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Eisagwgh kai Oria
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1/2011 1. : n\ ( ) ( ) ( )2 2 21 1 2 2 1 2 3 1: ... , : ... , : max ,n n i n ix x x x x x = + + + = + + + =x x x x . ( )1 2, ,..., nnx x x= x \ , ( )1 2, ,..., nnx x x= x \ , :
( ) ( ) ( ) ( )3 1 2 3n x x xn
x
. . \
, : ( )1 2 3, ,..., nx x x=x \
( ) ( ) ( )22 2 2 2 2 23 1 1 2 1 21 1max ... max ... ,i n ii n i n nx x x x x x x x + + + + + +x x
. : ( ) ( )
( )2 2 2
1 2 1 2 1 2
22 2 21 2 1 2
2 2 2 2 2 21 2 1 2
1
1
... ...
... ...
... ... 2
0 2 , .
n n
n n
n ni j n
j ji j n
x x x x x x
x x x x x x
j jx x x x x x x
x x <
<
+ + + + + + + + + + + + + + + + + + +
x x
x
:
( ) ( ) ( )2 3 1 2 1... max , .n ii nn x x x n x + + + x x ( ) ( ) ( ) ( )3 1 2 3n x x x
( )x , ,
. ,
nx \
( )1 2, x x n\0
,
, ,
> , ( ) ( ) (1 )x1 2a x x , . nx \
2. .
1dn\
( ) ( )( )1 1,
, :1 ,
dd
d= +
x yx y
x y,
. n\ : d
2
. ( ) ( )( )1 1,
, : 01 ,
dd
d= +
x yx y
x y, , nx,y \ ( )1 , 0d x y , , nx y \ .
:
( ) ( )( ) ( )1 11,
, : 0 ,1 ,
dd d
d= = +
x yx ,=y x y x y
x y
. 1dn\
.
( ) ( ) ( )( )( )( ) ( ) ( )1 1 1 11 1
, ,, , , , ,
1 , 1 ,d d
d d d dd d
= = =+ +x y y x
x y y x x y y xx y y x
n x,y \ . III.
( ) ( ) ( ) ( )( )( )( )
( )( )
( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )
1 1 1
1 1 1
1 1 1 1 1 1
1 1 1
1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
, , ,, , ,
1 , 1 , 1 ,
, 1 , 1 , , 1 , 1 ,
, 1 , 1 ,
, , , , , , , ,
, , , , , , 2 ,
d d dd d d z y
d d d
d d d d d d
d d d
d d d d d d d d
d d d d d d d d
+ ++ + + + + + +
+ + + + + + + + + +
x y x z z yx y x z
x y x z z y
x y x z z y x z x y z y
z y x y x z
x y x y x z x y z y x y x z z y
x z z y x y x z x y z y x z ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1
1 1 1 1 1 1 1 1
,
, , 2 , , ,
, , , 2 , , 2 , , ,
d d d d d
d d d d d d d d
+ + + + +
z y
x y z y x y x z z y
x ,y x z z y x z z y x y x z z y
. 3. , :
( )2
11 3 1 1, 1 , 1,2,3,..., , , , 1, 2,3,...2 1
nn
n n
nn n
n n n nn
= = = = + x y
: 1lim 0n n+
= 33lim 1n
ne
n
+ = , ,
, :
( )31 3 1 3lim lim , 1 lim , lim 1 0,n nnn n n n en nn n + + + + = = =
x .
( )1lim2 1
n
n
nn+
+ , .
, 12
,
12
.
4. ( ){ } 2,0 :n n = ] \ .
() .
3
() , ; () ( ),0n ,
( )( ),0 , , 0D n < ( ) ( ) ( ),0 0,0 ,0n n n M = = < , , . n .
5. fD
( ) ( )( ), ln lnf x y x y x= . fD ; fD fD .
( ) ( ){ }( ) ( ) ( ){ }( ){ }( ){ }
2
2
2
2
, : 0 ln 0
, : 0 0, ln 0 0, ln 0
, : 0 0, 1 0 0, 1
, : 0, 1 0, 1 .
fD x y y x x y x
x y y x x y x x y x
x y y x x y x y x x y x
x y x y x x x y x
= > >= > > > < > > > < > + < < < +
\\\\
1
4
6. : 2 2 2
2 2 2
2 2 2 2
2 2 2 2 2
2 2 2 2 2
0 ( 2) ((2 ) 4 9
2 ( ) 0 2 ( ) 02 0 ( 2) 4 4
1.
y z x y zx z y z y z
x y z x y x y z x yx y z x y zz y x x y z
= + + =+ = + = = + =+ = + + == + =
() ()() ()() ()() ()() ()
1) 4
4
() x x () (2,0,1) = 2. () : 2 0,x z y z+ = + = 0.() x x . () ( )( )2 2 2 ( ) 0 2 0 0 2 0x y z x y x y x y z x y x y z = + = = + = , 0, 2 0x y x y z = + = . () 2 , ,x y z , ( )0,0,0 . () , . 2 2z x y= + 2
() : 2
2 22
( 2) 12
x y z + + = , 2,1,1. () . ()
7. ( 0)xOy z = 2 2 2
2 2
9, 02 0, 0
x y z y zx y z x z+ + = =+ = =
()() .
9 )
() z
2 22x y+ = ,
( 0xOy z =2 2
2 2222 9, 0 1,3 3
2
x yx y z z+ = = + = = 0
1
0
.
() z( )22 2 22 0 1x y x x y+ = + =
( 0)xOy z = ( )2 +,
. 21 1,x y z= = . xOy . , () :
5
[ ]3 33cos , sin , sin , 0,22 2
x t y t z t t = = = . () :
[ ]1 cos , sin , 1 cos , 0,2x t y t z t t = + = = + 8. ()
2 2
2 2
( , ) 2 ( , ) 1
( , ) ( , , )4
( , , ) ( , , )x y z
f x y x y f x y x y
y xf x y f x y zzx
2
y
f x y z e f x y z x y z+
= + = += =
= =
() ()
() ()
() () +
() ( , ) 2 2,f x y x y c x y c c= + = + = \ ( ) () ( , ) 1 1 , 1f x y x y c x y c c= = + = < ( ) () ( , ) ,yf x y c y c x c
x= = = \ ( )
() 2 2
2 2( , , ) 44
x yf x y z c x y czz+= = + = (
) () ( ) ( , , ) ln , 0x y zf x y z e c x y z c c+ = = + = >() 2 2 2( , , )f x y z x y z c= + = ( )
9. 2 2 2
2 2 2 2
2 2 2
, , 0( , )
,
y x x yx y x yx y
x y
+ + > + += +
6
( ),x y 2 2 4x y 2+ =
e12
.
2 (0,0) , . 10.
: 2 2
12 2 3 3
2 2 2 2( , ) (0,0) ( , ) (0,0) ( , ) (0,0) ( , ) (0,0)
2
2 2 2 2 2 2( , , ) (0,0,0) ( , , ) (0,0,0) ( , , ) (0,0,0)
sin( )lim lim lim lim
lim lim lim
yx y
x y x y x y x y
x y z x y z x y z
x y x y e xx y x y
xyz xy x yzx y z x y z x
+
+ + +
+ + + +
(i) (ii) (iii) (iv)
(v) (vi) (vii)
2 2 2y z+ +
(i) : ( ) 22 0sin, 1F += , 1 .
( ) 2 2, :2
D x y x y = +
7
( ) 2 2 2 22 2 2 2cos 1 2sin 22 2
x y x y 2x y x + ++ = = +
y ,
( )2 2cos 1x y + < , 2 2 2 2x y x y + < + <
= .
2 2
2 2( , ) (0,0)
sin( )lim 1x y
x yx y
+ =+ (ii) : ( ) ( ) ( )3 3 0, cos sinF += 0
( ) ( )2 2
2 22 23 3
2 2 2 2 2 2
2 2 2 2
2
3 2 32
x yx y x yx y x xy yx yx y x y x y
x y x y
++ + + + + = + + += + = +
,
3 3
2 2
x yx y
> = +
+ < + <
,
:
0 > , , (0,1 ) 1 0ln
= > ,
2 2 1ln
x y + < = , 2 2
1x ye + < .
1 , 0 > 2 2
1x ye + < . . : 2 2
1
( , ) (0,0)lim 0x y
x ye +
= .
8
(iv) ( ), yf x y x= . (1 1, 0n n
= nx ),0
( )1
11 1 1 1, lim 1
n nnn n
n
f nn nn
+ + = = = = nx ,
(1 1, 0ne n = ny ),0 , :
( ) ( )1
11 1 1 1 1.
n
nnn n
fe ee
+ = = = ny e
. .
(1 1, 0,na n = nz )0
, 1a >
( ) ( )1
11 1 1 1 1.
n
nnn n
fa aa
+ = = = nz a
(v) (1 1 1, , 0,0,0n n n
= nx )
( ) 32
11 03 3 n
nfn
n
+= = nx .
0 .
: ( ) 2 2 2, , 0 xyzf x y z x y z =
9
( ) 22
11 1
3 3 3nnf
n
+= = nx , ( ) 22
22 2
11 11 11 3nnf
n
+1= = nx .
, . (vii)
(1 1 1, , 0,0,0n n n
= nx ) ( )1 2 3, , 0,0,0n n n
= ny
( )2
0 0 03 nf
n
+= = nx , ( ) 22
55 5 011 11 11n
nf
n
+
= = nx .
, .
( ) ( ) ( ) ( ), , , , , 0,0,0 0t t t = r > ,
( )( ) 2 202 2 2 2 2tf t 2 = + + + +r , . ., . 11. :
(i)
2
2 22
2 2
2 , ( , ) (0,0)( , )
0, ( , ) (0,0)
xx yx e x yf x y x y
x y
+ = + =
(ii) , 0
( , ), 0
x y x yx yf x ya x y
+ += + = , .a\
(i) ( , )f x y ( ) ( ), 0,x y 0 . ( )0,0 ( ) ( )
( , ) (0,0)lim , 0,0 0
x yf x y f = = .
, ( )2 2: 2x k y k + = ( ) ( ) [ ]cos , sin , 0,t k k t k t t = r ( ) ( )0 0,0=r ,
( )( ) 2 21 101 1k ktf t e ek k
= r ,
10
, ( ),f x y
. ( )0,0(ii) ( , )f x y ( ) ( )0 0 0, , ,x y x x x \ . ( )0 0 0, ,x x x \ , 0x y+ = :
( ) ( )0 0
0 0( , ) ( , )lim , ,
x y x xf x y f x x a = = .
, ( ) ( )0 0, , 0,t x t x t t = + + + r 0( ) ( )0 00 ,x x= r
( )( ) ( )( ) 00 00
, 02
, 0t
xx tf t
t x
=+ += +
r
( ),f x y ( )0 0 0, ,x x x \ .
1/2011