Introduction

Calculus is a classical branch of mathematics, dealing with the study of functions. A function is essentially a rule for association of one or more input variables with an output value. For instance, we might be interested in relating a decision, say, how much to produce, to the business outcome, say, profit. We have seen that this is needed, for instance, to figure out the best mix of products. Building a relationship linking managerial levers and the resulting outcome is essential in tackling any decision problem. In other cases, our aim is somewhat more instrumental. In statistical model building, we want to find a mathematical representation that yields the best fit between the empirically observed data and the predictions of the model. To accomplish this task, first we need to choose a functional form depending on some unknown parameters, and then we must choose another function expressing the lack of fit, which in turn is related to prediction errors. Empirical model building calls for the minimization of such lack of fit. In real life, we deal with functions of many variables, but in this chapter we just deal with functions of one variable, i.e., rules mapping one input value x to an output value y = f(x). We may introduce all of the required concepts in this simplified setting. Later, precisely at the end of the next chapter, we will generalize to functions of multiple variables.

Given a function, the first task that comes to our mind is plotting it, in order to get an intuitive feeling for the relationship between x and y, and to analyze the sensitivity of the output to variations in the input. Typical questions are as follows:

• How will a slight variation in interest rates affect an investment decision?
• How will a change in unit cost of some raw material affect total profit from manufacturing a product?

To deal with these important issues, we introduce a fundamental concept: the derivative of a function. The same concept is also essential in optimizing a function, i.e., in finding decisions maximizing profit or minimizing cost. Studying the related issues, such as convexity and concavity of functions, paves the way for later dealing with optimization models. We will also hint at some other important topics such as sequences, series, and integrals, even though our treatment will be quite brief, and limited to what is really essential to understand a few topics.