QABE_Notes16 - QABE Lecture 16 Differentiation III:...

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Unformatted text preview: QABE Lecture 16 Differentiation III: Optimization in One Variable School of Economics, UNSW 2011 Contents 1 Introduction 1 2 Extrema of Functions 2 2.1 Classification by Observation . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Classification by the Second Derivative . . . . . . . . . . . . . . . . . . . . 4 2.3 Concavity & Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Points of Inection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Applied to the Problem 7 1 Introduction After the past two weeks of introducing the various techniques of differentiation (or going over old ground for some) you may be keen to see your new knowledge applied to the real world. Not only is differentiation of interest to give us access to measurement of rates of change (e.g. the speed or acceleration of a car), but with some insight we can get at a similarly powerful concept that of optimization . Optimization is (literally) the process of finding the optimum. The highest, the lowest; the maximum, the minimum; the top, the bottom; the ... you get the picture. It turns out that the work we have done in finding the slope of a function, which is equivalent to, say, finding the gradient of a hill, helps us to identify when the ground beneath our feet (figuratively speaking) is at and when it is at, if you think about it for a moment, you know that you have reached a point that is a little higher, or a little lower, than the surrounding ground. In some cases, it will be the highest or lowest point for miles into the distance. That is exactly the same logic we apply to find maxima and minima of functions. However, we need to pay some care to exactly what we have found when we use our new-fangled optimization methods have we found the highest point? ... or perhaps just a bump along the way. To get our thinking on the practical track, well start and finish with our nascent micro-coffee roasting business, The Bean House , to see if we can help our business partner to find the optimum quantity of Boutique Blend packs to produce in order to yield maximum profits. This is similar in avour to the work we did in linear program- ming a few weeks back, but as youll see, optimization by differentiation drills down on the functional form of the problem itself. For now, to the hills!...
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This note was uploaded on 09/23/2011 for the course ECON 1202 taught by Professor Lorettiisabelladobrescu during the One '11 term at University of New South Wales.

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QABE_Notes16 - QABE Lecture 16 Differentiation III:...

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