Toan Cao Cap Gioi Han Va Lien Tuc

7/21/2019 Toan Cao Cap Gioi Han Va Lien Tuc

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Trng i hc Bch khoa tp. H Ch MinhB mn Ton ng dng

-------------------------------------------------------------------------------------

Gii tch hm nhiu bin

Chng 1: Gii hn v lin tc

Ging vin Ts. ng Vn Vinh (2/2008)

[email protected]


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Mn hc cung cp cc kin thc c bn ca gii tch hm nhiu bin.Sinh vin sau khi kt thc mn hc nm vng cc kin thc nn tng:hm nhiu bin, gii hn kp v lin tc, o hm ring v vi phn, o

hm theo hng, khai trin Taylor, Maclaurint ca hm nhiu bin, ngdng ca o hm ring: phng trnh mt phng tip din, php vct,ng dng tm cc tr; cch tnh tch phn bi: bi 2, bi 3; tch phnng: loi 1, loi 2; tch phn mt: loi 1, loi 2 v cc ng dng hnhhc, c hc ca cc loi tch phn ny; tch phn suy rng ph thuctham s; trng vct.

Mc tiu ca mn hc Ton 3


3/63

Gii hn v lin tc

o hm theo hng

ng dng ca o hm ring

Tch phn kp

Tch phn ng loi 1 v loi 2

Tch phn mt loi 1 v loi 2Trng vct

Tch phn bi ba

Tch phn ph thuc tham s


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Nhim v casinh vin.

i hc y (vng 20% trn tng s bui hc b cm thi!).

Lm tt c cc bi tp cho v nh.

c bi mi trc khi n lp.

nhgi, kimtra.

Thi gia hc k: hnh thc trc nghim (20%)

Thi cui k: hnh thc t lun + in kt qu (80%)


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Ti liutham kho

1. Cng Khanh, Ng Thu Lng, NguynMinh Hng. Gii tch nhiu bin.

NXB i hc qucgia

2.Ng Thu Lng, NguynMinh Hng. Bi tpton cao cp3.

4. James Stewart. Calculus, second edition, 2000.

5.www.tanbachkhoa.edu.vn

3. Cng Khanh. Giitch nhiu bin. NXB i hc qucgia


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Ni dung---------------------------------------------------------------------------------------------------------------------------

0.1Hm hai bin

0.2Cc khi nimtp trong Rn

0.4Gii hn

0.5lin tc

0.3Cc mt bchai


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I. Hm hai bin---------------------------------------------------------------------------------------------------------------------------

Nhit T ti mt imtrn b mttri t ti mt thi imtcho trc ph thucvo kinh x v v y ca imny. Chngta c thcoi T l mthm theo haibinx v y, k hiu

T = T(x,y)

V d

Thtch V ca mtbnh hnh tr ph thucvo bn knh yr v

chiucao h. Thc tta bit . Khi V l mthm haibintheo r v h:

2

V r h

2( , ) .V r h r h

V d

I H h i bi


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I. Hm hai bin---------------------------------------------------------------------------------------------------------------------------

D c gil minxc nh caf.

Cho . Hm haibinl mtnh x2D R

nh nghahm haibin

:f D R( , ) ( , )x y f x y

K hiu: ( , ).f f x y

{ | ( , ) : ( , )} E a R x y D a f x yMingi tr caf:

Nuf chobi biu thc i s: Minxc nhl tp hp tt ccc

gi tr cax v y, sao chobiu thcc ngha.

Mingi trl tp hp tt ccc s thcm hm c th nhn c.

I H h i bi


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I. Hm hai bin---------------------------------------------------------------------------------------------------------------------------

Minxc nh:

Hm haibinV d. 1

( , ) x y

f x yx y

{ }2( , ) | 1 0,D x y R x y x y

3 2 1(3,2) 6

3 2f

Hm haibinV d. 2 2

( , )f x y x y

Minxc nh: 2D R

Mingi tr: [0, )fE R

2 2 2 2( , ) ( ) ( ) 2( )f x y x y x y x y x y

2 2 2( , ) 2f x x x x x

I H h i bi


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I. Hm hai bin---------------------------------------------------------------------------------------------------------------------------

Minxc nh:

Hm haibinV d. ( , )1

xf x y

y

{ }2

( , ) | 1D x y R y

Hm haibinV d. 1

( , )1

f x yy

Minxc nh: { }2

( , ) | 1D x y R y

Mingi tr: {0}\fE R

Mingi tr: fE R

Hm haibinV d. neu

, neu

2 2

1

, ( , ) (0,0)( , )

0 ( , ) (0,0)

x ye x yf x y

x y

Minxc nh: 2D R

Mingi tr: [0,1)fE

II Tp trong R2


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II. Tp trong R2---------------------------------------------------------------------------------------------------------------------------

{ ) }2

0 0( , ) ( , ) | ( ,B M r M x y R d M M r

Hnh trn mtm , bn knh0 0 0( , )M x y 0r

{ }2 2 20 0( , ) | ( ) ( )x y R x x y y r

C mtln cn caM0 nm trntrong A, nghal ch cha nhng im caA. Khi M0 c gil imtrong ca tp A.

Hnh trn mny cng gil mtr-ln cn caM0v mi tpcon caR2 chamtr-ln cnno caM0 gil mtln cn caM0.

Xt mt im v mt tp . C th xyra ba trng hp loitrnhau sau y:

2

0M R

2

A R

C mtln cn caM0 nm trnngoi A, nghal hon ton khng chaimno ca A. Khi M0l mt imtrong ca phnb ca A.

Bt kln cnno caM0 cngc c nhng im caA v nhng imkhng thuc A. Khi M0l mt imbin ca A.

II Tp trong R2


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II. Tp trong R2---------------------------------------------------------------------------------------------------------------------------

Ch . 1) imtrong ca A l mt im thuc A.

2) imbin ca A c th thuc hockhng thuc A.

Mt tp hp c gil mnu mi im thucn ul imtrong can.

Mt tp hp c gil ng nu mi imkhng thucn ul imtrongca phnb can.

Mt tp hpl ng nu phnb can l m.

Mt tp hpl m nun khng cha imbin no can.

II Tp trong R2


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II. Tp trong R2---------------------------------------------------------------------------------------------------------------------------

imM0 c gil im t caA, nu miln cn caM0 u chav sim ca A.

imM0l im t ca tpA, nu miln cn can c chat nht mtim ca A khc viM0.

Ch . 1) im tc th thuc A, c thkhng thuc A.

2) C nhng tp hpkhng l tp ng, cngkhng l tp m.

Mt tp hpl ng nun cha tt ccc imbin can.

II Tp trong R2


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II. Tp trong R2---------------------------------------------------------------------------------------------------------------------------

V d.

Xt tp hpcc imtrong mt phng. Cho tp hp A

2 2 2

, 1A x y R x y

4. Tp A l tp m.

1.Tt ccc imtrong ca A: 2 2 2, 1x y R x y

2. Tt ccc imbin ca A: 2 2 2, 1x y R x y

3. Tt ccc im t ca A: 2 2 2, 1x y R x y

II Tp trong R2


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II. Tp trong R---------------------------------------------------------------------------------------------------------------------------

3. Tm tt ccc im t ca A.

V d.

Xt tp hpcc imtrong mt phng. Cho A l tp hpcc im nmtronghnh trn n vm ta cc iml cc s hu t.

{ }2 2 2( , ) | 1A x y xQ y

4. Tp A nghay m.

1. Tm tt ccc imtrong ca A.

2. Tm tt ccc imbin ca A.

p s: 1) Khng c imtrong

2) Tp imbin v im t bngnhau

{ }2 22( , ) | 1RA x y x y Tp imbin

4) A khng ng,khng m.

II Tp trong R2


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II. Tp trong R---------------------------------------------------------------------------------------------------------------------------

3. Tm tt ccc im t ca A.

V d.

Xt tp hpcc imtrong mt phng. Cho tp hp A

2n+3,n+1

211A Rn

4. Tp A nghay m.

1. Tm tt ccc imtrong ca A.

2. Tm tt ccc imbin ca A.

p s: 1) Khng c imtrong

2) C mt imbin l (1,2).

4) A khng ng,khng m.

3) C mt im tl (1,2).

III Cc mt bc hai


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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------

T chngtrnh ton A2, v mt bchai:

Phngtrnh tngqut ca mt bchai trong h ta Descartes 0xyz l

2 2 2 2 2 02Ax By Cz Dxy Ex Gxz Fyz Hy Kz L

1) a dngtonphng(mu ) v dngchnh tc bng bin i trcgiao

2) Tm phpbin i,

xcnh trc ta mi

.

3) Vhnh.

III Cc mt bc hai


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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------

Tp hp tt ccc im(x,y) ca minxc nhDf, sao cho f(x,y) = k c gil

ng mc,trong k l hng scho trc.

k= 0

k= 1

k= 2

k= 3

k= 4

Xt th cahm s: 2 2z x y

III Cc mt bc hai


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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------

Mtparaboloid elliptic2 2

2 2

x yz

a b

III Cc mt bc hai


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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------

Mtparaboloid elliptic2 2

2 2

x yz

a b

III Cc mt bc hai


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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------

Mtparaboloid elliptic 2 2( 1) ( 3) 4z x y

III. Cc mt bc hai


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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------

Mtparaboloid elliptic 2 2y x z

III. Cc mt bc hai


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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------

Mtellipsoid2 2 2

2 2 2 1

x y z

a b c

III. Cc mt bc hai


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III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------

MtParaboloid hyperbolic2 2

2 2

x yz

a b

III. Cc mt bc hai


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. Cc t bc a---------------------------------------------------------------------------------------------------------------------------

MtParaboloid hyperbolic

III. Cc mt bc hai


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---------------------------------------------------------------------------------------------------------------------------

MtParaboloid hyperbolic 2 2y z x

III. Cc mt bc hai


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---------------------------------------------------------------------------------------------------------------------------

MtHyperboloid 1 tng2 2 2

2 2 2 1

x y z

a b c

III. Cc mt bc hai


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---------------------------------------------------------------------------------------------------------------------------

MtHyperboloid hai tng2 2 2

2 2 2 1

x y z

a b c

III. Cc mt bc hai


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---------------------------------------------------------------------------------------------------------------------------

Ta thy vi mik, ng mclun l ngtrn bn knhbng1.

k= 0

k= 1

k= 2

k= -2

k= -1

Xt th cahm s: 2 2 1x y

III. Cc mt bc hai


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---------------------------------------------------------------------------------------------------------------------------

Mt tr: trongphngtrnh thiu hocx, hocy, hocz.

2 2

2 2 1

x y

a b

III. Cc mt bc hai


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---------------------------------------------------------------------------------------------------------------------------

Mt tr: 2 2 4x z

III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------


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---------------------------------------------------------------------------------------------------------------------------

Mt tr 2y x

x

z

III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------


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Mt tr 2z x

III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------


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Mt tr 22z x

III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------


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Mt tr 22z x

III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------


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Mtnn hai pha2 2 2

2 2 2

x x x

a b c

III. Cc mt bc hai---------------------------------------------------------------------------------------------------------------------------


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Mtnn hai pha

IV. Gii hn---------------------------------------------------------------------------------------------------------------------------


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nh ngha gii hnkp

Cho hm haibin , sao cho l im t caDf.( , )f f x y 2

0 0 0( , )M x y R 0M

Ta ni gii hn cafkhi (x,y) dn n imM0bng , nugi tr caf(x,y) tingn n ty thchbngcch ly nhng im(x,y) gn imM0, nhngkhngtrng viM0.

aa

0 0( , ) ( , )lim ( , )

x y x yf x y a

2 20 0 0 00, 0: ( , ) ,( , ) ( , ), ( ) ( )fx y D x y x y x x y y

Khi ( , ) .f x y a

K hiukhc ca gii hnkp: 00

lim ( , )x x

y x

f x y a

. n------------------------------------------------------------------------------------------------------------------------


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---

Tnh cht ca gii hn

( , ) ( , ) ( , ) ( , ) ( , ) ( , )1. lim [ ( , ) ( , )] lim limx y a b x y a b x y a bf x y g x y f g

( , ) ( , ) ( , ) ( , ) ( , ) ( , )2. lim [ ( , ) ( , )] lim lim

x y a b x y a b x y a bf x y g x y f g

( , ) ( , )

( , ) ( , ) ( , ) ( , )( , ) ( , )

lim ( , )( , )

lim , lim 0( , ) lim ( , )

3. neu

x y a bx y a b x y a b

x y a b

f x yf x y

gg x y g x y

( , ) ( , ) ( , ) ( , ) ( , ) ( , )

( , ) ( , ) ( , )lim lim , lim .

4. neu vath

x y a b x y a b x y a b

f x y g x y h x yf h M g M

. n---------------------------------------------------------------------------------------------------------------------------


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---

V d

Tm gii hn nu tn ti hoc chngminh khng tn ti.

( , ) (0,0)1lim sin

x y

I x yx

1 10 | ( , ) | sin | | sin | | f x y x y x y x y

x x

0

( , ) (0,0)

1lim sin 0.

x y

x yx

. n---------------------------------------------------------------------------------------------------------------------------


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V d

Tm gii hn nu tn ti, hoc chng t gii hnkhng tn ti.

2

2 2( , ) (0,0)

3lim

x y

x yI

x y

2 2

2 2 2 2

30 | ( , ) | 3 | |, v 1.

x y xf x y y

x y x y

0

2

2 2( , ) (0,0)

3lim 0.

x y

x y

x y

. n---------------------------------------------------------------------------------------------------------------------------


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V d

Tm gii hn (nuc) hoc chng tkhng tn ti.

2 22 2( , ) (0,0)

2lim

x y

x yIx y

Chndy 1

( , ) ,0 (0,0)

nn nx y

n

Khi 1

( , ) ,0 1.

n nf x y fn

Chndy thhai ' ' 1

( , ) 0, (0,0)

nn nx y

n

Khi ' ' 1( , ) 0, 2.

n nf x y fn

Vy tn tihai dy dn n(0,0) nhnggi tr cafti nhng im tin nhai skhc nhau, suy ra khng tn ti gii hn cho.

. n------------------------------------------------------------------------------------------------------------------


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V d


2 2( , ) (0,0)lim

x y

xyIx y

Chn y kx

Khi ( , ) , .1

kf x y f x kx

k

f(x,y) l mt i lng ph thucvok, mkthay inn khng tn ti gii hn.

Ch . Chny =kx, tcl tin n(0,0)bng nhng ng thng.

Phngphp ny khng thdng tm gii hn cady.

. n------------------------------------------------------------------------------------------------------------------


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V d


32 6( , ) (0,0)

lim

x y

xyIx y

Chndy 1

( , ) ,0 (0,0)

nn nx y

n

Khi 1

( , ) ,0 1.

n nf x y fn

Chndy thhai ' '3

1 1( , ) , (0,0)

n

n nx ynn

Khi ' ' 31 1 1( , ) , .2n nf x y f

nn

Vy tn tihai dy dn n(0,0) nhnggi tr ca fti nhng im tin nhai skhc nhau, suy ra khng tn ti gii hn cho.

. n------------------------------------------------------------------------------------------------------------------


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V d


2 22 2 2( , ) (0,0)

lim( )x y

x yIx y x y

Chndy 1

( , ) ,0 (0,0)

nn nx y

n

Khi 1

( , ) ,0 0.n nf x y fn

Chndy thhai ' '2

1 1 1( , ) , (0,0)

nn nx y

n n n

Khi ' ' 21 1 1 1( , ) , .2n nf x y f

n n n

Vy tn tihai dy dn n(0,0) nhnggi tr ca fti nhng im tin nhai skhc nhau, suy ra khng tn ti gii hn cho.

. n------------------------------------------------------------------------------------------------------------------


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V d


3( , ) (0,0)lim

1 1x yxyI

xy

t t xy Khi 0t

30lim

1 1t

tI

t

3

. n------------------------------------------------------------------------------------------------------------------


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V d

Tm gii hn (nuc) hoc chng tkhng tn ti.2

2( , ) (0,0)lim

9 3x yx yI

x y

t 2t x y Khi 0t

0lim

9 3t

tI

t

6

.

n------------------------------------------------------------------------------------------------------------------


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V d

Tm gii hn (nuc) hoc chng tkhng tn ti.4

2 2 2( , ) (0,0)lim

( )x yxyI

x y

S dng h ta cc, t cos ; sinx r t y r t

4 4

40

cos sinlimr

r t r t I

r

4

0lim( cos sin )r

I r t t

Khi th0; 0x y 0r

0I

.

n c------------------------------------------------------------------------------------------------------------------


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nh ngha

Hm sf(x,y) c gil lin tc ti , nu0 0( , )x y

0 0

0 0( , ) ( , )

lim ( , ) ( , )x y x y

f x y f x y

Hm c gil lin tc nun lin tc ti mi imm n xc nh

Tng, hiu,tch cahai hm lin tcl lin tc.

Thng cahai hm lin tcl lin tc nuhm mukhc 0.

Hp cahai hm lin tcl lin tc (ti nhng imthch hp).

.

n c------------------------------------------------------------------------------------------------------------------

h h


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nh nghaCc hm sau y c gilhm s cp c bn:

1) Hm hng; 2) hm m; 3) hm ly tha; 4) hm lnggic;5) hm lnggic ngc; 6) hm logarit.

nh ngha

Hm thu c tcc hm s cp c bn bng hu hncc phpton: cng, tr,nhn, chia, khai cn c gilhm s cp.

nhl

Hm s cplin tc ti nhng imm n xc nh.

. n c---------------------------------------------------------------------------------------------------------------------------


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V d

Tm cc imgin on cahm sau

( , ) xy

f x yx y

Suy ra nhng imgin on cahm sl ng thngx + y = 0.

yl hm s cp c bnnn lin tc ti nhng imm n xc nh

.

n c---------------------------------------------------------------------------------------------------------------------------

V d


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V d

Khost tnh lin tc cahm sau:

3 3

2 2sin( ) , ( , ) (0,0)

( , )

0, ( , ) (0,0)

x y x yf x y x y

x y

3 3 0

3 3

sin( ) sin1

tx y t

tx y

3 3 3 3 3 3

2 2 3 3 2 2

sin( ) sin( )x y x y x y

x y x y x y

3 3

2 20 | | | |

x yx y

x y

( , ) (0,0)lim ( , ) 1.0 0 (0,0)

x yf x y f

Vyhm cho lin tc ti mi imtrong mt phng.

Suy raf lin tc ti(0,0).

.

n c---------------------------------------------------------------------------------------------------------------------------


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V d

Tm tt ccc gi tr caa hm slin tc ti im(0,0)

2 2

2 2, ( , ) (0,0)

( , )

, ( , ) (0,0)

x yx y

f x y x y

a x y

Ta c khng tn ti.( , ) (0,0)

lim ( , )x y

f x y

Vyhm khng lin tc ti(0,0). Khng tn tia.

VI. Bi tp----------------------------------------------------------------------------------------------------------------


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1. Tm minxc nhv mingi tr cacc hm

2 21) ( , ) 4f x y x y

2

2) ( , )x y

f x y e

2 2

1

3) ( , ) x yf x y e

24) ( , ) ln( 4 8)f x y y x

5) ( , ) arcsiny

f x yx

2

16) ( , )f x y

x y

VI. Bi tp----------------------------------------------------------------------------------------------------------------


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2. Vcc mt bchai sau:

1) 3z

2 22) 2z x y

2 23) 1z x y

2

4) z x

25) 1z y

2 2

6) 1

9 4

x y

27) 1y x

2 28) 1z x y

2 29) 1z x y

10) 1x y z

11) 2z x

2 212) 2 2x y x y

2 213) 2x y z

2 2 214) 82x y z

VI. Bi tp----------------------------------------------------------------------------------------------------------------


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3. Vcc khisau:

2 21) ; 4x y z z

2 22) ; 1.x y z x z

2 23) 1; 1; 4.x y z z

2 2 2 2 2 2

4) 1; ; 4x y x y z x y z

25) 1 ; 0; 0; 2.y x y z z

2 22 2 2

6) 1; 4; 0.

9 4

x y

x y z z

2 27) 4 ; 2; 0; 4.z y z y x x

VI. Bi tp----------------------------------------------------------------------------------------------------------------3 khi ( i h )


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3. Vcc khisau (tiptheo)

2 28) ; 2; 2x y x x z x z

2 2 2 29) 2 ; ; 1; ;2 .y x y x x z x y z x y

210) 1 ; 3 ; 5; 0y x z x y z

2 2 2 211) ; 1 1z x y z x y

2 2 212) ; ; 1; 0.z x y y x y z

13) ; 2 ; 0; 6.y x y x z x z

2 2 2 214) 1; 2; 4; 1.x x y z y z

VI. Bi tp----------------------------------------------------------------------------------------------------------------4 T ii h k


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4. Tm cc gii hnkp2

2 4( , ) (0,0)

1) limx y

x y

y x

( , ) (0,0)

12) lim os

x yy c

y x

3

3( , ) (0,0)

3) limx y

x y

x y

( , ) (0,0)

1 14) lim sin sin

x yx y

y x

2 2

2 2( , ) (0,0)5) limx y

x y xy

x xy y

2 2 2

2 2( , ) (0,0)

( )6) lim

1 os( )x y

xy x y

c x y

VI. Bi tp----------------------------------------------------------------------------------------------------------------4 T ii h k (ti th )


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4. Tm cc gii hnkp (tiptheo)

2 2

1/

( , ) (0,0)

7) lim 1 x y

x y

xy

2 2

1/( )2 2

( , ) (0,0)8) lim os

x y

x yc x y

( , ) ( , )

9) lim sinx y

xy

xy

2 2 2 2

4 4 2 2 2 26( , ) ( , )

610) lim

2(1 )x y

x y x y

x y x y x y

2 2

( , ) (0,0)

111) lim ( )sin

x y

x y

xy

( , ) (0,0)12) lim

x y

x

x y

VI. Bi tp----------------------------------------------------------------------------------------------------------------4 T ii h k (ti th )


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4. Tm cc gii hnkp (tiptheo)2 2

2

( , ) (2,1)

413) lim

2 2 4x y

x y

x x xy y

( , ) ( , )14) lim (1 )

x

x y k

y

x

2 2( , ) ( , )

15) limx y

x y

x y

( , ) (0,2)

sin( )16) lim

x y

xy

x

2 2( , ) (0,0)

17) limx y

x y

x xy y

2 2 2

( , ) (0,0)18) lim ln( )

x yx x y

VI. Bi tp----------------------------------------------------------------------------------------------------------------5 Tm cc im gin on


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5. Tm cc imgin on

2 21) lnz x y

2 2

12)

1z

x y

2

13)

( )

z

x y

1

4) coszxy

3

2 25)

xz

x y

2

2 26)

xz

x y

VI. Bi tp----------------------------------------------------------------------------------------------------------------5 Tm cc im gin on (tip theo)


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5. Tm cc imgin on (tiptheo)

2 2

2 2

2 2

, 07)

0, 0

x

x yx yz

x y

3 3

, 08)

3, 0

x y

x yz x y

x y

2

sin( ), 0

9)

, 0

xyzz

zu

x z

VI. Bi tp----------------------------------------------------------------------------------------------------------------6 Tm tt c cc gi tr m hm s lin tc ti (0 0)


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6. Tm tt ccc gi trm hm slin tc ti(0,0).

3 22 2

2 2

2 2

, 01)

, 0

x xy

x yx yz

m x y

22 2

4 2

2 2

, 02)

, 0

x yx y

x yz

m x y

2 2

2 2

2 2

2

, 03)

, 0

xy

x yx yz

m x y

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