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Calculus of Optimization Review

Calculus of Optimization Review - Calculus of Optimization...

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Calculus of Optimization Review for AAE 320 Dr. Paul D. Mitchell (608) 265-6514, [email protected] This document provides a quick review of calculus-based optimization for AAE 320. It assumes a basic understanding of calculus, but that you need a quick “refresher.” It goes through the general rules, then works through an example, then gives sample problems (Think Breaks). A useful review of calculus based optimization is available at http://www.swlearning.com/economics/mcguigan/mcguigan9e/web_chapter_a.pdf . This pdf is a web chapter supplement for the text Managerial Economics (9 th ed.) by McGuigan, Moyer, and Harris. Note that I would like students to know univariate and multivariate unconstrained optimization, and so for AAE 320 you do not need to know the constrained optimization methods described in the web chapter. However, you may want to know them for other AAE courses. Part 1: Review of Derivatives Before explaining optimization, first let’s review derivatives. A derivative is the slope of a function. For some functions the slope is constant, say 1.5, for other functions, the slope changes depending where on the function you are, and so the slope if a function as well, say 1.5 + 3x. Notation for Derivatives Three different notations are used for 1 st and 2 nd derivatives of y = f(x) 1) dy dx and 2 2 d y dx Newton 2) f (x) and f ′′ (x) Leibniz 3) f x (x) and f xx (x) Leibniz They all mean the same thing, but sometimes one is easier to use than the other. Finally, not that the second derivative is just the derivative of the derivative Rules of Differentiation (Web Chapter A, Table A.1, page 12) Constant Function If y= f(x) = a f (x) = 0 Power Function If f(x) = ax b f (x) = bax b – 1 If f(x) = 7x f (x) = 7 If f(x) = 7 x 0.34 f (x) = 7(0.34)x 0.34 – 1 = 2.38x –0.66 Sum of Functions If y = f(x) + g(x), then dy dx = f (x) + g (x) If h(x) = 3 + 4x – 7x 2 , then h (x) = 4 – 2(7)x 2-1 = 4 – 12x If k(x) = 56 + 44x 0.5 – 17x, then h (x) = 0.5(44)x 0.5-1 – 17 = 22x -0.5 – 17
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Review Rules of Differentiation Find the first and second derivatives of the following functions.
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