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.

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

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 Working with derivatives

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

4. Applications of the Derivative

4.1 Maxima and minima

4.2 What derivatives tell us

4.3 Graphing functions

4.4 Optimization problems

4.5 Linear approximation and differentials

4.6 Mean Value Theorem

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

6.8 Logarithmic and exponential functions revisited

6.9 Exponential models

6.10 Hyperbolic functions

7. Integration Techniques

7.1 Basic approaches

7.2 Integration by parts

7.3 Trigonometric integrals

7.4 Trigonometric substitutions

7.5 Partial fractions

7.6 Other integration strategies

7.7 Numerical integration

7.8 Improper integrals

7.9 Introduction to differential equations

8. Sequences and Infinite Series

8.1 An overview

8.2 Sequences

8.3 Infinite series

8.4 The Divergence and Integral Tests

8.5 The Ratio, Root, and Comparison Tests

8.6 Alternating series

9. Power Series

9.1 Approximating functions with polynomials

9.2 Properties of Power series

9.3 Taylor series

9.4 Working with Taylor series

10. Parametric and Polar Curves

10.1 Parametric equations

10.2 Polar coordinates

10.3 Calculus in polar coordinates

10.4 Conic sections

11. Vectors and Vector-Valued Functions

11.1 Vectors in the plane

11.2 Vectors in three dimensions

11.3 Dot products

11.4 Cross products

11.5 Lines and curves in space

11.6 Calculus of vector-valued functions

11.7 Motion in space

11.8 Length of curves

11.9 Curvature and normal vectors

12. Functions of Several Variables

12.1 Planes and surfaces

12.2 Graphs and level curves

12.3 Limits and continuity

12.4 Partial derivatives

12.5 The Chain Rule

12.6 Directional derivatives and the gradient

12.7 Tangent planes and linear approximation

12.8 Maximum/minimum problems

12.9 Lagrange multipliers

13. Multiple Integration

13.1 Double integrals over rectangular regions

13.2 Double integrals over general regions

13.3 Double integrals in polar coordinates

13.4 Triple integrals

13.5 Triple integrals in cylindrical and spherical coordinates

13.6 Integrals for mass calculations

13.7 Change of variables in multiple integrals

14. Vector Calculus

14.1 Vector fields

14.2 Line integrals

14.3 Conservative vector fields

14.4 Green’s theorem

14.5 Divergence and curl

14.6 Surface integrals

14.6 Stokes’ theorem

14.8 Divergence theorem

Appendix A. Algebra Review

Appendix B. Proofs of Selected Theorems

D1. Differential Equations (online)

D1.1 Basic Ideas

D1.2 Direction Fields and Euler’s Method

D1.3 Separable Differential Equations

D1.4 Special First-Order Differential Equations

D1.5 Modeling with Differential Equations

D2. Second-Order Differential Equations (online)

D2.1 Basic Ideas

D2.2 Linear Homogeneous Equations

D2.3 Linear Nonhomogeneous Equations

D2.4 Applications

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