Themata panelladikon periigitis_new
87
Θέματα Πανελλαδικών Γ΄Λυκείου 2000 -2005 1 ΜΑΘΗΜΑΤΙΚΑ ΠΡΟΣΑΝΑΤΟΛΙΣΜΟΥ ΘΕΤΙΚΩΝ ΣΠΟΥΔΩΝ ΣΠΟΥΔΩΝ ΟΙΚΟΝΟΜΙΑΣ ΚΑΙ ΠΛΗΡΟΦΟΡΙΚΗΣ Θέματα Πανελλαδικών Εξετάσεων 2005-2015
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Transcript of Themata panelladikon periigitis_new
-
2000 -2005
1
2005-2015
-
2000 -2005
2
30 2000
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)
.
. , R, f x0 = 5.
. , x lim f(x)
.
4
. f(t)
t
, t 0 . f(t) 8 - 2 t 1
) f(t).
) t, ,
;
) t = 8
, t = 10
. ( ln11 2,4).
-
2000 -2005
3
15 2000
3
f f(x) = 2x - 3x 2
x - , .
. , f
x = 4.
. ,
f (1,0) (-2,3).
. > 2, x0 (1,2) ,
f x0
xx.
4
, 1.000
. .
4 25 .
200 .
4.000 . ,
10.000
.
. (x)
, : (x) = 10 (x + 16 x
+ 40) x
.
. , ;
. .
.
-
2000 -2005
4
9 2000
3
f(x) = 2x
x 1.
) x 1
lim f (x)
.
) f .
4
600 .
, .
) (x)
(x) = -2x2 + 600x ( 0 < x < 300).
) x (x)
) .
x xE(x)
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2000 -2005
5
12 2000
1o
A1.A f ' x0 ,
f
(x0, f(x0)).
2. , f ' x0
, .
1.
.
. f x0, f x0.
. f x0, f x0.
. f x0, f x0.
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.
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2000 -2005
6
3
f [0,1] f(x)>0
x(0,1). A f(0)=2 f(1)=4, :
. y=3 f '
x0(0,1).
. x1(0,1),
1
1 2 3 45 5 5 5
4
f f f ff x
. x2(0,1), f
(x2,f(x2)) y=2x+2000.
4
t=0 ' .
f(t)= 2 1
tt
,t0
t .
15 6
.
. .
. ,
12 ,
.
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2000 -2005
7
16 2000
1o
A. f, .
. f(x)0 x , f
.
. f(x)0 x ,
f ;
.1.
.
. f(x) =e1-x
. f f(x) = -2x+ 21
x + 3, x
2,)
.
. f(x) = g(x) + 3 x, h(x)=f(x)-g(x)
.
.2. f
-2,6.
f
.
-2 1 3 6x
y
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2000 -2005
8
3
f, , : 2x
x 0
f(x)- e 1 2xlim
= 5.
. f(0).
. f x0=0.
. -xh x e f(x) ,
f h (0,f(0)) (0,h(0)) .
4
( ) , t ,
P(t) = 4 + 2
t-6 25t 4
.
. .
. , .
. .
.
,
.
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2000 -2005
9
12 2000
: 2x x , x 1x 1
f (x) x 2 3, x 1 .
. f x0 = 1
. x 2 x 2
lim f (x), lim f (x).
4
100 , x
, :
3 21 1f (x) x x x 10 , 9 3
1< x < 5.
. x ,
.
. x 1 = 2
x 2 = 4 ( 2.000 4.000 ).
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2000 -2005
10
2 2001
2
f : 2
x-3
x , x 3 f(x) 1-e , x 3
x 3
. f , = 1/9.
. Cf f
(4, f(4)).
.
f, xx x=1 x=2.
3
f,
R, : f3(x) + f2(x) + f(x) = x3 2x2 + 6x 1
x R, , 2 < 3.
. f .
. f .
. f(x) = 0 (0,1).
4
f, R,
o :
i) f(x) 0, x R
ii)f(x) = 12 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)
. g .
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2000 -2005
11
. f : 21f(x)
1 x
.
. x
lim
(x f(x) 2x).
-
2000 -2005
12
5 2001
1o
A.1. f . F
f ,
:G(x)=F(x)+C, CR
f
G f : G(x)=F(x)+C, CR
.2.
.
. ( ) .....f x dx
. ( ) ( ) .....f x g x dx
. ( ) ( ) .....f x g x dx , ,R f,g [,]
.1. f, f(x)=6x+4, xR
(0,3) 2.
.2.
. xe x dx
. 24
1
3x dxx
. 2
0
2 x 3 x dx
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2000 -2005
13
3
x , x 1
f (x) 1 e ln(x 1), x 1, 2x 1
, R. .
. x 1
1 elimx 1
x 1
. R, f xo=1.
. =-1 (1,2) ,
f (,f()) xx.
4
f, (0,+) :
1 ln xf (x) , x 0x
:
. f.
. f.
.
f, xx x=1, x=e.
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2000 -2005
14
25 2001
2
f(x) = x2 - 4x + 3, x R .
) f
xx yy.
) f
(3, f(3)) .
) f.
3
f: RR, 2 - x4 f(x) 2 + x4,
x R . :
) f(0) = 2
) H f x0 = 0 .
) f x0 = 0 .
4
625 km
x km .
90 km . 160 ,
2x5,5
200
2000 .
) (x) :
1800000K (x) 500 x ,x
0 x 90 .
) .
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2000 -2005
15
6 2001
2
f(x) = x2 - kx + 1, x R .
) k, f
(1,0).
) f
(0, f(0)), k=17.
4
21f (x) 1 x , x R
) f(x).
) f.
) ( )
f.
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2000 -2005
16
30 2002
1o
A. f ' [, ]. G
f [, ],
f (t) dt G() G()
.1. f(x) = x. f
R f(x) = x .
.2. ,
.
. f [, ] (, ], f [, ] . . , 1-1 , .
. f x0 x x0lim f(x) 0 ,
x x0lim f(x) 0 .
. x x0
lim f(x) 0 ,
f(x) > 0 x0 .
3
f, g R .
fog 1-1.
. g 1-1.
. :
g(f(x) + x3 - x) = g(f(x) + 2x -1)
.
4
. h, g [, ]. h(x) > g(x) x
[, ],
h(x)dx g(x)dx .
. R f, :
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2000 -2005
17
f (x)f (x) e x 1, x R f(0) = 0 .
i) f f.
ii) f(x) x f (x) ,x2 x > 0.
iii) f,
x = 0, x = 1 xx, 1 1 E f (1)4 2
.
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2000 -2005
18
8 2002
1o
. ,
.
.
f (x)dx 0 , f(x) 0 x[,].
. f()
f .
. f IR. ,
[, ] , f
Rolle.
. f [, ]
x0[, ] f .
f(x0)=0.
. f [, ] x0(, )
f(x0)=0, f() f()0.
2
x
x
e 1f x , xe 1
. f f 1 .
. f 1 (x) = 0 .
. 1212
f x dx
4
f, R. ,
: f(x)f(x) + (f(x ))2 = f(x)f(x) , xR. f(0) = 2f(0) = 1.
. f.
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2000 -2005
19
. g [0,1],
x
20
g t2x dt 1
1 f t
[0,1].
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2000 -2005
20
5 2002
1
. f, g xo, f+g
xo :(f+g) (xo) = f (xo)+g (xo)
. ,
, , , , .
1. f ' xo ,
.
2. f ' xo ,
.
3. f ' f(x) = 0
x , f .
4. f ' f(x) > 0
x , f .
5. f g xo , :
o o ox x x x x x
lim f x g x lim f x lim g x( ) ( ) ( ) ( )
6. f g xo , :
o o ox x x x x x
lim f x g x lim f x lim g x( ) ( ) ( ) ( )
3
f(x) = x3- 6x2+9x-2 .
. f . . f
A , f ( )1 1 . . f(x) = 0 (0 , 1) . 4o
:
3
2
x 4x , x 2x 2f x
x k , x 2
( )
kR. :
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2000 -2005
21
. k, f x0 = 2 ,
. x 1lim
f(x) ,
. f x0 = 4
. f xg xx 3( )( )
.
-
2000 -2005
22
29 2003
1o
A. , f
x0 , .
.
;
. ,
.
. f
. f(x)>0
x , f .
. f ,
f
.
. f x0
. f x0 f(x0)=0, f
x0 .
3
f(x) = x5+x3+x .
. f
f .
. f(ex)f(1+x) xIR.
. f
(0,0) f
f 1.
.
f 1, x x=3.
-
2000 -2005
23
4
f [, ]
(, ). f() = f() = 0 (, ), (, ),
f()f() 0.
. f.
-
2000 -2005
24
8 2003
1o
A. f . F
f , :
. G(x) = F(x) c ,c R
f
. G f G(x) = F(x) c ,c R .
. ,
.
. f ' (, ),
x0 , f .
f (x) > 0 (, x0) f (x) < 0 (x0 , ), f (x0)
f .
. f : R 1 1 ,
x1 , x2 A : x1 = x2 , f(x1) = f(x2
. x = x0 f ;
3 2x) = x 1 - x .
. x lim f(x) 0
.
. f,
x . . 2 f (x) x 1 f(x) 0 .
. 1 0 21 dx ln 2 1 x 1 .
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2000 -2005
25
4
f IR ,
:
2f x f x f (x) 0 x IR .
. f .
. f(x) = 0 .
. f(x) g(x) f (x)
. g
xx, 45 .
-
2000 -2005
26
4 2003
1
. f(x) = x. f
R1 = IR {xx = 0} f(x) = 21
x.
. ,
, (), , (), .
1. x, y y = f(x), f
x0, y x
x0 f(x0) .
2. f (, ),
x0, f . f(x) > 0 (, x0) f(x) < 0 (x0, ),
f(x0) f .
3. f g x0, :
0
0
0
x x
x xx x
lim f xf (x) lim g(x) lim g x
( )( )
, 0x x
lim g(x) 0
.
2
2x - 3xf(x) x - 2
, x IR {2} .
. x 0
f(x)lim x
.
. y = x 1
f + .
. f (2, +).
3
2x , x 5
f(x) 10x - 25, x 5
x0 = 5 .
. f x0 = 5.
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2000 -2005
27
. f x0 = 5 f(5) .
. f (5,
f(5)).
. f .
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2000 -2005
28
27 2004
1
. f ' x0
. f x0
, f ( x0 ) =0
. f x0
;
.
.
. 0x x
lim f (x) l
, 0x x
lim f (x)
0x x
lim f (x) l
. f , g x0 ,
fg x0 : ( fg) ( x0 ) = f ( x0 ) g ( x0 ) . f, .
f ( x)>0 x , f
.
. f [ , ] . G
f [, ] ,
f(t)dt G() G()
2
f f( x) =x 2 lnx .
. f,
.
. f
.
. f.
3
g( x)=e x f( x) , f
IR f(0 )=f( 32
) = 0.
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2000 -2005
29
. (0, 32
)
f ( ) =f( ) .
. f( x) =2x 2 3x, () = 0
g(x)dx ,
a
. lim ()
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2000 -2005
30
5 2004
1o
A. f .
f
f(x) = 0 x ,
f .
. ,
.
. f x0 ,
. .
f, g IR fog gof,
.
C C f f1
y = x xOy xOy.
f x0, 0 0
k kx x x xlim f(x) lim f(x)
, f(x) 0 x0,
k k 2.
. f (, )
[, ].
2
f: IR IR f(x) = 2x + mx 4x 5x, m IR , m > 0.
. m f(x) 0 x IR .
. m = 10,
f, xx x = 0 x = 1.
4
f [0, +) IR , 2 1
20
xf(x) 2xf(2xt)dt2
.
. f (0, +).
. f(x) = ex (x + 1).
. f(x) [0, +).
. xlim f(x)
xlim f(x)
.
-
2000 -2005
31
4 2004
, , ,
, , ( ) ,
, () , .
. f, g . f, g f( x) = g ( x) x ,
c , x :
f( x) = g(x) + c. . f
, x1 , x2 x1 < x2 :
f( x1 ) < f( x2 ) . . f( x) = x . H f
(0 ,+) 2f (x)x
. , , (x0 ,
f( x0 ) ) , C f f,
x0 = f( x0 ) .
2
, 24x 3, x 1
f(x)6x k, x 1
, k IR .
. k, f x0 = 1 . . f
(1, f(1)) .
. , :
f(5) + f(5) + 34 = 0.
-
2000 -2005
32
3
f( x) = 2x3 3x2 + 6x + , x IR ,
. f x0 = 2 f(2) = 98. . = 6 = 54 .
. f .
. f.
. f(x) = 0
(1, 2) .
-
2000 -2005
33
31 2005
1
.1 f, [, ].
f [, ]
f() f()
f() f() ,
x0 (, ) , f(x
0) = .
.2 y = x +
f +;
. ,
.
. f [, ] f() < 0 (, ) f()
= 0, f() > 0. .
0x xlim f(x) g(x)
0x x
lim f(x)
0x x
lim g(x)
. f f - 1 f
y = x,
f - 1 .
. 0x x
lim f(x)
= 0 f( x ) > 0 x 0 , 0x x
1limf(x)
. f
, x f(t) dt f(x) f() x .
-
2000 -2005
34
. f
, x x ,
.
3
f f(x) = e, > 0.
. f .
. f,
, y = ex. .
. () ,
f, yy,
() = e 22 .
. 2
()lim 2
.
4
f IR ,
2 f(x) = ex f(x) x IR f(0) = 0.
. : f(x) = x1 eln
2
.
. : x
0
x 0
f(x t) dtlim
x .
. : h(x) = x 2005x t f(t) dt
g(x) =
2007x2007
.
h(x) = g(x) x IR .
. x 2005x t f(t) dt
=
12008
(0 , 1).
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2000 -2005
35
6 2005
1
.1 f f(x) x . f
(0,+) : 1f (x)2 x
.2 f: A IR 1 -1 ;
. ,
.
. , f
0,
f .
. f (, )
x o . f ( , x o)
(x o , ) , ( x o f( x o ) )
f.
f , g fog go f,
fog go f.
3
f, IR f( x)0
x IR .
. f 1 - 1. . C f
f (1, 2005)
( -2, 1) , 1 2f -2004 f(x 8) 2 .
. Cf,
Cf
( ) : 1y x 2005668
.
4
f: IR IR ,
-
2000 -2005
36
2x 0
f(x) xlim 2005x
.
. : i . f( 0)=0 i i . f( 0) =1.
. IR , : .
22
22x 0
x f(x)lim 3
2x f (x)
. f IR
f ( x)>f( x) x IR , : i . xf( x) >0 x0.
i i . 1
0
f(x)dx f(1) .
-
2000 -2005
37
8 2005
1
. 1. , f
x, .
.
(), , (), .
1. f : R. 1-1, x1, x2
: x1 x2, f(x1) f (x2).
2. f xA ()
, f(x), f(x) < f (x) xA.
3. f, g x f(x) g (x)
x, 0x x
lim f(x)
> 0x x
lim g(x)
4. f [, ]
(, ) , , (, ) , :
f() = f()-f()
-
2000 -2005
38
3
f(x) = x3+kx2+3x-2, xR , kR ,
(1,1). :
. k = -1.
. f .
. f(x) = 0 (0, 1).
4
22 x kx 2
f(x)x 3
, k R x 3.
. y = x f +, = 1 k = 3.
. (1, 2),
f xx. . f
x = 1.
-
2000 -2005
39
8
2005
1
. 1. f . f
f ( x) = 0 x ,
f .
2. R.
;
.
(), , (), .
3. x 0, 2x 01lim x
.
4. f( x) = x. f R 1
= R. {x / x = 0}
2
1f (x) x
:.
5. f x0 R, :
o ox x x x
lim k f(x) k lim f(x)
k R .
3
f :
3
4
, x 1x 1f(x)
x 1 , x 1
. f .
. f.
. , f
Ro l le [1,2] .
-
2000 -2005
40
4
2kx xf(x)
4
, x R,
(0,0) = 1. . k = 4.
. f ,
.
. (2,4 ) ,
f
, (2, f (2) ) ( 4, f(4) ) .
-
2000 -2005
41
27 2006
1o
A.1 f, .
:
f ( x) >0 x , f
.
f ( x)
-
2000 -2005
42
i i .
f f - 1 .
4
x 1f(x) ln xx-1
.
. f.
. f( x) =0 2
.
. g( x)= lnx (, ln) >0 \
h( x) =e x ( ,e ) I R ,
f( x) =0. . g
h .
-
2000 -2005
43
5 2006
1o
A.1 : ( x) =x, xI R .
.2 f . f ; B. ,
.
. f , g x g( x ) 0,
fg
x : .
o o o oo 2
o
f(x )g (x ) f (x )g(x )f xg g(x )
. x0 1ln x x
2
. f: I R 11, y
f( x)=y x
. f [ , ] . G
f [, ] ,
f(t)dt G() G()
2
x
x 1
1 ef(x)1 e
, xI R .
. f IR .
. 10
1 dxf (x)
. x
-
2000 -2005
44
. i . : 1ln(x 1) lnx , x 0x
.
i i . f (0, +) .
. x
1lim xln(1 )x
.
. ( 0 ,+) :
(+1) = + 1 .
-
2000 -2005
45
31 MA 2006
1
.
,
, , .
1. f x 0.
f( x)0 x. 0x x
lim f(x)
, 0x x
1limf(x)
.
2 f x0 , x0 .
3 f(x) x = [0, +) , 1f (x)x
x (0 , +).
4 0x x
lim f(x)
, 0x x
lim f(x)
+ ,
0x x f. 5 f , g . f , g f ( x) = g ( x) x ,
c , x : f( x) = g( x) + c
3 :
2
3 x , x 14f(x)
x 8x 4 , x 14x
R.
-
2000 -2005
46
. R. f
x0 =1. . =0 . f R .
.
f + .
4o
kR 3 2f(x) 2x kx 10 xR
. kR
f (1, f(1 ) ) xx.
. k = 3
. f .
. f (, 0] .
. ( 14, 15) f( x) = 5
(0, 1) .
-
2000 -2005
47
24 2007
1
.2 f, g ;
.3 y
f +;
B. ,
, ,
, , .
. f [, ] x [, ]
f(x) 0
f(x)dx 0 .
. f
x . f
, f(x) > 0 x .
. f x0 g
x0 , gof x0 .
. f
, g(x)
f(t)dt=f g(x) g (x)
.
. > 1 xxlim 0
.
3
f(x) = x3 3x 22 IR
+2
, Z
. f ,
.
. f(x) = 0
.
-
2000 -2005
48
. x1 , x2 x3
f, (x1 , f(x1)), B(x2 , f(x2)) (x3 ,
f(x3)) y = 2x 22.
.
f y = 2x 22.
-
2000 -2005
49
4
2007
1
. 1. : f,g x0
,
f + g x0
: (f+g) (x0) = f(x
0) + g(x
0).
2. f g ;
.
(), , (), .
1. +i +i
.
2. f , xx,
f.
3. f, g, h h (g f), (hg) f h (g f) = (hg) f.
4. 2 .
3
:
2
2
1 1x , x 28 2
f(x) x 5x 6 , x 22 x 1
. f x0=2.
. f
(0,f(0)).
. y = 12
x-2
f +.
-
2000 -2005
50
3 2008
1o
A. [, ]. G f
[, ],
f(t)dt G()- G()
. ; 5
. ,
, , ,
.
. 11, . 2
. f ,
f ,
.
.
f(x)dx xx xx. . (, x0) (x0, ) . :
0 0x x x xlim f (x) lim (f (x) ) 0
3
f(x)=x2 2lnx, x > 0.
. : f(x)1 x>0.
. f.
. :
ln x , x 0f(x)
g(x)
k , x 0
i. k g .
ii. 1k2
, g , , (0,e).
-
2000 -2005
51
28 2008
1 f x . f
x ;
2 f
[, ]
(, )
f() = f()
, , (, ) , : f () = 0.
3 f :
2
1 x , x 1f (x)
(x-1) , x 1
A. f :
. x = 1
. x = 1.
. f
(2, 1).
4o
f 2x 2x kf (x)
x
, k .
. f.
. f (1, f(1))
xx, k.
. k = 1,
. f.
. f [1, +).
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2000 -2005
52
20 2009
1o
. f . f
x , f
.
. f x0 ;
. ,
, , ,
.
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f(x)f(x0) xA
. x 0
x 1lim 1x
. f
.
. f [, ] f(x)-1 >0 1
A. f (x) 1 x>-1 = e
. = e,
. f .
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53
. f 1,0 0,
. , 1 0 0, , , f () 1 f () 1 0x 1 x 2
(1, 2)
-
2000 -2005
54
9 2009
1o
A. f(x) = x . f (0 , +)
:
1f (x)2 x
B. f xo . f
xo
;
. ,
, , ,
.
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f .
. ox x
lim f (x) 0
f(x) < 0 xo
ox x
1limf (x)
. f(x) = x. H f
1R R x / x 0 21f (x)
x
3
f(x)=ln[(+1)x2+x+1] - ln(x+2), x > 1
-1.
. , xlim f (x)
.
. = -1
. f .
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. f(x) + 2
= 0
0
4
f:[0,2] R
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2000 -2005
55
2xf (x) 4f (x) 4f (x) kxe , 0 x 2 , f (0) 2f (0) , f(2) = 2 f(2)+12 e4, f(1) = e
2
k .
. 2 2xf (x) 2f (x)g(x) 3x , 0 x 2
e
Rolle [0,2].
. (0,2) , f () 4f () = 6 e2
+ 4
. k = 6 g(x) = 0 x [0,2].
. 3 2xf (x) x e , 0 x 2
. 2 2 1f(x) dxx
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2000 -2005
56
25 MA 2010
1. , f
x0 , .
2. f
;
3. (5) , . . ,
,
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C f .
. f
c, : cf (x) f (x) , x .
. f [, ] [m, M], m . .
0x xlim f (x)
f(x)
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57
2. f.
3. f
(1, f(1)).
4. (, f()), >0, C f f,
Cf
(1,
f(1)), B(3, f(3)).
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2000 -2005
58
8 2010
1. f, [, ]. f [, ] f()f(), f()
f() x0(, ) f(x
0)=.
2. f ; 3. ,
, , ,
.
) f, g xo, ( ) ( )f x g x xo, :
0 0
lim ( ) lim ( )x x x x
f x g x
) f, g xo g(xo)0, fg
xo :
0 0 0 00
0
x x x xx
x
2
f g f gfg g
) P(x), Q(x) . P(x)Q(x)
,
P(x) ,
.
f(x) = (x+3) 29 x
1. .
2. f:
. (3, 3)
. xo = 3
3. f.
4. f.
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2000 -2005
59
16 2011
A1. f x0
. f x0
, : f (x0) = 0
A2. f R . y= x+
f + ;
A3. , , , , .
) f:A R 1-1, x1,x2A
: x1 x2, f(x1) f(x2)
) x R1= R {x | x=0} : (x)= 21
x
) : x
xxlim =1
) C C f f1
y=x xOy x Oy .
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
2. f .
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60
3. f
.
4. xln e x x
0,2
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61
6 2011
1. ( )f x x
x ( x) = x
A2. f, . f .
A3. , , ,
, .
i) f 0x A ()
0f x , 0f x f x x A
ii) f , 1-1
.
iii)
0x x
= 0lim f(x) f(x)>0 x0,
0
x x=1lim
f(x)
iv) f x0
.
y = x , x0.
(0, 1) xy
, .
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2000 -2005
62
t, t0
x(t)=16m/min.
1. , t, t0
: x(t)=16t
2.
(4, 2) , ,
.
3.
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4. t0
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xy.
f : , 3 , :
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x 0
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ii) f(0) < f(1) f(0)
iii) f(x) 0 x
1. f x0=0.
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2000 -2005
63
2. f .
g(x)=f(x) x, x :
3. g : 0x
xlimxg(x)
4. 2
0f x dx > 2
5.
g, xx x=0 x=1 ()=e 52
1
0f x dx (1, 2) ,
0( ) 2f t dt
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2000 -2005
64
( )
16 2011
1. f x0 .
f x0 ,
:
f(x0) = 0
3. , , , , .
i. 21x =
x
ii. :
x
xlim 1x
iii. C C f f-1
y=x xOy xOy.
2 2f(x) = xx
, x0
1. f .
2. f
A 2,f(2) .
3. f.
4. :
2x 1
1f 3xlim
x 1
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2000 -2005
65
f:, f(0)=0,
f(x) xf (x)=x x.
1. g(x) = xf(x) x , x .
2. : 1 xf(x) =x
, x x0
3. 1 x xx
, 32 2
4. (0,) :
222
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2000 -2005
66
6 2011
1. f(x)=x x (x) = x
2. (x,y) z=x+yi . z
3. , , , , .
i. f x A ()
f(x0), f(x) f(x0) x A
ii. f , 11
.
iii.
0x x
= 0lim f(x) f(x)>0 x0,
0
x x=1lim
f(x)
iv. f x0
.
y = x , x0. (0, 1) xy , .
t, t0 x(t)=16m/min.
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2000 -2005
67
1. , t, t0 : x(t)=16t 2. (4, 2) , , .
3. y(t) t, t>0 4m/min.
4. t0
10,4
d=() . xy.
2
1f(x) =x x
, .
f
52,12
518
.
1. =1 =4.
2. f . 3. f. 4. :
x3+(14)x2x+4=0 (1) f(x)=, , , (1) .
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2000 -2005
68
14 2012 A1. f [, ]. G f
[, ], :
f (t)dt
=G () G().
A2. (...)
A3. f [, ]
;
A4. ,
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) ox x
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) : x x x R
) : x 0
x 1lim 1x
) f
f .
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
2. f (g(x)) = 1
-
2000 -2005
69
13 2013 1. f x0, f
.
2. Fermat.
3. f .
f;
4. , ,
, , ,
, .
) f 1 1 ,
f .
) 0x x
lim f (x)
, 0x x
lim f (x)
) f, g x0 :
(fg)(x0) = f(x0)g(x0) f(x0)g(x0)
) f ,
f .
f: :
22xf (x) x f (x) 3 f (x) x
1f (1)2
1. :
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2000 -2005
70
3
2
xf (x) , xx 1
f
2. f 1.
3. :
2 3 2 2f 5(x 1) 8 f 8(x 1)
4. , , (0, 1) , :
3
2 30
f (t)dt 3 1 f ( )
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2000 -2005
71
27 2013
A1. f , [, ]. :
f [, ]
f () f ()
, f() f()
x0 (, ) ,
f(x0) =
A2. (...) A3. f [, ]
;
A4. , , , , .
i. 0
0 .
iv.
g(x) f(t)dt f g(x) g (x)
.
ln xxe , x 0f (x)
0 , x 0
1. f x0 = 0 2. f
3. i) , x > 0, f(x) = f(4) x4 = 4x
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ii) N x4 = 4x, x > 0, , x1 =2 x2 = 4
4. , (2,4) , :
2
f ( ) f (t)dt f ( ) 2 f ( )
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
2. f -1 . ,
f -1 ,
f -1
yy , x = 1
3. x 1A x,f (x) , 1B f (x), x
f -1 f .
) , x,
f -1 f A B , 1
-
2000 -2005
79
) x A, B ,
.
-
2000 -2005
80
2 IOYNIOY 2014
1. f . f f x 0 x , f
. 2. f
. f ;
3. f A . f 0x A () , 0f x ;
4. , ,
, , , ,
.
) ox x
lim f x
ox x1lim 0
f x
) f () , . ) 2 . ) f . f , . 2f x x 3 x 1 , x R
1. f
f .
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) y = 4x + 3
) f
.
3. g x x 1 f x , x R
.
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2000 -2005
81
h 2x x 2h(x)x 1
x1 .
y= x - 2 h +,
1. = 1.
2. ) y = x - 2
h - .
) h .
3. 4x 3
h x 0x
( 1, 0)
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2000 -2005
82
25 2015
1. f, [,] .
f [,] f() f() ,
f() f()
x0(,), f(x0) = .
2. f x0 . f x0;
3. f A. f
xo ;
4. , , , , .
i. f, g fog gof, fog=gof.
ii. x (x)= x.
iii. f [, ]. f(x)0
x[, ] f ,
f (x)dx 0
iv. ox x
lim f (x) 0
f(x)>0 xo, ox x
1limf (x)
.
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2000 -2005
83
x
2ef (x)
x 1
, x.
1. f (0, +).
2.
2
3 x 2 ef e (x 1)5
. 3.
2x
4xf (t)dt 2xf (4x)
x>0.
f: :
f(x)[ ef(x) + e - f(x)] = 2 x
f(0)=0.
1. 2f (x) n x x 1 , x. 2.) f
f.
)
f, y=x x=0 x=1.
3. 0 02 2x 2 x1 3 f (t )dt 8 3 f (t)dt
x 3 x 2
= 0
(2,3).
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2000 -2005
84
12 2015
1. f . F f
,
:
G(x) = F(x) + c, c f ,
G f G(x) = F(x) + c, c . 2. f : A 1 1;
3. 0x x
f;
4. ,
,
, , .
) f, g x0 f(x) g(x) x0,
0 0x x x x
imf(x) im g(x)
)
0x x
imf(x) , f(x) > 0 x0.
) 2,
.
) f [, ] G
f [, ] :
f(t )dt G() G()
x 1f(x) e nx, x 0, 1. f .
2. g
h( x ) 2
1g(x) t 1 dt,
2h(x) f(x 1) f(2) 1
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85
3.
1f f (x) 12
1 2x , x
4. 1 2x , x 3 1 2x x ,
1 x , 1 , f
, f ( )
3 0,2
f : 0, : 2(x x) f (x) x f(x) 1, x 0,
1. N
nx , 0 x 1f(x) x 1
1 , x 1
-
2000 -2005
86
12 2015
1. f (x) x
1 {x | x 0} 21( x)
x
2. f : A 1 1;
3. 0x x
f;
4. ,
,
, , .
) f, g x0 f(x) g(x) x0,
0 0x x x x
imf(x) im g(x)
)
0x x
imf(x) , f(x) > 0 x0.
) 2,
.
) x x x
2 21f(x) x x 0,x
1. f .
2. g, g(x) f(x) 2
3.
3f f(x) 2, x 0,2
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87
4.
1 , 12
,
f , f ( )
5 0,2
4 3 2f(x) 3x 4x x , x ,
. f 0x 1 , :
1. N = 12
2. f
, f (x) x
3.
3f(x)g(x) x 0,x 1
4. :
v 2x
f(x) 1im x x
.
