AN OVERVIEW OF THIS BOOK

Problems! Problems! Problems! Life has many facets, including family, work, relationships, activities, and so much more. One dimension of life is the existence of problems and our attempts to solve them. A problem involves the awareness of a challenge or an opportunity, a belief that a situation could be improved. Solving a problem usually means finding some way to improve the situation, to the greatest extent possible. Frequently the best solutions to a problem are the most creative. Russ Ackoff tells a story of a company that was having a serious problem with employees complaining about long waits for elevators.¹ They solved it by installing mirrors! The idea of problem solving is closely related to the basic ideas of decision making and working toward goals. Some people might feel that the word problem has too many negative connotations, but we will use it because it is so deeply rooted in our vocabulary. We don't mean to imply anything negative, we are using it in the broadest, more inclusive sense.

In this book, we will focus on problems that can be represented or modeled in a continuous way; that is, the quantities involved are continuous by nature, or close enough. This book may not help you with all of your problems, but it is intended to help you to solve many problems that you are likely to encounter in your other college courses, in your career, and in your personal life.

Traditional math texts and courses are somewhat analogous to teaching someone to fly at 5000 feet (do the straight math needed to solve a problem); our emphasis in this book is to take time to teach you how to take off (set up a problem) and land (make sure it makes sense in the real world) as well.

In order to get a quick feel for the kinds of problems we will be studying in the two volumes of this book, here are some typical examples:

• Your soccer team has decided to sell T-shirts to raise money to be able to attend a special all-day skills clinic in the state capital. You have been chosen to organize the entire effort. You want to be sure that you make money on this fundraiser - the last one was a disaster! How many T-shirts should you order? How much should you charge? Where do you begin?

• You play a sport or a musical instrument and want to figure out the ideal (optimal) amount of warm-up or practice before a match or performance. Or you want to find the optimal angle or distance for shots in a sport you play regularly.

• You are working for a small start-up firm with a radical new concept for cold-weather gloves. The firm has the initial capital to produce only one size glove. Fortunately, the design is such that, whatever actual size is produced, it should also fit people with a glove size that is up to 2 more or 2 less than the actual size produced. What single size, possibly fractional, should be produced to capture the largest possible market?

• We all have to learn how to manage our time. This can be very difficult if you are a first-year college student and this always had been done for you before now. You have to balance your need for sleep, exercise, studying, social life, extracurricular activities, and so forth. How can you get the maximum satisfaction with the way you divide your time? What is the best time of the day to do your math homework to get the highest grades? How many hours of exercise a week is best for you? How much sleep should you get every night to feel your best? How much should you study?

• You have some money you would like to invest. Your friend Rupert owns a newspaper delivery business. He is planning to go on a year-long walkabout in the Australian bush with an Aboriginal guide. He will not be in communication with the States, so he is looking for someone to take temporary ownership of the business (and keep the profits) while he is away. He is asking for an initial investment. You also have another investment opportunity to help another friend, Bonnie, who is a musician and wants to make a first CD. You and Bonnie have great confidence in the success of the album, and she agrees to give you a guaranteed amount one year from now. Which investment is better for you? (You don't have enough cash for both.)

• You are planning to mail a package and want to be sure that the post office will accept it. You called and they gave you information concerning the restrictions on the size of a package that they will accept. What dimension box should you get to stay within the regulations and still hold the greatest amount?

• You have decided to start a physical fitness/health/diet routine tomorrow. After consulting nutrition books and with your doctor, you are now working out guidelines for yourself on what to eat for breakfast. You decide that you want to get from your breakfast, on average, a certain limited amount of fat and calories, at least a fixed amount of fiber, and a specific amount of protein. You like to have just cereal for breakfast, and you have two favorite cereals. What should you eat for breakfast?

The first three examples above are typical of Volume 1 (Chapters 1-4), which focuses on situations where you want to find the optimal value and there is one major quantity over which you have control. The last three examples fall squarely in Volume 2 (Chapters 5-9), which focuses on situations where you want to optimize and there are two or more quantities over which you have control. The fourth example, about time management, could fall into either category, depending on the number of quantities over which you have control.

For some of these problems above, the connection to mathematics is easy to see. For others the connection is not so obvious, but it is definitely there. The power of mathematics can help us find solutions to these and many other problems, whether they arise in business, social science, or our personal lives. After you have worked with this book, we hope you will have substantially widened the categories of problems you are able to analyze quantitatively, including many problems you might never have guessed could be treated in that way.

The problems we will study in this book have a number of common characteristics. To distinguish this course from other quantitative courses, we will describe a number of these characteristics now.

Numerical quantities can be divided into two categories: discrete and continuous. Discrete means "composed of distinct parts." For example, you can order one hot dog or two hot dogs, but you normally can't order half of a hot dog. You can buy one T-shirt or two or three, but you usually can't buy one third of a T-shirt. A quantity being continuous (we will also use the term divisible as a synonym²) means exactly what it says: The set of possible values is an unbroken stream. You could eat one and a half hot dogs, or you can walk 1.235 miles or lose 5.3 pounds while on a diet. Some quantities are technically discrete, such as the price to charge for a T-shirt, which must be in whole cents, but are close enough to being continuous that we can think of them as continuous. The Census Bureau reports that the average family now has 1.20 children.³ Of course, this sounds ridiculous, but we understand what they are trying to tell us. In fact, we will assume throughout the book that all variables are divisible unless we specify otherwise. We will also assume that all graphs and relationships are smooth unless we specify otherwise.

Look back over the problems listed above. Convince yourself that they do indeed involve quantities that can usefully be represented as divisible, and that the graphs related to them would be smooth. Note also that some of them involve concepts of probability and statistics (especially the problem about making the gloves, which involved knowing the relative probability or proportion of people with different glove sizes). In this book, we will talk about aspects of probability and statistics that are continuous in nature, for example, probabilities related to bell curves, such as for the distribution of SAT scores, and fitting curves to data. However, we will not discuss discrete aspects of probability, such as calculating probabilities of poker hands (sorry about that, poker players!). Most of the problems we will discuss will be making an assumption of certainty. This means we assume that the relationships and equations hold exactly as they are written, without any fluctuations or uncertainty. For instance, for the problem about choosing breakfast cereals, we can reasonably assume that the numerical values and costs are stable and known. They may change slightly from time to time, but not so much that the uncertainty would be a major factor in solving the problem.

PREFACE TO VOLUME 2

Introduction to Chapters 5-9

In the first four chapters of this book, we worked with problems that involved a single dependent (output) variable that was a function of a single independent (input) variable. We went over the steps of the problem-solving process: saw how to formulate and fully define models from verbal descriptions and from data, learned how to define and interpret different kinds of rates of change (such as the derivative), and used this knowledge to find optimal solutions to problems involving minimization and maximization. We also saw how reversing the derivative process (finding an antiderivative) can help to calculate areas under curves, such as in calculating probabilities of continuous random variables given their probability density functions, finding net distance from velocity, or finding total accumulated net profit from a profit per unit time function.

In the last five chapters of the book, we essentially follow the same process for functions in which a single output variable is a function of two or more input variables. In Chapters 5 and 6, we see how the notion of a function generalizes to two or more input variables, how to fully define multivariable models involving such functions, and how to formulate such models, both from verbal descriptions ("word problems") and from data (using least-squares regression). In Chapter 7, we learn how to use matrices (rectangular tables of numbers) in various useful situations, including solving systems of linear equations using the idea of the inverse of a matrix. In Chapter 8, we use these matrix skills to help maximize or minimize certain kinds of multivariable functions (such as minimizing the SSE to find the best-fit regression model), based on a generalization of the idea of setting the derivative equal to 0 to find local extrema. Finally, in Chapter 9, we discuss optimizing a linear function subject to linear constraints (restrictions), called linear programming, utilizing the concept of a partial derivative from Chapter 8 and the matrix theory of Chapter 7.

TO THE INSTRUCTOR

In recent decades, the first-year course in mathematics for business and social science undergraduates has normally consisted of some combination of calculus and "finite math." The finite math usually includes matrices, linear programming, some topics in probability and statistics, and possibly specific business topics such as compound interest, present value, break-even analysis, and so on. Faculty and students on both sides (math and business/social science) have become increasingly dissatisfied with the course for two reasons: It has been perceived as having no real connection to the students' lives, curriculum, or careers, and it has seemed to be a random jumble of unrelated topics. Students tend to see the course as a hazing to be endured and forgotten as soon as possible after the trauma - a torture that is an initiation ritual into the "club" of being a business or social science graduate. Faculty and administrators have tended to see the course as an academic filter, to weed out weaker students from academic programs.

This does not need to be the case! As a former chief financial officer at Dupont has said, "Mathematics is the language of business." Whether it is profit and loss, risk, net present value, internal rate of return, forecasts, probabilities, expected values, market research, hypothesis testing, maximizing profit or efficiency, or minimizing cost, business is all about numbers, estimates, calculations, and optimization. Calculus can be thought of as the language of continuous change. If there is anywhere in the world that is continuously changing, it is the business world, especially now with the advent of global competition. In fact, most of the topics in most traditional versions of this course do have relevance to business and social science, but the connections are not communicated to the students (and often not to the math faculty teaching the course).

The Villanova Project text takes a very different approach to this course. Our goal is to address the causes of the dissatisfaction. Our goal is to show the connections between the mathematical topics in the course and the student's world (present, and future) in as many ways as possible, and to communicate the connections between the topics themselves. Our primary focus is on the process of problem solving in the real world. In the Introduction to Chapter 1, we give both brief and detailed outlines of this process and suggest the metaphor that most math texts teach students how to fly an airplane at a cruising altitude of 5000 feet (solve mathematical problems) but do not really teach them how to take off (formulate a real-world problem) or how to land (validate their model and solution). Our purpose in this book is to teach all of the steps needed to use mathematics to solve real problems in the real world. This includes clarifying a problem, making ballpark estimates, collecting data, defining variables clearly and unambiguously (including units), fitting models to data, understanding concepts in order to select appropriate methods and technologies, finding solutions, verifying (double-checking) calculations, validating assumptions of the models, performing sensitivity analysis (seeing the effect on the solution of plausible changes in the data), estimating a rough margin of error, and writing up conclusions.

To give students experience with this entire process from beginning to end, we strongly recommend the use of student-generated projects. The idea is to have students select topics in which they have a strong personal interest and apply the problem-solving process to the topic. Examples of successful topics from our experience include deciding the quantity to order and the price to charge for T-shirts being sold to raise money for a student group; determining the optimal number of hours to study (or sleep, or exercise, or the optimal combination of two or more of these) in an average week to maximize personal satisfaction; determining the optimal speed in a car to maximize fuel efficiency; finding the optimal combination of breakfast foods to eat to meet nutritional goals at minimum cost; and determining the optimal angle and/or distance for a shot in basketball (or hockey or water polo or\x85) to maximize the scoring percentage. When the students see how math can really help them, it makes all the concepts of the course come alive. After internalizing the process of problem solving with a topic they care about, they can then apply it to all kinds of problems. In fact, in their future careers, problems from work will have the same intensity and interest to them as personal topics do now, as beginning college students.

Because of our emphasis on solving real-world problems, we focus heavily on mathematical modeling with real data, usually by fitting curves. Thus, instead of giving students a function to be optimized (they typically cannot imagine how this would be given to them in the real world, and they are right!), we teach them how to take realistic data (which they can imagine being given to them, or getting themselves) and fit a model to those data points. We explain the idea of least squares from the beginning and teach that choosing a model depends on how the model will be used as well as on the data (e.g., one model may be great for interpolation but terrible for extrapolation). We explain that the Sum of the Squared errors (SSE) or coefficient of determination (R²) should not be the main criteria in choosing a model, but only used to choose between models which both reflect the shape of the data.

Given that our world is technology oriented, we seek to make the most of this reality. This is the career environment in which these students will find themselves. Although we expect students to be able to solve simplistic problems by hand to show mastery of the concepts, it does not make sense for them to try to solve realistic problems by hand. Technology now makes it possible to solve real problems relatively quickly.

We focus on two technologies that are commonly used by almost everyone in business: calculators and spreadsheets. (Other technologies, such as computer algebra systems, can also be used.) The calculator we recommend is the TI-83 (an old TI-82 will work fine for this course as well) or an equivalent calculator, for its curve-fitting ability and general level (not too elementary and not needlessly advanced or expensive). The TI-83 includes business functions (such as present value) and statistical programs, so this can be the only calculator the student will need for many years. The graphing capabilities make many calculus topics much less important (such as curve sketching and the First and Second Derivative Tests). On the other hand, spreadsheets are quite interchangeable these days, but we recommend one with some kind of optimization function (both linear and nonlinear), such as Quattro pro or Excel. Many calculators and spreadsheets have similar features, so either technology could work alone, but we believe the combination is most powerful: graphing calculators for the single-variable calculus and matrices and spreadsheets for the multivariable regression and constrained optimization.

Technology reduces the need for some mathematical and algorithmic skills but increases the need for many other skills. Near the top of this list is the ability to translate between words (or the real world) and algebraic symbols, as in problem formulation. Students typically have a lot of trouble with this (hence their common aversion to "story problems"). We distinguish two major types of formulation: The first is to take verbal descriptions and put them together into relevant equations and functions such as formulating a linear program, and the second is to take or gather data and find a model by fitting curves. The latter type comes up with a modeling approach and focuses more on unambiguous definitions of variables and functions, as well as least-squares regression models. In both cases, translating back from symbols (and computer/calculator output) to the problem or real-world situation is also necessary. We have segments in the book to focus on all of these processes.

Other skills essential to problem solving include the ability to make ballpark estimates by hand or in your head (to know if a solution makes sense), verifying solutions (double-checking, for example, by a different technique or technology), sensitivity analysis, and validation, as discussed above. Again, we have sections of the book devoted to these topics, and the students can gain hands-on experience with them on their projects.

The topics of the course and book have worked out as a result of exhaustive discussions among faculty in mathematics, business, and the social sciences. The idea of the course is to teach the fundamentals of problem solving and modeling listed above and to cover the mathematical topics which are the foundation of quantitative concepts that arise in these disciplines. The unifying mathematical theme is the analysis of quantities that can usefully be modeled as being continuous (as opposed to discrete). Unlike most traditional finite math courses, we do not try to teach probability and statistics in themselves (we leave this for a true statistics course), but we do use problems in these areas to motivate and apply the basic concepts of the single-variable calculus material. For example, people in business and social science use hypothesis testing, means medians, modes, and variances all the time. To understand these ideas mathematically, one must have some understanding of the concept of an integral. We start by calculating the mode of a distribution as an example of single-variable optimization. We then use the idea of calculating probabilities for continuous random variables as the main (but not only) motivation for finding the area under a curve, after which we use the calculation of expected values and medians as applications of integration.

We do cover a small amount of multivariate calculus (partial derivatives and optimization) which is often not covered in finite math. For all of our calculus material, we focus more on intuitive understanding of the concepts graphically, numerically, algebraically, and verbally (sometimes called the "rule of four"), rather than on theoretical details that are less relevant for these disciplines. For example we do not get into calculating limits in general or focus on discontinuous functions. We also do not cover implicit differentiation, related rates, most techniques of integration, or Taylor series, because our business/social science colleagues felt these were not a high priority for their first-year students. We plan to develop a more advanced third semester course for students in majors such as economics to cover these and other topics. One of the unifying and culminating topics for this full two-semester course is the mathematical derivation of the method of least squares, including the use of matrices to calculate the regression parameters. Since both semesters use the idea of fitting models to data, this is a perfect capstone to the calculus material and is one of the reasons why we consider it so important to include the material on partial derivatives.

The sequence of the course is designed to maximize the connections and interactions among the topics. We strongly recommend the full-year sequence: The first semester focuses on single-variable analysis and optimization, and then the second semester focuses on multivariate analysis and optimization. This means that students with AP Calculus credit can jump into the second semester. With in the second semester, we use an original sequence by covering matrices before multivariate optimization (so matrices can be used to solve a system of linear equations when optimizing a quadratic objective). We then cover linear programming after multivariate calculus so that the idea of a partial derivative can be used to better understand the simplex method and the idea of a shadow (dual) price.

We describe the development in the book as problem driven (sometimes called the "way of Archimedes"). The book as a whole, each chapter, and each section start off with a number of typical problem types to motivate and give students a feel for the usefulness of what follows. It is our experience that students have quite extraordinary intuition with real problems involving real numbers. Thus we use these realistic problems to develop that intuition and then generalize it to understand the major concepts of the unit. This way the theory comes out of the applications, and the applications are always there to make the theoretical abstractions concrete and comprehensible.

A number of sections, and some topics within sections, are not essential to any later topics and so can be considered optional. These have been indicated with asterisks. For optional topics within a section, the exercises at the end of the section that relate to them are also marked with asterisks. This material is purely at the discretion of those who determine your syllabus.

Technically, the material in Chapter 5 (multivariable functions and models, including compound interest and present value and future value) and Chapters 7 and 9 (matrices and linear programming) could be covered independently at any time. As discussed above, however, we strongly recommend the sequence as presented here to optimize motivation and interconnectedness of topics.

An additional feature of the Villanova Project is a collection of brief videos of faculty and nonacademics from business and social science showing how the math of this course underlies what they do in the concurrent and subsequent courses they teach and in their other work. We recommend that each school actually film its own faculty (we can provide scripts to help make this easier), so the students will be watching familiar faces, and names they know they will encounter later in their degree programs. We will also have videotapes available for general use. These should be available by mid-1998. This is one more way to keep students from believing "I'll never see or need this stuff again."

In summary, this book is a model of a truly collaborative service course. It has been designed to supply the concepts and skills that colleagues in business and social science want for their students, while maintaining mathematical integrity with an innovative selection and sequence of topics. It emphasizes the interrelationships among the mathematical topics, as well as the connections to students' curriculum, careers, and personal lives via realistic problems, videos, and student-generated projects. It is problem driven, uses technology to minimize drudgery, and makes it possible to solve real-world problems. It adopts a modeling approach and emphasizes critical thinking and understanding of concepts. It focuses on all of the steps in the process of problem solving (including translating between the real world, mathematical symbols, and computer/calculator output), reinforced by student-generated projects.

BASIC ORGANIZATION OF THE BOOK
AND GENERAL TOPICS

Volume 1 (Chapters 1-4) focuses on problems with a single decision variable (one thing to be decided, like the amount of exercise to get each week), while Volume 2 (Chapters 5-9) focuses on problems with two or more decision variables (such as choosing between two breakfast cereals). In each part, the initial chapters (Chapters 1, 5, and 6) focus on the process of problem solving and modeling with appropriate categories of functions including how to formulate models both from data and from verbal description. The other chapters in each part then analyze the models. Chapters 2, 3, and 8 explore rates of change of one variable with respect to another (such as velocity or marginal profit), and how these can be used to optimize the types of functions being studied. This is done both by hand and using technology. The major mathematical subjects we will explore are single and multivariable calculus (Chapters 2-4 and 8), which you can think of as the language of change (for continuous quantities), linear (matrix) algebra (Chapter 7), which studies how to solve systems of linear equations (useful for multivariable optimization), and linear and nonlinear programming (Chapter 9), which involve multivariable optimization with constraints. We will also discuss continuous probability and descriptive statistics (in Chapters 4 and 6), compound interest and time value of money (in Chapter 5), and the method of least squares regression (in Chapters 1, 5, 6, 7, and 8).

CORRECTIONS, COMMENTS, AND SUGGESTIONS

If you have any corrections, comments, or suggestions related to this material, we would greatly appreciate hearing them! You can mail hard copy to use at the Department of Mathematical Sciences, Villanova University, Villanova PA 19085, send us E-mail at brucepj@ucis.vill.edu, call us at (610) 519-6926, or fax us at (610) 519-6928. Thanks!

³Statistical Abstract of the United States, 1995. U.S. Bureau of the Census.