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Calculus Review

# Calculus Review - UNC-Wilmington Department of Economics...

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UNC-Wilmington ECN 321 Department of Economics and Finance Dr. Chris Dumas Calculus Review Calculus Calculus is the study of the rates of change of interrelated variables. It turns out that just about all of economics is about the rates of change of interrelated variables; so, just about all of economics is about calculus. There are two main parts of calculus: differentiation and integration . We will focus on differentiation in this handout. Differentiation Differentiation is the process of finding the rate of change of one variable with respect to changes in another. Suppose we have two variables, x and y, related together by function f: y = f(x). The derivative of y with respect to changes in x, denoted dx dy , gives the change in y that results from a small change in x. If we graph the relationship of y and x, the change in y that results from a small change in x is the slope of the graph ; hence, the derivative dx dy gives the slope of the graph of the relationship between y and x . IT TURNS OUT THAT MANY IMPORTANT THINGS IN ECONOMICS ARE SLOPES OF GRAPHS !!!! THUS, MANY IMPORTANT THINGS IN ECONOMICS ARE DERIVATIVES !!!! Amazingly , marginal cost, marginal revenue, marginal product, marginal utility, marginal benefit, marginal profit, and even part of the formula for elasticity are all derivatives!!!!!! 1 x y y = f(x) dy/dx is the slope of the y = f(x) graph

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UNC-Wilmington ECN 321 Department of Economics and Finance Dr. Chris Dumas Differentiation Rules for Common Functions of One Independent Variable : In the rules below, dependent variable y is a function of one independent variable x. In addition, a, b and c are parameters, e is the base of natural logarithms, and f is a function. A Constant (The derivative of a constant is zero.) If y = a, then dy/dx = 0 Examples: If y = 6, then dy/dx = 0 If y = 1/2, then dy/dx = 0 If y = 0, then dy/dx = 0 Linear Function If y = ax, then dy/dx = a Examples: If y = 6x, then dy/dx = 6 If y = 4x + 3z, then dy/dx = 4 Exponent Power Function If y = ax b , then dy/dx = b[ax (b-1) ] (Note: b can be a negative constant, or a fractional constant.) Examples: If y = 3x 5 , then dy/dx = 5∙[3x 4 ] = 15x 4 If y = 20x 3 , then dy/dx = 3∙[20x 2
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