QABE_Notes20

QABE_Notes20 - QABE Lecture 20 Multivariable Calculus The...

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Unformatted text preview: QABE Lecture 20 Multivariable Calculus: The Partial Derivative School of Economics, UNSW 2011 Contents 1 Introduction 1 2 Functions of two-variables 2 2.1 Seeing it Graphically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Derivatives in this Context . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 The Partial Derivative Method 5 3.1 First order partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Second order partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Seeing it graphically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4 Methods of Partial Differentiation 7 4.1 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 Total Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1 Introduction Up until now you may have been wondering why we have only dealt with (mainly) functions of just one variable (e.g. f ( x )). The reason is that this is the simplest case, and therefore is a good place to begin. However, as you are no doubt aware, much of the useful quantitative work that gets done is on functions of more than one variable — multivariable functions (e.g. f ( x, y )). (In fact, if you can remember back to linear programming, we were actually dealing with functions of this sort, but only in very straight-forward, but still useful, ways). What does a function of more than one variable look like? Given the usefulness to business and economic problems of being able to find the maxima and minima of functions, can we do the same in the many variable case? In regards to the first questions, if the function is just of two-variables then we can make a representation of it in three- dimensional space. But anything more than this, and we don’t really have the means to show it graphically. For the purposes of this course, however, we’ll stick to functions of two-variables. Moreover, the techniques we develop on two-variable functions are all applicable to any multivariable functions. In terms of the second question – about finding maxima and minima — this requires us as a first step to be able to deal with rates of change in the multivariable case. Here, 1 ECON 1202/ECON 2291: QABE c School of Economics, UNSW we do a similar thing to what we did in the single variable case, but we have to make sure we are careful that we are precise about the input/output rate-effect that we are analysing. Since in this many-variable world, we have more than one input variable that might cause the output variable to change. Therefore, when we try to look at the slope of the function, we must look at one variable’s effect at a time. This technique is calledof the function, we must look at one variable’s effect at a time....
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QABE_Notes20 - QABE Lecture 20 Multivariable Calculus The...

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