: 8 2015
:
http:// lisari.blogspot.com
2
00
0
20
14
30 2000
TEX :
1
. ) ; 2,5
) (x,y) , u (,) (x,y)
u , x ,y
u . 5
) ; N
. 5
1. I
.
1 :
x x
2 :
/2
1. 1 1
1 1
2. 0 1
1 0
3. 1 0
0 1
3
2. = 12 - 21 , 1 , 2
1 , 2 , 1 .
) . 4,5
) : 2x - y + 5 = 0 .
5
2
. 5 i
z 2 3i
) z + i , , R. 4
) z . 5
), )
.
) = Argz , iz :
.
- 4
.
2
.
- 2
. + 3
) 4 z : . 4 . 4 i . - 4i . -4 3
. , z ,
:z - 1
1 z - i
. 10
3
f :
2
2 2 5-x
x - 8x 16 , 0 x 5 f(x)
( ) ln(x - 5 e) 2( 1) e , x 5
. , x 5
lim f(x)
, x 5
lim f(x)
. 6
. , R , f x0 = 5. 10
. , x lim f(x)
. M 9
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
15 2000
TEX :
1
.1. z1 , z2 . : ______ __ __
1 2 1 2z . z z . z 5
.2. z1 = 1 (1 + i1) z2 = 2 (2 + i2)
,
, :
. 1
2
z
z 1. 12(1+2)+i(1+2)
. z1z2 2. 1 [(1)+i(1)]
. 1z 3. 1
2
(1+2)+i(1- 2)
4. 1
2
(1- 2)+i(1- 2)
5. 12(1+2)+i(1+ 2)
6.
1 [(1)-i(1)]
7,5
.1. .
:z1 = 2 (2 2
i 3 3
) z2 = 5 5
i 3 3
.
1
2
z z
: : 2 : 2i : -2 : -2i : 2 (1-i)
4,5
B.2. z = 1 + i . z1 6 . 8
2
. (2 - 2 - 1) ( + 2 - 3 )
( + 1) (3 - 2 + 2), .
. ,
5
. ,
10
. = 0 1
1 0
. : . 2 = -, 2
2 . 4
. 22004 + 2001 + 1999 = 2 , 2 2 . 6
3
f f(x) = 2x - 3x 2
x -
, .
. , f
x = 4. 5
. , f
(1,0) (-2,3). 10
. > 2, x0 (1,2) ,
f x0 xx. 10
4
,
1.000 .
. 4 25 .
200 .
4.000 .
,
10.000 .
. (x)
, : (x) = 10 (x + 16
x
+ 40) x
. 13
. ,
; 8
. .
. 4
9 2000
( )
1
. z= + i ,R .
z. K
A. Re(z)
. z
. z
. z z
. z + z
. z - z
Z. Im(z)
1. 2 2
2.
3. i
4. - i
5. 2 + 2
6.
7. 2
8. 2i
9. - + i
, ,
. 14
. z = 3-4i . :
) Re(z) Im(z) z
3
) z 4
) z z. 4
2
= 1 2
0 1
, = 1 0
0 1
.
) 2 = 2 - . 9
) (2 - ) = . 8
) X 2 X - = 2 . 8
3
f(x) = 2
x x 1
.
) x 1
limf (x)
. 12
) f . 13
4
600 .
, .
) (x)
(x) = -2x2 + 600x ( 0 < x < 300). 6
) x (x)
. 14
) . 5
x xE(x)
12 2000
:
1o
A1.A f ' x0 ,
f
(x0 , f(x0)). M 4
2. , f ' x0
, . 8,5
1.
.
. f x0 , f x0 .
. f x0 , f x0 .
. f x0 , f x0 . 4,5
2.
x0 .
. f(x)=3x3 , x0=1
1. y=-2x+
. f(x)=2x, x0=
2 2. y=
1 4
x+1
. f(x)=3 x , x0=0
3. y=9x-6
. f(x)= x , x0=4
4. y=-9x+5
5.
8
2
f(z)=2z i
, z C z -2i, z 2i
z o z.
. :
w1=f(9-5i) . . 6
. 1
1
02 M
3 0
w
w
1 w
w1 .
.
:
. 4
. x x
. yy
. y=x
. 2
. 3
5
. , :
=
. 2
8
3
f [0,1] f(x)>0
x(0,1). A f(0)=2 f(1)=4, :
. y=3 f '
x0(0,1). 7
. x1(0,1), f(x1)=(1 / 5) (2 / 5) (3 / 5) (4 / 5)
4
f f f f 12
. x2(0,1), f
(x2 ,f(x2)) y=2x+2000. 6
4
t=0 ' .
f(t)=2
1
t
t
, t0
t
. 15 6
.
. . 15
. ,
12 ,
. 10
16 2000
:
1o
A. f, .
. f(x)0 x , f
. M 8
. f(x)0 x , f ;
4,5
.1.
.
. f(x) =e1 -x
. 2,5
. f f(x) = -2x+2
1
x + 3, x
2,)
. 2,5
. f(x) = g(x) + 3 x, h(x)=f(x)-g(x)
. 2,5
.2. f
-2,6.
f
. 5
2
. z1 , z2 z2+2z+2=0, 20
1z - 20
2z = 0.
12
. z1 . ,
, 1z
. 8
-2 1 3 6x
y
.
z1 z2 . 5
3
f , ,
: 2x
x 0
f(x)- e 1
2xlim
= 5.
. f(0). 7
. f x0=0. 9
. -xh x e f(x) ,
f h (0,f(0)) (0,h(0)) .
9
4
( ) , t
, P(t) = 4 + 2
t-6
25t
4
.
. .
2
. , .
10
. .
8
.
,
. 5
12 2000
1
. . , :
= -1X A 6,5
. ( 1...1
1...2) , .
1.
.
0
. = = 0 .
0
.
0
. = = 0 3
2.
A
D
. ,
, :.-1
- 1 A
- D
.
-1 -
A -
.
-1 -
A D -
.
-1 1
A D
.
-1
A
3
. 2 3
A -2 4
1 0
B . 1 1
. 3
1 2 2
A 1 1
6
. = 6,5
2
z 1 = 7 + 8 i z 2 = 4 - 5 i .
. z 1 z 2 . 8
. 1
2
z
z. 8
. 1 2 z z z z
z4 . 9
3
:
2x x, x 1
x 1
f (x)
x 2 3, x 1 .
. f x0 = 1
13
. x 2 x 2
lim f (x), lim f (x).
12
4
100 , x
, 3 21 1
f (x) x x x 10 , 9 3
1< x < 5.
. x ,
. 13
. x 1 = 2
x 2 = 4 ( 2.000 4.000
) . 12
2 2001
:
1o
A.1. z1 , z2 . : z1 z2 = z1 z2 .
7,5
.2. ,
.
z :
. 2
z z z . 2 2 z z . z - z . z z . i z z 5
.1. 1 2 z 3 4 i z 1 - 3 i,
,
.
1. 1 2 z z . 4
2. 2
1 z . 2
3. 2
2 z . 25
4. 1 z . 5
5. 2 i z . 2
. 5
. 10
7,5
.2. z z 1, 1
z z
. 5
2
f :
2
x-3
x , x 3
f(x) 1-e , x 3
x 3
. f , = 1/9. 9
. Cf f
(4, f(4)). 7
.
f , x x x=1 x=2. 9
3
f , R,
: f3(x) + f2(x) + f(x) = x3 2x2 + 6x 1 x R , ,
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 2
0 1 - 2 x t f (xt)dt , x R.
g 21
g(x) - x f(x)
, x R.
. 2 f (x) - 2xf (x) 10
. g . 4
. f :2
1 f(x)
1 x
. 4
. x
lim
(x f(x) 2x). 7
E
5 2001
:
1o
A.1. f . F
f ,
:G(x)=F(x)+C, CR f
G f : G(x)=F(x)+C, CR
6,5
.2.
.
. f (x)dx . . . . . . f (x) g(x) dx
. . . . . . [f (x) g(x)]dx
. . . . .
,R f,g [,] 6
.1. f , f(x)=6x+4, xR
(0,3) 2. 6,5
.2.
. xe x dx
2
. 4 2
1
3xdx
x
2
. 2
0
2 x 3 x dx
2
2
. z
: z 16 4 z 1 9
. z
: z 1 z i 9
.
() (). 7
3
x , x 1
f (x) 1 e ln(x 1), x 1,2x 1
, R. .
. x 1
1 elim
x 1
x 1
7
. R f xo=1. 11
. =-1 (1,2) ,
f (,f()) xx.
7
4
f , (0,+) :
x2
1
tf t1f x dt x 0
x x
. f (0,+). 3
. f :1 ln x
f (x) , x 0x
7
. f. 6
. f. 4
.
f , x x x=1, x=e. 5
25 2001
:
1
. ) z1 = 1 ( 1 + i 1) z2 = 2 ( 2 + i 2)
, :
z1 z2 = 1 2 [ (1+2)+i (1+2)] 6,5
) z = + i , R , ,
,
.
A. Re(z)
. Im(z)
. -z
. z
. z
. z z
1. - - i
2. - i
3. +
4.
5. 2 2
6. 2 + 2
7.
6
. z1 = 1 + i z2 = i .
) z 1 z2 . 8
) z1 z2 . 4,5
2
f(x) = x2 - 4x + 3, x R .
) f xx
yy. 7
) f
(3, f(3)) . 9
) f. 9
3
f: RR , 2 - x4 f(x) 2 + x4 ,
x R . :
) f(0) = 2 6
) H f x0 = 0 . 9
) f x0 = 0 . 10
4
625 km x km .
90 km .
160 , 2x
5,5200
2000 .
) (x) : 1800000
K (x) 500 x ,x
0 x 90 . 12
) .
13
6 2001
:
1
. ) , z1 = + i z2=+ i , , , , R
, 1 2 1 2 z z z z 6,5
) z = + i, , R , ,
,
.
A. z z
. z z
. z z
. z
1.
2. 2
3. 2i
4.
5. 2 2
6. 22
6
. ) z1 = k + 15i z2 = 5+i, k , R .
k , z1=5 2 z . 6
) z , z z + ( z z ) = 5 +2i.
6,5
2
f(x) = x2 - kx + 1, x R .
) k, f
(1,0). 12
) f
(0, f(0)), k=17. 13
3
2
z 1 i
) z z=x+yi , x, y R. 8
) z. 8
) z
(2,0) = 2 . 9
4
2
1f (x)
1 x
, x R
) f(x). 5
) f. 12
) ( )
f. 8
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
. f x0 x x
0
lim f(x) 0 ,
x x
0
lim f(x) 0 .
1
. f R , f (x)dx xf (x) xf (x)dx
1
. x x
0
lim f(x) 0 ,
f(x) > 0 x0 . 1
2
z f() = i z , IN*.
. f(3) + f(8) + f(13) + f(18) = 0 . 7
. z= Arg(z) = , f(13) = i 2 2
.
8
. z= 2 Arg(z) = 3
,
0, z
f(13). 10
3
f, g R .
fog 1-1.
. g 1-1. 7
. :g(f(x) + x3 - 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)f (x) e x 1, x R f(0) = 0 .
i) f f. 5
ii) f(x) x f (x) ,x2
x > 0. 12
iii) f,
x = 0, x = 1 x x , 1 1
E f (1)4 2
. 6
E
8 2002
:
1o
A. z1 = 1(1 + i1) z2 = 2(2 + i2) ,
: z1 z2 = 12 [(1+2) + i(1+2)]. 15
. ,
.
.
f (x)dx 0 , f(x) 0 x[,]. 2
. f()
f . 2
. f IR. ,
[, ] , f
Rolle. 2
. f [, ]
x0[, ] f . f(x 0)=0.
2
. f [, ] x 0(, )
f(x0)=0, f() f()0. 2
2
x
x
e 1f x , x
e 1
. f f 1 .
10
. f 1 (x) = 0 . 5
. 1
21
2
f x dx 10
3
f, R. ,
22
22
x z x zf x
x z
z z = + i, ,R , 0.
. x x
f x , lim f (x)lim
. 8
. f, z 1 z 1 . 9
. f. 8
4
f, R. ,
: f(x)f(x) + (f(x ))2 = f(x)f(x) , xR. f(0) = 2f(0) = 1.
. f. 12
. g
[0,1],
x
2
0
g t2x dt 1
1 f t
[0,1]. 13
5 2002
1
. f, g x o ,
f+g x o :(f+g) (xo) = f (xo)+g (xo) 9
.
, , , , ,
.
1. f ' x o ,
. 2
2. f ' x o ,
. 2
3. f ' f(x ) = 0
x , f . 2
4. f ' f(x) > 0
x , f . 2
5. f g x o , :
o o ox x x x x x
lim f x g x lim f x lim g x( ) ( ) ( ) ( )
2
6. f g x o , :
o o o
x x x x x x
lim f x g x lim f x lim g x( ) ( ) ( ) ( )
2
7. z i :2 2z 2
8. i : i 4 = 1 . 2
2
z 1 = -1+i , z2 = 3-4i
. z 1+5z2 6
. 2
1
z
z 6
. z 1 : rg(z1)=3
4
6
. 81
z . 7
3
f(x) = x3- 6x2+9x-2 .
. f .
10
. f
A , f ( )1 1 . 5
. f(x) = 0 (0 , 1) .
10
4o
:
3
2
x 4x , x 2x 2
f x
x k , x 2
( )
kR . :
. k, f x0 = 2 , 7
. x 1lim
f(x) , 5
. f x 0 = 4 5
. f x
g xx 3
( )( )
.
8
29 2003
:
1o
A. , f x0 ,
. 8
. ;
7
. ,
.
. z _z , z z z
2
. f
. f(x)>0 x , f
. 2
. f , , f (x)dx f (x) c
,cIR 2
. f ,
f
. 2
. f x0
. f x0 f(x0)=0, f
x0 . 2
2
z=+i, ,IR w=3z _
i z +4, z
z.
. Re(w)=3+4 , m(w)=3. 6
. , w
y=x12, z y=x2.
9
. z ,
y=x2, . 10
3
f(x) = x5+x3+x .
. f f
. 6
. f(ex)f(1+x) xIR. 6
. f (0,0)
f
f 1. 5
.
f 1, x x=3. 8
4
f [, ]
(, ). f() = f() = 0 (, ), (,
), f()f() 0. 9
. f. 8
8 2003
:
1o
A. f . F f
, :
. G(x) = F(x) c ,c R f
. G f G(x) = F(x) c ,c R .
10
. ,
.
. z1 , z2 ,
1 2 1 2 1 2 z z z z z z . 2
. f ' (, ),
x0 , f .
f (x) > 0 (, x0) f (x) < 0 (x0 , ), f (x0)
f . 2
. f : R 1 1 , x1 ,
x2 A : x1 = x2 , f(x1) = f(x2) . 2
. f, g , :
f(x) g (x) dx f(x) g(x) f (x) g(x) dx . 2
. x = x0
f ; 7
2
. () z
: z 2 m (z) 0 . 12
. , z (),
1 4
w z 2 z
x x . 13
3
2f(x) = x 1 - x .
. x lim f(x) 0
. 5
. f , x
. 6
. 2 f (x) x 1 f(x) 0 . 6
. 1
2 0
1 dx ln 2 1
x 1
. 8
4
f IR ,
: f(x) = f (2 x) f (x) 0 x IR .
. f . 8
. f(x) = 0 . 8
. f(x)
g(x) f (x)
.
g xx ,
45 . 9
4 2003
:
1
. f(x) = x . f
R1 = IR {xx = 0} f(x) = 2
1
x. 10
.
, , (), , (),
.
1. z = x + yi , x, y ,
2 2 z x y .
2. x, y y = f(x ), f
x0 , y x
x0 f(x 0) .
3. f (, ),
x0 , f . f(x) > 0 (, x0) f(x) < 0
(x0 , ), f(x0) f .
4. z = x + yi , x, y ,
_
z = x + yi .
5. f g x 0 ,
0
0
0
x x
x x
x x
lim f xf (x)
lim g(x) lim g x
( )
( )
, 0x x
lim g(x) 0
. 15
2
2x - 3x
f(x) x - 2
, x IR {2} .
. x 0
f(x) lim
x. 7
. y = x 1
f + . 8
. f (2, +). 10
3
2x , x 5
f(x) 10x - 25, x 5
x0 = 5 .
. f x0 = 5. 5
. f x0 = 5 f(5) . 8
. f
(5, f(5)). 4
. f . 8
4o
z = x + yi , x, y i (i z)
w i z
z i . : .2 2
2 2 2 2
2x 1 - x - y w i
x (y 1) x (y 1)
, 8
. w , z
(0 , 0) 1 = 1 8
. z , w
(0 , 0) 2 = 1 . 9
27 2004
:
1
. f ' x 0
. f x 0
, f (x 0 )=0 10
. f x 0
; 5
.
.
.
. 2
. 0x x
lim f (x) l
, 0x x
lim f (x)
0x x
lim f (x) l
2
. f , g x 0 , f g
x 0 : ( f g) (x 0 ) = f (x 0 ) g(x 0 ) 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 ) , f IR
f (0)=f(3
2) = 0 .
. (0, 3
2) f ( )=f( ) .
8
. f (x )=2x 2 3x, () = 0
g(x)dx , IR
8
. lim ()
9
4
f : IR IR f (1)=1. x IR ,
3 x
1
1g(x) z f (t)dt 3 z (x 1) 0
z z=+iC, , IR * , :
. g IR
g . 5
. N 1
z zz
8
. Re(z 2 ) = 1
2
6
. A f (2)=>0, f (3)= >, x 0 (2 ,3)
f (x 0 )=0. 6
5 2004
:
1o
A. f .
f
f(x) = 0 x ,
f . 9
. ,
.
. f x 0 ,
. 2
. .
2
. f , g IR fog
gof , . 2
. C C f f 1
y = x xOy xOy. 2
. f x 0 , 0 0
kk
x x x xlim f(x) lim f(x)
, f(x) 0
x0 , k k 2. 2
. f
(, ) [, ]. 6
2
f: IR IR f(x) = 2 x + mx 4x 5x , m IR , 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) |.
1
z f ()z
2 22
1z f ()
z , :
. |z |= 1 11
. f2() < f2() 5
. x 3f() + f() = 0 (1, 1).
9
4
f [0, +) IR , 12
2
0
xf(x) 2xf(2xt)dt
2 .
. f (0, +). 7
. f (x) = ex (x + 1). 7
. f(x) [0, +). 5
. xlim f(x)
xlim f(x)
. 6
4 2004
:
1
. + i, +i , , , , IR +i 0,
: 2 2 2 2
i i
i
9
. ,
( ).
I
. i1
B. i2
. i3
. i4
1. i
2. + 1
3. i
4. 1
5. 0
6. 4
,
. 4
, , ,
, , ( ), ,
( ), .
. f , g . f , g
f (x) = g(x) x ,
c , x : f(x) = g(x) + c. 3
. f
, x 1 , x2 x 1 < x 2 : f(x 1 ) < f(x 2 ) .
3
. f(x) = x . H f (0,+)
2
f (x)x
3
. , , (x 0 , f (x 0 )) ,
C f f , x 0
= f (x 0 ) . 3
2
, 24x 3, x 1
f(x)6x k, x 1
, k IR .
. k, f x 0 = 1. 10
.
f ( 1, f(1)) . 8
. , : f(5) + f (5) + 34 = 0.
7
3
f(x) = 2x 3 3x2 + 6x + , x IR ,
. f x 0 = 2
f(2) = 98.
. = 6 = 54 . 6
. f . 9
. f . 4
. f(x) = 0
(1, 2) . 6
4
z = x + yi , x , y ,
IR :
2 2
z z z zi (1 )i
2 2i
:
. Im(z) = 0, = 1. 5
. = 0, z 2 + 1 = 0. 5
. : 0 1 . 7
. z
, . 8
7 2004
:
1
. 1z = + i 2z = + i , ,,, IR , ,
: 1 2 1 2z z z z
,,,
(), , (), .
. 1(, ) 2(, ) + i + i
, +i
+i . 3
. z = + i, , IR, z = + i . 3
. f(x) = x , x IR . H f
f (x) = x. 3
. f .
f
f(x) = 0 x ,
f . 3
. f, . f (x) < 0
x , f .
3
2
: 2f x 4x 1 2x , x IR .
) :
i)
x 0
f x 1lim 2
x
10
ii) f (0)= 2 f (0). 5
) : xlim f x
10
3
2x x 3
f xx 2
, xIR{2} , ,
. Cf f (1, 4)
f(3) + 3 f(1) = 0.
) = 1 = 0. 9
) f
(1,4) . 8
) y = x + 2
f + . 8
4
z = x + yi , x, y ,
k IR : x = 3k y = 2k+1 .
:
) 3 Re(z) + 4 Im(z) = 3, k = 2. 9
) z 1 5 , z 10 . 10
) z
, . 6
31 2005
:
1
.1 f, [, ].
f [, ]
f() f()
f() f() ,
x0 (, ) , f(x
0) = . 9
.2 y = x +
f +; 4
. ,
.
. f [, ] f() < 0 (, ) f() = 0,
f() > 0. 2
. 0x x
lim f(x) g(x)
0x x
lim f(x)
0x x
lim g(x)
2
. f f - 1
f
y = x ,
f - 1
. 2
. 0x x
lim f(x)
= 0 f (x ) > 0 x 0 , 0x x
1lim
f(x)
2
. f
, x f(t) dt f(x) f() x . 2
. f ,
x x ,
. 2
2
1z , 2z , 3z
1 2 3z z z 3 .
. : 1
1
9z
z 7
. 1 2
2 1
z z
z z . 9
. : 1 2 3 1 2 2 3 3 1
1z z z z z z z z z
3 . 9
3
f f(x) = e , > 0.
. f . 3
. f,
, y = ex.
. 7
. () ,
f, yy,
() =e 2
2
. 8
. 2
()lim
2
. 7
4
f IR ,
2 f(x) = ex f ( x )
x IR f(0) = 0.
. : f(x) = x1 e
ln2
. 6
. :
x
0
x 0
f(x t) dtlim
x
. 6
. : h(x) = x
2005
x t f(t) dt
g(x) =
2007x
2007 .
h(x) = g(x) x IR . 7
. x
2005
x t f(t) dt
=
1
2008 (0 , 1).
6
6 2005
:
1
.1 f f(x) x . f
(0,+) : 1
f (x)2 x
9
.2 f : A IR 1 -1; 4
. ,
.
. , f
0 , f .
2
. f (, )
x o . f ( , x o ) (x o , )
, (x o f (x o ) )
f . 2
.
. 2
. f , g fog gof ,
fog gof . 2
. z , z
x x . 2
. f IR * , :
f(x)dx f(x)dx 2
2
. 1z , 2z 1z + 2z =4+4i 2 1z - 2z
= 5+5 i , 1z , 2z . 10
. z ,w
z 1 3 i 2 w 3 i 2 :
i . z , w , z = w
10
i i . z w . 5
3
f , IR f (x )0
x IR .
. f 1 -1. 7
. C f f (1 ,2005)
( -2,1) , 1 2f -2004 f(x 8) 2 . 9
. Cf,
Cf
( ) : 1
y x 2005668
. 9
4
f : IR IR , 2x 0
f(x) xlim 2005
x
.
. :
i . f (0)=0 4
i i . f (0)=1. 4
. IR , :
22
22x 0
x f(x)lim 3
2x f (x)
. 7
. f IR f (x )>f(x )
x IR , : i . xf(x )>0 x0. 6
i i .
1
0
f(x)dx f(1) . 4
8 2005
:
1
. 1. , f x ,
. 12
2. (x,y) z = x+yi .
z; 3
.
(), , (), .
1. f : R. 1-1, x1, x2
: x1 x2, f(x1) f (x2). 2
2. f xA () , f(x),
f(x) < f (x) xA. 2
3. f, g x f(x) g (x) x ,
0x xlim f(x)
> 0x x
lim g(x)
2
4. z1 z2 , 1 2 1 2z z z z 2
5. f [, ]
(, ) , , (, ) , :
f() = f()-f()
2
2
: z = 2- 2 + (3-2)i , R w = k+4i, k > 0.
z, w : Re(z) + Im(z) = 0 w = 5.
. z = -1+i. 8
. k = 3. 8
. R , z z 3i w 9
3
f(x) = x 3+kx2+3x-2, xR , kR ,
(1,1). :
. k = -1. 5
. f . 10
. f(x) = 0 (0, 1). 10
4
22 x kx 2
f(x)x 3
, k R x 3.
. y = x
f +, = 1 k = 3. 10
. (1, 2),
f xx.
8
. f
x = 1. 7
8 2005
:
1
. 1. f . f
f (x ) = 0 x , f
. 12
2. R.
; 3
.
() , , () ,
.
1. z = x+yi , x , y R. , : z z . 2
2. z = +i , : z z , , R . 2
3. x 0 , 2x 0
1lim
x . 2
4. f (x ) = x. f
R 1 = R. {x / x = 0} :
2
1f (x)
x . 2
5. f x0 R, :
o ox x x xlim k f(x) k lim f(x)
k R . 2
2
x 3i
z2 i
, x R .
. x , z . 10
. x = 6, z . 6
. x = 4 , z . 9
3
f :
3
4
, x 1x 1
f(x)
x 1 , x 1
. f . 6
. f . 10
. , f Rol le
[ 1,2] . 9
4
2kx x
f(x)4
, x R,
(0,0) = 1 .
. k = 4 . 7
. f , .
8
. (2 ,4) ,
f
, (2 , f (2) ) (4 , f (4) ) . 10
27 2006
:
1o
A.1 f , .
:
f (x )>0 x , f
.
f (x )
i i .
f f - 1 . 7
3
1 2 3z ,z ,z 1 2 3z z z 1 1 2 3z z z 0
. :
i . 1 2 3 1 2 3z z z z z z . 9
i i . 2
1 2z z 4 21Re(z z ) 1 8
. 1 2 3z ,z ,z ,
. 8
4
x 1
f(x) ln xx-1
.
. f . 8
. f (x )=0 2 .
5
. g(x )=lnx
(, ln) >0
h(x )=e x
( ,e ) IR ,
f (x )=0. 9
. g h
. 3
5 2006
:
1o
A.1 : (x)= x , x IR . 10
.2 f .
f ; 5
B. ,
.
. z 1 , z 2 , : 1 2 1 2z z z z 2
. f , g x g(x )0 ,
f
g x
:
o o o o
o 2
o
f f(x )g (x ) f (x )g(x ) x
g g(x )
.
2
. x0 1
ln x x
2
. f : IR 1 1 , y
f (x )=y x . 2
. f [ ,] . G
f [ , ] ,
f(t)dt G() G() 2
2
x
x 1
1 ef(x)
1 e
, x IR .
. f IR . 9
. 1
dxf(x)
. 9
. x
z , (4 z)1 0
= z 1 0
f f (x ) = x2
+x+, IR .
. z x=2.
7
. ( ) f
x=2 yy y =3,
i . ( ) . 9
i i .
f , ( ) , x x
3x
5 . 9
4
f(x) xln(x 1) (x 1)lnx x>0.
. i . : 1
ln(x 1) lnx , x 0x
.
i i . f (0 ,+). 12
. x
1lim xln(1 )
x . 5
. (0 ,+)
(+1)
= + 1 . 8
31 MA 2006
:
1
. 1 2z ,z , : 1 2z z = 1 2z z . 7
.
, , ,
.
1. f x 0.
f (x )0 x. 0x x
lim f(x)
0x x
1lim
f(x) . 3
2. , . (, )
(,) z i z i
. 3
3. f x 0 ,
x 0 . 3
4. f(x) x = [0 , +) , 1
f (x)x
x (0 , +) . 3
5. 0x x
lim f(x)
,0x x
lim f(x)
+ ,
0x x f . 3
6. f , g .
f , g
f (x ) = g(x ) x ,
c , x : f (x ) = g(x ) + c .
3
2
x 2
4x + 13 = 0 (1)
. (1) . 9
. 1 2z ,z (1) ,
2 2006
1 1 2 2 z 2 z z 13 z i . 9
. z1
= 2+3i ,
z : 1z z 5 7
3
2
3x , x 1
4f(x)
x 8x 4, x 1
4x
R.
. R. f
x 0 =1. 10
. =0
. f R . 7
. f
+ . 8
4o
k R 3 2f(x) 2x kx 10 x R
. k R
f (1 , f (1)) x x .
5
. k = 3
. f . 8
. f ( , 0] . 5
. ( 14 ,15) f (x ) = 5
(0 ,1) . 7
24 2007
:
1
A.1 z1 , z2
, : 1 2 1 2z z = z z . 8
.2 f, g ; 4
.3 y f
+; 3
B. ,
, , ,
, .
. f [, ] x [, ] f(x) 0
f(x)dx 0 . 2
. f
x . f
f(x) > 0 x . 2
. f x 0 g x 0 ,
gof x 0 . 2
. f ,
g(x)
f(t)dt=f g(x) g (x)
. 2
. > 1 xxlim 0
. 2
2
2 i
z 2i
.
. z (0,0)
=1. 9
. z1 , z2
2 i
z 2i
= 0 =2
.
i . z 1 z2
. 8
ii . 2v v
1 1z z . 8
3
f(x) = x3 3x 22 IR
+2
, Z
. f ,
. 7
. f(x) = 0 .
8
. x1 , x2 x 3
f, (x 1 , f(x1)), B(x2 , f(x2)) (x 3 , f(x3))
y = 2x 22. 3
.
f y = 2x 22. 7
4
f [0, 1]
f(0) > 0. g [0, 1]
g(x) > 0 x [0, 1].
: F(x) =x
0f(t)g(t)dt , x [0, 1], G(x) =
x
0g(t)dt , x [0, 1].
. F(x) > 0 x (0, 1]. 8
. : f(x) G(x) > F(x) x (0, 1]. 6
. : F(x) F(1)
G(x) G(1) x (0, 1]. 4
. :
2x x2
0 0
xx 0 5
0
f(t)g(t)dt t dt
lim
g(t)dt x
7
3 2007
:
1
A.1 f x0,
. 10
.2 Rolle ; 5
. ,
.
. 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
. f
(, ), (,) =
x a
lim f x
= x lim f x
2
. f, g . f , g
f(x) = g(x) x , f(x) = g(x)
x. 2
2
2
3x, x 0
x
f x
x x x ,x 0
. x 0
lim f x 3
8
.
f ' 2
f x
0=0,
= = 3. 9
. = = 3,
0
f(x)dx . 8
3
f(x) = ex
e lnx, x > 0.
. f(x) (1, + ).
10
. f(x) e x > 0. 7
.
2 2
2 2
x 2 x 2 4
2x 1 x 3
f(t)dt = f(t)dt f(t)dt
(0, +). 8
4
z1
= +i 121
2 zz
2 z
, IR 0.
z2
z1 IR .
. z2
z1
= 1. 9
. z1
. 6
. 21z >0, 1z
2020
1 1z 1 i z 1 i 0 10
30 MA 2007
:
1
. 1. f x x , {0,1}. f
R 1f x x 10
2. N f
. 5
.
(), , (), .
1. z z z z . 2
2. f 1-1, ( xx)
. 2
3. f x0R
0x xlim f x 0
, f(x)0.
.
i) x
f ' xlim
f x ii)
2
x 2
xf xlim
x 2 8
. N f (0,0)
. 9
. N f,
y=-2x+6. 8
4
f, R. A x0 xf(x)=x+2x, :
. f(0). 7
. f(x)
4 2007
:
1
. 1. : f,g x0
, f + g
x0
: (f+g) (x0) = f(x
0) + g(x
0). 12
2. f g ; 5
.
() , , () ,
.
1. +i +i
. 2
2. f , xx,
f. 2
3. f, g, h h (g f), (h g) f h (g
f) = (h g) f. 2
4. 2 . 2
2
z z 1 i iz .
. i) M z. 10
ii) (0,0) 5 . 10
. Re(z)=0, z=i. 5
3
:
2
2
1 1x , x 2
8 2f(x)
x 5x 6, x 2
2 x 1
. f x0=2. 12
. f
(0,f(0)). 6
. y =1
2x-2
f +. 7
4
f, R, 3 3 2x f x 8x 12x 8x 2f ,
xR.
. f 1-1. 8
. f(x)=0 (0, 1). 9
. g: RR 2f g x 3x f x 2 , xR ,
x0 g . 8
24 2008
:
1
A.1 f (x) ln x , x IR* IR*
: 1
ln xx
10
A.2 f [, ];
5
. ,
, , ,
, .
. f:A IR 11, f -1
:
1f (f (x)) x , x A 1f (f (y)) y , y f(A) 2
. f
f . 2
. z 2+z+=0 ,,IR 0 ,
C . 2
. f IR
, f( x ) > 0 x.
2
. 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 . 6
. w . 7
. w 6
. z w 6
3o
x ln x, x 0
f (x)0, x 0
. f 0. 3
. f
. 9
.
xx e
. 6
. f(x+1) > f(x+1)f(x) , x > 0 . 7
4
f 2
3
0f (x) (10x 3x) f (t)dt 45
. f(x)=20x 3+6x45 8
. g IR .
h 0
g (x) g (x h)g (x) lim
h
4
. f () g ()
2h 0
g(x h) 2g(x) g(x h)lim f (x) 45
h
g(0)=g(0)=1,
i . g(x)=x 5+x3+x+1 10
ii . g 11 3
3 2008
:
1o
A. [, ]. G
f [, ],
f(t)dt G()- G() 10
. ;
5
. ,
, ,
, .
. 11, . 2
. f ,
f
, . 2
.
f(x)dx
xx
xx. 2
. , , : +i=0 =0 =0 2
. (, x 0) (x0 , )
. : 0 0x x x x
lim f (x) lim (f (x) ) 0
2
2
1
1 i 3z
2
z 2+z+=0,
.
. =1 =1. 9
. 31z 1 8
. w,
: 11w z z 8
3
f(x)=x2 2lnx, x > 0.
. : f(x)1 x>0. 6
. f. 6
.
ln x , x 0
f(x)g(x)
k , x 0
i . k g . 6
i i . 1
k2
, g , , (0,e).
7
4
f [0, +) f(x) > 0
x 0. :x
0F(x) f(t)dt ,x [0, +), x
0
F(x)h(x)
tf (t)dt
,x(0,+)
. 1
t 1
0 e [f (t) F(t)]dt F(1) 6
. h (0, +).
8
. h(1)=2, :
i . 2
0 f (t)dt
2
02 t f (t)dt 6
ii . 1
0
1 F(t)dt F(1)
2 5
2 8 2 00 8
:
1
.1. z1 = + i z 2
= + i ,
1 2 1 2z z z z . 7
2. f x . f
x ; 6
.
, , ,
.
1. z1 , z2 , : 1 2 1 2z z z z . 3
2. x IR : (x) = x. 3
3. f ,
x x,
. 3
4. f
[, ]
(, )
f() = f()
, , (, ) , : f () = 0. 3
2
3z 2
+ z + = 0, , .
. z1 = 1 + i , = 6, = 6
z 2 . 14
. : . 2 2
1 2z z 0 6
.2008 2008 1005
1 2z z 2 5
3
f 2
1 x , x 1f (x)
(x-1) , x 1
A. f :
. x = 1 8
. x = 1. 10
. f
(2, 1). 7
4o
f 2x 2x k
f (x)x
, k .
. f. 3
. f (1, f(1))
xx, k. 8
. k = 1,
. f. 8
. f [1, +). 6
20 2009
:
1o
. f . f
x , f . 10
. f x0 ; 5
. ,
, , ,
.
. z1, z
2 , 1 2 1 2z z z z 2
. f () x0A,
f(x)f(x0) xA 2
. x 0
x 1lim 1
x
2
. f
. 2
. f [, ] f(x)
xf (x) a ln(x 1), x>-1 >0 1
A. f (x) 1 x>-1 = e 8
. = e,
. f . 5
. f 1,0
0, 6
. , 1 0 0, , , f () 1 f () 1 0x 1 x 2
(1, 2) 6
4
f [0, 2] 2
0t 2 f (t)dt 0
x
0
x
0
2
2t o
H(x) tf (t)dt, x 0,2
H(x)f (t)dt 3, x 0,2
xG(x)
1 1 t6lim , x=0
t
. G [0, 2]. 5
. G (0, 2)
2
H(x)G (x) , 0
9 2009
:
1o
A. f(x) = x . f (0 , +) :
1
f (x)2 x
9
B. f xo . f
xo ; 6
. ,
, , ,
.
. z z z 2
. f 1-1, f
. 2
. ox x
lim f (x) 0
f(x) < 0 xo
ox x
1lim
f (x) 2
. f(x) = x. H f 1R R x / x 0
2
1f (x)
x 2
. f, , f (x)dx f (x) c , x c
. 2
2
z : (2 i)z (2 i)z 8 0
. N z = x+yi
. 10
. N 1z 2z
. 8
.
2 2
1 2 1 2z z z z 40 7
3
f(x)=ln[(+1)x2
+x+1] - ln(x+2), x > 1 -1.
. , xlim f (x)
.
5
. = -1
. f . 10
. f 6
. f(x) + 2
= 0 0
4
4
f:[0,2]R
2xf (x) 4f (x) 4f (x) kxe , 0 x 2 , f (0) 2f (0) , f(2) = 2 f(2)+12 e4
, f(1) = e2
k .
. 22x
f (x) 2f (x)g(x) 3x , 0 x 2
e
Rolle [0,2]. 4
. (0,2) , f () 4f () = 6 e2
+ 4 6
. k = 6 g(x) = 0 x [0,2]. 6
. 3 2xf (x) x e , 0 x 2
5
. 2
2 1
f(x)dx
x 4
26 MA 2009
:
1
. 1. x = x0
f ; 5
2. f , g x 0 ,
f + g x 0 :(f+g)(x0)=f(x0)+g(x0)
8
B.
, ,
, .
1.2 2z = z , z. 3
2. +i, ,
( ,). 3
3. 0
limx
x=0
x. 3
4. f [, ]
(, ), (, )
, : f() - f()
f () =-
. 3
2
1z =2+3i 20092
2z =(1- i) +3i +1
. z 2 = 1 + i . 8
. 1 2z z . 7
. 1
2
z
z +i , , . 10
3
2x + , x 1f(x) =
2x +3 , x >1 , .
. f x0 = 1, + =5. 5
. f x0 = 1 , = 1
= 4. 10
. = 1 = 4 ,
f(x)
g(x) =x
, x0 + . 10
4
3 2f(x) =x +x -3x +1, x .
I. f x 0 = 1 ,
. 4
. = 0
. f . 8
.
f y = 9x. 8
. 0f(x) - x
( 0 , 1 ) . 5
( )
19 2010
:
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
.
) f
. f ,
.
) f
(, ), (,),
x A lim f (x)
x B lim f (x)
) (x)= x, x
) 0xx
lim f (x) 0
, f (x) 0 x 0 10
2
z 2z
z z 0
B1. z 1 z2 . 7
B2. z1 2 0 1 0 +z2 2 0 10 =0 6
B3. w 1 2w 4 3i z z
w . 7
B4. w 3 , 3 w 7
5
f(x)=2x+ln(x 2+1), x
1. f. 5
2. :
2
2
4
3x 2 12 x 3x 2 ln
x 1
7
3. f
f .
6
4.
1
1
xf (x)dx
7
f : x :
f (x) x x
0
tf (x ) x 3 dt
f ( t ) t
1. f
f (x)f (x) , x
f (x) x
5
2. 2
g(x) f (x) 2xf (x) , x , .
7
3. 2f (x) x x 9, x 6
4.
x 1 x 2
x x 1
f (t)dt f (t)dt, x
7
7 2010
:
A1. f(x) = x, xIR, IR
(x)= x. 8
A2. f [, ]
; 4
A3. f x0A () , f(x0);
3
4. ,
, , ,
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z 1 , z2
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= 5
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5
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w
(x+1) 2
+ y2
= 4 8
B3. w 2
2 Re(w) + Im(w) = 0 6
B4. w1 , w2 w 2
|w1 w2 |=4, |w1 + w2 |=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. x1 , x2 3 x1
< x2 ,
(x1 , x2) , f() f() = 0
f (, f())
. 9
f: IR IR IR f(0)=1
f(0)=0
1. f(x) 1 x 4
2.
13
03x
x f(xt)dt +xlim
x0 f(x) + 2x = 2x(f(x)+x2) ,
x, :
3. 2x 2f x =e x- , x 8
4. x+2
xh(x) = f(t)dt , x 0
2
2
x +2x+3 4
6x +2x+1
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( )
25 MA 2010
:
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. 10
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f (x) 1lim
x
5
2x 3
f (x) 2x, x 0x
. :
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2. f. 8
3. f
(1, f(1)). 4
4. (, f()), >0, C f f,
Cf
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8 2010
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f
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0
0
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x
2
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Q(x),
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f: [, ] , ,
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2(+) 5
z 26z+=0 ,
z1 , z2 Im(z1) > 0 |z1 | = 5.
1. =25. 8
2. =25, . 5
3. w |w z1 | = |w z2 | , w .
6
4. (z123i)8 + (z24+5i)8 . 6
f(x) = (x+3) 29 x-
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2. f:
. (3, 3) ( 3)
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3. f. 9
4. f. 6
( ) 16 2011
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f x0 ,
: f (x0) = 0 10
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f + ; 5
A3. ,
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, .
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: x1 x2, f(x1) f(x2)
) x R1= R {x | x=0} : (x)=2
1
x
) : x
x
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=1
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y=x xOy x Oy . 10
z w z3i , :
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z 3i
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z 3iz 3i
4
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f : R R , R , f(0)=f(0)=0,
: xe f x f x 1 f x xf x x R.
1. : xf x ln e x x R 8
2. f . 3
3. f .
7
4. xln e x x
0,
2
7
f, g : R R, x R
:
i) f(x) > 0 g(x) > 0
ii)
2tx
2x 0
1 f x edt
e g x t
iii)
2tx
2x 0
1 g x edt
e f x t
1. f g R f(x) = g(x)
x R . 9
2. : f(x) = ex, x R 4
3. :
x 0
ln f x
1f
x
lim
5
4.
x
2
1
F x f t dt x x y y x=1.
7
6 2011
:
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( x) = x 10
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f . 5
A3. ,
, , ,
.
i. z = + i, , z z =2
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f(x) f(x0) xA
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.
iv.
0x x
=0lim f(x) f(x)>0 x0,
0
x x=
1lim
f(x)
v. f x0
. 10
z, w, : z-i =1+m(z) (1)
w w 3i i 3w i (2)
B1. z
21
y = x4
7
B2. w
(0, 3) =2 2 . 7
B3. ,
z, w z =w. 5
B4. N , ,
u ,
, , , . 6
y = x , x0.
(0, 1) xy
, .
t, t0
x(t)=16m/min.
1. , t, t0 :
x(t)=16t 5
2.
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6
3.
. 6
4. t0
10,
4 d=()
. 7
xy.
f : , 3 , :
i)
x 0
=f(x)
lim 1 f(0)x
ii) f(0) < f(1) f(0)
iii) f(x) 0 x
1. f
x0=0. 3
2. f . 5
g(x)=f(x) x, x :
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xlim
xg(x) 6
4. 2
0f x dx >2 5
5. g,
xx x=0 x=1 ()=e5
2
1
0f x dx (1, 2) , 6
( ) 16 2011
:
1. f x0 . f
x0 , :
f(x0) = 0 10
2. f . y=x+
f +; 5
3. ,
, , ,
.
i. z 0 z0=1
ii. f:A 1-1, x1, x2A
: x1x2, f(x1) f(x2)
iii. x 1= {x/x=0} : 21
x = x
iv. :
x
xlim 1
x
v. C C f f-1
y=x xOy xOy. 10
z w z3i, :
z 3i z 3i
1w =z 3i
z 3i.
B1. z 7
B2.
1z 3i =
z 3i 4
B3. w 2 w 2 8
B4. : z w z 6
22
f(x) =xx
, x0
1. f . 6
2. f
A 2,f(2) . 6
3. f. 6
4. :
2x 1
1f 3
xlim
x 1 7
f:, f(0)=0, f(x) xf (x) =x
x.
1. g(x) =xf(x) x , x . 6
2. : 1 x
f(x) =x
, x x0 6
3. 1 x xx
, 3
2 2
6
4. (0,) : 22
2
7
6 2011
:
1. f(x)=x
x (x) = x
10
2. (x,y) z=x+yi
.
z
5
3. ,
, , ,
.
i. z=i, , zz=2
ii. f
x0A () f(x0), f(x) f(x0) xA
iii. f
, 11 .
iv.
0x x
=0lim f(x) f(x)>0 x0,
0x x
=1
limf(x)
v. f x0
.
10
z, w,
:
z-i =1+m(z) (1) w w 3i i 3w i (2) 1.
z 21
y = x4
7
2.
w (0, 3)
=2 2. 7
3. ,
z, w z =w.
5
4. u=i ,
, , , .
6
y = x, x0.
(0, 1)
xy ,
.
t, t0 x(t)=16m/min.
1. ,
t, t0 : x(t)=16t
5
2.
(4, 2) ,
, .
6
3. y(t)
t, t>0
4m/min.
6
4. t0
10,
4
d=()
. 7
xy.
2
1f(x) =
x x , .
f
52,
12
5
18.
1. =1 =4.
5
2. f
.
6
3. f.
7
4. :
x3+(14)x2x+4=0 (1)
f(x)=, , , (1)
. 7
( )
28 2012
:
A1. f . f(x) > 0 x ,
f
7
A2. f [, ];
4
A3. f . f
x0A ; 4
A4. , ,
, , , ,
.
i.
ii. f 1-1,
y f(x)=y
x.
iii.
0x x
lim f(x)= , f(x)
B3. w
2 2x y
19 4
|w|. 6
B4. z,w (1) (2) :
1 |z w| 4 6
f(x) = (x 1) nx 1, x>0
1. f
1=(0,1]
2=[1,+). f. 6
2. x-1 2013x e , x>0 .
6
3. x1, x1 x1< x1
2, x 0( x1, x2) ,
f( x0) + f(x0) = 2012
6
4.
g(x) = f(x) + 1
x>0, xx x=e.
7
f:(0,+), x>0 :
f(x) 0
2 2x -x+1
1
x xf(t)dt
e
x
1
nt tnx x= dt e f(t)
f(t)
1. f
.
10
f(x) = ex(nx x), x>0, :
2. :
2
x 0
1lim f(x) f(x)
f(x)
5
3. nxx1, x>0,
x
F(x) f(t)dt , x>0
>0, ( 2).
:
F(x) + F(3x) > 2F(2x), x>0 ( 4).
6
4. >0.
0(,2) : F() + F(3) = 2F()
4
( ) 28 2012
:
1. f
. f(x) > 0 x
, f
7
2. f
[, ];
4
3. f . f
x0A ; 4
4. ,
,
, ,
, , .
i.
ii. f 1-1,
y f(x)=y
x.
iii.
0x x
lim f(x)= , f(x)
9
3.
w
= 1. 8
22
f(x) x x
, x>0 ,
1.
14 2012
:
1. f (,
), x 0, f
. f(x)>0 (, x0) f(x)
4. u,
i
u ui = ww
, w0 x2y2=1
6
f:, : xf(x)+1= ex, x.
1.
x
e 1
, x 0f(x)= x
1 , x =0
6
2. o f 1
.
6
3.
f (0,f(0)). ,
f ,
2f(x)=x+2, x .
8
4.
x 0
lim x( nx) n f(x)
5
f:A A=(0,+) :
f() = (,0]
f (0,+ ),
x
f(x) f(t)
1
1 12f(x) x e = e f (t) t dt 2
x t
x
1F(x) f(t)dt , x>0
1.
2
2xf(x) n
x 1, x>0
8
2. F
(x0,F(x0)), x0>0,
.
(x0,) >x0, F M(,F())
: F()x(1)y2012(1)0 6
3. >1,
35F() 1 )f() x 1) x 10
x 1 x 3
, x, (1,3)
5
4.
2 x x
x 1
tf dt tf t dt
x, x>0
6
14 2012
:
1. z1 , z2 , :
|z1z2|=|z1||z2| 7
2. f g ;
2
3. Rolle.
6
4. ,
,
, ,
, , .
i. f , xx, f.
ii. +i
+i .
iii. P(x)=x+-1x-1+1x+0
0 : 0xlim P x
iv. f x0,
x0
v. f (, ),
x0, f
. f(x)>0 (,x0) f(x)
f : ,
3
2
x , x 1f x
x , x 1
,.
x0=1
1. 2 2 = 1 6
2. 11, f(x)=0
[1,1] 8
3. f x0 = 1, a
6
4. =5
4 =
1
2
f (1, f (1))
5
(1,1) f
f(0) = 3 21
g x f x1 x
= , x(1,1) g(x)
x3, x(1,1), f f
(1,1 ).
1.
f g x0=0
.
6
2. g(0)=
f g
x0=0 y=x3 8
3. f (x)=, x(1,1), 0
4
4. f(x) x3, x(1,1) 7
( )
27 2013
A1. f [, ] . G
f [, ] , : f (t)dt
=G () G(). 7
A2. (...) 4
A3. f [, ] ;
4 A4. , , , , .
) oz z , > 0 K (z o)
2 , z, zo .
) ox x
lim f (x) 0
< , f (x) < 0 x o
) : x x xR
) : x 0
x 1lim 1
x
) f f .
10
z :
(z 2)( z 2) + z 2 = 2.
B1. z , K(2,0) = 1. ( 5) , z ,
z 3 . ( 3)
8 B2. z 1 , z2 w2 + w + = 0, w , , R ,
1 2Im(z ) Im(z ) 2
: = 4 = 5 9
B3. o , 1 , 2 1 . v :
v3 + 2 v2 + 1 v + 0 = 0 : v 4
8
f,g :R R , f :
(f (x) + x) (f (x) + 1) = x , x R
f (0) = 1
g (x) = x3 + 23x
2 1
1. : f (x) = 2x 1 x , x R 9
2. f (g(x)) = 1 8
3. x 0 0,4
, :
o
0
x4
f (t)dt = f (x0 4
) x0
8
f : (0, + ) R :
f (0, + )
f (1) = 1
h 0
f (1 5h) f (1 h)lim 0
h
g (x) = x f (t) 1
dtt 1
, x (1, + ) > 1
:
1. f (1) = 0 ( 4), f x 0 = 1
( 2).
6
2. g ( 3), ,
R 2 4
2 4
8x 6 2x 6
8x 5 2x 5g(u)du g(u)du
( 6)
9
3. g ,
( 1) x f (t) 1
dtt 1
= (f( ) 1) (x ) , x > 1
.
10
13 2013 :
1. f x 0 , f
. 7
2. Fermat. 4
3. f . f;
4
4. , , , , , , .
) z z z
( 2)
) f 1 1 , f .
( 2)
) 0x x
lim f (x)
, 0x x
lim f (x)
( 2)
) f, g x 0 : (f g)(x0) = f(x0)g(x0) f(x0)g(x0)
( 2)
) f , f .
( 2) 10
z w
22x w 4 3i x 2 z , x
, x = 1 1. z
1= 1, w (4,3) 2= 4.
8
2. , .
5
3. z, w 1 :
z w 10 z w 10
6
4. z 1 , :
22z 3z 2zz 5
6
f: :
22xf (x) x f (x) 3 f (x) x
1
f (1)2
1. 3
2
xf (x) , x
x 1
f 6
2. f 1.
4 3. :
2 3 2 2f 5(x 1) 8 f 8(x 1) 7
4. , , (0, 1) , :
3
2 3
0f (t)dt 3 1 f ( )
8
f: [0,+) , ,
[0,+), :
2
1 1
u(f (t))
dt
f (t)
duf (x) x
x > 0
f (x)f (x) 0 x > 0 f (0) = 0
:
f (x)g(x)
f (x)
x>0
3h(x) f (x) x0
1.
2
f (x)f (x) 1 f (x) x>0
4
2. . f f (0,+) ( 4)
. f(0) = 1 ( 3)
7
3. g (0, +) , :
. g(x) 2 x x(0, +)
( 2)
. 1
0(2 x)f (x)dx 1
( 4)
6 4.
h, x x x = 0 x = 1 8
( )
27 2013
:
A1. f , [, ].
:
f [, ] f () f ()
, f () f()
x 0 (, ) , f(x0) =
7 A2. (...)
4 A3. f [, ]
;
4 A4. ,
,
, , .
a) |z z0 |=, > 0 K (z o)
2 , z, zo .
b) 0
0 (z0) , z0, z .
( 2)
ii. f (,x0)(x0,)
o 0 0
x x x x x xlim f x lim f x lim f x
( 2)
iii. 0 < < 1 , xxlim 0
.
( 2) iv. f
. f , f (x) > 0 .
( 2)
v.
g(x)
f(t)dt f g(x) g (x)
. ( 2)
10
z w :
2z i
w2z i
,
iz
2
w
1 . z,
=1
2,
1
M 0,2
10
2 . z, 1,
|w|= 1.
8
3. 1
z2
, w 4 + i w7 = 0
7
ln x
xe , x 0f (x)
0 , x 0
1. f x 0 = 0
4
2. f
7
3. i) , x > 0, f(x) = f(4) x4 = 4x
( 2)
ii ) N x 4 = 4x , x > 0, , x 1 =2
x2 = 4
( 6)
8
4. , , (2,4) ,
2
f ( ) f (t)dt f ( ) 2 f ( )
6
f: , = (0,+)
f (A) = , ,
f (x) 2e f (x) 2f (x) 3 x 1 . N f
f -1 f .
2 3
1 x 2f (x) e x 2x 3 , x 7
2 . f -1 . ,
f -1 , f -1
yy , x = 1
9
3. x 1A x,f (x) , 1B f (x), x f -1 f .
) , x,
f -1 f A
B , 1
( 3)
) x A, B
, .
( 6)
9
( )
2 IOYNIOY 2014 :
1. f . f
f x 0 x , f
.
8
2. f
. f
;
4
3. f A . f
0x A () , 0f x ;
3
4. , ,
, , ,
, . ) z C z z 2Im z
) ox x
lim f x ox x
1lim 0
f x
) f () ,
.
) 2
.
) f
. f ,
.
10
2
2 z z z i 2 2i 0, z C
1. .
9
2. 1z 1 i 2z 1 i ,
39
1
2
zw 3
z -3i
8
3. u
1 2u w 4z z i w, z1 , z2
2
8
2
f x x 3 x 1 , x R
1. f
f .
8
2. f
) y = 4x + 3
) f
.
8
3. g x x 1 f x , x R
.
9
h 2x x 2
h(x)x 1
x1 .
y= x - 2 h +,
1 . = 1.
7
2. ) y = x - 2
h - .
) h .
9
3.
4x 3
h x 0x
( 1, 0)
9
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