380 L2-Mathematical ReviewR

380 L2-Mathematical ReviewR - Mathematical Review (Really...

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Mathematical Review (Really More than a Review) Economics 380 R. Pope Introduction Economists aren’t mathematicians but we use mathematics along with most sciences. Here are a few quotes to show that it is a process to accomplish the mathematical skill and comfort we seek. "Mathematics is the gate and key to the sciences." -- Roger Bacon [Statement without proof but I agree] "I hear and I forget. I see and I remember. I do and I understand." -- Chinese Proverb. [Some don’t hear and forget, some do and don’t understand] "In mathematics, you don't understand things. You just get used to them." -- Johann von Neumann [I’ve been around a long time so maybe I’m used to things that aren’t comprehensible to you-stop me at those times] "There are two ways to do great mathematics. The first is to be smarter than everybody else. The second way is to be stupider than everybody else -- but persistent." -- Raoul Bott [What mathematics I know likely came through persistence] "Black holes are where God divided by zero." -- Steven Wright [Just for fun] "I recoil with dismay and horror at this lamentable plague of functions which do not have derivatives. -- Charles Hermite [Most of the time, we will assume not only continuity but differentiability. However, non-differentiable examples often teach us something important] Most important central ideas of modern economics can be accessed without much mathematics but do require deductive mathematical reasoning. However, these ideas can’t be made rigorous and can’t be empirically tested without quantification. Economics 380 formalizes much of the micro-theory that one learns in Economics 110. Interesting theoretical conclusions arise that weren’t evident in Economics 110. In order to proceed with more formalism, we need to make sure we have the necessary algebraic and calculus tools. We proceed from review to a few ideas that most of you
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have not had in math classes: calculus of several variables. We focus most on differential calculus but integral calculus is important in many areas of economic theory and application. It will be fun. Sets (We use notions of sets quite superficially) A set is a clearly specified collection of elements. It may contain a finitely or infinite number of elements. The most common set of elements for this class is R the set of real numbers. A set without any elements is usually denoted or the null or empty set. The symbol is generally used to denote membership in a set. Thus, -2 R . A subset of R is the set of nonnegative numbers denoted R + . The set of positive numbers is often denoted R ++ . Thus, 2 R  while zero is not in ( ) R  . An example of a subset is an interval in R . It is open if it doesn’t contain it’s endpoints (a,b) or e.g., all numbers between 1 and 2 but not including a or b. It is a closed interval (subset of R) if it includes a and b written [a,b]. For example, all numbers between 1 and 2 including these two endpoints. An interval can be half open or closed (a,b] as well. There are a number of
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380 L2-Mathematical ReviewR - Mathematical Review (Really...

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