επαναληπτικά θέματα μαθηματικά γενικής παιδείας 2015

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  • 1. - 1 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499

2. - 2 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 23 , TEDx Athens, designwars street artist ino " , ." , 1911-1996 3. - 3 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 O !! f(x) , . ( !). mm ! ( ). ! , - . , . . : 3 146 , 45 46 : 34 2 98 2 99 3, , , 1[2 ( 1) ] 2 v a v S + = 1 1 1 vS a = . : http://mathhmagic.blogspot.gr/2011/03/blog-post_30.html : http://cutemaths.wordpress.com/2014/03/03/%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE %B1%CF%84%CE%B9%CE%BA%CE%B1-%CE%B3%CF%80-%CE%B3- %CE%BB%CF%85%CE%BA%CE%B5%CE%B9%CE%BF%CF%85-%CF%84%CE%BF- %CE%B1-%CE%B8%CE%B5%CE%BC%CE%B1/ mathematica http://mathhmagic.blogspot.gr/2012/05/mathematica.html . ! 4. - 4 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 - 1. . 1) . 2) . 3) . 4) c , c. 5) f f '(x) 0< x f . 6) 1 P( ) 5 = , 4 P(B) 5 = , 3 P(A B) 5 = . 7) ' B P( ) P(B) 1 + < . 8) P( ) 0.5 = P(B) 0.6= , . 9) . 10) . 11) . 12) , 68% ( x 2s , x 2s+ ) 13) CV 10% 14) R . 15) . 16) X . 5. - 5 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 17) . 18) [ , ) - 2 . 19) f , 1x , 2x 1x < 2x f( 1x ) ) () , : i) () , . ii) 10.500 , . 27. 2 ( ) 10 11,f x s x x x x= + + , x s . f (-1,f(-1)) : 2011y = : ) f . B) . ) f . ) f 1 : i) . ii) f . 28. 20 : 20 1 100i i t = = 20 2 1 1000i i t = = Y , 2 5Y X= + . . 29. 1 2, ,..., vx x x . ) 1 2, ,..., vx x x 20% 25 . i) . ii) 2 2 2 1 2, ,..., vx x x . iii) 10 , , . ) 2 , . 30% 60% 50% , : 16. - 16 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 i) . ii) , , . ) . 1 . 0.8% . ( 2007) 30 ( ) ) , CV . i) . ii) s CV x = 0x < . iii) CV . iv) , 50%CV = . v) CV , 100%CV . vi) 0CV < . vii) , . 31. . P(x) ( ) x 3 2 ( ) 15 600 300,0 30P x x x x x= + + < < ) 10 . ) . ) 10x = . ) . 32.) , . A B ( ) 0.2P A = 2 2 ( ) 4 ( ) ( ) lim 2x x P A B P A P B x = , ( ')P B ( )P B A . ) }{1,2,..,1.000 = . , : 2 16[ ( )] 25 ( ) ( ) 10 0(1)P B P B P + = : i) ( ), ( )P B P ii) . . 17. - 17 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 33. IQ x 2 s . IQ 85 16% 130 2.5% , : ) , . ; ) IQ 145. 34. 210 gr . . %if [ )195 197 10 [ )197 199 10 [ )199 201 55 [ )201 203 20 [ )203 205 5 ) 200 gr. ) . ) . 200 gr. ) 8 gr. . . 35. 8 1 10. ) R. B) x 8 [ ]1,10x . ) 10x = 8 . 18. - 18 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 36. { }1, 2 3 4, , = . 1 2 3 4 1 1 1 1 ( ) , ( ) , ( ) , ( ) 4 4 4 4 P P P P + + + + 1 9 . : 2 2 2 2 2 1 2 3 4 1 1 1 1 2 ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( ) 4 4 4 4 9 P P P P + + + = CV . 37. , 6 . . 1 2 1 5 . 1400 2000 , : ) . ) s , 20% 3000 . ) 80% , , s 300 . 38. ... 15%. , 10% . 1000 ( ), : ) . ) ...; 19. - 19 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 39. 6 , . x , 0 x 3< < , : ) x : 2 f(x) 4x(3 x) ,0 x 3= < < ( ,, V = ). ) x . ) x 0 f(x 2) 8 lim x + . ) i iy f(x ),i 1,2,3,4,5= = 1 2 3 4 51 x x x x x 2= < < < < = , y 12= , ys 2= yCV . R iy , i 1,2,3,4,5= . 12 0 < < , iy CV y R CV 2CV 12 = + . ) , . A , B A B , : 2 P(A) 3 P(B) P(B) 3 P(A) 20. - 20 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 40. {1,2,3,4,5,6} = . ( ) (4 ) 4,= + x f x e x x . )N '( ), ''( )f x f x ) () f (0, (0))M f 2 (5 )= y x ) ={ / () () 1 2014 4 = +y x } ={ / 'f } E) 1 = : i) '(0), (0)f f ii) 0 3 3 lim + h h e h h . 41.( ) A) 1) . 2) 50% 50% 3) 3 0x s= , . 4) , . B) . i) ; 70 90 75 85 21. - 21 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ii) ; iii) ; 42. {1,2,3,....,2 } = . R 1,2,3,..,2 2 40R + = . : ) R,,. ) {1,2,3,...., }A R= ) 2 ( ) ln( 5 )f x x x = + + . ) 20 2 1 2870i i x = = 1,2,3,,2 ( ) 33.25s = . 43. ( ) 39,x f x e x = + . ) f ( 0,f(0)) xx . ) 1 2 100( ), ( ),..., ( )f x f x f x 1 2 100( ) ( ) ... ( )f x f x f x< < < x s. i) f 40. ii) 40 > iii) 4s > iv) 3 2 ( ) 6 3 ,g x x x sx x x= + + + . 44. 2 ( ) 1 ln( ), 0f x x a a= + + > . ) fC (1,f(1)) xx 45o . ) =1 f . ) , f(P(A))=P(B). . 22. - 22 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 45. 1) , , A B ( ) ( )P A P B . 2) %iF ix . 3) . 4) . 5) . 46.( ) ) . x y , : x y z + = + ) 2 s : 2 22 1 v i i i x s x = = ) ( * v ) x 1 0, 2 s 2 1 4 s ( : ()). ) 1 2, ,...., vx x x x xs , 0 1, i i x x x y s = , 0y = 1ys = ( 0xs ). ) ( * v ) x CV : Do or do not there is no try. Yoda 23. - 23 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 1 3 CV < . 47) ( ) fC (1,1) 1 2 9....x x x< < < 1 2 9, ,..,M M M -2 -1. ) 1 2 9, ,..,M M M . ) 1 2 9, ,..,M M M . ) 0 (1 ) (1) lim h f h f h + 1 x1 x2 x9 A(1,1) Cf M1 M2 M9 1 120 , . . . , , . William Paul Thurston Fields 1982 24. - 24 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 48) 2 2 21 ( ) ( 20 ( 1) ( )), 60 g x x x x = + + + + ,, . g yy A(0, 1 3 ) : 20 (1 ) lim 1 x x e x x x = + ) ,,. ) =1,=0 =1. 30 1 2 3, ,x x x , 1 2 3, , 1 2 3, ,f f f . : ) 1 2 3( ) ( ) ( ) 1g g g + + = ) 1 2 3 1 ( ) ( ) ( ) 30 g f g f g f+ + = ) x s ,: i) 1 1 2 2 3 3( )g v x v x v x x+ + = ii) 2 1 1 2 2 3 3 20 ( ) ( ) ( ) 2 s v g x x v g x x v g x x + + + = ) h ( 1) 7h = . ( ) (180 ( 2) 20) (2 5),f x g x h x x= : ) Cf (2, (2))A f . ) Cf (2, (2))A f . ) ''(2)f . 25. - 25 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 49. , . 1 ( ) 4 P B = 1 ( ) 6 P A B = ,: ) ( )P B A , ( ')P B . ) 1 ( ' ') ( ) 12 P A B P B A = = ) 11 ( ) 12 P A 50. . 1. lim( ) ox x x x = 2. ( )( ) ' '( )cf x cf x= 3. . 4. 10%. 5. , A B 51. , . ) : ( ) ( ) ( ) ( )P A B P A B P A P B + = + ) ( ') 0.4P A ( ') 0.5P B : i) ( ) 0.6P A ( ) 0.5P B ii) ( ) ( ) 1.1P A B P A B + iii) A B . 52. 1 2,x x 1 2, , : i) s 1 2 1 2 1 2 v v s x x v v = + ii) 1 2v v= 1 2 1 2 x x CV x x = + May the Force be with you. Yoda 26. - 26 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 53.i)( ) ln ( ) , 0 x f x x x = > . iii) ,A B ,A B A B ( )3 2 ( ) ( )1 ( ) ( ) ( ) 1 1974 3 P A B P A B g x x x P A B P A B x = + + + . (: (i)) iii) 3 3 4 4 5 5 1 1 2 (2), ( ), ( ), ( ),...., ( ) 2 2 3 3 4 4 v f f f f f + + ln 2014 x = . . 54. 1 2, ,...,t t t x , s : 2 1 ( ) ( )i i f x t x = = ) f 'y y 22 (0, ( ))A s x + ) f 'x x , x = . ) f x 2 ( )f x s . 55. 1 2, ,...,t t t .... 2014. 3 3 3 1 2 1 ( ) [( ) ( ) ... ( ) ] 3 f x t x t x t x= + + + ) 2 '( )f x s = , 2 s x . ) ''(2 ) 3 5f x a= , 1 i i t = 3 2 21 1 lim ( 3 2)x x x x a x + = + ) 3 1 6042i i t = = , f (0, (0))f 22 ( ) 2014y s x x= + 27. - 27 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) 1 2, ,...,t t t , ( < ) .... 2014 16% 2029 . i) . ii) 13 . iii) .... 8 2014 29 . 56.) 2 ( ) (0.6 ) ,0 0.6g x x x x= . f . ) . 4 . i) 2 3 4 0.032f f . ( : ()) ii) 4 0.3f = . ; . iii) , : ) [ )16,20 ) [ )17,19 ) [ )12,15 [ ) if 12-14 0.1 14-16 0.3 16-18 3f 18-20 4f . 28. - 28 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 57. 5 c, : 3F 5F : 2 5 8 3 ,x x + ) =1 =10. ) 1 2 3 4% 10, % 30, % 20, % 30f f f f= = = = 5 % 10f = . ) 25% 16 25% 24 , =10 c =4. . ) 22 800, . 58. . ,. , , ,A B A B A B B A : ( ) ( ) ( ) 2 2 P B P A P A B< < , 1 ( ) 8 P A B = , 5 16 x = 1 4 = , : ) A B . ) ,A B . ) ,A B . ) ( ), ( ), ( ), ( )P A B P A P B P B A " . !!" John H.Conway 29. - 29 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 58.!!!! - - ,, . . 150% , 10% . 200% . . i) . ii) . iii) . 59. {0,1,2,3} = 4 (0) (1) 2 (2) 4 (3)P P P P= = = i) . ii) 2 23 ( ) ( 3 5) 666 2 f x x x = + + , , x . { / 1}f x = = iii) : 2 1,1,6, ,3,3,2,6 3 . x { / 2.5}B x = iv) : ( ), ( ), ( ), ( ), ( ' ), ( ' ), ( ' '), ( ' ')P A B P A B P A B P B A P B A P A B P A B P A B 30. - 30 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 60. , 1 0 2 1, 1 2 + = . .. , , 1 ( )P = , 2 ( )P B = . : ) ' = . ) ( )P . ) ( ) ( )P P . ) , ( )P . 61. ( ) , , 1x x f x e e x = > ) '( ), ''( )f x f x ) ''( ) ( 1) '( ) ( )f x f x f x = + ) fC (0,f(0)). ) f . ) 1x x e e + + x . 62. , , . fi % : A(8,0),B(10,10),(12,20),(14,y),(16,y) Z(18,10),H(20,0) y , y AB. A) y , y , B) fi%. ) ( fi, fi %Fi, Fi%) . !! . 31. - 31 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) x . ) 15000 . . ) 80. . 63. ) . 1. 2 ( ) ln 2f x x= + , '( )f x : . 1 2 2 x + . x . 2x . 1 2 2 x x + 2. ( ) x x f x e + = : ( ) '( )f a f a= , . 1 2 a = . 0 = . , 2 = + . , = 3. 2 ( ) x f x e= (1, (1))f : . 2y x e= + . 2y ex e= + . 2y ex e= . 2y x e= + 4. 2 2 ( )f x x = . fC 1 (2, ) 2 A : .-2 . 1 2 . 1 2 . 1 . 2 5. 1 ( ) ( ) x f x e + = '(1)f = .0 . e . e . e . 1 32. - 32 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) , . . 1. . . 2. . . 3. . . 4. . . ( )' 5. . ' 6. . 7. . 64. 2014 ( ) ,f x ax x= + , , . ) fC y x= (0,0) (1,1) , ,. B) =1 =0 , i) fC (3,f(3)). ii) 2014 2014 0 (3 ) 3 lim ( 2014)h h h h + + iii) fC (): 2014y x= iv) fC (0,-2013). 33. - 33 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 65. 4 . R=16. ) c=4 =20. ) : 2 , , %, , %, ,i i i i i i i i ix f f F F x f x f ) x , s x s x s+ . ) 4 . . fi% [ ) 2 a [ ) a [ ) 10 3 2 a [ ) 2a G.H.Hardy "" . , . . , . , . (, ) , , , . 34. - 34 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 66. , 3 21 9 1 ( ) 2014, 2 40 20 f x x x x x= + + ( ), ( ), ( ), ( )P A P B P A B P A B f ( ), ( ), ( ), ( ), ( ), ( )P A P B P A B P B A P A B P A B x f. ( ), ( )P A B P A B . 67 , A . ) 2 ( ) 2 2 1,f x x x x= + . ) : ( ), ( '), ( ), ( )P A P A P P i) . ii) : 2 21 (2 ( ) 2 ( ) 1) 4 s P A P A= + iii) 2 2 CV ( ) ( ')P A P A= 68.A) x s 30 1 2 30, ,....,x x x . 30 2 2 1 3030i i x s = = . ) , , 'A B B A . : 3 240 ( ) 2 ( ) ( ) 2, 3 f x CVx P A B x P A B x x= + + , CV () 23 ( ) 666, 2 g x x ax x= + , . i) f . 35. - 35 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ii) Cf (0,f(0)). iii) 4 ., 1 ( ) 2 P A B = . iv) 0 (1 ) (1) lim 2 h g h g h + = . v) g ( )x P A B= ( )P B . 69.. 0.46 A1. =16. A2. x . B. 2 ( ) 1,x f x e x = + ( )3 { / 3.31} 2 x = + . 1) ', ''f f . 2) '(0)f ; ix iN 1 2 30 3 50 4 100 . . 36. - 36 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 3) , : { / '(0) (0)}A f f= > { / ''(0) 256}B f= < : i) ( )P A B ii) ( )P A B iii) ( )P B A iv) ( )( ) ( )P A B B A ) x 5 , 6 ,3 ,10 , 1 { / 1} 1 x x = > + . 70. 1 2, ,..., x x x 0x s. f ( )21 ( ) 1 8 f x xx s x= + . . Cf (1, f(1)) 2y x= + , CV . . 2 lim ( ) 2 x s f x = , x s. . 4 1x s= = 2 12 2 1 1 i i i i X s X = = = , 1 2( ) ( ) ... ( )f x f x f x+ + + , . . , (2, 5) R . 37. - 37 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 71. 1 2 12x = 4. 1 1 , 2 10%. . . . ; . 1 4325, 1. . 1 , 14 . . 1 15 11, 1. 72. f ( ) ( )2 2 1, 0,f x x x x R = + + + . . () fC ( )( )0, 0f 1) () x x y y ( ) ( ) 2 1 4 E + = . 2) . 3) . 4) . 10 () 4 1 4 10 10%. 38. - 38 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 73. ( ) 2 ( ) 1 2 9,P P + = + ( ) (2 3) 2 x a e f x x = + + , x , . ) ,. ) 4, 5 = = . i) f . ii) A,B 1( )P A x= 1( ) ( ) 6 f x P B e = f 1x , ( ), ( )P A P B . iii) 1 2 ( ) , ( ) 2 3 P A P B= = , . iv) 1 2 ( ' ') 6 3 P A B . 74. 2 ( ) 1, 1 x f x x x = + + 1 2 3 4{ , , , } = 1 1 = , 2 0 = 3 41 < < . , 1 ( ) ( ) 3 i iP f = , 1,2i = 3 1 1 '( ) ( ) lim 6 1x f x P x = . ,, { / '( ) 0}A f = , { / ( ) 1}f = > 2 1 { / , } 4 x x x = + i) 1 2 3 4( ), ( ), ( ), ( )P P P P ii) ( ), ( ), ( )P P P ( )P B . () f , 45 . . ( , )y , 1,2,3,4 = (): 1y x= + 39. - 39 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 2 y = 5kyR = 3 4 , : y : kyR : 75. ={1,2,,}, * : 3 3 3 (2014 100 ) (40 1000) (60 1014) 0 + + = i) ( ) 25 = . ii) , : ={ 3 4} ={ 3 4} ( ), ( )P P . Bertrand A. W. Russell , . : . , , . 40. - 40 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 76) 5 dm. 20 dm x dm 0 + + R ( ), 1,2,..,15i iy E x i= = ( 2014) 41. - 41 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 77) 100 m. , x m. 1. , . i) , x : ( )2 ( ) 2 200 10000 , 0,100E x x x x= + ii) x ( )E x . x=50 , , 1,2,..,il i = , 1,2,..,ix i = , 1,2,..,ix i = 2x = 0,2s = : 2. =25. 3. il ix , 1,2,..,25i = . : 2 12 2 1 1 i i t s t = = = 4. il , 1,2,..,25i = . . { , 1,2,..,25il i = = , i 3 4} 42. - 42 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 78) x1=, x2=+5, x3=+10 x4=+35, . : 7 3 i i F = , 1, 2, 3, 4i = , . . =25. . f1 , f2 , f3 f4 . . . 19x = , . 79) 1AX = .000 125AS = . 800EX = 90ES = . ) . ) 250 . 20%. . ) ; (2005 .) 43. - 43 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 80) x 1 , x 2 , x 3 , x 4 =72 () 1 , 2 , 3 , 4 , 4 = 3 3 . x 1 x 2 50 30. . i , i=1,2,3,4 . x 3 x 4 . x 1 3. 10 R+72 x =52 R, x , , . (2009 ) 81) 3 2 4 ( )f x x x = + , x (0,1), >2 A. . f f . . f f(x) 3 2 x (0,1) B. = {1, 2, ..., } , ( ) 3 2 2 4 ( ) 3 ( ) + = ()= 2 98 () () . 1 ( ) 5 P = . 1 ( ) 6 P B = , .( (2009 ) 44. - 44 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 82) = {1,2,3,4,5,6,7} m x , 5 e x, x + 4, 7x, 1 x . g(x) = m x2 2 x + 3 , xR . m=2. . ={ / g (1, g(1)) xx}. N E. . , , , h (x)= 3 2 P ( A B ) 1 2 x 3 + P ( A ) 2 x 2 + x +2008 . 83) ) f , : 0 ( ) ( ) '( ) lim o o o h f x h f x f x h + + = ) f ( x,f(xo)) f(xo) f(x) x=xo. ) A B . ) . ) . ! . 45. - 45 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 84) , c. , : ) c . ) f1%=25 , . ) R, x s . ; ) 2 1 27000i i i x = = , . ) ; 85)) () : t1 = 0, t2 = 0, t3 = 1, t4 = 2, t5 = 4, t6 = 5 . ) x . ) f(x)=(t1x)2+(t2x)2+(t3x)2+(t4x)2+(t5x)2+(t6x)2,: i) f ( x ) = 0 ii) f( x ) = 6s2 , s2 iii) f ( x , f( x )). 50 fi 5 45 46. - 46 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 86) O , ( min), , 4 6,10,7,7. 2 2 2 2 1 2 3 4( ) 6( ) 10( ) 7( ) 7( )f x x x x x x x x x= + + + x=7 134 1 2 3 4, , ,x x x x . ) . ) . ) 4 ; . 87) 3 2 ( ) 3 2 x ax f x x = + + + , , . , ={1,2,3,4}. o: i) = { , /a 1 1 lim '( ) 3x f x } ii) = { f } 88) ( ) 2 1 2 0( ) , , x f x x x = + . ) f . ) 1 2 ( )f . ) 3 4 1( )f x < . ) 2 3 2 1 2 2 1 1 '( ) lim x x f x x xa x + + = . ) : 2 2( ) '( )xf x x f x = . 47. - 47 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 89) f : 2015 ( ) 1 1 ,f x x x= ) : i) f (1,f(1)). ii) . ) : ( ) 2020 6 ( ),g x f x x= . g , . ) 2 (1)f ,4,6,2 2,6,4, . g. 4, . ) , ,3,4a 1 2 3 4, , ,f f f f . : i) x . ii) s iii) 1 2 3 4, , , : 1 2 3 42 3 4 120 + + + =+ + + =+ + + =+ + + = 90) 0 ( ) 1P A< < . , : 3 2 ( ) 4 6 ( ) 3 ( ') 2018,f x x P A x P A x x= + ) : i) . ii) 2015 ( ( )) 2019 3 ( )f P A P A< < ) : i) f (0,f(0)) ii) . 48. - 48 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 iii) P(A) 20152. ) 1 1 1 2 2 2 20 20 20( , ), ( , ),..., ( , )A x y A x y A x y 20 2000y = 200yS = . 1 2 20, ,...,A A A . : 2 12 2 1 1 i i x s x = = = Andrew Wiles ,o Fermat , . : . . , . , , , , , !! 49. - 49 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 50. - 50 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 1.1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 2.) ( ) 2 2 2 2 2 22 26 10 2 1 0 25 10 1 2 0 (5 1) 0 5 1 0 0 1 5 + + = + + + = + = = = = = 1 2 3 1 1 ( ) ( ) ( ) ( ) 2 2 1 1 1 1 1 2 5 5 2 10 P P P P a = + + = + + = + + = = ) 3 4 2 4 ( ) ( ) ( ) 3 ( ) g x P x g x P x = = g (1,g(1)) =y x (1) 1g = 4 4 1 3 ( ) 1 ( ) 3 = =P P 1 2 3 4 5 5 5 5 5 ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 5 ( ) 1 ( ) 1 5 5 10 3 6 5 1 ( ) 1 ( ) 6 6 P P P P P P P P P + + + + = + + + + = + = = = ) 1 2 5 4( ) ( ) { , , , }K A B B A = = 3 1 9 ( ) 1 ( ) 1 10 10 P K P = = = 'A B = = 1 ( ) ( ) 2 P P = = 3. 51. - 51 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 : ) 23 16% . 11 14 17 20 23 26 29 99,7% 95% 68% 34% 34% 13.5% 50% 16% 2.5% 0.15% 13.5% 50% 16% 2.5% 0.15% 2.35%2.35% 15 17 19 21 23 25 27 99,7% 95% 68% 34% 34% 13.5% 50% 16% 2.5% 0.15% 13.5% 50% 16% 2.5% 0.15% 2.35%2.35% 20Ax = 3As = 21Bx = 2Bs = 52. - 52 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) 17 16% 2,5% . 4.) : 240 340 ( ) , ( ) 400 400 P B P A= = : 240 340 580 ( ) ( ) ( ) 1 400 400 400 P B A P B P A = + = + = > . ) ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) 1 ( ) 340 60 3 ( ) 1 ( ) ( ) 400 400 20 P B A P B P A P B A P B P B A P A P B A P B A P B A = + = = = = = ) ) 3 3 ( ) ( ) ( ) 20 20 3 240 3 ( ) ( ) ( ) 20 400 20 180 9 ( ) ( ) 400 20 P B A P B P B A P B A P B P B A P B A P B A = = = = = = 340 9 8 ( ) ( ) ( ) .. 4 2 2 500 0 20 P A B P A P B A = = = = = ) (( ) ( )) ( ) ( ) ( ) ( ) ( ) ( ) 9 11 ( ) ( ) 1 20 20 P A B B A P A B P B A P A P B A P B P B A P B A P B A = + = = + = = = = 5.) ( ) ( )( )2 2 2 2 2 2 2 2 2 2 2 ( ) 1 11 lim lim 22 1 2 1 1 1 1x x x x x x x x xx xx x x x x x x + + = = = + + + + ( ) ( )2 2 2 2 2 2 2 ( ) 1 11 lim 2 1 1 x x x x x x x x x + + = = + ( ) 2 2 2 2 2 ( ) 1 11 lim 2 1 1x x x x x x x x x + + + ( ) ( )2 2 2 2 2 2 2 2 2 2 ( ) 1 11 1 1 lim 1 1 2 2 2 2x x x x x x x x x x x x x x + + = + + = = 53. - 53 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) i) 1 1 1 1 1 ( ) 2 2 2 P x > > > , 1x , 1x 2x . 1 1 ( ) 2 P x . ii) 1 2 4 1 2 4 1 1 2 2 4 4 1 2 41 2 4 ( ) ( ) ( ) ( ) ( ) ( ) x x x x P x x P x x P x A P x P x P x + + + + = = = + + + + 3 3 3 3 3 3 3 31 1 2 2 4 4 3 1 2 4 3 3 3 (1 ) 1 1 1 x x x x f x x f x fx f x f x f x f f f f f f = + + = = = = = + + 6.A) ix i i ix 1 3 ( )P B A 3 ( )P B A 2 2 ( )P B 4 ( )P B 3 3 ( ) 2P A + 9 ( ) 6P A + 4 3 12 3 ( ) 2 ( ) 3 ( ) 5v P B A P B P A= + + + 3 ( ) 4 ( ) 9 ( ) 18P B A P B P A + + + 3 ( ) 4 ( ) 9 ( ) 18 3 ( ) 3 ( ) 4 ( ) 9 ( ) 18 3 3 ( ) 2 ( ) 3 ( ) 5 3 ( ) 3 ( ) 2 ( ) 3 ( ) 5 1 1 7 ( ) 3 9 ( ) 18 7 ( ) 9 ( ) 18 6 23 3 1 1 5 ( ) 3 3 ( ) 5 5 ( ) 3 ( ) 5 6 2 3 15 ( ) 9 ( ) 15 7 ( 2 P B A P B P A P B P B A P B P A x P B A P B P A P B P B A P B P A P B P A P B P A P B P A P B P A P B P A P + + + + + + = = + + + + + + + + + + = = + + + + + + = 1 1 ) 9 ( ) 18 8 ( ) 14 18 8 ( ) 4 ( ) 2 2 B P A P B P B P B + + + = = = 54. - 54 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 B) 1 (( ) ( )) 2 1 ( ) ( ) ( ) ( ) 2 1 1 1 1 ( ) 2 ( ) ( ) ( ) 2 2 3 2 1 ( ) 3 P B A A B P B P B A P A P B A P B P B A P A P A P A = + = + = + = = 1 1 1 (( ) ( ) ( ) 2 6 3 P B A P B P B A = = = : ix i 1 1 2 1 3 3 4 3 8 1 4 . )=3 ix i ix x ( ) 2 ix x ( ) 2 i iv x x 1 1 1 3 2 = 4 4 2 1 2 3 1 = 1 1 3 3 3 3 0 = 0 0 4 3 4 3 1 = 1 3 8 8 2 8 1 1 8 s s= = = 1 3 CV = . 7.) 1 2 20 1 x + = = 2 2 1 1 2 2 2 x = = = (1) 2 1 + = (2). (1) ,(2) 2 1 2 1 22 + = = . 55. - 55 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) 2 2 2 2 1 2 1 2 1 2 2 1 2 1 1 1 1 1 11 ( ) (0 ) (1 ) 12 2 4 4 4 4 4 v x x s + + + + = = = = = = 2 1 1 4 2 s s= = ) 1 2 1 1 2 s CV x = = = , . ) 2 2 21 ( ) 2 ( ) 4 f x s x xx f x x x= = , 1 '( ) 1 2 f x x= x 2 + ( )f x - + '( )f x ) : ( ) ( ) ( ) ( ) 2 2 2 i i i 2 2 i 12 2 2 2 2i 1 i 1 i i 1 t t t 1 s t x x s x x v = = = = = = = = (3) (3) 2 2 2 2 21 1 1 1 5 x x x 4 2 16 4 16 = + = = . 8. ) 2 2 2 1 2 3 10 50 2 6 15v v v v a + + = + + + = ( ) ( ) ( ) 50 15 35 2 2 2 25 9 1 2 2 2 2 5 25 2 3 9 2 1 0 5 3 1 0 5 0 5 3 0 3 1 0 1 = = = + + + + + + + = + + = = = = = = = Minf=f(2)=-1 56. - 56 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 . 2 2 10 50 1 10 5 50 1a + = + = 2 2 2 3 2 1 9 2 7a = = = 2 2 6 5 6 3 25 18 7 = = = ) 3 1 60 4 4 15 i i i x v x x v = = = = = 15v = 1 15 1 16 8 2 2 2 v i + + = = = = , 8 4 4t = = = , x = 9. 1 2 6 1 2 6 ... 15 15 ... 90 (1) 6 x x x x x x x + + + = = + + + = ( ) 26 6 2 2 6 12 2 1 1 2 2 2 2 2 2 2 2 2 21 6 1 6 1 2 6 2 2 2 1 2 6 1 9 9 9 6 6 6 6 ... ... 9 15 9 ... 15 9 6 6 6 ... 234 6 (2) i i x i xi xi xi S xi x x x x x x x x x x x = = = = = = + + + + = = + + = + + + = 7x 7, : ix iv 11 1 3 7 4 7 15 ix iv i ix v iN 3 7 21 7 1-7 4 7 28 14 8-14 11 1 11 15 15 15 60 57. - 57 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 (1) 1 2 6 7... 90 8 98 14 14 (3) 7 7 7 x x x x y y + + + + + = = = = = ( ) 2 27 7 7 2 7 12 2 2 1 1 1 2 2 2 2 2(2),(3)2 2 2 2 21 2 6 7 1 7 7 7 7 ( ... ) 234 6 8 96 96 14 7 7 7 7 i i i y y i y y y y yi yi yi S yi S x x x x S y S S S = = = = = = + + + + + = = = = x 14 15 100% 100% 6,67% 15 = = = 6,67%. 10. { } 0 ln( 1) ln3 ln1 ln( 1) ln3 1 1 3 2 4 2,3 2,3 x x x x x x A < < < < = = ( )( ) ( ) ( )( ) ( ) ( )( ) { } 2 2 2 2 5 1 6 1 1 5 6 1 0 1 5 6 0 1, 5 6 0 2,3 1,2,3 x x x x x x x x x x x x x x x = + = + = = + = = = ) A B ( ) 0P A B = { } { } ( ) 1,4,5 1,2,3,4,5 ( ) 1 A B A P B A P = = = = ) { } { }4,5 1,4,5B A B = = ( ) (1) (4) (5) (1)P A B P P P = + + , 1 1 ( ) (2) (3) 4 4 P A P P= + = (1) 1 (1) (2) (3) (4) (5) 1 (1) (4) (5) 1 4 1 4 1 3 ( ) 1 4 4 4 P P P P P P P P P A B + + + + = + + = = = = 58. - 58 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 { }1 1 1 ( ) (1) 8 8 1 1 ( ) (2) (3) 4 4 B A P B A P P A P P = = = = + = ) { } { }2,3 1,2,3A B= = A X B = , : { } { } { } { }1 1,2 1,3 1,2,3X X X X= = = = , 1 min ( ) (1) 8 P x P= = 1 1 3 max ( ) (1) (2) (3) 8 4 8 P X P P P= + + = + = 11. ) 7 5 2 5 8 6 5 3 6 6 41 66 25 11 a x a a + + + + + + + + + + = = + + + = + + = 6 = , 6 5 5. , 2,3,5,5,5,6, , , , , , 7, , ,8, 6. , 8 max min 8 max 2 8 max 10R = = = = , 10 = , < < . ( ) 2 2 2 2 22 2 2 22 2 2 2 2 2 2 25 10 25 15 15 10 217 117217 117 15 117 225 30 117 0 2 30 108 0 15 54 0 54 6 9 15 9 6 6, 9 P S + + = + + = + = = + + = + =+ + = + = + = + + = + = + = = = = < = = = = = ) ( ) ( ) ( ) 2,3,5,5,5,6, 6 ,7,8, 9 ,10 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 6 3 6 5 6 3 6 6 2 7 6 1 8 6 9 6 10 6 11 16 9 3 0 1 4 9 16 58 58 11 11 11 S S S + + + + + + + = + + + + + + + = = = 59. - 59 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) 1 1 1 2 2 2 1 2 1 2 11 11 1 2 ... y x C C y x C C Y X C C y x C C = + = + = + = + : 1 1 1 6z x C z C x C= = = 1 1Sz C Sx C Sx= = , 1 21 22 2 2 2 1 1 1 9 69 6 9 6 2 3 2 2 C CC Cy z C y z C C C Sy C Sx Sx Sx C CSy Sz = += += + = + = + = = = == 12. ) 1F = , N v = , i = ( ) ( ) 2 2 2 22 2 2 2 2 10 10 1 1 1 1 10 1 10 10 0 ( 10) 0 0 10 0 10 N N a v v a F F a a v v a a v v a v v v v a v v v + + + = + = + = = + = = = = = ) i) 2 2 2 1 12 2 2 1 2 2 1 1 10 10 10 i i i ii i i ii i i i x v x vx v s x v s = == = = = DEN EINAI K 2 2 2 2 2 2 21 1 1 1 2 10 0 0 10 10 10 10 i i i i i i i i i i i i x v x v x v x v s s s s = = = = = = = = ii) ( ) ( ) ( ) 2 2 2 1 1 2 22 ... 0 0 x x v x x v x x v s v + + + = = 60. - 60 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ( ) ( ) ( ) 2 1 1 1 2 22 2 1 2 2 0 0 ... ...... 0 x x v x x x xx x v x x x x x x x v = = = = = = = = = 13. 1 1 ( ) 1 f x x x = + ( ) ( ) ( ) ( ) [ ] 2 3 ( ) (1) 9 (2) 22 (2) 9 (3) 22 (3) 9 (2) (3) ... ( ) 22 (2) (3) ... ( ) ... 9 ( ) 22 ( ) 1 1 1 1 1 1 1 1 1 1 9 1 22 ... 2 3 3 4 4 5 1 1 1 1 9 22 2 1 v P f P f P P P v f f f v P v f v v v v v v = = + = = = + + + = + + + = = + + + + + + = + 1 2 ( 1) 9 22 9 11 2( 1) 1 9 9 11 11 2 20 10 v v v v v v v v + = = + + + = = = 14. ) 2 2, 20x s= = ( ) 2 ( ) 3 2f x x = + : ( ) ( ) ( )1 2 10, ,...,f x f x f x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 2 10 1 2 10 2 2 2 2 1 2 10 2 ... 3 2 3 2 ... 3 2 10 10 ... 3 3 3 20 60 60 10 x f x f x f x x x x x x x x x x x s x = + + + + + + + + + = = + + + = = = = ) ( ) 6( 2)f x x = + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 10 1 2 10 1 2 10 ... 6 2 6 2 ... 6 2 6 ... 20 6 10 20 6 10 ( 2) 20 6 20 20 6 0 0 f x f x f x x x x x x x x + + + = + + + + + + = + + + + = + = + = + = = ) : 1 2 10, ,...,y y y 61. - 61 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ( ) ( ) ( ) ( ) ( )1 2 101 2 10 6 2 ...... 0 0 0 10 10 10 i i iy x f x f x f x f xy y y y y = + = + + ++ + + = = = = = CV. 15. ) ( )( ) 2 2f x x = ( ) 0 2f x x = = ) ( ) ( ) 2 2 1 101 2 10 2 ... 2... 10 10 x xy y y y + + + + + = = = ( ) ( ) 2 2 1 10 2 2 ... 3 9 9 10 x x x x s y + + = = = = = ) ( )( ) 2 2f x x = = ( ) ( ) ( ) 1 1 1 2 2 2 10 10 10 ( ) 2 2 ( ) 2 2 .... ( ) 2 2 f x x f x x f x x = = = = = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 10 1 2 10 1 2 10 ... 2 2 2 2 ... 2 2 10 10 2 10 2 102 ... 2 10 30 20 10 2 2 2 10 10 10 10 f x f x f x x x x xx x x + + + + + + = = + + + = = = = = ) 1 2 10... 2x x x< < < 10 15 5R x x= = , ( )( )2 2 2 2 10 1 10 1 10 1 10 115 15 15x x x x x x x x= = + = ( )10 1 10 15 15 3 (1)x x x x + = + = 1 1 2 2 10 10( ), ( ), ...., ( )y f x y f x y f x= = = 62. - 62 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ( ]1 2 10, ,..., ,2x x x f 1 2 10 1 2 10... ( ) ( ) ... ( ) f x x x f x f x f x< < < > > > . , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 10 1 10 1 10 1 10 1 10 1 10 ( ) ( ) 2 2 2 2 2 2 4 5 3 4 5 7 35 R f x f x x x x x x x x x x x = = = + + = = + = = = 16. ) ( ) ( ),0 0,fD = + 2 2 2 9 9 ( ) 1 ( ) 0 3 x f x f x x x x = = = = ) 1 9 (1) 8 1 f = = (1) 1 9 10 (1) (1) ( 1) 10 8( 1) 8 18 f y f f x y x y x = + = = = = + ) 1 18 18y x= + 2 2 10 10 8 18 ... 8 18 y x y x = + = + 8 18 8 8 32 8 4 i i i i z x y x z x z x S S = + = = = = = 18 32 18 14 18 4 4 4 2 28,57% 14 14 7 i i y z y y y z y z S S CV CV = + = + = = + = = = = = = 63. - 63 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 CV , 0y < . , 0 14 4 4 4 10% 0,1 0,1 4 1,4 0,1 26, 26 14 14 y y S S CV = > = = = + + ) i) , 5 5 2x x = = (0,3) ( )f x 1 2 9 1 2 9... ( ) ( ) ... ( )x x x f x f x f x< < < > > > 5 9 4 9 13 ( ) (2) 2 2 2 2 f x f + = = = + = = ii) 10 1 1 10 1 10 1 10 1 10 9( )9 9 9 2 18 4 62 ( ) ( ) 2 2 2 5 5 5 4 x x R f x f x x x x x x x = = + = + = + = + = 17. ) 1 2 100( ) ( ) ... ( ) 1 1 100 100 100 P P P x x + + + = = = ) ( )( ) ( )( ) ( )( ) 2 2 2 1 2 100( ) 3 3 ... 3f x P x P x P x = ( ) ( ) 2 2 2 1 2 100 2 2 1 100 2 2 1 1 1 ( ) ( ) ... ( ) 100 100 1001 3 100 100 100 ( ) ... ( )1 1 3 100 3 100 300 100 100 100 P P P f P x P x f s f s + + + = + + = = = ) 1 75 y = . f g = ( )0 0, ( )x g x ( )0 0 0 0 0 0 0 1 ( ) ( ) ( ) ( ) ( ) 75 y g x g x x x y g x x g x x g x y = = + = 0 0 0 0 0 0 0 ( ) 0 ( ) 0 1 1 ( ) ( ) ( )1 75 100( ) ( ) 75 g x f x g x f x f x f g x x g x = = = = = = + = 2 2 1 100 2 1 1 ( ) ... ( ) 100 100 100 P P s + + = 64. - 64 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 , 2 2 21 1 1 1 1 1 300 100 75 75 300 75 22500 150 f s s s s = = = = = 1 100 2150 1 150 3 100 s CV x = = = = 18. ) 1( ) ... ( ) 1 1 9 9 vP P x v v v + + = = = ) 1 9( ),..., ( )P P . ( )iP 1 , 9, : ( )5= P . 0,2 > . 9 5v = = . , 4 0,2. , 1. ) 2 1 ( ) 3 0 9 f x x f = > ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 9 1 1 2 2 ( ) 1 9 2 2 2 9 9 1 3 ( ) ... ( ) 1 12( ) 3 ( ) 9 ( ) ... ( )1 ... 3 9 12 9 1 1 1 1( ) 3 ( ) 279 12 12 27 18 P x P x f P P P x P x f P P s s s + = + + = + + = = = = = , 1 9 1 118 1 18 2 2 9 s CV CV x = = = = = 19. ) ( ) ( ) 22 2 (1) (3 1 2) (1 1 1) (1) (1) 1 1 0g f f f f= + + = + = + = 2 2 1 ( ) 2 (3 2) (3 2)(3 2) ( 1)(2 1) ( ) 2 (3 2) (3 2) 3 ( 1)(2 1) (1) 2 (1) (1) 3 (1) 1 2( 1) 1 3 1 1 5 x g x f x f x x f x x x g x f x f x f x x x g f f f = = + + = + + = + = + = , ( )(1) (1) 1 0 5( 1) 5 5y g g x y x y x = = = + ) 5 5y x= + : 5z x= , 5 5 400 2000z x z= = = 65. - 65 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 5 5 2000 5 1995y z y z y= + = + = + = ( ) ( ) ( ) ( ) 2 2004 2 2 2 2 1 22 2 2 2 2 2 2 2 2 1 200 200 2004 2004 200 200 2004 2004 200 400 200.000 i i i i i x s x x x x x x x = = = = = = + = 20. ) ( )0,fD = + ( )f x : 1 1 ( ) 1 ( ) x f x f x x x = = ( ) 0 1 0 1f x x x = = = ( ) ( ]0,1 ( ) 0 ( ) 0,1x f x f x x > ( ) [ )1, ( ) 0 ( ) 1,x f x f x x + < + (1) 0f = , ( )f x () ( ) ( )1, (1) 1,2010A f A (1) ln1 1 2011 0 2010 (1) 2010f f= + = + = ) x f , 1x = . 2 10x = . : ( ) 2 2 2 2 12 2 2 2 2 21 1 1 1 10 1 v v v i i iv i i i i i t t t s t s s x x s v v v v = = = = = = = = 2 9 3s s= = . , 3 300% 1 s CV x = = = 66. - 66 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) ( )3 2 ,..., 1,v vt t + , , ( ) ( )1,+ , ( )1, + . 1x = , x < . 21. ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 1 2 23 3 ... 3 ( ) 3 v vt x t x t x t x t x t x f x v + + + = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 2 2 2 2 1 2 2 2 2 1 2 3 3 ... 3 ( ) 3 ... ( ) 3 3 ... ( ) (1) v v v t x t x t x f x v t x t x t x f x v t x t x t x f x v = + + + = + + + = : ( ) ( ) ( ) 2 2 2 1 2( ) ... vg x t x t x t x= + + + ( )g x ( ), ( )K x g x . , ( ) ( ) 1 (1) 1 1 ( ) ( ) ( ) ( ) v g x g x x g x g x f x f x v v ( )f x ( ), ( )K x f x , f ( )4, 4A ( )f x , A K , 4x = . ( ) ( ) ( ) ( ) 2 2 2 (1) 1 2 2 2 ... 4 4 4 4 2 vt x t x t x f x s s s v + + + = = = = = , 2 1 50% 4 2 s CV x = = = = . 67. - 67 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) ( )( ) ( )( ) ( )( )1 1 2 22 2 ... 2 ( ) v vt x t x t x t x t x t x f x v + + + = ( ) ( ) ( ) ( )1 2 1 22 2 ... 2 ... ( ) ( ) 2 2 ( ) ( ) 2 ( ) ( ) 2( ) (2) v vt x t x t x t t t v x f x f x v v vx vx v x x f x f x f x x x v v + + + + + + = = = = = fC ( )2,4B , : ( )(2) (2) 2y f f x = (2) (2) 4, (2) 2( 2) 2(4 2) 4f f x = + = = = , ( )4 4 2 4 8 4 4 4 ( )y x y x y x = = + = ) ( )1 2 9, ,...,M M M . , 1 1 2 2 9 9 4 4 4 4 : 7 2 ... 4 4 y y x y x y S y x = = = = = ix 1 1 2 2 9 9 1 1 4 1 1 1 14 4 ... 1 1 4 x y x y X Y x y = + = + = + = + : 1 4 z y= , 1 1 7 7 4 4 4 z y= = = 1 1 2 1 2 4 4 4 2 z yS S= = = = , 1x z= + 7 7 4 11 11 1 1 4 4 4 4 x z x + = + = + = = = 1 1 2 2 x z xS S S= = = ( ). 68. - 68 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ( ) 2 29 9 9 2 9 2 12 2 2 2 21 1 1 2 2 2 2 2 1 9 9 9 9 2 7 4 49 53 i i i i i i y i y y i y y y S y S S y y y y y = = = = = = = = + = = 22.) }{ ,2, , 3 = +x y x 2 3< < < +x y x , 2 2.5 2.5 3 2 + = = = y y 2 3 3 2.5 2.5 1 4 + + + + = = = x x x x ) : }{1,2,3,4 = , }{2,4,8 = , : (,) : 2 4 8 1 (1,2) (1,4) (1,8) 2 (2,2) (2,4) (2,8) 3 (3,2) (3,4) (3,8) 4 (4,2) (4,4) (4,8) ={ (1,2), (2,2), (3,2), (4,2), (1,4), (2,4), (3,4), (4,4), (1,8), (2,8), (3,8), (4,8)}. ) 2 0 2 2 2 2 2 2 2 2 2 20 2 2 22 2 2 2 2 2 2 2 2 2 : , 2 2 2 2 ( 2 )( 3 2 ) ( 2 )( 3 2 ) lim lim lim ( 3 4 )3 2 ( 3 2 )( 3 2 ) ( 2 )( 3 2 ) lim ( ) + + + + + + + + = = = + + + + + + + + = x x x x a x x ax a x ax a x a a x ax a x a a x a ax a a x a a x a a x ax a x a a x a 2 2 ( )( 2 ) ( ) lim = + = x a x x x x 2 2 ( 2 )( 3 2 ) ( ) + + + x x a a x a 2 2 ( 2 )( 3 2 ) (3 )(4 ) lim 6 ( ) (2 )( ) + + + = = = ++ x x x a a a x a ax a 2 3 2 2 ( )( ) lim lim lim 2 2 2( ) = = x x x xx x x x ( ) 2 ( ) + x x 2 = 2 2 2 3 2 2 2 2 lim lim 6 2 23 2 + + x x x ax a x xx a a : = { (1,2), (2,2), (3,2), (4,2),(3,4), (4,4)}. 6 ( ) 12 =P 69. - 69 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 23. A) 1 2 10'( ) ( ... ) 10= + + + f x t t t x : 1 2 10 1 2 10 ... '( ) 0 ( ... ) 10 0 10 + + + = + + + = = = t t t f x t t t x x t x + '( )f x + - ( )f x max ) ( ) 810 ( ) 810g a g t= = 2 2 2 1 2 10 2 ( ) ( ) .... ( ) 810 81 9 + + + = = = t t t t t t s s : 1 2 10'( ) 2( ) 2( ) .... 2( )= g x t x t x t x 1 2 10 1 2 10 1 2 10 1 2 10 1 2 10 '(0) 2000 2 2 .... 2 2000 2( .... ) 2000 ....2000 1000 ( .... ) .... 1000 2 10 10 100 = = + + = + + + + = + + = = = g t t t t t t t t t t t t t t t t 9 0.09 100 = = = s CV t 9%s x x s 100% 100% 5% 20 = = = s s CV sx . ) 20 '( ) 0 20 0 0 ( 1) 0 1 = = + = + = + = = x s f x s x x x x x x x x 1< x '( ) ( 1) 0= + x '( ) ( 1) 0= + >f x x x f [ )1, + . 72. - 72 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 f 1= x '( 1) 10 ( 1) 11 11 11 2 2 = + + = + = x x f s x x ) i) f 1 . : 11 1 20 2 = = x x , 20 1 20 20 = = = x s . ii) (-1,1) '( 1) 0= =f f (-1,1) 1y = . 28. : 20 1 100 5 20 20 = = = = i i t x : 2 12 2 1 1 1 10000 1 500 1000 (1000 500) 25 20 20 20 20 20 = = = = = = = i i x i i t S t 5=xS 2 5= +Y X : 2 5 10 5 15= + = + =y x 2 2 5 10= = =y xS S . 29.. i) 20%=CV : 0.2 0.2= = s s x x (1) 25 ' 25= +x x . 10%=CV ,: 0.1 25 = + s x 0.1 2.5= +s x (2) (1) (2) : 0.2 0.1 2.5= +x x 25x = , 5s = . ii) 2 2 2 1 2, ,..., vx x x 2 2 2 2 21 2 1... =+ + + = = i v i x x x x x 2 2 2 2 2 2 1 2 3 1 ( ) ( ) ( ) ..... ( )1 ( ) = + + + + = = i i x x x x x x x x s x x 25=x 5s = , = +Y aX , : = +y x =y xS S 73. - 73 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 2 2 2 1 1 2 2 2 2 2 1 2 1 2 2 2 2 1 2 1 2 2 2 2 1 2 1 2 2 50 625 50 625 ..... 50 625 25 .. 50 50 ..... 50 625 625 ... 625 25 .. 50( ..... ) 625 25 .. 50( ..... ) 625 25 25 5 + + + + + + = + + + + + + + = + + + + + + + = + + + + + + = + = 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 2 2 0 625 25 1250 625 650+ = + =x x x iii) ( ) 6 30= =R s . , max min 30 =X X max 10 30 =X max 40=X . 40 . ) . = = P(A)=0.3 P()=0.6 P()=1-P()=1-0.6=0.4. 50% ( ) 0.5 =P A B . i) A B , ( ) ( ) ( ) ( ) 0.20 = + =P A B P A P B P A B . 20%. ii) -, ( ) ( ) ( ) 0.10 = =P A B P A P A B 10% . - , ( ) ( ) ( ) 0.20 = =P B A P B P A B , 20% . . ) . 1 : 1 0.8%= v 1 0.008= v 1 0.008= v 125=v . 30 . i) , 0x = , . ii) . s CV x = 0x > . , s CV x = , 0x . iii). CV , . iv) . , 10%CV . v) , 0 x s< < , 1 s CV x = > , 100%CV > . 74. - 74 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 vi) . CV , 0 s CV x = . vii) . : 7, 7, 7, 7, 7. 7 7, . 31.) 3 2 (10) 10 15 10 600 10 300 6200= + + =P . ) : 3 2 2 '( ) ( 15 600 300)' 3 30 600= + + = + +P x x x x x x 2 2 '( ) 0 3 30 600 0 10 200 0 10( ) .. 20= + + = + + = = =P x x x x x x x : '( ) 0P x > 20x ( )P x x=20 , 3 2 (20) 20 15 20 600 20 300 8000 6000 12000 300 9700= + + = + + =P . ) : 3 2 2 '( ) ( 15 600 300)' 3 30 600= + + = + +P x x x x x x x=10 : 2 '(10) 3 10 30 10 600 600= + + =P . ) : 2 ''( ) ( 3 30 600)' 6 30P x x x x= + + = + ''( ) 0 6 30 0 6 30 5P x x x x= + = = = ''( ) 0P x > 5x . 32. ) A B A B = A B = .: 2 2 2 2 2 2 2 ( ) 4 ( ) ( ) 4 ( ) ( )( 4) lim lim lim 2 2 2 ( ) ( 2) lim = = = x x x x x P A B P A x P A P A P A x x x x P A x ( 2) 2 + x x 2 lim( ( )( 2)) 4 ( ) 4 0.2 0.8 = + = = = x P A x P A ( ) 0.8P = , ( ) ( ) 0.8 = =P P ( ') 1 ( ) 1 0.8 0.2 = = =P P ) 2 2 2 2 16[ ( )] 25 ( ) ( ) 10 0 16[ ( )] 24 ( ) ( ) ( ) 10 0 16[ ( )] 24 ( ) ( ( ) ( )) 10 0 16[ ( )] 24 ( ) 10 ( ) ( ) + = + = + + = + = + P B P B P P B P B P B P P B P B P B P P B P B P B P ( ) ( ) ( ) 1 + = P P P .: 75. - 75 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 2 2 2 16[ ( )] 24 ( ) 10 1 16[ ( )] 24 ( ) 9 0 3 (4 ( ) 3) 0 4 ( ) 3 0 ( ) 4 + + = = P B P B P B P B P B P B P B ( )P B (1) : 1 ( ) 4 =P A . ( ) 1 ( ) ( ) ( ) 4 ( ) = = N A N A P A N N . ( ) 250=N A . ( ) 750 =N . A B = ( ) ( ) 1000 ( )+ = = N A N A = . 33. ) : 85 2 130 = + = x s x s 100 15 = = x s 15 15% 100 = = = s CV x >10% . . ) : x s x s x s x x s x s x s + + +3 2 2 3 99,7% 95% 68% 34% 34% 13.5% 50% 16% 2.5% 0.15% 13.5% 50% 16% 2.5% 0.15% 2.35%2.35% 76. - 76 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 IQ 145 0.15%. 34.) ix if i ix f [ )195 197 196 0.1 19.6 [ )197 199 198 0.1 19.8 [ )199 201 200 0.55 110 [ )201 203 202 0.2 40.4 [ )203 205 204 0.05 10.2 1 200 : 1 1 1 1 = = = = = = i i i i i i i i i v x v x x f x . 1 200 = = = i i i x f x . ) : x s x s x s x x s x s x s + + +3 2 2 3 99,7% 95% 68% 34% 34% 13.5% 50% 16% 2.5% 0.15% 13.5% 50% 16% 2.5% 0.15% 2.35%2.35% 55 70 85 100 115 130 145 77. - 77 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 if % %iF [ )195 197 10 10 [ )197 199 10 20 [ )199 201 55 75 [ )201 203 20 95 [ )203 205 5 100 100 : 30 6 12 5 12 6 1.09 2 25 2 5 11 = = = = x x x x x gr x x =199+1.09=200.09 gr. ) : 75 1 95 75 20 47.5 20 1 2 = = = = x x x x x x 78. - 78 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 200 gr 47.5%. , p=47.5%=0.475. ) 8 gr , . if % %iF [ )203 205 10 10 [ )205 207 10 20 [ )207 209 55 75 [ )209 211 20 95 [ )211 213 5 100 100 79. - 79 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 : : 95 1 95 75 85 75 1 = = = x x x x x 100-85=15%. q=15%=0.15. 35. ) 1 2 7 8, ,..., ,x x x x . 10 1 9= =maxR 1 10. ) 80. - 80 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 [ ] 1 2 ( ) 3 1 2 3 7 1 2 3 8 8 1 10 1 10 1 10 ..8 80 8 .. 8 10 8 8 8. . 1 10 1 10 1,10 + + + + + + + + + x x x x x x x x x x x x x x ) 10=x 8 10 1 2 3 8.. 10= = = = =x x x x 36. 1 2 3 4 1 2 3 4 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 4 4 4 4 4 4 4 1 1 2 1 4 4 2 + + + + + + + + + + + = = = + = = P P P P P P P P x 1 9 21 1 ( ) 9 81 = : 2 2 2 2 1 2 3 4 2 2 2 2 1 2 3 4 1 1 1 1 1 1 1 1 ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) 4 2 4 2 4 2 4 2 4 1 1 1 1 ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) 14 4 4 4 4 81 + + + + + + + = + + + = P P P P P P P P : 2 2 2 2 2 1 2 3 4 1 1 1 1 2 ( ( ) ) ( ( ) ) ( ( ) ) ( ( ) ) ( ) 4 4 4 4 9 + + + =P P P P . CV : 1 29 22.2% 1 9 2 = = = = s CV x . 37. ) ()=++6 : : ()= : ()=6 : ()= : 1 ( ) 2 6 6 6 2 = = = + + = + + P A (1) 1 ( ) 5 6 4 6 6 5 = = = + + = + + P (2) 81. - 81 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 (1) (2) =10 =4. ) : 10 1400 6 3000 4 2000 14000 18000 8000 40000 2000 20 20 20 + + + + = = = =x : 1 1 1 20% 400 5 5 2000 5 = = = = = s s CV s x . ) ,18 6 . x , : 18 1400 6 25200 6 ' 24 24 + + = = x x x (3) : 1 1 1 20% ' 5 5 300 1500 5 5 5'' = = = = = = = s s CV x s xx (4) (3) (4) : 18 1400 6 1500 .. 1800 24 + = = x x . 38. A) iy ix .... : 1,15= i iy x a 1,15= y x a 1,15=y xS S : 1,15 1,1 1,1 1,1 1,15 1,15 1 1,15 1,1 1,1 1,15 1000 1000 1150 1000 1150 1,1(1150 ) ... 104,55 = = = + = = + + = + = y x x x y x S S S S CV CV y x x a x a a a a . B) 15% 15 1000 150 100 = . 150-104,55= 45,45 . 39.) : 6 2x, 6 2x,g x = = = . : 2 f(x) (6 2x)(6 2x)x 2(3 x)2(3 x)x 4(3 x) x,0 x 3= = = < < 82. - 82 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) 2 '( ) (4 (3 ) ) ... 12(3 )(1 )f x x x x x= = = x 0 1 3 f '(x) + f(x) x=1 m. ) x 0 x 0 f(x 2) 8 f (x 2) f(2) lim lim f '(2) 12 x x + + = = = ) [ ]f (x) 1,3 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 x x x x x 2 f (1) f (x ) f (x ) f(x ) f (x ) f (x ) f(2) 16 y y y y y 8 = < < < < = = > > > > = = > > > > = R=16-8=8 y y S 1 CV 6y = = iy y i , : y y 12 0 S S 2 = + = + > = = s 2 CV 12 = = + y R 2 1 8 CV 2CV 2 ... 10 12 12 6 12 = + = + = + ) [ ]f 0,1 2 2 0 P( ) P( ) 1 f (P(A)) f(P(B)) 4 P(A)(3 P(A)) 4 P(B)(3 P(B)) < P( ) 0 > ( ) 2 3 ( ) 0 >P ( ) ( ) 22 2 3 P(B)P(A) P(A) 3 P(B) P(B) P(B) 3 P(A)3 P(A) 83. - 83 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 40.) f ,f : '( ) (4 ) x f x e = + 2 ''( ) (4 ) x f x e = B) 2 '(0) (4 ) 5f = + = , (0) (4 ) 4f = = ( ) ( )2 2 (0) '(0)( 0) 5 5y f f x y x y x = = = + ) 1 1 1 4 4 = = = 2 5 = 2 2 4 5 5 4 0 1 5 = + = = = 2 1 ( ) 6 3 P = = ''( ) 0f x > x , 2 2 (4 ) 0 (4 ) 0 4x e > > < 3 1 ( ) 6 2 P = = ) 1 = : ( ) 3 4x f x e x= + , '( ) 3 1x f x e= + 0 '(0) 3 1 4f e= + = , 0 (0) 3 0 4 1f e= + = . , 0 0 0 0 (0 ) (0) ( ) (0) 3 4 ( 1) 3 3 '(0) lim lim lim lim 0 x x h x x x f h f f x f e x e x f h x x x + + + = = = = 0 3 3 lim 4 x x e x x + = 41.i) 80 , 80A Bx x= = ii) A Bs s> iii) 80 80 A B A B A B A B A B s s s s s s CV CV x x > > > > . 42. ) 2 1 10 ( 1) ... 19 2 10.5 2 40 R R R = = + + = = =+ = ) 10( ) 19 ( ) ( ) 20 N A P A N = = = ) 2 5 0,x x + + > x , 2 1 0 1 00 7,8,9,...,2025 0 5 4 0 4 a >>> = < > < , 14 ( ) 20 P = 84. - 84 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) 2 220 220 1 12 2 2 2 2 1 1 2 2 2 1 1 1 210 ( ) ( ) (2870 ) 20 20 20 20 1 665 (2870 2205) 33.25 33.25 20 20 i i i i i i i i x x s x s x s v s s s s = = = = = = = = = = = ( ) 1 20 1 2 ... 20 1 20 210 2 = = + + + = + = i i x 43.A) '( ) x f x e = , () fC xx 0 '(0) 0 0 1f e = = = = . ( ) 39x f x e x= + ) i) '( ) 0 .... 0f x x= = f [ )0,+ , f ( ],0 f x=0 . x : ( ) (0) ( ) 40f x f f x ii) 1 2 100( ) ( ) ... ( ) 40 40 ... 40 100 40 40 40 100 100 100 f x f x f x x x + + + + + + = > = = (1) x = , 40 > . ( ) iii) (1) 1 1 10 40 10 40 4 10 10 s CV s x s s x > > > > > > (2) iv) 2 2 '( ) 3 12 3 3( 4 )g x x x s x x s= + + = + + 16 4 4(4 ) 0s s = = < '( ) 0g x > x g . 44.) () fC (1,f(1)) xx 45o 2 2 '( ) x f x x a = + , 0 2 '(1) 45 1 1 1 1 f a = = = = = + . 2 ( ) 1 ln( 1)f x x= + + . ) 2 2 '( ) 0 0 ... 0 1 x f x x x = = = + , '( ) 0f x > ( )0,x + , '( ) 0f x < ( ),0x . f 0 0=x f [ )0,x + f ( ],0x ) 0 ( ) 1P A 0 ( ) 1P B f [ )0,+ : 0 ( ) 1 (0) ( ( )) (1) 1 ( ) 1 ln 2P A f f P A f P B + ( ) 1P B = ( ) 2 ( ( )) 1 1 ln( ( ) 1) 1 ( ) 0f P A P A P A= + + = = . 45. 1) 2) 3) 4) 5) 85. - 85 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 46.) 1 2, ,...,t t t 1 1 2, ,...,t t t 2 . : ( )1 2 1 2 ... ... 1 t t t x t t t vx + + + = + + + = 1 2 1 2 ... ... (2) t t t y t t t y + + + = + + + = ( ) ( ) (2) 1 2 1 2 (1) ... ...t t t t t t x y z + + + + + + + + = + += ) 2 2 2 2 212 2 1 1 1 1 1 v v v v i i i i i i i iv i i i i i i i x x x x s x x = = = = = = = = ) 1 1 0x = 2 2 1x = 1 2 + = . 1 1 2 2 1 2 20 1x x x + + = = = 2 2 22 1 2 2 v i i i x s x = = = . 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 4 4 4 4 v s ( ) 22 2 2 2 2 2 2 2 24 4 0 4 4 0 2v v + , . ) i iz x x= 0z x x= = z xs s= 1 i i x y z s = 1 1 0 0 x x y z s s = = = 1 1 1y x z x x s s s s s = = = ) 3x s , 1 1 3 0 3 3 3 s x s x s CV x > > > > . 86. - 86 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 47) ) 0 120 3 = = (1,1) : 3 1 3 1 1 3y x = + = + = + : 3 1 3y x = + + 3 1 3 3 ( 2) 1 3 3 3 1y x= + + = + + = + ) (*) 3 1 3 3 ( 1) 1 3 2 3 1y x = + + = + + = + ) 0 (1 ) (1) lim '(1) 3 h f h f f h + = = = (*) 1 2, ,....,x x x , . ) 1 2, ,....,y y y 1 2, ,....,x x x c , : y c = + . ) 1 2, ,....,y y y 1 2, ,....,x x x c , : y c = . : , 1,2,...,i iy x c i = + = , y : y k k xy x c c = = + = + . ky , 1ky + 1 1 1 12 2 2 2 2 k k y x y y x c x c x x c x x c c + + + ++ + + + + + + = = = = + = + y c = + . : , 1,2,...,i iy c x i = = , y : y k k xy c x c = = = . ky , 1ky + 1 1 1 1( ) 2 2 2 2 k k y x y y c x c x c x x x x c c + + + ++ + + + = = = = = 87. - 87 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 y c = 48)) 0/0 2 20 0 (1 ) (1 ) lim lim 1 1 1 x x x x e x e x x x x x = = = + + 20 0 (1 )(1 ) lim lim xx x x e xe x x x = = + (1 )x x 0 lim 1 x x e x = = H Cg (0, 1 3 ) 1 (0) 3 g = . 2 2 2 2 2 ( ) ( ) 0 ( ) 0 ( ) 0 2 2 1 1 (0) 0 (0 20 ( 1) (0 1)) 60 3 ( 1) 0 0 1 x x x x g + = = = = + + + + = + = = = 21 ( ) ( 20), 60 g x x x= + ) 2 ( ) 60 30 x x g x = = ) 3 1 2 31 2 1 2 3 30 ( ) ( ) ( ) 1 30 30 30 30 30 g g g + + + + = + + = = = ) 1 2 3 1 2 3 1 ( ) ( ) ( ) 30 30 f f f g f g f g f + + + + = = ) i) 1 1 2 2 3 3 1 1 2 2 3 3( ) 30 v x v x v x g v x v x v x x + + + + = = ii) 88. - 88 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 1 1 2 2 3 3 2 2 2 1 1 2 2 3 3 22 2 3 3 31 1 1 2 2 2 2 2 2 1 1 2 2 3 3 1 2 3 2 1 1 ( ) ( ) ( ) 1 1 1 (( ) 20) (( ) 20) (( ) 20) 60 60 60 ( ) 20( ) 20 ( ) 20 60 60 60 60 60 60 ( ) ( ) ( ) 20 20 20 60 60 ( ) v g x x v g x x v g x x v x x v x x v x x v x x vv x x v v x x v v x x v x x v x x v v v v x x + + = + + + + + = + + + + + = + + + + + = + 2 2 2 2 3 3 1 2 3 2 2 2 1 1 2 2 3 3 2 2 ( ) ( ) 20( ) 60 60 ( ) ( ) ( )1 20 30 2 30 60 20 20 2 2 2 v x x v x x v v v v x x v x x v x x s s + + + + = + + + = + + = )) 21 ( ) (180 ( 2) 20) (2 5) (180 (( 2) 20) 20) (2 5) 60 f x g x h x x h x= = + = 2 2 (3( 2) 20) 20) (2 5) 3( 2) (2 5)x h x x h x= + = ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 '( ) 3( 2) (2 5) ' 3( 2) ' (2 5) 3( 2) (2 5) ' 6( 2)( 2)' (2 5) 3( 2) '(2 5)(2 5)' 6( 2) (2 5) 3( 2) '(2 5) 2 6( 2) (2 5) 6( 2) '(2 5) f x x h x x h x x h x x x h x x h x x x h x x h x x h x x h x = = + = + = + = + cf (2,f(2)) '(2)f = 2 '(2) 6(2 2) (2 2 5) 6(2 2) '(2 2 5) 0f h h= + = ) 2 (2) 3(2 2) (2 2 5) 0f h= = . : (2) '(2)( 2) 0y f f x y = = ) ( )2 ''( ) 6( 2) (2 5) 6( 2) '(2 5) 'f x x h x x h x= + = 2 6 (2 5) 6( 2) '(2 5) 2 12( 2) '(2 5) 6( 2) ''(2 5) 2h x x h x x h x x h x + + + =. 2 6 (2 5) 12( 2) '(2 5) 12( 2) '(2 5) 12( 2) ''(2 5)h x x h x x h x x h x + + + = ''(2) .. 6 ( 1) 6 7 42f h= = = = 49.) 1 1 1 ( ) ( ) ( ) 4 6 12 P B A P B P A B = = = 89. - 89 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 1 3 ( ') 1 ( ) 1 4 4 P B P B= = = ) [ ]( ' ') ( ') ( ' ') 1 ( ) ( )' 1 ( ) (1 ( )) 1 ( ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 12 P A B P A P A B P A P A B P A P A B P A P A B P A P A B P A P A P B P A B P B P A B P B A = = = = + = + = + + = = = ) 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) 4 6 12 P A B P A P B P A B P A P A = + = + = + 1 11 ( ) 1 ( ) 1 ( ) 12 12 P A B P A P A + 50. 1. 2. 3. 4. 5. 51.) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )P A B P A P B P A B P A B P A B P A P B = + + = + )i) ( ') 0.4 1 ( ) 0.4 ( ) 0.6P A P A P A (1) ( ') 0.5 1 ( ) 0.5 ( ) 0.5P B P B P B (2) ii) (1) +(2): ( ) ( ) 0.5 0.6 1.1P A P B+ + = () ( ) ( ) 1.1P A B P A B + iii) A B = , : ( ) ( ) ( ) 1.1 ( ) 1.1P A B P A P B P A B = + , . : A B . 52. i) 1 1 2 2 1 2 x v x v x v v + = + (1), 1 1 2 1 1 1 2 2 2 1 2 1 1 2 1 2 ( )v x v x v x v x v x x x x v v v v + = = + + 1 2 1 2 1 2 ( ) ... v x x x x v v = = + 2 2 2 21 1 2 2 1 2 1 2 2 1 2 1 2 ( ) ( ) ... ( ) ( ) x x x x s x x + = = = + + (1) 90. - 90 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ii) (1) : 2 1 1 1 2 2 1 1 2 1 2 1 2 1 ( ) 2 2 v v x v x v v x x x x x v v v = + + + = = = + (2) : 2 2 1 22 2 21 2 1 1 2 1 2 1 22 2 1 2 1 ( ) ( ) ( ) ( ) (2 ) 4 2 x xx x s x x x x s = = = = + : 1 2 1 2 1 2 1 2 2 2 x x x xs CV x x x xx = = = + + 53.i) H f 0x > .: 2 2 1 ln ln 1 ln '( ) ' x x x xxf x x x x = = = 2 1 ln '( ) 0 0 1 ln 0 x f x x x e x = = = = ( ) ln 2 1 ln '( ) 0 0 1 ln 0 ln ln h x x x f x x e x e x x = > > > > > ( ) ln 2 1 ln '( ) 0 0 1 ln 0 ln ln h x x x f x x e x e x x = < < < < < f 0x e= 1 ( )f e e = f ( ]0,e , f [ ),e + ii)H g ( )3 2 ( ) ( ) 2 ( ) ( ) 1 '( ) ( ( ) ( ) 1 1974)' 3 2 ( ) ( ) 1 P A B P A B P A B P A B g x x x P A B P A B x x x P A B P A B = + + + = = + + ( ) ( ) ( ) ( ) ( ) ( ) 4 4( ( ) ( ) 1) 4 4( ( ) ( ) ) 4 4( ( ) ( ) ) P 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 P A B P A B P A B = + = = ( ) ( ) ( ) ( )P A B P A B P A B P A B 91. - 91 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 A B A A B A = = ( ]0, ( ) ln (1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ln ( ) ln ( ) 0 ( ) ( ) 1 ( ( )) ( ( )) ( ) ( ) ( )ln ( ) ( )ln ( ) ln ( ) ln ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 f e h x x P B P A P B P A P A B P A B P A B P A B P A P B P A P B f P A f P B P A P B P B P A P A P B P A P B P A P B P A B P A B P A B P A B = < < < < < < < < < > 0, 1 0 < = > '( ) 0g x > x g . iii) 3 3 4 4 5 5 1 1 3 4 5 1 2 (2) ( ) ( ) ( ) .... ( ) ln 2 ln ln ln .... ln 2 2 3 3 4 4 2 3 4 v f f f f f x x + + + + + + + + + + + + + = = ln2014 3 4 5 1 ln 2ln 2 .... ln 20142 3 4 x x = + = = 3 2 4 3 5 4 1 .... + ln 2014 ln( 1) ... 2013 + = = 54.) : 2 2 1 1 (0) ( 0) (1)i i i i f t t = = = = 2 2 2 2 2 2 21 12 2 1 1 1 1 2 1 1 i ii i i i i ii i i i i i t tt t t t s t x = == = = = = = = = = ( ) 2 (1)2 2 2 22 2 2 2 2 21 1 1 (0) i i i i i i t s x s t x t s x f s x = = = = = = + = + f 'y y 22 (0, ( ))A s x + . ) f 'x x : 1( ,0)x 2 2 2 2 1 1 1 1 2 1 1 1 2 1 1 ( ) 0 ( ) 0 ( ) ( ) .... ( ) 0 .....i i f x t x t x t x t x t t t x = = = + + + = = = = = : 1 2 1 1 .....t t t t x t + + + = = = - 92. - 92 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 1 1 1 1 2 2 2 t t t t + = = = - 1t = x = . ) : 2 2 2 2 1 2 1 ( ) ( ) ( ) ( ) ... ( )i i f x t x t x t x t x = = = + + + 1 1 2 2 1 2 1 2 1 2 '( ) 2( )( )' 2( )( )' ... 2( )( )' 2( ) 2( ) ... 2( ) 2( ... ) 2( ( ... )) f x t x t x t x t x t x t x t x t x t x t t t x x t t t = + + + = + = + + + = + + + : 1 2 1 2 ... '( ) 0 2( ( ... )) 0 t t t f x x t t t x x x + + + = + + + = = = 1 2 1 2 ... '( ) 0 2( ( ... )) 0 t t t f x x t t t x x x + + + < + + + < < < '( ) 0 ..f x x x> > f x x= x 2 2 2 1 2( ) ( ) ( ) ( ) ( ) ... ( )f x f x f x t x t x t x + + + 2 2 2 21 2( ) ( ) ... ( ) ( ) ( ) t x t x t x f x f x s + + + 55. ) 3 3 3 2 2 2 1 2 1 1 2 2 2 2 2 2 2 2 1 2 1 2 1 1 '( ) [( ) ( ) ... ( ) ] ' [3( ) ( )' 3( ) ( )' ... 3( ) ( )'] 3 3 1 3[( ) ( ) ... ( ) ] ( ) ( ) ... ( ) 3 f x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x = + + + = + + + = + + = + + 2 2 2 21 2( ) ( ) ... ( )'( ) t x t x t xf x s + + = = ) 93. - 93 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 3 2 2 2 2 2 2 21 1 1 1 2 2 2 2 2 2 21 1 1 1 ( 1) ( 1) ( 1)( 1) ( 1)( 1)( 1) lim lim lim lim ( 3 2) ( 3 2) ( 3 2) ( 3 2) ( 1) ( 1) ( 1) ( 1)( 3 2) ( 1) ( 1)( 3 lim lim lim ( 3 2) ( 3 2) ( 3 2) x x x x x x x x x x x x x x x x x x a x x x x x x x x x x x x x x x + + = = = = = + + + + + + + + + + + = = + + + + ( )( ) ( )( ) ( ) ( ) 2 2 2 2 2 2 2 221 1 2 2 2 2 21 1 2) 3 2 3 2 ) ( 1) ( 1)( 3 2) ( 1) ( 1)( 3 2) lim lim 3 43 4 ( 1) ( 1)( 3 2) lim lim( 1)( 3 2) 2( 1 3 2) 2 16 32 1 x x x x x x x x x x x x xx x x x x x x = + + + + + + + + + = = + + + + + = + + + = + + = = f ( )2 2 2 1 2 1 1 2 2 1 2 1 2 1 2 ''( ) ( ) ( ) ... ( ) ' 2( )( )' 2( )( )'... 2( )( )' 2( ) 2( )... 2( ) 2( ... ) 2( ... ) f x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t t t x = + + = + + + = = + + = + + 1 2''(2 ) 2( ... 2 ) 2( 2 ) 2f x t t t x x x x = + + = = 1 1 91 ''(2 ) 3 32 5 2 91 45.5 45.5 2 i i i i t f x x x t = = = = = = = ) 3 3 3 3 3 3 3 1 2 1 2 1 1 1 1 1 (0) [( 0) ( 0) ... ( 0) ] ( ... ) 6042 2014 3 3 3 3 i i f t t t t t t t = = + + + = + + + = = = 2 1 '(0) i i f t = = : 2 2 2 2 2 2 2 212 2 2 21 1 1 1 1 ... ( ) i i i i i i i i i i t t s t s x s t vx t s x v = = = = = = = = = + 22 '(0) ( )f s x= + . : 2 22 2 (0) '(0)( 0) 2014 ( ) ( ) 2014y f f x y s x x y s x x = + = + = + ) 1 2, ,...,t t t x s CV=16%, 16 100 s s CV x x = = (1) 2029 1 215, 15,..., 15t t t+ + + ' 15x x= + 's s= CV=10%, 10 ' 10015 15 s s CV x x = = + + (2) 94. - 94 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 (1) ,(2) 25x = 4s = 6 24R s R R = 24= 13 =37 : 25 4 29x s+ = + = . 16% 29x s+ = : 16 8 50 100 v v= = . 56. A) 0 0.6x ( )2 2 '( ) (0.6 ) ' (0.6 ) 2 (0.6 ) (0.6 )(0.6 2 ) (0.6 )(0.6 3 )g x x x x x x x x x x x= = = = 0 0.2x '( ) 0g x < g [ ]0,0.2 0.2 0.6x '( ) 0g x > g [ ]0.2,0.6 f 0.2x = : ( ) (0.2) ( ) 0.032g x g g x )i) 4 1 1i i f = = 4 3 30.6 , 0 0.6 (1)f f f= 2 3 3(0.6 ) 0.032f f ) 3( ) 0.032g f 30 0.6f . 13 17 21 25 29 33 37 99,7% 95% 68% 34% 34% 13.5% 50% 16% 2.5% 0.15% 13.5% 50% 16% 2.5% 0.15% 2.35%2.35% 95. - 95 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ii) 4 4 1 1 ... 16.4i i i i i i x x f x f = = = = = %iF 16.5 = . x . iii) . A={ [ )16,20 } 4 3( ) 0.6P A f f= + = ={ [ )17,19 } 34 ( ) 0.3 2 2 ff P B = + = ={ [ )12,15 } 2 1( ) 0.25 2 f P f = + = 57.) 5 1F = 5 % 100F = vieta : 5 3 5 1 3 3 5 8 3 5 ... 5 3 1 5 F F F F F F = + = = == 1 2 2 5 % 100 3 30 100 3 30 100 10 7 F = = + = + = = = 1% 0,F = 10 = [ ) ix if i ix f iF % 12-14 13 0.1 1.3 10 14-16 15 0.3 4.5 40 16-18 17 0.4 7.6 80 18-20 19 0.2 3.4. 100 96. - 96 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) 1 1 2 2 2 1 3 3 3 3 2 2 4 4 4 3 5 5 4 % % 10 % 3 10 40 % % % 40 10 30 % 100 60 % % % 60 40 20 % 2 10 90 % % % 90 60 30 % % % 100 90 10 f F F f F F F F f F F F f F F f F F = = = = + = = = = = = = = = = + = = = = = = = ) 2 1 1 4 5 4 (1) 4 2 (2) % 25% % ... 16 (1) 2 % 25% % ... 24 (2) 2 2 2 8 4 f f x f f x x x c c c = + = = + = = = = 1 [ ), 4 ,a a + 2 [ )4, 8 ,a a+ + 2 4 8 4 8 2 12 16 16 16 6 10 2 2 2 a a a a a x a a + + + + + + + = = = = + = ix %if iF %iF [ )10,14 12 10 0.1 10 [ )14,18 16 30 0.4 40 [ )18,22 20 20 0.6 60 [ )22,26 24 30 0.9 90 [ )26,30 28 10 1 100 100 ) 40% (f4%+ f5%) 22. 40% 800 97. - 97 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 100% 40 800 100 ... 2000 = = 57.) : 5 ( ) ( ) ( ) ( ) 5 16 4 16 P A B P A B P B A P B A x + + + = = 5 ( ) ( ) ( ) ( ) 4 P A B P A B P B A P B A + + + = 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 P A P B P A B P A B P A B P B P A B+ + + = 5 5 5 2( ( ) ( ) ( )) 2 ( ) ( ) 4 4 8 P A P B P A B P A B P A B+ = = = ) : ( ) ( ) ( ) ( ) 2 ( ) ( ) 2 2 P B P A P A B P B P A B P A< < < < ( ) ( ) 2 ( ) ( ) ( ) ( )P B P A B P A B P A B P A P A B < < ( ) ( ) ( )P B A P A B P A B < < (1) ( ) ( )A B A B P A B P A B (2) (1) ,(2): ( ) ( ) ( ) ( )P B A P A B P A B P A B < < 1 ( ) ( ) 1 1 1 ( ) ( ) ( ) ( ) ( ) 4 2 4 2 2 1 1 ( ) ( ) ( ) ( ) 2 2 P A B P A B P A B P A B P A B P A P A B P A B P A P A B P A + = = + = + = + = = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5 1 1 1 ( ) ( ) 8 2 8 4 P A B P A P B P A B P B P A B P A P A B P B P B = + = + = + = ) (( ) ( )) ( ) ( ) ( ) ( ) ( ) ( )P A B B A P A B P B A P A P A B P B P A B = + = + = 1 1 1 1 ( ) ( ) 2 ( ) 2 2 4 8 2 P A P B P A B+ = + = 98. - 98 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) 1 2 3 4 5 1 1 1 ( ) , ( ) , ( ) , ( ) , 8 2 4 8 x P A B x P A x P B x P A B= = = = = = = = 4 1 5 1 1 1 3 8 2 4 8 2 ix = + + + = 2 2 2 224 1 5 1 1 1 23 8 2 4 8 32 ix = + + + = 24 24 12 1 1 5 4 4 128 i i x s x = = 58. :{ } :{ } :{ } :{ } ,,, . : 150 3 ( ) ( ) ( ) 100 2 N A N B N B= = 10 1 ( ) ( ) ( ) 100 10 N N N = = ( ) 1 ( ) ( ) 10 N P = = 200 ( ) ( ) 2 ( ) 100 N N N = = ( ) ( ) ( ) ( ) ( )N + + + = 3 1 ( ) ( ) ( ) 2 ( ) ( ) 2 10 B N B B + + + = 9 9 2 ( ) 2 ( ) ( ) ( ) ( ) 2 10 10 ( ) 10 2 ( ) 10 B B B P B = = = = 99. - 99 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 3 ( ) 3 ( ) 3 2 3 ( ) ( ) ( ) 2 ( ) 2 ( ) 2 10 10 N A N N A N B P N N = = = = ) 3 1 4 ( ) ( ) ( ) 10 10 10 P A P A P = + = + = ii) 1 ( ) 10 P = iii) : ( ') ( ) ( ') ( ') ( ) 1 ( ) ( ) ( ) 1 ( ) ( ( ) ( )) 2 8 ( ) 1 ( ) ( ) 0 1 ( ) 1 10 10 P P P P P P P P P P P P P P P = + = + = + = = + + = = = 59.i) 4 (0) (1) 2 (2) 4 (3)P P P P = = = = : (0) , (1) , (2) , (3) 4 2 4 P P P P = = = = 1 (0) (1) (2) (3) 1 1 ... 4 2 4 2 P P P P + + + = + + + = = 1 1 1 1 (0) , (1) , (2) , (3) 8 2 4 8 P P P P= = = = ii) 2 23 ( ) ( 3 5) 666 2 f x x x = + + , x 2 '( ) 3 ( 3 5)f x x = + . f 1x = : 2 3 5 '( ) 0 .. 3 f x x + = = 2 3 5 '( ) 0 3 f x x + < < 2 3 5 '( ) 0 3 f x x + > > . f , 2 3 5 1 ... 1 2 3 + = = = 100. - 100 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 {1,2} = , 1 1 3 ( ) (1) (2) 2 4 4 P P P = + = + = iii) 2 1,1,6, ,3,3,2,6 3 . : 2 2 1 1 6 3 3 2 6 3 3 22 2.5 8 8 x + + + + + + + + = = > 2 2 3 22 20 3 2 0 .. 2 1 + > + > > < 3 0 = = {0,3} = . iv) {1,2} = , {0,3} = , {0,1,2,3} = , : = , = , = , = , ' = , ' = , ( ) 0P = , ( ) 1P = , 3 ( ) ( ) (1) (2) 4 P P P P = = + = 1 1 1 ( ) ( ) (0) (3) 8 8 4 P B A P B P P = = + = + = ( ' ) ( ) ( ) 0P A B P B B P = = = ( ' ') ( ) ( ) 0P A B P B A P = = = 1 ( ' ') ( ) ( ) 4 P A B P B A P B = = = 60.) : 1 1( ) ( )P P = = (1) 2 2( ) ( )P B P = = (2) (1) +(2): 1 2 ( ( ) ( )) ( ( ) ( ))P P B P P B + = + = + ( ) ( ) 1 ( ) ( ) ' B B P P B P B P B B = = + = = = = ) 1 20 1 ( ) ( ) v v P B x P v v + = = = . ) : ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 1 2 1 1 2 2 2 2 2 2 2 2 1 2 1 2 ( ) ( ) 1 1 1 1 ( ) ( ( ) ) (0 ( )) (1 ( )) 1 1 1 (0 ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i i i i i i P P B s x x P B x P B P B P B P P B P P P B P B P P P B P B P P P B P P B P P B = = + = = = = + = + = + = + = + = + = . , - - . : 1) A B , A B ( ) ( )P A P B< 2) ( ) 1P A = A = 3) ( ) 0P A = A = . 101. - 101 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) ( ) ( ) 1P P B + = 2 2 ( ) ( ) ( )(1 ( )) ( ) ( )s P P B P P P P= = = 2 ( )f x x x= , 0 1x , 0 1 2 x = . 1 ( ) 2 P A = 61.) ( )'( ) ' ( )x x x x x x f x e e e e e e = = = ( ) 2 ''( ) ( ) 'x x x x f x e e e e = = ) : ( ) 2 2 2 2 2 2 2 :( 1) '( ) ( ) ( 1) ( ) ( ) ( ) ( ) ( ) ''( ) x x x x x x x x x x x x f x f x e e e e e e e e e e e e f x + = + = + + + = + + = = ) (0) 1f = , ( )0,1 . f 0 0 '(0) ( ) 0f e e = = = . y x = + 0 = y= ( )0,1 1 = , 1 , 1y = > . ) 1 '( ) 0 ( ) 0 ( 1) 0 0x x x x f x e e e e x x x x > = = = = = = 0x < 1 > 0 '( ) 0x x x x x x e e e e f x < < < < 0x > 1 > 0 '( ) 0x x x x x x e e e e f x > > > > f ( ],0 [ )0,+ . f 0x = (0) 1f = ) f 1 ( ) 1f x x , 1 1x x x x e e e e + + x . 102. - 102 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 62.) . . E// xx f3 %= f4 % ,: f1 %+ f2 % + f3 %+ f4 % + f5 %=100 40+2f3 %=100 f3 %=30% . y = y=30 ) . ) , [ ) [ ), ,..., 4 , 5a a c a c a c+ + + : 10 22 ... 4 5 9 18 2 c c c c + + = = + + + = = . ( fi, fi %Fi, Fi%) : [ ) ix if iF if % iF % 9-11 10 0.1 0.1 10 10 11-13 12 0.2 0.3 20 30 13-15 14 0.3 0.6 30 60 15-17 16 0.3 0.9 30 90 17-19 18 0.1 1 10 100 1 100 ) 1 1 2 2 3 3 4 4 5 5 .. 14.2x x f x f x f x f x f= + + + + = = . fi% Z H 103. - 103 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 15 13 60 30 ... 14,3 13 50 30 = ) 15000 : 4 5% % 30% 10% 40%f f+ = + = ) 80 , c , =80. : 80(40/100)+32. 63. 1. 2. 3. 4. 5. 6. ) 1. 2. 3. 4. 5. 6. 7. 64.A) 2014 (0) 0 0 0 0fC f a = + = = 2014 (1) 1 1 1 1fC f a a = + = = Fi % 9 11 19171513 100 10 50 60 30 20 90 104. - 104 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) i) 2014 ( )f x x= , 2013 '( ) 2014f x x= . fC (3,f(3)) 2013 '(3) 2014 3f = . 2014 2014 0 (3 ) 3 '(3) lim h h f h + = 2014 2014 2013 0 (3 ) 3 lim 2014 3 h h h + = . ii) : 2014 2014 2014 2014 2013 0 0 (3 ) 3 (3 ) 3 1 1 lim lim 2014 3 ( 2014) 2014 2014h h h h h h h h + + = = = + + iii) 0 0( , ( ))x f x fC // 2013 2013 0 0 0 0'( ) 2014 2014 2014 1 1f x x x x = = = = = . (1, (1))f (1,1) : .... 2014 2013y x y x = + = iv) : y x = + 2013 0 2013 = + = 2013y x= 0 0( , ( )x f x : 2014 2013 0 0 0 0 0 0 0 2014 2014 2014 2014 0 0 0 0 0 '( ) 2013 ( ) '( ) 2013 2014 2013 2014 2013 2013 2013 1 1 y f x x f x f x x x x x x x x x x = = = = = = = (1, (1))f ( 1, ( 1))f , . 65.A) [ ), c + [ )3 , 4c c + + ( 4 ) 4R c c = + = 4 16 4c c= = . 1 2 3 4 3 % % % % 100 2 100 ... 20 2 2 a f f f f a a a + + + = + + + = = ) [ )2 , 3c c + + 10 42 3 2 20 10 10 .... 0 2 2 cc c =+ + + + = = = [ ) [ ) [ ) [ )0,4 , 4,8 , 8,12 , 12,16 105. - 105 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 fi fi% Fi Fi% i ix f 2 i ix f [ )0 4 2 0.1 10 0.1 10 0.2 0.4 [ )4 8 6 0.2 20 0.3 30 1.2 7.2 [ )8 12 10 0.3 30 0.6 60 3 30 [ )12 16 14 0.4 40 1 100 5.6 78.4 - 1 10 - - 10 116 . : 4 1 10i i i x x f = = = . : ( ) 2 24 4 4 4 4 2 1 12 2 2 2 2 1 1 1 1 116 100 16 i i i i i ii x i i i i i i i i x x s x x f x x = = = = = = = = = = = 4xs = . 10 4 6xx s = = , 10 4 14xx s+ = + = : 2 4 3 % % 20 40 % 30 % 60% 2 2 2 2 f f f + + = + + = ) [ )0,4 [ ) [ ) [ )4,8 , 8,12 , 12,16 2 3 4, , . : 2 2 2 1 11 1 ' 1 f f f = = = , 3 3 3 2 11 1 ' 1 f f f = = = , 4 3 1 ' 1 f f f = : 106. - 106 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 3 32 4 2 1 3 2 4 3 2 3 4 1 1 1 1 2 2 3 3 4 4 1 ' ' ' ' ' 1 1 1 1.2 3 5.6 12.25 1 1 0.2 i i i ff f x x f x f x f x f x x x f f f x f x f x f f = = = + + = + + = + + + + = = = 66. 3 2 21 9 1 9 1 '( ) 2014 ' 2 40 20 20 20 f x x x x x x = + + = + 1 2 1 1 '( ) 0 ... , 4 5 f x x x= = = . : x 1 5 1 4 + f '(x) + f(x) ( ), ( ), ( ), ( )P A P B P A B P A B = 1 4 ( ), ( ), ( ), ( ), ( ), ( )P A P B P A B P B A P A B P A B 1 5 x = . ( ) ( ) ( ) ( )P A B P A P B P A B ( ) ( ) ( ) ( )P A B P B P A P A B ( ) ( ) 2 P B P A + = . ( ) ( ) 1 1 ( ) ( ) 2 4 2 P B P A P B P A + = + = (1) 107. - 107 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 (1) ( ) ( ) ( ) ( ) ( ) ( ) 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 6 5 1 3 2 ( ) 3 ( ) 3 ( ) 2 ( ) 1 12 6 5 6 5 3 4 ( ) 1 3 4 ( ) 12 15 20 ( ) 12 6 5 12 5 20 ( ) 3 ( ) P A P B P A B P B A P A B P A B x P A P B P A B P B A P A B P A P B P A B P A B P A P B P A B P A B P A B P A B P A B P A B + + + + + = + + + + + + + = + = = = = = = 3 20 = : 1 3 7 ( ) ( ) ( ) ( ) 2 20 20 P A B P A P B P A B = + = = 67)A) ( )2 '( ) 2 2 1 ' 4 2f x x x x= + = , 1 '( ) 0 4 2 0 2 f x x x= = = '( ) 0f x < 1 2 x < f 1 , 2 '( ) 0f x > 1 2 x > f 1 , 2 + f 1 2 x = 1 1 ( ) .. 2 2 f = = ) i) : 4 1 ( ) ( ') ( ) ( ) 1 0 1 1 4 4 4 2 i i t P A P A P P x = + + + + + = = = = : ( ) 0, ( ), ( '), ( ) 1P P A P A P = = ( A ( ) 0P A ( ') 1P A 0 ( ) 1P A< ) ( ) ( ) ( ') ( )P P A P A P < 0 ( ) ( ') 1P A P A< . 108. - 108 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ( ) ( ') 1 2 2 P A P A + = = ii) ( ) 4 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 [( ( ) ) ( ( ') ) ( ( ) ) ( ( ) ) ] 4 4 2 2 2 2 1 1 1 1 1 1 1 1 1 1 [( ( ) ) (1 ( ) ) ( ) ( ) ] [( ( ) ) ( ( )) ] 4 2 2 2 2 4 2 2 4 4 1 1 1 1 1 1 1 1 [2( ( ) ) ] [2( ( ) ( ) ) ] [2 ( ) 2 ( ) 4 2 2 4 4 2 4 2 i i s t x P A P A P P P A P A P A P A P A P A P A P A P A = = = + + + = + + + = + + + = + = + + = + + 2 1 ] 2 1 [2 ( ) 2 ( ) 1] 4 P A P A = + iii) 2 1 ( ( ( )) 4 s f P A= () 1 ( ) 2 f x x f 1 2 , : 21 1 1 1 1 1 ( ( )) ( ( )) 2 4 8 8 8 2 2 f P A f P A s s s 1 ( ) 2 P A = . 1 1 22 2 1 1 22 2 2 s s CV x = = = = . 2 2 CV 1 ( ) 2 P A = , 1 ( ') 1 ( ) 2 P A P A= = ( ') ( )P A P A= 68.A): 2 2 2 2 2 2 212 2 21 1 1 1 1 1 i i i i i i i i i i i i x x x x x s x x s x = = = = = = = = = = 109. - 109 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 30 2 2 2 02 2 2 22 2 2 2 21 2 3030 1 1 1 101 100 30 30 100 10 10 i x i x s s s s x s x s s x x s CV xx > = = = = = = = = . .i) () 3 2 3 2 3 2 40 40 1 ( ) 2 ( ) ( ) 2 ( ) 2 ( ) ( ) 2 3 3 10 4 ( ) 2 ( ) ( ) 2 3 f x CVx P A B x P A B x f x x P A B x P A B x f x x P A B x P A B x = + + = + + = + + 3 2 24 '( ) 2 ( ) ( ) 2 ' 4 4 ( ) ( ) 3 f x x P A B x P A B x x P A B x P A B = + + = + 2 '( ) 0 4 4 ( ) ( ) 0f x x P A B x P A B= + = : 2 16 ( ) 16 ( ) 16 ( )( ( ) 1)P A B P A B P A B P A B = = , ( ) 0, ( ) 0 0 ( ) 1 ' ' A B P A P B P A B B A B A < < 0 < x '( ) 0f x > x f x . ii) (0) '(0)( 0) 2 ( ) ( ) 2y f f x y P A B x y P A B x = = = + iii) xx : 2 0 ( ) 2 0 ( ) y P A B x x P A B = + = = 2 ( ,0) ( ) A P A B yy : 0 2x y= = (0,2)A : 1 2 2 2 2 ( ) ( ) E P A B P A B = = . 2 1 4 4 ( ) ( ) 2 E P A B P A B = = = 110. - 110 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 iv) 23 '( ) 666 ' 3 2 g x x ax x a = + = 0 (1 ) (1) '(1) lim 2 3 1 2 1 h g h g g a a h + = = = = 23 ( ) 666 2 g x x x= + . v) 1 '( ) 0 3 1 0 3 g x x x= = = 1 1 '( ) 0 3 1 0 , , 3 3 g x x x g x > > > + 1 1 '( ) 0 3 1 0 , , 3 3 g x x x g x < < < g 1 3 x = , 1 ( ) 3 P A B = : ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 ( ) ( ) ( ) ( ) 2 3 6 P A B P A P B P A B P A B P B P A B P B P A B P A B P B = + = + = = = 69 1, 60 2 60 200 320 3.2 100 100 100 x + + + = = = 3 4 3.5 2 + = = , x > 0.46 3.5 3.2 0.46 0.3 0.46 30 46 16 100 100 x = + = + = + = = ix iN i i ix 1 2 30 30- 60-2 3 50 20 60 4 100 50 200 111. - 111 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 2. 16 3.2 3.2 0.16 3.04 100 x = = = 1. x : ( )2 2 2 '( ) 1 'x x f x e x e = + = ( )2 2 2 4 ''( ) 'x x f x e e = = B2. 2 2 0 2 '(0)f e = = . 2 ( )g = '( ) 2 1g = 1 '( ) 0 2 1 0 2 g = = = 1 '( ) 0 2 1 0 2 g > > > 1 '( ) 0 2 1 0 2 g < < < '(0)f 1 2 = 3) ( )3 3 3 { / 3.31} { / 0.46 3.31} { / 0.46 3.31} 2 2 2 x = + = + = + { / 4} { / 4 4} { 4, 3, 2, 1,0,1,2,3,4} = = = 2 2 '(0) (0) 2 2 0 1 2f f > > > < > { 4, 3, 2,3,4} = 4 4 ''(0) 256 256 256 4 4 4f < < < < < < { 3, 2, 1,0,1,2,3}B = i) ( ) 3 1 ( ) ( ) 9 3 A B P A B = = = ii) { 4,3,4}A B = ( ) 3 1 ( ) ( ) 9 3 A B P A B = = = iii) { 1,0,1,2}B = ( ) 4 ( ) ( ) 9 N P N = = 1 2 + '( )g + ( )g 112. - 112 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 iv) ( ) ( ) { 4,1,0,1,2,3,4} = , : (( ) ( )) 7 (( ) ( )) ( ) 9 N P N = = ): 5 6 3 10 12 3 4 4 x + + = = = .: 1 3 1 1 1 1 .... 3 1 31 x x > > < ++ { 4, 3, 2, 1} = , ( ) 4 ( ) ( ) 9 N P N = = 70. ) ( ) 1 f (x) x x s 1 4 = + . Cf (1, f(1)) 2y x= + 1 1 f (1) 1 x 1 (s 1) 1 x s x 4s 4 4 = + = = = (1). : s s s 1 CV 25% 10% x 4s 4s 4 = = = = = > , . ) ( ) ( ) ( ) 2 222 2 2 2 2 2 2 1 1 lim ( ) lim 1 1 2 8 8 = = + + = x s x s xs s s sf x xx s x x s s 2 2(1) 2 2 3 2 2 4 2 2 2 2 2 2 2 2 2 2 0 2 2 lim ( ) 2 = = + == x s xs s s s s s s s s sf x ( ) ( ) ( )( ) ( ) ( ) 22 2 2s s 1 2 s 1 0 2 s 1 s 1 0 2 s 1 s 1 0 s 1 s 1 = = + = = = s 0 s 1= (1) x 4= . ) 21 2 2 ( ) = f x x x , 2 2 1 2 1 1 1 1 1 1 ( ) ( ) ... ( ) ( ) 2 2 2 2= = = = + + + = = = i i i i i i i i i f x f x f x f x x x x x (2) : i i 1 i i i 1 i 1 x x x x x 4 = = = = = = (3) 113. - 113 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ( ) 2 2 2 2 212 2 2 21 1 1 1 1 = = = = = = = = i i i i i i i i i i x x x x s x s s x ( ) ( ) 2 22 2 2 2 i i i 1 i 1 s x x x s x = = = = + , ( ) 22 2 i i 1 x 1 4 17 = = + = (4). (2), (3) (4) 1 2 1 1 ( ) ( ) ... ( ) 17 2 4 2 2 + + + = =f x f x f x . ) R 6 s 6 = . ( )2,5 ( )x 2s,x s + 95 68 68 % 81,5% 2 + = . 71.) 1 1 , y x 1 12 1 13= + = + = ys s 2= = . 2 10% , i i i i 10 t x x 1,1 x 100 = + = , i 1,2,...,= . t 1,1 x 1,1 12 13,2= = = ts 1,1 s 1,1 2 2,2= = = . ) - 1 y y s 2 CV y 13 = = - 2 t t s 2,2 2 CV t 13,2 12 = = = y t 2 2 CV CV 13 12 < < , 1 . ) y 13= , ys 2= 2 i i 1 y 4325 = = . : ( ) 2 2 2 2 212 2 2 21 1 1 1 1 = = = = = = = = i i i i i i i i y i y y i y y y y s y s s y ( )( ) ( )22 2 2 2 1 2 13 4325 173 4325 25 = + = + = = = y i i s y y . 114. - 114 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) 1 12x = s 2= , 14 x s= + 100 68 % 16% 2 = , 16 25 4 100 = . ) 25 i 25 i 1 i i 1 y y 13 13 y 325 25 = = = = = . 15 11, : 25 i i 1 1 y 11 15 325 4 329 y 13,16 25 25 25 = + + = = = = . 2 i i 1 y 4325 = = , 2 2 4325 11 15 4429 + = , : 1 2 2 y 1 329 s 4429 3,9744 25 25 = = . 72. . ( )f x 2 x 2 = . () fC ( )( )0, 0f f (0) 2 = = , (): y 2 x= + . ( )( ) ( )0, 0 f , f (0) 2 0 1= + + = . (): ( )y 2 x 1, 0,= + + + . . 1) () x 0= y 1= + y 0= 1 x , 0 2 + = > . () x x 1 A ,0 2 + y y ( )B 0, 1 + . () : ( ) 1 1 1 1 1 1 ( ) ( ) 1 1 , 0 1 0 0 2 2 2 2 2 2 + + + = = + = + > + > > ( ) ( ) 2 1 ( ) , 0, 4 + = + . 2) ( )0, + : ( ) 2 2 2 2 2 2 2 2 2 2( 1) 1 4 ( 1) 4 8 8 4 8 4 4 4 1 ( ) 16 16 44 + + + = = = = , 2 2 2 2 1 ( ) 0 0 1 0 1 1 1( . 0) 4 = = = = = = > 115. - 115 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 0 1 + ( )E - 0 + ( )E ( ]0,1 [ )1, + . 3) 1 = ( ) 2 1 1 4 (1) 1 4 1 4 + = = = 4) ( )0, + : ( ) ( ) 2 22 3 3 22 4 4 32 2 4 1 81 8 8 8 8 1 ( ) 0 4 16 16 24 + = = = = = > ( )0, + . (x) ( )0,+ , . . 1 2 10x ,x ,...,x () x 4= 1 s 2 = . i iy 2 x 1, i 1,2,...,10= + + = , y 2 x 1 2 4 1 7 1= + + = + + = + 2 0 y 1 s 2 s 2 2 < = = = . 1 7 : y y s 1 CV 10% 10 7 1 y 7 1 7 1 10 = = = = = + + + ( ) 1 1 7 1 10 7 1 10 . 0 17 3 + = + = = = > 116. - 116 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 73.A)0 ( ) 1 2 ( ) 2 1 2 2 ( ) 2 1 ( ) 2 2 ( )P P P P P = 0 ( ) 1 1 ( ) 1 2 ( ) 1 ( ) 1P P P P + + = + ( ) 2 ( ) 1 2 9 2 ( ) ( ) 1 2 9 1 2 ( ) 2 9 2 ( ) 2 8 ( ) 4 ( ) 4 P P P P P P P P + = + = + = + = + = + = 0 ( ) 1 0 4 1 4 5 5 4P -5 -4, 4, 5 = = , ) ( 4) ( ) (2 5 3) ( ) 2 (2 3) 2 x xe f x x f x e x = + = i) ( )'( ) 2 (2 3) ' 2 (2 3) 2 (2 3)' 2 (2 3) 2 2 2 (2 3) 4 2 (2 3 2) 2 (2 1) x x x x x x x x x f x e x e x e x e x e e x e e x e x = = + = + = + = + = 1 '( ) 0 2 (2 1) 0 2 1 0 2 x f x e x x x= = = = 1 '( ) 0 ... 2 f x x f> > 1 , 2 x + 1 '( ) 0 ... 2 f x x f< < 1 , 2 x f 1 2 x = 1 ( ) 4 2 f e= ii) 1 1 2 x = 1 ( ) 2 P A = , 4 2 ( ) 36 e P B e = = iii) , ( ) 0A B P A B = = 1 2 7 ( ) ( ) ( ) ( ) ( ) 1, 2 3 3 P A B P A P B P A B P A B = + = + = > . , . iv) ( )' ' ' ' ' ' 'A B A B A B B A B A = = = = : 2 2 ( ) ( ) ( ) ( ' ') 3 3 B A B P B A P B P B A P A B 117. - 117 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 : 1 1 1 ( ' ') ( ) ( ) ( ) 6 6 6 P A B P B A P B P B A 1 2 2 1 1 ( ) ( ) ( ) ( ) ( ) 6 3 3 6 2 P B A P B A P B A P B A P A . 74.. i) 1 1 1 1 ( ) ( ) ... 3 6 P f = = = 2 2 1 2 ( ) ( ) ... 3 3 P f = = = ( ) ( ) 2 2 2 22 2 2 1 (2 ) 1 '( ) 1 ' 1 1 1 x x x x x f x x x x + + = + = = + + + ( ) ( ) ( ) 2 2 22 2 3 21 1 1 1 2 1 ( 1)( 1) 1 11 '( ) 1 1 1 ( 1) 1 ( ) lim lim lim lim 6 1 6 1 6 1 6 121x x x x x x x x xf x x P x x x x + + + + + = = = = = + 4 1 2 3 1 2 1 1 ( ) 1 ( ) ( ) ( ) 1 6 3 12 12 P P P P = = = ii) 22 1 0 2 2 2 1 '( ) 0 0 1 0 ... 1 1 ( 1) f + > + 3 4{ 1, , } = 3 4 1 1 1 1 ( ) ( 1) ( ) ( ) 6 12 12 3 P P P P = + + = + + = 2 1 0 2 2 ( ) 1 1 1 0 0 1 1 f + > > + > > > + + , 3 4{ , } = 3 4 1 1 1 ( ) ( ) ( ) 12 12 6 P P P = + = + = 2 21 4 4 1 0 4 x x x x + + + x . 2 0 1 0 .. 1 1 { 1,0} = 118. - 118 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 1 2 5 ( ) ( 1) (0) 6 3 6 P P P = + = + = { 1} = 1 ( ) ( 1) 6 P P = = . 2 0 0 0 02 2 0 1 '( ) 45 1 ... 0 ( 1) x f x x x = = = + '(0) 1, (0) 1f f= = : : (0) '(0)( 0) .. 1y f f x y x = = + . 3 41,0, ,i = 1 1 1 1 1 1 0M y = + = + = 2 2 2 1 0 1 1M y = + = + = 3 3 3 1M y = + 4 4 4 1M y = + 3 41 < < 1 2 3 4y y y y< < < 4 1 4 45 ( 1) 0 .. 4yR y y = = + = 3 30 2 2 + = = 3 31 1 2 2 2 y + + + = = 3 3 3 2 2 2 ... 2 2 2 y + = = = 75.i) Euler : 3 3 3 3 (2014 100 ) (40 1000) (60 1014) 0 3(2014 100 )(40 1000)(60 1014) 0 2014 1002014 100 0 1000 40 1000 0 25 40 60 1014 1014 60 + + = = = = = = = = ( ) 25 = . ii) ( ) 12 ( ) 2 ( ) , ( ) ( ) 25 ( ) 25 P P = = = = 119. - 119 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 76)1. , : =20 2 2 20 10 10 ,x z x z z x+ = + = = 0 10x< < 1 2 3 2 2 ( ) 2 2 ( ) 2(5 ) 2(5 ) ( ) 2(5 ) 2 5(10 ) (10 ) ( ) 10 100 10 10 ( ) 10 100,0 10 E x E E E E x x z xz E x x x x x E x x x x x E x x x x = + + = + + = + + = + + = + + < < ( )2 '( ) 10 100 ' 2 10 '( ) 0 2 10 0 5 E x x x x E x x x = + + = + = + = = 5x = . 2.) 1 1 1 8 0,8 10 10 8 10 10 s s CV s s x > > > > > 2 2 5 2 0s s + = , 2s = 1 2 s = . 2s = . 120. - 120 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 ) 2 2 2 2 212 2 1 1 1 1 1 i i i i i t t t t s t x = = = = = = = = 2 2 2 22 2 21 1 1 2 8 68 i i it t t s x = = = = + = + = 2 ix 1,2,..,15i = 68. 3. [ ]( ) 5,9 1 2 15 1 2 155 ... 9 (5) ( ) ( ) ... ( ) (9) E x x x x E E x E x E x E= < < < = = > > < = 1 2 14...y y y> > > (5) 125E = , (9) 109E = 1 15 125 109 16R y y= = = 2 2 1 2 14 4 9 1 ( ) 4 9 16 1 10 100 4 9 16 1 14 45 0 .... 5 9 { , ,..., } i i i i i i i i i i y x R E x x x x x x x x B A A A > + + > + + + + > + + + > < < = ( ) 13 ( ) ( ) 15 N B P B N = = 77)1.i) =x , =100-x 2 2 2 2 2 ( ) (100 ) 10000 200 2 200 10000 ,E x x x x x x x x = + = + = + + = + 0 0 100 100 0 x x x > < < > ii) ( )2 '( ) 2 200 10000 ' 4 200E x x x x= + = , '( ) 0 4 200 0 50E x x x= = = =50. 2. 1 1 ( ) 50 25 2 2 2 i i i i x x x x x = = = = = = = = 3. 121. - 121 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 2 2 2 2 21 12 2 2 21 1 2 1 2 2 22 2 2 2 21 1 1 1 0,2 2 4,04 4,04 i ii i i i i i x xx x s x s s x x x E s x m E m = == = = = = = = = = = + = + = = 4. i 3 4, 1,2,..,25i = 3i = 4i = 6i = 8i = 9i = 12i = 15i = 16i = 18i = 20i = 21i = 24i = 3 4 6 8 9 12 15 16 18 20 21 24{ , , , , , , , , , , , }l l l l l l l l l l l l = , ( ) 12 ( ) ( ) 25 N P N = = 78). 4 25 1 1 25F = = = . =25 : . 1 1 1 4 0,16, 0,16 25 F f F= = = = , 2 2 2 1 11 0,44, 0.28 25 F f F F= = = = 3 3 3 2 18 0,72, 0.28 25 F f F F= = = = , 4 4 4 31, 0.28F f F F= = = . 4 1 1 2 2 3 3 4 4 1 19 0,16 0,28( 5) 0,28( 10) 0,28( 35) 19 0,16 0,28 1,4 0,28 2,8 0,28 9,8 19 14 5 i i i x x f x f x f x f x f a a a a a a a a a a = = = + + + = + + + + + + = + + + + + + = + = 79)) 125 100% 100% 12,5% 1000 A A A s CV x = = = 90 100% 100% 11,25% 800 E E E s CV x = = = A ECV CV> , . ) 250 : ' 250 1000 250 1250A Ax x= + = + = ' 125A As s= = 122. - 122 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 20% 1,2. : ' 1,2 800 1,2 960E Ex x= = = ' 1,2 90 1,2 108E Es s= = = ) ' 125 ' 100% 100% 10% 1250' A A A s CV x = = = ' 108 ' 100% 100% 11,25% 960' E E E s CV x = = = : ' 'A ECV CV< , . 80)) 0 0 0 01 1 1 1 1360 50 360 5 50 10 72 a = = = = 0 0 0 02 2 2 2 2360 30 360 5 30 6 72 a = = = = 1 2 3 4 + + + = 3 3 310 6 3 72 14 + + + = = 4 42 = ) 0 0 03 3 14 360 360 70 72 a = = = 0 0 04 4 42 360 360 210 72 a = = = ) 4 1R x x= 4 1 1 2 3 410 6 14 42 72 i i i x v x x x x x = + + + = = 72 72 1 36 372 2 4 4 4 2 2 2 x x x x x x x + + + + = = = = .: 123. - 123 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 1 2 3 4 4 1 4 1 1 2 3 4 4 2 3 4 4 10 6 14 42 10 72 10( ) 72 72 10 10 10 6 14 42 52 6 14 52 6( 7) 14 3 52 52 x x x x R x x x x x x x x x x x x x x + + + + = + = + + + + = + + = + + = = 81) .. ( ) 3 3 3 2 3 3 3 3 3 8 8 '( ) ( 4 )' 8 , 0,1 x f x x x x x x x = + = = = ( )0,1x : 33 3 3 3 3 3 3 3 8 8 2 2 '( ) 0 0 8 0 x f x x x x x x > > > > > > f 2 ,1 2 0, . f 2 x = 3 2 2 2 2 2 2 2 2 4 2 2 4 4 1( ) 2 2 2 3 4 42 f = + = + = + = + = ( )0,1x : 22 ( ) ( ) ( ) 3f x f f x . . ( ) 0P A > - ( ) 1P A = 3 2 2 4 ( ) 3 ( ) P A P A + = 3 2 3 2 4 3 3 4 0 1 2 + = + = = = 2 = ()=-22 ,. - 0 ( ) 1P A< < 3 2 2 2 4 ( ) 3 ( ( )) 3 ( ) P A f P A P A + = = 22 2 0 ( ) ( ( )) ( ) ( ( )) 3 f P A f P A f f P A < < > > , . 22 2 ( ) 1 ( ) ( ( )) ( ( )) 3 f P A f f P A f P A < < < > , . 124. - 124 hhttttpp::////mmaatthhhhmmaaggiicc..bbllooggssppoott..ccoomm// :: -- -- -- 4499 2 22 ( ) 2 ( ) ( ) 2 9 8 2 9 10 0P A = = = = = : 10 = 1 = ( ) 2 1 ( ) 10 5 P A = = . ( ' ) ( ') ( ) ( ' )P A B P A P B P A B = + = 1 ( ) ( ) ( )P A P B P B A + = 1 1 29 1 ( ) ( ) ( ( ) ( )) 1 ( ) ( ) 1 5 6 30 P A P B P B P B A P A P B A= + = = = 82). 5 4 7 1 5 5