Đề thi CH trường ĐH Vinh (các môn)
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Đặng Xuân C)ơng - Cao học 12 - Giải tích - Đại học Vinh
K nim hố 2005
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
4I"-5$"-JKL3"M$35"):&"56)"3NA"OPPP
QR3S""T$U$"-V)5
@#+35S"/&%3
Thời gian làm bài: 180 phút
"8WJOX!! "!#$%!&'!()*! RRf
2
: !!+(,! $!+/01!2(3+
( )
=+
+
+
=
0 0
0
,
22
22
22
2
yx
yx
yx
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!045
045
6"!782!290(!:$;0!2<+!+=6! f 2>;0!
2
R ?
-"!782!290(!@(%!A$!+=6!()*! f 2B$!C$D*!
( )
0,0 ?
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!!! !!!!!!!!!
n
n
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+
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1
12
1
0
8WJ"YX!K9!($L5!
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1,0
C !:)!@(/01!1$60!CX0(!+(5Y0!+V+!()*!&Z!2([+!:$;0!2<+!2>;0!
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1,0 !AO$!+(5Y0!&5\
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1,0
C
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C !-$40! x !2()0(!
Ax
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2
= !AO$!*_$! x
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C !A)!
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6"!P(301!*$0(!>Q01!]!:)!V0(!`B!25W40!290(!:$;0!2<+?!F90(! A
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[ ]
( )
1,0
CA !:)!@(/01!1$60!+,0!Cb01!+=6!
[ ]
1,0
C ?
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Đặng Xuân C)ơng - Cao học 12 - Giải tích - Đại học Vinh
K nim hố 2005
E
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
4I"-5$"-JKL3"M$35"):&"56)"3NA"OPPP
QR3S""4.$"M\
@#+35S"/&%3
Thời gian làm bài: 180 phút
8WJ"OX!#_$!
1+n
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n
!AO$!(L!&Z!2([+?!F>,01
1+n
E !+(,!+V+!C6!2(3+!
( )
xu
k
!AO$!
nk
0
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6"!P(301!*$0(!>Q01!+V+!C6!2(3+!
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n
k
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!:e\!2()0(!*I2!+i!&.!+=6!
1+n
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-"!jkW!+(301!2a!2l0!2B$!m5W!0(f2!*I2!\(8\!-$40!Cn$!25W40!290(!
!+=6!
1+n
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1
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C$H5!@$L0!
( )
k
k
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K= ?!o)! !:)!*I2!&,01!V0(?
+"!7V+!CX0(!V0(!`B!!
^
1+n
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1+n
E ! $!C$H5!@$L0!
( )
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xpxpxp += 1 N!
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pxE
+
?
jkW!+(301!*$0(!
!:)!*I2!V0(!`B!25W40!290(!?!FG*!0(p0!A)!%0(!+=6
?!FG*!+V+!C6!2(3+
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xu
k
N nk ,,2,1,0 K= ?
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+f5!AO$!#?
! +"!P(301!2a!>Q01!*_$!0(b*!+,0!+f\!(s5!(B0!015W;0!2Z!CH5!:)!0(b*!7W+:$+?
8WJ"ZX"F6!1_$!*I2!2>Tt01!:)!015W;0!2Z!045!0b!@(/01!+(36!*I2!2>Tt01!+,0!2([+!&[!0),?
6"!P(301!*$0(!>Q01!2>Tt01!+V+!&&b!(s5!2g!
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p
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8WJ"[X!#$%!&'!\(8\!-$40!Cn$!25W40!290(!
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x
!CZ$!!AO$!+i!&.!Ci0!AX!+b!*6!2>0!:)^
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
815
231
411
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=
6"!FG*!1$V!2>X!>$;01!A)!A8+!2i!>$;01!+=6! ?
-"!FG*!*I2!+i!&.!+=6!w
x
!*)!CZ$!AO$!0b!*6!2>e0!+=6!
!+b!mB01!26*!1$V+!?!o$42!*6!2>e0!Cb?
+"!#$V!2>X!>$;01!+=6! !+b!2(6W!Cn$!@(/01!@($!26!2(6W!Cn$!+i!&.?
Đặng Xuân C)ơng - Cao học 12 - Giải tích - Đại học Vinh
K nim hố 2005
x
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
4I"-5$"-JKL3"M$35"):&"56)"3NA"Y]]]
QR3S""T$U$"-V)5
@#+35S"/&%3
Đề số 1
Thời gian làm bài: 180 phút
8WJOX!!P(,!()*!&Z!!
( )
=+
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=
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0
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22
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2
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yx
yx
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K(%,!&V2!290(!:$;0!2<+!A)!290(!@(%!A$!+=6!()*!&Z!Ck!+($!2>;0!*$H0!`V+!CX0(!+=6!0b?
8WJ"YX"FG*!*$H0!(I$!2<!A)!290(!2n01!+=6!+(5J$!()*!!
( )
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=
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K= N!
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1
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8WJ"^X!!K9!($L5!
[ ]
1,0
C !:)!@(/01!1$60!2f2!+%!+V+!()*!:$;0!2<+!!
[ ]
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yx,
[ ]
1,0
C !26!Cv2!
( )
[ ]
( ) ( )
0,1
,sup
t
dxyxtyt
=?!!P(301!*$0(!>Q01
6"!á0(!`B!
[ ] [ ]
1,01,0
: CCf !+(,! $!
()
[ ]
() ()
dssxtxf
t
=
0
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[ ]
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2<+?!F90(!+(5Y0!+=6!f.
-"!
[ ]
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1,0
!@(/01!\(%$!:)!@(/01!1$60!+,*\6+2?
Đặng Xuân C)ơng - Cao học 12 - Giải tích - Đại học Vinh
K nim hố 2005
{
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
4I"-5$"-JKL3"M$35"):&"56)"3NA"Y]]]
QR3S""T$U$"-V)5
@#+35S"/&%3
Đề số 2
Thời gian làm bài: 180 phút
8WJ"OX!6"!K(%,!&V2!&[!(I$!2<!+=6!+(5n$^!!
1
(1)
ln
n
n
n
=
?
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1
2
n
n
x
n
=
?
+"!F90(!2n01!+=6!+(5n$!:qW!2(c6^!
2
1
(1)
n
n
nnx
=
+
8WJ"YX!K|!($L5!
{ }
2
2
1
:
nn
n
lxx
=
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nn
n
pxyxy
=
N
( )
1
2
2
1
,
nn
n
dxyxy
=
=
!AO$!
{ }
n
xx = N!
{ }
n
yy = !2(5I+!
2
l
6"!P(301!*$0(!>Q01!\R!m!:)!+V+!*z2>$+!2>;0!
2
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-"!á0(!`B!Cl01!0(f2!
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()
f(x) f(x)n
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n
fx
=
045
045
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f
n
f !(?!@?!0
P(301!*$0(!>Q01! lim()
AnA
x
IfdLIfd
àà
= ?
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E!+(36!2f2!+%!+V+!C$D*!1$O$!(B0!+=6!0b?
8WJ"[X"á0(!`B!~^!E!!!F!2c!@(/01!1$60!CX0(!+(5Y0!E!A),!@(/01!1$60!CX0(!+(5Y0!F!CTU+!1_$!:)!-X
+(v0!045!2l0!2B$!*I2!(Q01!&Z!P!>!!&6,!+(,!!
()fxC !AO$!*_$!`!!E!*)! 1x ?!P(301!*$0(
>Q01!CD!~^!E!!!F!!-X!+(v0R!!C$H5!@$L0!+S0!A)!C=!:)!~!:$;0!2<+?
§Æng Xu©n C)¬ng - Cao häc 12 - Gi¶i tÝch - §¹i häc Vinh
Kỷ niệm hè 2005
5
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
4I"-5$"-JKL3"M$35"):&"56)"3NA"Y]]]
QR3S""4.$"M\
@#+35S"/&%3
Thêi gian lµm bµi: 180 phót
"8WJ"OX!#$%!&'!o!:)!@(/01!1$60!A8+!2i!2([+!0!+($H5!A)! VVf
→: !:)!V0(!`B!25W40!290(?
6"!P(301!*$0(!
( ) ( )
nfimf =+ kerdimdim !?
-"!#$%!&'! f !Ci0!+f5?!P(301!*$0(!! f :)!2[!Cr01!+f5!+=6!o?
+"!#$%!&'!! ff =
2
?!P(301!*$0(! Vfimf =⊕ ker ?
m"!#$%!&'!*_$!A8+!2i!@(V+!@(/01!+=6!o!CH5!:)!A8+!2i!>$;01!+=6! f ?!P(301!*$0(!>Q01! f
CTU+!`V+!CX0(! $!!
( )
xxf α= !uα !:)!&Z!2([+!+(,!2>TO+"?
8WJ"YX"#$%!&'!7!:)!0(b*!7W+:$+!+f\!*!A)!d:)!0(b*!7W+:$+!+f\!0?!P(301!*$0(!>Q01^
6"!N(b*!+,0!+=6!0(b*!7!!:)!0(b*!7W+:$+?
-"!7!+(g!+b!*I2!&Z!(s5!(B0!0(b*!+,0?
+"!7
≅ d!@($!A)!+(g!@($!*M0?
m"!7
×d!:)!0(b*!7W+:$+!+f\!*×0!@($!A)!+(g!@($!!u*R0"M ?
8WJ"ZX!#$%!&T!7!:)!*I2!A)0(!1$6,!(,V0!+b!Ci0!AX!?!MI2!IC;60!]
≠ !7!+=6!7!CTU+!1_$!:)!IC;60!2Z$
CB$!!045!+A)!+(g!045!+V+!IC;60!+=6!7!+(36!]!+(90(!:)!7!A)!-%0!2(p0!]?!MI2!IC;60!P!+=6!7!CTU+
1_$!:)!015W;0!2Z!045!A)!+(g!045!AO$!5RA!
X∈ !2(G!29+(!5?A P∈ !@8,!2(z,!5 P∈ !(,v+!A P∈ ?!#$%!&'!I
:)!IC;60!+=6!7?!P(301!*$0(!>Q01^
6"!7/I!:)!*I2!*$H0!015W;0!@($!A)!+(g!@($!I!:)!!IC;60!2Z$!CB$?
-"!!7/I!:)!*I2!2>Tt01!@($!A)!+(g!@($!I!:)!!IC;60!2Z$!CB$!?
+"!N45!I!:)!!IC;60!2Z$!CB$!2(G!I!:)!!IC;60!2Z$!CB$?
§Æng Xu©n C)¬ng - Cao häc 12 - Gi¶i tÝch - §¹i häc Vinh
Kỷ niệm hè 2005
6
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
4I"-5$"-JKL3"M$35"):&"56)"3NA"Y]]O
QR3S""T$U$"-V)5
@#+35S"/&%3
§Ò sè 1
Thêi gian lµm bµi: 180 phót
8WJ"OX!P(,!+(5n$!()*^!!
( )
( )
1
1
21
3
n
n
n
x
n
∞
=
−
−
∑
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6"!FG*!*$H0!(I$!2<!+=6!+(5J$!u "
-"!F90(!2n01!+=6!+(5n$!u "!2>,01!@(,%01!(I$!2<!+=6!0b?
8WJ"YX"P(,!()*!&Z!
( )
1
y cos 0
,
x
0 0
x
fxy
x
≠
=
=
045
045
6"!FG*!2f2!+%!+V+!C$D*!1$V0!C,B0!+=6!~?
-"!Fe\!+V+!C$D*!1$V0!C,B0!+=6!~!@(/01!Cb01!2>,01!w
E
!0(T01!*.!2>,01!2e\!
{ }
(0,):yy∈¡ ?
8WJ"ZX"P(,!mkW!()*
!!!!!!!!!!!!!!!!!!!!!
()
[ ] [ ]
[ ]
K,2,1,
1,0 0
1,0
1
=
∉
∈
= n
x
xnx
n
xf
n
!045
045
P(301!*$0(!>Q01
6"
( )
lim
n
x
fxx
→∞
= !AO$!!
[ ]
1,0∈∀x
-"!
1
lim
2
n
x
If
→∞
= !2>,01!Cb!
n
If !:)!29+(!\(p0!hi-z!+=6!
n
f !2>;0!wR!
[ ]
nx !:)!\(S0!015W;0!+=6! nx ?
8WJ"[X"#$%!&'!!
∞
l !:)!2e\!2f2!+%!+V!mkW!&Z!2([+!-X!+(v0!N!
0
c !:)!2e\!2f2!+%!+V+!mkW!&Z!2([+!(I$!2<!2O$
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6"!P(301!*$0(!>Q01!+/01!2(3+
!!!!!!!!!!!!!
sup
n
n
xx
∈
=
N
!AO$!
{ }
n
xx = !∈!
∞
l !`V+!CX0(!*I2!+(5Y0!2>;0!
∞
l ?
-"!P(301!*$0(!>Q01!
0
c !:)!@(/01!1$60!+,0!Cb01!2>,01!
∞
l AO$!+(5Y0!0b$!2>;0?
+"!P(,!V0(!`B!! Rlf →
∞
: `V+!CX0(! $!+/01!2(3+!
()
1
3
n
n
n
x
fx
∞
=
=
∑
!R!AO$!*_$!!
{ }
n
xx = !∈
∞
l R!jkW!+(301!*$0(!>Q01!f!!:)!*I2!\($4*!()*!25W40!290(R!:$;0!2<+!2>;0!
∞
l !A)!290(!
f ?
8WJ"^X"#$%!&'!E!:)!@(/01!1$60!CX0(!+(5Y0!(s5!(B0!+($H5R!B!:)!(G0(!+S5!Ci0!AX!Cb01!2>,01! E?
P(301!*$0(!>Q01!AO$!*_$!`!∈!ER!CH5!2l0!2B$!W!∈!B!&6,!+(,! xy− !M!mu`R!B"?
§Æng Xu©n C)¬ng - Cao häc 12 - Gi¶i tÝch - §¹i häc Vinh
Kỷ niệm hè 2005
7
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
4I"-5$"-JKL3"M$35"):&"56)"3NA"Y]]O
QR3S""T$U$"-V)5
@#+35S"/&%3
§Ò sè 2
Thêi gian lµm bµi: 180 phót
8WJ"OX!FG*!*$H0!(I$!2<!+=6!+(5J$!()*!
( )
( )
∑
∞
=
+
−
1
1
1
n
n
n
xn
?u "
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8WJ"YX! "!782!290(!:$;0!2<+!+=6!()*!&Z!
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2
R !0(T01!*.!2>,01!_?
8WJ"ZX!P(,!mkW!()*
!! !!!!!!!!!!!!!!!!!!!
()
[ ] [ ]
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1,0 0
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= n
x
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n
xf
n
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045
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lim
n
x
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= !AO$!!
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= !2>,01!Cb!
n
If !:)!29+(!\(p0!hi!-z!+=6!
n
f !2>;0!wR!
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nx !:)!\(S0!015W;0!+=6
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Đặng Xuân C)ơng - Cao học 12 - Giải tích - Đại học Vinh
K nim hố 2005
8
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
4I"-5$"-JKL3"M$35"):&"56)"3NA"Y]]O
QR3S""4.$"M\
@#+35S"/&%3
Thời gian làm bài: 180 phút
8WJ"OX!P(,!o!:)!@(/01!1$60!2f2!+%!+V+!C6!2(3+!*I2!Y0!+b!-e+!
n
!AO$!(L!&Z!2([+!A)! ^
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!:)
V0(!`B!-$40!*J$!C6!2(3+!2()0(!CB,!()*!+=6!0b?
6"!P(301!*$0(!>Q01!
!:)!*I2!\(8\!-$40!Cn$!25W40!290(!+=6!@(/01!1$6!A8+!2i!o?
-"!FG*!1$V!2>X!>$;01!A)!A8+!2i!>$;01!+=6!
?
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23
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paa
p
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8WJ"^X"782!0(b*! Ô "+V+!&Z!(s5!2g!AO$!\(8\!+I01!2(/01!2(Tt01?
6"!P(301!*$0(!>Q01!Ô !@(/01!\(%$!:)!0(b*!7W+:$+?
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Đặng Xuân C)ơng - Cao học 12 - Giải tích - Đại học Vinh
K nim hố 2005
9
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
4I"-5$"-JKL3"M$35"):&"56)"3NA"Y]]Y
QR3S""T$U$"-V)5
@#+35S"/&%3
Thời gian làm bài: 180 phút
8WJ"OX!782!&[!(I$!2<!CH5!+=6!!+(5J$!()*!
( )
=
+
1
22
1
n
xnn
x
?
8WJ"YX"P(,!()*!&Z
( )
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=
+=
0,0, 0
0,0,
1
cos
,
22
3
yx
yx
yx
x
yxf
!045
045
6"782!290(!@(%!A$!+=6!()*! f 2B$!C$D*!
( )
0,0 ?
-"!782!290(!:$;0!2<+!+=6!+V+!CB,!()*!>$;01!+=6! f 2B$!C$D*!
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n
P(301!*$0(!>Q01!^
6"!PV+!+/01!2(3+!
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n
nn
yxyxd R!
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yxyxd =
sup, !AO$!
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1
l N
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1
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2
n
n
n
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:$;0!2<+?!F90(!
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nn
xx sup=
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n
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{ }
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XA = !2(G!AO$!*_$! n !2(G!
I
=
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Đặng Xuân C)ơng - Cao học 12 - Giải tích - Đại học Vinh
K nim hố 2005
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
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Thời gian làm bài: 180 phút
+$"OX !6"!!P(,!\(8\!-$40!!Cn$!25W40!290(!
!+=6!
3
Ă !CZ$!AO$!+i!&.!Ci0!AX!+b!*6!2>e0!:)^
815
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-"!P(301!2a!>Q01!045!
A !:)!*6!2>e0!A5/01!\(S0!2'!2([+!2(a6!*k0!
2
0AI+=!2(G! A !@(/01
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2
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+$"YX!!P(,!0(b*!#!A)!]52#!:)!0(b*!2f2!+%!+V+!2[!Cr01!+f5!+=6!#!AO$!\(8\!2,V0!0(p0!V0(!`B?
oO$!*J$!6!
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+"!P(301!*$0(!>Q01!*I2!0(b*!+,0!j!+=6!#!:)!TO+!+(5Y0!+=6!#!@($!A)!+(g!@($!~
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-"!FG*!Cv+!&Z!+=6!2>Tt01!7?
+$"[X" 6"!P(301!*$0(!>Q01!045!K!:)!*I2!2>Tt01!2(G!A)0(!C6!2(3+!K[`]!:)!*I2!A)0(!+(90(?
-"!P(301!*$0(!>Q01!*$H0!015W;0!P!@(/01!\(%$!:)!2>Tt01!2(G!P[`]!@(/01!:)!A)0(!+(90(?
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Đặng Xuân C)ơng - Cao học 12 - Giải tích - Đại học Vinh
K nim hố 2005
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
4I"-5$"-JKL3"M$35"):&"56)"3NA"Y]][
QR3S""T$U$"-V)5
@#+35S"/&%3
Thời gian làm bài: 180 phút
8WJ"OX!!P(,!()*!&Z
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=
+
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0y
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2
2
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x
[ ]
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1,0
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Đặng Xuân C)ơng - Cao học 12 - Giải tích - Đại học Vinh
K nim hố 2005
E
!"#$%&"'()"*+",+&" &
/0123#"4.$"56)"7$35
8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
4!)"BCD"E"/F"'&"E"G.35"D5H)
4I"-5$"-JKL3"M$35"):&"56)"3NA"Y]][
QR3S""4.$"M\
@#+35S"/&%3
Thời gian làm bài: 180 phút
8WJ"OX!P(,!0!:)!&Z!015W;0!mTi01R!P
0
uw"!!:)!2e\!(U\!2f2!+%!+V+!C6!2(3+!Y0!`!AO$!(L!&Z!2([+!+b!-e+
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2>,01!A)0(!Cb?
§Æng Xu©n C)¬ng - Cao häc 12 - Gi¶i tÝch - §¹i häc Vinh
Kỷ niệm hè 2005
x
!"#$%&"'()"*+",+&" &
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8!3#"59:";<"5!$")5="3#5>:"7$?-"@:A
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Thêi gian lµm bµi: 180 phót
8WJ"OX "!782!290(!:$;0!2<+!A)!@(%!A$!+=6!()*!&Z^
( )
322
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(;)(0;0)
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