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Θέματα και απαντήσεις στα Μαθηματικά Γ΄Λυκείου Γενικής Παιδείας 2000-2012
Citation preview
1 & . / 2000
1
. ) F(x) = f(x) + g(x). f, g , :
F(x) = f(x) + g(x) ( 8)
) :
cf(x), f(x)g(x), f(x)/g(x) g(x) 0
c . ( 4,5)
.) .
. x2 + 3 1. 1 - x
. x + x 2. 3x2 - 8x . xx 3. 2x + 3
. x3 - 4x2 4. x - xx 5. 2x
6. 3x2 - 4x 7. x + xx
( 8)
) . :
0, x,x
ef(x)
x
=
:
, e :Ax
,x
xee :B
2
xx
,x
xee :
2
xx+
,x
exe :
2
xx
x
exe :
xx
( 4,5)
2: . ) f, g , :
h
g(x)-h)g(xlimg(x
h
f(x)-h)f(xlimf(x
0h0h
+=
+=
:
F (x + h) F (x) = f(x + h) + g(x + h) f(x) g(x) =
= f(x + h) f(x) + g(x + h) g(x)
=+++
=+
h
g(x)-h)g(xf(x)-h)f(xlim
h
F(x)-h)F(xlim
0h0h
g(x)f(x)h
g(x)-h)g(xlim
h
f(x)-h)f(xlim
0h0h
+=+
++
=
) [cf(x)] = cf(x), c R. [f(x)g(x)]= f(x)g(x) + f(x)g(x).
0g(x)(x)g
f(x)g(x)-f(x)g(x
g(x)
f(x)2
=
. ) (x2 + 3) = (x2) + 3= 2x + 0 = 2x (x + x)= 1 x. (xx)= (x)x + x(x)= x + x x. (x3 4x2)=3x2 8x. :
5 1 7 2. ) x 0 :
2
xx
2
xx
x
e-xe
x
(x)e-)x(ef(x) ==
.
3 2 . .
x i v i fi
fi(%)
Nix ivi x i
2x i
2v i
1 10 10 1 10
2 35 4
3 9
v = 50 1 100 - -
( 16) . .
( 4) . : s2 = 0,49. :
=
=
=
k
1i
2k
1i
ii
i
2
i
2
v
vx
vxv
1s
( 5) : . :
10vx9 x,4 x,1 x10,v x35,N 10,v1
2
1
2
3
2
2
2
11121=======
: 2 = v1 + v2 35 = 10 + v2 v2 = 25.
:
v = 50 v1 + v2 + v3 = 50 10 + 25 + v3 = 50 v3 = 15.
:
3,050
15
v
vf 5,0
50
25
v
vf ,2,0
50
10
v
vf 3
3
2
2
1
1=========
:
v1 = N1 =10 v = 3 = 50.
4: x2v2 = 2 25 = 50, x3v3 = 3 15 = 45
.105455010vx
3
1i
ii=++=
=
, :
135153 v x100,252 v x2
3
2
3
2
2
2
2====
.24513510010vx
3
1i
i
2
i=++=
=
:
xi
vi
fi
fi(%)
Ni
xivi xi2
xi2vi
1 10 0,2 20 10 10 1 102 25 0,5 50 35 50 4 1003 15 0,3 30 50 45 9 135
v = 50 1 100 - 105 - 245 . :
1,250
105
v
vx
x
3
1
ii
===
2. :
22
22
2
xx
2120=
+=
+
=
=
=
=
=
=
50
105245
50
1
v
vx
vxv
1s .
2k
1i
2k
1i
ii
i
2
i
2
=1/50(245-(11.025/50)) = 1/50(245-220,5) = 24,5/50 = 0,49.
5 3 120 , 24 , 20 12 . . : : .
( 8) : .
( 8) : .
( 9) : 120 , () = 120. : : : . : : . : () = 24/120 = 0,2 () = 1 () = 1 0,2 () = 0,8 () = 20/120 = 1/6 () = 1 () = 1 (1/6) () = 5/6 ( ) = 12/120 = 0,1. , : ( ) = () + () ( ) = = (24/120) + (20/120) (12/120) = 32/120 () = 4/15. : ( ) ( ) : [( ) ( )] = ( ) + ( ) = = () ( ) + () ( ) = = () + () 2( ) = = (24/120) + (20/120) 2 (12/120) = 20/120 () = 1/6. , ( ), [( )] = 1 ( ) = 1 (4/15) = 11/15.
6 4 100 . :
[ - )
fi (%)
0 - 5 105 - 10 1510 - 15 1215 - 20 1520 - 25 1825 - 30 1830 - 35 12
. 15 ;
( 5) . 35 : ) 12,5 ; .
( 10) ) , .; .
( 10) : . 15 15 + 18 + 18 + 12 = 63 . . ) , , . : ( ) , 12,5 22,5 , 22,5 [20, 25), 18, 9 ( ), 9 + 18 + 12 = 39 . : , , . 12,5 22,5 . 22,5 [20, 25), 18 + 12 = 30
7 18 + 12 + 18 = 48 . : () (22,5, ) 22,5 , 100 .
[ - )
fi (%)
Fi(%)
0 - 5 10 105 - 10 15 2510 - 15 12 3715 - 20 15 5220 - 25 18 7025 - 30 18 8830 - 35 12 100
(20, 52) (25, 70). y = x + . :
=
=
=
+=
=
+=
+=
+=
3,6
-20
3,6
2052
518
2052
2570
2052
: y = 3,6x 20 x = 22,5 : = 3,6 22,5 20 = 61 : 100 61 = 39 .
8) 30 35 , 12 .
1 & 2001
1
.1. :
() = () (). 8,5
.2.
, (=, , ) : . () ... 1()
2 . () ... ().
2 .1.
. . . () + () < 1. . () = () 2() = ().
4 .2. .
, () = 1/4 () = 5/12 () : . 1/4 . 5/12 . 2/3 . 1/6.
2,5 .3.
, . :
() = 1/3, () =1/4 () =1/5.
. () 1. 1/20
. ( ) )AB( 2. 2/15
. ( ) )BA( 3. 4/5
4. 1/12
5. 19/20
6
: .1. () U () = A
: P(A) = P() + P()
: P() = P() P()
2A.2. . P(A') = 1 P(A) . B : P() P() B.1. .
' P(A') P(B) 1 P(A) P(B) P(B) + P(A) 1
: .
. :
P(A) = P(A') P(A) = 1 P(A) 2P(A) = 1 2P(A) = P()
: .
.2. :
B = :
P() = P(A) = 1/4 E:
P(AUB) = P(A) + P(B) P() = 1/4 + 5/12 - 1/4 = 5/12 :
.
.3. P(A B) = P(A) P() = 1/3 - 1/5 = 2/15 P((B - A)') = 1 P(B A) = 1 [P(B) P()] = = 1 P(B) + P() = 1 - 1/4 + 1/5 = 3/4 + 1/5 = 19/20 P((A)') = 1 P(A) = 1 - 1/5 = 4/5 :
. 2 . 5 . 3
2
f(x) = x+x. A. f(x) + f''(x) = 0.
8
. f (0,1). 8
. IR :
2
f
2 2
f
= 2.
9
: . f 2 R : f (x) = ( x + x) = x + x. f '' (x) = ( x + x) = x x
3 : f (x) + f '' (x) = x + x x x = 0. B. = x + Cf (0,1). : f (0) = 1 = . f (x) = x + x, : f (0) = 0 + 0 = 1 : y = x + 1 . :
1222
=+=
f .
1222=+=
f .
:
=
2
22
2
ff
= 212)1(
22 =
:
= 4
3
80 . 4 .
[ )
Fi
45-55 0,2
55-65 0,5
65-75
75-85
4. , .
8 . .
9 . 80 .
. 65 . 4
. 55 75 .
4
: A. f3 f1 = F1 = 0,2
: f3 = 2 f1 = 20,2 = 0,4
: F2 = f1 + f2 0,5 = 0,2 + f2 f2 = 0,3
f1 + f2 + f3 + f4 = 1 :
0,2 + 0,3 + 0,4 + f4 = 1 f4 = 1 0,9 = 0,1 :
f1 = 0,2, f2 = 0,3, f3 = 0,4 f4 =0,1 :
F3 = f1 + f2 + f3 = 0,9 F4 = 1 . :
ii
i
ifvv
v
vf ==
= 80, i i = 1, 2, 3, 4 : 1 = 800,2 = 16 2 = 800,3 = 24 3 = 800,4 = 32 4 = 800,1 = 8 :
[ ) xi i fi Fi ixi45 55 50 16 0,2 0,2 5016 = 80055 65 60 24 0,3 0,5 6024 = 144065 75 70 32 0,4 0,9 7032 = 224075 85 80 8 0,1 1 808 = 640
v = 80 1
:
64512080
1
)64022401440800(80
1x
1x
4
1
i
==
=+++== =i
iv
v
5. . " 65 " P(A) :
2
1
80
40
80
2416P(A) ==
+=
. " 55 75
" P(B) :
10
7
80
56
80
2342P(B) ==
+=
4
, , 50% 12 , 16% 10 . . .
. 6
. , .
6
. 4.000, 14 16 .
6
. , , 5 . (CV).
7
: . 50% 12 , :
12=x .
16% 10 , 10 12 (50 16)% = 34% = (68/2)% .
:
2101210 ===xxx
sssx
. :
6
1
12
2
1===
x
sCV
x 16,6%.
16,6% > 10%, .
. :
12=x
14=+x
sx
162 =+x
sx
6 14 16 :
%5,13%2
6895=
:
5405,1340100
5,1340000 ==
. 5 3 99
. :
: 175125 =+=+= x
:
2==x
SS
:
17
2
2==
s
CV 11,7%.
5%.
1
2002
1
. A x1,x2,,xk ,
, k,
k .
. i , xi , i = 1,2,,k;
3
. fi xi ,
i = 1,2,,k;
3
. :
i) 0 fi 1 i = 1,2,,k
ii) f1 + f2 + + fk = 1.
4 .1. ,
: ( ) = () + ().
8
.2. .
.
5
.
:
i) P() ii) ().
2
2: ) , i, xi . ) fi i xi .
:
f
i
i= i = 1, 2, , .
) i) 0 i i = 1, 2, ,
1
0
i .
0 fi 1 i = 1, 2, , . ii)
f1 + f2 + + f = 1
...
...
2121
==
+++
=+++
. 1. 1. . 150 . . .2 . .
()
()
P(A) ==
.2..
(i) P() = 1. (ii) P() = 0
3 2
f(x) =
1x
x2
+
.
. f.
4
. )x(flim3x
.
4
. f. 7
. f
y = 2x + 5. 10
: () x+1 0, x -1 Af= -{-1}
() 2
3
4
6
1
2lim)(lim
33
==
+
= x
xxf
xx
() =+
++=
+=
2
'
)1(
)'1(2)1()'2(
1
2)('
x
xxxx
x
xxf
=+
+=
+
+
22 )1(
222
)1(
2)1(2
x
xx
x
xx
2)1(
2
+x
() }1{
ox 2)(' =
oxf
: 2)1(
2)('
+
=
xxf
:
+==+
2
2)1(222
)1(
2o
o
x
x
=+=+ 01)1(02)1(2 22oo
xx
=+=+++ 0)2(0)11)(11(oooo
xxxx ( 0=o
x 2=o
x )
(0,f(0)) = (0,0) B(-2,f(-2)) = (-2,4).
4 : (0,0)
)0)(0(')0( = xffy
xy 20 =
xy 2=
(-2,4) )2)(2(')2( += xffy
)2(24 += xy
424 += xy
82 += xy
:
( ) : y = x+ f
(0,0). : = f '(0) = 2 0 = 20 - = 0 y = 2x y = 'x+' f
B(2,4). : ' = f '(-2) = 2 4 = 2(-2)+ = 8 y = 2x+8 3 10 , :
8, 10, 13, 13, 15, 16, 18, 14, 14, 9. . ,
. 6
. ,
. 6
.
10%, .
13
5:
xi i ixi 8 1 8 9 1 9 10 1 10 13 2 26 14 2 28 15 1 15 16 1 16 18 1 18 10 130
)
1. 1310
130
10
x
x
8
1i
ii
===
=
2. : 8 9 10 13 13 14 14 15 16 18
: 13,52
1413
2
tt 65 =
+=
+
=
3. = 13, 14. ) R = 18 - 8 = 10. s2 :
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]=+++++++= 222222222 1318131613151314213132131013913810
1s
[ ] 910
902594291625
10
1==++++++=
3ss 2 ==
13
3
x
sCV
1==
23%. ). yi, i = 1, 2, , 10 10% 0,9. xy 9,0= , sy =
0,9 sx
1
xx
2CV
x
s
x0,9
s0,9CV ==
=
.
6 4 , () + () 2( ). :
f(x) = (x - P(AB))3 - (x - P(AB))3 , xR. . P(AB) P(AB).
5
. f(x)
2
)B(P)A(P x
+
=.
13
. , , f(P(A)) = f(P(B)).
7 : ) : P(A)+P(B) 2P(A B) . P(A)+P(B) - P(A B) P(A B) P(A B) P(A B)
) : ( ) ( ) = xB)P(Ax3B)P(Ax3(x)' 22f
: f '(x)=0 ( ) ( )22 B)P(Ax3B)P(Ax3 = 0
B)P(AxB)P(Ax
B)P(AxB)P(Ax
+=
=
P(B)P(A)B)P(AB)P(A2x
B)P(AB)P(A
+=+=
=
2
P(B)P(A)x
+=
: f '(x)>0 ( ) ( ) 0B)P(Ax3B)P(Ax3 22 > ( ) ( ) 0B)P(AxB)P(Ax B)P(Ax-B)P(Ax >++ ( ) ( )[ ] 0B)P(AB)P(A2 B)P(A-B)P(A >+ x ( ) ( )[ ] )1(0P(B)P(A)2 B)P(A-B)P(A >+ x
7 : A B A B P(A B) P(A B) : P(A B) P(A B) : P(A B) < P(A B) : P(A B) - P(A B) < 0 : (1) 2x < P(A)+P(B)
2
P(B)P(A)x
+<
A : f '(x)
f max 2
P(B)P(A)x
+=
) A B = P(A B)=0 (1) P(A B) = P(A)+P(B) (2)
: ( ) [ ] [ ]33 B)P(AP(A)B)P(AP(A)P(A) =f
[ ] [ ]33(2)(1),
P(A)P(B)P(A)P(A) =
= -P3(B) - P3(A)
( ) [ ] [ ]33 B)P(AP(B)B)P(AP(B)P(B) =f
[ ] (B)PP(B)P(A)P(B) 33(2)(1),
=
= -P3(A) - P3(B) : f(P(A)) = f(P(B)).
1
2003
1
. f(x) = x f(x) = 1. 8
. f ;
6
. () . 6
. , .
. .
. .
. (f(g(x))) = f (g(x)) . g (x)
f , g .
. , = .
. .
5 : . : f(x)=x, . 28 . . : . 13 . . : . 87 . . - - - - -. 2 55% , 40% 30% . .
:
. 5
. 5
. 7
. . 8
2: : : : 55% , : P()=0,55. 40% , : P()=0,40. 30% ,
: P( )=P()=0,30. : . P()=P()+P()-P()=0,55+0,40-0,30=0,65. . P()=P()-P()=0,55-0,30=0,25. . , : P()=P()-P()=0,40-0,30=0,10. . , : P() = P()+P()-P() = = 1-P()+P()-P()+P() = = 1-P() +P() = 1-0,55+0,30=0,75.
3
1x
xf(x)
2
=
. . : . R . (-1,1) . R- {-1,1} . (1, + )
5
. f(x)
3. f -{-1,1}
=
=
=
=
22
22
22
22
2 )1x(
x21x
)1x(
)1x(x)1x(x
1x
x)x(f
0)1x(
1x
)1x(
1x22
2
22
2
CV .
5. yi i = 1,2,3,4,5,6
20%,
ii
i
iixx
xxy 2,1
100
201
100
20=
+=+= .
i i = 1,2,3,4,5,6 5 , i = xi + 5.
3, 99
652,1x2,1yA
===
13585x =+=+=
.
3
232,1S2,1S Ay == .
3
41SSB== .
:
30,0CVx2,1
S2,1
y
SCV A
A
Ay
A =
==
08,0507
41
13
3
41
SCV
B==
= .
CV > CV, .
1
2004
1 . f(x)=c 0.
8 . f x0
. 5
.
.
. xi . . 95%
s),xs,x( + x s
. . i
, fi xi. 6
.
. .
.
,
1
2 -
3
4
B)(A
BA
BA
. 6
: . : f = R f(x+h) - f(x) = c - c = 0.
h 0 0)()(=
+
h
xfhxf.
2 0)()(
lim0
=+
h
xfhxf
h
(c) = 0 . f x0
)()(lim0
0
xfxfxx
=
. . . . . . 4 . 2 . 1 2
f 3x
34xxf(x)
2
+= .
. f. 10
. f(x)lim3x
15 :
. (i) 0x
(ii) 3x
3x
),3()3,0[ +=fA .
B. ),3()3,0[ +x :
( )( )
( )( )=
+
+=
+=
33
3)3)(1(
3
342
xx
xxx
x
xxxf
( ) ( )3)1(3
3)3)(1(+=
+= xx
x
xxx
( ) ( ) ( )[ ]=+=
31limlim33
xxxfxx
( ) 3433)13( =+= 3 " " 200 , 5 45 . :
3
.
xi
i .
fi %
Ni
.
. .
Fi%
[5, 15) 60
[15, 25) 68
[25, 35) 180
[35, 45)
200 .
. 10
. (xi , fi%) .
5 . x .
5 . 25
. 5
: .
xi i fi % Ni Fi%
[5, 15) 10 60 30 60 30
[15, 25) 20 76 38 136 68
[25, 35) 30 44 22 180 90
[35, 45) 40 20 10 200 100
200 100 .
.
=+++
== =
200
2040443020766010x
1x
4
1i
ii
Km2,21200
4240
200
80013201520600==
+++=
. 3 + 4 = 44 + 20 = 64 .
4 4
f 10xx2
52xf(x) 23 ++= .
() () x, f .
. 2
1P(A) =
3
1P(B) = .
9
. (), () 3
2B)P(A = ,
:
i. P(A B) ii. P(A - B) iii. P[(A B)] iv. P[(A - B) (B - A)].
16 : . f R
156)(' 2 += xxxf
===+=
2
1
3
101560)(' 2 xxxxxf
( )3
1BP
2
1)( ==AP
. ( )3
2)(
3
1BP ,
2
1)( === BAPAP :
i. 6
1
3
2
3
1
2
1)()()()( =+=+= BAPBPAPBAP
ii. 3
1
6
2
6
1
2
1)()()( ==== BAPAPBAP
iii. ( )[ ]6
5
6
11)(1' === BAPBAP
5iv. -, - 2 . 153 . .
( ) ( )[ ] ( ) ( ) ( ) ( ) =+=+= )()( BAPBPBAPAPABPBAPABBAP
2
1
3
1
3
1
2
1
6
2
3
1
2
1)(2)()( =+=+=+= BAPBPAP .
1
2005
1
. :
P(AB) = P(A) + P(B) P(AB)
.
10
. . ;
3
. ;
4
. , .
. f f(x)>0 , f .
2
. :
( )2)(
)(')()()('
)(
)(
xg
xgxfxgxf
xg
xf +=
f, g .
2
. .
2
. P(A) > P(B).
2
2 2
i :
. :
/
[ )
xi
i
fi
i
. .
Fi
[4, 8)
[8,12)
[12,16)
[16,20)
11
. .
8
. 10;
6
v i
25
20
15
10
5
4 8 12 16 20 0
3 3
, , : (i) ,
7/8 (ii) P(B) , P(AB)
=4
5,
2
1,kX , :
56
153lim
25 +
=
xx
xk
x
. k
5
. P(B), P(AB) .
8
. :
(1)
6
(2)
6
4
f ),0(,1
)( += xx
xf
. f (1,1).
7
. (x, y) f xx yy,
x, Oy . , .
10
. ()
5=x sx = 2. y
sy .
8
4
1
.
3 : :
P(A B) = P(A) + P(B) P(A B)
( ) + () + () ( ), (1)
() + () A B . (1) () :
)(
)(
)(
)(
)(
)(
)(
)(
+
=
P(A B) = P(A) + P(B) P(A B). (additive law).
.
. . .
.
(, ).
.
2
.
/
[ )
xi
vi
fi
Ni
..
Fi
[4, 8) 6 5 0,1 5 0,1
[8, 12) 10 10 0,2 15 0,3
[12, 16) 14 25 0,5 40 0,8
[16, 20) 18 10 0,2 50 1
50 1
. 13,2.50
660
50
18101425101065x ==
+++=
5. 10 5 + 5 = 10 .
3
. 4
3
15
3
1x
3lim
5)1)(x(x
5)3(xlim
56xx
153xlim
5x5x25x
=
=
=
=
+
=
.
. 4
3 = ,
=4
5,
2
1,
4
3X .
14
5> ,
4
5
P (A B), P (B).
=2
1,
4
3P(B)}B),{P(A .
BBA P (A B) P (B) P (A B) P (B)
P (A B) < P (B).
2
1B)P(A = ,
4
3P(B) = .
.
(1) P (A B) = P (A) + P (B) P (A B).
2
1
4
3P(A)
8
7+= ,
8
5P(A) = .
(2) :
8
1
2
1
8
5B)P(AP(A)B)P(A === .
4
. 1 :
x
1f(x) = (0, +),
2x
1f(x) = .
f (1, 1) = f(1) = -1.
y = -x + .
(1, 1) ,
1 = -1 + = 2. y = -x + 2.
2 :
f
(1, 1) : y f(1) = f(1) (x 1).
1f(1) = 11
11
2==)(f .
y 1 = 1(x 1) y = x + 1 + 1 y = x + 2.
6
.
M(x,y)B
AOx' x
y'
y
1
2
(x, y) x
1f(x) = 1, 2
xx yy.
= 2x + 2y = 2(x + y) (1)
x
1y = , (1) :
+=x
1x2 .
+=x
1x2(x) x (0, +).
(x) (0, +)
( )( )22
2
2 x
1x1x2
x
1x2
x
112(x)
+=
=
= .
(x) = 0 :
( )( )1x1x0
x
1x1x2
2=+==
+.
x = -1 x (0, +).
7 :
' (x)
(x)
0 1
- +
x +
.
(1) = 4
x = 1, (x) .
(1, 1).
. :
32xy =+=
== .
1
' 2006
1o A. f R. c .
(cf(x))=cf(x), x R .
10 B.. , ;
3 . f ;
4 . , . . f ,
x0A, f(x) f(x0) x x0. 2
. A , ,
, . 2
. x 0 : 2
11
xx
=
'
.
2 . .
2
2 2 50 . , :
xi
i
0 +4
1 5+8
2 4
3 -1
4 2
50 . .
3 : . .
7 . .
7 . 3 .
8 3o x (x+4)2 . . ,
. x .
7
. 19
1
100 , , .
8 . , , ;
10
3 4
f(x) = -2x2+kx + x4 + 10, x 0.
. A (1,f(1)) xx, k = 2 .
5 .
)1(fx = 13
)4(2 fs = . ,
, 8. (i)
(10,16). 10
(ii) ,
. > 0, , .
10
4 1 . . 30. . . 142. . . 16. . - - - - 2
.
3
3913
50214854
=
=
=++++++
a
a
aaaaa
.
xi vi xivi Ni
0 7 0 7
1 23 23 30
2 12 24 42
3 2 6 44
4 6 24 50
50 77
50
77
50
24624230=
++++=x .
. 12
11
2
2625=
+=
+
=
tt
. 3 .
25
4
50
8)( ==AP .
5 3
. . () = x + (x + 4)2.
() = x.
2
4)()(
)()(
++
=
=
xx
x
N
ANAP , Rx
( ). x 0x .
2)4(0 ++ xxx 1)4(
02
++
xx
x
.
. )()(
)( 820161019
1
419
1 2
2===+=
++
= xxxxxx
xAP
x = 8 () = 8 + (8 + 4)2= 152 > 100. x = 8 .
x = 2 () = 2 + (2 + 4)2= 38 < 100. x = 2 . ,
() = (2 + 4)2 = 36, 19
18
38
36==
=
)(
)()(
N
KNKP
. 04
2
++
= xxx
xxf ,
)()( .
f :
[ ]0
4
16
22
2
++
= x
xx
xxf ,
)(
)( .
x 4
f
f
+ -
0
1
17
+
f x = 4 17
14 =)(f .
() f.
6 () = {f(1), f(2),
f(3),f(4),f(5), ... }, f(A). )4()( fxf ),0[ +x f(A)
17
1)4( =f .
() 17
1
4. 4
. 01042 2 +++= xxkxxxf ,)(
x > 0 x
kxxf2
4 ++=)(
Cf (1, f(1)) xx 202401 ==++= kkf )( .
k = 2 10422 2 +++= xxxxf )(
f(1) = 14 (1, f(1)) (1,14). Cf , y = 14. . 14)1( == fx
2134 ===13
2(-13)-s,)(f
(i)
14=x s = 2 :
X
18 20
13,50% 2,35% 0,15%
8 10
0,15% 2,35% 13,50% 34% 34%
12 14 16
3 8
20003100
150==
,
(10, 16) 81,5% = 2000 ,
16302000100
581=
,
.
(ii)
10,014,07
1
14
2cv >===
x
s
7 . > 0 ,
+14
2.
660101041210014
2+
+
,,,,,
.
= 6.
1 '
2007
1
.
( ) = () ( ). 8
.. f x0
; 4
. () , .
3
1. , , , , .
. ,
Fj xj.
2
. f, g , :
(f(g(x))) = f(g(x)) g(x). 2
. f f(x0) = 0 x0 (, ), f(x) > 0
(, x0) f(x) < 0 (x0, ), f (, ) x = x0 .
2
2.
:
f1(x) = x,
f2(x) = ln x, x > 0
f3(x) = x , x > 0
f4(x) = x, x 4
2 2
f (x) = x ex + 3, x .
. f(x) = f (x) + ex 3 10
. xx
exfx
x
2
0
)(lim
15
3
= { 1, 0, 1, 2, 3, 4, 5 } :P(1) = P(0) = P(1) = P(2) = 2 P(3) = 2 P(4) = 2 P(5).
: = { 1, 3, x2 x 3 }, B = { 2, x + 1, 2 x2 + x 2, 2 x + 1 }
x .
. , :
P(1), P(0), P(1), P(2), P(3), P(4), P(5). 7
. x = { 1, 3 } 8
. x = 1 :P(A) = 5/11, P(B) = 7/11, P( ) = 3/11
P(A B) P(A B'). 10
4
2 : : 12, 18, t3, t4, , t25 : 16, 14, t3, t4, , t25
t3 + t4 + + t25 = 345.
. A
x , B
x
A
x = B
x = 15.
7
. 2A
s 2B
s
, 2A
s 2B
s = 16/25
8
. CVA = 1/15,
CVB . 10
3
1
. ( . . 152)
.
. ( . . 22). ( . . 87)
1. , ,
2.
f1(x) = x 1, x
f2(x) = 1/x, x > 0
f3(x) = ,2
1
x
x > 0
f4(x) = x, x
2
. f :
f(x) = (xex + 3) = ex + xex = ex + f (x) 3
f (x) = xex + 3 xex = f (x) 3
. =
+=
xx
e3(x)fe
xx
e(x)f2
xx
0x
2
x
0x
limlim
=
=
+=
=
1)(xx
ex
xx
33ex
xx
3(x)f x
0x
2
x
0x
2
0x
limlimlim
110
e
1x
e0x
0x
lim =
=
=
. ( 1x
e(x)g
x
=
{1} .)
3
. = {1, 0, 1, 2, 3, 4, 5},
() = 1 (1) + (0) + (1) + (2) + (3) + (4) + (5) = 1.
(1) = (0) = (1) = (2) = 2(3) =2 (4) = 2(5) = .
(1) = (0) = (1) = (2) = , (3) = (4) = (5) = /2.
4
+ + + + (/2) + (/2) + (/2) = 1
11
22112381
2
34 ===+=+ .
(1) = (0) = (1) = (2) = 2/11 (3) = (4) = (5) = 1/11.
. {1, 3} {1, 3, x2 x 3}
1 {1, 3, x2 x 3}.
x2 x 3 = 1 x2 x 2 = 0 x = 2 x = 1.
x = 2 : = {2, 3, 8, 3}
= {3} {1, 3} x = 2 .
x = 1 : = {2, 0, 1, 3}
= {1, 3} x = 1 .
. x = 1 = {1, 3, 1 } = {2, 0, 1, 3}.
() = (1) + (3) + (1) 11
5
11
2
11
1
11
2=++= .
() = (2) + (0) + (1) + (3) 11
7
11
1
11
2
11
2
11
2=+++= .
( ) = (1) + (3) 11
3
11
1
11
2=+= .
( ) = () ( ) 11
2
11
3
11
5== .
( ) = () + () ( )=
= () + 1 () [() ( )] =
= 1 () + ( ) 11
7
11
3
11
71 =+=
4
.
1525
34530
25
...18122543
=+
=
+++++
=
tttxA
1525
34530
25
...14162543
=+
=
+++++
=
tttxB
5.
[ ]225
2
3
222 )15(...)15()1518()1512(25
1++++= ttS
A
[ ]225
2
3
222 )15(...)15()1514()1516(25
1++++= ttS
B
( )25
161133
25
1 222222=+=
BASS .
.
( ) ( )
==22
2
2
2
22
1525
16
25
16
B
B
A
A
BA
x
S
x
SSS
( ) ( ) ( )
=
=2
2
B2
2
B
2
A
1525
16CV
225
1
1525
16CVCV
( ) ( )
=
=
25
161
225
1CV
22525
16
225
1CV
2
B
2
B
( )25
1CV
515
3CV
25225
9CV
BB
2
B=
=
= .
1 &
2008
1o
A. f(x)=c ( x
) 0, (c)= 0.
8
B.
x, , 0>x , 0 0, ( )x
x
2
1=
2
. x
o
xx
xx
o
lim =
2
. ,
.
2
2
xe
xxf
1)(
= , x .
. 1
)(lim 2
1 x
xfe x
x
7
. xxfex = 2)(
9
. f(x).
9
2 3
5 .
( )
:
A B
20 26
26 32
24 19
22 20
18 23
.
.
5
. 38
40 , ; (
).
5
. SA
SB
.
7
.
. 3,311
8
4
50% , 30%
.
. , ,
;
7
. :
: , .
10
7
5
1 P(B)
9
. :
xBPxxxf )(2
1)(
23+=
x
. f(x) .
9
3
1
. : ( cxf =)( ). . 28 .
.
. : . 96 . .
. -
-
-
-
- .
2
. 1
11
11
)(22
+=
=
xe
x
x
e
x
xfex
xx
2
1
1
1lim
1
)(lim
12
1
=
+
=
xx
xfx
e
xx
.
. =
=
=x
xx
x e
exex
e
xxf
2
))(1()1()
1()(
xx
x
x
xx
e
x
e
xe
e
xee =
+=
=
2)11()1(22
: xe
xexfe
x
xx
=
= 22
)( .
. ex
> 0, x R, :
i) f (x) = 0 x = 2
ii) f (x) > 0 2 x > 0 x < 2
iii) f (x) < 0 2 x < 0 x > 2
:
4x
f
f -+
1
e2
2 +-
f (, 2] [2, +),
0)2( =f .
f x = 2, 2
1)2(
ef = .
3
. 225
110
5
1822242620==
++++=
Ax
245
120
5
2320193226==
++++=
Bx
. / 11
19
22
38= ,
/ 3
5
24
40=
11
19
3
5< .
. ( ) ( ) ( ) ( ) ( )[ ]=++++= 222222 221822222224222622205
1
AS
[ ]=++++= 22222 )4(02)4()2(5
1
( ) 85
40164164
5
1==+++=
228 ==A
S
( ) ( ) ( ) ( ) ( )[ ]=++++= 222222 242324202419243224265
1
BS
[ ]=++++= 22222 1)4()5(8)2(5
1
( ) 225
11011625644
5
1==++++=
511222 ==B
S
. 11
2
22
22
x
SCV
A=== .
24
112
x
SCV
B
B
B== .
AB
CVCV > 24111111
1
24
11
11
2
24
112>>> ,
3,311 11 3,3 > 24 36,3 > 24.
.
4
.
: 5,0)( =AP , 3,0)( = BAP .
. )( BAP .
=+= )()()()( BAPBPAPBAP =+ )()()(1 ABPBPAP
=+= )]()([)()(1 BAPBPBPAP =+ )]()(1 BAPAP
== )]()([1 BAPAP10
77,03,01)(1 === BAP .
. )()( BAPBPBAB10
7)( BP .
3,0)( = BAP :
== 3,0)(5,03,0)()( BAPBAPAP5
12,0)( == BAP .
)(5
1)()( BPBPBAPBBA .
. )(3)( 2 BPxxxf += .
f ).(121 BP=
5
1)( BP
5
12)(12 BP 0
5
7
5
121)(121 xf Rx , f
.
1 &
2009
1o
A.
( ) = ( ) + ( )
10
B. x1, x2,, x X
( ), fi
xi, i = 1, 2,, .
5
. ,
.
. f, g
(f (x) g (x)) = f (x) g(x) + f (x) g (x) 2
. A , ,
A B = A 2
. f (x) = x
( x) = x 2
.
.
2 . .
2
2o xi, i = 1, 2, 3, 4
i, i = 1, 2, 3, 4. 2
x2 = 3 . 4=x .
2
xi i
2 6
3 ;
5 3
8 4
. 2 = 7.
9 . 4,9.
9 . X .
2,29,4 .
7
3o f (x) = x3 6 x2 + x 7, ,
2 f (x) + f (x) + 15 = 3 x2, x . = 9
7
. 1
)(lim 2
1 x
xf
x
.
8 . f,
y = 3 x.
10
4o
02,62
ln)( 2 >++= xx
xxf
.
.
. f f .
6 . f .
6
3. f (2), f (4), f (8), f (3) f (5) .
. R ,
4
1ln3+=R 64ln 2 +=
7
. = {1, 2, 3, , 100} . A
,
= { | R + < 2}
6
4
1
. 150 . .
. , 65 . .
. , , , ,
2
)
+++
+++=
+++==
=436
48353624/
2
244332211
4
1 v
v
v
xvxvxvxv
xxvxi
i
i
7359452359)13(413
32153124
22222
2
2=+=++=+
+
+++= vvvvv
v
v
.
)
=
+++
=
=
=
v
xxvxxvxxvxxv
v
xxv
Si
ii 2
44
2
33
2
22
2
11
4
1
2
2)()()()(
)(
=+++
=
+++
+++=
20
164131746
4376
)48(4)45(3)43(7)42(6 2222
9,420
98
20
643724==
+++= .
) 4
2,2
4
9,4CV
2
===
x
S
x
S%10%55
20
11
40
22>=== ,
.
5 3
) f = R.
f R .
: f (x) = 3x
2 12x + ,
f (x) = 6x 12. :
2f (x) + f (x) + 15 = 3x2 2(6x 12) + 3x
2 12x + + 15 = 3x
2
12x 24 12x + + 15 = 0 = 9.
) x 1:
1
9123
1
)(2
2
2
+=
x
xx
x
xf
1
93
)1)(1(
)3)(1(3
)1)(1(
)34(3 2
+
=
+
=
+
+=
x
x
xx
xx
xx
xx
.
: 32
6
1
93lim
1
)(lim
112
=
=
+
=
x
x
x
xf
xx
.
) (xo, f (xo)) .
xy 3=
3)(3 o == xf =+ 39123 o2
oxx
044012123o
2
oo
2
o=+=+ xxxx .20)2(
o
2
o== xx
(2, f (2)).
5729262)2( 23 =+=f (2, -5).
: += xfy )2( , . += xy 3 .
.1235 =+=
: 13 += xy .
6 4
. . x
x
x
x
xxf
2
2
2
2
2
11)(
=
== .
x > 0 :
20)( == xxf
200)( >>> .
:
=++++== )2648(ln)162(ln)8()2( 22 ffR
.34
1ln38ln2ln +=+=
, :
.64ln2624ln)4( 22 +=++=f
. =
1
&
2010
1. t1, t2, ..., t
, x .
1 2
, , ...,v
t x t x t x .
.
7
2. x1, x2, , x X w1, w2, ..., w
(), .
4
3. . .
4
4. ,
, , , .
) f , g x0 ,
0 0 0
( ( ) ( )) ( ) ( )lim lim limx x x x x x
f x g x f x g x
=
) x > 0 ( )1
x
x
= .
) x = f (t), t0 (t0) = f (t0).
) f
, x1, x2 x1 < x2 f (x1) < f (x2).
) ,
.
10
2
2( ) 2 1 1f x x x= + , x
1. 1
( ) 1lim
1x
f x
x
.
10
2. f x0 = 0.
10
B3. xx.
5
, , 160 ,
, 5 , :
xi
i
[0 ...) ... 20
[... ...) 6 40
[... ...) ... 45
[... ...) ... 30
[... ...) ... 25
160
1. c 4.
6
2. ,
x s.
8
3. .
5
4. ,
: 7
14 .
6
3
2
12 2
1
1
k
i ik
i
i i
i
x
s x
=
=
=
, (), ()
21( ) ln( ( )) ( ( )) ( ), ( )2
f x x P A x P A P B x P A= + >
1. f .
13
2. f 0
5
3x = f (x0) = 0,
:
2( )
3P A =
1( )
2P B =
2
2 5
( )6
P A B = ,
:
3. , .
5
4. , .
5
4
1. :
( ) ( ) ( ) ( )1 2 1 2... ( ... )vt x t x t x t t t vxv
+ + + + +
= = 1 2...
0v
t t tx x x
v
+ + = = .
2. 86, 87, .
3. . 140, .
.
.
4.
1. 1x : 2 2 2( ) 1 2 1 1 1 2 1 2 2( 1 1)
1 1 1 1
f x x x x x x x
x x x x
+ + + = = =
=
=2 2 2
2 2
2( 1 1)( 1 1) 2( 1 1)
( 1)( 1 1) ( 1)( 1 1)
x x x x x x
x x x x x x
+ + + + =
+ + + +
=
2
2 2 2
2( ) 2 ( 1) 2
( 1)( 1 1) ( 1)( 1 1) 1 1
x x x x x
x x x x x x x x
= =
+ + + + + +
21 1
( ) 1 2lim lim 1
1 1 1x x
f x x
x x x
= =
+ +
.
B2.
2 2
2 2 2
1 1 2 1( ) (2 1 1) 2 ( 1) (2 1)
2 1 1 1
xf x x x x x x
x x x x x x
= + = + = =
+ + +
f x0 = 0 :
2
2 0 1(0) 1
0 0 1f
= =
+
.
3. xx
: = f (0) = 1, 0 < , 3
4
= .
5
1. c, [0, c) [c, 2c).
2 6, 2
6 42
c c
c
+= = .
2.
i
i
[0, 4) 2 20
[4, 8) 6 40
[8, 12) 10 45
[12, 16) 14 30
[16, 20) 18 25
160
5
1
1 1 1600(2 20 6 40 10 45 14 30 18 25) 10
160 160 160i i
x x
=
= = + + + + = = .
5
2 2 2 2 2 2 2
1
1 1( ) 20(2 10) 40(6 10) 45(10 10) 30(14 10) 25(18 10)
160 160i i
i
s x x
=
= = + + + +
14000 25
160= = . s = 5 .
3. 5
CV 50%10
s
x
= = = >10%. .
4. 2 3 4
1 1 1 140 45 30
( ) 70 74 2 4 2( )( ) 160 160 160 16
N AP A
N
+ + + +
= = = = =
.
: s 2
:
5
5
2 2 2 21
1
1 1( ) (20000) ( ) 125 100 25
160 160 160
i i
i
i i
i
x v
s x v x=
=
= = = =
. s = 5 .
6
1. ( )
2
1 ( )1( ) ( ( )
( ) ( )
x P Af x x P A
x P A x P A
= = =
(1 ( )) (1 ( ))
( )
x P A x P A
x P A
+ +
.
1
( ) 0 ( ) 1f x x P A= = + 2
( ) 1x P A= .
x1 > P(A) () + 1 > () 1 > 0 ,
x2 < P(A) () 1 < () 1 < 0, x2 .
( )f x :
) x > P(A) x P(A) > 0
) x > P(A) x P(A) > 0 x P(A) + 1 > 1 > 0.
) ( ) 0 1 ( ) 0 1 ( ) ( ) 0f x x P A x P A f x > + > < + <
1 ( ) 0 1 ( ).x P A x P A + < > +
f :
P( )x
f (x)
f(x)
+
-+
1+P( )
f ((), 1 + ()]
[1 + P(A), +). f x = 1 + ()
( ) ( ) ( )21
1 ( ) ln 1 ( ) ( ) 1 ( ) ( ) ( )2
f P A P A P A P A P A P B+ = + + + =
21 1ln1 1 ( ) ( )2 2
P B P B= + = .
2. f x0 = 5/3, 1 :
1 + () = 5/3 () = 5/3 1 () = 2/3.
f (x0) = 0 1
1 1(1 ( )) 0 ( ) 0 ( )
2 2f P A P B P B+ = = = .
3. : ( ) 1 ( )P A B P A B = .
( ) ( ) ( ) ( )P A B P A P B P A B = +
( ) ( ) ( ) ( )P A B P A P B P A B = + = 2 1 5 1
3 2 6 3+ = .
1 2
( ) 13 3
P A B = = .
4. : [ ]( ) ( ) ( ) ( ) 2 ( )P A B B A P A P B P A B = + .
[ ]2 1 1 2 1 2 1
( ) ( ) 23 2 3 3 2 3 2
P A B B A = + = + = .
1
&
2011
1. :
( ) = ( ) ( ).
7
2. , ;
4 3. fi xi .
4
4. , ,
, , ,
, .
) .
2
) ,
6R x .
2 )
(f (g (x))) = f (g (x)) g(x) 2
) .
2
) , 10%.
2
, . .
P (M) = 1
4, P (A) = 4 2
P (K) = 7
54
+ , . ()
64 < () < 72,
2
1. () = 68.
6 2. .
8
3. , .
6
4. . .
5
, ,
. fi % :
(8, 0) (10, 10) (12, 20) (14, y) E (16, yE) (18, 10) (20, 0)
y, y
.
1. y
y
,
14200
7 2. fi %.
3
3. fi %
.
7
4. 15000 .
. 4
5.
80. 4 .
4
3
1 11 22
3 10 5( )
x x x
f x e
+ = , x
1. f .
8
2. , ( ), ( )
f
( ), ( ), ( ), ( ).
8 3.
21 3 1
5 2 3
( )
x
x x
h x e
= , x
) f (x) = h (x).
3
) A x1 < x2 < x3 vi = 2 xi + 1, i = 1,2,3 xi .
6
4
1. , . 152 . 2. . ,
.A B =
3. , . 65 . fi xi ,
ii
vf
v= , i xi
. , 100 xi, .
4. ) , ) , ) , ) , ) .
1. ( ), ( ), ( ) ( ), ( )
( ) . 1
( )4
P M = , : ( ) 1
( ) 4 ( )( ) 4
N MN N M
N= =
.
64 < ( ) < 72 64 < 4 ( ) < 72 16 < ( ) < 18. ( ) , ( ) = 17.
( ) = 4 17 = 68.
2. = , ( ) = () = 1 (1),
= , = , = , , , , .
(1) ( ) + ( ) + ( ) = 1.
2 21 7 1
4 5 1 4 5 1 0 1 .4 4 4
+ + = + = = =
= 1 ( ) = 4, = 1 0 ( ) 1.
1
4 =
1( )
4P A = ,
1( )
2P K = ,
1( )
4P M = .
1
4 =
.
5
3. 1 ( ) 1 1 1
( ) ( ) ( ) 68 17.4 ( ) 4 4 4
N MP M N M N
N= = = = =
, 1 ( ) 1 1 1
( ) ( ) ( ) 68 17.4 ( ) 4 4 4
N AP A N A N
N= = = = =
' 1 ( ) 1 1 1( ) ( ) ( ) 68 34.2 ( ) 2 2 2
N KP K N K N
N= = = = =
4.
. . ,
, :1 1 1
( ) ( ) ( )4 4 2
P A M P A P M = + = + = .
1. : 7
1
8 0 10 0,1 12 0,2 14 16 18 0,1 20 0100 100
E
i i
i
y yx x f
=
= = + + + + + +
14,2x = y = y
14,2 = 1 + 2,4 + 30100
y + 1,8 14,2 5,2 = 30y
100
y =
9
30 .
: y = y = 30
2 : y + y = 60
// xx y = y. y = y = 30.
2. :
30
20
10
fi%
xA
B
E
Z
H
8 10 12 14 16 18 20( )
i
6
3. :
[ - ) ix %if
9 - 11 10 10
11 - 13 12 20
13 - 15 14 30
15 - 17 16 30
17 - 19 18 10
100 4. 3,
40%. 5. = 80,
. ,
4, : 80 40% = 32.
1. f ,
2
'1 11 2
3 10 5'( )x x x
f x e
+
= =
2'1 11 2
3 23 10 51 11 2
3 10 5
x x x
e x x x
+
+ =
2 21 11 2 1 11 2
3 23 10 5 3 10 51 11 2 11 2
33 5 5 15 15
x x x x x x
e x x x e x x
+ +
= + = +
21 11 2
23 10 511 2
'( ) 0 015 15
x x x
f x e x x
+
= + =
21 11 2
3 10 5( 0 )x x x
e x R
+
2 11 20
15 15x x + =
2 1 215 11 2 0 .
3 5x x x x + = = =
:
x
f
f
- +1/3
- +
2/5
+
7
f :
1,3
,
1 2,
3 5
,
2,
5
+
.
2. f 1 :
. 1
1
3x =
. 2
2
5x = .
( ) ( ), 1
( )3
P A = 2
( )5
P B = .
, = .
:
1
( ) ( )3
P A B P A = = .
1 1
( ) ( ) ( ) 03 3
P A B P A P A B = = = .
1 2 1 2
( ) ( ) ( ) ( )3 5 3 5
P A B P A P B P A B = + = + = .
2 1 1
( ) ( ) ( )5 3 15
P B A P B P A B = = = .
3. )
2
21 11 2 1 3 1 1 1( ) ( )3 10 5 5 2 3( ) ( )
xxe
x x x x x
f x h x e e
+
= =
( )
2 2
2 2
2
2
2
2
2 2
1 2 3
1 11 2 1 3 1 11 2 3 15 3
3 10 5 5 2 3 10 5 2 3
11 2 3 15 3 0
10 5 2 3
11 95 2 3 1 0
2 2
1 53 0 5 6 0 0, 2, 3.
2 2
x x
x x x x x x x x x x
x
x x x x x
x
x x x x
x
x x x x x x x x
+ = + =
+ =
+ + + =
+ = + = = = =
8
) :
1 1
2 1 2 0 1 1.x = + = + =
2 2
2 2 2 2 1 5.x = + = + =
3 3
2 2 2 3 1 7.x = + = + =
:
x i v i x i v i
x 1 = 0 v 1 = 1 0
x 2 = 2 v 2 = 5 10
x 3 = 3 v 3 = 7 21
v = 13
0 10 21 31
13 13x
+ +
= = .
1
& & ..
23 2012
1. f, g ,
(f (x) + g (x)) = f (x) + g(x), x . 7
2. .
4 3.
X, 0x > , 0x < ; 4
4. , , , , , , . )
( 2). ) f x0 y = f (x) x,
x = x0 ( 2). ) , , () > ()
( 2). ) ,
( 2). )
00lim
x xx x
= , x0 ( 2).
10 ( ) [5,45) . .
2
50%
F %
5 15 25 35 45
F %3
F %1
i
0
1. , .
4 2. , = 8
( 3) ( 5).
() x i v i f i% i F i%
[5, ) + 4[ , ) 3 6[ , ) 2 + 8
[ , 45) 2 8
3. x s .
( : 84 9,17) 8
4. 37 .
5 . 3,
231v
v +
3
221
v
v
+
+
211
v
v
+
+
( )221 2 3 2limx xx x + + . 1.
. 7
2. = 3. 6
3. .
6 4. 32,
. 6
21 ln( ) ,xf x x+
= x > 0. 1. f .
5 2. (x, f (x)), x > 0 f.
yy Ox (x, 0) xx Oy (0, f (x)). O , , .
7 3. : y = x + , 10,
f (1, f (1)). (xi, yi), i = 1,2,,10 , xi 10x = sx = 2.
yi .
8 4. ,
B , f (()) + f (( )) 2 f (( ))
5
1
&
& ..
23 2012
1. , . 31 . .
2. , . 148 . .
3. , . 96 . .
4. ) , ) , ) , ) , ) .
B
1. 50% pi, pi
= 25.
2. 50% = 25
+ 4 + 3 6 = 2 + 8 + 2 + 3 2 = 4 + 6 + 8 2 = 8
pi
(pi) xi vi fi% Ni Fi%
[5-15) 10 12 20 12 20[15-25) 20 18 30 30 50[25-35) 30 24 40 54 90[35-45) 40 6 10 60 100 60 100
B3.
4
110 12 20 18 30 24 40 6 1440
2460 60
=
+ + +
= = = =i i i ii i
i
v x
x
v
pi.
pi
( ) ( ) ( ) ( ) ( )4
22 2 2 2
1 1 2 2 3 3 4 42 1=
+ + + = = =
i ii
v x x v x x v x x v x x v x xS
v v
( ) ( ) ( ) ( )2 2 2 2
12 10 24 18 20 24 24 30 24 6 40 24
60
+ + +
= =
12 195 18 16 24 36 6 256 5040 50484
60 60 6
+ + + = = = .
2
pi pi 2
84 9,17S s= = pi.
4.
pi 35 45 10% pi x% pi
pi pi 37 45 . :
45 35 10 10 1010 80 8%.
45 37 8x x
x x
= = = =
1. , pi,
:
( ) =P
2 2 2
22 2
1 1 1
2( 3 2) 2( 3 2)( 3 2) 2( 1)( 1)
( 1)( 3 2) ( 1)( 3 2)lim lim limx x x
x x x x x
x x x x x x x x
+ + + + += = = =
+ + + + + + +
2 21 1
2( 1) 2( 1)1.
( 3 2) ( 3 2)lim lim
= = =
+ + + +x x
x x
x x x x
pi 2
.
2. 2
3( )
1P
=+
, 2
2( )
1P I
+=
+
, 2
1( )
1P
+ =
+, ( ) 1P = .
( ) ( ) ( ) ( )P P P I P = + , :
2 2
2 2 2
3 2 11 1 3 2 1 3 0 0 3
1 1 1
+ += + + = + + = = =
+ + +
.
pi 3 pipi 3 = .
3. pi :
( ) ( ) .
9 5 4 6 3
(( ) ( )) ( ) ( ) 2 ( ) 210 10 10 10 5
P P P P = + = + = = .
4. 4 2
( )10 5
P = = . ( )
( )( )
P
=
.
( ) 2 32 2
( ) 80( ) 5 ( ) 5
= = =
.
3
1. H f pi (0, +) pi pi pi
,
21 ln x
f (x)x
+ = =
2
2
2 2
1x 2 ln x 1 ln x
2ln x 1 ln xx
x x
= =
2 2
2 2
ln x 2ln x 1 (ln x 1)
x x
+ =
pi f(x) < 0 x (0, e) (e, +), f (e) = 0, pipi f
(0, +).
2. : 2
21 ln xE(x) x f (x) x 1 ln x
x
+= = = + .
E(x) pi (0, +) :
2 2ln xE (x) (1 ln x) 2ln x(ln x)x
= + = = .
2ln xE (x) 0 0 ln x 0 x 1
x = = = = .
x
E (x) +1
+0
E(x)
0
min
21 ln 1
E(1) 1 11
+ = = . x = 1, f(1) = 1, pi () = (),
pi .
3. pi : y = x + pi pi Cf
(1, f(1)), : = f(1) = 1.
: y = x + , 10.
pi pi xi x 10= yi = (1)xi + pipi
:
y 10= + y x
S | 1| S 2= =
pi yi i = 1,2 , 10 pipi:
yS
0,1| y | .
4
yS
0,1| y |
2 10x 20
x 100 +
+
( )10 20 10 20 10 20 + + + 10 30 .
: ( , 10] [30, ) + .
4.
(i) ( ) ( )P A P A B pi f
( ( )) ( ( ))f P A f P A B (1)
(ii) ( ) ( )P A B P A B pi f
( ( )) ( ( ))f P A B f P A B (2)
(1) (2) :
( ( )) ( ( )) 2 ( ( ))f P A f P A B f P A B+ .
___2000_2011200020012002200320042005200620072008200920102011
2012ggen_ap_2012
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