1. FUNCTIONS AND LINEAR MODELS 

Introduction 

Functions from the Numerical and Algebraic Viewpoints 

Functions from the Graphical Viewpoint 

Linear Functions 

Linear Models 

Linear Regression 

Case Study: Modeling Spending on Internet Advertising 

Optional Internet Topic: New Functions from Old: Scaled and Shifted Functions 

2. SYSTEMS OF LINEAR EQUATIONS AND MATRICES 

Introduction 

Systems of Two Equations in Two Unknowns 

Using Matrices to Solve Systems of Equations 

Applications of Systems of Linear Equations 

Case Study: The Impact of Regulating Sulfur Emissions 

3. MATRIX ALGEBRA AND APPLICATIONS 

Introduction 

Matrix Addition and Scalar Multiplication 

Matrix Multiplication 

Matrix Inversion 

InputOutput Models 

Case Study: The Japanese Economy 

4. LINEAR PROGRAMMING 

Introduction 

Graphing Linear Inequalities 

Solving Linear Programming Problems Graphically 

The Simplex Method: Solving Standard Maximization Problems 

The Simplex Method: Solving General Linear Programming Problems 

The Simplex Method and Duality (Optional) 

Case Study: Airline Scheduling 

5. THE MATHEMATICS OF FINANACE 

Introduction 

Simple Interest 

Compound Interest 

Annuities, Loans, and Bonds 

Case Study: Saving for College 

6. SETS AND COUNTING 

Introduction 

Set Operations 

Cardinality 

The Addition and Multiplication Principles 

Permutations and Combinations 

Case Study: Designing a Puzzle 

7. PROBABILITY 

Introduction 

Sample Spaces and Events 

Estimated Probability 

Empirical Probability 

Probability and Counting Techniques 

Probability Distributions 

Conditional Probability and Independence 

Bayes' Theorem and Applications 

Case Study: The Monty Hall Problem 

8. RANDOM VARIABLES AND STATISTICS 

Introduction 

Random Variables and Distributions 

Bernoulli Trials and Binomial Random Variables 

Measures of Central Tendency 

Measures of Dispersion 

Normal Distributions 

Case Study: Spotting Tax Fraud with Benford's Law 

Optional Internet Topics: Sampling Distributions and the Central Limit Theorem 

Confidence Intervals 

Hypothesis Testing 

9. MARKOV SYSTEMS 

Introduction 

Markov Systems 

Distribution Vectors and Powers of the Transition Matrix 

LongRange Behavior of Regular Markov Systems 

Absorbing Markov Systems 

Case Study: Predicting the Price of Gold 

10. NONLINEAR MODELS 

Introduction 

Quadratic Functions and Models 

Exponential Functions and Models 

Logarithmic Functions and Models 

Logistic Functions and Models 

Case Study: Checking up on Malthus 

Optional Internet Topics: Inverse Functions 

Linear and Exponential Regression 

Using and Deriving Algebraic Properties of Logarithms 

11. INTRODUCTION TO THE DERIVATIVE 

Introduction 

Average Rate of Change 

The Derivative: Numerical and Graphical Viewpoints 

The Derivative : Algebraic Viewpoint 

Derivatives of Powers, Sums, and Constant Multiples 

A First Application: Marginal Analysis 

Limits: Numerical and Graphical Approaches 

Limits and Continuity 

Limits and Continuity: Algebraic Approach 

Case Study: Reducing Sulfur Emissions 

Optional Internet Topics: Sketching the Graph of the Derivative 

Proof of the Power Rule 

Continuity and Differentiability 

12. TECHNIQUES OF DIFFERENTIATION 

Introduction 

The Product and Quotient Rules 

The Chain Rule 

Derivatives of Logarithmic and Exponential Functions 

Implicit Differentiation 

Case Study: Projecting Market Growth 

Optional Internet Topic: Linear Approximation and Error Estimation 

13. APPLICATIONS OF THE DERIVATIVE 

Introduction 

Maxima and Minima 

Applications of Maxima and Minima 

The Second Derivative and Analyzing Graphs 

Related Rates 

Elasticity 

Case Study: Production Lot Size Management 

14. THE INTEGRAL 

Introduction 

The Indefinite Integral 

Substitution 

The Definite Integral as a Sum: A Numerical Approach 

The Definite Integral as Area: A Geometric Approach 

The Definite Integral: An Algebraic Approach and the Fundamental Theorem of Calculus 

Case Study: Wage Inflation 

Optional Internet Topic: Numerical Integration 

15. FURTHER INTEGRATION TECHNIQUES AND APPLICATIONS OF THE INTEGRAL 

Introduction 

Integration by Parts 

Area Between Two Curves and Applications 

Averages and Moving Averages 

Continuous Income Streams 

Improper Integrals and Applications 

Differential Equations and Applications 

Case Study: Estimating Tax Revenues 

16. FUNCTIONS OF SEVERAL VARIABLES 

Introduction 

Functions of Several Variables from the Numerical and Algebraic Viewpoints 

Three Dimensional Space and the Graph of a Function of Two Variables 

Partial Derivatives 

Maxima and Minima 

Constrained Maxima and Minima and Applications 

Double Integrals 

Case Study: Modeling Household Income 

17. TRIGONOMETRIC MODELS 

Introduction 

Trigonometric Functions, Models, and Regression 

Derivatives of Trigonometric Functions and Applications 

Integrals of Trigonometric Functions and Applications 

Case Study: Predicting Cocoa Inventories 

Appendix A: Algebra Review 

Introduction 

Real Numbers 

Exponents and Radicals 

Multiplying and Factoring Algebraic Equations 

Rational Expressions 

Solving Polynomial Equations 

Solving Miscellaneous Equations 

OPTIONAL INTERNET CHAPTERS 

G 

Game Theory 

Introduction 

TwoPerson Zero Sum Games; Reduction by Dominance 

Strictly Determined Games 

Solving Games using the Simplex Method 

Expert OpinionHarvesting Forests 

L 

Introduction to Logic 

Introduction 

Statements and Logical Operators 

Logical Equivalence, Tautologies and Contradictions 

The Conditional and the Biconditional 

Tautological Implications and Tautological Equivalences 

Rules of Inference 

Arguments and Proofs 

S 

Calculus Applied to Probability and Statistics 

Introduction 

Continuous Random Variables and Histograms 

Probability Density Functions: Uniform, Exponential, Normal, and Beta 

Mean, Median, Variance and Standard Deviation 

Case Study: Creating a Family Trust 

Appendix A: Real Numbers 

Appendix B: Table: Area Under a Normal Curve 

Answers to Selected Exercises 
