12.1Functions - 12.1FunctionsofSeveralVariables Onevariable

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Unformatted text preview: 12.1FunctionsofSeveralVariables Onevariable Thestatementy=f(x)or"yisafunctionofx"meansthaty dependsonx. x=independentvariable y=dependentvariable Domain={x|f(x)isdefined} Range={f(x)|xisamemberofthedomain} Twovariables z=f(x,y)or"zisafunctionofxandy"meansthatzdepends onbothxandy. xandy=independentvariables(Valuesareassigned independentlytoxandy.) z=dependentvariable Domain={(x,y)|f(x,y)isdefined} Range={f(x,y)|(x,y)isanelementinthedomain} Threeormorevariablescan'tbegraphed,buttheconcepts staythesame.Forexamplew=f(x,y,z):wisthedependent variablewhilex,y,andzaretheindependentvariables. Ex. f (x, y) = y 2 ! x 2 Domain= Range= f (x, y) = x 2 + y 2 Ex. Domain= Range= f (x, y) = x 2 + y Ex. Domain= Range= Ex. ! y$ f (x, y) = arctan # & " x% Domain= Range= Ex.Belowisachartofelevationabovesealevelatanypoint (x,y)infeet. xy 6 4 2 0 2 4 6 4 152 186 209 218 209 186 152 2 193 236 266 277 266 236 193 0 209 256 288 300 288 256 209 2 193 236 266 277 266 236 193 4 152 186 209 218 209 186 152 f(4,2)= Thevalueofxwheny=0sothattheelevationis300is Thevalueofxwheny=0sothattheelevationis256is f (x, y) = x 2 + y 2 . Ex.Sketchthegraphof Ex.Sketchthegraphof f (x, y) = y 2 ! x 2 . Do:Findthedomainandrangeofproblems1and2and sketchthedomain. 1. x!y g(x, y) = x+y 2. h(x, y) = ln ( x + y ! 1) 3. Graph f (x, y) = cos y LevelCurves Alevelcurve(orcontour)forafunctionz=f(x,y)isthecurve ofintersectionofthesurfacez=f(x,y)andthehorizontal planez=k. f (x, y) = x 2 + y 2 Ex. y g(x, y) = 2 x Ex. Note: 1. Theequationofacontour(levelcurve)isf(x,y)=k, plottedinthexycoordinatesystem. 2. Allpointsonaparticularcontourhavethesamezvalue. Thatis,fisconstantonalevelcurve. 3. Contourscannotintersect. Do:1.Drawanddescribethelevelcurvesfork=0,1,2of f (x, y) = x .Whatisthedomainandrangeofthis function? 2.Drawanddescribethelevelcurvesfork=0,2,5for f (x, y) = 4x 2 + y 2 + 1 . ...
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