2000-2011
2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 2010 . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 3 7
2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 11 2009 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 15
2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 19 2008 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 23
2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 27 2007 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 31
2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 34 2006 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 38
2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 41 2005 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 44
2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 48 2004 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 52
2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 55 2003 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 58
2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 61 2002 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 65
2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 68 2001 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 71
2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 75 2000 . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 79 2000 . . . . . . . . . . . . . . . .
. . . . . . . . . . . 84
2011
1
( ) 16 2011 : : (4) A1. f x 0 . f x 0 , : f (x 0 ) = 0 10 A2. f
. y=x+ f + ; 5 A3. , , , , . ) z 0 z 0 =1 ) f:A 1-1, x 1 , x 2 A :
x 1 x 2 , f(x 1 ) f(x 2 ) ) x 1 ={x|x=0} : (x ) = ) : x =1 x + x
lim
1 2 x
1 4 3
2011
2
) C C f f 1 y=x xOy xOy. 10 z w z 3i , : z 3i + z + 3i = 2 w = z
3i +
1 z 3i
B1. z 7 B2. z + 3i =
1 z 3i 4
B3. w 2 w 2 8 B4. :
zw = z 6
f : , , f (0 ) = f (0) = 0 , :e x (f ( x ) + f ( x ) 1) = f ( x
) + xf ( x )
x.
2 4 4
2011
3
1. : f ( x ) = ln(e x x ), x 8 2. f . 3 3. f . 7 4. ln(e x x ) =
x 0, 2 7 f, g : , x : i) ii) f(x)>0 g(x)>0
1 f (x) e2x1 g( x )
x
=
0
e2t dt g(x + t )e2 t dt f (x + t)
x
iii)
e
2x
=
0
1. f g f(x) = g(x) x . 9 2. : f(x) = e x , x 4 3 4 5
2011
4
lim ln f ( x ) 1 f x
3. :
x 0
5 4. F( x ) =
1
x
f ( t 2 ) dt
xx yy x=1. 7
1. 2.
3. 4. 5. 6. 7. 8.
( ) (, ). . . . . . . , . . . : (3) . : 10.00 ..
4 4 6
2010
1
7 2010 : : (5) A1. f(x) = x, x, ( x ) = x 8 A2. f [,] ; 4 A3. f
x 0 A () , f(x 0 ); 3 4. , , , , . ) f(x) = x , > 0, x = x x
1
( )
) f o g g o f, f o g = g o f ) lim f ( x ) = + , limxx 0
x x 0
1 =0 f (x)
1 5 7
2010
2
) f [,] f(x) 0 x[,],
f (x)dx 02
) zC z = z z 10 z 1 , z 2 z 1 +z 2 = 2 z 1 z 2 = 5 B1. z 1 , z 2
5 B2. w w z1 + w z 22 2
= z1 z 2
2
w (x+1) 2 + y 2 = 4 8 B3. w 2 2 Re(w) + Im(w) = 0 6 2 5 8
2010
3
B4. w 1 , w 2 w 2 w1 w 2 = 4 ,
w 1 + w 2 = 2 6 f(x) = (x2)lnx + x 3, x > 0 1. f 5 2. f (0,1]
[1, + ) 5 3. f(x) = 0 . 6 4. x 1 , x 2 3 x 1 < x 2 , (x1, x2) ,
f() f() = 0 f (, f () ) . 9
3 5 9
2010
4
f: f(0) = 1 f(0) = 0 1. f(x) 1 x 41
x f ( xt )dt + x 32. lim0 x0
x
3
= + 6
f(x) + 2x = 2x f ( x ) + x 2 , x, :
(
)
3.
f(x) = e x x 2 ,
2
x 8
4. x +2
h(x) =
f (t)dt , x 0x4
x 2 + 2x +3 x 2 + 2 x +1
f (t)dt + f (t)dt < 06
7
4 5 10
2010
1
( ) 19 2010 : : (4) A1. f . F f , : G(x)=F(x)+c, c f G f
G(x)=F(x)+c, c
6 A2. x=x 0 f ; 4 A3. f . f ; 5 4. , , , , . ) +i +i . 1 4
11
2010
2
) f . f , . ) f (,), (,), A = lim +f ( x ) B = lim f ( x )x
x
) (x)=x, x ) lim f ( x ) < 0 , f(x) -1
. , lim f(x) .x +
5 . = -1 . f . 10 f 6
.
. f(x) + 2 = 0 0 4
3 5 17
2009
4
4
f: [0 , 2] f ( x ) 4 f ( x ) + 4 f ( x ) = k x e 2 x , 0 x 2
f (0) = 2 f (0) , f (2) = 2 f(2)+12 e 4 , f(1) = e 2 k . . f ( x
) 2 f ( x ) , 0 x 2 g(x) = 3x 2 e2x Rolle [0,2]. 4 (0,2) , f ( ) +
4 f ( ) = 6 e 2 + 4 f ( ) 6 . k = 6 g(x) = 0 x [0,2]. 6 f ( x ) = x
3 e 2 x , 0 x 2 5 .
.
.
1
2
f (x) x2
dx
4
4 5 18
2009
1
( ) 20 2009 : : (5)
1 o . f . f x f ( x ) = 0 , f . 10 . f x 0 ; 5 . , , , , . . z 1
, z 2 , z1z 2 = z1 z 2
2 . f () x 0 A, f(x)f(x 0 ) xA 2 1 5 19
2009
2
.
x 1 =1 x 0 x lim
2 . f . 2 . f [, ] f(x) 0 1 x > 1,
A. f ( x ) 1 x > 1, =e 8 . =e, . f . 5 . f (1, 0] [0, + ) 6 .
, (1, 0) (0, + ) , f () 1 f ( ) 1 + =0 x 1 x2
(1, 2)
6
3 5 21
2009
4
4 f [0, 2]
0 (t 2)f (t )dt = 0 H( x ) =
2
0 t f (t )dt,
x
x [0, 2],
x H( x ) f ( t )dt + 3, x 0 G(x) = 1 1 t2 , 6 lim t 0 t2
x (0, 2] x=0
. G [0, 2]. 5 . G (0, 2) G( x ) = H( x ) x2
,
00. 6 . f. 6 .
ln x f (x) g( x ) = k i.
, x >0 , x=0
k g . 6
1 ii. k = , g , 2 , (0, e).
7 4 f [0, + ) f(x) > 0 x 0. : F(x) =h(x) =
x 0
f(t) dt ,
x [0, + ), x (0, + ).
F( x )
x 0
,
t f ( t ) dt
3 4 25
2008
4
.
1 0
e t 1[f ( t ) + F( t )]dt = F(1)
6 . (0, + ). h
8 . h(1)=2, : i.
2 0
f(t) dt < 2
2
tf(t)dt0
6 ii. 1 0
1 F( t ) dt = F(1) 2
5
1. 2. (, , ). . , . . . . . , . . : (3) . : 10.00 .
3. 4. 5. 6. 7.
K 4 4 26
2008
1
24 2008 : : (5) 1 o A.1 f(x) = ln x , x* * :
( ln x ) =.2
1 x
10 f [,]; 5 B. , , , , , . . f:A 11, f 1 : f 1 ( f ( x ) ) = x,
x A f ( f 1 ( y )) = y, y f ( A ) 2 . f f . 2 1 5 27
2008
2
. z 2 +z+=0 ,, 0 , . 2 . f , f( x ) > 0 x.
2
. A f ,, f(x)dx = f(x)dx + f(x)dx
2
2
z w (i + 2 2 )z = 6 w (1 i) = w (3 3i)
: . z . . w . 6 7 . w
6 . zw
6 2 5 28
2008
3
3 x lnx , x > 0 f(x) = 0 , x=0
. f 0. 3 . f . 9 . x = e . 6 . f(x+1)>f(x+1)f(x) , x > 0 .
4 f 7 x
f(x) = (10x 3 + 3x) 0 f(t)dt 45. f(x)=20x 3 +6x45 8
2
3 5 29
2008
4
.
g
. g(x) = lim g(x) g(x h) h 0 h
4 . f () g () lim g(x + h) 2g(x) + g(x h) h2 = f(x) + 45
h 0
g(0)=g(0)=1,
10
i. g(x)=x 5 +x 3 +x+1
ii. g 11
3
( ) 1. (, , ). . 2. , . . . 3. .
4 5 30
2007
1 3 2007 : : (4)
1o A.1
f x 0 , . 10
.2
;
Rolle
5
B.
, , , , , . . f() f . 2 . f, g, g [,],
f ( x )g( x )dx = f ( x )dx g( x )dx .
2 . f , x f ( t )dt = f(x)
x. 2
1 31
2007
2
. f (,), (,) = lim f ( x ) = lim f ( x ) .x +
x
2 . f, g . f, g f(x) = g(x) x , f(x) = g(x) x. 2 2
3x , x f (x) = 2 x + x + x, . lim f ( x ) = 3 .x 0
x 0.
. f(x) (1, +). 10 . f(x) e x > 0. 7 . x 2 +2 x2 +2 x 2 +3
4
x 2 +1
f ( t )dt =
f ( t )dt + 2 f ( t )dt
(0, +). 8 4 z 1 = +i z 2 = R R , I 0. z 2 z 1 I . . z 2 z 1 = 1.
9 . z 1 . 6 . z 1 z12
2 z1 2 + z1
,
>0,
(z1 + 1 + i)20 (z1 + 1 i)20 = 0 .
10
3 33
2007
1
24 2007 : : (5) 1 o A.1 z 1 , z 2 , :z1 z 2 = z1 z 2 .
8 .2 f, g ; 4 .3 y = f +; 3 , , , , , . . f [,] x[, ] f(x) 0
f(x) dx > 0 . 2 . f x . f f(x) > 0 x . 2 1 34
B.
2007
2
. f x 0 g x 0 , g o f x 0 . 2 . f , g(x) f(t) dt = f (g(x) ) g
(x)
(
)
. . > 1
2
x
lim x = 0 .
2 2 z= 2 + i + 2i
IR .
. z (0,0) =1. 9 . z 1 , z 2 z= 2 + i + 2i
= 0 = 2 . i. z 1 z 2 . 8 2 35
2007
3
ii. :(z1 ) 2 = ( z 2 )
. 8 3 : f(x) = x 3 3x 2 2 IR + , Z. 2 . f , . 7 . f(x) = 0 . 8 .
x 1 , x 2 x 3 f, B ( x 2 , f(x 2 ) ) ( x 3 , f(x 3 ) ) ( x 1 , f(x
1 ) ) , 2 y = 2x 2 . 3 . 2 f y = 2x 2 . 7
3 36
2007
4
4 f [0, 1] f(0) > 0. g [0, 1] g(x) > 0 x [0, 1]. : F(x) =
0 f(t) g(t) dt , G(x) = 0 g(t) dt ,x x
x [0, 1], x [0, 1]. 8
. F(x) > 0 x (0, 1].
. N : f(x)G(x) > F(x) x (0, 1]. 6 . N :F(x) F(1) G(x) G(1) x
(0, 1].
4 . : x f(t) g(t) dt x 2 t 2 dt 0 0 . lim x0+ x g(t) dt x 5 0 7
4 37
2006
1 5 2006 : : (4)
1o A.1 : (x)=x, xI . R 10 .2 f . f ; 5 , . . z 1 , z 2 , :
B.
z 1 z 2 z1 + z 2 . 2 . f, g x o f g(xo)0, g xo :
f(x ) g (xo ) f (xo ) g(xo ) f (xo ) = o [g(xo )] 2 g. x0 [ln x
] =
. 2
1 . x 2
1 38
2006
2
. f: I 11, R y f(x)=y x . 2 . f [,]. G f [,], f(t)dt = G() G() .
2
21 + ex f(x) = , xI . R x +1 1+ e . f I . R 9
.
1 dx . f(x) 9
. x0 . x ii. f (0,+). : ln(x + 1) lnx < 12
.
1 lim xln(1 + ) . x + x 5
.
(0,+) (+1) = +1 . 8
3 40
2006
1
27 2006 : : (4) 1 o A.1 f, . : f(x)>0 x , f . f(x) 0 ,
xx0
f ( x ) > 0 2
x 0 .
1 41
2006
2
. H f() f . 2 . (3 x ) = x 3 x - 1 , x IR . 2 . f
(x)g(x)dx=[f(x)g(x)] f (x)g(x)dx, f,g
[,]. 2
2 f(x) =2+(x-2) 2 x2. . f 1-1. 6 . f -1 f . 8 . i. f f -1 y=x. 4
ii. f f -1 . 7
2 42
2006
3
3 z1 , z 2 , z 3 z1 = z 2 = z 3 = 1 z1 + z 2 + z 3 = 0 . . : i.
z1 z 2 = z 3 z1 = z 2 z 3 . 9 ii. z1 z 2 .2
4 Re (z1 z 2 ) 1 .
8 z 1 ,z 2 ,z 3 , . 8
4 f(x)=x+1 lnx. x1
. f. 8 . N f(x)=0 2 . 5 . g(x)=lnx (,ln) >0 h(x)=e x ( ,e )
IR , f(x)=0. 9 . g h . 3
3 43
2005
1 6 2005 : : (4)
1 o A.1 f f (x) = x . f (0,+) :f(x) = 1 2 x
. 9
.2 B.
f:A IR 1-1;
4
, . . , f 0, f . 2 . f (,) x o. f (,x o ) (x o ,) , (xo , f (xo
)) f. 2 1 44
2005
2
. . 2 . f,g fog gof, fog gof. 2 . z, z xx. 2 . f IR *, :
f (x)dx = f (x)dx . 2
2 . z 1 , z 2 z 1 +z 2 =4+4i 2z1 2 = 5 + 5i , z z 1 , z 2 .
10
. A z,w z 1 3i 2 w 3 i 2 : i. z, w , z=w 10 ii. z w. 5
2 45
2005
3
3 f, IR f(x)0 x IR . . f 1-1. 7 . C f f (1,2005) (-2,1), f 1 (
2004 + f (x2 8)) = 2 . 9 . C f , C f 1 x + 2005 . (): y = 668 9 4
f: IR IR ,
limx0
f (x) x = 2005 . x2
. : i. f(0)=0 ii. f(0)=1. 4 4
3 46
2005
4
. IR , :
x2 + (f (x)) lim 2 2 = 3. x0 2x + (f (x))2
7 . f IR f(x)>f(x) x IR ,
:i. ii.
xf(x)>0 x 0.
6
f (x)dx < f (1) .0
1
4 ( ) 1. (, , ). . 2. , . . , . 3. . 4. . 5. : (3) . 6. : 10.30
. K 4 47
2005
1
31 2005 : : (4) 1 o A.1 f, [, ]. f [, ] f() f() f() f() , x 0 (,
) , f(x 0 ) = . 9 .2 y = x + f +; 4 , . . f [, ] f() < 0 (, )
f() = 0, f() > 0. 2 . lim (f(x) + g(x)) , x x0
B.
lim f(x) x x0
x x0
lim g(x) .
2
1 48
2005
2
. f f 1 f y = x, f 1 . 2 . limx x0
lim f(x) = 0 f(x) > 0 x 0 ,
1 =+ . x x 0 f(x)
2 . f , x x . f (t)dt = f(x) - f()
(
)
2 . f , x x , . 2 2 z 1 , z 2 , z 3 z 1 =z 2 =z 3 = 3. 9 . . :
z1 = z1 7 z1 z2 + . . z2 z1 9 1 . : z 1 + z 2 + z 3 = z 1 z 2 + z 2
z 3 + z 3 z 1 . 3 9
2 49
2005
3
3 f f(x) = e x , . f . 3 . f, , y = ex. . 7 . () , f, yy, e -2 .
() = 2 8 > 0.
2 () . lim + 2 +
. 7
4 f IR , 2 f(x) = e x f(x)
x IR f(0) = 0.
1 + ex . . : f(x) = ln 2
6 . N :
x0
lim
0
x
f(x - t) dt x
. 6
3 50
2005
4
. : h(x) =
x x
t
2005
x2007 . f (t)dt g(x) = 2007 7
h(x) = g(x) x IR .
. (0 , 1).
1 x 2005 x t f (t)dt = 2008
6
( ) 1. (, , ). . 2. , . . , . 3. . 4. . 5. : (3) . 6. : 10:30 .
K 4 51
2004
1 5 2004 : : (4)
1o A. f . f f(x) = 0 x ,
f . 9 . , . . f x 0 , . 2
. . 2 . f, g IR fog gof, . 2
1 52
2004
2
. C C f f 1 y = x xOy xOy. 2 . f x 0 , x x0
lim
k
f (x) = k lim f(x) , f(x) 0 x x0
x 0 , k k 2. 2 . f (, ) [, ]. 6 2 f: IR IR f(x) = 2 x + m x 4 x
5 x , mIR, m > 0. . m f(x) 0 x IR . 13 . m = 10, f, xx x = 0 x =
1. 12 3 f: [, ] IR [, ] f(x) 0 x [, ] z Re(z) 0, m(z) 0 Re(z)
>Im(z).
2 53
2004
3
z +
1 1 = f() z 2 + 2 = f 2 (), : z z 11
. z= 1
. f 2 () < f 2 () 5 . x 3 f() + f() = 0 (1, 1). 9 4 f [0, +)
IR , 1 x2 f(x) = + 2 2xf(2xt) dt . 0 2
. .
f (0, +). 7 f(x) = e x (x + 1). 7
. .
f(x) [0, +). 5 x+
lim f(x)
x
lim f(x) . 6
3 54
2004
1
27 2004 : : TE (4) 1o A. f ' x 0 . f x 0 , f(x 0 )=0 10 . f x 0
; 5 . . . . . lim f(x) = ,
2x x 0 x x 0
lim f(x) = lim f(x) =+ x x 0
2 . f, g x 0 , fg x 0 : (fg)(x 0 ) = f(x 0 ) g(x 0 ) 2
1 55
2004
2
. f, . f(x)>0 x , f . 2 . f [,]. G f [,],
f (t)dt = G () G () 2 2 f f(x)=x 2 lnx . . f, . 10 f . 8 f.
7
.
.
3
g(x)=e x f(x), IR f(0)=f( 3 )=0 . 2
f
. f()=f().
(0, 3 ) 2 8
2 56
2004
3
. f(x)=2x 2 3x, I() = .
g(x)dx ,lim I()
0
IR 8
-
9 4 f: IR IR f(1)=1. x IR , g(x)=
1
x3
z f (t)dt 3 z +
1 (x 1) 0 , z
z=+iC, , IR *, : . g IR g. 5 . N z = z + 1 z 8 . 1 Re(z 2 ) = 2
6 . A f(2)=>0, f(3)= >, x 0 (2,3) f(x 0 )=0. 6
3 57
2003
1 8 2003 : : (4)
1o A. f . F f , : .
G(x) = F(x) + c ,
c R
f . G f
G(x) = F(x) + c ,.
c R . 10
, . . z 1 , z 2 , z1 z2 z1 + z2 z1 + z2 . 2 . f ' ( , ), x 0 , f
. f (x) > 0 ( , x 0 ) f (x) < 0 ( x 0 , ), f (x 0 ) f.
2 1
58
2003
2
. f : R 1 1 , x 1 , x 2 A : x 1 = x 2 , f(x 1 ) = f(x 2 ) . 2 .
f, g , :
f(x) g (x) dx.
= f(x) g(x) f (x) g(x) dx . 2
x = x 0 f ; 7
2 . ( ) z :
z = 2
m (z) 0 . 12
. , z () ,
1 4 z + 2 z xx .
w =
13
2
59
2003
3
3 .
f(x) =x+
x2 + 1 x .
lim f(x) = 0 . 5 6
. f , x . . f (x) x + 1 + f(x) = 0 . . 2
0
1
1 x +12
dx = ln
( 2 + 1)
6 . 8
4 f IR , : f(x) = f (2 x) f (x) 0 x IR . . f . 8 . f(x) = 0 . 8
.
g(x) =
f(x) . f (x)
g xx, 45 . 9 3
60
2003
1
29 2003 : : 1o A. , f x 0 , . 8 . ; 7 . , . . z z , z = z = z .
2 . f . f(x)>0 x , f . 2
_
1
61
2003
2
. ,
f,
f (x)dx = f (x) + c
, c IR . 2
. f , f . 2 . f x 0 . f f(x 0 )=0, f x 0 x 0 . 2 2
z=+i, , IR _ _ w=3z i z +4 , z z . m(w)=3. 6
. Re(w)=3+4
. , w y=x12 , z y=x2 . 9 2
62
2003
3
. z , y=x2 , . 10 3
f(x) = x 5 +x 3 +x .
. f f . 6x . f(e )f(1+x) x IR.
6 . f (0,0) f
f 1.
5 . 1 f , x x=3 . 8 4 f [,] (,) . f() = f() = 0 (,), (,), f()f()
z1 . 9 . f. 8
4 f, R. , :
f (x)f(x) + (f (x )) 2 = f(x)f (x) , x R. f(0) = 2f (0) = 1. .
f. 12
. g [0,1], x g( t )0 [0,1].
2x
1 + f 2 (t)
dt = 1
13
3
67
2002
1 30 2002 :
1o A. f ' [, ]. G f [, ],
f (t ) dt = G() G() . 12 .1. f(x) = x. f R f(x) = x . 8 .2. , .
. f [,] (,], f [,] . 1 . , 1-1 , . 1
1
68
2002
2
. f x 0 lim f(x) = 0 , x x 0lim f(x) = 0 . x x 0
1 . f R ,
f (x)dx = xf (x) xf(x)dx . 1 . lim f(x) > 0 , f(x) > 0 x 0
. x x 0
1 2 z f() = i z, . f(3) + f(8) + f(13) + f(18) = 0 . 7 . z=
Arg(z) = , f(13) = + + i + . 2 2
IN*.
8 . z = 2 Arg(z) = , 3 0, z f(13). 10
2
69
2002
3
3 f, g R . fog 1-1. . g 1-1. 7 . : g ( f(x) + x 3 - x ) = g (
f(x) + 2x -1 ) . 18 4 . h, g [, ]. h(x) > g(x) x [, ], h(x)dx
> g(x)dx . 2 . R f, :
f (x) e f ( x ) = x 1,
x R
f(0) = 0 . 5 12
) )
f f. x < f(x) < x f(x) , 2
x > 0.
) f, x = 0, x = 1 xx, 1 1 < E < f (1) . 4 2 6
3
70
2001
1 E 5 2001 :
1o A.1. f . F f , : G(x)=F(x)+ C , C R f G f : G(x)=F(x)+ C , C
R 6,5 .2. . . f (x)dx = ..... (f (x) + g(x) ) dx = ..... .
.
[f (x) + g(x)] dx = .....
,R f,g [,] 6 1
71
2001
2
.1. f, f(x)=6x+4, xR (0,3) 2. 6,5 .2.
.
1 0
ex + x dx 2
.
4 3x2 1
x
dx 2
.
2
(2x + 3x) dx 2
0
2 . z :
z + 16 = 4 z + 1 9 . z :
z1 = zi 9 . () (). 7 2
72
2001
3
3
x 1 x + , f (x) = x +1 ln( x 1), x (1,2] 1 e
(
)
R.. .
1 e x +1 lim x1 x 1
7
. R f x o =1. 11 . =-1 (1,2) , f ( ,f() ) xx. 7 4 f, (0,+) :
t f( t ) 1 dt x > 0 f (x) = + 2 x 1 x. f (0,+). 3 . f :
x
f (x) = 1+ ln x , x > 0 x
7
3
73
2001
4
. f. .
6
f. 4 f, xx x=1, x=e. 5
.
4
74
2001
1
2 2001 :
1o A.1. z 1 , z 2 . : z 1 z 2 = z 1 z 2 . 7,5 .2. , . z : . . .
. .
z = zzz2 = z2 z=- z
2
z=z iz=z 5
.1. z1 = 3 + 4 i z2 = 1 - 3 i, , .
1
75
2001
2
1. 2. 3. 4. 5.
. .2
z1 z22 z1
4 2
z2
. 25 . 5 . . 2 5 7,5
z1i z2
. 10 .2. z =
z z = 1,
1 . z 5
2 f :
x 2 , f(x) = 1 - ex -3 , x3
x3 x>3
. f , = 1/9. 9 . Cf f (4, f(4)). 7
2
76
2001
3
. f, xx x=1 x=2. 9 3 f, R, : f 3 (x) + f 2 (x) + f(x) = x 3 2x 2
+ 6x 1 , 2 < 3. . f . 10 . f . 8 . f(x) = 0 (0,1). 7 4 f, R, o :
i) f(x) 0, x R ii) f(x) = 1 - 2 x2
x R,
0
1
t f 2 (xt) dt , x R. x R.
g
g(x) =
1 - x2 , f(x)
3
77
2001
4
.
f (x) = - 2xf 2 (x)
10 . g . 4 . f :
f(x) =.
1 . 1 + x2(x f(x) 2x).
4 7
x +
lim
4
78
2000
1
12 2000 :
1o A1. A f ' x 0 , f (x 0 , f(x 0 )). M 4 2. , f ' x 0 , . 8,5
1. . . f x 0 , f x 0 . . f x 0 , f x 0 . . f x 0 , f x 0 . 4,5
1
79
2000
2
2. x 0 . . f(x)=3x 3 , x 0 =1 . f(x)=2x, x 0 = . f(x)=3 x , .
f(x)=x,
1. y=-2x+
2 x 0 =0
1 x+1 4 3. y=9x-62. y= 4. y=-9x+5 5.
x 0 =4
2 f(z)=
8
z o z. . : w 1 =f(9-5i) 2 w2 = f (9 5i ) 3 2004
2z + i , z C z -2i, z 2i
6 Mo 6
2
80
2000
3
.
M=
2 w1 3 0
0 w1
w1 w 1 . . : . = 4 . xx . yy . y=x . 2 . = 3 5 . , : = = . 2
8
3
81
2000
4
3 f [0,1] f(x)>0 x(0,1). A f(0)=2 f(1)=4, : . y=3 f ' x 0
(0,1). 7 x 1 (0,1), f(x 1 )=
.
f (1 / 5) + f (2 / 5) + f (3 / 5) + f (4 / 5) 4
12
.
x 2 (0,1), f (x 2 ,f(x 2 )) y=2x+2000. 6
4 t=0 ' . t ,t 0 f(t)= 2 t 1+ t . 4
82
2000
5
15 6 . . . . 15 , 12 , . 10
5
83
2000
1
30 2000 TEX :
1 . ) ; 2,5
)
(x,y) , u = (, ) (x,y) u , x,y u . 5 ; N . 5
)
1. I .
1
84
2000
2
1. 1 1 1 1 2. 0 1 1 0 2 : /2 3. 1 0 0 1 3 2. = 1 2 - 2 1 , 1 , 2
1 , 2 , 1. ) . 4,5 ) : 2x - y + 5 = 0 . 5 2
1 : xx
. z =
5+ i 2 + 3i ) z + i, , R. 4 ) z . 5 ), ) . 2
85
2000
3
) = Argz, iz : - . + . . + . 4 2 2 3 ) z4 : . 4 . 4i . - 4i . -4
3 . , z, : z -1 =1 . z-i 10 3 f : x2 - 8x + 16 , 0< x < 5
f(x) = 2 2 5-x x5 ( + ) ln(x - 5 + e) + 2( + 1) e , . , 6 . , R, f
x 0 = 5. 10 . , lim f(x) .x +
x 5
lim f(x) ,
x 5 +
lim f(x) .
M 9
3
86
2000
4
4 . f(t) t , t 0 . f(t) 8 -2 t +1 ) f(t). 6 ) t, , ; 6 ) t = 8 ,
t = 10 . ( ln11 2,4). 13
4
87
