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QABE_Notes14 - QABE Lecture 14 Dierentiation Responding to...

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QABE Lecture 14 Differentiation: Responding to Change School of Economics, UNSW 2011 Contents 1 Introduction 1 2 Limits 2 3 Rates of change 2 3.1 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.2 Approximating the solution . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Applying the approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 Differentiation 6 4.1 Rules on One Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2 Rules on Multiple Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.3 Rules on Functions of Multiple Variables . . . . . . . . . . . . . . . . . . . 8 1 Introduction So far we have concerned ourselves mostly with functions as mappings (telling us for a given input, or set of inputs, what the output will be) – drawing them, solving them, finding feasible solutions by using them as boundaries – we haven’t necessarily been interested in the way that these outputs change in response to changes to the inputs . This is therefore, a natural topic for us to consider now. Indeed, the nature of change in the real-world is an ever-present phenomenon. Whether it be changes in fish populations due to over-fishing or pollution, or changes in the birth-rate, or changes in the supply of certain raw materials or finished goods, change is all around us. The trick is, that many of the elements that change are ac- tually inputs to our neatly constructed equations, which aimed at representing certain processes occurring in the real world. Hence, it becomes necessary to consider what a change in an input might do to an output. Of course, this subject is not at all confined to economics and business inquiry. We owe a great debt on this score to our friends in physics, astronomy and mathematics, who for many millennia (OK, about two), have been interested in thinking about the cause-effect nexus of change in the world. In approaching this very large field of inquiry, normally taught as one pillar of the twin pillars of calculus ( integration being the other pillar), we shall first ask what we mean by a derivative ? Or rather, what we mean by working out the rate of change of a function at a given point on the function’s domain. Then, once we’re happy about the 1
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ECON 1202/ECON 2291: QABE c School of Economics, UNSW
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