£¨ÕûÀí£©Êýѧ·ÖÎö¿Î±¾£¨»ªÊ¦´óÈý°æ£©-ϰÌâ¼°´ð°¸µÚÊ®ÆßÕÂ

ÄÚÈÝ·¢²¼¸üÐÂʱ¼ä : 2025/10/14 20:54:37ÐÇÆÚÒ» ÏÂÃæÊÇÎÄÕµÄÈ«²¿ÄÚÈÝÇëÈÏÕæÔĶÁ¡£

¾«Æ·Îĵµ

8. ÇóÇúÃæZ=arctg

y???ÔÚµã?1,1,?´¦µÄÇÐÆ½Ãæ·½³ÌºÍ·¨Ïß·½³Ì.

4?x?9. ÇóÇúÃæ3x2+y2-Z2=27ÔÚµã(3,1,1)´¦µÄÇÐÆ½Ãæ·½³ÌÓë·¨Ïß·½³Ì.

10. ÔÚÇúÃæZ=xyÉÏÇóÒ»µã,ʹÕâµãµÄÇÐÆ½ÃæÆ½ÐÐÓÚÆ½Ãæx+3y+Z+9=0,²¢Ð´³öÕâÇÐÆ½Ãæ·½³ÌºÍ·¨Ïß·½³Ì.

11. ¼ÆËã½üËÆÖµ:

(1) 1.002¡Á2.0032¡Á3.0043; (2) sin29¡ã¡Átg46¡ã.

12. Éè԰̨ÉÏϵ׵İ뾶·Ö±ðΪR=30cm, r=20cm¸ßh=40cm. ÈôR,r,h·Ö±ðÔö¼Ó3mm,4mm,2mm.Çó´Ë԰̨Ìå»ý±ä»¯µÄ½üËÆÖµ.

13. Éè¶þÔªº¯ÊýfÔÚÇøÓòD=[a,b]¡Á[c,d]ÉÏÁ¬Ðø (1) ÈôÔÚintDÄÚÓÐfx¡Ô0,ÊÔÎÊfÔÚDÉÏÓкÎÌØÐÔ? (2) ÈôÔÚintDÄÚÓÐfx=fy¡Ô0,fÓÖÔõÑù?

(3) ÔÚ(1)µÄÌÖÂÛÖÐ,¹ØÓÚfÔÚDÉϵÄÁ¬ÐøÐÔ¼ÙÉè¿É·ñÊ¡ÂÔ?³¤·½ÐÎÇøÓò¿É·ñ¸ÄΪÈÎÒâÇøÓò?

x2?y214. ÇóÇúÃæZ=ÓëÆ½Ãæy=4µÄ½»ÏßÔÚx=2´¦µÄÇÐÏßÓëOZÖáµÄ½»½Ç.

415. ²âµÃÒ»ÎïÌåµÄÌå»ýv=4.45cm3,Æä¾ø¶ÔÎó²îÏÞΪ0.01cm3,ÓÖ²âµÃÖØÁ¿W=30.80g,Æä¾ø¶ÔÎó²îÏÞΪ0.018,ÇóÓɹ«Ê½d=

wËã³öµÄ±ÈÖØdµÄÏà¶ÔÎó²îÏ޺;ø¶ÔÎó²îÏÞ. vdZ; ?x16.ÇóÏÂÁи´ºÏº¯ÊýµÄÆ«µ¼Êý»òµ¼Êý: (1) ÉèZ=arc tg(xy),y=ex,Çó

(2) ÉèZ=

x?yexy22x2?y2xy,Çó

?Z?Z,; ?x?y?Z; dtu?Z?Z(4) ÉèZ=x2lny,x=,y=3u-2v,Ç󣬣»

v?u?v(3) ÉèZ=x2+xy+y2,x=t2,y=t,Çó(5) Éèu=f(x+y,xy),Çó

?u?u,; ?x?y(6) Éèu=f???xy??u?u?u,Çó,,. ,???yZ??x?y?Z17.Çóº¯Êýu=xy2+z3-xyzÔÚµã(1,1,2)´¦ÑØ·½ÏòL(Æä·½Ïò½Ç·Ö±ðΪ60,¡ã45¡ã,60¡ã)µÄ·½Ïòµ¼Êý. ¾«Æ·Îĵµ

¾«Æ·Îĵµ

18.Çóº¯Êýu=xyzÔÚµãA(5,1,2)´¦Ñص½µãB(9,4,14)µÄ·½ÏòABÉϵķ½Ïòµ¼Êý.

19.Çóº¯Êýu=x2+2y2+3z2+xy-4x+2y-4zÔÚµãA(0,0,0)¼°µãB(5,-3,

z)´¦µÄÌݶÈÒÔ¼°ËüÃǵÄÄ£. 320.É躯Êýu=ln??,ÆäÖÐr=µãÉϳÉÁ¢µÈʽgradu=1.

?1??r??x?a?2??y?0?2??z?c?2 ÇóuµÄÌݶÈ;²¢Ö¸³öÔÚ¿Õ¼äÄÄЩ

z2x2y221É躯Êýu=2?2?2,ÇóËüÔÚµã(a,b,c)µÄÌݶÈ.

cab22.Éèr=r2?y2?z2,ÊÔÇó: (1)grad r; (2)grad

1. r23.Éèu=x3+y3+z3£­3xyz,ÊÔÎÊÔÚÔõÑùµÄµã¼¯ÉÏgrad u·Ö¼ÓÂú×ã:(1)´¹Ö±ÓÚZÖá,(2)ƽÐÐÓÚZÖá(3)ºãΪÁãÏòÁ¿.

24.Éèf(x,y)¿É΢,LÊÇR2ÉϵÄÒ»¸öÈ·¶¨ÏòÁ¿,ÌÈÈô´¦´¦ÓÐfL(x,y)?0,ÊÔÎʴ˺¯ÊýfÓкÎÌØÕ÷? 25.ÇóÏÂÁк¯ÊýµÄ¸ß½×Æ«µ¼Êý:

(1) Z=x4+y4£­4x2y2,ËùÓжþ½×Æ«µ¼Êý; (2) Z=ex(cos y+x sin y),ËùÓжþ½×Æ«µ¼Êý;

?3z?3z(3) Z=xln(xy),,; 22?x?y?x?y(4) u=xyze

x+y+z

?p?q?zu,; ?xp?yq?zr(5) Z=f(xy2,x2y),ËùÓжþ½×Æ«µ¼Êý; (6) u=f(x2+y2+x2),ËùÓжþ½×Æ«µ¼Êý£» (7)Z=f(x+y,xy,

x),zx, zxx, Zxy. y26.ÇóÏÂÁк¯ÊýÔÚÖ¸¶¨µã´¦µÄÌ©ÀÕ¹«Ê½: (1) f(x,y)=sin(x2+y2)ÔÚµã(0,0)(µ½¶þ½×Ϊֹ); (2) f(x,y)=

xÔÚµã(1,1)(µ½Èý½×Ϊֹ); y(3) f(x,y)=ln(1+x+y)ÔÚµã(0,0); ¾«Æ·Îĵµ

¾«Æ·Îĵµ

(4) f(x,y)=2x2¨Dxy¨Dy2¨D6x¨D36+5ÔÚµã(1,£­2).

27.ÇóÏÂÁк¯ÊýµÄ¼«Öµµã: (1) Z=3axy¨Dx3¨Dy3 (a>0); (2) Z=x2+5y2¨D6x+10y+6; (3) Z=e2x(x+y2+2y).

28.ÇóÏÂÁк¯ÊýÔÚÖ¸¶¨·¶Î§ÄÚµÄ×î´óÖµÓë×îСֵ. (1) Z=x?y,?x,y?x2+y?4£»

222??(2) Z=x?xy?y,?x,y?x?y?1;

22??(3) Z=sinx+sing£­sin(x+y),?x,y??x,y?x?0,x?y?2? 29.ÔÚÒÑÖªÖܳ¤Îª2PµÄÒ»ÇÐÈý½ÇÐÎÖÐ,Çó³öÃæ»ýΪ×î´óµÄÈý½ÇÐÎ.

30.ÔÚxyÆ½ÃæÉÏÇóÒ»µã,ʹËüµ½ÈýÖ±Ïßx=0,y=0,¼°x+2y£­16=0µÄ¾àÀëÆ½·½ºÍ×îС. 31.ÒÑÖªÆ½ÃæÉÏn¸öµãµÄ×ø±ê·Ö±ðÊÇ

??A1?x1,y1?,A2?x2,y2?,¡­An?xn,yn?.

ÊÔÇóÒ»µã,ʹËüÓëÕân¸öµã¾àÀëµÄƽ·½ºÍ×îС.

1 1 1 32.Éè u= x y z

x2 y2 z2Çó(1)ux+uy+uz; (2)xux+yux+zuz; (3)uxx+uyy+uzz.

33.Éèf(x,y,z)=Ax2+By2+Cz2+Dxy+Eyz+Fzx,ÊÔ°´h,k,LµÄÏÂÕýÕûÊýÃÝÕ¹¿ªf(x+h,y+k,z+L).Èý¡¢

Èý¡¢¿¼Ñи´Ï°Ìâ

1. Éèf(x,y,z)=x2y+y2z+z2x,Ö¤Ã÷ ¾«Æ·Îĵµ

¾«Æ·Îĵµ

fx+fy+fz=(x+y+z)2.

2. Çóº¯Êý

?x3?y322,x?y?0?22f(x,y)??x?yÔÚÔ­µãµÄÆ«µ¼Êýfx(0,0)Óëfy(0,0),²¢¿¼²ìf(x,y)ÔÚ(0,0)µÄ¿É΢

?0,x2?y2?0?ÐÔ.

1 1 ? 1 x1 2 x ? n x3. Éè u?x1 2 x ? n x

222????n?1?1?1x1 n? n2x nx

?u?0; (2) Ö¤Ã÷: (1)??xk?1kn?xkk?1n?un(n?1)?u. ?xk2

4. É躯Êýf(x,y)¾ßÓÐÁ¬ÐøµÄn½×Æ«µ¼Êý:ÊÔÖ¤º¯Êýg(t)=f (a+ht,b+kt)µÄn½×µ¼Êý

dng(t)???????h?k??x?f(a?ht,b?kt). dtn?y???2? ?x f ? yÇó2. 5. Éè ?(x,y,z)?d?z e?xg?y h ? z k ?x

na?x b ? y c ? z?3¦µ 3 (y)Çó6. Éè ¦µ(x,y,z)?g1(y ) g. 2(y ) g?x?y?zh1(z ) h 2(z ) h3(z)7. É躯Êýu=f(x,y)ÔÚR2ÉÏÓÐuxy=0,ÊÔÇóu¹ØÓÚx,yµÄº¯Êýʽ.

8. ÉèfÔÚµãp0(x0,y0)¿É΢,ÇÒÔÚp0¸ø¶¨ÁËn¸öÏòÁ¿Li(i=1,2,¡­n).ÏàÁÚÁ½¸öÏòÁ¿Ö®¼äµÄ¼Ð½ÇΪ

f1(x ) f 2 (x ) f 3 (x)2¦Ð£¬Ö¤Ã÷ n¾«Æ·Îĵµ

¾«Æ·Îĵµ

?fi?1nLi(p0)?0.

9. Éèf(x,y)Ϊn´ÎÆë´Îº¯Êý,Ö¤Ã÷

??????x?yf?n(n?1)?(n?m?1)f. ??x??y??

10. ¶ÔÓÚº¯Êýf(x,y)=sin

my,ÊÔÖ¤ x

??????x?yf=0.??x??y??m¾«Æ·Îĵµ

ÁªÏµ¿Í·þ£º779662525#qq.com(#Ìæ»»Îª@) ËÕICP±¸20003344ºÅ-4 ceshi