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 meanswe 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 thew nutritional 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.

TO THE STUDENT

Most developments in mathematics, like that of our numeration system, have come about because people wanted to solve a problem, either theoretical or practical. Powerful mathematical tools have been developed from the time of the ancient Greeks through the present to help us solve many types of problems. In this book, we want to show you how to use these powerful tools to help you solve the problems that you face, both now and in the future. In order to use these mathematical tools, we must be able to express a problem in mathematical terms, we must have a mathematical model of the problem.

Traditionally, students in a math class are given equations that represent problems and are asked to solve them: Find the optimal solutions or to forecast from them. Occasionally, they are asked to formulate models from verbal descriptions, the dreaded word problems. In the real world, there will rarely be anyone to hand you these models on a silver platter, and if they did, how much confidence could you place in them? If you job depended on the analysis that you did on some model, you would probably want to know where that model came from. The most that you will probably have, particularly in this age of computers, is a lot of data and information about different quantities (variables). One of the skills that you will learn in this course is how to take data values and turn them into a mathematical expression of the relationship between variables. For example, you will learn how to analyze market research (the relationship between the selling price of something and the quantity that consumers will buy) to estimate a demand function. Some models can be constructed from definitions and basic principles, such as the fact that the revenue from selling a product equals the selling price times the quantity sold, and profit equals revenue minus cost. You will develop skills in recognizing and formulating this kind of model as well. You will start to learn how to solve realistic problems based on realistic information.

In this book we will explain how the power of mathematics can be used to take some of the guesswork out of decision making. We will study some of the mathematical concepts and techniques that have been developed to solve such problems. Most traditional mathematics courses proceed deductively. They introduce the theory first, then discuss its applications. In this book we start with problems that other students, business and social science faculty, and people in the nonacademic work world have found interesting and useful and then develop the theory needed to solve them. This way you can use the examples to build your intuition and understanding during the development of various topics.

This book emphasizes the understanding of the underlying concepts involved in these mathematical techniques. When you really understand where a formula comes from or what a process means, you will know how and when to use it effectively. You will learn how to find something called a "derivative" in this course, but it is not enough to know "how to take the derivative." You have to know the meaning of the derivative that you have found in order to use it to solve problems. Knowing the proper way to use a tool (like a power saw or a sewing machine) is good, but we also want to know how to use that tool to actually do something. Our goal is for you to learn about the mathematical concepts involved so that you can solve simplistic problems by hand. However, working out solutions to real-world problems by hand can be a waste of time when we have the technology to "crunch the numbers," so we want you to be able to solve larger and messier problems by using this technology.

When you have finished this course, you should be able to solve many types of problems. When presented with a problem, you should be able to express it clearly and concisely, set it up mathematically, solve it mathematically, validate your model and assumptions, and finally reach conclusions and implement your solution. Each of these steps is equally important. For example, it is virtually worthless to be able to set up the problem and solve it mathematically if you do not know how valid you assumptions are. Computer people are fond of saying "Garbage In, Garbage Out"; a solution is only as good as the information on which it is based.

Many traditional textbooks in mathematics focus on only one or two of these five stages (solving the problem mathematically, and maybe setting it up). These are certainly very important, but not enough in themselves to solve real-world problems. This book will discuss each stage, and will help you practice them individually and then all together. This will help you solve problems in future courses, in your career, and in your life in general.

This is a course in applied mathematics. It is designed to prepare you to work with real data from the real world. In order to deal with real data, we will use all the modern technology available to us as tools. We will use the technology to draw pictures - graphs - of the numerical data. It is often very difficult to see a trend in numerical data. A graph of the data makes it much easier to see a trend, if there is one. We will use the technology and our common sense to help us determine a mathematical expression to represent these data for the intended purpose and context - to find a mathematical model. Once we have the mathematical model, we may use the mathematical technique of differentiation to study how our data are changing or to find an optimal solution. Or we might use an optimization program in a spreadsheet to determine prices to charge to maximize profit or a product mix to meet certain conditions and minimize costs. Using technology frees us from much of the tedious calculation commonly associated with doing mathematics and allows us to concentrate on the formulation of the problem and the interpretation of the results.

Many students don't see the usefulness of the mathematics they are studying, or any connection to other courses they are taking or will be taking in the future. We have spent a great deal of time and study with business and social science faculty and students to determine exactly what mathematical tools you will need to succeed in your business and social science courses, careers, and personal lives. The topics presented in this book are the results of those studies. You will be using this math and studying topics that are based on it, so it is important that you understand it thoroughly.

When you decide to learn to play a new game or a sport like tennis or basketball, it is important to spend some time learning the rules and the vocabulary of the game. And if you want to excel, you typically must spend quite a lot of time practicing the techniques involved in playing the game: for instance, forehand shots and backhand shots and serves for tennis, ball handling, shooting baskets, and rebounding for basketball. You cannot learn to play a sport simply by watching good players. If this were true we could all be pros! Mathematics is like that - it is not a spectator sport. If you want to be able to do mathematics, to use the power of mathematics to help solve problems, you must first learn the rules and the terminology and then practice, practice, practice! Shooting hoops for several hours may be more fun than doing practice problems in mathematics, but the sense of achievement when the skills are mastered can be quite similar. We wish you well in your efforts!

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