οδηγίες διδασκαλίας μαθηματικών

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Οδηγίες για τη διδακτέα ύλη και τη διδασκαλία των ΜΑΘΗΜΑΤΙΚΩΝ του ΓΕΝΙΚΟΥ ΛΥΚΕΙΟΥ κατά το σχολικό έτος 2007 - 2008
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Transcript of οδηγίες διδασκαλίας μαθηματικών

  • 1. 2007 - 2008

2. 2 , . 3. 3 2007 - 2008 - 4. 4 (1997) : 1. , .. 2. , .. 3. o, .. 4. /, .. 5 . .. 6. , . , .. 7. , .. 8. , .. (1998) - (1999) - (2000) (2001) - (2002) - (2003) () 2007 , : .. . .. .. .. .. .. - : , . Cabri II . .. .. - .. .. - .. .. - .. . 5. 5 .............................................................................. 7 . ............................................... 9 1 . ............................................................ 11 2. ................................................... 15 3. ........ 30 . ......................................................................... 35 . ........................... 69 . & ....................................................... 79 .......................................................................... 89 . .......................... 89 . & ......................................................... 123 . I - I I I ............................. 139 ............................................................ 165 6. 6 7. 7 - . - , , , 1997 - . , , - , - . , - . , , , - . , . - - . , , - . , - 3 ( 35 ) 8. 8 - - . , - . - , The Geometers Sketchpad, Cabri II Function Probe - . , - ( ) - . - , , - . 9. 9 . TOY : ) , , , , , , . ) , - , , , , , , - , . ) - , , - , - . ) , , , , . ) H - - . , - , : ) , - . ) - , - 10. 10 , . ) , , - : ) . ) - - , ) - , - . ) - . 11. 11 B1. - . , , , - . - , - . , . ' - . , , - . , : 1. - - . 2. , - , , - . , , , . 3. - . , 12. 12 - . - (- - ) - - . , ' - . 4. , . , - , . , . . . - - . 5. , .. - , , , . , , - , . , . 6. - ' , - , , . . - . , - , . . 7. , 13. 13 . , . - , , , ( - , , , .) 8. - , - , . - . , , - . , , . - . 9. - , . - , . - , - , . , - . ' - , - - . . 14. 14 10. . - . , . ( ) , . 11. - . - - . - . 12. - - . - - . - , - , - . - - ' . 13. , (calculators), - . , , . , - , , . 15. 15 B2. 1. . '80, , , - , - - , , , , - , . - , - - . , , - . , : - , . - , - . - : - - . , - . - , , 16. 16 . , , , . , , - , , : - , , - . - . , - : . , , . , , - , - . , , - ! . Lakatos: - - . - - . . . - ; , 17. 17 ; : - ; 2. . . , - . - , : . - , , - : - - ( ). - , - . - : . , - - , - - . - . - , , . - , - ( ). 18. 18 - - -. ' , . , - , - , , , - . , . , , - . - , - , , , - , - , , - . - , - , . - . . - : . - , . ( ). , - , . , - 19. 19 : , - . - . - . . - . : . . - . , - - . 3. - - - . , - . , - , , , - , , - . , , - 20. 20 , , , . . - , - . , - - - - . . - , - . , - , - - , - ! - , - . . , . - . . ' - . , - - , - 21. 21 , . - . , - . , . - . , . - , 300 , - 50 , . , - . , - , : . , - . , . , - , . 22. 22 4. . ; . , . - . , - , . , - - . , - . - : ; " , - , . ( ) - . - - . ' , : ( ). - , . . . , - , - . . 23. 23 - ; , - , , . , - . , , - . , - - , . , - , , ; .... , , - . . , , , : , , , , , . - . , , . , - . : : - 24. 24 , , , ; - - - , , . : - . , - , - , ( 2, 3 - ). , - , . , - - . , , , , . , - : - 1 : 1. : : - (t) 4 12 25. 25 ) : = [4,16] , ( )T t : ) ) ) ) 1t , 2t , - : 1 2t t> 1 2( )... ( )T t T t ) : ) ) ) - . , , ( ) 2 1f x x= + - : 1 2 1 2 1 2 1 2 2 ...2 2 1...2 1 ( )... ( ) x x x x x x f x f x < + + 26. 26 ( ) ,f x ax = = 0, > . 2) - . ( ) 2 1,f x x= + ( ) ,f x ax = + 0,a < - . 3) i) 3 92 ii) , - : ) ) iii) : ) ( ) ,f x x= ) ( ) 8 2 ,f x x= ) 1 ( ) 3,f x x = + (0, )+ ) 4 ( ) 3 ,f x x = (0, )+ i) 1 92 27. 27 2 : 1. T(t) 12 - 12 . ) ; ; ) ; ) : ( )... (4),T t T [ ]0,24t : , - . , - 2 ( ) 2 3,f x x= + (0) 3,f = : 28. 28 2 0x x 2 2 ...0x x 2 2 3...0x + x ( )... (0)f x f x ( ) 2 1 5f x x= . 2. - . 3. ) ( ) - : 29. 29 ) 4 92. 30. 30 B3. 1. . . - . 1. : : i) : (2)f : . , . , , . ( 2)f : . , . , . ( 1)f : . , . , . (0)f : . , . , . . (1)f : . , . , . 31. 31 ii) : ( ) 0f x = ......................................... ( ) 0f x > ......................................... ( ) 0f x < ......................................... 2. 2 ( ) 4 3f x x x= + i) :f = ............... ii) f ; . iii) f - x ; ; i) f : ( ) 0f x > ......................................... ( ) 0f x < ......................................... 3. 2 ( ) 2 2f x x x= + i) :f = ............... ii) f ; . iii) f - x ; ; i) f : ( ) 0f x > ......................................... ( ) 0f x < ......................................... 32. 32 4. 2 ( ) 4 4f x x x= + i) :f = ............... ii) f ; . iii) f - x ; ; i) f : ( ) 0f x > ......................................... ( ) 0f x < ......................................... 5. 2 ( ) 2 3f x x x= + + i) :f = ............... ii) f ; . iii) f - x ; ; 33. 33 iv) : ( ) 0f x > ......................................... ( ) 0f x < ......................................... 6. 2 ( ) 2 1f x x x= + i) :f = ............... ii) f ; . iii) f - x ; ; iv) f : ( ) 0f x > ......................................... ( ) 0f x < ......................................... 34. 34 7. 2 ( ) 2 2f x x x= + i) :f = ............... ii) f ; . iii) f - x ; ; iv) f : ( ) 0f x > ......................................... ( ) 0f x < ......................................... 35. 35 . 2 2007-2008 - . , . , . , . , . . - , , . - , - , , . . . , . , , - , (.. - ) . . , 2, 3 4 . . / 12 .1 1.1. 2 .2 1.2. 2 36. 36 .3 : 0ax + = 1.3. 2 .4 - 1.3. 3 .5 1.4. 2 .6 : 0ax + > 0ax + < 1.5. 1 10 .1 1.6. 3 .2 1.7. 3 .3 2 0,ax x + + = 0a 4.1. 2 .4 4.2. 1 .5 4.3. 1 7 .1 2.1. 1 .2 2.2. 2 .3 2.3. 2 .4 ( )f x ax = + 2.4. 2 7 .1 3.1. 2 .2 - - 22 3.2. 1 .3 - 3.3. 2 .4 4.3. 2 12 .1 2.5. 4 37. 37 .2 2 ( ) ,f x ax x = + + 0a 4.4. 4 .3 2 ( ) ,f x ax x = + + 0a 4.5. 2 .4 : 1 2( ) ( )... ( ) 0vP x P x P x 0 ( ) ( ) P x Q x 0 0. 4.5. 2 6 .1 5.1. 2 .2 5.2. 2 .3 1 5.3. 2 : 12 . . - , - - . 0,ax + = - . , - , 0,a > > - 0ax + > 0. + < - . 38. 38 A.1 ( 1.1): : i. - . ii. . iii. , . - . : 2 2 ( )( 1) 0x x x = , - 2 x x 2 1x , 2 0x x = 2 1 0x = (1) 1x = , 0x = 1x = - . 2 0x x = 2 1 0x = , , 1x = , . 1x = 2 2 1x = x a= 2 2v v x a= ( )*v . .1 i) 13 ' 16. .2 ( 1.2): : i. . ii. . iii. , . i. . .2 : 1. 1 2 1 ( )( ... )v v v v v a a = + + + 2. 1(iii) 18 3(i) 19. 39. 39 3. 5 ' 22 ' 23. , , : 1. N 2 2 ( ) ( )a a + - : 2 2 999 1000 999 1000 4. 1000 999 1000 999 + = 2. 2 ( 1)( 1)a a a + 2 2 1,3265 0,3265 2,3265 1 3,12345 2,12345 = 4,12345=1 3. , : 2 2 3 1 1 , 1 1 x x x x x + + + 3 2 2 2 , x x x x x + 2 2 ( ) 2 2 , 1 x x x x + 2 2 2 2 3 2 2 , 2 x x x x x x x x + + + 3 2 2 3 1 ( ) , ( 1) x x x x x + + ( 2) 1 . ( 2)( 1) x x x x + .4, .5 .4, . .3(1.3): 0.ax + = .3 : 25 , : : i) ( 1) 1, = ii) ( 1) , = iii) ( 1) 1 = 40. 40 . . : ) 0v v at= + t ) 1 2 1 1 1 R R R = + 1R ) 2 0 1 2 S v t at= + 0 ,at = + 0 . 2 v v S t + = 5 ' 28 - 900km/h . 2 3 ' - 28. .4(1.3): . .4: . : i. 2 ( 4) 2 ( 4) ( 4) 0x x x x x + + = ii. 2 3 2 ( 1) 0x x x x + = iii. 2 2 ( 1) 1 0x x+ + = iv. 2 ( 2) (2 )(4 ) 0x x x + = v. 2 2 ( 2) 4 4x x x x = + vi. 2 2 ( 4)( 1) ( 1)( 2)x x x x = vii. 3 2 2 2 0x x x + = viii. 3 2 2 (2 1)( 2) 0x x x x = 41. 41 ix. 2 1 2 4 x x x = + x. 2 1 1 x x x x = xi. 2 2 1 2 0 1 2 1 x x x x + + = + .5 ( 1.4): : i. - , . ii. - . iii. . .5: : ) 3 32 . ) 2 2 0a + 1 37, . 2 2 0 0a + = = 0 = 2 2 0 0a + > 0 1o 31, 4o 33 6 8 36 2 3 - 37. , , : ,x y 10 cm. ( : 9x cm= 1y cm= 8x cm= 2y cm= ... 5x cm= 5y cm= ) - : 1. 2 25cm 2. 2 50cm 42. 42 - ,x y 10 ,cm : y .x x 2 25cm . - x - 2 50cm . .6 ( 1.5): : i. 0ax + > 0ax + < ii. - . : 10 , , . - . - . , - . . : .1 ( 1.6): : i. . ii. . iii. . 43. 43 iv. . .1: , , a 0 ( ( ,0)a d a= ), 0a 0.a < , : ) : 7 ..., = 2 1 ..., = 3 ..., = 2 2 ... = ) x : 5 ...,x + = 2 ...,x = 5 2 ...x x+ + ,x < < < 0, > - : : 0,x x < < 0 0x x < 0,x< x < < 0 0x x . = : : 0a 0, 0, a = = 0a 0,< 0, ( )a a = = = 0a< 0, 0, ( )a a = = = 44. 44 0a< 0,< 0,> ( )( )a a = = = a = . a + + . , , a + + , : , , - 0, . 43. , , - , : ) - : i) 1 3x x = ii) 2 2 1x x = + ) 2 0,1x < 4 0,2y < : 45. 45 , : : ' , 1, 2 x , : ) 1,x < ) 1,x = ) 1 2,x< < ) 2,x = ) 2 x< ) 1) 1x 2x 1 2x x + 2) 1 2x x + ; 3) ; ) 1) 1 2 ;x x 2) - 1 2x x ; .2 ( 1.7): : ,( 0).v a a i. . ii. - - . iii. .v x a= .2: 6 36 4 51 - : , < : ) 2 a a + ) 2a a a + 46. 46 ) 2 2 a + + 5 6 ' - 51 52. .3 ( 4.1): : i. . ii. - . iii. , - , . i. . - ,v x a= 2v = 0,a > ,a a .3 - 2.ii) 122. .4 ( 4.2): : i. , - 0. ii. . .4 1o 1 iii) i), 4 ii) iii), 5 6 124 125. .5 ( 4.3): - : 2 0,ax + + = 0a 2 0,v v ax x + + = 0a . 47. 47 : 7 , . - , 1 3 - . - , ., . ,y = + , . - . : .1 (2.1): : i. Venn. ii. . iii. . i. : , , - VENN. .1 . .2 (2.2): : i. . ii. f(x). iii. f x. 48. 48 .3 (2.3): : - . 70 - {,} (,). i. A(x,y), , 1 3 . ii. . iii. , . i. . .4 (2.4): : i. , .y ax y ax = = + - 1 3 , , 3 1 3 y x y x y x= = = + ii. . .4: : 0ax + > 0ax + < x < x > : 2 4 0,x > 2 4 0,x + > 2x < 2.x > 49. 49 , - 4 76 1ii), 1iii) 3 78. , , : 20 cm . , - , . x ( )f x , ) f ) f ) x 2 ( ) 120f x cm= : 7 , , ' . , - , - . - - . - . 2x2 , 50. 50 . - . - - , - . - /. - , . - . : .1 (3.1): : i. - ax y + = 0a 0. ii. - . iii. . .2 (3.2): - . .2: : ax y y + = + + = 0,D ,x xD D x y D D = = 0,D = 51. 51 , , xD D .yD 2x2 , : i. : 3 4 4 3 y y = = ii. : 2 2 2 y y = = iii. - ; 2 2 4 5 y y + = + = . 6 1 . 109. .3 (3.3): : i. . ii. , . iii. . .3 1 2 ' . 114. 52. 52 .4 (4.3): - - . . , , , . : : (1,1) C R=(). , 0x y + = > . ) ) ) 2 2 d = ) : 2. > 2. = 0 2.< < 53. 53 ) ) M(x,y) C, 2 2 2x y+ = ) : 2 2 2x y x y + = + = : 12 - - - . , - , , - . - 2 ( )f x ax= ( ) . a f x x = 2 ( ) ,f x ax x = + = + 0a - , 2 0ax + + 0. , 1 2( ) ( ) ( ) ... ( ),vf x P x P x P x= - . - 1 2( ) ( ) ... ( ) 0vP x P x P x 0 ( ) 0 ( ) P x Q x 0 , : 54. 54 E.1 (2.5): : i. . ii. . iii. . iv. 2 ( )f x ax= ( ) , a f x x = 0a . .1 - . , , - . , - ( 24-29 ). .1 : 1) . 2) 3) 2 ( )f x x= 2 ( ) , 0.f x ax a= > 4) 2 ( ) , 0.f x ax a= < 5) 1 ( )f x x = ( ) , 0. a f x a x = > 6) ( ) , 0. a f x a x = < 1) 23 29. 2) - : C - , - 55. 55 ) - i. C yy ii. f 0 - x , x : x ( ) ( ).f x f x = , 0, . , (0, )+ f ) , ( ,0) - . , - x .x , 2 ( )f x x= 2 ( )f x ax= . ). 56. 56 3 ( )f x x= 3 ( )f x ax= 1 ( )f x x = ( ) a f x x = . - : i. 11 93. ii. ; 57. 57 iii. 13 93. iv. - ) ) . . 9 10 i), 10 ii) 12 - 93. 10 iii) 10 iv) 93. 3. 2 ( )f x x= : 58. 58 ) f y'y ) f [0, )+ . ) , f - . 4. 1 ( )f x x = . .1 2 92 ' 94. Function probe, a y x = . 80- 84 ( 2.5 . ) .2 (4.4): : i. 2 ( ) ,f x ax = + + 0a 2 ( ) 2 4 f x a x = + , , 1 2( ) ( ) ( ),f x a x x = 2 ( ) ( )f x a x = 2 ( ) 2 4 f x a x = + + 59. 59 (. - , .). ii. ( ) ( ) .f x c = iii. ( ) ( ).f x c = iv. 2 ( ) ,f x ax = + + 0a . 2 0,ax + + = 0a .2: ( ) ( ) ,f x c = 0,c > - c , - . , , ( ) ( ),f x c = 0,c > c - . ' , : ) : ( ) ,f x x= ( ) 1 ,g x x= ( ) 1.h x x= + ) : 2 ( ) 2 ,x = 2 ( ) 2( 3) ,f x x= 2 ( ) 2( 3) .g x x= + ; 60. 60 x 5 4 3 2 1 0 1 2 3 4 5 2 ( ) 2x = 50 32 18 8 2 0 2 8 18 32 50 2 ( ) 2( 3)f x x= 128 98 72 50 32 18 8 2 0 2 8 2 ( ) 2( 3)g x = + 8 2 0 2 8 18 32 50 72 98 128 [O f , g ] 135, - f(x), 2 ( ) 2( 3) 1,f x x= + 3x = , (3) 1.f = . 136 137 2 ( )f x ax = + + 2 y ax= - , 2 4 K a a : : o , - //. ()=4, ()=3 ()=x, (x) , ) : 61. 61 3(4 ) ( ) 4 x MN = 23 3 ( ) 8 2 E x x x= + ) (x) - . (x); Function probe, - . 40- 43 ( 4.4 . ). . 44-46 ( 4.4 . ), 2 y ax = + + . 50-52 ( 4.4 . ), . 48-51 ( 4.4 . ) .3 (4.5): - . - - . .4 (4.5): - 1 2( ) ( ) ( )... ( )vf x P x P x P x= : 1 2( ) ( ) ... 0vP x P x P x ( ) 0 ( ) 0 ( ) P x Q x 0 1 2( ) ( ) ( )... ( )vf x P x P x P x= : 1 2( ) ( ),..., ( )vP x P x P x, . - ( )f x o - 62. 62 1 2( ) ( ), ...P x P x, ( ).vP x ( )f x : - , - , - , , . , 2 2 2 ( ) ( 4)( 3 2)( 1)f x x x x x x= + + + + -2, 1 2 () - - , ( )f x : x 2 1 2 ( )f x 0 + 0 0 ( )f x [ ] { }2,1 2x .4 152. , , - : 1 1 2x < 1 1 . 2x > 63. 63 : 6 . ( - ). - . , , - . - 1 . , - : .1: : i. - . 64. 64 ii. - 360o . iii. ' . .2: - : i. - ii. . - - . .3: i. - . 180 o 180 o 90 o ii. - - 0o 90o . 2007-2008 - ., ., - ., . . , - . - 1-8. - - . 1: ( 1 ) 65. 65 2: ( 4-5 ). . 3: ( 16-18 - ). : 3.2,3.3,3.4 3.5 I &II 3.6 3.10 3.12 4 3.12 3.13 3.14 . 70 Cabri II, . 15 ( 3.8 3.9 . ) .19 ( 3.4 . ) The Geometer's Sketchpad, (--) . 74 ( 3.2 . ) 4: ( 6-7 ). : IV 4.2 . The Geometer's Sketchpad, . 54-55 ( 4.5 . ), . 59-60 ( 4.5 . ) 5: ( 12-14 ). : 5.8 . 66. 66 The Geometer's Sketchpad, - .61-62 ( 5.3 . ), - - . 15 ( 5.3 . ), . 63-64 ( 5.4 . ), . 52-53 ( 5.7 . ), - . 56-58 ( 5.8 . ), . 54-55 ( 5.12 . ), - . 59-60 ( 5.12 . ) 6: ( 5-6 ). 6.2 (i) : 2 6.3 6.4 6.6 3 6.6 1,2,4 6.7 The Geometer's Sketchpad, . . 43-44 46-47 ( 6.4-6.7 . ) 7: ( 5-6 ). 7.1 7.6 - 149 150. : 7.7 7.9 Cabri II, 1 2 . 43-44 ( 7.7 . ) 67. 67 8: ( 4 ). : II III 8.2 1 3 8.2 Cabri II, 1 . 45 ( . . 8 . ) : . 48 1,2 . 58 2,3,4 . 83 1,3,4 . 88 3,4,5,6 . 100 1,4,5 . 104 1,2 . 111 2,4,6,7,8 . 115 3,4,5 . 130 2,3 . 134 1-2-3-4 . 140 1-2-3 . 157 1-2-3-4-5 . 163 1-2-3-4-5 . 178 1-2-3 68. 68 69. 69 . : 2 . 10 - - . - . - - . , . , . , . . . . 1. 17 . - . 4 . - , - - . - , - - . - , ()= + . - . - , . . 70. 70 ( )f x = x + x . - : 1.1: : i) . ii) y = x, y = x, y =(x), y =(x) y = x. - , . The Geometer's Sketchpad. - . 28-30. Function probe. y = x y = x . 62-65. y = x y = x - . 67-69. 1.2: - : x=, x= x=, . - , ' . 1 ( 1.1) 13 ( 1.2) . 1.3 1.4: : i) . ii) - 2. 71. 71 iii) : ) , ) , ) . 1.5: . 1.6: : i) ( ) ( )f x = + ii) ( )f x = + ( ) ( )f x = + . iii) . 1.7: . 2. 12 . - . - . - - (.. , ) . , : 2.1: : i) - x, : , , . ii) : - - - - . 72. 72 iii) : ) - - . ) - . iv) , - . 2.2: : i) . ii) - . iii) ( )P x x : ( ) ( ) ( ) ( )P x x x P = + - : ) ( ):( )P x x ) : ( ) 0 ( ) ( ) ( )P P x x x = = Horner (- ) - . 2.2 1, 2, 4 5 73. 2.3: : i) - 2 , - ii) . iii) - () - . 73. 73 2.4: , - . 3. 10 . , , - . - - . , - , , - . , : 3.1: : i) . ii) - - . 3.1 93. 3.2 3.3: : i) 1v va a+ 1v v a a + . ii) . iii) - , , . 74. 74 iv) - , - - . 3.4: . 3.5: : i) : ) , ) Sv, + ii) - 1 , 1 a S = < 1 . iii) - - . 4. 12 . - . >0, - x y e= . , , . , - . . - , , , . , , . - 75. 75 - . , : 4.1: : i) - - . ii) . iii) -- . i) e . 4.2: : i) , 0, >alog ,=x a : logx aa x = = : 10 log = =x x lnx e x = = ii) : log 10 , = 10 ,=x log x ln ,e = ln ,x e x= x xlna a e= i) , . i) - , , - =a log log loga =a ln log lna 4.2 : - 10 e. 76. 76 4.3. : i) 10 x - , - e - x . ii) 10 e . iii) 10 e. 4.3 - 10 e. 2007-2008 - ., ., - ., . . , - . ' 9-13. 9-13 - 15 - . . 9: ( 10 ). 9.6 II 9.4, 2 9.4. 77. 77 Cabri II, . 49 ( 9.2 . ), . 49 ( 9.4 . ), . 57 ( 9.7 . ) 10: ( 7 ). 10.6 3 10.4. Cabri II, . 63 ( 10.3 . ), . 69 ( 10.3 . ), .73 ( 10.3 . ). 11: ( 8 ). 11.2 II 11.3. Cabri II, - . 79 ( 11.1 . ), . 75 ( 11.1-11.3 . ) . 81 ( 11.5 . - ), . 83 ( 11.7 . - ) 12: ( 11 ). , II III 12.5 II III 12.7. 12.6 - . 13: ( 10 ). 13.4-13.18 - . 13.19 . 78. 78 : : 9: . 186 4, 6 . 194 1, 2, 3 . 199 4, 5 . 204 3, 4 10: . 218 1, 5 . 221 1, 2 . 225 1, 2, 3, 4 11: . 237 1 . 238 2, 3 . 242 1, 2, 3 . 245 2 . 251 4 12: 13: . - , . : - (4-5 ). 79. 79 II. & : 3 - , - ., ., ., . . , , . . 1. 22 . - , , - . , - - - , . : - , , - . , , - . , ( ). =OA a =OB .AOB , = ,AOB 0 . - 80. 80 . - , - , - AB OB OA - . + +a , . : =// ( 0 ) (.. 6 1.3). , - . , 14 1.3. , - - . 1 1( , )a x y= 2 2( , ): = x y 1 1 2 2 0 x y a x y =// det 1 1 2 2 ( , ) = x y a x y - . , . - . 81. 81 , , - . : - , - (, .) , . - . , , - , . - . 1 1 2 1.3 . 2. 10 . , . : - - , . 0 0( , )A x y 0x x= 0 0( ), .y y x x = 1 1( , )A x y 2 2( , ),B x y 82. 82 1 2,x x 2 1 0 0 2 1 ( ), y y y y x x x x = . - 3x3, - . - . . - . - 1 1( , )A x y , 2 2( , )B x y 3 3( , )x y 3x3, : 2 1 2 1 3 1 3 1 1 1 ( ) det( , ) 2 2 x x x x = = : . . - . 2 . 3. 30 . - , , - . - , ( , , , .). : 83. 83 , . - 2 2 0.x y Ax By+ + + + = . . 1, 12 , 2 1, (- - ). . , - . , . - ' - . - . ( ) . - . - , . 2 2 0.Ax By y E+ + + + = - - . , 2 2 0 y y x y E = + + + + + = . : 84. 84 . . 3 : , . . . 1 96, 107 2 110. 4. 13 . - , - . , - , : i) , - , . ii) . iii) - . : 85. 85 - () =1 () (+1) - . - , - . - , - . 4. 7 . - 0. ... - (, ) = (, ), . - ... , ... (, ) [, ] = . ... - [, ] = [, ] ( . 159). - , . - . , . , , . ax y + = - ... . - 0x x t= + 0y y at= (, ) = 1. ( , )a ( , ) 1, ax y + = (,), - x y . - . 86. 86 - . , , : . . , - . 4 : 4.1 5 7 4.3. - . : 4.4 - 4.5 4.6 4.7 . 87. 87 1 1. (i) (ii) (iii) (iv) (v) (vi) 2. (i) , (ii) , (iii) , (iv) , (v) , (vi) , (vii) , (viii) 0, (ix) 0 3. i) , (ii) , (iii) 2 , (iv) , (v) 2 , 4. (ii) 5. (i)(3,2), (ii)(3,2), (iii)(3,2) (iv) (2,3) 6. (3,4), = ( 7 3), = ( 6,4), = (0, 4), = ( 9,0) = 7. 1 7 , , 2 2 7 3 , , 2 2 (0, 3), (0,0) 8. (i)4, (ii)4 9. (i)0 (ii)2 , (iii)0, (iv) 2 , 2 a (v) 2 , (vi) 2 10. (i)6, (ii)3 3, (iii)3, (iv)0, (v)3 (vi) 3 3, (vii)6 11. 12. 1. , 2. , 3. , 4. A, 5. O, 6. O 13. (iii) 2 1. 2. 2 =A x 3,B y = 3 2 0x y = 2 5 8 = x y 3 =y 3 2 0 =E x y 2Z x = 3. (3,2) 0B x+y=8 4. y=3x+1, y=3x2 1 8, 3 y x= + 1 10 3 y x= + 3y x= 1 3 y x= 5. 3 6. 88. 88 3 1. 2. A 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 2 2 2 a + = =0 =0 = = == 13. 14. 4 1. (i) (ii) (iii) 2. (i) (ii) (iii) (i) () 3. (i) (ii) (iii) (i) 4. (i) (ii) 5. (i) (ii) 6. (i) (ii) 7. (i) (ii) 8. (i) (ii) 9. (i) (ii) (iii) 1. 2. 3. 4. 5. 6. 89. 89 : 2 " - " ., ., . - , - . : 1 : 2 : 3 : . , - . - , - , - , , - . : - . , - . - . , - - , - . " - ", - . - . 90. 90 - . - - . , - , - , , - . 1. 15 . . 1.1 - . - ' - , , . . - - , . "". , , - , - , , . 0 0lim ( ) ( ) x x f x f x = 91. 91 ( )f x 0( ),f x x 0.x x . 1.2 - . - . - . ' (, R) - AM, - . f ( )0 0, ( )A x f x = 0 0 0 ( ) ( ) lim . h f x h f x h + - - f ( )0 0, ( ) .x f x , - , , - . - . - . , , , , - ., 0 0 0 ( ) ( ) lim . h f t h f t h + , , - f 0.t , - . 92. 92 1.3 () .f ,f x ,f , . f (t)=x(t) (t)=x(t) - . , , - . . x, x x x (rad). , x= . 180 x = , 0 0 ( ) ( ) ( ) . 180 180 = = = = - . 1.3 , . , . , - . , 39 , , . , t , h(t)=20t5t2 , (t)=h(t)= 2010t=0. t=2 , h(2)=4020=20. , , - . - , - o- . 93. 93 . , - . " ", , - . - , - . - . , - , . . , - , - . , - , . , : . . . . 94. 94 2 16 . , , , - , , , - , , , - . - - , - . , - , - , , - - , , . . , - . , , , . , . 2.1 , - , (, ), . . , - " " , , , , , , , , , , ,, . - , - . , 95. 95 . 2.2 . - - . . - , . : () i, xi, - xi . fi ( %) xi, - . , . . , ,=i iv f v Ni - Fi, - , xi. , . - - . - , , . - , 1 () () - - 16 - . . 96. 96 1 () (), - y, x - y 30. . - , . , 9 . 80 , , - . - , - . . - 97. 97 Sturges: =1+3,32logv, . , - . , . , - . - , . - , - ... - 1973 2 . () , () ( - ). () - ... "" , - . - 25.000$ , 7.000$. ... . (). 98. 98 $ ( )if % * i i i f v c = % 0 1 1 2 3 2 3 4 4 5 5 6 6 7 7 10 10 15 15 25 25 50 1 2 3 4 5 5 5 15 26 26 8 1 2 3 4 5 5 5 5 5,2 2,6 0,32 99. 99 2 2.3 . - . . - , . , - , - . . - . , x . - . .x - , , : 100. 100 . - , 1 ( ) 0. = = v i i x x . - : - . 0x " " . , xi. vi. . 1 1 ( ) 0, k i i x x v = = x v1, v2,..., x1,x2,...,x. - , - " ". 0x i, xi x , , .. "". 101. 101 . (- 2, 98). . , . - . - , 80 17 50 15, - 80+50=130 17 2110 16.23. 130 x = = 80+1550 80+15 50% 50% . , . - , ( ) : 1 2 i i i i v N L c v = + i vi Ni Fi% 1 2 3 3 5 6 156-162 162-168 168-174 174-180 180-186 186-192 2 8 12 11 5 2 2 10 22 33 38 40 5,0 25,0 55,0 82,5 95,0 100,0 102. 102 Li i ci 1iN , . , , 9 73 , - , - v/2=20 . , 1 40 10 2 2168 6 173 , 12 i i i i v N L c cm v = + = + = () . . , - - . 14 91 - . - , Li, - , 1 2 , c - , 0 1 0 2( ) + = i i M L c M L 0 - : 1 0 1 2 .i c M L = + + , . - . , 70. ; , . , : 103. 103 - ; ; ; . P k% (100- k)% - . ( = 50), 1 25(Q P= 3 75 )Q P= 1 10 2 20 9 90, ,..., .D P D P D P= = = - , .. , . (geometric mean) 1 2, ,..., vt t t , 1 2 ...v vG t t t= 1 2 1 2 ...= kvv vv kG x x x . , , (harmonic mean). 1 2, ,..., vt t t 1 2 1 1 1 ... v v H t t t = + + + 1 2 1 2 , ... = + + + k k v H v v v x x x . , , 5 - , 10 6 , ( ) 3 1 1 1 5 10 6 + + 6,4 . ; , . , , 104. 104 , . , - . , . - , . - ( ) . - - . - , . , , - , - , - . , . , - , . - , . - . , . - . 2 ( ) N t N i i 1= 2 2 = (1), = = N i it N 1 1 , ( ) 2* s 105. 105 ( ) ( ) 2 2* 1 1 = i i= t - x s - (2). ( ) 2* s , 2 . (2) - . - ( ) 2* s (2), 2 . , , ( ) x-t s i= i = 1 2 2 , 2 . , ( ) x-t s i= i = 1 2 2 , . - . - . , - 3 ( 99 ), axy += , xy += xy ss = . - , 900=Ax 150=As . ),( sxsx + , )2,2( sxsx + . - , - 106. 106 15 95. , , , Chebyshev, - ( , ) +x s x s , 1 , 2 1 1 . , )2,2( sxsx + 75% - , )3,3( sxsx + - 89% . , , - , : 68% (750,1050) 95% (600,1200) 99,7% (450,1350), , , 0%, 75% 89%. - , - . . - . x s 3 : 1 900=Ax s 150= 900=Bx Bs 200= 2 900=Ax s 150= 3000=Bx Bs 250= 3 900=Ax s 150= 2000$=Bx Bs 420$= 107. 107 1 , , - . - , . , - , . s 150= 900=Ax , 250=Bs - 3000=Bx . - , . - , . , - . , = s CV x . , - . - . . - . - . - . . , - , . - . - . . 108. 108 . - . . . . . . - . , - - . . - . - - . - . - . - . - . . - . - , - . - . - . - . 68%, 95%, 99,7% - . - - . - - 109. 109 ,x s 2x s 3x s - . - . . - , . - . - - . 2.4 . - 1 1 2 2 3 3( , ), ( , ), ( , ),..., ( , )v vx y x y x y x y , . 1( , ),ix y 1,2,3,...i v= ( ). , 1( , ), 1,2,3... .=ix y i v ( ), - ,y a = + ( , ),i ix y 1,2,3...i v= - 2 1 ( ) v i i i y a = ( ). (5) (6) 110 , - a - , : 2 1 ( ) v i i i S y a = = - , S a S . 110. 110 1 2 ( ) v i i i S y a a = = (1) 1 2 ( ) v i i i i S y a = = (2). (1) (2) , - , : i ia v y + = 2 i i i ia x x y + = . 2.4 : - y a = + , - . y a = + - , , . , - , = +x y y a = + - ( , ).x y . ( , )i ix y , . = +y = 0, . . 111. 111 , , , - . . 2 ( 4) 1y x= + ( - ) . 1 2 3 4 5 6 7 10 5 2 1 2 5 10 - , - . , : y ae = = +lny lna * , * , * = = =y lny a lna - * * * .= +y a x 2.5, r . - (, ), - . , - . , ,= +y a . 112. 112 2 2 ( )( ) ( ) ( ) = i i i i x x y y r x x y y 1 r 1, : ( ) 2 2 2 2 2 2 2 2 2 2 1 (( ) ( )) 0 ( ) ( ) 2 ( )( ) 0 ( ) ( ) 2 ( )( ) 0 ( ) 2 ( )( ) ( ) 0. + + + + + + + i i i i i i i i i i i i i x x y y x x y y x x y y x x y y x x y y y y x x y y x x ,R 2 4 0 : 2 2 2 4 ( ( )( )) 4 ( ) ( ) i i i ix x y y y y x x 2 2 2 ( )( ) ( ) ( ) i i i i x x y y y y x x 1 113. 113 2 1r r r 1 1 1. r , - - . , , - . ( - ) . , - Y , . - . Y. Y . , Salk , - . - ' - - . - ( ) ( ). , - . , - . ; , . . , . , . 114. 114 ( ) - ( ). , : , - - , . , - . , . - , . 2 : ) (. 74) ) (. 89) (. 92) ) (. 90, 91) ) , 2.4 ) , 2.5. ) 4 . 81. - , . - - . , c vi, , d i d v c . , , 5 . 103 5 6 1 14 7. 2 = 2 4 93 & 94 . 115. 115 3. 19 . , - . - , , - - . 3.1 , . , , - . , - . - . - - - , , Venn ., . - , - . , - , : ( ) ( ) ( ) ) ) ) ). = = ( ), =( )( )=( ( ) =( ( ( 116. 116 3.2 , . , (. 65 ). - . , , . ' , "" . . - "" ; - , , , - , ; 0, - 1 ( 0%, 100%). 0 1. ; "" - , - , ( ) ,P A = - . () - , : 0 ( ) 1P A ( ) 1P = ( ) 0P = , 117. 117 ; - , .. Kolmogoroff. - , D , :P D R : 0 ( ) 1,P A A D ( ) 1P = ( ) ( ) ( ),P A B P A P B= + A B = Kolmogoroff - , - . - . . , - 149, , Kolmogoroff. 3.2 - , - . . 3.3 , - . - , "" . - . 118. 118 = ( 164, 5, ) 1 1 1 = + ( 174 3, ) . - , , , - - , 1 , 1 - , 1 . 3.4 - . - - , - . , - B - .A B , - ( ) ( )/ ( ) ( ) ( ) . ( ) ( )/ ( ) ( ) = = = N A B N A B N P A B P A B N B N B N P B B , . : A B B 0 ( ) ( ),P A B P B ( ) 0 ( ) P A B P B 1, 0 ( ) 1P A B . 119. 119 , = ( ) ( ) ( ) 1, ( ) ( ) P B P B P B P B P B = = = ( ) 1P B = . 1 2A A = 1 2( ) ( ) ,A B A B = 1 2 1 2( ) ( ) ( ),A A B A B A B= 1 2 1 2 (( ) ) ( ) ( ) P A A B P A A B P B = 1 2( ) ( ) ( ) P A B P A B P B + = 1 2( ) ( ) ( ) ( ) P A B P A B P B P B + = + 1 2( ) ( )P A B P A B= + , 1 2 , 1 2 1 2( ) ( ) ( )P A A B P A B P A B= + . - ( ) ( ) ( )P A B P A P B A= = ( ) ( )P B P A B - . , - . ( ) ( )P A B P A= ( ) ( ),P B A P B= ( ) =P A B ( ) ( ),P A P B - , ( ) 0P A > ( ) 0.P B > - . 2 169 - Bayes () (1720-1761): 120. 120 1 2, ,..., vA A A ( ,iA 1,2,...,i v= ), - 1 2( )vB B A A A B= = ... 1 2( ) ) ),vA B A B A B= ( ...( ,iA B 1,2,...,i v= . - , 1 2( ) ( ) ( ) ... ( ).vP B P A B P A B P A B= + + + 1 1 2 2( ) ( ) ( ) ( ) ( ) ... ( ) ( )= + + + v vP B P A P B A P A P B A P A P B A (1), . , i ( ) ( ) . ( ) = i i P A B P A B P B - () (1) ( ) ( ) ( ),i i iP A B P A P B A+ : 1 1 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ... ( ) ( ) = + + + i i i v v P A P B A P A B P A P B A P A P B A P A P B A Bayes. , "" - . , : - . . 121. 121 - . - . 3 3.4 - : - . " " 1 1. 9. (0,0), (1,-1), (3,3) 2. 10. (i) 0, (ii) 0, (iii) 2. 3. (iv) , () 0, 4. (vi) 3, (ii) 1 5. 11. (i) -2, (ii) 3, (iii) 2 /6 6. 12. ()-1, ()-11, () 8, ()-8 7. 13. 28 8. 14. (1)() (2)() (3)() (4)() 122. 122 2 1. 17. 2. 18. 3. 19. 4. 20. 5. 21. 6. 22. 7. (/ 23. c ) 24. 8. 25. () (i), () (iii), () (ii) 9. 26. () (i), () (i), () (iii) 10. 27. (), (), ()(i), (), (), () (ii) 11. 28. () (ii), () (iii) () (i), 12. ( 0 > 29. () (ii), () (iii), () (i),() (ii), () (ii) 0 < ) 30. (i) (), (ii) (), (iii) (), (i) () 13. 31. () (iii), () (i), () (ii) 14. 32. () (ii), ()(i), () (iii) 15. 33. () (i), () (i), () (iii), () (ii) 16. 34. () (i), () (iii), () (ii) 35. () , () , () , () 3 1. : "" 9. "" 10. 11. , . 12. 2. 13. , 14. ( ) 0,P A B ( ) 0,8,P A B ( ) 0,P A B / ( / ) 0P B A . 15. ( ) 0,12,P A B 3. () ( ) 0,68,P A B () 3 6. ( ) 0,6,P A B / ( / ) 0,2P B A 4. () 16. , 5. () ( ) 1,3,=P A B 6. () . 7. () 8. 123. 123 . &I : 5 " ' " : ., ., ., ., - . . . : 1. . 2. 12 . - - 2 . - - . - . - , - . , - , 2 - . - , - - , . . - - . 124. 124 - : 0z z a = 0( )K z 1 2 = z z z z 1( )A z 2( ).B z , - , - . . , 1,v z = , - ,v z a= - - . : : ( ),a a i = + 0, 0 , , 0 > = < : 0 ,vva a i z = + = ,v oz a i = + 0, 0 , , 0 > = < v z a= : 0 v v v z a z z= = 0 1 v z z = 0 z z 0 , 0,1,2,..., 1 z z = 0 , 0,1,2,..., 1kz z z k = = = 125. 125 z0. , - : 4 16,z = 4 0 0 16 2, 4 4 oz i = + = , : 2 , 0,1,2,3kz k = = 2 2 4 4 i i = + = 4 16,z = 2 , 4 4 oz i = + , : 2 , 0,1,2,3, 4 4 k kz i k = + = 2 2 . 4 4 i i = + = , . - : 1. : ) ) . 2. : ) , , . ) - . 3. : ) . ) . 4. - : ) . ) - ( De Moivre). 126. 126 5. v z a= a . 2.4 - 2.5 . 1. 24 . : ) , ) { }0 ,x + ) . ) - . - . - , "" - . . - , - , . , , . , , . , - , . 2 . ) - - 127. 127 f 0x . 0x ' 0 0( , ) ( , )a x x 0( , )a x 0( , ).x , - , , - , . - . ' 0 .x R - . De L' Hospital, - (. 2) . - 0x . - 0 ,x . . 0 lim ( ( )), x x f g x 0( ) ,g x u 0 ,x , ,f g 0, 0 ( ) ( ) . 1, 0 = = = x f x g x x 1, 0 ( ( )) , 0, 0 = = x f g x x , 0 0,x = 0 lim x x ( ( ))f g x = 0 lim x ( ( ))f g x = 0 lim1 1 x = 128. 128 0u = 0 lim x x ( )g x = 0 lim x ( ) 0,g x = 0 lim u u ( )f u = 0 lim u ( )f u = 0 lim0 0. u = , 0 lim x x ( ( ))f g x 0 lim u ( )f u 0( ) 0,g x u= = 0.x ) - 0x . Bolzano - f - [ ]( ) 0, ,f x x a = ( , ).a , . , : 1. : ' 0.x 2. - . 3. , , - . 4. "11", . , - , - - . 5. , , . 129. 129 6. 0 ,x . 7. . 8. . 9. + . 10. . 11. . 12. 0x . 13. f , . 14. , , , . 15. : Bolzano, - , - - , - , . 2. 35 . , , - : . - ( )y f x= ' - . - ( )y f x= ' , - 130. 130 , , .. , . ' - , . , - - . , : 1. ' 0x . 2. , , , - . 3. - . 4. : 0x - - . 5. Rolle, Fermat - . 6. : : ) ) . 7. ( ) 0.f x = 8. de L' Hospital . 131. 131 9. - . 10. - . 2 : ) . ) 262 ) ( . 264). , - , , - . 3. 32 . - - . , , . , , - . , , - ''- ". : ) - ) . - : 1. - . 132. 132 2. - . 3. - . 4. . 5. . 6. - . 7. - - . 8. . 3 : ) 3.3 : ) 3.6 : . 133. 133 1 I. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. i) ii) . 1. 2. 3. 4. 5. . 1 , 2 ,a 3 2 1. i) , ii) 2. , 3. z i z i xx = + 1 1z z yy = + 1z z i y x = = 1z z i y x+ = + = 4. 45 , 45 , 225 , 135k ki k ki k ki k ki+ + 5. A, B, , 6. 2 2 2 2 2 2 2 2 : , : , : , : 2 2 2 2 2 2 2 2 A i B i i i+ + 7. ,z 1 , 2 z E 1 , z ,z z B 1 I. 1. , 2. 3. 4. 5. , 6. 7. 8. 9. 10. 11. 12. . 1. 2. 3. 4. . 1. 2. , , 3. 134. 134 2 I. 1. 2. 3. 4. , 5. , 6. 7. 8. 9. , 10. , , , 11. , , 12. . 1. 2. 3. 4. 5. 6. 7. 8. . 1. a E , , 2. 1 , 2 , 3 3 . 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. . 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. . 1. 2. , 3. F 1 ( ) ,f x x = 1 1 1dx dx x x = + 1 2( ) 1 ( ) ,F x c F x c+ = + + 1 2 1c c = , 0=1. . 4. x 0, 1 0.x u = 5. F (0) = 0, F (2) = 2, F (3) = 4, F (4) = 6, F (6) = 12 135. 135 6 / / 1.2 , - : (ii): ,f g gD D= = ( ) .fg D ( ) ( ,0].g = R : ( ) ( ,0].fD g = ( ) ( ,0],y g = 2 ( ) ,y g x x= = .gx D = 2 ( ) ( ( )) 1 1f y f g x x y= = + = , ( ) ( ,0]g = f - ( ) 1 .f x x= ( )fD g f - . , f . : 1 , ( ,0] ( ) , ( ), ( ,0] = x x f x h x x A fA D= ( ,0] h ( ,0].A , fg: : ( ) g f x D g x D 2 . x x x A .f gD = : 2 2 ( )( ) ( ( )) 1 ( ) 1 ( ) 1 ,= = = = +f g x f g x g x x x 136. 136 2 ( ) 0,g x x= .gx D , , f g f ( ,0].A (iii): f - . - : , ( ) , , S f x S = S . : 2 2 ( ) ,f x = x : ( )f x = x ( )f x = x . , , : 2 2 ( ) ( ),f x g x= ,x A : ( ( ) ( ),f x g x= )x A ( ( ) ( ),f x g x= ),x A ,B A : ( ), ( ) ( ), g x f x g x = x B x A B ( )f x x= ( ) ,g x x= : 2 2 ( ) ( ),f x g x= x . , - , : ( ), ( ) ( ), g x f x g x = [0, ) ( ,0) x x + 137. 137 7 / / 1.8. , - , , : ( ) ( ) 0,f x g x = ,x A : ( ( ) 0,f x = )x A ( ( ) 0,g x = ),x A ,B A : ( ( ) 0,f x = )x B ( ( ) 0,g x = ).x A B : 0, 0 ( ) , 0 x f x x x < = , 0 ( ) 0, 0 x x g x x < = : ( ) ( ) 0,f x g x = .x , - . 3 / A / 1.3 , - : ( ) f y x= - f y x= (: 1 ,3 4 ). - f y x= (: 4 ). 3 / / 2.6 & 5 / / 2.8 2) - ( ) - : : - ( ), (t), |(t)|, ( - ). 138. 138 : (t) , (t)= (t)= g0 (t) , |(t)| , , , (t)