Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 1Tp ti liu ny do ti bin son cho cc SV ca mnh, ch lu hnh ni b v khng c mc ch thng mi. Ngoi cc bi tp ti bin son, mt s khc tham kho t cc ti liu sau: 1) Liasko, Boiatruc, Gai, Golobac, Gii tch ton hc. Cc v d v cc bi ton. 2) Demidovich, Problems in mathematical analysis. 3)Mendelson, 3000 solved problems in Caculus. 4) N..Tr, T.V.nh, N.H.Qunh, Bi tp ton cao cp. 5) .C.Khanh, N.M.Hng, N.T.Lng, Bi tp ton cao cp. CHNG 1: HM NHIU BIN I.TP TRONG Rn , GII HN V LIN TC. Ta cL y x fy yx x=) , ( lim00 vi mi dy) , ( ) , (0 0y x y xnn n thL y x fnn n ) , (vy + tnh ) , ( lim00y x fy yx x ta xt mt dy) , ( ) , (0 0y x y xnn n ty v kim tra lun c L y x fn nn=+ ) , ( lim + chng minh) , ( lim00y x fy yx x ta ch ra hai dy) , ( ) , (0 0y x y xnn n , ) , ( ) ' , ' (0 0y x y xnn n m+ ) , ( limn nny x f ) ' , ' ( limn nny x f+ Vi mt s gii hn bng 0, ta c th dng gii hn kp. V d: Tnh 1sinlim) 0 , 0 ( ) , (yy xexy Xt mt dy) 0 , 0 ( ) , ( nn ny xty ( 0 , 0 nnnny x ). Ta c0 0 . 1 . 1 )1)(sin( lim1sinlim = ==+ + nynn nn nnyn nnxeyy xy xey xn n. Vy01sinlim) 0 , 0 ( ) , (=yy xexy. V d: Kho st tnh lin tc ca =+=) 0 , 0 ( ) , ( ,21) 0 , 0 ( ) , ( ,) , (4 43y xy xy xxyy x fti (0,0). Ta kim tra 21lim ) 0 , 0 ( ) , ( lim4 43) 0 , 0 ( ) , ( ) 0 , 0 ( ) , (=+ = y xxyf y x fy x y x (?). Ta xt dy) 0 , 0 ( )1,2( ) , ( = nn nn ny x , ta c34 42 217 17nn n nn nxyx y= + Tc21lim4 43) 0 , 0 ( ) , (+y xxyy x , hay ) , ( y x fgin on ti (0,0). Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 21.1.Tm v biu din hnh hc tp xc nh ca cc hm sau trn cc khng gian tng ng a)( ) y x y x y y x f + = 2 ln 2 ) , (2 2 b)( ) ( )2 2422 ln ) , ( y x x y x y y x f + = c)( ) ( ) ( ) y x y x y x f + + = arccos arcsin , d)( )2 2 2 2 2ln 2 ) , , ( y x z z y x z y x f + = 1.2. Vit phng trnh mt tr a) Qua giao tuyn 2 mt 2 2 2 2, z x y z x y = + = +v c phng song song vi Oz. b) Qua giao tuyn 2 mt 2 2, 1 y x z x y z = + + + =v c phng song song vi Oy. 1.3.Cho hm 2 2( , , ) f xyz x xy yz = + + . Tnh a) f(1,1,2)b) f(z,x-z,y) 1.4. Kho st s tn ti ca cc gii hn v tnh ( nu c) a) y xy xy x+ + +1 1) sin(lim) 0 , 0 ( ) , ( b) y xxy x+ ) 0 , 0 ( ) , (limc) 2 22) 0 , 0 ( ) , (limy xxyy x+ d)2 22) 0 , 0 ( ) , () (limy xy xy x++e)2 2limy xy xyx+++ + f) y xy xyx3 311lim HD: a,f : tn ti,dng nh ngha c,e : tn ti, dng gii hn kp b,d : khng tn ti 1.5. Kho st tnh lin tc ca hm s ti (0,0) a) =+ =) 0 , 0 ( ) , ( , 0) 0 , 0 ( ) , ( ,) ( ) , (2 2 2y xy xy xxyy x f b) =+ =) 0 , 0 ( ) , ( , 0) 0 , 0 ( ) , ( ,1sin) , (2 2y xy xy xxy x fHD: a) gin on, b) lin tc (dng gii hn kp). Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 3II.O HM V VI PHN. 1.6. Tnh cc o hm ring cp 1 ca cc hm s sau a)( ) ( )2, sin f xy xy = b) ( )21, ,1xyf xyzxz+=+ 1.7. Dng nh ngha ch ra cc hm sau khng c o hm ring ti (0,0) a) 2 2) , ( y x y x f + = b) 3sin ) , ( y x y x f + = HD:( )( ) ( )( )( ) ( )' '0 0, 0 0, 0 0, 0, 00, 0 lim , 0, 0 lim0 0x yx yf x f f y ff fx y = = . 1.8. Tnh ' '(0, 0), (0, 0)x yf fvi a) =++=) 0 , 0 ( ) , ( , 0) 0 , 0 ( ) , ( ,1sin ) () , (2 24 2y xy xy xy xy x fb) 32 2, ( , ) (0, 0)( , )0 , ( , ) (0, 0)x yxyf xy x yxy += += HD: Phi dng nh ngha : a) ' '(0, 0) 0, (0, 0) 0x yf f = = b) ' '(0, 0) 1, (0, 0)x yf f = 1.9. Tnh) 2 , 1 ( ), 2 , 1 (yfxf bit +=2 2220) , (y xtdt e y x f 1.10. Tnh) 2 , 1 ( ), 2 , 1 (yfxf bit ++=2 22 1cos) , (2y xxdttty x f HD:S dng cng thc( ) ( ) ( )' '( )( ) , ( ) ' ( )u x xa ax xf t dt f x f t dt u x f ux| | | |= = || |\ \ 1.11. Tnh(1,1), (1,1)f fx y bit ( )22 2( , ) 1xyf xy x y = + + HD: S dng cng thc o hm ring ca hm hp vf u =vi 2 2 21 , u x y v xy = + + = . 1.12. Tm hm( ) , f xynu bit rng 2fx yx= v2fy xy= . HD:( ) ( )32,3f xx y f xy yx g yx= = +, kt hp gi thit th hai. 1.13. Chng minh hm ( )2 2ln ) , ( y x y y x f = tha mn phng trnh 2.1.1yfyfy xfx=+ Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 41.14. Chng minh hmy xxyxy x f1 12 2) , (2 + + =tha mn phng trnh

yxyfyxfx32 2=+. 1.15. Vi gi thitf, gl cc hm kh vi, chng minh hm ) ( ) ( y x yg y x xf u + + + =tha mn phng trnh 0 222 222=+ yuy xuxu. 1.16. Chng minh rng hm2 2 21( , , ) uxyzx y z=+ + tha mn phng trnh Laplace 2 2 22 2 20u u ux y z + + = . 1.17. Tnh vi phn ton phn ca cc hm sau a)( , ) ln 1 sin xf xyy| |= + |\ b) ( ) ( ) , ,zf xyz xy = 1.18. Tnh a)f d2 vi y xe y x fsin) , ( = b) 21,2d f | | |\ vixy y x f sin ) , ( = 1.19. Tnh 3d fnu( )3 3, 3 ( ) f xy x y xyx y = + + . 1.20. Cho( )2 2 2, , u xyz x y z = + + . Chng minh 20, , , du dx dy dz . 1.21. Dng vi phn ton phn tnh gn ng a)( ) ( )( )2 3ln 2, 98 1, 99 b) 3 3) 97 , 1 ( ) 02 , 1 ( + c)) 01 , 1 ln( ) 02 , 1 (99 , 1+ d) ( )5 4ln 3 1, 02 2 0, 99 4 + e)02 , 3) 97 , 0 (f) 2,97sin1,49arctan0,022 vi1, 572 HD: f) Xt hm( ) , , 2 sin arctanxf xyz y z = . 1.22. Cho z l hm n cax,y xc nh bi 2 2 21 x y z + + = . Chng minh rng

1 z zx y zx y z + = 1.23. Tnh vi phn cp 1 ca hm n z=z(x,y) xc nh bi cc phng trnh tng ng a)3 22 z xz x y + = + b)( 2 3 ) x y zx y z e + ++ + = Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 51.24. Tnh vi phn cp 1 v cp 2 ca hm n z=z(x,y) xc nh bi cc phng trnh2ln 2x zz y| |= + |\ 1.25. Khai trin Taylor ti bc hai vi phn d Peano ca hm( )1, f xyx y= ti (2,1). 1.26. Khai trin Maclaurin ti bc hai vi phn d Peano ca hma)( ) , ln(1 2 ) f xy x y = + +b)( ) ,2xf xyx y=+ +. 1.27. Tm a thc xp x bc 2 trong ln cn ca (0,0) ca cc hm s sau a) ( )2, cosxf xy e y =b) ( ) ( ) , ln 1 2 sin f xy x y = + 1.28. Cho hm hai binf(x,y) kh vi trn R2 c cc o hm ring b chn ( ) ( ) ( )2, , , ; ,x yf xy M f xy M xy R . Chng minh ( ) ( ) ( ) ( ) ( )21 1 2 2 1 2 1 2 1 1 2 2, , , , , , f x y f x y M x x y y x y x y R + . III.O HM THEO HNG, VECT GRADIENT. Xt hm hai bin( ) , f xyti( )0 0, Mx y . Vct gradient caftiMl ( ) ( ) ( )0 0 0 0, , ,f ff M x y x yx y| | = | \ Vi hng( ) , u a b =r th( ) ( ) ( )2 2 2 2. .f f a f bM M Mx y ua b a b = + + +r hay ( ) ( ) ( ) .cos .cosf f fM M Mx y u = + r vi( ) ( ), , , u Ox u Oy = =r r. Ta c ( ) ( )1.fM f M uu u| | |= |\ ur rr r ( ) ( ) ( ) ( )( ) ( ) ( ) ( )max 0min 0fM f M u k f M kufM f M u k f M ku= = >= = k b. Vi hm ba bin) , , ( z y x fta kim tra im dng c l im cc tr hay khng bng cch xt duf d2:dng bin i Lagrange a v tng bnh phng hoc dng tiu chun Sylvester xt du dng ton phng (nu c nh thc con chnh bng 0 th phi xt trc tipf d2). V d:Tm cc tr ca 2 2 4 42 4 2 ) , ( y xy x y x y x f + + = Phng trnh im dng = += += + == + =000 4 4 40 4 4 433 33 '3 'y x yy xy x y fy x x fyx

Suy rafc 3 im dng ) 2 , 2 ( ), 2 , 2 ( ), 0 , 0 ( N M O Ti N M,thft cc tr v0 > . TiO th 0 = , ta ch ra khng t cc tr tiO. Xt V lmt ln cn ty ca O, vi )`< < 2 ,2min 0kthV k k P k k P ) , ( ), , (2 1, nhng ta c 0 ) 4 ( 2 8 2 ) (2 2 2 41< = = k k k k P f,0 2 ) ( , 0 ) (42> = = k P f O f Vy ) ( ) ( ) (2 1P f O f P f < < , tcfkhng t ln nht hay nh nht tiO trong Vm Vl ln cn chn ty , vyfkhng t cc tr tiO. Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 82.1. Chng minh hm 4 24( , ) f xy x y = + khng c cc o hm ring ti) 0 , 0 ( nhng vn t cc tr ti . HD: Dng nh ngha ch ra ' '(0, 0), (0, 0)x yf f , dng bt ch ra t CT ti (0,0). 2.2. Tm cc tr ca cc hm s sau a)( ) ( )50 20, 0, 0 f xy xy x yx y= + + > > b)( )3 2 2, , 12 2 f xyz x y z xy z = + + + +c) ( )4 4, 4 f xy x y xy = + +d)( ) ( )2 22, , , , 04y zf xyz x xyzx y z= + + + >2.3. Tm cc tr ca cc hm s sau a) 2 23 5 z x y xy x y = + + + b)yz xy y y x z y x f 2 2 3 2 ) , , (2 2 2 + + = c)yyxxz + + =8 d) y xyx z142+ + = f)z x xy z y x z y x f 2 ) , , (2 2 2 + + + =g)( )( ) xy y x y x f = 1 ) , ( 2.4. Tm cc tr ca cc hm s sau a) 2 3 3x y x z + + =b) 3 2 2 2 33 3 3 3 y y xy x x x z + + + = HD: y l cc bi c0 = b) == = + =010 ] ) 1 [( 32 2 'yxy x zx v xt 2 1, P Pdng) 0 , 1 ( k vi0 > k b. II.CC TR C IU KIN tm cc tr c iu kin ca( ) , f xyvi iu kin( ) , 0 xy =xt hm ph Lagrange ( ) ( ) ( , ) , , Lxy f xy xy = +Tm im dng ( )' ' '' ' '00, 0x x xy y yL fL fxy = + == + ==.Ti im dng (x0,y0) ng vi 0 , kim tra 2 " 2 " " 22xx xx yydL Ldx Ldxdy L dy = + +xc nh dng hay xc nh m ta s c (x0,y0) l im C hay CT c iu kin ca ( ) , f xy . Nu cha c ngay xc nh dng hay m ta ch rng buc ca dx,dy ti (x0,y0) :( ) ( )' '0 0 0 0, , 0x yx y dx x y dy + = . Hm ba bin hon ton tng t. 2.5. Tm cc tr ca a)( ) , f xy x y = +vi iu kin( )2 22 21 , 0x ya ba b+ = > b)( )1 1, f xyx y= +vi iu kin( )2 2 21 1 10 ax y a+ = > c) ( )2 2, 12 2 f xy x xy y = + +vi iu kin 2 24 25 x y + = d) ( ) , , 2 2 f xyz x y z = +vi iu kin 2 2 21 x y z + + =Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 92.6. Tm hnh ch nht c ng cho bng a cho trc m c din tch ln nht. HD: Gi di hai cnh k nhau ca hnh ch nht l, xyth iu kin l2 2 2x y a + = . III.GI TR LN NHT, GI TR NH NHT. tm cc min, max ca ftrnD ta ch cn tm cc im ti hn (l cc im dng trong trng hpc ' ',y xf f ) caftrong Dv tm cc im min,max caftrn bin ca D ri so snh cc gi tr cafti cc im . Khi xt cc im trn bin (thng l ng cong 0 ) , ( = y x ) ta rt y theo x hoc x theo y hoc tham s ha ng cong binafv mt bin v tm GTLN, GTNN nh ca hm mt bin thng thng. 2.7. Tm min,max ca cc hm) , ( y x f z =trn cc minD tng ng a) 2xy z =trn} 14: ) , {(22 + = yxy x D . b)y x y x z + + =2 23trnD gii hn bi cc min 1 , 1 , 1 + y x y x . c)) cos( cos cos y x y x z + + + =trn}4, 0 : ) , {( = y x y x D . d)xy y x z 33 3 + =trn} 2 1 , 2 0 : ) , {( = y x y x D . e)xy y x z 33 3 =trn{ } 2 0, 0 2 D x y = . f)y x z + = trn} 1 : ) , {(2 2 + = y x y x D . g)y x xy z = 2 trnD gii hn bi cc min 2 , , 0 ; 02 + y x x y y x . HD: a) Cc im) , ( y x trn bin= D } 14: ) , {(22= + yxy xtha( ) 2 24122 = xxy f) Cc im) , ( y x trn bin= D } 1 : ) , {(2 2= + y x y xc dng

==t yt xsincos ) 2 0 ( t 2.8. Trn mt phngOxyxt min kn tam gicOABxc nh bi cc trcOy Ox,v ng 0 1 = + y x. Tm cc im) , ( y x M thuc min tam gic sao cho a) Tng cc bnh phng khong cch t M ti ba nhB A O , ,l ln nht, nh nht. b) Tng cc khong cch t M ti ba nhB A O , ,l ln nht, nh nht. 2.9. Tm khong cch b nht ca hai ng thng: 2 1 1 2 1 2v4 7 1 2 1 3x y z x y z + + = = = = . HD: Vit pt hai ng thng dng tham s ri dng cng thc khong cch. Ch nu c ''0 v0( 0); ( , )xxf xy > > < th CT (C) cng chnh l GTNN (GTLN). Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 10IV.NG DNG HNH HC. vit phng trnh tip tuyn ca ca ng cong (C) l giao ca hai mt (S1) v (S2) ti ( ) M C , ta vit phng trnh hai tip din ca hai mt ti M, khi tip tuyn cn tm l giao ca hai tip din va tm c. 2.10. Vit phng trnh ca mt phng tip din vi mt 3 33xyz z a =ti im ng vi0, x y a = = . 2.11. Vit phng trnh tip din ca a) Paraboloid elliptic2 2z x y = +ti(1, 2, 5) b) Nn 2 2 2016 9 8x y z+ =ti (4,3,4). 2.12. Tm tip din ca ellipxoit 2 2219 4y zx + + =song song vi mt phng x+y+z=1. 2.13. Vit phng trnh tip tuyn v php din ca ng a) 2 2sin , sin cos , cos x a t y b t t z c t = = =ti im ng vi 4t= . b) 2 3, , x t y t z t = = =ti im ng vi3 t = . 2.14. Vit phng trnh tip tuyn ca ng cong giao bi hai mt2 2z x y = +v2 2 22 x y z + + = tiM(1,0,1). 2.15. Vit phng trnh tip tuyn ca ng cong giao bi hai mt 2 21 x y + =v z x y = + ti( )2, 2, 2 2 M . 2.16. Vit phng trnh tip tuyn v php din ca ng cong giao bi hai mt 2 210 x y + =v2 210 y z + = ti ( ) 1,1, 3 M . 2.17. Chng minh cc mt2 2 28 8 6 24 0 x y z x y z + + + =v2 2 23 2 9 x y z + + = tip xc nhau ti ( ) 2,1,1 M . HD: Ch ra hai mt c cng tip din ti M. 2.18. Chng minh tip din ca mt1/ 2 1/ 2 1/ 2 1/ 2x y z a + + = ti mt im ty s chn cc trc ta bi cc on c tng di bnga . Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 11 CHNG 3: TCH PHN HM NHIU BIN I. TCH PHN KP. tnh tch phn kp ta dng cng thc Fubini a v tch phn lp. Ch hai dng min ca nh l Fubini, trong nhiu trng hp kh tnh theo dng min ny nhng li n gin theo dng min kia. C th dng i bin. Nu min ly tch phn l hnh trn hoc cc hnh lin quan ti hnh trn (hnh vnh khn,hnh qut,) ta dng php i bin ta cc ==sincosr yr x.Nu hnh trn c tm khng ti gc ta ta c th di trc a gc ta v tm hnh trn trc sau mi i bin ta cc. Din tch ca min kn D:( )DS D dxdy = 3.1. Tnh tch phn kp a) +5132) 2 ( dx y x dy b) +32) (2 210xxdy y x dxc) ydx x y x dy02 221 d) 2410yydxxydy 3.2. Tnh tch phn kp ca hm cho trn minD a)y x y x f sin . ln ) , ( =vi y x D 0 , 2 1 : . b)1 2 ) , ( + = x y x f vix y x x D 2 , 2 0 :2 . c)xy y x f = ) , (viy x y y D , 1 0 : . d)x y y x f = ) , (vi 2 24 , , 0 : x y x y x D . e)x y y x f = ) , (vix y x y x D , 2 , 0 :2. f)y x y x f2) , ( =vi2 , 0 , 0 : + y x y x x D . g)( , ) f xy x y = viD gii hn bi cc ng2 23v 4 y x y x = = . h)( , ) f xy x y = +viD gii hn bi cc ng 0, , 2 y y x x y = = + = . i) 2( , ) f xy y =viD gii hn bi cc ng 2 , 5 v 2. y xy x x = = = j) 21( , )2f xyy y= vi D l gc phn t th nht gii hn bi ng24 2 . x y = k) 3( , )xf xy e = vi D gii hn bi cc ng2, 3vy=0. y x x = = 3.3. i th t cc tch phn kp sau a) 2020) , (xdy y x f dx b) ydx y x f dy010) , (c) 22 /40) , (ydx y x f dyd) 21001) , (xdx y x f dx Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 123.4. Tnh tch phn kp a) Dydxdy xsin viD l tam gic vi cc nh) 0 , 0 ( O ,) , 0 ( A ,) , ( B . b)( )+Ddxdy y x viD l hnh thang vi cc nh) 1 , 0 ( M ,) 0 , 1 ( N ,) 0 , 2 ( P v) 1 , 3 ( Q . c) Ddxdy xy2viD l gii hn bi cc ng4 , 4 ,2 2= = = y x y x y . d) +Ddxdy y x ) cos( viD l tam gicOABvi ) 0 , 0 ( O ,) 0 , ( A ,) , 0 ( B . e) Ddxdy y x ) ( viD l gii hn bi cc ng 22 , x y x y = = . f) 12202yxdx e dy . g) Dyxdxdy e viD l gii hn bi cc ng1 , 0 ,2= = = y x x y .

HD:d) 2 1D D D =:}2, , 0 : ) , {(1 + = y x y x y x D, ' '2'2 2D D D =

2= + y x = + y x'2D

1D ' '2Df)1 2 / , 2 0 : x y y D. a v dng min th nht v i th t tch phn. g) a v dng min th hai (hai trc ngang). 3.5. Bng phng php i bin ta cc hy tnh cc tch phn sau a)( )+ +Ddxdy y x2 21 ln viD l hnh trn12 2 + y x . b)( )+Ddxdy y x232 2viD l min gii hn bi hai ng 12 2= + y x , 42 2= + y x . c) Ddxdy y x2 21viD :x y x +2 2. d) +Dy xdxdy e2 2 viD :0 , 0 , 12 2 + y x y x . e) + + Ddxdyy xy x2 22 211 viD :0 , 0 , 12 2 + y x y x . Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 13 f)( )+Ddxdy y xviD :x y y x + , 0 , 2 12 2. g) Ddxdybyax22221viD l min gii hn bi ellipse12222= +byax. h)( )+Ddxdy y x2 2 viD:0 2 , 0 1 22 2 2 2 + + + + x y x x y x . HD:g) i bin ta cc m rng:cos , sin x ar y br = = h) i trc ri i bin ta cc: 1 cos , sin x r y r = + = 3.6. Bng phng php i bin tng qut cc hy tnh cc tch phn sau a) +Ddxdy y x y x4 7) ( ) ( viD l gii hn bi cc ng1 , 1 , 3 , 1 = = = + = + y x y x y x y x . b) Ddxdy y x ) 2 ( viD l gii hn bi cc ng 0 2 , 1 2 , 2 , 1 = = = + = + y x y x y x y x . c) +Ddxdy y xviD l min gii hn bicc ng 1 , 1 , 1 , 0 = = = + = + y y y x y x . d)( )+Dy xdxdy e y x2 22 viD:1 , 0 , 0 + y x y x . HD: a) i biny x v y x u = + = ,b) i biny x v y x u = + = 2 , c) i biny v y x u = + = ,d) i biny x v y x u = + = , 3.7. Tnh din tch min phng gii hn bi ng cong( ) ( )2 22 3 3 4 1 100 x y x y + + + = . HD: i bin1 4 3 , 3 2 + = + = y x v y x ua min phng v dng hnh trn. 3.8. Cho( ) { }2 2 2, :1 2 D xy R x y x = + (nm ngoi hnh trn 2 21 x y + = , nm trong hnh trn ( )221 1 x y + = ) a) Tnh din tch ca min D. b) Tnh tch phn Dydxdy. HD: i bin ta cc. Trong ta cc th cc ng trn 2 21 x y + = ,( )221 1 x y + =ln lt c phng trnh l1, 2cos r r = = , ch trong D th 3 3 . 3.9. ChoD l min gii hn bi cc ng 3 , x y y x = =v( )221 1 x y + = . a) Tnh din tch ca min D. b) Tnh tch phn Dxdxdy. HD: i bin ta cc. Trong ta cc th cc ng3 , x yy x = = ,( )221 1 x y + =ln lt c phng trnh l,6 4 = =v2cos r = . Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 143.10. Tnh tch phn2 211Ddxdyx y , trong ( ) { }2 2 2, : D xy R x y y = + . HD: Dng i bin ta cc (nu dng php di trc ri mi i bin ta cc th sao?) 3.11. Tnh Dxdxdyvi( ) ( )2 2: 1 2 1 D x y + . HD: Dng i bin ta cc di trc: 1 cos , 1 sin x r y r = + = + .

II.TCH PHN BI BA. tnh( ) , , f xyz dxdydz ta xc nh:( ) ( ) ( )1 2: , , , , xy D Oxy xy z xy th ( )( )( )( )21,,, , ( , , )xyxyDf xyz dxdydz f xyzdz dxdy= c bit, nu ( ) ( ) ( ) ( )1 2 1 2: , , , , a x b x y x xy z xy th ( )( )( )( )( )2 21 1,,, , ( , , )b x xya x xyf xyz dxdydz dx dy f xyzdz = Ch D l hnh chiu ca ln Oxy. Mt s trng hp c bit: + gii hn bi cc mt( ) ( )1 2, , z xy v z xy = = : nu khi c dng n gin th D l min gii hn bi hnh chiu ca ng giao tuyn hai mt ln Oxy. Gi s trong D ta c( ) ( )1 2, , xy xy th( ) ( ) ( )1 2: , , , , xy D Oxy xy z xy . + gii hn bi mt tr( ) , 0 xy =v cc mt( ) ( )1 2, , , z xy z xy = = : nu trong min D Oxy gii hn bi ng chun( ) , 0 xy = ca mt tr m ( ) ( )1 2, , xy xy th D chnh l hnh chiu ca ln Oxy v ( ) ( ) ( )1 2: , , , , xy D Oxy xy z xy . Hon ton tng t trong trng hp ta chiu ln Oxz hoc Oyz. i vi c hnh chiu dng hnh trn ta c th dng php i bin ta tr, cn nu c dng hnh cu ta c th dng i bin ta cu. Th tch ca:( ) V dxdydz=. 3.12. Tnh( ) , , f xyz dxdydz vi a)( )2 2, , f xyz x y = + v:1 2, 0 2, 1 1 x y z . b)( ) , , f xyz z = v 2 2: 0 1/ 4, 2 , 0 1 x x y x z x y . c)( )2 2, , f xyz x y = + v 2 2: 1, 0 1 x y z + . Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 15 d)( )2 2 21, , f xyzx y z=+ + v 2 2: 1, 1 3 x y z + . e)( )( )22 2 2, ,zf xyzx y z=+ + v 2 2: 2 , 0 1 ( 0) x y ax z a + > . HD: d) ( )2 22 21ln z z a Cz a= + + ++ e) a v tch phn kp v i bin ta cccos , sin x r y r = = . 3.13. Tnh( ) , , f xyz dxdydz vi a)( ) , , f xyz xy =v gii hn bi cc mt 2 2 2 21, 0, x y z z x y + = = = + . b)( ) ( )3/ 22 2, , f xyz z x y = +v gii hn bi cc mt 2 2 2 2, z x y z x y = + = + . c)( )2 2 2, , f xyz x y z = + +v gii hn bi mt ellipxoit( )2 2 24 4 x y z + + = . d)( ) , , f xyz y =v gii hn bi mt nn ( )2 2, 0 y x z y aa = + = > . e)( )2 2, , f xyz z x y = +v gii hn bi cc mt 2 24 , 0, 1 y x x z z = = = . f)( )2, , f xyz z =v( )2 2 2 2 2 2 2: 2 , 0 x y z azx y z a a + + + + > . g)( )2, , f xyz z =v 2 2 2 2: x y z R + + . h)( )2, , f xyz z =v 2 2 2 2 2: 4, 2 x y z x y x + + + . HD: a),b),c): hnh chiu D ca ln Oxy l hnh trn 2 21 x y + hoc dng ta tr. d):hnh chiu D ca ln Oxz l 2 2 2x z a + . e):hnh chiu D ca ln Oxz l hnh trn( )22 22 2 x y + hoc dng ta tr. f):hnh chiu D ca ln Oxz l hnh trn 22 232ax y| |+ | |\ . g):hnh chiu D ca ln Oxz l hnh trn 2 2 2x y R + hoc dng ta cu. h): hnh chiu D ca ln Oxz l hnh trn( )221 1 x y + hoc dng ta tr.

3.14. Tnh( ) , , f xyz dxdydz vi a)( ) , , 1 f xyz x y z = v: 0, 0, 0, 1 x y z x y z + + b)( ) , , f xyz x y z = + +v gii hn bi cc mt0, 0, 1, 0, 1 x y x y z z = = + = = = . c)( )( )31, , f xyzx y z=+ + v gii hn bi cc mt3, 2, 0, 0, 0 x z y x y z + = = = = = . d)( ) , , f xyz xy =v gii hn bi cc mt 2, 0 4 y x z v y z = = + = . HD: a),b): hnh chiu D ca ln Oxy l tam gic0, 0, 1 x y x y + . c):hnh chiu D ca ln Oxz l tam gic0, 0, 3 x z x z + . d):hnh chiu D ca ln Oxz l min gii hn bi 2 4 y x v y = = Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 163.15. Tnh th tch vt th gii hn bi cc mt cong a) 2 2 2 230, 1, 1 z x y z x y = + = = + + . b)y x z y x y z2 21 , 1 , , 0 + = = = = . c)1 , , 02= + = = z y x y z . d) 2 2z x y = +v2 2 26 x y z + + = . e) 2 2z x y = +v2 22 z x y = . f) 2 2z x y = +v2 2z x y = + . g) 2 22 z x y = v2 2z x y = + . h) 2 22 z x y = v2 2 20 x y z + = . 3.16. Tnh th tch ca vt gii hn bi mt ( ) ( ) ( )22 2 2 2 2 2 20 x y z a x y z a + + = + > HD: Dng ta cu. III.TCH PHN NG LOI MT. tnh tch phn ng loi mt ( , )ABf xy ds ta tham s ha cung AB:

( ): ( )( )x xtAB a t by yt= = Khi ( ) ( ) ( )2 2( , ) ( ), (( ) '( ) '( )bAB af xy ds f x t y t x t y t dt = + c bit Nu : ( )( )x tAB y f xy f t= = =,nu( ) cos: ( )( )sinx rAB r ry r = = = di ng cong AB:( )ABl AB ds = 3.17. Tnh tch phn ng loi mt ( , )Cf xyds vi a) 2 21( , )4f xyx y=+ + vi C l on thng ni(0, 0) Ov(1, 2) A . b)( , ) f xy x y = +vi C l na trn ng trn 2 2 2x y a + = . c)( , ) f xy xy =vi C l ellip 2 22 21x ya b+ = gc th nht. d)( , ) f xy x =vi C l hnh trn 2 22 x y x + = . e)( , ) f xy y =vi C l hnh trn 2 22 x y y + = . Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 173.18. Tnh tch phn ng loi mt ( , , )Cf xyzds vi a) ( )3/ 22 2( , , ) f xyz z x y = +vi C l phn ng cong cos , sin , (0 2 ) x a t y a t z bt t = = = , a>0, b>0. b)( , , ) f xyz x y = +vi C l phn t ng trn 2 2 2 2x y z Rx y + + ==,nm trong gc phn tm th nht. c)( , , ) f xyz xyz =vi C l phn t ng trn 2 2 2 222 24x y z RRx y + + =+ =, nm trong gc phn tm th nht. IV. TCH PHN NG LOI HAI. tnh tch phn ng loi hai( ) ( ) , ,ABPxy dx Qxy dy + ta tham s ha cung AB:

( ): ( , )( ) A Bx xtAB t a t by yt= = ==

(c th a>b) Khi ( ) ( ) ( ) ( ) , , ( ), ( ) '( ) ( ), ( ) '( )bAB aPxy dx Qxy dy Px t y t x t Qx t y t y t dt + = +( Nu C l ng cong kn gii hn min D ta c th dng cng thc Green a tch phn ng loi hai v tch phn kp ( ) , ( , )C DQ PPxy dx Qxy dy dxdyx y| | + = | \ (C ly theo hng dng) Ch trong trng hp ny thng th P(x,y)cha 1 hm phc tp c lp theo bin x,Q(x,y)cha mt hm phc tp c lp theo bin y (gii thch v sao?). Nu Q Px y = (tch phn khng ph thuc ng i) tnh ( ) ( ) , ,ABPxy dx Qxy dy +ta chn ng i n gin t A ti B. 3.19. Tnh ( ) ( )2 22 2Cx xy dx xy y dy + + vi C l cung ni t A(1,1) ti B(2,4) dc theo cung 2y x = . 3.20. Tnh 2 2Cydx x dy + vi C l na trn ellip 2 22 21x ya b+ =theo chiu kim ng h. 3.21. Tnh 2( )Cydx y x dy + vi C l phn cung parabol 22 y x x = nm pha0 y v theo chiu ngc kim ng h. Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 183.22. Tnh( )21Cxy dx xydy + vi C ng i t A(1,0) ti B(0,2) theo cc ng a) ng thng ni A v B. b) ng parabol 214yx = . 3.23. Tnh 2Cxdx ydy zdz + + vi C l cungcos , sin , x a t y a t z bt = = =(a>0, b>0) i t A(a,0,0) n B(a,0,2b). 3.24. Tnh 2 2 2Cx dx y dy zdz + + vi C l ng cong giao tuyn 2 mt 2 2 24 x y z + + =v 2 2z x y = +i theo chiu ngc chiu kim ng h nhn theo trc Oz. 3.25. Tnh ( )2 2 22 ( )Cx y dx x y dy + + + vi C l cung bin tam gic A(1,1),B(2,2),C(1,3) theo chiu dng bng 2 cch a) Tnh trc tip. b) Dng cng thc Green. 3.26.Tnh ( )2 2( )xCex xy dx x y dy + + vi C l ng trn 2 22 x y y + =theo chiu dng. 3.27.Tnh ( )2 2sin ( 2 )yCx x y dx y e xy dy+ + + vi C l na trn ellip 2 22 21x ya b+ =lytheo chiu ngc chiu kim ng h. 3.28.Tnh ( )2 2 2arctan ( 2 )yCx x y dx y e xy x dy+ + + + vi C l na trn ellip 2 22 21x ya b+ =ly theo chiu ngc chiu kim ng h. HD:Ni on A(-a,0),B(a,0) thnh ng cong kn v dng cng thc Green. Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 19 CHNG 4: PHNG TRNH VI PHN I.PHNG TRNH VI PHN CP 1. Ch cng thc dxdyy= ' a phung trnh v dng vi phn hay dng o hm.Dng ' ( ) y f ax by c = + + a c v dng c bin phn li bng php tz ax by c = + + . Thng thng ta tm nghim dng) (x y y =nhng trong mt s trng hp, n gin, ta tm nghim dng) ( y x x =(l hm nguc ca hm) (x y y = ). Khi ch ''1 1yxxdydxdxdyy = = =V d: Tm nghim ca phng trnh( ) ( ) ' cos 2 cos 2 y x y x y + + = vi iu kin( ) 04y=a v 0 2 sin sin 2 = y xdxdy +) (20 2 sin Z kky y = = khng l nghim v khng tha 4) 0 (= y+) (20 2 sin Z kky y a v dng c bin phn li

C xyyC xyy dC xdxydyxdxydy= ++= + = = cos 21 2 cos1 2 cosln41cos 22 cos 12 cos21sin 22 sin0 sin 22 sin2 iu kin 4) 0 (= y 2 0 cos 2142 cos142 cosln41= = ++ C C Vy nghim ring cn tm l2 cos 22 cos 12 cos 1ln41= ++xyy V d: Gptvp ' 2 sin '3xy y y x y = + Ta tm nghim dng) ( y x x =ch ''1yxxy = . * NuC y y = = 0 ' thay vo pt ta c0 = C . Vy0 = yl mt nghim ca pt. * Nu0 ' y , phng trnh tng ng vi

) 1 (2sin21'2sin21'1' 2 sin '333yy xxyxyy xxy yxy y y x y = = = + y) ( , ''y x x x xy= = . Phng trnh (1) l phng trnh Bernoulli theoxl hm ca y . a v dng (ch (?) 0 x) Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 20 ) 2 (sin 1'2sin212)' (2sin21 '2223yyzyzyyxyxyyxy xx= + = = vi 2 = x z , (2) l ptvp tuyn tnh cp 1 nn c NTQ yC yxyC yC dyyye e zdyydyy+ = + = + =cos cos)sin(21 1

Vy ptvp c cc ngim+ y=0 + 0cos 12=++yC yx 4.1. Gii cc ptvp a)0 ln = xdy x tgydxb)y y x = ' . cos c)) 0 ( cos ' > > + = a b b y a yd)1 ) ( ' = + y x y HD: L cc pt loi tch bin. 4.2. Tm cc nghim ring ca ptvp tha mn iu kin tng ng a)0 ) 5 ( ) 5 (2 3 3 2= + + + dy y x dx y x tha1 ) 0 ( = y . b)dx e dy y ex x= +2 2) 1 (tha0 ) 0 ( = y . c) 0 1 ) 1 (2 2= + + + dy x y xydx tha1 ) 8 ( = y . d) ) ( 21 3 3'y xy xy+ + = tha2 ) 0 ( = y . e)y tgx y = 'tha1 )2( =y . HD: y l cc pt tch bin c, tm NTQ ri sau tm nghim ring tha iu kin tng ng 4.3. Gii cc ptvp tuyn tnh cp 1 a) 22 1 2 ' x xy y = b)) 2 2 (1'x xe xe yxy + = c)0 2 ) 1 ( ' ) 1 (2 2= + + x x y y x xd)arctgx y y x = + + ' ) 1 (2 4.4. Gii cc ptvp a)y y x y = + ) ( '2b)0 ) 3 2 (2= + dx y dy xy c)dy x y ydx ) 2 ( 23 =d)0 ) sin (2= + dy y y x ydx HD: Coi x l hm theo y 4.5. Gii cc ptvp a)y xxxyy =+21' b)0 ) 1 ln 2 ( ' = x y y xy c)0 sin 2 '2 2= + x y ytgx y d) ) cos ( '3tgx x y y y + = HD: L cc pt Bernoulli. Bi tp Ton A3 H Ngc K, H Nng Lm Tp.HCM Created: 05/05/10. Last modified: 25/05/11. ---------------------------------------------------------------------------------------------------------------------------- 21II.PHNG TRNH VI PHN CP 2. Cc ptvp dng0 ) ' ' , ' , ( & 0 ) ' ' , ' , ( = = y y y F y y x Fta u a c v ptvp cp 1bng php t' y z = . Ch l dng th nht th a v cp 1 theoz x, cn dng th 2 th a v cp 1 theoz y, . V d: Gii ptvp( )2'' 2 ' ln ' 0 yy yy y y =t dydzz y z y = = ' ' ', phng trnh tr thnh 0 ln 22= z y yzdydzyz+) 0 ( 0 ' 0 > = = = C C y y zl nghim pt. +C y z 0a pt v y zyzyln 21'= y l ptvp tuyn tnh cp 1 i viz (theo bin) y .

4.6. Gii cc ptvp a)0 4 ) ' ( " 22= + y y b)) 1 (1'' ' = x xxyytha mn iu kin1 ) 2 ( ' , 1 ) 2 ( = = y y . c) 3) ' ( ' ' ' y y y + = d)' ' ) ' ( 12yy y = + e)y y y yy ln ) ' ( "2 2= f)0 ' sin ) ' ( cos "2= + y y y y ytha iu kin2 ) 1 ( ' ,6) 1 ( = = y y. g)) 1 ' ( ' " + = y y yy HD:a), b) : pt gim cp c khng cha y;c)-g) : pt gim cp c khng cha x.