QABE_Notes14 - QABE Lecture 14 Dierentiation: Responding to...

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QABE Lecture 14 Diferentiation: Responding to Change School of Economics, UNSW 2011 Contents 1 Introduction 1 2 Limits 2 3 Rates of change 2 3 . 1 Th ep rob l em . ................................. 2 3 . 2 App ro x im a t ingth es o lu t i on . ......................... 4 3 . 3 App ly eapp x a t i on. 5 4 DiFerentiation 6 4 . 1 Ru l e sonOn eF un c t i ............................ 6 4 . 2 Ru l e sonMu l t ip l c t i on s.......................... 6 4 . 3 Ru l e sonF c t i so fMu l t l eV a r i ab l e s................... 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, ±nding 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 ±sh populations due to over-±shing or pollution, or changes in the birth-rate, or changes in the supply of certain raw materials or ±nished 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 con±ned 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-e²ect nexus of change in the world. In approaching this very large ±eld of inquiry, normally taught as one pillar of the twin pillars of calculus ( integration being the other pillar), we shall ±rst 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 o fEconomics, UNSW
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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_Notes14 - QABE Lecture 14 Dierentiation: Responding to...

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