Category: Calculus

Functions are rules that map input values to output values in a well determined way. They come in many guises, depending on what is mapped on what. Generally, a function is specified as where D is the domain of the function, i.e., the set of possible input values on which the function is defined, and I is the image or range of the function, i.e.,…

Many practical problems involve permutations and combinations of objects. A first question is: Given a collection of n objects, in how many ways can we permute them? For instance, let us consider the set {a, b, c}. Since the set is quite small, we can enumerate all of the possible permutations systematically. First we consider permutations beginning…

Consider an expression like We will meet similar expressions quite often in the book, and a nice shorthand notation for this expression is which should be read as the sum of “x subscript i,” for i ranging from 1 to 4. Sometimes, the sum limits can be symbolic, as in We may even consider an infinite sum like In…

Inequalities like a ≤ x ≤ b, where a and b are arbitrary real numbers such that a < b, define intervals on the real line. The inequality above defines an interval that includes its extreme points. In such a case, we use the notation [a, b] to denote the interval, and we speak of a closed interval. On the contrary, inequalities a < x < b define the open interval (a, b). For…

If we order cars from a car manufacturer, we cannot order 10.56986 cars; we may order either 10 or 11 cars, but any value in between makes no sense. It should be intuitively clear what we mean by an integer number; integer numbers are used to measure variables that have a intrinsically discrete nature. A real number is…

As we have already pointed out, the reader is assumed to be equipped with a basic mathematical background about sets as well as integer and real numbers. In this section we briefly recall a few basic concepts for convenience.

In plotting the function, we have ignored the purchase cost component cd, which is constant and would just push the graph up a bit. This is not relevant to us, since what we are interested in is finding an order size Q* minimizing total cost. Indeed, since the function goes to infinity for very small and very…

Having figured out a relationship between the order size and the average total cost per year, it would be useful to plot the function in order to see the effect of Q and to figure out a good decision. There are plenty of powerful software packages that, given a range of the independent variable Q, compute the corresponding…

In order to express average total cost per unit time as a function of the order size Q, we should consider all of the factors contributing to the overall cost. The first one that comes to mind is purchase cost. If the unit item cost is c, measured in money per item, we have to pay cQ whenever…

Before getting into formal details of calculus, it is essential to arm ourselves with some motivation for doing so. In management science, we often want to relate decisions to cost or profit. This is necessary in order to find an “optimal” decision yielding the best performance in some well-specified sense. We should quote “optimal,” because…