[MUSIC] Hello everyone, and welcome to the third week of the Quantitative Foundations for International Business MOOC. This week is devoted to the concept of differentiation. A very important question in many scientific disciplines, including decision making and economics, is the study of how rapidly quantities and variables change. In order to compute the position of a planet in the future, in order to predict the population growth of a species, or to estimate the demand for a commodity, we have to have information about the rates of change. The tool used to describe the rates of change of a function is its derivative. This is a central tool in mathematical analysis. This week, we will define the derivative of a function, and present some of the important rules for calculating it. Isaac Newton and Gottfried Wilhelm Leibniz discovered most of these general rules, interestingly independently of each other. This work initiated differential calculus which is the foundation of the development of modern science. It has been of crucial importance to the theoretical development of economics as well. Even though in economics we're usually interested in the derivative as a rate of change, we begin this week, with a geometric motivation for the concept. When we study the graph of a function, we would like to have a precise measure of the steepness of the graph at a particular point. In the previous week, we gave you a rather vague definition of the tangent to a curve at a point. All we said was that it is a line which just touches curve at that particular point. Now we give a more definition of the same concept. The term increasing and decreasing functions have been used of course previously, to be described the behavior of a function as we travel from left to right along its graph. In order to establish a definite terminology, we introduce a number of definitions. We assume that F is defined in an interval L. And that X1 and X2 are numbers from that interval. The derivative of a function at a particular point, was defined as the slope of the tangent line to it's graph at that particular point. Economists interpret the derivative in many important ways. Starting with the rate of change of an economic variable. Many economic models involve functions that are defined implicitly by one or more equations. In some simple but economically relevant cases, we show how to compute derivatives of such functions. The word continues is common even in everyday language. We use the in particular, to characterize changes of a gradual rather than sudden. This user, is closely related to the idea of a continuous function. We discussed this concept, and explained its close relationship with the limit concept. Limits and continuity are key ideas in mathematics, and also very important in the application of mathematics to economic problems. The preliminary discussion of limits were necessarily rather sketchy. We take a closer look at this concept, and extend it in several directions. Next we present the intermediate value theorem, which makes precise the idea that the continuous function has a connected graph. This makes it possible to prove that certain equations have solutions. Thank you very much. [MUSIC]