Lec2.pdf - Calculus II Lecture 2 Outline 1 Recall of...

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Calculus IILecture 2
Outline1Recall of Lecture 1ObjectContinuityPartial derivatives2Tangent planes3The Chain Rule4Directional Derivatives and the Gradient VectorTangent plane to level surface5Maximum and Minimum6Lagrange Multipliers
Function of several variables
Main objectA functionfof two variablesis a rule that assigns to each orderedpair of real numbers(x,y)in a setDa unique real number denotedbyf(x,y). The setDis the domain offand its range is the set ofvalues that takes on, that is,{f(x,y) : (x,y)D}.
GraphIffis a function of two variables onD, then thegraphoffis(x,y,z)R3:z=f(x,y),(x,y)D.E.g. forf(x,y) =p9-x2-y2
Figure:Topographics maps of mountain
Figure:Weather map of the world indicating the average January temperatures
DiscontinuityForL16=L2andIff(x,y)L1as(x,y)(a,b)along a pathC1,Iff(x,y)L2as(x,y)(a,b)along a pathC2,then lim(x,y)(a,b)f(x,y)does not exist.
Examples
Examples
Partial derivativesDefinitionThepartial derivativeoff with respect to xat(a,b)isfx(a,b) =limh0f(a+h,b)-f(a,b)h.

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