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math_review - Chapter 0 Mathematical Review We review here...

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Chapter 0 Mathematical Review We review here some of the basic mathematics required to study and learn microeconomics. Please note that I will update this as well as make any revisions and corrections as we progress through the course. 0.1 The Derivative The calculus is the most widely used mathematical technique for microe- conomic theory and analysis. 0.1.1 Maximization of a function of one variable A function of one variable can be written simply as: y = f ( x ) When x changes from some initial value x 0 by a some amount 4 x to ( x 0 + 4 x ), the value of the function y = f ( x ) changes from f ( x 0 ) to f ( x 0 + 4 x ). The change in y per unit of change in x can be represented by a difference quotient : 4 y 4 x = f ( x 0 + 4 x ) - f ( x 0 ) 4 x (1) 1
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2 CHAPTER 0. MATHEMATICAL REVIEW Frequently, we are interested in the rate of change of y when 4 x is very very small. That is, as 4 x 0, we can identify the limit (if it exists) of the difference quotient 4 y/ 4 x as the derivative of the function y = f ( x ) which we denote as ∂y/∂x or f 0 ( x ). That is, ∂y ∂x f 0 ( x ) lim 4 x 0 4 y 4 x lim 4 x 0 f ( x 0 + 4 x ) - f ( x 0 ) 4 x
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