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.

1.1 Review of functions

1.2 Representing functions

1.3 Trigonometric functions and their inverses

Review

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

3. Derivatives

3.1 Introducing the derivative

3.2 Rules of differentiation

3.3 The product and quotient rules

3.4 Derivatives of trigonometric functions

3.5 Derivatives as rates of change

3.6 The Chain Rule

3.7 Implicit differentiation

3.8 Derivatives of inverse trigonometric functions

3.9 Related rates

Review

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

Review

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

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 Hyperbolic functions

Review

7. Logarithmic and Exponential Functions

7.1 Inverse functions

7.2 The natural logarithm and exponential functions

7.3 Logarithmic and exponential functions with general bases

7.4 Exponential models

7.5 Inverse trigonometric functions

7.6 L'Hôpital's rule and growth rates of functions

Review

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 Other integration strategies

8.7 Numerical integration

8.8 Improper integrals

Review

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 differential equations

9.5 Modeling with differential equations

Review

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 The Ratio, Root, and Comparison Tests

10.6 Alternating series

Review

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

12. Parametric and Polar Curves

12.1 Parametric equations

12.2 Polar coordinates

12.3 Calculus in polar coordinates

12.4 Conic sections

Review

13. Vectors and Vector-Valued Functions

13.1 Vectors in the plane

13.2 Vectors in three dimensions

13.3 Dot products

13.4 Cross products

13.5 Lines and curves in space

13.6 Calculus of vector-valued functions

13.7 Motion in space

13.8 Length of curves

13.9 Curvature and normal vectors

Review

14. Functions of Several Variables

14.1 Planes and surfaces

14.2 Graphs and level curves

14.3 Limits and continuity

14.4 Partial derivatives

14.5 The Chain Rule

14.6 Directional derivatives and the gradient

14.7 Tangent planes and linear approximation

14.8 Maximum/minimum problems

14.9 Lagrange multipliers

Review

15. Multiple Integration

15.1 Double integrals over rectangular regions

15.2 Double integrals over general regions

15.3 Double integrals in polar coordinates

15.4 Triple integrals

15.5 Triple integrals in cylindrical and spherical coordinates

15.6 Integrals for mass calculations

15.7 Change of variables in multiple integrals

Review

16. Vector Calculus

16.1 Vector fields

16.2 Line integrals

16.3 Conservative vector fields

16.4 Green's theorem

16.5. Divergence and curl

16.6 Surface integrals

16.6 Stokes' theorem

16.8 Divergence theorem

Review

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