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Lecture 3

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University of Ottawa
Leslie Shiell

ECO 3145 Mathematical Economics I Lecture 3 Implicit FunctionsChiang ch 8586I Representation of Functions1 explicit function notationxxfyn1 y is explicitly expressed as a function of the xs 2 general function notationwhere c is some constant possibly 0 cxxyFn1 strictly speaking it is the lefthand side which is the functionwe say that the equation defines a generalfunction relationship between y and the xs3 implicit function if a relationship is expressed in the generalfunction form it may be of interest to know whether one variable can be expressed as a function of the othersfor example using the notation above does there exist some functionxxgyn1such thatcxxxxgFn1n1 if the answer is yes thenis said to be an implicit function defined by xxgyn1 cxxyFn1 the implicit function has a closedform or explicit solution if an exact expression can be written forie if it is possible to isolate y from all other variables xxgn1 note the existence of an implicit function does not guarantee a closedform solution exercises Do the following equations implicitly define y as a function of the xs If yes 1does the implicit function have a closedform solutioniyx6y4x32211yx1yxii2210y11x1yxiiiyxyx0xyeeivcomments notice the importance of specifying the domain of the general function domain and range of the implicit function 1e See Gabriel Archinard and Bernard Guerrien Analyse Mathmatique pour conomistes 3 dition ParisEconomica 1988 p 301 for a discussion of equation iv 31453doc Shiell 1
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