Calculus

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 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 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 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 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 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 Revisited and Growth Rates of Functions

 

8. Integration Techniques

8.1 Integration by Parts

8.2 Trigonometric Integrals

8.3 Trigonometric Substitutions

8.4 Partial Fractions

8.5 Other Integration Strategies

8.6 Numerical Integration

8.7 Improper Integrals

8.8 Introduction to Differential Equations

 

9. Sequences and Infinite Series

9.1 An Overview

9.2 Sequences

9.3 Infinite Series

9.4 The Divergence and Integral Tests

9.5 The Ratio, Root, and Comparison Tests

9.6 Alternating Series Review

 

10. Power Series

10.1 Approximating Functions with Polynomials

10.2 Power Series

10.3 Taylor Series

10.4 Working with Taylor Series

 

11. Parametric and Polar Curves

11.1 Parametric Equations

11.2 Polar Coordinates

11.3 Calculus in Polar Coordinates

11.4 Conic Sections

 

12. Vectors and Vector-Valued Functions

12.1 Vectors in the Plane

12.2 Vectors in Three Dimensions

12.3 Dot Products

12.4 Cross Products

12.5 Lines and Curves in Space

12.6 Calculus of Vector-Valued Functions

12.7 Motion in Space

12.8 Length of Curves

12.9 Curvature and Normal Vectors

 

13. Functions of Several Variables

13.1 Planes and Surfaces

13.2 Graphs and Level Curves

13.3 Limits and Continuity

13.4 Partial Derivatives

13.5 The Chain Rule

13.6 Directional Derivatives and the Gradient

13.7 Tangent Planes and Linear Approximation

13.8 Maximum/Minimum Problems

13.9 Lagrange Multipliers

 

14. Multiple Integration

14.1 Double Integrals over Rectangular Regions

14.2 Double Integrals over General Regions

14.3 Double Integrals in Polar Coordinates

14.4 Triple Integrals

14.5 Triple Integrals in Cylindrical and Spherical Coordinates

14.6 Integrals for Mass Calculations

14.7 Change of Variables in Multiple Integrals

 

15. Vector Calculus

15.1 Vector Fields

15.2 Line Integrals

15.3 Conservative Vector Fields

15.4 Green’s Theorem

15.5 Divergence and Curl

15.6 Surface Integrals

15.6 Stokes’ Theorem

15.8 Divergence Theorem