Category: Calculus

In this section we try to further motivate the use of definite integral, at least conceptually, for business management problems. To do so, we use the EOQ model of Section 2.1 once again. There, we have claimed that the contribution of inventory holding cost to average total cost per unit time is In the reasoning, a key…

So far, we have considered the integral of a continuous function on a bounded interval. The idea can be generalized to unbounded intervals and to functions featuring certain types of discontinuity. In fact, the integral might not exist, because the function has pathological behavior; in other cases, it could go to infinity, which may well…

Using the definition above to compute an integral is cumbersome, to say the least. It may work in some simple cases, but we certainly need something more handy. Luckily, the following theorem, which really deserves the name fundamental,27 provides us with a practical way to compute definite integrals. THEOREM 2.22 (Fundamental theorem of calculus) Let F(x) be a function…

Consider a function f on interval [a, b]. If the function assumes nonnegative values on that interval, it will define a region below its graph; this is illustrated as the shaded region in Fig. 2.31. Now imagine that we are interested in the area of that region. If the function were constant or linear, we would get the…

The last section of this chapter deals with definite integrals. The concept of integral plays a fundamental role in calculus and applied mathematics and, as we shall see, it is in a sense the opposite operation with respect to taking derivatives. In the book, we use definite integrals essentially to deal with continuous random variables…

Series are another important topic in classical calculus. They have limited use in the remainder, so we will offer a very limited treatment, covering what is strictly necessary. To motivate the study of series, let us consider once again the price of a fixed-coupon bond, with coupon C and face value F, maturing at time T. If we discount…

One of the most fruitful application fields of quantitative methods is revenue management. Revenue management is actually a group of techniques that can be applied in quite diverse settings, such as pricing of aircraft seats or perishable products. In this section we consider an idealized case in which a manufacturer has to find an optimal price…

Convexity and concavity play a major role in optimization. Consider a one-dimensional optimization problem, ; this problem is unconstrained, since x can be any point on the real line. Furthermore, assume that f is convex on the whole real line  and that x* is a stationarity point. Property 2.18 applies to x*: for any , but this implies that x* is a global minimizer. We have proved the following theorem.…

Convexity can be easily generalized to functions by applying the idea of convexity for sets to the epigraph of the function. For functions of a single variable, which can be plotted on a plane, the epigraph of the function is just the set of points lying above the function graph. The idea generalizes to an arbitrary number…

Convexity can be introduced as a fairly intuitive concept that applies to n-dimensional subsets of . Spaces with multiple dimensions will be the subject of next chapter, but we can visualize things on a plane, which is just the set  of points with two coordinates. We use boldface characters when referring to a point , with coordinates (x1, x2). Subscripts…