Odigies Did Math Lyk

8/3/2019 Odigies Did Math Lyk

1/166

2007 - 2008


2/166

2

, .


3/166

3

2007 - 2008

-


4/166

4

(1997)

:

1. , ..2. , ..3. o, ..4. /, ..5 . ..6., . , ..7., ..

8. , ..

(1998) - (1999) - (2000)

(2001) - (2002) - (2003)()

2007

, :

.. . .. .. .. .. ..

-:

, . Cabri II

. ..

.. -.. .. -

..

.. -.. .


5/166

5

.............................................................................. 7

. ............................................... 9

1 . ............................................................ 112. ................................................... 153. ........ 30

. ......................................................................... 35

. ........................... 69. &

....................................................... 79.......................................................................... 89

. .......................... 89. &

......................................................... 123. I - II I ............................. 139

............................................................ 165


6/166

6


7/166

7

-

. -, , , 1997 -.

, , -, - . , - .

, , , - . , . - -

. , , - .

, - 3 ( 35 )


8/166

8

- -. , -. -, The Geometers Sketchpad, Cabri II Function Probe - . , -

( ) - . - , , -.


9/166

9

. TOY

:) ,

, , , ,

, .

) , -, , , , , , -, .

) -, , - , -.

) , ,, , .

) H - -.

, -, :

) , - .

) - , -


10/166

10

, .

) , , -:

) .

) - -,

) - , -.

) - .


11/166

11

B1.

- . , , , -

. -, -.

, . ' -.

, , -.

, :1. -

-.

2. , -, , -

. , , ,.

3. - . ,


12/166

12

-. - (--) - -. ,' -.

4. ,

. , -, . , . . . - - .

5. , .. -, , , . , , -, . , .

6. -' , -, ,. .-

. , -

, . .

7. ,


13/166

13

. , . -, , , ( - , , , .)

8. -, -

, . -., , - . , , . -.

9. -, . - , . -, - , ., -.

'-, - - . .


14/166

14

10. . - . , . ( ) , .

11. - . -

- . - .

12. - - . - - . -, - , -. - -' .

13. , (calculators), -. ,

, ., -, ,.


15/166

15

B2.

1. . '80, ,

, -

, - -, , , , -, .

- , - -.

, , - .

, : - , . - , -

. -: -- . , - .

- , ,


16/166

16

. , , , . , , -, , : - , , -.

-

. , - :

. , , .

, , -, -.

, , -!

. Lakatos: -

-. -

- . . .

-; ,


17/166

17

; : - ;

2. .

. ,

-.

- , : . -

, , -: -- ().

- , -.

-: .

, - - , - -. - .

- , , . - ,- ().


18/166

18

- - -.

' , . , -, - , , , -.

, . , , - . -, -, , , - , -, , -. -, -, . -

.. - :

. -, . (). ,

-, . , -


19/166

19

: , -. -. -. . -.

: .

. -

. , -

-.

3.

- - -. , - . , -, , , -

, , - . , , -


20/166

20

,

, , . .

-, - . , --

- - . . -, -., -, - - , - !

- , - . . , .

-

. . ' -. ,-- , -


21/166

21

, .

-. , - . , . - . ,

. -, 300 , - 50 , . , -. , - , : .

, -

. , . , -, .


22/166

22

4. .

; . , . -

. , -

, . , -- . ,- . -: ;

", -, . () - . - -. ' , : (

). -

, . . . , -

, -.

.


23/166

23

-; , -, ,. , -.

, , -. , - -

, .

, -, , ; .... , , - . . , , , : ,

, , , , . -.

,

, ., - . :

:-


24/166

24

, , , ; -- -, , .

:- . , -

, - , ( 2, 3-). , -, ., - - . , , , , .

, -:

-

1 :

1. :

: -(t) 4 12


25/166

25

) := [4,16] , ( )T t :) ) )

) 1t , 2t , -

:

1 2t t> 1 2( )... ( )T t T t

) :) ) )

-.

, , ( ) 2 1 f 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

.

2) -

. ( ) 2 1, f x x= +

( ) , f x ax = + 0,a < -.

3)

i) 3 92ii) , -

:) )

iii) :

) ( ) , f x x= ) ( ) 8 2 , f x x=

)1

( ) 3,f xx

= + (0, )+ )4

( ) 3 ,f xx

= (0, )+

i) 1 92


27/166

27

2 :

1.

T(t) 12 - 12 .

) ;;

) ;

) :

( )... (4),T t T [ ]0,24t

: , -.

, - 2( ) 2 3, f x x= +

(0) 3,f = :


28/166

28

2 0x x 22 ...0x x

22 3...0x + x ( )... (0) f x f x

( ) 2 1 5 f x x= .

2.

-

.

3.

) () -:


29/166

29

) 4 92.


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30

B3.

1. .

. -

.

1. :

:i): (2)f : . , . , , .

( 2)f : . , . , . ( 1)f : . , . , . (0)f : . , . , . . (1)f : . , . , .


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31

ii): ( ) 0f x = ......................................... ( ) 0f x > ......................................... ( ) 0f x < .........................................

2. 2( ) 4 3 f x x x= + i) :f = ...............ii) f ; .iii) f -

x ; ;i)

: ( ) 0f x > ......................................... ( ) 0f x < .........................................

3. 2( ) 2 2 f x x x= + i) :f = ...............ii) f ; .iii) f -

x ; ;i)

: ( ) 0f x > ......................................... ( ) 0f x < .........................................


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32

4. 2( ) 4 4 f x x x= + i) :f = ...............ii) f ; .iii) f -

x ; ;i)

f : ( ) 0f x > ......................................... ( ) 0f x < .........................................

5. 2( ) 2 3 f x x x= + + i) :f = ...............ii) f ; .iii) f -

x ; ;


33/166


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34

7. 2( ) 2 2 f x x x= + i) :f = ...............ii) f ; .iii) f -

x ; ;iv)

: ( ) 0f x > ......................................... ( ) 0f x < .........................................


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35

.

2

2007-2008 - . , ., . , . , . . -, , .

-, -, , .

. . , .

, , -

, (.. - ) . . , 2, 3 4. .

/

12

.1

1.1. 2

.2

1.2. 2


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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


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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 x0 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. + <

-

.


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38

A.1 ( 1.1): :i. -

.ii.

.iii. ,

. -. :

2 2( )( 1) 0 x x x = , - 2x x 2 1x

, 2 0x x = 2 1 0x = (1)

1x = , 0x = 1x = - .

2 0x x = 2 1 0x = , , 1x = , .

1x = 2 21x =

x a= 2 2v vx a= ( )*v ..1 i)

13 ' 16.

.2 ( 1.2): :i.

.ii.

.iii. ,

.

i. ..2 :1. 1 2 1( )( ... )v v v v va a = + + + 2. 1(iii) 18 3(i) 19.


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39

3. 5 ' 22 ' 23.

, , :

1. N 2 2( ) ( )a a + -:

2 2999 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 22

3

1( ) ,

( 1)

x xx

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 =


40/166

40

.

. :) 0v v at = + t

) 1 2

1 1 1

R R R= + 1R

)

2

0

1

2S v t at = +

0 ,at = +

0 .2

v vS t

+=

5 ' 28 - 900km/h .

2 3 ' - 28.

.4(1.3):

. .4: .

:

i. 2 ( 4) 2 ( 4) ( 4) 0 x x x x x + + =

ii. 2 3 2( 1) 0 x x x x + =

iii. 2 2( 1) 1 0x x+ + =

iv. 2( 2) (2 )(4 ) 0 x x x + =

v. 2 2( 2) 4 4 x x x x = +

vi. 2 2( 4)( 1) ( 1)( 2) x x x x =

vii. 3 22 2 0 x x x + =

viii. 3 22 (2 1)( 2) 0 x x x x =


41/166

41

ix.2

1

2 4

x

x x=

+

x.2

1

1

x

x x x=

xi.2 2

1 20

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. ( : 9 x cm=

1 y cm= 8 x cm= 2 y cm= ... 5 x cm= 5 y cm= ) -:1. 225cm 2.

250cm


42/166

42

- ,x y 10 ,cm : y .x x

225cm . -

x - 250cm .

.6 ( 1.5): :

i. 0ax + > 0ax + <

ii. -.

: 10

,

, . -.

- .

, - . .

:

.1 ( 1.6): :i.

.ii. .iii.

.


43/166

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,x x <


44/166

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 <

:


45/166

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. .vx a=

.2: 6 36 4

51 - :,


46/166

46

)2

2

a

+

+

5 6 ' - 51 52.

.3 ( 4.1): :i.

.ii.

- .iii. , -

, .i.

. -

,vx 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 vax x + + = 0a

.


47/166

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.


48/166

48

.3 (2.3): : -. 70 - {,} (,).

i. A(x,y), , 1 3.

ii. .iii.,

.i.

.

.4 (2.4): :i. , . y ax y ax = = + -

13 , , 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 >


49/166

49

, - 4 76 1ii), 1iii) 3 78.

, , :

20 cm . , -, .

x ( )f x ,

) f ) f

) x 2( ) 120 f x cm=

: 7 , ,

' .

, - , -. - - .

- . 2x2 ,


50/166

50

.

- . - - , -. - /.

-,

.

-.

:

.1 (3.1): :i. -

ax y + = 0a 0.

ii.-

.iii. .

.2 (3.2): - .

.2:

:

ax y

y

+ = + + =

0,D ,x xD D

x yD D

= =

0,D =


51/166

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.


52/166

52

.4 (4.3): - - . .

, , , . :

:

(1,1) C R=(). , 0x y + = > .) )

)

2

2d

=

) :

2.> 2.= 0 2.< <


53/166

53

) ) M(x,y) C,

2 2 2x y+ =

) :

2 2 2x y

x y

+ =

+ =

: 12 -

- -. , -, , - . -

2( ) f x ax= ( ) .a

f xx

=

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

, :


54/166

54

E.1 (2.5): :i.

.

ii. .iii. .

iv. 2( ) x ax= ( ) ,a

f xx

= 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 xx

= ( ) , 0.a

f x ax

= >

6) ( ) , 0.a

f x ax

= <

1) 23 29.

2) -:

C -

, -


55/166

55

) -

i. C yyii. f 0 -

x , x :

x ( ) ( ). x f x =

, 0, . , (0, )+ f ) , ( ,0) -. , - x .x

, 2( ) f x x= 2( ) f x ax=

.

).


56/166

56

3( ) f x x= 3( ) f x ax=

1

( )f xx

= ( )a

f xx

=

. -

:

i. 11 93.

ii. ;


57/166

57

iii. 13 93.

iv. -) ) .

. 9 10 i), 10 ii) 12 - 93.

10 iii) 10 iv) 93.

3. 2( ) f x x= :


58/166

58

) f y'y

) f [0, )+ .

) , - .

4.

1

( )f x x= .

.1 2 92 ' 94.

Function probe, ayx

= . 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

= + +


59/166

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= +

;


60/166

60

x 5 4 3 2 1 0 1 2 3 4 52( ) 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 = + + 2ax= -

,2 4

Ka a

:

:

o , - //. ()=4, ()=3 ()=x, (x) ,) :


61/166

61

3(4 )( )

4

xMN

= 2

3 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( ) ( ) ( )... ( )vx P x P x P x=

:

1 2( ) ( ),..., ( )vP x P x P x,

.

- ( )f x o -


62/166

62

1 2( ) ( ), ...P x P x, ( ).vP x

( )x : -, -, -, , .

,

2 2 2( ) ( 4)( 3 2)( 1) f x x x x x x= + + + +

-2, 1 2 () - - , ( )f x :

x 2 1 2( )x 0 + 0 0

( )f x [ ] { }2,1 2x

.4 152. , , -

:

1 1

2x<

1 1.

2x>


63/166

63

: 6 . ( -).

-.

, , - .

- 1.

, -:

.1: :

i. -.


64/166

64

ii. - 360o.

iii. ' .

.2: -:

i. -

ii. . -

-.

.3: i. -

. 180 o 180 o 90 oii.

- - 0o 90o.

2007-2008 -., ., -., . . , -

. - 1-8. - -.

1: (1 )


65/166

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 .


66/166

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. )


67/166

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


68/166

68


69/166

69

.

: 2 .

10 --. -. - -

. , . , ., . . .

.

1. 17 .

-.

4 . -, - - . - , --.

- , ()= + . -

.- , .

.


70/166

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.


71/166

71

iii):)

,) ,) .

1.5:.

1.6: :i) ( ) ( )f x = + ii) ( )f x = +

( ) ( )f x = + .iii)

.

1.7:.

2. 12 .

-.

- .

-- (.. , ) ., :

2.1: :

i) - x, : , , .

ii) : - - - -.


72/166

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) - () -.


73/166

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) - , , .


74/166

74

iv) -, - - .

3.4:.

3.5: :i):

) ,) Sv, +

ii) -

1 ,1

aS

= 0, -

xy e=

., , .

, -.

. -, , , . ,, . -


75/166

75

-.

, :

4.1: :i) -

- .

ii) .

iii) --.

i) e .

4.2: :i) , 0, >alog

,=xa :

logx aa x = =

:

10 log = =x x lnxe x = =

ii) :log10 , = 10 ,=xlog x ln ,e = ln ,xe x= x xlnaa e=

i) , .

i) -, , -

=a

loglog

loga

=aln

log

lna

4.2 : -

10 e.


76/166

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.


77/166

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 .


78/166

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 ).


79/166

79

II. &

: 3

-

, -., ., ., . . , , .

.

1. 22 .

-, , - . , - - - , . :

-, , -. , , -

. , ().

=

OA a =

OB

.AOB , = ,AOB 0 . -


80/166

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

0x y

ax y

= //

det 1 12 2

( , ) =

x ya

x y -

.,

. -.


81/166

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


82/166


83/166

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

= +

+ + + + =

.

:


84/166

84

.

.

3 :

, .

.

.

1 96, 107 2 110.

4. 13 .

-, -.

, -, :i) , -

, .

ii) .iii) -

.

:


85/166

85

-() =1 () (+1) -.

- , - . - , - .

4. 7 .-

0.... -

(, ) = (, ), . - ... , ... (, ) [,] = . ... - [, ] = [, ] (. 159).

-, . - . , . , , .

ax y + = -... . - 0 x x t = + 0 y y at = (, ) = 1.

( , )a ( , ) 1, ax y + = (,), - x y .

- .


86/166

86

- .

, , :

. .

, -

.

4 : 4.1 5 7 4.3.

- .

: 4.4 -

4.5 4.6 4.7 .


87/166

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. O13. (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

8,3y x= +

1

103y x= + 3y x=

1

3y x=

5. 3

6.


88/166

88

3

1. 2.A 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 2 2 2a + = =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.


89/166

89

: 2

"-" ., ., . - , -

.

:1: 2: 3: .

, -. - , -, -, , -. : -

. , -

.

- . , - -

, -. "-

", -. - .


90/166

90

-. - - .

, -, - , ,

-.

1. 15 .

.

1.1 -. - ' -, , .

. --, . "".

, , -, -, , .

0

0lim ( ) ( )x x

f x f x

=


91/166

91

( )f x 0( ),f x x

0.x x

.

1.2 -. -.

-. '

(, R) - AM, - .

f ( )0 0, ( )A x f x

= 0 00

( ) ( )lim .h

x h f x

h

+ -

-

f ( )0 0, ( ) . x f x , -,

, -. -. - .

, ,,, -.,

0 0

0

( ) ( )lim .h

f t h f t

h

+ , , -

0.t

, - .


92/166

92

1.3 () . ,f

x ,f , . f (t)=x(t)(t)=x (t) -. , , -. .

x,xxx (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-.


93/166

93

. , -. "", , -

. - ,-.

- . , -, . . , - , -. , - , .

, :

. .

.

.


94/166

94

2 16 .

, , , -, , , -, , , -. - -, -

. , - , -, , -- ,, .

. , - .

, , , . , .

2.1 , - , (, ), . . , -

"" , , , ,, , , , , , ,, .

- , - . ,


95/166

95

.

2.2 . -- . . - , .

:

() i, xi, -xi.

fi ( %)xi, -. , . . ,

,=i iv f v Ni -

Fi, -, xi.

, . - - . - , ,

. -, 1 () () -- 16 -. .


96/166

96

1

() (), - y, x - y 30. . -,. , 9 .80 ,

, -. - , -. . -


97/166

97

Sturges:=1+3,32logv, .

,

-

. ,

.

,

-

. -, . -

, -

... -

1973 2

. ()

, ()

(

-

).

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... "" ,-

. -

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7.000$. ... .

().


98/166

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$

( )if%

* iii

fv

c=

%

0 11 23 23 44 55 6

6 77 1010 1515 2525 50

123455

51526268

123455

55

5,22,60,32


99/166

99

2

2.3 .

- .

. - , . , -, - . .

-

. , x . - .

.x - , , :


100/166

100

. -

, 1

( ) 0.=

=v

i

i

x x

. - : -. 0x "" .

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1

1

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i

i

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=

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101/166

101

. (- 2, 98). .

, . - . -

, 80 17 50 15, - 80+50=130

17 211016.23.

130x = =

80+1550

80+15

50% 50% . ,

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12

i

i i

i

vN

L cv

= +

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156-162

162-168168-174174-180180-186186-192

2

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52

2

1022333840

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25,055,082,595,0100,0


102/166

102

Li i ci

1iN , ., ,

9 73 , -, - v/2=20 . ,

1

40

102 2168 6 173 ,12

i

i i

i

v

N L c cm

v

= + = + =

() .

. , - -. 14 91 - . -

, Li, -, 12, c -,

0 1

0 2( )

+ =

i

i

M L

c M L 0 -

: 101 2

.ic

M L

= + +

, . - . , 70. ; , . , :


103/166

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

vH

t t t

=+ + +

1 2

1 2

,

...

=+ + + k

k

vH

v v v

x x x

., , 5 -

, 10 6 , ( )

31 1 1

5 10 6+ +

6,4 .

; , . , ,


104/166

104

, . , - . , . -, . - ( ) . - -

. -, . , , - , - , - .

, . , -, .

-, . -. , . - .

2

( )

N

t

N

ii

1=

2

2

= (1),

=

=N

iit

N

1

1 ,

( )2

*s


105/166

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 + . -, -


106/166

106

15 95. , , , Chebyshev, -

( , ) + s x s , 1 , 2

11

. ,

)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$=


107/166

107

1 , , - . - ,., - ,.

s 150=

900=Ax , 250=Bs -

3000=Bx . -, .

- , . , - .

, =s

CV

.

,

-

. -

. . -

.

-.

-.

., -, .

-.

- .

.


108/166

108

.

- .

.

.

.

.

.

-.

, - -.

. -

. -

-

.

-.

-. -.

-

.

. - .

- , - .

-.

-

. -.

68%, 95%, 99,7%

-

. - - .

--


109/166

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... .=i x 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

.


110/166

110

1

2 ( )v

i i

i

Sy a

a

=

=

(1)

1

2 ( )v

i i i

i

Sy a

=

=

(2).

(1) (2) , -, :

i ia v y + =

2

i i i i

a x x y + =

.

2.4 : -

y a = +

, -

. y a = + -

, , . , -

, = +x y

y a = + -

( , ).x y

. ( , )i ix y , . = +y = 0, . .


111/166


112/166

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 i x x y y y y x x

2

2 2

( )( )

( ) ( )

i i

i i

x x y y

y y x x

1


113/166

113

2 1r

r

r

1

1 1.

r

, - -. , , -

. (-

) . , -Y, . - . Y. Y.

, Salk, -. -' --. - ( ) (). , -. , -. ;

, . . , . , .


114/166

114

() -().

, : , -

-, .

, -.

, .

- , .

2 :

) (. 74)) (. 89)

(. 92)) (. 90, 91)) , 2.4) , 2.5.) 4 . 81.

-, .

- -. , c vi, ,

di

dv

c. ,

, 5 . 103 5 6

1

14 7.2

=

2 4 93 & 94 .


115/166

115

3. 19 .

, - . - ,

, --.

3.1, . ,, -. , -. - . - - -, , Venn., .

-, -.

, - , :

( ) ( ) ( )

)

) ) ).

= =

( ),

=( )( )=( ( )

=( ( (


116/166

116

3.2, . , (. 65 ).

- . , , . ' ,

"" . . - "" ; -, , , -,;

0, - 1 ( 0%, 100%). 0 1. ; "" - ,

-

, ( ) ,P A

= -

. () - , : 0 ( ) 1P A

( ) 1P = ( ) 0P =

,


117/166

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 , -. -, "" . -.


118/166

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 BN B N B N P B

B , . : A B B

0 ( ) ( ),P A B P B ( )

0( )

P A B

P B

1,

0 ( ) 1 P A B .


119/166

119

, = ( ) ( )

( ) 1,( ) ( )

P B P BP 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 21 2(( ) )

( )( )

P A A BP 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= +

, 12 ,

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):


120/166

120

1 2, ,..., v A A A ( ,iA 1,2,...,i v= ), -

1 2( )v B B A A A B= = ...

1 2( ) ) ),v 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( )

( ) .( )

= iiP A B

P A BP 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.

, "" -

., : -

.

.


121/166

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) 15. 11. (i) -2, (ii) 3, (iii) 2 / 6 6. 12. ()-1, ()-11, () 8, ()-87. 13. 288. 14. (1)() (2)() (3)() (4)()


122/166


123/166

123

. &I

: 5

" ' " :., ., ., ., -. .

.

:

1..

2. 12 .

- - 2 . - - . - . -, -.

, -, 2- . -, -

- , .

. - -.


124/166

124

- :

0 z z a = 0( )K z

1 2 = z z z z

1( )A z 2( ).B z

, - , -.

. , 1,vz = , - ,vz a= - -. :

:

( ),a a i = + 0, 0

,, 0

> =

=


125/166

125

z0. , -:

4 16,z = 40 0

16 2,4 4

oz i = + =

, :

2 , 0,1,2,3kz k= =

2 2

4 4i i

= + =

4 16,z = 2 ,4 4

oz i

= +

, :

2 , 0,1,2,3,4 4

k

k z i k

= + =

2 2.

4 4i i

= + =

, .

-

:1. :) ) .

2. :) , , .) - .

3. :) .) .

4. - :) .) - ( De Moivre).


126/166

126

5. vz a= a .

2.4 -

2.5 .

1. 24 .

:) ,

) { }0 ,x + ) .

)- . -

. -, "" - .

. -, -, .

, , .

, , .

, -, .2 .

)- -


127/166


128/166

128

0u =0

limx x

( )g x =0

limx

( ) 0,g x =

0

limu u

( )f u =0

limu

( )f u =0

lim0 0.u

=

,

0

limx x

( ( )) f g x 0

limu

( )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. , , .


129/166

129

6.

0 ,x .

7. .

8. .

9. + .

10. .

11..

12. 0x .

13. f, .

14. , , , .

15. : Bolzano, - , -

-, -, .

2. 35 .

, , -: . - ( ) y f x=

' -. - ( ) y f x= ' , -


130/166


131/166

131

9. - .

10. - .

2 :

) .) 262

) (. 264). , -

, ,-.

3. 32 .

- -. , , . ,, - . , , - ''-". : ) -) .

- :1.

-

.


132/166

132

2. - .

3. - .

4. .

5.

.6.

-.

7. - -.

8. .

3 :) 3.3 : ) 3.6 :

.


133/166

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 x x = + 1 1 z z yy = +

1 z z i y x = = 1 z 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.


134/166

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 xx

=

1 1

1dx dxx x= + 1 2( ) 1 ( ) ,F x c F x c+ = + + 1 2 1c c =

, 0=1. .4.x 0,

10.x

u=

5. F(0) = 0, F(2) = 2, F(3) = 4, F(4) = 6, F(6) = 12


135/166

135

6 / / 1.2 , -

: (ii):

, f g g D D= = ( ) .fg D

( ) ( ,0].g = R :

( ) ( ,0].fD g =

( ) ( ,0],y g = 2( ) , y g x x= =

.gx D =

2( ) ( ( )) 1 1 f y f g x x y= = + =

, ( ) ( ,0]g = f -

( ) 1 . f x x= ( )fD g f -

. , f

. :

1 , ( ,0]( ) ,

( ), ( ,0]

=

x xf x

h x x A

fA D= ( ,0] h

( , 0].A

, fg:

:( )

g

f

x D

g x D

2.

xx

x A

.f gD =

: 2 2( )( ) ( ( )) 1 ( ) 1 ( ) 1 ,= = = = + f g x f g x g x x x


136/166

136

2( ) 0, g x x= .gx D , ,

f g f

( , 0].A

(iii): f -

. -:

,( ) ,

,

Sf 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 xf x

g x

=

x B

x A B

( ) f x x= ( ) , g x x=

: 2 2( ) ( ), x g x= x . , -

, :

( ),( )

( ),

g xf x

g x

=

[0, )

( ,0)

x

x

+


137/166

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

xf x

x x


138/166

138

:

(t) , (t)= (t)= g0 (t) , |(t)| , ,

, (t)


139/166

139

III. :

, -. , - . - -

, , -. , , , , . -, , .

, , . -, ,

, . , .

, , -, ., , - , -, ..

, - , , ,

. , , - , , - -.


140/166

140

, , : --

. -

, .

.

.

, , .

-.

-- ., :

, .

, , -.

, , , .

1.

( 8 )

1.

: : , ,

.


141/166

141

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.

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.

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: : , . .

. -

, , , .

4.

: : . -

.


142/166

142

. . -

.

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: : .

6.

: : .

. . .

. -

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: :

. -

.

.

. .


143/166

143

8.

: :

. ,

.

.

. . . -. .

II:( 18 )

1. , ,

: : . . -

. . .

.

-, , -.

2. ,

: :


144/166

144

. -

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.

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. . .

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.

. - -, .

.

.


145/166

145

5.

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.

. . -

.

.

.

. .

6.

: : ,

.

. .

.

.

. .

7.

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146/166

146

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.

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.

.

.


147/166

147

9.

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) . - -

. -

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10.

: : . -

.

.

11. , ,

: : , -

. .

.

12.

: :


148/166

148

.

13. ,

: : ,

. (-

).

.

.

14.

: : . -

. .

.

15.

: : -

.

. -

. modus po-

nens, modus tollens, -.


149/166

149

.

16.

: :

.

.

17. -

: : -

.

.

18. II -

: : (

) . .

.


150/166

150

:( 15 )

1.

: :

. .

.

2. ,

: : -

. ,

.

.

3.

: : , , . .,

.

-.

4.

: :


151/166


152/166

152

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-

. .

7.

: :

- - , , -, .

-.

. .

8.

: : / -

.

-

. -

.

9.

: :

.


153/166

153

.

10.

: :

.

.

IV: ( 13 )

1. : -

: :

. - -

. -

.

-

-. .

2. : po - : - -

: :

.


154/166

154

-.

.

.

-

. , .

. - . , - .

3. : - - - -

: : -

. -

. . .

,

. -

, -.


155/166

155

4. : - - - -

: :

, -, -.

II.

. - . . -.

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: :

. -

.

: .

-, , .

6. : - -


156/166


157/166

157

32 3.

, . (- ). - . , -. , ,.

32 4.

, , , -. '' ". , , . , . , . -. , . - , . , .

33 5. : "

". , . , . . -.

-.

47 2.

. - - -. .

47 3. , -

: -


158/166

158

. , , . -. , . ( - ). , , -. , , . , - -

. , -. , ., . , . , -.

51 1.: ""

.,

. - "-" -. -. -. . . . -""- . - -" "--.

51 2.. - - " " - . , .


159/166

159

51 3.

: " ". , . "" . , . . , -. "" -. "-

". .

58 1.

" ", ., , . .

59 2. .

-. - " -" . .

59 3.,

, , . - , " "

. , -. . - . . -.


160/166

160

59 4.A, -

. , - . . -.

.

60 1.., -

. , - . .

, -. , .

. .

60 2. .

-.

, - -. -

, - ., -

. , . .


161/166

161

. -.

69 2.i) -

, . - , .

ii) .

. -.

iii) . --. .

82 2.i) -

, .

ii) , . (). -, .

iii) . , . , .

iv) .

82 3.i) .

, . ( )x . , . ( )x . ( )x -, . .

ii) .


162/166


163/166

163

. 90, 1: = . 91, 3: Modus Ponens = Modus Ponens (-

, ). 92, . 5: "" . 92, . 10: = . 94, . 3: = . 94, : -

, -"".

. 96, . 3: = . 96, . 4. : Pones =Ponens

. 96, 5: .. 97, . 7: "". 98, . 10: , = . 98. . 5 : ; -

, .

. 102, . 2: , = . 103, .3

= . 107, . 6: = . 107, . 12: = . 110, . 5: = "". 110, . 3: = . 112. . 6: . 113, . 3: (). 120. . 12: = ,. 122, . 17: (5) = (1) ((1")). 124, . 2: = . 124, . 10: "" "" "" "". 126. . 16: = ". 126: . 18: () = (). 129, . 2: "" = (-

, )

. 159, . 10: = . 162, .12: = . 172, 1 9: "x" "x". 173, 3 3: "".. 175, . 4: = . 175, . 4: 3, 4 p3, p4. 176, . 15: , =


164/166

164

. 177, : p16.

. 177 . 16: ". -" .


165/166

165

7.

: 2/1

-, :

1.( 20 . )2. ( 15 . )

3. ( 8 . )

: 1/2

-, 1 5 .

: 1

-, :

4. ( 14 .)

5. ( 10 . )

: 1

-, 6 8

.


166/166

166

.

.

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