David Ellenbogen has taught math at the college level for twenty-two years, spending most of that time in the Massachusetts and Vermont community college systems, where he has served on both curriculum and developmental math committees. He has also taught at St. Michael's College and the University of Vermont. Professor Ellenbogen has been active in the Mathematical Association of Two Year Colleges since 1985, having served on its Developmental Mathematics Committee and as a delegate, and has been a member of the Mathematical Association of America since 1979. He has authored dozens of publications on topics ranging from prealgebra to calculus and has delivered lectures at numerous conferences on the use of language in mathematics. Professor Ellenbogen received his BA in mathematics from Bates College and his MA in community college mathematics education from the University of Massachusetts at Amherst. A cofounder of the Colchester Vermont Recycling Program, Professor Ellenbogen has a deep love for the environment and the outdoors, especially in his home state of Vermont. In his spare time, he enjoys playing keyboard in the band Soularium, volunteering as a community mentor, hiking, biking, and skiing. He has two sons, Monroe and Zack.

Scott Surgent received his B.S. and M.S. degrees in mathematics from the University of California–Riverside, and has taught mathematics at Arizona State University in Tempe, Arizona, since 1994. He is an avid sports fan and has authored books on hockey, baseball, and hiking. Scott enjoys hiking and climbing the mountains of the western United States. He was active in search and rescue, including six years as an Emergency Medical Technician with the Central Arizona Mountain Rescue Association (Maricopa County Sheriff’s Office) from 1998 until 2004. Scott and his wife, Beth, live in Scottsdale, Arizona.

R. Functions, Graphs, and Models

R.1 Graphs and Equations

R.2 Functions and Models

R.3 Finding Domain and Range

R.4 Slope and Linear Functions

R.5 Nonlinear Functions and Models

R.6 Mathematical Modeling and Curve Fitting

1. Differentiation

1.1 Limits: A Numerical and Graphical Approach

1.2 Algebraic Limits and Continuity

1.3 Average Rates of Change

1.4 Differentiation Using Limits of Difference Quotients

1.5 Differentiation Techniques: The Power and Sum-Difference Rules

1.6 Differentiation Techniques: The Product and Quotient Rules

1.7 The Chain Rule

1.8 Higher-Order Derivatives

2. Applications of Differentiation

2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs

2.2 Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs

2.3 Graph Sketching: Asymptotes and Rational Functions

2.4 Using Derivatives to Find Absolute Maximum and Minimum Values

2.5 Maximum-Minimum Problems; Business and Economic Applications

2.6 Marginals and Differentials

2.7 Implicit Differentiation and Related Rates

3. Exponential and Logarithmic Functions

3.1 Exponential Functions

3.2 Logarithmic Functions

3.3 Applications: Uninhibited and Limited Growth Models

3.4 Applications: Decay

3.5 The Derivatives of a^x and log _a x

3.6 An Economics Application: Elasticity of Demand

4. Integration

4.1 Antidifferentiation

4.2 Antiderivatives as Areas

4.3 Area and Definite Integrals

4.4 Properties of Definite Integrals

4.5 Integration Techniques: Substitution

4.6 Integration Techniques: Integration by Parts

4.7 Integration Techniques: Tables

5. Applications of Integration

5.1 An Economics Application: Consumer Surplus and Producer Surplus

5.2 Applications of Integrating Growth and Decay Models

5.3 Improper Integrals

5.4 Numerical Integration

5.5 Volume

6. Functions of Several Variables

6.1 Functions of Several Variables

6.2 Partial Derivatives

6.3 Maximum-Minimum Problems

6.4: An Application: The Least-Squares Technique

6.5 Constrained Optimization

6.6 Double Integrals

7. Trigonometric Functions

7.1 Basics of Trigonometry

7.2 Derivatives of Trigonometric Functions

7.3 Integration of Trigonometric Functions

7.4 Inverse Trigonometric Functions and Applications

8. Differential Equations

8.1 Differential Equations

8.2 Separable Differential Equations

8.3 Applications: Inhibited Growth Models

8.4 First-Order Linear Differential Equations

8.5 Higher-Order Differential Equations and a Trigonometry Connection

9. Sequences and Series

9.1 Arithmetic Sequences and Series

9.2 Geometric Sequences and Series

9.3 Simple and Compound Interest

9.4 Annuities and Amortization

9.5 Power Series and Linearization

9.6 Taylor Series and a Trigonometry Connection

10. Probability Distributions

10.1 A Review of Sets

10.2 Probability

10.3 Discrete Probability Distributions

10.4 Continuous Probability Distributions

10.5 Mean, Variance, Standard Deviation, and the Normal Distribution

11. Systems and Matrices (online)

11.1 Systems of Linear Equations

11.2 Gauss-Jordan Elimination

11.3 Matrices and Row Operations

11.4 Matrix Arithmetic: Equality, Addition and Scalar Multiples

11.5 Matrix Multiplication, Multiplicative Identities and Inverses

11.6 Determinants and Cramer’s Rule

11.7 Systems of Linear Inequalities and Linear Programming

12. Discrete Probability (online)

12.1 Compound Events and Odds

12.2 Combinatorics: The Multiplication Principal and the Factorial

12.3 Permutations and Combinations

12.4 Conditional Probability and the Hypergeometric Distribution Model

12.5 Independence, Bernoulli Trials and the Binomial Probability Model

12.6 Bayes’ Rule

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