exercises

3. Derivatives

• 3.1 Introducing the Derivative
• 3.2 The Derivative as a Function
• 3.3 Rules of Differentiation
• 3.4 The Product and Quotient Rules
• 3.5 Derivatives of Trigonometric Functions
• 3.6 Derivatives as Rates of Change
• 3.7 The Chain Rule
• 3.8 Implicit Differentiation
• 3.9 Derivatives of Logarithmic and Exponential Functions
• 3.10 Derivatives of Inverse Trigonometric Functions
• 3.11 Related Rates

Review Exercises

4. Applications of the Derivative

• 4.1 Maxima and Minima
• 4.2 Mean Value Theorem
• 4.3 What Derivatives Tell Us
• 4.4 Graphing Functions
• 4.5 Optimization Problems
• 4.6 Linear Approximation and Differentials
• 4.7 L’Hôpital’s Rule
• 4.8 Newton’s Method
• 4.9 Antiderivatives

Review Exercises

5. Integration

• 5.1 Approximating Areas under Curves
• 5.2 Definite Integrals
• 5.3 Fundamental Theorem of Calculus
• 5.4 Working with Integrals
• 5.5 Substitution Rule

Review Exercises

6. Applications of Integration

• 6.1 Velocity and Net Change
• 6.2 Regions Between Curves
• 6.3 Volume by Slicing
• 6.4 Volume by Shells
• 6.5 Length of Curves
• 6.6 Surface Area
• 6.7 Physical Applications

Review Exercises

7. Logarithmic, Exponential, and Hyperbolic Functions

• 7.1 Logarithmic and Exponential Functions Revisited
• 7.2 Exponential Models
• 7.3 Hyperbolic Functions

Review Exercises

8. Integration Techniques

• 8.1 Basic Approaches
• 8.2 Integration by Parts
• 8.3 Trigonometric Integrals
• 8.4 Trigonometric Substitutions
• 8.5 Partial Fractions
• 8.6 Integration Strategies
• 8.7 Other Methods of Integration
• 8.8 Numerical Integration
• 8.9 Improper Integrals

Review Exercises

9. Differential Equations

• 9.1 Basic Ideas
• 9.2 Direction Fields and Euler’s Method
• 9.3 Separable Differential Equations
• 9.4 Special First-Order Linear Differential Equations
• 9.5 Modeling with Differential Equations

Review Exercises

10. Sequences and Infinite Series

• 10.1 An Overview
• 10.2 Sequences
• 10.3 Infinite Series
• 10.4 The Divergence and Integral Tests
• 10.5 Comparison Tests
• 10.6 Alternating Series
• 10.7 The Ratio and Root Tests
• 10.8 Choosing a Convergence Test

Review Exercises

11. Power Series

• 11.1 Approximating Functions with Polynomials
• 11.2 Properties of Power Series
• 11.3 Taylor Series
• 11.4 Working with Taylor Series

Review Exercises

12. Parametric and Polar Curves

• 12.1 Parametric Equations
• 12.2 Polar Coordinates
• 12.3 Calculus in Polar Coordinates
• 12.4 Conic Sections

Review Exercises

13. Vectors and the Geometry of Space

13.1 Vectors in the Plane

• 13.2 Vectors in Three Dimensions
• 13.3 Dot Products
• 13.4 Cross Products
• 13.5 Lines and Planes in Space
• 13.6 Cylinders and Quadric Surfaces

Review Exercises

14. Vector-Valued Functions

• 14.1 Vector-Valued Functions
• 14.2 Calculus of Vector-Valued Functions
• 14.3 Motion in Space
• 14.4 Length of Curves
• 14.5 Curvature and Normal Vectors

Review Exercises

15. Functions of Several Variables

• 15.1 Graphs and Level Curves
• 15.2 Limits and Continuity
• 15.3 Partial Derivatives
• 15.4 The Chain Rule
• 15.5 Directional Derivatives and the Gradient
• 15.6 Tangent Planes and Linear Approximation
• 15.7 Maximum/Minimum Problems
• 15.8 Lagrange Multipliers

Review Exercises

16. Multiple Integration

• 16.1 Double Integrals over Rectangular Regions
• 16.2 Double Integrals over General Regions
• 16.3 Double Integrals in Polar Coordinates
• 16.4 Triple Integrals
• 16.5 Triple Integrals in Cylindrical and Spherical Coordinates
• 16.6 Integrals for Mass Calculations
• 16.7 Change of Variables in Multiple Integrals

Review Exercises

17. Vector Calculus

• 17.1 Vector Fields
• 17.2 Line Integrals
• 17.3 Conservative Vector Fields
• 17.4 Green’s Theorem
• 17.5 Divergence and Curl
• 17.6 Surface Integrals
• 17.7 Stokes’ Theorem
• 17.8 Divergence Theorem

Review Exercises

D2 Second-Order Differential Equations ONLINE

D2.1 Basic Ideas

• D2.2 Linear Homogeneous Equations
• D2.3 Linear Nonhomogeneous Equations
• D2.4 Applications
• D2.5 Complex Forcing Functions

Review Exercises

Appendix A. Proofs of Selected Theorems

Appendix B. Algebra Review ONLINE

Appendix C. Complex Numbers ONLINE

Answers

Index

Table of Integrals

1. Functions
2. Limits
3. Derivatives
4. Applications of the Derivative
5. Integration
6. Applications of Integration
7. Logarithmic, Exponential, and Hyperbolic Functions
8. Integration Techniques
9. Differential Equations
10. Sequences and Infinite Series
11. Power Series
12. Parametric and Polar Curves
13. Vectors and the Geometry of Space
14. Vector-Valued Functions
15. Functions of Several Variables
16. Multiple Integration
17. Vector Calculus