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