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 University. He teaches undergraduate and graduate courses throughout the mathematics curriculum with a special interest in mathematical modeling and differential equations as 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 Trigonometric Functions

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 

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 Related Rates

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

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

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

7. Logarithmic and Exponential Functions

7.1 Inverse Functions

7.2 The Natural Logarithmic and Exponential Functions

7.3 Logarithmic and Exponential Functions with Other Bases

7.4 Exponential Models

7.5 Inverse Trigonometric Functions

7.6 L’ Hôpital’s Rule and Growth Rates of Functions

7.7 Hyperbolic Functions

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

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