多元函数微分学部分习题解析.pdf

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1、õ¼ê©ÆS)!Àq1x′§f(x,y)=xy+(y−1)arcsiny,fx(x,1)=())ÛµÏǑf(x,1)=x,¤±fx(x,1)=1p2,xyu=z,du(1,1,1)=()1(lny−lnz))Ûµu=exux(1,1,1)=0,uy(1,1,1)=1,uz(1,1,1)=−1,¤±du=dy−dz3,¼êf(x,y)3:(x0,y0)?)Û5§(·´£A)ëY´ê37^£B)ê3ëY¿©^£C)ê3´¿©7^(D)´ëYê3¿©^)ÛµD(2∂u=x,2∂u4§u=u(x,y)§y=x§ku(x,y)=19∂

2、xy=x(x6=0)§∂y=(),A.1,B.−1,C.0D.122∂u1222121)Ûµd∂x=x,=⇒u=2x+ϕ(y)5¿u(x,x)=1=⇒ϕ(x)=1−2x,=⇒ϕ(y)=1−2y121∂u1¤±u(x,y)=2x+1−2y,=⇒∂y=−22sin(xy),xy6=0xy,′(0,1)=()5.f(x,y)=fxx,xy=0√A.0,B.1,C.2D.Ø3f(x,1)−f(0,1)sinx2)Ûµfx(0,1)=limx→0x=limx→0x2=1y∂z∂z6.

3、^Cþu=x,v=xò§x∂x+y∂y=zzǑ#§()√A.u∂z=z,B.

4、v∂z=z,C.u∂z=z,D.v∂z=z∂u∂v∂v∂u1∂z∂z∂zy)Ûµ∂x=∂u−∂vx2∂z=∂z1∂y∂vx∂z∂z∂zu´z=x∂x+y∂y=u∂u7.∂z∂zd§F(cx−az,cy−bz)=0(½¼êz=f(x,y),F(u,v)äkëYê§a∂x+b∂y=()A.a,B.b,C.c,D.1∂z)−Fb∂z=0,∂zcF1)ÛµF1(c−a∂x2∂xu´∂x=bF2+aF1∂z∂zqF1(−a∂y)+F2(c−b∂y)=0∂z=cF2∂yaF1+bF2∂z∂zacF1+bcF2u´a∂x+b∂y=aF1+bF2=cx=asin2t8.

5、πmΓ:y=asintcost3:t=4?{²¡7()z=ccos2tA.²1ux¶B.²1uy¶,C.²1uz¶,D.n
þπ)ÛµΓþǑ{asin2t,acos2t,−csin2t}u´:t=4{²¡{þǑ{a,0,−c}acu´{²¡§Ǒa(x−2)−c(z−2)=0²1uy¶x=acosτcost9§Γ:y=asinτcost(a,τ)Ǒ~êz=asintA.½:B.½ØÏ:,C.´ÄÏ:t0k',D.´ÄÏ:τk'′′′)Ûµx(t)=−acosτsint,y=−asinτsint,z=acost{²¡−

6、acosτsint0(x−x0)−asinτsint0(y−y0)+acost0(z−z0)=0{²¡¥~êǑx0acosτsint0+ay0sinτsint0−az0cost0222=asint0cost0(cosτ+sinτ−1)=0,={²¡½:10.¼êf(x,y)3:(0,0),k½Â§fx(0,0)=3,fy(0,0)=−1§k2A.dz

7、(0,0)=3dx−dyB¡z=f(x,y)3:(0,0,f(0,0)){þǑ(3,−1,1)z=f(x,y)C3:(0,0,f(0,0))þǑ(1,0,3)y=0z=f(x,y)D3:(0,0

8、,f(0,0))þǑ(3,0,1)y=0)Ûµdu¼êؽ§¤±Ø±Ñ©§²¡{þAǑ{3,−1,−1}§¤±AǑx=x,y=0,z=f(x,0)þǑ{1,0,3}11.32z=x−3x+y,§3:(1,0)?()A.4§B.4§C.Ø4§D.´Ä4§ØU(½2)Ûµzx=3x−3,zy=2y7:Ǒ(1,0)A=f(1,0)=6,B=f(1,0)=0,C=f=2,AC−B2>0,A>0xxxyyy412.eª¥½Ø´,¼ê©´¤£A.ydx+xdy,B.ydx−xdy,C.xdx+ydy,D.xdx−ydy)Ûµ

9、ÏǑA,C,D©´¼ê©§ÏdkBØ´,¼ê©213.∂z∂x∂y=2x−y¤á¼ê´A.z=x2y−1xy2+ex+y,B.z=x2y−1xy2+ex22C.z=x2y−1xy2+sin(xy),D.z=x2y−1xy2+exy+322)ÛµBf(a+x,b)−f(a−x,b)14.f(x,y)3:(a,b)?ê3§lim=()x→0xA.0,B.fx(2a,b),C.fx(a,b),D.2fx(a,b))ÛµD315.π¡z=f(x,y)²¡y=y03:(x0,y0,f(x0,y0))?x¶¤¤ÆǑ6