Bai Tap Lon Giai Tich 2 Bo Mon Toan (1)

BI TP LN MN GII TCH 2

Trng i hc Bch Khoa TP HCMKhoa Khoa hc ng dng, b mn Ton ng dng

TP. HCM 2011.

(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 1 / 82

Yu cu:

Dng phn mm MatLab gii nhng bi tonsau y.Sinh vin c th tham kho Bi ging in t -Ton gii tch 2 ca thy ng Vn Vinh.


Nhm 1.

NHM 1


Nhm 1. Mt Paraboloid elliptic

Mt Paraboloid elliptic

Cu 1.

1 V mt Paraboloid elliptic z = x2

a2+y 2

b2vi a, b

nhp t bn phm.2 V mt Paraboloid elliptic y = x2 + z2


Nhm 1. o hm ring cp cao

Cu 2.

Nhp hm s u(x , y) t bn phm. Tm10u

x10(1, 2).


Nhm 1. Tm cc tr t do

Cu 3.

Nhp hm s f (x , y) t bn phm. Tm cc tr tdo ca hm f (x , y). V th minh ha trn ch ra im cc tr nu c.


Nhm 1. Tch phn kp

Cu 4.

Nhp ta ca 3 nh ca tam gic, hm sf (x , y). Tnh I =

D

f (x , y)dxdy , vi D l tam

gic c 3 nh cho. V min D.


Nhm 1. Tch phn bi 3

Cu 5.

Tnh th tch vt th E gii hn bix2 + y 2 + z2 6 4, x2 + y 2 + z2 6 4z . V vt thE . V hnh chiu ca E xung Oxy t xc nhcn ly tch phn.


Nhm 1. Tch phn ng

Cu 6.Nhp hm f (x , y) t bn phm. TnhI =C

f (x , y)d` vi C l ng trn

x2 + y 2 = 2x , x > 1. V ng cong C .


Nhm 1. Tch phn mt

Cu 7.Tnh I =

C

(x + y)dx + (2x z)dy + ydz vi C lgiao ca mt cong z = y 2 v x2 + y 2 = 1 ngcchiu kim ng h theo hng ca trc Oz bngcch dng cng thc Stokes. V giao tuyn, phpvc t vi mt cong chn trong cng thcStokes ti im M0(x0, y0, z0) nhp t bn phm.


Nhm 2.

NHM 2


Nhm 2. Mt ellipsoid

Mt ellipsoid

Cu 1.V mt ellipsoid

x2

a2+

y 2

b2+

z2

c2= 1

vi a, b, c nhp t bn phm.


Nhm 2. Mt phng tip din

Cu 2.

Nhp hm z = z(x , y) v im M0 thuc mtcong z = z(x , y) t bn phm. Tm phng trnhmt phng tip din v phng trnh php tuynvi mt z = z(x , y) ti im M0. V mt congz = z(x , y), mt phng tip din, php tuyn vimt cong ti im M0.


Nhm 2. Tm cc tr c iu kin

Cu 3.

Nhp hm f (x , y), iu kin l 1 ellip ty . Tmcc tr ca hm f (x , y) vi iu kin (x , y) thamn phng trnh ellip. V th minh ha trn ch ra im cc tr nu c.


Nhm 2. Tch phn kp

Cu 4.Nhp hm y = y1(x), y = y2(x) t bn phm saocho th ca 2 hm ny ct nhau ti 2 imphn bit. Cho D l min gii hn bi 2 ngcong y1, y2. Nhp hm f (x , y). TnhI =D

f (x , y). V min D


Nhm 2. Tch phn bi 3

Cu 5.

Tnh th tch vt th E gii hn bix2 + y 2 + z2 = 1, x2 + y 2 + z2 = 4, z >

x2 + y 2.

V vt th E . T xc nh cn ly tch phn.


Nhm 2. Tch phn ng

Cu 6.

Nhp hm s f (x , y) t bn phm. TnhI =C

f (x , y)d` vi C l giao ca x2 + y 2 = 4 v

x + z = 4. V giao tuyn C .


Nhm 2. Tch phn mt

Cu 7.TnhI =C

(3x y 2)dx + (3y z2)dy + (3z x2)dzvi C l giao ca mt phng 2x + z = 2 v mtparaboloid z = x2 + y 2 ngc chiu kim ng htheo hng ca trc Oz bng cch dng cngthc Stokes. V giao tuyn, php vc t vi mtcong cha (C ) ti im M0(x0, y0, z0) nhp t bnphm.


Nhm 3.

NHM 3


Nhm 3. Mt Paraboloid Hyperbolic

Mt Paraboloid Hyperbolic

Cu 1.

1 V mt Paraboloid Hyperbolic z = x2

a2 y

2

b2vi

a, b nhp t bn phm.2 V mt Paraboloid Hyperbolic y = z2 x2


Nhm 3. o hm ca hm hp

Cu 2.

Nhp hm f (u), hm u = u(x , y) v imM0(x0, y0) ty . Tm o hm ring ca hm hpf = f (u), u = u(x , y) ti im M0. Tinh f x , f y tiM0(x0, y0). V th minh ha ngha hnh hcca o hm ring ti 1 im.


Nhm 3. Tm cc tr c iu kin

Cu 3.Nhp hm f (x , y). Tm cc tr ca hm f (x , y)vi iu kin (x , y) tha mn phng trnhparabol ty nhp t bn phm. V th minhha trn ch ra im cc tr nu c.


Nhm 3. Tch phn kp

Cu 4.

Nhp hm x = x1(y), x = x2(y) ct nhau ti 2im phn bit. Nhp hm f (x , y). TnhI =D

f (x , y)dxdy , vi D c gii hn bi

x = x1(y), x = x2(y). V min D


Nhm 3. Tch phn bi 3

Cu 5.

Nhp hm s f (x , y , z). Tnh tch phn bi 3I =E

f (x , y , z)dxdydz , vi E l vt th gii

hn bi z = 1, x2 + y 2 + z2 = 2z , z 6 1 bngcch i sang h ta cu. V vt th E . T xc nh cn ly tch phn.


Nhm 3. Tch phn ng

Cu 6.Nhp hm f (x , y). Tnh I =

C

f (x , y)d` vi C l

giao ca x2 + y 2 + z2 = 4, x + y + z = 0. Vng cong C .


Nhm 3. Tch phn mt

Cu 7.TnhI =S

(y + z)dydz + (x z)dzdx + (z + 1)dxdyvi S l phn mt hng pha trn ca na mtcu z =

4 x2 y 2 bng cch dng cng thc

Ostrogratxki-Gauss. V giao tuyn (C ), php vct vi mt cong cha (C ) ti im M0(x0, y0, z0)thuc mt cong nhp t bn phm.


Nhm 4.

NHM 4


Nhm 4. Mt Hyperboloid

Mt Hyperboloid

Cu 1.

1 V mt Hyperboloid 1 tng x2

a2+

y 2

b2 z

2

c2= 1

2 V mt Hyperboloid 2 tng x2

a2+y 2

b2 z

2

c2= 1



Nhm 4. Tm o hm ca hm hp

Cu 2.Nhp hm f = f (u, v), u = u(x), v = v(x). Tmo hm f (x) ti im x = x0 vi x0 nhp t bnphm. Tinh f x , f y ti M0(x0, y0). V th minhha ngha hnh hc ca o hm ring ti 1im.


Nhm 4. Tm gi tr ln nht, gi tr nh nht

Cu 3.

Nhp hm f (x , y). Nhp min D l hnh trn tmI (x0, y0) bn knh R t bn phm. Tm GTLN,GTNN ca hm f (x , y) trn min D. V thminh ha trn ch ra im t GTLN, GTNNnu c.


Nhm 4. Tch phn kp

Cu 4.

Nhp y = y1(x), y = y2(x) sao cho th cachng khng ct nhau trong khong x [a, b].Nhp hm f (x , y) Tnh I =

D

f (x , y)dxdy , vi

D c gii hn biy = y1(x), y = y2(x), x = a, x = b. V min D.


Nhm 4. Tch phn bi 3

Cu 5.Nhp hm f (x , y , z) t bn phm. Tnh tch phnbi 3 I =

E

f (x , y , z)dxdydz , vi

E : z > 0, x2 + y 2 + z2 6 2y bng cch i sangh ta cu. V vt th E . T xc nh cnly tch phn.


Nhm 4. Tch phn ng

Cu 6.

Nhp ta 3 im A,B ,C trong mt phngOxy . Nhp hm f (x , y), g(x , y). TnhI =C

f (x , y)dx + g(x , y)dy vi C l bin ca

tam gic ABC ngc chiu kim ng h bngcng thc Green. V ng cong C .


Nhm 4. Tch phn mt

Cu 7.Tnh I =

S

z2dydz + xdzdx zdxdy vi S l mtxung quanh, hng pha ngoi ca vt th gii hnbi cc mt z = 4 y 2, z = 0, x = 1, x = 0 bngcch dng cng thc Ostrogratxki-Gauss. V mtcong (S), php vc t vi mt cong ti imM0(x0, y0, z0) nhp t bn phm.


Nhm 5.

NHM 5


Nhm 5. Mt tr

Mt tr

Cu 1.

1 V mt tr ellipse x2

a2+

y 2

b2= 1, z R, vi a, b

nhp t bn phm.2 V mt tr parabol y = x2, z R


Nhm 5. Tm o hm ring ca hm hp

Cu 2.Nhp hm f = f (u, v), u = u(x , y), v = v(x , y).Tm f x , f y ti im M0. Tinh f x , f y ti M0(x0, y0).V th minh ha ngha hnh hc ca o hmring ti 1 im.



Cu 3.Nhp hm f (x , y). Tm GTLN, GTNN ca hmf (x , y) trn min D : |x | + |y | 6 1. V thminh ha trn ch ra im t GTLN, GTNNnu c.


Nhm 5. Tch phn kp

Cu 4.

Cho hm s x = x1(y), x = x2(y) khng ct nhautrong khong y [a, b]. Nhp hm f (x , y). TnhI =D

f (x , y)dxdy , vi D c gii hn bi

x = x1(y), x = x2(y), y = a, y = b. V min D.


Nhm 5. Tch phn bi 3

Cu 5.Nhp hm f (x , y , z). Tnh tch phn bi 3I =E


hn bi z >x2 + y 2, x2 + y 2 + z2 6 z bng

cch i sang h ta cu. V vt th E . T xc nh cn ly tch phn.


Nhm 5. Tch phn ng

Cu 6.

Nhp ta 3 im A,B ,C trong mt phngOxy . Nhp hm f (x , y), g(x , y). TnhI =C

f (x , y)dx + g(x , y)dy vi C l bin ca

tam gic ABC theo chiu kim ng h. V ngcong C .


Nhm 5. Tch phn mt

Cu 7.Tnh I =

S

x3dydz + y 3dzdx + z3dxdy vi S l

phn mt hng pha ngoi ca mt cux2 + y 2 + z2 = 4 bng cch dng cng thcOstrogratxki-Gauss. V mt cong S , php vc tvi mt cong ti im M0(x0, y0, z0) nhp t bnphm.


Nhm 6.

NHM 6


Nhm 6. Mt nn 2 pha

Mt nn 2 pha

Cu 1.

x2

a2+

y 2

b2=

z2

c2



Nhm 6. Tm o hm ring ca hm hp

Cu 2.

Nhp hm f = f (x , y), y = y(x), im M0 ty .Tm f

x,df

dxti im M0. Tinh f x , f y ti

M0(x0, y0). V th minh ha ngha hnh hcca o hm ring ti 1 im.



Cu 3.Nhp hm f (x , y). Tm GTLN, GTNN ca hmf (x , y) trn min D l tam gic ABC bt k vita nhp t bn phm. V th minh ha trn ch ra im t GTLN, GTNN nu c.


Nhm 6. Tch phn kp

Cu 4.

Nhp hm s z = z(x , y). Tnh din tch phnmt cong z = z(x , y) nm trong hnh tr c yl hnh trn tm I (x0, y0) bn knh R bt k nhpt bn phm. V hnh minh ha.


Nhm 6. Tch phn bi 3



hn bi 2y = x2 + z2, y = 2. V vt th E v hnhchiu ca E xung Oxz , t xc nh cn lytch phn.


Nhm 6. Tch phn ng

Cu 6.Nhp hm f (x , y), g(x , y). TnhI =C

f (x , y)dx + g(x , y)dy vi C l na trn

ng trn x2 + y 2 = 2x cng chiu kim ng hbng cch dng cng thc Green. V ng congC .


Nhm 6. Tch phn mt

Cu 7.

Tnh I =S

(x + z)dxdy vi S l phn mt

z = x2 + y 2, b ct bi mt phng x + z = 2, phadi theo hng trc Oz . V mt cong S , phpvc t vi mt cong ti im M0(x0, y0, z0) nhpt bn phm.


Nhm 7

NHM 7


Nhm 7 Tnh gn ng gi tr ca hm nhiu bin

Cu 1. Tnh gn ng gi tr ca hm nhiu bin

Nhp hm f (x , y). Nhp im M0(x0, y0). Tnhgn ng gi tr f (x0 + x , y0 + y) vi x ,y nh, s dng cng thcf (x , y) f (x0, y0) + f x (x0, y0)x + f y (x0, y0)y


Nhm 7 Tm o hm ca hm n

Cu 2.

Nhp hm F (x , y). Tm y (x), y (x) bity = y(x) l hm n xc nh t phng trnhF (x , y) = 0. V th minh ha ngha hnh hcca o hm y (x) ti im M0 nhp t bn phm.


Nhm 7 Cc tr c iu kin

Cu 3.Nhp hm f (x , y). Tm cc tr ca hm f (x , y)vi iu kin |x | + |y | = 1. V th minh hatrn ch ra im cc tr nu c.


Nhm 7 Tch phn kp

Cu 4.

Tnh din tch min phng gii hn bix2 + y 2 = 2y , x2 + y 2 = 6y , y > x

3, x > 0. V

hnh min phng cho. T xc nh cn lytch phn.


Nhm 7 Tch phn bi 3



hn bi z = x2 + y 2, z = 2 + x2 + y 2, x2 + y 2 = 1.V vt th E v hnh chiu ca E xung Oxy , t xc nh cn ly tch phn.


Nhm 7 Tch phn ng

Cu 6.

Cho P(x , y) = xexy , Q(x , y) = 1 x

y. Tm hm

g(xy ) tha g(0) = 1 v biu thcg(xy )P(x , y)dx + g(

xy )Q(x , y)dy l vi phn ton

phn ca hm u(x , y) no . Vi g(xy ) va tm,tnh I =

L g(

xy )P(x , y)dx + g(

xy )Q(x , y)dy ,

trong L l phn ng cong y = cosh x 14t

im A(1, 34

) n B(ln 2, 1). V ng ly tchphn L v 2 im A,B .


Nhm 7 Tch phn mt

Cu 7.TnhI =S

(2x+y)dydz+(2y +z)dzdx+(2z+x)dxdy

vi S l phn mt phng x + y + z = 3 nm tronghnh tr x2 + y 2 = 2x , pha di theo hng trcOz . V mt cong S , php vc t vi mt cong tiim M0(x0, y0, z0) nhp t bn phm.


Nhm 8

NHM 8


Nhm 8 Cng thc Taylor, Maclaurint

Cng thc Taylor, Maclaurint

Cu 1.

Nhp hm f (x , y), s n. Tm khai trin Taylor ncp n ca f (x , y) trong ln cn ca imM0 = (x0, y0) nhp t bn phm. V mt cong f vmt cong ca hm khai trin Taylor ti im M0.


Nhm 8 Tm o hm ring ca hm n

Cu 2.

Nhp hm F (x , y , z). Tm z x , z y bit z = z(x , y)l hm n xc nh t phng trnh F (x , y , z) = 0.V th minh ha ngha hnh hc ca o hmz x , z

y ti im M0 nhp t bn phm.


Nhm 8 Cc tr c iu kin

Cu 3.Nhp hm f (x , y). Tm cc tr ca hm f (x , y)vi iu kin l phng trnh hyperbolx2

a2 y

2

b2= 1, a, b nhp t bn phm. V th

minh ha trn ch ra im cc tr nu c.


Nhm 8 Tch phn kp


D

f (x , y)dxdy , vi

D l min phng gii hn bi(x a)2 + (y b)2 6 R2, x > a, a, b,R nhp tbn phm, bng cch i sang h ta cc mrng. V min D.


Nhm 8 Tch phn bi 3



hn bi z = 4, z = 1 x2 y 2, x2 + y 2 = 1 bngcch i sang h ta tr. V vt th E v hnhchiu ca E xung Oxy , t xc nh cn lytch phn.


Nhm 8 Tch phn ng

Cu 6.Cho P(x , y) = x2y 3, Q(x , y) = x(1 + y 2). Tmhm h(x , y) = xy, , l cc hng s sao chobiu thc h(x , y)P(x , y)dx + h(x , y)Q(x , y)dy lvi phn ton phn ca hm u(x , y) no . Vih(x , y) va tm, tnhI =L

h(x , y)P(x , y)dx + h(x , y)Q(x , y)dy , trong

L l phn ng cong y = arcsin x +

1 x2t im A(1, pi

2) n B(0, 1). V ng ly tch

phn L v 2 im A,B .(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 65 / 82

Nhm 8 Tch phn mt

Cu 7.Tnh I =

S

(x + y + z)ds vi S cho bi

x + y + z = 1, z > 0, x > 0, y > 0. V mt congS v hnh chiu ca n xung Oxy , php vc tvi mt cong ti im M0(x0, y0, z0) nhp t bnphm.


Nhm 9

NHM 9


Nhm 9 Cng thc Taylor, Maclaurint

Cu 1.

Nhp hm f (x , y), s n. Tm khai trinMaclaurint n cp n ca f (x , y). V mt cong fv mt cong ca hm khai trin Maclaurint.


Nhm 9 o hm theo hng

Cu 2.

Nhp hm f (x , y), im M0(x0, y0), vc tu = (u1, u2). Tm o hm ca f (x , y) ti imM0 theo hng ca vc t u . V hnh minh ha ngha hnh hc ca o hm theo hng ti imM0.


Nhm 9 Cc tr t do

Cu 3.Kho st cc tr t do ca hmf (x , y) = x4 + y 4 x2 2xy y 2. V thminh ha trn ch ra im cc tr nu c.


Nhm 9 Tch phn kp


D

f (x , y)dxdy , vi

D l min phng gii hn bix2 + y 2 6 2x , x2 + y 2 6 2y bng cch i sang hta cc. V min D.


Nhm 9 Tch phn bi 3



hn bi x = y 2, z = x , z = 0, x = 1. V vt th Ev hnh chiu ca E xung Oxy , t xc nhcn ly tch phn.


Nhm 9 Tch phn ng

Cu 6.Nhp hm f (x , y , z), g(x , y , z), h(x , y , z). TnhI =C

f (x , y , z)dx + g(x , y , z)dy + h(x , y , z)dz

vi C l ng congx = a cos t, y = a sin t, z = bt, 0 6 t 6 2pi, a, bnhp t bn phm. V ng cong C .


Nhm 9 Tch phn mt

Cu 7.

Tnh I =S

zds vi S l phn ca mt paraboloid

z = 2 x2 y 2 trong min z > 0. V mt congS v hnh chiu ca n xung Oxy , php vc tvi mt cong ti im M0(x0, y0, z0) nhp t bnphm.


Nhm 10

NHM 10


Nhm 10 o hm ring

Cu 1.

Nhp hm f (x , y) v im M0(x0, y0). Tm ohm ring f x , f y ti im M0. V th minh ha ngha hnh hc ca o hm f x , f y ti im M0.


Nhm 10 o hm theo hng

Cu 2.

Nhp hm f (x , y) v im M0(x0, y0). Tm hngm o hm ca f theo hng ti M0 c gi trbng 1. V hnh minh ha ngha hnh hc cao hm theo hng ti im M0.


Nhm 10 Cc tr t do

Cu 3.Nhp hm f (x , y , z). Kho st cc tr t do cahm f (x , y , z).


Nhm 10 Tch phn kp


D

f (x , y)dxdy , vi

D l min phng gii hn bix2 + y 2 = 1, x2 + y 2 = 4, y > 0, y 6 x bng cchi sang h ta cc. V min D.


Nhm 10 Tch phn bi 3



hn bi y = 1 x , z = 1 x2 v cc mt phngta . V vt th E v hnh chiu ca E xungOxy , t xc nh cn ly tch phn.


Nhm 10 Tch phn ng

Cu 6.Nhp hm f (x , y , z), g(x , y , z), h(x , y , z). TnhI =C

f (x , y , z)dx + g(x , y , z)dy + h(x , y , z)dz

vi C l giao cax2 + y 2 + z2 = 4, y = x tan, 0 < < pi, ngcchiu kim ng h nhn theo hng trc Ox . Vng cong C .


Nhm 10 Tch phn mt

Cu 7.

Tnh I =S

(x2 + y 2 + z2)ds vi S l phn ca

mt nn z =x2 + y 2 nm gia hai mt phng

z = 0 v z = 3. V mt cong S v hnh chiu can xung Oxy , php vc t vi mt cong ti imM0(x0, y0, z0) nhp t bn phm.


Nhm 1.Mt Paraboloid elliptico hm ring cp caoTm cc tr t doTch phn kpTch phn bi 3Tch phn ngTch phn mt

Nhm 2.Mt ellipsoidMt phng tip dinTm cc tr c iu kinTch phn kpTch phn bi 3Tch phn ngTch phn mt

Nhm 3.Mt Paraboloid Hyperbolico hm ca hm hpTm cc tr c iu kinTch phn kpTch phn bi 3Tch phn ngTch phn mt

Nhm 4.Mt HyperboloidTm o hm ca hm hpTm gi tr ln nht, gi tr nh nhtTch phn kpTch phn bi 3Tch phn ngTch phn mt

Nhm 5.Mt trTm o hm ring ca hm hpTm gi tr ln nht, gi tr nh nhtTch phn kpTch phn bi 3Tch phn ngTch phn mt

Nhm 6.Mt nn 2 phaTm o hm ring ca hm hpTm gi tr ln nht, gi tr nh nhtTch phn kpTch phn bi 3Tch phn ngTch phn mt

Nhm 7Tnh gn ng gi tr ca hm nhiu binTm o hm ca hm nCc tr c iu kinTch phn kpTch phn bi 3Tch phn ngTch phn mt

Nhm 8Cng thc Taylor, MaclaurintTm o hm ring ca hm nCc tr c iu kinTch phn kpTch phn bi 3Tch phn ngTch phn mt

Nhm 9Cng thc Taylor, Maclaurinto hm theo hngCc tr t doTch phn kpTch phn bi 3Tch phn ngTch phn mt

Nhm 10o hm ringo hm theo hngCc tr t doTch phn kpTch phn bi 3Tch phn ngTch phn mt

Documents

Bai Tap Lon Giai Tich 2 Bo Mon Toan (1)