可微分局部线性数calc 13

13.4

Differentiability,Differentialsand

LocalLinearity

DifferentiabilityThegraphofy=f(x)hasnonverticaltangentlineatthepoint.fmaybeclosedapproximatedbyalinearfunctionnear.fiscontinuousat.

differentiabilityIfafunctionf(x,y)isdifferentiableatwewantittobethecasethatthesurfacez=f(x,y)hasanon-verticaltangentplaneatthispoint,Thevaluesoffatthispointcanbeveryclosedapproximatedbythevaluesofalinearfunction;fiscontinuousatthispoint.

Onecouldreasonablyconjecturethatafunctionfshouldbecalleddifferentiableatapointifallthefirst-orderpartialderivativesofthefunctionexistatthatpoint.Butwehaveseenaexampleinlastsection,thatthemereexistenceofbothfirst-orderpartialderivativesisnotsufficienttoguaranteethecontinuityofthefunction.

differentiability

IncrementSimilarly,letz=f(x,y).Thatis,?xand?yaretheincrementsofxandy,andtheincrementofzisgivenby

differentiability

differentiability

Example2–Showthatthefunctiongivenbyf(x,y)=x2+3yisdifferentiableateverypointintheplane.Solution:Lettingz=f(x,y),theincrementofzatanarbitrarypoint(x,y)intheplaneis?z=f(x+?x,y+?y)–f(x,y) I=(x2+2x?x+?x2)+3(y+?y)–(x2+3y)=2x?x+?x2+3?y

Example2–Solution=2x(?x)+3(?y)+?x2=fx(x,y)?x+fy(x,y)?y+?x2itfollowsthatfisdifferentiableateverypointintheplane.Figure13.34

Differentiability

DIFFERENTIALS

Thisdefinitioncanbeextendedtoafunctionofthreeormorevariables.Forinstance,ifw=f(x,y,z,u),thendx=?x,dy=?y,dz=?z,du=?u,andthetotaldifferentialofwisDIFFERENTIALS

LINEARIZATIONThelinearfunction,namely L(x,y)=f(a,b)+fx(a,b)(x–a)

+fy(a,b)(y–b)iscalledthelocallinearapproximationoffat(a,b).Equation3

DIFFERENTIALSThefigureshowstherelationship

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