sg cilo DUc vA DAo rAo
D4.I HQC HUE
Ho vd
tEn
thl sinh:
56 bdo
danh.
ri'rm rUYEN SINH
sAU DAI
HQC
NAM
2012
(
Egt
1)
Mdn
thi: GIAI
TiCH
@anh
cho
cao hqc)
Thdi
gian lam bdi: 180
Phut
CAu
1.
a) Khao
s6t
tinh kha
vi cua
hdm
, \ ( 9'n6u (x;
y)
+(o;
o)
f(x,!)=l*,*y' \'r
[o n6u
(x;
!) = (0;
o).
b) Tim mi0n
hQi tu cua chu6i nam
lfly
thua
@
Y (-2)"
(x
_1,), .
L n*L
c) Tinh tich phdn
ducrng
I G.siny + 2xy)d.x * (x' + e' cosY)dY
|
\ r /'
trong
d,6 L lii cung
cua
parab
ol x - y2 chAy
tu di6m
0
(0;0) d6n A(L; 1).
Ciu 2. Cho A ld tdp con kh6c rSng
trong kh6ng gian metric (X,
d). Chrmg
minh
ring, him sO;, X + IR x6c dinh
bdi
f (r) = d(x;
A)
: IEId(a;
x)
liOn
tuc trOn X vd
tap hap
M - {* . X:
0 s (d(*; A))' + d.(x; A) s 2} dong
trong X.
)-
Ciu 3. Xdt
tAp X g6m
c6c hdm
thgc x = x(t) liOn
tpc
tr0n [0;
+*) sao
cho
"i'Tl*'
erlx(t)l
<
+*'
a) Chtmg
minh (X; ll ll) h khdng
gian
dinh
chuAn
voi
llrfl
- sup etlx(t)l
,vx
e
X.
re
[o;+m)
b) Xet phi6m ham f , X + IR, sao
cho f (x) - Io** tx(t) dt. Hdy
ki6m
tra sy x6c
clinh
cna
f Q) vd chrmg
minh
f tuy}ntinh li€n tpc.
Tinh ll/ll.
Cffu
4.
a) Chokh6ng
gianHilbert
H vitM ldt4p
contrum4ttrong
cuaH.
Gid su x e H vit
(x,")n
ld ddy
bi ch4n trong
H sao
cho
v6i m6i
y e M thi limrl-+@(xn;!l
= (x;yl.
Chimg minh ring ddy
(x) r, hQi
tq y6u d€n
x.
b) Tr€n mQt
khdng
gian
Hilbert
H, vsimoi a * 0 chrmg
minh
ring, phi6m ham tuy6n
rinh li€n
t.uc
fr(x) = (x; al ldmQt
todn
6nh
vd suy
ra
fo cingld mOt
6nh
x4 mo.
Ghi chti: Cdn bQ coi thi khong
giai thich
gi th€m.