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8/9/2019 Tong Hop de Thi Cao Hoc Toan

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Luyn gii thi cao hc mni s 1

GII THI TUYN SINH CAO HC THNG 8/2008MN C BN: I S V GII TCH

Bi 1:Cho nh x tuyn tnh f : R4 R3 xc nh bif(x1,x2,x3,x4)=(x1+x2,x2+x3,x3+x4) vi mi x=(x1,x2,x3,x4) R

4

M={ (x1,x2,x3,x4) R4: x1-x2=0 v x3-x4=0}

a. Tm ma trn f trong c s chnh tc ca R4v R3 . xc nh Imf v Kerf

b. CM f(M) l khng gian vect con ca R3

. tm dimf(M)Gii :

Tm ma trn f trong c s chnh tc ca R4v R3Trong R4 ta c e1=(1,0,0,0),e2=(0,1,0,0),e3=(0,0,1,0),e4=(0,0,0,1)

Trong R3 ta c e1=(1,0,0),e2=(0,1,0),e3=(0,0,1)Ma trn f trong c s chnh tc l

1100

0110

0011

4321

4321

4321

)(),/( 34

cccc

bbbb

aaaa

Aeef

m f(e1)=(1,0,0)=a1e1+b1e2+c1e3ta tm c (a1,b1,c1)=(1,0,0)f(e2)=(1,1,0) (a2,b2,c2)=(1,1,0)

f(e3)=(0,1,1) (a3,b3,c3)=(0,1,1)

f(e4)=(0,0,1) (a4,b4,c4)=(0,0,1)

Xc nh Imf,Kerf

Kerf ={ xR4: f(x)=0 }

Ta c h

Rx

xx

xx

xx

xx

xx

xx

4

43

42

41

43

32

21

0

0

0

h c nghim tng qut l (-a,a,-a,a)

H nghim c bn l (-1,1,-1,1)Vy dimKerf=1, c s ca Kerf =(-1,1,-1,1)

Tm Imf

Ta c f(e1)=(1,0,0),f(e2)=(1,1,0), f(e3)=(0,1,1),f(e4)=(0,0,1)Nn Imf=

Ta c

000

100010

001

...

100

110011

001

vy c s ca Imf l f(e1),f(e2),f(e3) v dimf=3b.

Bi 2:Gii v bin lun h phng trnh


2/25



1

1

1

4321

4321

4321

xmxxx

xxmxx

xxxmx

Gii: lp ma trn cc h s

mmmm

mm

m

m

m

m

m

m

m

A

1.1200

0.0110

1.111

....

1.111

1.111

1.111

1.111

1.111

1.111

2

vy ta c

1

0)1()1(

1)1()2)(1(

4321

32

43

xmxxx

xmxm

mxmxmm

Bin lun:Vi m=1 h c v s nghim ph thuc 3 tham s x2,x3,x4nghim ca h l (1-a-b-c,a,b,c) a,b,c Rvi m=-2 h c v s nghim ph thuc tham s x3

nghim ca h l (a,a,a,1) a Rvi m khc 1,-2 h c v s nghim ph thuc tham s x4v m

nghim ca h l

ax

m

ax

m

ax

m

ax

2

1

2

1

2

1

a R

Bi 3:Cho chui lu tha

1

1

2.

)2()1(

nn

nn

n

x

a.

Tm min hi t ca chuib. Tnh tng ca chui

Gii:

a.

ta cn

nn

nn

xxU

2.

)2()1()(

1

tnh Cxx

nxU

nn

n

nn

2

2

2

2.

1)(

limlim

theo tiu chun csi nu chui hi t khi C0 v

0yx,0

0,1sin),(

22

22

2 yxyx

xyxf

a


3/25



Tu theo gi tr ca a>0 xt s kh vi ca f ti (0,0), s lin tc ca fx,fyti (0,0)Gii : Tnh cc hr

ti x2+y2>0

aax

yxyx

x

yxxf

2222

3

22

' 1cos2

)(

1sin2

ay yxyx

yxf )(

1cos

22222

2'

ti x=y=0

t

ftff

tx

)0,0()0,(lim

0

'

t

ftff

ty

)0,0(),0(lim

0

'

xt s kh vi ca f ti (0,0) Cn xt : ),(lim0,

tsts

Vi tfsfftsf

tsts

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

1),( ''

22

Nu ),(lim0,

tsts

=0 th hm s kh vi ti (0,0) ngc li th khng kh vi

xt s lintc ca fx,fyti 0(0,0)nu : )0,0(),( ''

0,lim xx

yx

fyxf

, )0,0(),( ''

0,lim yy

yx

fyxf

th hm s khng lin tc ti

(0,0) ngc li th lin tcBi 5: Cho (X,d ) l khng gian Metric A X khc rngCho f: X R nh bi f(x)=d(x;A)=inf{d(x,y): yA}

a.

CM: f lin tc iu trn Xb. Gi s A l tp ng , B l tp compc cha trong X v AB =t d(A,B)= inf{ d(x,y),x A,y B }CM : d(A,B)>0

Gii :

a.

CM f lin tc iu trn X cn CM )',()'()( xxdxfxf

ta c d(x,y) d(x,x)+d(x,y) ly inf 2 v d(x,A)-d(x,A) d(x,x)tng t thay i vai tr v tr ca x v x nhau ta suy ra PCMvy f lin tc ti x, do x tu nn f lin tc iu trn X

b.

Gi s tri li d(A,B)=0Khi ta tm c cc dy (xn) A, (yn)B sao cho limd(xn,yn)=0Do B compc nn (yn) c dy con knky )( hi t ve y0B

Ta c ),(),(),( 00 yydyxdyxd kkkk nnnn

M 0),(0),(),( 00 limlimlim

yxdyydyxdkkkk n

kn

knn

k

Do A l tp ng dy knkx )( A, 0)( yx knk nn y0A

iu ny mu thun vi gi thit AB =.Vy d(A,B)>0


Bi 1: Tm min hi t ca chui lu tha


4/25



nn

n

xn

n 2

0

232

1

Gii : t X=(x-2)2 k X 0

Ta tm min hi t ca chui nn

n

Xn

n

0 32

1 xt

32

1

n

nun

Ta c 2

1

32

1

limlim

n

nul

n

n nn

21

l

R nn khong hi t l (-2,2)

Xt ti X= 2, X= -2

Ta c chui

n

n

n

n

n

n2

32

1)1(

0

n

n

n

n

n

0 32

22)1(

0132

22limlim

n

nu

n

nn

n

nn chui phn k

vy min hi ttheo X l (-2,2)min hi t theo x l 222222 xx

Bi 2:Cho hm s

0yxkhi0

0yxkhi1

sin)(),(

22

22

22

yxyx

yxf

Chng t rng hm s f(x,y)c o hm ring fx,fykhng lin tc ti 0(0,0)Nhng hm s f(x,y)kh vi ti 0(0,0).Gii :

Tnh cc hr ti (x,y) (0,0) va ti (x,y)=(0,0)

Ti (x,y) (0,0)

Ta c

222222

' 1cos21

sin2yxyx

x

yxxfx

222222

' 1cos21

sin2yxyx

y

yxyfy

Ti (x,y)=(0,0)

1

t

1sindo0

1sin

)0,0()0,(2

0

2

2

00

'

limlimlim

t

t

tt

t

ftff

ttt

x

1t

1sindo0

1sin

)0,0(),0(2

0

2

2

00

'

limlimlim

tt

tt

t

ftff

ttty

CM : fx,fykhng lin tc ti 0(0,0) Ta CM : 0'0,

lim

xyx

f v 0'

0,lim

y

yx

f

Hay CM : )0,0(),( ''

0,lim xx

yx

fyxf

, )0,0(),( ''

0,lim yy

yx

fyxf

Ta c :


5/25



Do

0 xkhi,221

cos.2

11

cos

0xkhi,021

sin2,1yx

1sin

,1

cos.21

sin.2),(

22222222

22

22220,

220,

'

0,limlimlim

xyx

x

yxyx

x

yx

xyx

x

yxyx

x

yxxyxf

yxyxx

yx

nn )0,0(),( ''

0,lim xx

yx

fyxf

tng t ta CM : c )0,0(),( ''0,

lim yyyx

fyxf

vy fx,fykhng lin tc ti 0(0,0) Ta CM : f(x,y)kh vi ti 0(0,0). Cn CM : 0),(lim

0,

tsts

Vi tfsfftsfts

ts yx )0,0()0,0()0,0(),(1

),( ''22

)1ts

1sin(do01sin.),(2222

22

0,0,limlim

ts

tstststs

vy f(x,y)kh vi ti 0(0,0)Bi 3: Cho RR*1,0: l mt hm s lin tcCMR : Hm F: C[0,1]R xc nh bi

1

0

))(,()( dttxtxF khi x(t) 1,0C l hm s lin tc trn C[0,1]

Gii: C nh x0, CM f lin tc ti x0

t M=1+sup )(0 tx , t 1,0C

Cho 0

lin tc trn tp compac D= [0,1]*[-M,M] nn lin tc u trn Dtn ti s 1 >0 sao cho

)','(),(',')','(),,( 11 ststssttDstst

t ),(1,0:),1min( 01 xxdx m MMtxtxtx ,)(1)()( 00

)()())(,())(,())(,())(,( 01

0

00 xFxFdttxttxttxttxt

ta CM c ))(),((),(:0,0 00 xFxFdxxd vy F lin tc ti x0

Bi 4:Cho nh x tuyn tnh 34: RRf xc nh bif(x1,x2,x3,x4)=(x1-2x2+x4,-x1+x2+2x3,-x2+2x3+x4)

1.

Tm c s v s chiu ca kerf, Imf2.

f c phi l n cu , ton cu khng?

Gii : 1. Tm c s v s chiu ca kerf

Vi x=( x1,x2,x3,x4)


6/25



Ta c : 0)(:ker 4 xfRxf

f(x1,x2,x3,x4)=(x1-2x2+x4,-x1+x2+2x3,-x2+2x3+x4)=0

02

02

02

432

321

421

xxx

xxx

xxx

lp ma trn

00001210

1021

12101210

1021

12100211

1021

A

vy Rank(A)=2

ta c

Rxx

xxx

xxx

43

432

421

,

2

2

nn dimKerf=2

nghim c bn l (1,1,0,1),(4,2,1,0) v l c s ca Kerfdo dimKerf =2 0 nn f khng n cu Tm c s , s chiu ca Im fIm f l khng gian con ca R3sinh bi h 4 vectf(e1)=(1,-1,0) vi e1=(1,0,0,0)f(e2)=(-2,1,-1) vi e2=(0,1,0,0)f(e3)=(0,2,2) vi e3=(0,0,1,0)f(e4)=(1,0,1) vi e4=(0,0,0,1)ta tm hng ca 4 vect trn

xt ma trn

000

000

110

011

110

220

110

011

101

220

112

011

B

Rank(B)=2, , dim Imf =2 , c s ca Imf l f(e1),f(e2)Do , dim Imf =2 nn f khng ton cuBi 5:Cho '':,': VVgVVf l nhng nh x tuyn tnh sao cho gf kerker Hn naf l mt ton cu . CMR tn ti duy nht mt nh x tuyn tnh ''': VVh sao cho h.f=g

Gii:

Bi 6: Cho dng ton phng trn R3f(x1,x2,x3)= 3121

2

3

2

2

2

1 222 xaxxxxxx

a.

a dng ton phng v dng chnh tc bng phng php Lagrangeb.

Vi gi tr no ca a th f xc nh dng, khng mGii : a. f(x1,x2,x3)= 3121

2

3

2

2

2

1 222 xaxxxxxx =

= 2322

32

2

321

61

62

3

4

22 x

ax

ax

axxx


7/25



t

33

322

3211

33

322

3211

6

32

6

42

yx

ayyx

ayyyx

xy

axxy

axxxy

ta c c s f chnh tc l u1=(1,0,0),u2=(-1/2,1,0),u3=(-a/3,a/6,1)

ma trn trong c s chnh tc l

1006

10

32

11

a

a

Tu

b. f xc nh dng khi 6606

12

aa

f xc nh khng m khi 60612

aa


Bi 1:Cho u=u(x,y), v=v(x,y) l hm n suy ra t h phng trnh

021

.

012.

xv

uey

uvex

vu

vu

tm vi phn du(1,2), dv(1,2) bit u(x,y)=0, v(x,y)=0Gii :l thuyt : cho hm n

0),,,(

0),,,(

vuyxG

vuyxF xc nh bi u=u(x,y), v=v(x,y)

Tnh cc o hm ring ca hm nT h trn ta c

0

0

''''

''''

vvuuyyxx

vvuuyyxx

dGdGdGdG

dFdFdFdF

v

u

vvuuyyxx

vvuuyyxx

d

d

dGdGdGdG

dFdFdFdF

''''

''''

Tnh

)2,1(

)2,1(

v

u

d

d

Ta c :

Bi 2: Tm min hi t ca chui lu tha

2

2)1(

)(ln

1

n

nxnn

Gii :t X= x+1 ta c

22)(ln

1

n

nXnn

Xt212

))1)(ln(1(

1

)(ln

1

nn

u

nn

u nn

Ta c : 2

21

)1ln()1(

)(lnlimlim

nn

nn

u

uL

nn

n

n


8/25



Tnh )1ln(

ln.

1

1

1).1ln(.2

1.ln.2

)1ln(

)(lnlimlimlim

tan

2

2

nn

n

n

nn

nn

n

n

nn

lopi

n

Tnh 1

11

1

)1ln(

lnlimlim

tan

n

n

n

n

n

lopi

n

Nn 11

L

R , khong hi t l (-1,1)

Ti X= 1 ta c chui

2

2)1(

)(ln

1

n

n

nn

T ta c

1)1ln()1(

)(ln2

21

limlimnn

nn

u

uL

nn

n

n

Chui phn k , MHT theo X l (-1,1)

MHT theo x l (-2,0)Bi 3:Cho X l khng gian metric compac f: XX thod(f(x),f(y))0Khi g(f(x0))=d(f(x0),f(f(x0)))< d(x0,f(x0))=g(x0)iu ny mu thun vi s kin g(x0)=min(g(x))

Vy g(x0)=d(x0,f(x0))=0 hay x0=f(x0)CM tnh duy nht ca x0.Gi s c y0X sao cho y0=f(x0)Khi d(x0,y0) =d(f(x0),f(y0))


9/25



Hn na do A1=f(X)X nn A2=f(A1) f(X)=A1Gi s An+1 An ta c An+2=f(An+1) f(An)=An+1Vy An+1 NnAn , nA l h c tm cc tp ng trong khng gian compac

Theo tnh cht phn giao hu hn ta c A=

n

n

A1

CM: f(A)=A cn CM : f(A)A (1) , f(A) A (2)

CM : f(A)A (1)

Do A Annn f(A) f(An)=An+1vi mi n, l dy gim nn

f(A) AAnn

1

1

f(A) A (2)

ly tu xA cn CM x f(A)v x An+1=f(An) vi mi n=1,2 tn ti xnAn: x=f(xn)do X compact nn c dy con (x

nk)

k: ax

knk

lim

khi xxfkn

k

)(lim , do f lin tc nn afxf knk

()(lim

) ta cn CM a A

c inh n ta c nnnnn AxAAx kkk khi nk n

do Anng nnk

Aaxk

lim

vy a An vi mi n=1,2 do a A, x=f(a) f(A)

vy ta CM c f(A)=A

Bi 4:Gii v bin lun h

1

1

1

4321

4321

4321

xmxxx

xxmxx

xxxmx

Gii :Ta c ma trn m rng

1.1111.111

1.111

mm

m

A i ch d1, d3, bin i ma trn v dng

1.1)2)(1(00

0.0110

1.111

mmmm

mm

m

A

bin lun nu m=1 h c VSN ph thuc 3 tham s x2,x3,x4v RankA=1

nghim ca h l x1=1-a-b-c, x2=a,x3=b,x4=c

nu m=-2 h c VSN ph thuc tham s x3v RankA=3nghim ca h l x1=x2=x3=a,x4=1

nu m 1v m -2 th h c VSN ph thuc vo tham s x4va tham s m


10/25



nghim ca h l2

11

m

ax ,

2

12

m

ax ,

2

13

m

ax , Raax ,4

Bi 5:Trong R3cho c s :u1=(1,1,1), u2= (-1,2,1), u3=(1,3,2)

cho nh x tuyn tnh f: R3 R3xc nh bif(u1)= (0,5,3), f(u2)=(2,4,3), f(u3)=(0,3,2)

tm ma trn ca f trong c s l ma trn cho ho cGii :b1. Tm ma trn ca f trong c s u

Ta c h

)3()(

)2()(

)1()(

3322113

3322112

3322111

ucucucuf

ubububuf

uauauauf

T (1) ta c (0,5,3)=a1(1,1,1)+a2(-1,2,1)+a3(1,3,2)

1

1

0

02

032

0

3

2

1

321

321

321

a

a

a

aaa

aaa

aaa

Tng t t ( 2) ta c b1=1,b2=0,b3=1Tng t t (3) ta c c1=1,c2=1,c3=0

Vy ma trn A trong c s f l

011

101

110

333

222

111

)(/

cba

cba

cba

A ufA

B2. Tm GTR- VTR ca A v ca f (GTR ca A chnh l GTR ca f)

Xt ma trn t trng2

)(1023

11

11

113

m

kepmmm

m

m

m

A c 2 gi tr ring, nn f c 2 gi tr ring m=-1, m=2Tm VTR ca A t suy ra VTR ca f

vi m=-1 ta c 0000

000

111

111

111

111

VTR ca A c dng

bx

ax

baxxx

Rxx

xxx

3

2

321

32

321

,

0 a,bR

Dng VTR ca A l (-a-b,a,b)Vy A c 2 VTR (-1,0,1),(-1,1,0)T VTR ca f c dng n= x1u1+x2u2+x3u3=(-a-b)u1+au2+bu3=

=(-a-b)(1,1,1)+a(-1,2,1)+b(1,3,2)=(-2a,a+2b,b)

vy f c 2 VTR LTT vi a=1,b=0 :VTR l n1=(-2,1,0)vi a=0,b=1: VTR l n2=(0,2,1)

vi m=2 ta c 0000

330

211

112

121

211

211

121

112


11/25



VTR ca A c dng

ax

axx

aaaxxx

Rx

xx

xxx

3

32

321

3

32

32122

033

02

aR

Dng VTR ca A l (a,a,a),Vy A c VTR (1,1,1)

T VTR ca f c dng n= x1u1+x2u2+x3u3=au1+au2+au3==a(1,1,1)+a(-1,2,1)+a(1,3,2)=(a,6a,4a)vy f c VTR l n3=(1,6,4)

b3 : KL vy f c 3 VTR LTT n1,n2,n3do 3 VTR n1,n2,n3lm thnh 1 c s caR

3v ma trn ca f trong c s l ma trn cho ho c

ta c :

200

010

001

)/(nfA


Bi 1:Cho

0yx,0

0y x,1

sin),(

22

22

2

yx

yxyxf

a. Xt s kh vi ca f ti (x,y)R2c bit ti (0,0)b.

Xt s lin tc ca cc HR '' , yx ff ti (0,0)

Gii :

Ti (x,y) (0,0) Ta c

22222

3

22

'

22222

2'

1cos.

)(

21sin.2

1cos.

)(

21

yxyx

y

yxyf

yxyx

xyf

y

x

Do '' , yx ff lin tc ti mi (x,y) (0,0) nn f kh vi ti mi (x,y) (0,0)

Ti (x,y)=(0,0)Ta c

1)0,0()0,()0,0( lim0

' t

ftfft

x

1)t

1sin(do0

1sin

.)0,0(),0(

)0,0(2

22

00

'

limlim

t

ttt

ftff

tty

Tnh ),(lim0,

tsts

Ta c 22

2

22

''

22

1sin..

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

1),(

stt

tstfsfftsf

tsts yx

2222

2

0,0,

1sin.),( limlim tststts

tsts

0


12/25



do 11

sin22

ts

nn f kh vi ti (0,0)

b.Xt s lin tc ca cc HR '' , yx ff ti (0,0)

Xt s lin tc ca cc HR '' , yx ff ti (0,0) ta tnh ),(),,('

0,

'

0,limlim yxfyxf y

yxx

yx

nu )0,0(),(),0,0(),( ''

0,

''

0,

limlim yyyx

xx

yx

fyxffyxf

th '' , yx ff lin tc ti (0,0)

'', yx ff khng lin tc ti (0,0)

chn )0,0(0,1,

nyx nn ta c

0)0,1(

1)0,1(

'

0,

'

0,

lim

lim

nf

nf

yyx

xyx

chn )0,0(2

1,

2

1','

nnyx nn ta c

),(

),(

'''

0,

'''

0,

lim

lim

nny

yx

nnxyx

yxf

yxf

vy '' , yx ff khng lin tc ti (0,0)

Bi 2:Cho (X,d )l khng gian mtric compac, f: XX tho mn:d(f(x),f(y))0khi h(f(x0))=d(f(x0),f(f(x0)))


13/25



Nn gnlin tcDo Xxxgxfxdxffxfdxfxdxg n

nnn

n ),())(,()))((),(())(,()( 001

01 (gn(x))dy

gim khng m nn hi tt a= limgn(x)Gi s a>0, do X compac dy fn(x)cha dy con hi t k

nxf k )(

t )(lim

xfy kn

k

Ta ckn

xgk

)(1 l dy con ca nn xg )( nn )()( 1limlim xgxga kk nk

nk

0),())(,()( 00limlim

yxdxfxdxga kk

n

kn

k

)(1lim xga knk

ayxdyfxfdxfxdk

n

k

k

),())(),(())(,( 001

0 limlim

mu thun vy Xxxgnn

,0)(lim

c. CM (gn)nhi t iu v 0 trn Xvi 0 t ),()(: 1 nnn gxgXxG l tp m

do gn(x) >gn+1(x)nn GnGn+1ta c nn

GX

1

do X compac nn c n0:0

0

1

nn

n

n

GGX

vy 0nn,)(0 khiXxxgn vy (gn)nhi t iu v 0 trn XBi 3 Cho V l khng gian vect , f: V V l nh x tuyn tnh tho mn f2=f CM:Kerf+Imf=V v 0Imker ff

Gii

CM: Kerf+Imf=V ta cn CM Kerf+ImfV (1), Kerf+Imf V (2)

CM Kerf+ImfV (1) hin nhin

CM: Kerf+Imf V (2)

Ly tu xV cn CM x Kerf+ImfTa c x= x-f(x)+f(x) m f(x) Imf cn CM (x-f(x)) Kerf cn CM f(x-f(x))=0Xt f(x-f(x))=f(x)-f2(x)=f(x)-f(x)=0 nn (x-f(x)) Kerf hay xKerf+Imf

Vy Kerf+Imf VT (1),(2) ta c Kerf+Imf=V

CM 0Imker ff

Ly y tu y: y ff Imker cn CM y=0Do y ff Imker khi c xV : f(x)=y v f(y)=0Do f2=f nn y=f(x)=f2(x)=f(f(x))=f(y)=0

Vy y=0 hay 0Imker ff Bi 4: Cho f: R4 R3nh bif(x1,x2,x3,x4)=(x1-x2+x3,2x1+x4,2x2-x3+x4)

a. Tm c s v s chiu ca Kerf, Imf

b.

Tm u R4 sao cho f(u)=(1,-1, 0)

Gii :a.

Tm c s s chiu ca KerfVi x=(x1,x2,x3,x4)


14/25



02

02

0

0)(:

432

41

321

4

xxx

xx

xxx

xfRxKerf

ta c ma trn m rng

0.1120

0.1002

0.0111

0.0100

0.1220

0.0111

bin i ta c h

ax

x

ax

ax

Rx

x

xxx

xxx

2

00

022

0

4

3

2

1

4

3

432

321

l nghim tng qut ca h

ta c dimKerf =1

c s ca Kerf l (1,1,0,2) Tm c s v s chiu ca Imf

ta c f(e1)=(1,2,0), f(e2)=(-1,0,2), f(e3)=(1,0,-1), f(e4)=(0,1,1)Imf=(f(e1),f(e2),f(e3),f(e4))

Ta c

110

101

201

021

000

100

200

021

Nn dim Imf =3

Vy c s ca Imf l (f(e1),f(e2),f(e3))

b. Tm u

R

4

sao cho f(u)=(1,-1, 0)ta c : f(u)=(1,-1, 0) =(x1-x2+x3,2x1+x4,2x2-x3+x4)

ta c h

ax

x

xx

xx

xxx

xx

xxx

2

1

2

1

2

1

2

1

2

1

02

12

1

4

3

42

41

432

41

321

(a R)

lp ma trn m rng bin i gii h trn ta c u=(x1,x2,x3,x4)Bi 5

: Tm GTR- VTR v cho ho ma trn

A=

221

221

115

Gii :Xt a thc t trng

3

6

0

0189

221

221

11523

a

a

a

aaa

a

a

a

vy A c 3 GTR a=0, a=6, a=3

tm VTR


15/25



vi a=0 :ta c

000

990

221

115

221

221

221

221

115

ta c h

ax

ax

x

ax

xx

xxx

3

2

1

3

32

321 0

099

022

suy ra VTR (0,a,a) vi a=1 th VTR (0,1,1)

vi a=6: ta c

000

330

111

421

241

111

c h

ax

ax

ax

ax

xx

xxx

3

2

1

3

32

321 2

033

0

suy ra VTR (-2a,-a,a) vi a=1 th VTR (-2,-1,1)

vi a=3: ta c

000330

121

112221

121

121211

112

c h

ax

ax

ax

ax

xx

xxx

3

2

1

3

32

321 3

033

02

suy ra VTR (3a,a,a) vi a=1 th VTR (3,1,1)

ma trn cn tm l T=

111

111

320

v T-1

AT=

300

060

000


Bi 1:Cho hm s0y xkhi0

0yxkhi1

sin)(),(

22

22

22

yx

yxyxf

CMR hm s f(x,y ) c cc o hm ring '' , yx ff khng lin tc ti (0,0) nhng f(x,y)

kh vi ti (0,0)Gii :

Tnh cc hr '' , yx ff

ti (x,y) (0,0)

ta c222222

' 1cos21

sin2yxyx

x

yxxfx

222222

' 1cos21

sin2yxyx

y

yxyfy

ti (x,y)=(0,0)

0

1sin

)0,0()0,( 22

00

'

limlim

t t

t

t

ftff

ttx


16/25


17/25



a.CMR : M l tp ng khng rng v b chn trong khng gian metric C([0,1]) vimtric d(x,y)=max{ )()( tytx : t 1,0 } vi x(t),y(t) )1,0(C

b. xt RCf )1,0(: xc nh bi f(x)= 1

0

2 )( dttx

CM : f lin tc trn M nhng f khng t c GTNN trn M t suy ra M khngphi l tp compc trong C([0,1])Gii : a.

CM : M l tp ngLy dy (xn) M : limxn=x cn CM xMTa c 1)1(,1,0,1)(0 nn xttx

Cho n ta c 1)1(,1,0,1)(0 xttx nn xMVy M l tp ng

b.

CM f lin tc trn M

Xt tu x )1,0(C , (xn) M : limxn=x cn CM limf(xn)=f(x)Ta c Nxxdxxdtxtxtxtxtxtxtx nnnnn ),().,()(2)()(.)()()()(

22

Vi N= 1,0,)(2sup ttx

Nxxdxxdtxtxxfxf nnnn ),().,()()()()(1

0

22

do limd(xn,x)=0 nn t y ta c limf(xn)=f(x)vy f lin tc trn M

CM f khng t GTNN trn M

Trc tin ta CM inff(M)=0, nhng khng tn ti x

M f(x)=0t a= inff(M) ta c f(x) Mx ,0 nn a 0 Vi xn(t)=t

nta c xnM

n0121

12)()(

1

0

121

0

2

1

0

2 khinn

tdttdttxxfa

nn

nn

vy a= inff(M)=0 khng tn ti xM f(x)=0

gi s tn tai xM f(x)=0 ta c )(,0)(,0)( 221

0

2 txtxdttx lin tc trn [0,1] suy

ra x(t)=0 vi mi t [0,1] iu ny mu thun vi x(1)=1 vi mi xMvy khng tn ti xM f(x)=0t y ta suy ra M khng l tp compcgi s nu M l tp compc , f lin tc th f t cc tiu trn M tc l c x0M saocho f(x0)=inff(M)=0 iu ny mu thun vi khng tn ti xM f(x)=0vy M khng l tp compcBi 4: Cho 33: RRf l mt php bin i tuyn tnh xc nh bif(u1)=v1, f(u2)=v2, f(u3)=v3

u1=(1,1,1),u2=(0,1,1), u3=(0,0,1)

v1=(a+3,a+3,a+3),v2=(2,a+2,a+2), v3=(1,1,a+1)a.tm ma trn f vi c s chnh tc e1=(1,0,0), e2=(0,1,0), e3=(0,0,1)

b. Tm gi tr ca a f l mt ng cu


18/25



c. khi f khng l mt ng cu hy tm c s v s chiu ca Imf v Kerfd. vi a=-3 f c cho ho c khng trong trng hp f cho ho c hy tm mtc s ma trn f voi c s c dng cho .Gii :

Bi 5:Cho dng ton phng323121

2

3

2

2

2

1321 2222),,( xxxaxxxxxxxxxf

a. a dng ton phng v dng chnh tc

b. Vi gi tr no ca a th f l xc nh dng v na xc nh dngGii : a. ta c

2322

32

2

321

323121

2

3

2

2

2

1321

)22()1()(................

.......2222),,(

xaaxaxaxxx

xxxaxxxxxxxxxf

t

33

322

3211

33

322

3211

)1(

)21(

)1(

yx

yayx

yayyx

xy

xaxy

axxxy

c s f chnh tc l u1=(1,0,0),u2=(-1,1,0),u3=(1-2a,a-1,1)

ma trn

100

1102111

aa

Tu

b.f xc nh dng khi -2a2+2a>0 10 a f na xc nh dng khi -2a2+2a=0 1,0 aa


Bi 1: Tm min hi t ca chui hm lu tha

nnn

n

xn

n )1(

1 1

2

Gii :

Xt

nn

nn

u

1

1

11

Ta c L= en

u

n

n

nn

n

1

1

11limlim

NneL

R11

, khong hi t l

ee

1,

1

Xt tai 2 u mt x=e

1


19/25



Ta c chui n

n

n

nnnnn

n enen

n

11

1

11

1

1

2

1

1)1(

1

011.

1.

1

11

1

limlim

ee

enu

n

n

nn

n

vy MHT ca chui hm lu tha l

ee

1,

1

Bi 2 :Cho hm s f:R2R xc nh bi

(0,0)y)(x,khi0

(0,0)y)(x,khi2

),(22

yx

xy

yxf

a.

Xt s lin tc ca f trn R2b.

Tnh cc o hm ring ca f trn R2Gii :Ch : nu 0)0,0(),(lim

0,

fyxfyx

th hm s lin tc

Ti mi (x,y) (0,0) th hm s lin tc v l hm s cp

Xt s lin tc ca f trn R2ti (0,0)

Tnh22

0,0,

2),( limlim

yx

xyyxf

yxyx

Chn dy )0,0()1,1(),(

nn

MyxM nnnn khi n

Ta c 0111

12

)(

22

2

nn

nMf n , )0,0(012

22

),(

2

2

220,0,

limlimlim f

n

n

yx

xyyxf

nyxyx

vy hm s khng lin tc ti (0,0) Tnh cc hr '' , yx ff

Ti (x,y) (0,0)

ta c222

22'

)(

)2(2)(2

yx

xyxyxyfx

222

22'

)(

)2(2)(2

yx

xyyyxxfy

Ti (x,y)=(0,0)

ta c 0)0,0()0,(

lim0

'

t

ftff

tx

0)0,0(),0(

lim0

'

t

ftff

ty

Bi 3:Tnh tch phn D

dxdyyxI )2(

Vi D l na trn ca hnh trn c tm ti im (1,0) bn knh 1Gii :

Phng trnh ng trn tm I(1,0) bn knh R=1 l (x-1)2+y2 1 x2+y2 2x


20/25



i sang to cc

t

0,r

sin

cos

ry

rx1 chu k

Ta c x2+y2 2x ta c r2

2rcosnn cos2r

Vy ta c

20

cos20

r

Vi

rdrddxdy

ry

rx

sin

cos

Vy

2

0

cos2

0

22

0

cos2

03

2

16...)sincos2()sincos2()2(

drrdrdrdrrdxdyyxID

Bi 4:Cho tp hp cc s t nhin N vimi m,n N

t

nmneu0

nmneu1

1),( nmnmd

a. CM d l metric trn N

b. CM (N,d ) l khng gian metric y Gii :a. d l metric trn N

d(m,n) Nnm ,,0

d(m,n)=0 m=n

),(0

11

nmneu0

nmneu1

1

),( mndmnnmnmd

CM d(m,n) d(m,l)+d(l,n) (1) Nnml ,,

TH1 : nu m=n,m=l,n=l th (1) ng

TH2 : nu m n thnm

nmd

1

1),(

nu m l thlm

lmd

1

1),(

nu l n thnl

nld

1

1),(

th VT ca (1) 2 , VP ca (1) 2 nn (1) ngb. (N,d ) l khng gian metric y gi s (xn) l dy cauchy trong (N,d) ta CM xn x ddo (xn) l dy cauchy trong (N,d) nn ta c 0),(lim

,

nmnm

xx

),(:,:,0 00 nm xxdnnmn

chn 000 ,.0),(2

1),(,:,

2

1nnmxxxxdxxdnnmn nmnmnm

vy xxnnxxNx nn 0:: trn d


21/25



Bi 5:tnh nh thc

000021

000073

217667

125115

850042

640031

Gii :

Bi 6:Cho nh x tuyn tnh f: R4 R3c ma trn trong cp c s chnh tc l

3502

1132

1201

xc nh nhn v nh ca f , Hi f c n cu , ton cu khng? V sao?Gii :t mae trn tac nh x

f(x1,x2,x3,x4)=(x1+2x3+x4,2x1+3x2-x3+x4,-2x1-5x3+3x4)Xc nh nhn v nh ca f tc l tm c s v s chiu ca Imf, Kerf

Tm c s v s chiu ca Kerf

Ta c

0.3502

0.1132

0.1201

0.5100

0.1530

0.1201

Ta c h )(,

315

26

33

3

5

1

3

5

2

05

053

02

4

3

2

1

4

43

432

431

43

432

431

Ra

axax

ax

ax

ax

xx

xx

xx

xxx

xx

xxx

xxx

f c 1 n t do nn dimKerf = 1 v Kerf c c s l (-33,26,15,3)Vy f khng n cu v dimKerf = 1

Tm c s, s chiu ca ImfTa c

B=

0.3110.512

0.030

0.221

0.5100.150

0.030

0.221

0.1500.030

0.050

0.221

0.26000.1500

0.510

0.221

0.000

0.39000

0.510

0.221

Vy Rank (B)=3 nn dimImf=3 v Imf c 1 c s gm 3 vect(f(e1),f(e4),f(e2))f khng ton cu v dimImf=3

Bi 7:Cho ma trn

133

153

131

A

a. Tm GTR-VTR ca A


22/25



b. Tnh A2004

.

Gii :

a. Tm GTR- VTR ca A Tm GTR ca A

Xt a thc t trng

Ta c:210)2)(1(

133

153

131

2

aaaa

a

a

a

Vy A c 2 GTR a=1, a=2 Tm VTR ca A

Vi a=1 ta c

000

110

132

330

110

132

033

143

132

Ta c h

11

1

0

032

3

2

1

3

32

321

xx

x

axxx

xxx

vy c VTR (1,1,1)

Vi a=2 ta c

000

000

133

133

133

133

Ta c h

bx

ax

xxx

bx

ax

xxx

3

2

321

3

2

321 33033

Vy c 2 VTR (1,1,0), (-1,0,3)b.

ta c

301

011

111

Q

ma trn cho ca A l

200

020

0011AQQB

(Q

-1

AQ)

2004

=Q

-1

A

2004

Q

vy A2004=QB2004Q-1= 1

2004

200

020

001

QQ = 1

2004

2004

2004

200

020

001

QQ


Bi 1:

Bi 2:

Bi 3:Cho (X,d) l khng gian metric compc

a.Gi s Anl h cc tp con ng trong X v An+1 Anmi n N

CMR nu vi mi n N ,An th

1n

nA


23/25



b.Gi s NnRXfn ,: l cc hm lin tc vXxxfxfxf n ........,)(.........)()( 21 CMR nu Nnnn

n

fXxxf

)(,,0)(lim

H t u v 0 trn XGii :

a. gi s vi mi n N ,An CMR :

1n

nA

vi mi n N ly xn Ando nnpn AnenNpnAA pn x,,,

do X compc nn vi (xn)nX c dy con (kn

x )khi t

t x=kn

k

xlim

do nk k nn Ak l tp ng vi mi i dy nn

iikn AxAxx

k

1

vy

1n

nA

b. cn CM :1. 0)( xfn

2. )(xfn

ta c Xxxfxfxf n ........,)(.........)()( 21 v Xxxfnn

,0)(lim nn 0)( xfn

vi 0 cho trc t ),()(: 1 nnn fxfXxF do ),( l tp m, f lin tc nn Fnm

do fn+1(x) fn(x) suy ra fn(x) l dy gim nn 1 nn FF

do XFXxxf nn

nn

1

,0)(lim

do X compc nn c tp J hu hn trong N sao cho XFnJn

t n0=maxJ ta c c 0,)()(0 00 nnxfxfFXF nnnnJn

vy NnnfXx )(, hi t u v 0 trn X

Bi 4:b tm min hi t ca chui hm

1n

n

n

x

dn

n

Gii : xt

2n

ndn

nU

Ta cdn

n

n

n

nn

n e

n

dndn

nUL

11limlimlim

Bn knh hi t R=ed, khong hi t (-ed;ed)Xt ti 2 u mt x=ed,x=-ed

Ta c chui 011)( lim1 1

22

n

nnn n

d

n

nd

n

Uedn

nedn

n

Vy MHT ca chui l (-ed;ed)


24/25




Bi 1 :a. Cho hm s

(0,0)y)(x,khi0

(0,0)y)(x,khi)(

),( 22

22

yx

yxxy

yxf

Xt tnh lin tc ca f(x,y) v cc hr '' , yx ff trn tp xc nhGii :

Ti mi (x,y) (0,0) f(x,y) lin tc v l hm s cp Xt s lin tc ca f ti (x,y)= (0,0)

Nu 0)0,0(),(lim0,

fyxfyx

th hm s lin tc

Ta c :22

3

0,22

3

0,22

22

0,0,limlimlimlim

)(),(

yx

xy

yx

yx

yx

yxxyyxf

yxyxyxyx

Xt 02 22

3

0,

2

22

3

lim yxyxx

yxyx

yx

02 22

3

0,

2

22

3

lim

yx

xyy

yx

xy

yx

t 0)0,0(),(lim0,

fyxfyx

vy f lin tc Tnh cc hr '' , yx ff

Ti (x,y) (0,0)

'xf

'yf

Ti (x,y)=(0,0)

0)0,0()0,(

lim0

'

t

ftff

tx

0)0,0(),0(

lim0

'

t

ftff

ty

b.

Tnh tng ca chui hm

1n

nnx trong MHT ca n

Gii : ta tm c khong hi t l (-1,1)

Ta c

1

1 1..)(n

n

xxnxS

t

1

1

1 .)(n

nxnxS (1)

Ly tch phn 2 v ca (1) trn on [0,x] ta c

1 0 1

1

0

11

1.)(

n

x

n

nn

x

ttdttndttS (2) l CSN

o hm 2 v ca (2) ta c 21 )1(1)(x

xS


25/25


vy

1x:1

1x:)1.(

1

)( 2xxxS

Bi 2:

Bi 3:

Bi 4:b. Tm cc VTR- GTR ca ma trn

A=

702

052

226

Gii :Xt a thc t trng

9

3

6

0)2712)(6(

702

052

2262

a

a

a

aaa

a

a

a

Vy c 3 GTR a=6,a= 3, a= 9 Tm VTR

Vi a=6 ta c

102

012

220

220

012

102

000

110

102

Ta c h

ax

ax

ax

x

ax

xx

xx

2

2

2

2

0

02

3

2

31

3

32

31

C VTR l (-1,2,2)

Vi a=3 ta c

402

022

223

Ta c hC VTR l (,,)

Vi a=9 ta c

202

042223

Ta c hC VTR l (,,)

NH GH THM THNG XUYN: VIOLET.VN/NGUYENTUC2THANHMY

Tong Hop de Thi Cao Hoc Toan

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