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 | |
Input-Output 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 | |
Long-Range 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 | |
Two-Person Zero Sum Games; Reduction by Dominance | |
Strictly Determined Games | |
Solving Games using the Simplex Method | |
Expert Opinion--Harvesting 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 |