下冊(cè)-第九章試題解答

一、1、xyx2、xyyx2且x+y3、xyyx且x4、xyx2 5、x,yx2y21且 6、x,y2kx2y22k+1,k0,1, 7、xy1x2y24且y-x8、xyzr2x2y2+z29、x2x2x2x2x2x2 x2 x,y0,0x212 xy

xy11x, xy1sin

x,sin

xy11314

yx, x, 10x2y2 2

0,x 2 x,y,xy x,

xysin1cos1 16z3x2sin2yz2x3cos2 17

x0,y0

2x03y0

x0,y0

y

3x0218、u

x,uy,uzx2y2ux2y219f/0,0limf0x,0f0,0lim00 f/0,0limf0,0yf0,0lim00

y02x2

, 20、 2xlnxy

x2y2 x221zyexyz 22zexcosysinyxsiny,zexxcosysiny

2

23 1xy2z3 1xy2z324uyzxylnzuxzxylnzu

1xy2z325zy1zxx26

y yzu z sinsin, sinsin y y y1 27

uyxz

1xzlnx,u

xzlnx28

yxy 1y xy1x1x1

xyxy2 xy1xxy2yy y11xxy2yy29

z 1xyyxy 1 xy 21

1xy

1xyxy1xyz 1xyxxy 1 xy 21

1xy

1xyxy1xy30 2

yxy1

yy1xy2, y

ln2z

2 2

xlnx,xyyx lnx31、令r x2y2z2x2y2z2 x2y2u 2x xx2y2z2 x2y2 x2y2 r2 r2 rx2y2z22ux2y2z22x2y2z2 2ux2z2y22ux2x2y2z2

r , r , r

2u2u 2u2u 2u2u2 r4, r4, r4z1xexsiny,zxexcos33

2

2

2sin xexsin

2

；1xexcosz4x38xy2,

4y38x2y,2z12x28y22

2

2

34

12y28x2

yxz2sinaxbycosaxbyaasin2axbyz2sinaxbycosaxbybbsin2axby2

2a2cos2axby

2b2cos2axby2

2

2abcos2axby35、dz2xy3dx3x236、dz dx dy 3dx3x2 3x2 3x2y y y 1y 37、dzex dxe dy exdxdy x2 38

u1,u1cosy ,u y2z2 y2 2dudx1cosy dy 2 y y2 3940

x

e2,

2e2,dze2dx2a2x22a2x2y2

2x a2x2a2x2y2

,u a2a2x2y2a2x2y2a2x2y2du a2x2y241zyxyxy xxy xy xy 2xy

xy

xydz42

xy

y2dxx2dyx x 11y

y x2 x2x

1xy1xy 1x2xx ,dz43

x2xx2x

xxxdyx44

z2xy,zx22 dz2xydxx22ydz 1x2

dx 21x2

1dx2 fxyxyx01y04x0.08y

1,f/1,4y

4,f/1,4xyln

xy1.08396140.0800.04xy46設(shè)z x3y3x3xx3xx3yyxy

z

3x2xx3x3x3x3這里取x1,y2,x0.02,y1.02311.0231 213

30.02322

474設(shè)zln3x 1,4

4ln31.03 1zxzy x 44433 133 44334ln31.03 1 10.0310.024

111

fxyxy取x01,y02,x0.04,yy求出f1,2xyf12xylny

xx

1.04202120.0400.02fxyxy取x02,y01,x0.03,yy求出f2,1xyf2,1xylny

2,f/2,1yx20.693x

50

1.97105210.031.3860.0551

z 2x22x2y2x2y2arctan3

3,1,1,3,z 2 lnxy1,x2x,x2y2z1, 1 y 5253

zzuzveusinvyeucosv1exyysinxycosxy u vzzuzveusinvxeucosv1exyxsinxycosxy u vzfufv2xfyf u v zfufv2yfxf u v 54、uff/ 55zfufv u vzfufvf56

u v uf(s,2ufsftf/xyz,x2y2z22xf/xyz,x2y2z22 s t 2ufsftf/xyz,x2y2z22yf/xyz,x2y2z22 s t 2ufsftf/xyz,x2y2z22zf/xyz,x2y2z22 s t 57

zxzyzcos z y zzxzyzrsinzrcos x y z2r

1z 1 r2 cos sin rsin r z2z

r

x

y58

zyf/yf/,zxf/1f/ x22 x2 f yf 2

2 2 1

2 f/y2f//

f, x2

2 2 x

2

f2259

2z2z xyf11x3f22f1x2f2 Fx,ysinyexxy1,F/exy,F/cosy 2F/F/F//2F/F/F//F/2Fxy /2FxFydxcosy 2 /xFy 2exy exy2sin 60

xcos

xcosy

xcosy Fx,y,zy2xzz2ez4,F/z,F/2y,F/x2z z ,z x2zez x2z .2 z2x2zzez2ez .61

x2xx2zez

x2zezz2uuvu2vuv22uvv2uvv2uu22uv 3x2sinycosycosysinyzx3sin3ycos3y2sinycosysinycosy 62du eaxayzeaxacosxsinx a21 eax a2sinxacosxacosxsinxeaxsinx.a21 63、證明：令ux2z

f2xy2xy f2uz 2 f2ufuyu2yfu

f2u1z1z zx y yfu y2fu 64

y

f y

2nxnfu 2nxnfuxn x zxnf1xn1 u xzyznxnfuxn1yf

f65

x2y22xxyx2y2y x4y42x322z2xexyx2y2exy 22

x

x2 zy4x4

x2exy66

uf(s,2ufsft2xf/x2y2,exyyexyf/x2y2,exy2 s t 2ufsft2yf/x2y2,exyxexyf/x2y2,exy2 s t 67 /,u2yf/,u67 68uf(x,s,uffsftf/x,xy,xyzyf/x,xy,xyzyzf/x,xy,xyz s t ufsftxf/x,xy,xyzxzf/x,xy,xyz,uxyf/ s t 692 2x2ydy 2 x2x2x2x2 x2 70

xydyxdyy,dyxy xylnxxlny,dylnxy1lnyx1dy ylny x

lnyxy

xlnyxy1y71

dxlnxx xylnxy ylnxxyx1 f f f 2 rxdurxdxrydyrx

dx 2x22x2f1xdxydyr722cosxsinx2coszsinzz0,zsin2x sin73

zsin2y sin3x23z2z3yz3xy

x274

y2yxyz2

x0,xxyz2Fx,y,zezxyz,Fyz,Fezxy,zFx ,

z

ez ezxyyzez

y2 z

2y2zez2xy3zy2 75

ezxy ezxyFx,y,zz33xyz,F3yz,F3xz,F3z2 zFx z2xy,yFz2xy z

2 z

z

z2xyyz2z

xyy yy yx yz2xy z2xyzz42xyz2x2y2..76

z2xy2x2zz42z0,z2x,z2y z1 zdz 12xdx2z1 2cosx2y3z13z13z,2cosx2y3z23z2 x

z

兩式相加2cosx2y3z33x3y33x3 z

2cosx2y3z1 zz 78

Fx

z,y,z0,Fxx

0,xFy,yFz,zFx yz F xyzFyFzyz F x y z79x/asin,y/acos,z/ ak22 ,k,切點(diǎn)坐標(biāo) 2 2,422 2x- y2

z 2

42 22ax2a a2

a

k

y

kz

80

即xy

2kz2k

2

2 4 z/1,12a,z/1,1 切平面方程為:z-a+b=2ax-1+2by-1,即2ax+2by-z-a-x-

y1

zab81

x1cost,ysint,z

2 切向量T=1,1,2切點(diǎn)坐標(biāo)-1,1,22, zz22 2y1 1

1y1 2z220,即xy

82

2

2y1z/2t,切向量T=11,2切點(diǎn)坐標(biāo)1/ x /

切線(xiàn)方程為：2y2z1 41 1y22z10,即2x8y16z183

4 Fx,y,z3x2y2z227,F6x,F2y,F 法向量T=18,2,-切平面方程：18x-3+2y-1-2z-10,即9x+y-z-..84

Fx,y,zarctanyz,F ,F ,F 1 法向量T=-,,-

x2 x2

x1

1y-1-

0,即xy+2z

2 x4 4

y

z

,即2x-12y1z 85

Fx,y,zezzxy3,Fy,Fx,Fez 切平面方程：1x2+2y-1+0z-00,即x+2y4x y86

2zx/1,y/2t,z

3t2,切向量T=1,2t3t已知平面的法向量1，2，1，切線(xiàn)與平面平 3t=0,t=-1,-所求點(diǎn)的坐標(biāo)為1，1,1,111 ，， 3 2787

2y Fx,y,za2b2c21,Fxa2,Fyb2,Fzc22x0xx02y0yy02z0zz0 b xx yy zz 0 0000 88

f/x113x yx f/4yyx

,0,1,33f//x,y46x,f//x,y4,f//x,y

在,0處,B2AC80,A20,在,0處取得極小值 89

在10處B2AC80,在10z3ay3x23 3ax

得駐點(diǎn)a,a,其函數(shù)值為aAxx

6aCyy

,Bxy

B2AC9a236a20,A函數(shù)在a,a取得極大值90z2e2xxy22ye2xe2x2x2y24y1 ze2x2y2 ,-1 其函數(shù)值為-, Az//1e2x4x4y28y42e,Cz// 2e,Bz// xx,-

yy,-

xy,-

2,-1取得極小值-. 91zay2xyy2 axx22xy

aa 1得駐點(diǎn) ,,其函數(shù)值為 a,33 Az//aa2aC33

33

,Bz//aay,33

3 4BAC9992

aa 10,A0,函數(shù)在 ,取得極大值27a33z15x2y4x3y3x2y2x2y154x3y z5x3x42x3yx35x2y 得駐點(diǎn)3,1,其函數(shù)值為在邊界x+y=6上:z=-4x36x,0x6.dz4x318x22x22x9,z92187,z0z6 2 函數(shù)的最大值為27，最小值為218793

z2x

2