Example:The function fis defined by( ,)2f x yxxy=+=2(, )?f aa=22223(, )22f aaaa aaa=+=+11.3 The graph of a function of two variablesSuppose z= f(x,y) is a function of two variables over a domain Din the xy-plane.The graph of the function fis the set of all points (x, y, f (x,y)) obtained by letting (x,y) “run through” D.11.3 ExampleThe graph of z= x2+ y2The surface is called a paraboloid.11.1 ExampleDetermine the domain of the function given by the following formula.2211( ,)1ln(2)1f x yxxxyxy=-+++-+-++-++-+-+Domain: 1x≥0xy-≠-2210xy++≠++≠20x->-11.2 Partial derivatives with two variablesConsider the functionThe rate of change of zw.r.t. xis given by:The rate of change of zw.r.t. yis given by:Write2lnzxy=+2dzxdx=1dzdyy=and instead of zzxy∂∂∂∂and dzdzdxdx11.1 Functions of two variablesis called the partial derivative of z= f(x,y)w.r.t. x. You can also use the following notations:•••zx∂∂'xz' ( ,)xfx y1' ( ,)fx y
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311.2 ExampleFind the partial derivatives of the following function:3( ,)lnf x yx yxyy=+32(ln)3ln(holding constant)x yxyyx yyyxy∂+=+=∂33(ln)ln(holding constant)x yxyyxxyxyx∂+=+++++∂11.8 Partial elasticitiesThe definition of elasticity()'()()xxEl f xfxf x=⋅=⋅If fis differentiable in xand f(x) ≠0 we define the elasticity of fw.r.t. xas:1. Function of one variable: f(x)2. Function of 2 variables:z=f(x,y)
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