Calculus_and_Optimization_StudyGuide

Calculus_and_Optimization_StudyGuide - Environmental...

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Environmental Economics AEM 451/Econ 409, 2008 Calculus and Optimization: A Review This course requires use of simple calculus and optimization. The purpose of this handout 1 is to provide a brief refresher of derivatives and an introduction to constrained optimization--an important tool in economics. With an eye toward measuring Hicksian and Marshallian surplus values, basic elements of integral calculus will also be introduced. 1. Derivatives The derivative of a function Y = f(X) with respect to X at the point X = X 0 is defined as: ( 29 ( 29 ( 29 0 0 X X 0 X X X f X f lim X dX dY 0 - - = provided the limit exists. This limit is called the rate of change , or slope, of the function Y with respect to X at X = X 0 . In graphical terms, it is the tangent of the curve at X 0 . In most cases we will want to take the derivative of Y as a function of X, and not just the derivative of the function at a particular point. The derivative of a function Y = f(X) as a function of X is defined as: ( 29 ( 29 h X f h X f lim dX dY 0 h - + = As an example, suppose that the relationship between profit (π) and quantity (Q) produced is given by: π = - 4 2 Q Q One could try to graph this or try several guesses in order to find the level of Q that maximizes profit. However, the most straightforward approach is to take the derivative of the function, set the derivative equal to zero and then solve for Q: h ) Q Q (4 h) (Q h) 4(Q lim dQ 2 2 0 h - - + - + = Q 2 4 h Q 2 4 lim h h Qh 2 h 4 lim h Q Q 4 h Qh 2 Q h 4 Q 4 lim 0 h 2 0 h 2 2 2 0 h - = - - = - - = + - - - - + = d π /dQ = 0 when Q*=2. (* is often used to indicate the optimal level of the variable).
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Page Fortunately, this cumbersome limit approach does not have to be used each time we want to find the derivative of an equation or a line tangent to a curve. The effectiveness of calculus as a mathematical tool rests in part on the fact that derivatives of many functions can be computed without referring back to the definition of the derivative: they can be computed by application of the rules of differentiation. Some of these rules are complex, but most are very simple (all derivatives in this class will take a simple form; if more complex forms are required formulas will be provided). For example, the first derivative of a nonlinear function of the form Y cX n = (where c is a constant) is the following: ( 29 1 n ncX dY/dX - = Consider the following functions and their first derivatives: Y = X 2 dY/dX = 2X (2-1) = 2X Y = 7X dY/dX = (7) (1) X (1-0) = 7 Y = 5X 9 dY/dX = (9) (5) X (9-1) = 45X 8 Y = 55 + X -1 dY/dX = -1X (-1-1) = -X -2 (note: the derivative of a constant is zero) Y = 0.5X 0.5 dY/dX = (1/2) (0.5) X (0.5-1) = 0.25X -0.5 The second derivative is defined as the derivative of the first derivative. While the first derivative gives the slope of the function, the second derivative measures the rate of change in the slope of the function. The second derivative is calculated in the same way as the first derivative, except now we take the derivative of the first derivative instead of the derivative of the original function. Let d 2 Y/dX 2 denote the second derivative and consider the following
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