GII TCH HM NHIU BIN

CHNG I: O HM V VI PHN

CHNG II : TCH PHN BI

CHNG III: TCH PHN NG

CHNG IV: TCH PHN MT

CHNG V: CHUI S - CHUI LY THA

CHNG I: O HM V VI PHN

1: Cc khi nim c bn

2: o hm ring

3: Kh vi v Vi phn

4: o hm ring v vi phn hm hp

5: o hm ring v vi phn hm n

6: o hm theo hng Vecto gradient

7: Cng thc Taylor Maclaurint

8: Cc tr hm nhiu bin : Cc tr t do, cc tr c iu kin, GTLN-GTNN trong min ng

1 : Cc khi nim c bn

Min (Tp) xc nh ca hm l tt c cc gi trca (x,y) lm biu thc ca hm c ngha

Min (Tp) gi tr ca hm l tp cc gi tr m hm c th nhn c

Hm 2 bin f(x,y) l nh x f : D R

nh ngha hm 2 bin : Cho D l tp con ca R2

Hm 2 bin f(x,y) l nh x f : D R

nh ngha hm 2 bin : Cho D l tp con ca R2

( , ) ( , )x y f x y z

V d : Tm MX, MGT ca hm 2 2( , ) 9f x y x y

MGT l on [0,3]

MX l hnh trn 2 2 2( , ) : 9D x y R x y

MX 3

3

MGT

30

f(x,y)

(x,y)

1 : Cc khi nim c bn

1 : Cc khi nim c bn

Gii :

a. f(2,1) = 2

V d: Cho hm1

( , )1

x yf x y

x

Tnh f(2,1) v tm MX ca f

b. MX :

Ta ly na mt phng pha trn ng thng x+y+1 = 0 v b i ton b ng x = 1

th hm z = f(x, y) l phn mt cong S, khc

vi th hm 1 bin y = f(x) l phn ng cong.

Cho f(x, y) l hm 2 bin vi MX l D. th ca f l

tp tt c cc im M(x, y, z)R3, vi (x,y)D, z= f(x, y)

1 : Cc khi nim c bn

2

0 0

2 2 2

0 0

( , ) : ( , )

( , ) : ( ) ( )

B M r M R d M M r

x y R x x y y r

Hnh trn (khng

tnh ng trn) m ny cn cgi l mt r - lncn ca im M0

Hnh trn m tm M0(x0,y0), bn knh r k hiu B(M0,r) l tp

1 : Cc khi nim c bn

M0r

B(M0,r)

im trong : M1 gi l imtrong ca D nu tn ti tnht r1 > 0 sao cho r1- lncn ca M1 l B(M1,r1) nmhon ton trong D.

im bin : M2 gi lim bin ca D nu vimi r2 > 0, hnh cu mB(M2,r2) cha nhng imthuc D v nhng imkhng thuc D.

Cho tp D v 1 im M thuc R2. Ta nh ngha 2 loi im nh sau :

1 : Cc khi nim c bn

M1

M2

1 : Cc khi nim c bn Gii hn v lin tc

Tp D c gi l tp ng nu D cha mi im binca n. Tp cc im bin ca D gi l bin ca D

Tp D c gi l tp m nu R2\D l tp ng, khi ,mi im thuc D u l im trong, D khng cha btk im bin no

: ( , )r D B O r Tp D c gi l tp b chn nu n c cha trongmt hnh cu no , tc l

Nh vy, c nhng tp ch cha 1 phn bin mkhng cha ton b bin nn s l tp khng m,khng ng.

V d: Hnh v slide trc l tp khng ng, khng m

1 : Cc khi nim c bn

V d : Cho D l phn hnh cu

3 2 2 2( , , ) : 4D x y z R x y z

Bin ca D l ton b mt cu x2 + y2 + z2 = 4, do D khng cha bt k im bin no tc l mi im thuc D u l im trong. Vy D l tp m

V d : Cho hnh vnh khn

2 2 2( , ) :1 4D x y R x y

Bin ca D l 2 ng trn x2 + y2=1 v x2+y2 = 4 nm hon ton trong D nn D l tp ng

OB

A

Bin ca D l 3 on OA, OB, AB. Min D khng cha on AB tc l D khng cha mi im bin nn D khng l tp ng.

Tuy nhin, D khng l tp m v D cha cc im bin thuc on OA, OB

Tp D l tp b chn v ta c th ly ng trn ngoi tip tam gic OAB, ng trn ny cha min D tc l D nm hon ton trong 1 hnh cu m

1 : Cc khi nim c bn

V d : Trong R2 cho min D

2( , ) : 3, 0, 0D x y R x y x y

1 : Cc khi nim c bn

Gii hn hm : Hm f(x,y) vi min xc nh D c gi l c gii hn bng L khi M(x,y) dn n M0(x0,y0), k hiu

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

x y x yf x y L

nu 0, 0 : 0( , ) , ( , ) ,M x y D d M M

,f x y L

hoc 0

0

lim ,x xy y

f x y L

nh ngha ny tng t nh ngha gii hn hm 1 bin: Khong cch gia M v M0 gim dn th khong cch gia f(x,y) v L cng gim dn

1 : Cc khi nim c bn

ngha: Khi khong cch gia M v M0 gim dn th khong cch gia f(x,y) v L cng gim dn

1 : Cc khi nim c bn

Hm lin tc : Hm f(x,y) c gi l lin tc ti (x0,y0) nu f (x0,y0) xc nh v

0 00 0

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

x y x yf x y f x y

Hm lin tc trn min D nu n lin tc ti miim thuc min D

Tng, tch, thng ca 2 hm lin tc l hm lin tc

Cc hm s cp lin tc ti mi im thuc MX

Hp ca 2 hm lin tc l mt hm lin tc

2 : o hm ring

Ta c:

0

00

0

limx x

g x g xg x

x x

t g(x)=f(x,y0); nu hm g(x)c o hm ti x=x0 th ta gi l o hm ringtheo bin x ca hm f ti im (x0,y0) v k hiu l

0, x x x

0 00 0

0

0 00 0

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

xx

f y f yf x x xxf y yx

x x

0 00 0 00 0, , ,x xf

g x f y yx x f xx

yD

nh ngha o hm ring:Cho hm 2 bin f(x,y), 1 im (x0,y0) thuc minxc nh ca hm f.

Tng t, ta c nh ngha hr ca hm f theo bin y

Quy tc: Khi tnh o hm ring ca hm nhiu bin theo 1 bin no , ta coi cc bin khc l hng s

2 : o hm ring

Cc o hm ring ca hm n bin x1, x2, , xn (ni chung) li l cc hm n bin x1, x2, , xn

00 0 00 0, , ,y yf

f x x D f xy y yy

00

0 00x , ,limy

y y y

y

f f x

Nu f l hm nhiu hn 2 bin, ta cng nh ngha o hm ring tng t

2 : o hm ring 2 : o hm ring

V d: Trong nhng ngy nng, ta s cm thynng hn khi m trong khng kh thp (khhanh) v mt m hn khi m trong khng khcao hn.

,I f T H

Ngi ta t ra khi nim ch s nhit (K hiu l I) m t s kt hp gia nhit thc t (T) v m trong khng kh (H). Vy I l hm theo 2 bin T v H :

Di y l bng cc gi tr ca ch s nhit I theo nhit thc t T (0F) v m khng kh H (%)

Lu : Nhit ng bng ca nc l 00C=320Fv quy i 10C=1.80F

2 : o hm ring

T: Nhit thc t (0F)

H: m (%)

Cho H=70% c nh, ta xem xt tc thay i cach s nhit I khi c nh m H=70%. Lc ny I chcn ph thuc vo T

t : ,70g T f T, , th o hm ca g khi T=960F l tc thay i ca I khi T=960F=35,60C

0 0

96 96 96 ,70 96,7096 lim lim

h h

g h g f h fg

h h

2 : o hm ring

Ta c:

Tnh xp x g(96) bng cch s dng bng trn vi 2 trng hp h=2, h=-2

98 96

962

g gg

98,70 96,70

2

f f 4

94 96 94,70 96,70

962 2

g g f fg

3.5

Ly gi tr trung bnh, ta c gi tr xp x

96 3.75g

2 : o hm ring

96 3.75g

Tng t, ta nhn hng ng vi T=960, tc l gi c nh T, v t G(H)=f(96,H)

Tnh o hm G(70) bng cch cho h = 5 v h = -5, ri ly trung bnh cng nh trn, ta c kt qu:

70 0.9G

iu ny c ngha l khi nhit l 35.60C v m l 70% th ch s nhit tng khong cho mi phn trm m tng ln.

0 00.9 0.5F C

iu ny c ngha l khi m khng kh l 70% vnhit thc t l th ch s nhittng khong cho mi tng thc t

0 096 35.6F C0 03,75 2.1F C

2 : o hm ring

ngha hnh hc ca hr ca hm f(x,y) ti (a,b):

Tng t: fy(a,b) l h s gc ca tip tuyn T2 tc l h s gc ca mt S theo phng Oy ti P

Ta gi fx(a,b) l h sgc ca mt S theophng Ox ti P(a,b,c)

Xt trong mp y=b: C1 l

giao tuyn ca mp vimt S th pt C1 l f(x,b), T1l tip tuyn ca C1 tiP(a,b,f(a,b)) th o hmfx(a,b) l h s gc catip tuyn T1

Gi S l mt cong z=f(x,y)

2 : o hm ring

V d: Tnh o hm ring ti (x,y)=(1,1) ca hm 2 2( , ) 4 x 2f x y y

2 1,1 2x xf x f

H s gc ca tip tuynvi ng cong C1 ti(1,1,1) l - 2

4 1,1 4y yf y f

H s gc ca tip tuynvi ng cong C2 ti(1,1,1) l - 4

,v v hnh minh ha

2 : o hm ring

Gii : 2 2 2 2

,x yx y

f fx y x y

a.

V d: Tnh o hm ring ca cc hm sau: 2 2

cos

. f(x,y)=

. f(x,y)=e

. f(x,y,z)=ln(x+e )

x

y

y

a x y

b

c xyz

1. f ,f ,

y

x y zy y

ec yz xz f xy

x e x e

b.cos cos

2

1sin , sin

x x

y y

x y

x x xf e f e

y y y y

2 : o hm ring

0

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

00 mx

x

f x

x

ff

V d : Cho hm 3 33( , )f x y x y Tnh fx, fy ti (0,0)

Gii :

Nu tnh bng cch thng thng, ta s khng tnh c hr ti im c bit (0,0). Do , ta s tnh cc hr trn bng nh ngha

3

0

30

lim 1x

x

x

V vai tr ca x, y nh nhau trong hm f nn ta cng c fy(0,0) = 1

2 : o hm ring

V d : Tnh cc hr ca hm

Gii: Ta tnh 3 hr ca hm 3 bin

h theo x: y, z l hng s

h theo z: x,y l hng s

, ,z

xf x y z

y

1

. .

z

x

x

x xf z

y y

1

.

zz x

y y

h theo y: x, z l hng s

1

. .

z

y

y

x xf z

y y

1

2.

zzx x

y y

.ln

z

z

x xf

y y

2 : o hm ring

o hm cp 2 ca hm f(x,y) l o hm ca o hm cp 1:

o hm cp2 theo x:

o hm cp 2 theo y:

o hm cp 2 hn hp:

2

0 0 0 0 0 0( , ) ( , ) ( )( , )xy x yf

f x y xy x

y f f x y

2

0 0 0 0 0 02( , ) ( , ) ( )( , )yy y y

ff x y x y

yf f x y

2

0 0 0 0 0 02( , ) ( , ) ( )( , )xx x x

ff x y x y

xf f x y

2

0 0 0 0 0 0( , ) ( , ) ( )( , )yx x yf

f x y xx y

y f f x y

2 : o hm ring

nh l Schwarz : Nu hm f(x,y) v cc o hmring fx, fy, fxy, fyx tn ti v lin tc trong min mcha (x0,y0) th fxy(x0,y0) = fyx(x0,y0)

Ghi ch :

1.i vi cc hm s cp thng gp, nh lSchwarz ng ti cc im tn ti o hm

2.nh l Schwarz cn ng cho cc o hm ringt cp 3 tr ln. Tc l cc o hm ring hnhp bng nhau khi s ln ly o hm theo mibin bng nhau, m khng ph thuc vo th tly o hm theo cc bin

2 : o hm ring

Gii :

Hm 2 bin nn ta tnh 2 o hm ring cp 1

v 4 o hm ring cp 2

V d: Tnh o hm ring n cp 2 ca hm

( , ) sin( )x yf x y e e

cos( ), cos( )x x y y x yx yf e e e f e e e

cos( ) sin( ) ,x x y x x yxx

f e e e e e e

cos( ) sin( ) ,y x y y x yyy

f e e e e e e sin( )x y x y

xy yxf f e e e e

2 : o hm ring

Tng t, ta c cc o hm ring cp (n+1) l o hm ca o hm cp n

V d: Tnh o hm ring cp 3 ca hm:

f(x,y) = x2y 3ex+y

Gii:

2 o hm ring cp 1 : 22 3 , 3x y x yx yf xy e f x e

4 o hm ring cp 2 : 2 3 , 3 ,

2 3

x y x y

xx yy

x y

xy yx

f y e f e

f f x e

8 o hm ring cp 3: 3 , 2 3 ,

3 , 3

x y x yf e f e f fxxx xxy yxx xyx

x y x yf e f e f fyyy yyx yxy xyy

2 : o hm ring

Ghi nh: o hm ring cp cao hn hp bng nhaunu s ln ly o hm theo cc bin bng nhau(khng k n th t ly o hm theo tng bin)

V d: Cho hm f(x,y,z) = xcosy 2ysinz. Tnh o hm ring cp 2.

3 o hm cp 1:

cos , sin 2sin , 2 cosx y zf y f x y z f y z 9 o hm cp 2

0, sin , 0 ,

cos , 2cos , 2 sin

xx xy yx xz zx

yy yz zy zz

f f y f f f

f x y f z f f y z