Mathematical Connections

A Modeling Approach to Business Calculus and Finite Mathematics
The Villanova Project

Bruce Pollack-Johnson and Audrey Fredrick Borchardt

A completely redesigned course, teaching the Entire Process of Problem Solving Using Real-World Data and Technology.

  • Single Variable Calculus (Volume 1: A Modeling Approach to Business Calculus)
  • Multivariable Calculus and Finite Mathematics
Single Variable Problem Solving   Multivariable Problem Solving

Chapter 1: Problem Solving, Functions and Models

Chapter 2: Rates of Change

Chapter 3: Single-Variable Optimization and Analysis

Chapter 4: Continuous Probability and Integration

Instructors Guide and Solution Manual

Student Solution Manual

Technology Manuals: TI-83 Excel


Chapter 5: Multivariable Models from Verbal Descriptions: Interest, NPV, SSE

Chapter 6: Multivariate Models from Data: Regression and Statistics

Chapter 7: Matrices and Solving Systems of Equations

Chapter 8: Unconstrained Optimization of Multivariable Function

Chapter 9: Constrained Optimization and Linear Programming

Chapter 1


1.0 The Process of Problem Solving

1.1 Functions

1.2 Mathematical Mocels and Formulation from Verbal Descriptions

1.3 Linear Functions and Models

1.4 Functions with One Concavity: Quadratic, Exponential, Power

1.5 Functions with Changing Concavity: Cubis, Quartic, Logistic


Chapter 2

2.1 Average and Percent Rate of Change Over an Interval

2.2 Instantaneous Rate of Change at a Point

2.3 Derivative Notation and Interpretation, Marginal Analysis

2.4 The Algebraic Definition of Derivative and Basic Derivative Rules

2.5 Composite Functions and the Chain Rule

2.6 The Product Rule


Chapter 3

Single-Variable Optimization and Analysis


3.1 Analysis of Graphs and Slope Graphs

3.2 Optimization – Algebraic Determination of Maxima and Minima

3.3 Testing of Critical Points, Concavity and Points of Inflection

3.4 Post-Optimality Analysis

3.5* Per-Cent Rate of Change at a Point, Elasticity, Average Cost


Chapter 4

Continuous Probability and Integration


4.1 Continuous Probability Distributions

4.2 Approximating Area under Curves (Subinterval Methods)

4.3 Finding Exact Areas Using Limits of Sums

4.4 Recovering Functions from their Derivatives

4.5 The Fundamental Theorem of Calculus

4.6 Variable Limits of Integration and Medians, Improper Integrals

4.7* Consumer and Producer Surplus



Chapter 5

Multivariable Models from Verbal Descriptions: Interest, NPV,SSE


5.1 Multivariable Functions and Models, 3-D Graphs

5.2 Formulating Models from Verbal Descriptions

5.3 Interest and Investments

5.4 The Time Value of Money (Present Value and Future Value) and Loans

5.5 Formulating SSE in Terms of Model Parameters


Chapter 6

Multivariate Models from Data: Regression and Statistic


6.1 Multivariable Models from Data – Spreadsheets and Regression

6.2 Mean, Variance, Standard Deviation, MSE, Misuse of Statistics

6.3 R2, Standard Error, Misuse of Regression, Regression Assumptions

6.4* Investment Portfolios, Risk-Return Tradeoffs, Pareto Efficiency


Chapter 7

Matrices and Solving Systems of Equations


7.1 Introduction to Matrices and Basic Operations

7.2 Matrix Multiplication

7.3 Systems of Linear Equations and Augmented Matrices

7.4 Matrix Equations and Inverse Matrices

7.5* Markov Chains


Chapter 8

Unconstrained Optimization of Multivariable Functions


8.1 Rates of Change of Multivariable Functions

8.2 Finding Local Extrema of Multivariable Functions

8.3 Optimization using a Spreadsheet

8.4 Testing for Local and Global Extrema

8.5 The Method of Least Squares


Chapter 9

Constrained Optimization and Linear Programming


9.1 Optimization with Equality Constraints: Lagrange Multipliers

9.2 Solving Linear Programs Graphically

9.3 The Simplex Method

9.4 Linear and Nonlinear Optimization on Spreadsheets Summary

Unique Features of the Redesigned Course

  • Problem Driven
  • Connected Topics
  • Sequence of Topics
  • Technology as a Teaching Tool
  • Technology as a Calculating Tool
  • Mathematical Models
  • Emphasis on Connecting Topics to Students’ Academic, Personal and Professional Lives