bai tap lon giai tich 2 bo mon toan (1)
DESCRIPTION
Bai Tap Lon Giai Tich 2 Bo Mon ToanTRANSCRIPT
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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
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 2 / 82
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Nhm 1.
NHM 1
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 3 / 82
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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
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 4 / 82
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Nhm 1. o hm ring cp cao
Cu 2.
Nhp hm s u(x , y) t bn phm. Tm10u
x10(1, 2).
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 5 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 6 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 7 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 8 / 82
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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 .
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 9 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 10 / 82
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Nhm 2.
NHM 2
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 11 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 12 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 13 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 14 / 82
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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
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 15 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 16 / 82
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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 .
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 17 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 18 / 82
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Nhm 3.
NHM 3
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 19 / 82
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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
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 20 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 21 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 22 / 82
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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
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 23 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 24 / 82
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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 .
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 25 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 26 / 82
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Nhm 4.
NHM 4
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 27 / 82
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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
vi a, b, c nhp t bn phm.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 28 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 29 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 30 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 31 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 32 / 82
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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 .
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 33 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 34 / 82
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Nhm 5.
NHM 5
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 35 / 82
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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
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 36 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 37 / 82
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Nhm 5. Tm gi tr ln nht, gi tr nh nht
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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 38 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 39 / 82
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Nhm 5. Tch phn bi 3
Cu 5.Nhp hm f (x , y , z). Tnh tch phn bi 3I =E
f (x , y , z)dxdydz , vi E l vt th gii
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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 40 / 82
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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 .
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 41 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 42 / 82
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Nhm 6.
NHM 6
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 43 / 82
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Nhm 6. Mt nn 2 pha
Mt nn 2 pha
Cu 1.
x2
a2+
y 2
b2=
z2
c2
vi a, b, c nhp t bn phm.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 44 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 45 / 82
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Nhm 6. Tm gi tr ln nht, gi tr nh nht
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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 46 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 47 / 82
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Nhm 6. Tch phn bi 3
Cu 5.Nhp hm f (x , y , z). Tnh tch phn bi 3I =E
f (x , y , z)dxdydz , vi E l vt th gii
hn bi 2y = x2 + z2, y = 2. V vt th E v hnhchiu ca E xung Oxz , t xc nh cn lytch phn.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 48 / 82
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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 .
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 49 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 50 / 82
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Nhm 7
NHM 7
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 51 / 82
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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
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 52 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 53 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 54 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 55 / 82
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Nhm 7 Tch phn bi 3
Cu 5.Nhp hm f (x , y , z). Tnh tch phn bi 3I =E
f (x , y , z)dxdydz , vi E l vt th gii
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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 56 / 82
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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 .
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 57 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 58 / 82
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Nhm 8
NHM 8
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 59 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 60 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 61 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 62 / 82
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Nhm 8 Tch phn kp
Cu 4.Nhp hm f (x , y). Tnh I =
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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 63 / 82
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Nhm 8 Tch phn bi 3
Cu 5.Nhp hm f (x , y , z). Tnh tch phn bi 3I =E
f (x , y , z)dxdydz , vi E l vt th gii
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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 64 / 82
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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
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 66 / 82
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Nhm 9
NHM 9
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 67 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 68 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 69 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 70 / 82
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Nhm 9 Tch phn kp
Cu 4.Nhp hm f (x , y). Tnh I =
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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 71 / 82
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Nhm 9 Tch phn bi 3
Cu 5.Nhp hm f (x , y , z). Tnh tch phn bi 3I =E
f (x , y , z)dxdydz , vi E l vt th gii
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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 72 / 82
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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 .
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 73 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 74 / 82
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Nhm 10
NHM 10
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 75 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 76 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 77 / 82
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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).
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 78 / 82
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Nhm 10 Tch phn kp
Cu 4.Nhp hm f (x , y). Tnh I =
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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 79 / 82
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Nhm 10 Tch phn bi 3
Cu 5.Nhp hm f (x , y , z). Tnh tch phn bi 3I =E
f (x , y , z)dxdydz , vi E l vt th gii
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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 80 / 82
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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 .
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 81 / 82
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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.
(BK TPHCM) BI TP LN MN GII TCH 2 TP. HCM 2011. 82 / 82
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