8/3/2019 Odigies Did Math Lyk
1/166
2007 - 2008
8/3/2019 Odigies Did Math Lyk
2/166
2
, .
8/3/2019 Odigies Did Math Lyk
3/166
3
2007 - 2008
-
8/3/2019 Odigies Did Math Lyk
4/166
4
(1997)
:
1. , ..2. , ..3. o, ..4. /, ..5 . ..6., . , ..7., ..
8. , ..
(1998) - (1999) - (2000)
(2001) - (2002) - (2003)()
2007
, :
.. . .. .. .. .. ..
-:
, . Cabri II
. ..
.. -.. .. -
..
.. -.. .
8/3/2019 Odigies Did Math Lyk
5/166
5
.............................................................................. 7
. ............................................... 9
1 . ............................................................ 112. ................................................... 153. ........ 30
. ......................................................................... 35
. ........................... 69. &
....................................................... 79.......................................................................... 89
. .......................... 89. &
......................................................... 123. I - II I ............................. 139
............................................................ 165
8/3/2019 Odigies Did Math Lyk
6/166
6
8/3/2019 Odigies Did Math Lyk
7/166
7
-
. -, , , 1997 -.
, , -, - . , - .
, , , - . , . - -
. , , - .
, - 3 ( 35 )
8/3/2019 Odigies Did Math Lyk
8/166
8
- -. , -. -, The Geometers Sketchpad, Cabri II Function Probe - . , -
( ) - . - , , -.
8/3/2019 Odigies Did Math Lyk
9/166
9
. TOY
:) ,
, , , ,
, .
) , -, , , , , , -, .
) -, , - , -.
) , ,, , .
) H - -.
, -, :
) , - .
) - , -
8/3/2019 Odigies Did Math Lyk
10/166
10
, .
) , , -:
) .
) - -,
) - , -.
) - .
8/3/2019 Odigies Did Math Lyk
11/166
11
B1.
- . , , , -
. -, -.
, . ' -.
, , -.
, :1. -
-.
2. , -, , -
. , , ,.
3. - . ,
8/3/2019 Odigies Did Math Lyk
12/166
12
-. - (--) - -. ,' -.
4. ,
. , -, . , . . . - - .
5. , .. -, , , . , , -, . , .
6. -' , -, ,. .-
. , -
, . .
7. ,
8/3/2019 Odigies Did Math Lyk
13/166
13
. , . -, , , ( - , , , .)
8. -, -
, . -., , - . , , . -.
9. -, . - , . -, - , ., -.
'-, - - . .
8/3/2019 Odigies Did Math Lyk
14/166
14
10. . - . , . ( ) , .
11. - . -
- . - .
12. - - . - - . -, - , -. - -' .
13. , (calculators), -. ,
, ., -, ,.
8/3/2019 Odigies Did Math Lyk
15/166
15
B2.
1. . '80, ,
, -
, - -, , , , -, .
- , - -.
, , - .
, : - , . - , -
. -: -- . , - .
- , ,
8/3/2019 Odigies Did Math Lyk
16/166
16
. , , , . , , -, , : - , , -.
-
. , - :
. , , .
, , -, -.
, , -!
. Lakatos: -
-. -
- . . .
-; ,
8/3/2019 Odigies Did Math Lyk
17/166
17
; : - ;
2. .
. ,
-.
- , : . -
, , -: -- ().
- , -.
-: .
, - - , - -. - .
- , , . - ,- ().
8/3/2019 Odigies Did Math Lyk
18/166
18
- - -.
' , . , -, - , , , -.
, . , , - . -, -, , , - , -, , -. -, -, . -
.. - :
. -, . (). ,
-, . , -
8/3/2019 Odigies Did Math Lyk
19/166
19
: , -. -. -. . -.
: .
. -
. , -
-.
3.
- - -. , - . , -, , , -
, , - . , , -
8/3/2019 Odigies Did Math Lyk
20/166
20
,
, , . .
-, - . , --
- - . . -, -., -, - - , - !
- , - . . , .
-
. . ' -. ,-- , -
8/3/2019 Odigies Did Math Lyk
21/166
21
, .
-. , - . , . - . ,
. -, 300 , - 50 , . , -. , - , : .
, -
. , . , -, .
8/3/2019 Odigies Did Math Lyk
22/166
22
4. .
; . , . -
. , -
, . , -- . ,- . -: ;
", -, . () - . - -. ' , : (
). -
, . . . , -
, -.
.
8/3/2019 Odigies Did Math Lyk
23/166
23
-; , -, ,. , -.
, , -. , - -
, .
, -, , ; .... , , - . . , , , : ,
, , , , . -.
,
, ., - . :
:-
8/3/2019 Odigies Did Math Lyk
24/166
24
, , , ; -- -, , .
:- . , -
, - , ( 2, 3-). , -, ., - - . , , , , .
, -:
-
1 :
1. :
: -(t) 4 12
8/3/2019 Odigies Did Math Lyk
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
8/3/2019 Odigies Did Math Lyk
27/166
27
2 :
1.
T(t) 12 - 12 .
) ;;
) ;
) :
( )... (4),T t T [ ]0,24t
: , -.
, - 2( ) 2 3, f x x= +
(0) 3,f = :
8/3/2019 Odigies Did Math Lyk
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.
) () -:
8/3/2019 Odigies Did Math Lyk
29/166
29
) 4 92.
8/3/2019 Odigies Did Math Lyk
30/166
30
B3.
1. .
. -
.
1. :
:i): (2)f : . , . , , .
( 2)f : . , . , . ( 1)f : . , . , . (0)f : . , . , . . (1)f : . , . , .
8/3/2019 Odigies Did Math Lyk
31/166
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 < .........................................
8/3/2019 Odigies Did Math Lyk
32/166
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 ; ;
8/3/2019 Odigies Did Math Lyk
33/166
8/3/2019 Odigies Did Math Lyk
34/166
34
7. 2( ) 2 2 f x x x= + i) :f = ...............ii) f ; .iii) f -
x ; ;iv)
: ( ) 0f x > ......................................... ( ) 0f x < .........................................
8/3/2019 Odigies Did Math Lyk
35/166
35
.
2
2007-2008 - . , ., . , . , . . -, , .
-, -, , .
. . , .
, , -
, (.. - ) . . , 2, 3 4. .
/
12
.1
1.1. 2
.2
1.2. 2
8/3/2019 Odigies Did Math Lyk
36/166
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
8/3/2019 Odigies Did Math Lyk
37/166
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. + <
-
.
8/3/2019 Odigies Did Math Lyk
38/166
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.
8/3/2019 Odigies Did Math Lyk
39/166
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 =
8/3/2019 Odigies Did Math Lyk
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 =
8/3/2019 Odigies Did Math Lyk
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
8/3/2019 Odigies Did Math Lyk
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.
.
8/3/2019 Odigies Did Math Lyk
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 <
8/3/2019 Odigies Did Math Lyk
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 <
:
8/3/2019 Odigies Did Math Lyk
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 - :,
8/3/2019 Odigies Did Math Lyk
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
.
8/3/2019 Odigies Did Math Lyk
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.
8/3/2019 Odigies Did Math Lyk
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 >
8/3/2019 Odigies Did Math Lyk
49/166
49
, - 4 76 1ii), 1iii) 3 78.
, , :
20 cm . , -, .
x ( )f x ,
) f ) f
) x 2( ) 120 f x cm=
: 7 , ,
' .
, - , -. - - .
- . 2x2 ,
8/3/2019 Odigies Did Math Lyk
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 =
8/3/2019 Odigies Did Math Lyk
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.
8/3/2019 Odigies Did Math Lyk
52/166
52
.4 (4.3): - - . .
, , , . :
:
(1,1) C R=(). , 0x y + = > .) )
)
2
2d
=
) :
2.> 2.= 0 2.< <
8/3/2019 Odigies Did Math Lyk
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
, :
8/3/2019 Odigies Did Math Lyk
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 -
, -
8/3/2019 Odigies Did Math Lyk
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=
.
).
8/3/2019 Odigies Did Math Lyk
56/166
56
3( ) f x x= 3( ) f x ax=
1
( )f xx
= ( )a
f xx
=
. -
:
i. 11 93.
ii. ;
8/3/2019 Odigies Did Math Lyk
57/166
57
iii. 13 93.
iv. -) ) .
. 9 10 i), 10 ii) 12 - 93.
10 iii) 10 iv) 93.
3. 2( ) f x x= :
8/3/2019 Odigies Did Math Lyk
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
= + +
8/3/2019 Odigies Did Math Lyk
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= +
;
8/3/2019 Odigies Did Math Lyk
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) ,) :
8/3/2019 Odigies Did Math Lyk
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 -
8/3/2019 Odigies Did Math Lyk
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>
8/3/2019 Odigies Did Math Lyk
63/166
63
: 6 . ( -).
-.
, , - .
- 1.
, -:
.1: :
i. -.
8/3/2019 Odigies Did Math Lyk
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 )
8/3/2019 Odigies Did Math Lyk
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 .
8/3/2019 Odigies Did Math Lyk
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. )
8/3/2019 Odigies Did Math Lyk
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
8/3/2019 Odigies Did Math Lyk
68/166
68
8/3/2019 Odigies Did Math Lyk
69/166
69
.
: 2 .
10 --. -. - -
. , . , ., . . .
.
1. 17 .
-.
4 . -, - - . - , --.
- , ()= + . -
.- , .
.
8/3/2019 Odigies Did Math Lyk
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.
8/3/2019 Odigies Did Math Lyk
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) : - - - -.
8/3/2019 Odigies Did Math Lyk
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) - () -.
8/3/2019 Odigies Did Math Lyk
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) - , , .
8/3/2019 Odigies Did Math Lyk
74/166
74
iv) -, - - .
3.4:.
3.5: :i):
) ,) Sv, +
ii) -
1 ,1
aS
= 0, -
xy e=
., , .
, -.
. -, , , . ,, . -
8/3/2019 Odigies Did Math Lyk
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.
8/3/2019 Odigies Did Math Lyk
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.
8/3/2019 Odigies Did Math Lyk
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 .
8/3/2019 Odigies Did Math Lyk
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 ).
8/3/2019 Odigies Did Math Lyk
79/166
79
II. &
: 3
-
, -., ., ., . . , , .
.
1. 22 .
-, , - . , - - - , . :
-, , -. , , -
. , ().
=
OA a =
OB
.AOB , = ,AOB 0 . -
8/3/2019 Odigies Did Math Lyk
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 -
.,
. -.
8/3/2019 Odigies Did Math Lyk
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
8/3/2019 Odigies Did Math Lyk
82/166
8/3/2019 Odigies Did Math Lyk
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
= +
+ + + + =
.
:
8/3/2019 Odigies Did Math Lyk
84/166
84
.
.
3 :
, .
.
.
1 96, 107 2 110.
4. 13 .
-, -.
, -, :i) , -
, .
ii) .iii) -
.
:
8/3/2019 Odigies Did Math Lyk
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 .
- .
8/3/2019 Odigies Did Math Lyk
86/166
86
- .
, , :
. .
, -
.
4 : 4.1 5 7 4.3.
- .
: 4.4 -
4.5 4.6 4.7 .
8/3/2019 Odigies Did Math Lyk
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.
8/3/2019 Odigies Did Math Lyk
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.
8/3/2019 Odigies Did Math Lyk
89/166
89
: 2
"-" ., ., . - , -
.
:1: 2: 3: .
, -. - , -, -, , -. : -
. , -
.
- . , - -
, -. "-
", -. - .
8/3/2019 Odigies Did Math Lyk
90/166
90
-. - - .
, -, - , ,
-.
1. 15 .
.
1.1 -. - ' -, , .
. --, . "".
, , -, -, , .
0
0lim ( ) ( )x x
f x f x
=
8/3/2019 Odigies Did Math Lyk
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
, - .
8/3/2019 Odigies Did Math Lyk
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-.
8/3/2019 Odigies Did Math Lyk
93/166
93
. , -. "", , -
. - ,-.
- . , -, . . , - , -. , - , .
, :
. .
.
.
8/3/2019 Odigies Did Math Lyk
94/166
94
2 16 .
, , , -, , , -, , , -. - -, -
. , - , -, , -- ,, .
. , - .
, , , . , .
2.1 , - , (, ), . . , -
"" , , , ,, , , , , , ,, .
- , - . ,
8/3/2019 Odigies Did Math Lyk
95/166
95
.
2.2 . -- . . - , .
:
() i, xi, -xi.
fi ( %)xi, -. , . . ,
,=i iv f v Ni -
Fi, -, xi.
, . - - . - , ,
. -, 1 () () -- 16 -. .
8/3/2019 Odigies Did Math Lyk
96/166
96
1
() (), - y, x - y 30. . -,. , 9 .80 ,
, -. - , -. . -
8/3/2019 Odigies Did Math Lyk
97/166
97
Sturges:=1+3,32logv, .
,
-
. ,
.
,
-
. -, . -
, -
... -
1973 2
. ()
, ()
(
-
).
() -
... "" ,-
. -
25.000$ ,
7.000$. ... .
().
8/3/2019 Odigies Did Math Lyk
98/166
98
$
( )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
8/3/2019 Odigies Did Math Lyk
99/166
99
2
2.3 .
- .
. - , . , -, - . .
-
. , x . - .
.x - , , :
8/3/2019 Odigies Did Math Lyk
100/166
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 , ,.. "".
8/3/2019 Odigies Did Math Lyk
101/166
101
. (- 2, 98). .
, . - . -
, 80 17 50 15, - 80+50=130
17 211016.23.
130x = =
80+1550
80+15
50% 50% . ,
.-, ( ) :
12
i
i i
i
vN
L cv
= +
i vi Ni Fi%1
23356
156-162
162-168168-174174-180180-186186-192
2
81211
52
2
1022333840
5,0
25,055,082,595,0100,0
8/3/2019 Odigies Did Math Lyk
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. ; , . , :
8/3/2019 Odigies Did Math Lyk
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 .
; , . , ,
8/3/2019 Odigies Did Math Lyk
104/166
104
, . , - . , . -, . - ( ) . - -
. -, . , , - , - , - .
, . , -, .
-, . -. , . - .
2
( )
N
t
N
ii
1=
2
2
= (1),
=
=N
iit
N
1
1 ,
( )2
*s
8/3/2019 Odigies Did Math Lyk
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 + . -, -
8/3/2019 Odigies Did Math Lyk
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$=Bx
Bs 420$=
8/3/2019 Odigies Did Math Lyk
107/166
107
1 , , - . - ,., - ,.
s 150=
900=Ax , 250=Bs -
3000=Bx . -, .
- , . , - .
, =s
CV
.
,
-
. -
. . -
.
-.
-.
., -, .
-.
- .
.
8/3/2019 Odigies Did Math Lyk
108/166
108
.
- .
.
.
.
.
.
-.
, - -.
. -
. -
-
.
-.
-. -.
-
.
. - .
- , - .
-.
-
. -.
68%, 95%, 99,7%
-
. - - .
--
8/3/2019 Odigies Did Math Lyk
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
.
8/3/2019 Odigies Did Math Lyk
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, . .
8/3/2019 Odigies Did Math Lyk
111/166
8/3/2019 Odigies Did Math Lyk
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
8/3/2019 Odigies Did Math Lyk
113/166
113
2 1r
r
r
1
1 1.
r
, - -. , , -
. (-
) . , -Y, . - . Y. Y.
, Salk, -. -' --. - ( ) (). , -. , -. ;
, . . , . , .
8/3/2019 Odigies Did Math Lyk
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 .
8/3/2019 Odigies Did Math Lyk
115/166
115
3. 19 .
, - . - ,
, --.
3.1, . ,, -. , -. - . - - -, , Venn., .
-, -.
, - , :
( ) ( ) ( )
)
) ) ).
= =
( ),
=( )( )=( ( )
=( ( (
8/3/2019 Odigies Did Math Lyk
116/166
116
3.2, . , (. 65 ).
- . , , . ' ,
"" . . - "" ; -, , , -,;
0, - 1 ( 0%, 100%). 0 1. ; "" - ,
-
, ( ) ,P A
= -
. () - , : 0 ( ) 1P A
( ) 1P = ( ) 0P =
,
8/3/2019 Odigies Did Math Lyk
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 , -. -, "" . -.
8/3/2019 Odigies Did Math Lyk
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 .
8/3/2019 Odigies Did Math Lyk
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):
8/3/2019 Odigies Did Math Lyk
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.
, "" -
., : -
.
.
8/3/2019 Odigies Did Math Lyk
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)()
8/3/2019 Odigies Did Math Lyk
122/166
8/3/2019 Odigies Did Math Lyk
123/166
123
. &I
: 5
" ' " :., ., ., ., -. .
.
:
1..
2. 12 .
- - 2 . - - . - . -, -.
, -, 2- . -, -
- , .
. - -.
8/3/2019 Odigies Did Math Lyk
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
> =
=
8/3/2019 Odigies Did Math Lyk
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).
8/3/2019 Odigies Did Math Lyk
126/166
126
5. vz a= a .
2.4 -
2.5 .
1. 24 .
:) ,
) { }0 ,x + ) .
)- . -
. -, "" - .
. -, -, .
, , .
, , .
, -, .2 .
)- -
8/3/2019 Odigies Did Math Lyk
127/166
8/3/2019 Odigies Did Math Lyk
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. , , .
8/3/2019 Odigies Did Math Lyk
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= ' , -
8/3/2019 Odigies Did Math Lyk
130/166
8/3/2019 Odigies Did Math Lyk
131/166
131
9. - .
10. - .
2 :
) .) 262
) (. 264). , -
, ,-.
3. 32 .
- -. , , . ,, - . , , - ''-". : ) -) .
- :1.
-
.
8/3/2019 Odigies Did Math Lyk
132/166
132
2. - .
3. - .
4. .
5.
.6.
-.
7. - -.
8. .
3 :) 3.3 : ) 3.6 :
.
8/3/2019 Odigies Did Math Lyk
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.
8/3/2019 Odigies Did Math Lyk
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
8/3/2019 Odigies Did Math Lyk
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
8/3/2019 Odigies Did Math Lyk
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
+
8/3/2019 Odigies Did Math Lyk
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
8/3/2019 Odigies Did Math Lyk
138/166
138
:
(t) , (t)= (t)= g0 (t) , |(t)| , ,
, (t)
8/3/2019 Odigies Did Math Lyk
139/166
139
III. :
, -. , - . - -
, , -. , , , , . -, , .
, , . -, ,
, . , .
, , -, ., , - , -, ..
, - , , ,
. , , - , , - -.
8/3/2019 Odigies Did Math Lyk
140/166
140
, , : --
. -
, .
.
.
, , .
-.
-- ., :
, .
, , -.
, , , .
1.
( 8 )
1.
: : , ,
.
8/3/2019 Odigies Did Math Lyk
141/166
141
(, -) .
-
. . .
2.
: :
.
-
.
3.
: : , . .
. -
, , , .
4.
: : . -
.
8/3/2019 Odigies Did Math Lyk
142/166
142
. . -
.
5.
: : .
6.
: : .
. . .
. -
.
7. ,
: :
. -
.
.
. .
8/3/2019 Odigies Did Math Lyk
143/166
143
8.
: :
. ,
.
.
. . . -. .
II:( 18 )
1. , ,
: : . . -
. . .
.
-, , -.
2. ,
: :
8/3/2019 Odigies Did Math Lyk
144/166
144
. -
16 .
.
3. ,
: :
. . .
.
-
.
4.
: : ,
.
.
. -
.
. - -, .
.
.
8/3/2019 Odigies Did Math Lyk
145/166
145
5.
: :
.
. . -
.
.
.
. .
6.
: : ,
.
. .
.
.
. .
7.
: :
8/3/2019 Odigies Did Math Lyk
146/166
146
-.
.
.
.
. -
.
.
.
8.
: : , -
.
.
.
.
.
.
.
8/3/2019 Odigies Did Math Lyk
147/166
147
9.
: : . , (-
) . - -
. -
.
.
10.
: : . -
.
.
11. , ,
: : , -
. .
.
12.
: :
8/3/2019 Odigies Did Math Lyk
148/166
148
.
13. ,
: : ,
. (-
).
.
.
14.
: : . -
. .
.
15.
: : -
.
. -
. modus po-
nens, modus tollens, -.
8/3/2019 Odigies Did Math Lyk
149/166
149
.
16.
: :
.
.
17. -
: : -
.
.
18. II -
: : (
) . .
.
8/3/2019 Odigies Did Math Lyk
150/166
150
:( 15 )
1.
: :
. .
.
2. ,
: : -
. ,
.
.
3.
: : , , . .,
.
-.
4.
: :
8/3/2019 Odigies Did Math Lyk
151/166
8/3/2019 Odigies Did Math Lyk
152/166
152
-.
-
. .
7.
: :
- - , , -, .
-.
. .
8.
: : / -
.
-
. -
.
9.
: :
.
8/3/2019 Odigies Did Math Lyk
153/166
153
.
10.
: :
.
.
IV: ( 13 )
1. : -
: :
. - -
. -
.
-
-. .
2. : po - : - -
: :
.
8/3/2019 Odigies Did Math Lyk
154/166
154
-.
.
.
-
. , .
. - . , - .
3. : - - - -
: : -
. -
. . .
,
. -
, -.
8/3/2019 Odigies Did Math Lyk
155/166
155
4. : - - - -
: :
, -, -.
II.
. - . . -.
5. : -
: :
. -
.
: .
-, , .
6. : - -
8/3/2019 Odigies Did Math Lyk
156/166
8/3/2019 Odigies Did Math Lyk
157/166
157
32 3.
, . (- ). - . , -. , ,.
32 4.
, , , -. '' ". , , . , . , . -. , . - , . , .
33 5. : "
". , . , . . -.
-.
47 2.
. - - -. .
47 3. , -
: -
8/3/2019 Odigies Did Math Lyk
158/166
158
. , , . -. , . ( - ). , , -. , , . , - -
. , -. , ., . , . , -.
51 1.: ""
.,
. - "-" -. -. -. . . . -""- . - -" "--.
51 2.. - - " " - . , .
8/3/2019 Odigies Did Math Lyk
159/166
159
51 3.
: " ". , . "" . , . . , -. "" -. "-
". .
58 1.
" ", ., , . .
59 2. .
-. - " -" . .
59 3.,
, , . - , " "
. , -. . - . . -.
8/3/2019 Odigies Did Math Lyk
160/166
160
59 4.A, -
. , - . . -.
.
60 1.., -
. , - . .
, -. , .
. .
60 2. .
-.
, - -. -
, - ., -
. , . .
8/3/2019 Odigies Did Math Lyk
161/166
161
. -.
69 2.i) -
, . - , .
ii) .
. -.
iii) . --. .
82 2.i) -
, .
ii) , . (). -, .
iii) . , . , .
iv) .
82 3.i) .
, . ( )x . , . ( )x . ( )x -, . .
ii) .
8/3/2019 Odigies Did Math Lyk
162/166
8/3/2019 Odigies Did Math Lyk
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: , =
8/3/2019 Odigies Did Math Lyk
164/166
164
. 177, : p16.
. 177 . 16: ". -" .
8/3/2019 Odigies Did Math Lyk
165/166
165
7.
: 2/1
-, :
1.( 20 . )2. ( 15 . )
3. ( 8 . )
: 1/2
-, 1 5 .
: 1
-, :
4. ( 14 .)
5. ( 10 . )
: 1
-, 6 8
.
8/3/2019 Odigies Did Math Lyk
166/166
166
.
.
Top Related