mathhmatika kai stoixeia statistikhs-biblio mathiti
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1:
1.1 9
1.2 19
1.3 271.4 39
2:
2.1 58
2.2 62
2.3 83
2.4 104
2.5
117
3:
3.1 - 138
3.2 146
3.3 157
3.4 - 165
- 179
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-
,
1999-2000.
.
,
. ,
,
.
.
. . , -
-
. -
-
. -, -
.
, -
, ,-
.
.
. -
, .
, -
,
. -
, 396, 15310
.
1999
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1
17
,
.
. ,
t )(tfx ,
t
)(tfx t , ,
)(tfx . , ,
,
. Newton (1642-1727) Leibniz (1646-
1716),
,
. 17
, 18
Jacob Bernoulli (1654-1705) Johann Bernoulli (1667-1748), Euler
(1707-1783), , Lagrange (1736-1813)
. ,
19 Bolzano (1781-1848),
Cauchy (1789-1857) Weierstrass (1815-1897).
1.1
(function)
.
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10
fBA
,
, R
, R.
. f, g, h,, .
.
f .
Ax By ,
)(xfy
y f x. )(xf f x.
x, ,
, y, x
x, .
,
2
1)( xxf . : f
21)( xxf 21)( xxf
21 xy , , 21 x .
)(xf ,
R )(xf
. , 2
1)( xxf
01 2 x ,
]1,1[ , 2
3)(
x
xg
}2{ RA , R 2, 13)( xxh
R.
f
xy
,
1
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11
. , , 2
2
1)( gxxf 2
2
1)( gtts .
f, g,
:
gfS , )()()( xgxfxS , Ax gfD , )()()( xgxfxD , Ax gfP , )()()( xgxfxP , Ax
g
fR ,
)(
)()(
xg
xfxR , Ax 0)( xg .
, 1)(2 xxf 1)( xxg ,
)1(11)(2 xxxxxS
)1)(2(211)(22 xxxxxxxD
)1()1()1)(1()(22 xxxxxP
11
1
)(
)()(
2
x
x
x
xg
xfxR , 1x .
f .
f
Oxy ))((,( xfxM
Ax
. , ),( yxM f, )(xfy . )(xfy
),( yx
f
f.
.
.
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12
O
y
x
y=x
-2 -1 O 1 x
y
2
1
2
3
y=x2
()
xxf )(
13.
() 2)( xxf .
-2 -1
O 1
y x1
y
x2
-2
-1
1
2
-2 O-1 x1
1
2
3
y
y=ex
()
xxf
1)( .
() x
exf )(
xx , 0xe Rx .
O
-1
y
1
1 x
y=lnx
2
y=x
O x
y
2
y=x
O x
y
()
xxf ln)(
yy ,
0x .
() xxf )(
xxg )(
2.
2
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13
x
xf 1)(
0x . f .
-
2
y=x
3/2
/2 O x
y
xxf )( , ]2,0[ x 21 ,xx
2,0
21 xx 21 xx .
xxf )(
2,0
.
2,
23 .
21,xx
2
3,
2
21 xx , 21 xx .
xxf )(
2
3,
2
. :
f
, 21
,xx 21
xx
)()( 21 xfxf , ,
21 ,xx 21 xx )()( 21 xfxf .
.
,
3
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14
]2,0[ x 2
1 x 2
31 x . , ,
xxf )( (maximum) 2
x
(minimum) 2
3x .
g 4
1xx g
gx
1x , ,
1x .
g
1x . 3xx . )( 1xg )( 3xg
. , 4xx )( 4xg
gx
4x . g 4x .
2xx . )( 2xg )( 4xg
.
. ,
)( 1xg )( 4xg .
:
f :
Ax 1 , )()( 1xfxf x
1x , Ax 2 , )()( 2xfxf x
2x .
, ,
.
1
1)(
2
x
xxf , 1x .
f x 1.
x
y
O x4x3x2x1
y=g(x)
4
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15
)(xf x 1.
x 1 )(xf 1x )(xf
0,5
0,9
0,99
0,999
0,9999
1,500000
1,900000
1,990000
1,999000
1,999900
1,5
1,1
1,01
1,001
1,0001
2,500000
2,100000
2,010000
2,001000
2,000100
x
1 ( 1), )(xf 2.
, 1x
11
)1)(1(
1
1)(
2
x
x
xx
x
xxf ,
x 1
)1( x , 1)( xxf
2 )21( x . f
1 (limit) 2
2)(lim1
xfx
.
f
0x , ,
0x .
0x
. , 23
1)( xxf ,
R. 0x , 2)( xf ,
2)(lim0
xfx
. , 0lim2
0
x
x 0lim
0
x
x.
23
1 xy
2
y
x
O
x
y
y=x2
y x
y
x
() () ()
yx
x
2 1
1
x
2
1O
y
5
6
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16
fg 0x , 1)(lim
0
xfxx
2)(lim0
xgxx
1 2 ,
:
21))()((lim0
xgxfxx
1))((lim0
kxkfxx
21))()((lim0
xgxfxx
2
1
)(
)(lim
0
xg
xf
xx
xx
xf 1))((lim0
xx
xf 1)(lim0
.
, , 93)(2 xxxf
19649limlim3lim)93(lim)(lim22
2
2
2
22
xxxxxxxxxxf .
93)(2 xxxf )2()(lim
2fxf
x
.
f 20 x .
f,
Ax 0 )()(lim 00
xfxfxx
.
,
.
, ,
, , ,
.
0lim0
xxxx
, 0lim0
xxxx
0lim0
xxxx
( 0 0 x ).
:
i)x
x
x 4
53lim
5
ii) 14lim8
xxx
iii)3
9lim
2
3
x
x
x.
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17
i) 12020
54553
453lim
5 x
xx
ii) 53294184814lim8
xxx
iii) 6)3(lim3
)3)(3(lim
3
9lim
33
2
3
x
x
xx
x
x
xxx.
1. xxxf 3)(3 , )1(f , )2(f , )1(f .
2. 65)(2 ttt , )0( )1( .
t 0)( t ;
3. h )( , )0(h
2
h .
]2,0[ 0)( h ;
4. 2ln2
1)( xxf , )1(f )(ef .
5. )2)(1(
2)(
xx
xxf ;
6. x )7)(3()( xxxf ; )7)(3()( xxx ;
7. 123)( 2 xxxf 12)( xxg , )()( xgxf , )()( xgxf ,
)(
)(
xg
xf.
8. :
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18
i) )43(lim 20
xxx
ii) )]4)(12[(lim2
xxx
iii)
xx
x
1lim4
iv) )32(lim0
xxx
v) )3(lim
4
xx
x
.
9. :
i))2(3
4lim
2
2
x
x
xii)
1
5lim
2
2
1 x
x
xiii) ])1[(lim
0xx
x
iv)4
16lim
2
4
x
x
xv)
5
25lim
2
5
x
x
xvi)
2
232lim
2
2
x
xx
x.
1. xe
xf1
1
1)(
, 1)()( xfxf .
2. 100 m,
.
. x,
x.
3. , , 20 cm.
h cm, h.
r,
r.
4. 10 . , , .
5. i)
5
5lim
5
x
x
x 52
1 ii)
h
h
h
11lim
0
2
1 .
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19
1.2
),( RO
. .
,
o90 .
, ,
.
. ),( RO
, ,
.
-
.
f
))(,( 00 xfxA
C.
))(,( 00 hxfhxM C 0h .
C ,
0h , (tangent) C.
h
xfhxf
A
M
)()(
00 ,
C
h
xfhxf
h
)()(lim
00
0
.
B
xx0+hx0
f(x0)
y
f(x0+h)
C
7
8
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20
, S t sec (s)
2
2
1)( gttS ,
2
m/s81,9g .
, 5t s;
, s5t s1,5t . :
1,0
)5()1,5(
SS
m/s5405,491,0
5905,4)1,5(905,422
.
.
65 t 1,55 t
05,55 t
01,55 t
001,55 t
0001,55 t
00001,55 t
53,955
49,5405
49,29525
49,09905
49,054905
050491,49
050049,49
,
49,05 m/s.
5t s5t . 5 s m/s05,49 .
, 0t
O
B t0+h
A t0
9
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21
, h . 0t
2001
2
1)( gttSOAS
ht 0
)2(2
1
2
1)(
2
1)(
20
20
2002 hhtggthtghtSOBS .
, h
)2(
2
1 2012 hhtgABSSS
0t ht 0
ghgth
hhtg
tht
S
2
1)2(
2
1
)(0
20
00
.
h ,
, 0gt .
0t 0t .
, 0t
00
00
0lim
)()(lim gt
h
S
h
tShtS
hh
.
50 t , m/s05,49581,9 , -
.
,
, ,
t )(tfx .
O t0
A
t0+h
B
0t ,
0t ht 0 0h . h
)()( 0012 tfhtfxxx . ,
10
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22
h
h
tfhtf
h
x
)()( 00 .
,
0t
h
tfhtf
h
x
hh
)()(limlim 00
00
.
,
.
f x x0
, -
,
,
h
xfhxf
h
)()(lim 00
0
.
, ,
, , , ,
.
,
f 0x .
f x0, )( 0xf
f 0x . :
h
xfhxfxf
h
)()(lim)( 00
00
, 2
3)( xxf 4, :
)4()4( fhf :22
43)4(3)4()4( hfhf
)484(3222 hh
)8(3 hh
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23
0h h fhf )4()4( :
hh
hh
h
fhf324
)8(3)4()4(
.
h
fhf
h
)4()4(lim
0
:
24)324(lim)4()4(
lim00
h
h
fhf
hh.
, 24)4( f .
f 0x (rate of change)
)(xfy x, 0xx . ,
:
f ))(,( 00 xfx
)( 0xf , )(xf x 0xx .
)(tfx
0t )()( 00 tft ,
)(tf t 0tt .
. , ,
||)( xxf 00 x .
0h ,
h
fhf
h
)0()0(lim
0
1lim
0
h
h
h,
0h , 1lim)0()0(
lim00
h
h
h
fhf
hh,
h
fhf
h
)0()0(lim
0
.
O
y
x
y=|x|
11
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24
1. tttx 2)( , t .
) :
(i) ]2,0[ (ii) ]1,0[ (iii) ]5,0,0[ (iv) ]1,0,0[ .
) 0t .
) )(txx .
) )0,0(O
().
,
)(txx 0t .
)
i) m/s32
6
2
)0()2(
xx ii) m/s2
1
2
1
)0()1(
xx
iii) m/s5,15,0
)0()5,0(
xx iv) m/s1,1
1,0
)0()1,0(
xx
) 0t ,
m/s1)1(limlim)0()0(
lim0
2
00
h
h
hh
h
xhx
hhh.
)
t )(tx ,
4
1
2
1)(
2
2
ttttx ,
,
4
1,
2
1
2
1t . ,
.
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25
-1 1 2
1
2
3
4
5
(2,6)
t
x
6
x=t()
x=1,1t
x=1,5t
x=2t
x=t2+t
x=3t
) )0,0(O
3, 2, 1,5 1,1, tx 3 ,tx 2 , tx 5,1 tx 1,1 .
.
0t
0t , 1.
, tx , .
2.x
xf3
)( .(i) )3(f .(ii) f
))3(,3( f .
(i) h
hh
hh
fhf
33
3333
33)3()3( 0h
hhh
h
h
h
h
h
fhf
3
1
)3(
3)3()3( .
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26
)3(f31
31lim)3()3(lim
00
hhfhf
hh.
(ii) f
3x )3(f . ,
xy 3
1.
)1,3())3(,3( f ,
33
11
11
2 .
, 23
1 xy .
1.
i) 13)( xxf 3x
ii) 5)(2 xxg 2x
iii) xxx 2)( 2 4x
2. 1
1)(
ttf 1t .
3. i) L r rL 2 . L r, 3r .
ii) r 2rE . r, 2r .
y x 1
32
yx
3
x
y
O
3
1
654321
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27
4. i) x x 5x .ii) x
x, 10x .
5. :
i)2
)( xxf ))3(,3( fA
ii) xxf 2)( , ))4(,4( fA .
1.3
f , Ax f . ,
Bx h
xfhxfxf
h
)()(lim)(
0
.
() (derivative) f f ., 23)( xxf , :
)2(3)2(33)(3)()(22222 hxhxhxhxxhxxfhxf ,
0h
hxh
hxh
h
xfhxf36
)2(3)()(
.
, xhxxfh
6)36(lim)(0
.
, f 0x
. ,
23)( xxf 40 x xxf 6)(
40 x , 2446)4( f .
f f
f .
)(tx t,
)()( txt .
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28
, t,
)()( tt )()( txt .
f
h
xfhxfxf
h
)()(lim)(
0
.
.
cxf )(
0)()( ccxfhxf
0h ,
0)()(
h
xfhxf,
0
)()(
lim0
h
xfhxf
h .
0)( c . xxf )( hxhxxfhxf )()()( , 0h ,
1)()(
h
h
h
xfhxf.
11lim)()(
lim00
hh h
xfhxf.
1)( x . xxf )( 2)( xxf .
hhxxhxhxxhxxfhxf )2(2)()()(22222 ,
O
y
c
x
y=c
O
y
x
y=0
() ()
O
y
x
y=x
O
y
x
y= 1
()
()
12
13
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29
0h , hxh
hhxh
xfhxf 2)2()()( .
, xhxh
xfhxf
hh2)2(lim
)()(lim
00
.
xx 2)(2
O x
y
y=x2
O
y
x
y=2x
() ()
1)(
xx , ..
2
2111 11)(
1
xxxx
x
,
3
3122
2
222)(
1
xxxx
x
x
xxxx2
1
2
1
2
12
11
2
1
2
1
.
1
)( xx , .
x x .
xxf )( ( 15).
)(xf 0xx
f ))(,( 00 xfx ,
f .
f
x .
14
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30
y
y=x
y=(x)
x
x
y
2
2
/2
45o135
o135
o
45o
, xxf )(
xx )( . xxg )(
xx )( . xe xln , e,
xx ee )( xx 1)(ln .
)(xcf )()( xcfxF .
))()(()()()()( xfhxfcxcfhxcfxFhxF , 0h
hxfhxfc
hxfhxfc
hxFhxF )()()()(()()( .
)()()(
lim)()(
lim00
xfch
xfhxfc
h
xFhxF
hh
.
)())(( xfcxfc .
15
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31
4455
1553)(3)3( xxxx ,
22
414
14
4
xxxx
xx
xx61
6)(ln6)ln6( .
)()( xgxf )()()( xgxfxF .
))()(())()(()()( xgxfhxghxfxFhxF
))()(())()(( xghxgxfhxf ,
0h ,h
xghxg
h
xfhxf
h
xFhxF )()()()()()(
.
).()()()(
lim)()(
lim)()(
lim000
xgxfh
xghxg
h
xfhxf
h
xFhxF
hhh
)()())()(( xgxfxgxf 34)3()()3(
344 xxxxx
2
22 112
1)()(
1
xx
xxx
xxx
.
)()( xgxf )(
)(
xg
xf
:
)()()()())()(( xgxfxgxfxgxf
2))((
)()()()(
)(
)(
xg
xgxfxgxf
xg
xf
.
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32
xxxxxxxxx )()()(
34
2
4
22
22
22
2
)2(222
)(
)()1()1(1
x
x
x
xx
x
xxx
x
xxxx
x
x
x , x, x,x
e xln . , ,
23
)1( x 33 )1( x
)1(6)1(3)1(3))1)(1(())1((3232323323 xxxxxxxxx
23222333232333)1(93)1()1)(1(6])1()1[())1(( xxxxxxxxxx
1)(
2
xxF ; )(xF xxf )(
x 1)(2 xxg . , , ))((1)( 2 xgfxxF .
F g f.
:
)())(())(( xgxgfxgf . ))(( xgf ,
f g(x)
g.
,
23222332333)1(93)1(3))1()1(3))1(( xxxxxxx .
, , x
x2
1
, :
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33
12
12
1)1(12
1122
2
2
2
x
xxx
xx
x .
, )12(2)12()12())12(( xxxx .
.
0)( c
1)( x 1
)(
xx
xx
2
1)(
xx )(
xx )( xx
ee )(
xnx
1)(
)())(( xfcxcf
)()())()(( xgxfxgxf
)()()()())()(( xgxfxgxfxgxf
2))((
)()()()(
)(
)(
xg
xgxfxgxf
xg
xf
)())(())(( xgxgfxgf
1. i) xxf )( ii) xxf 3)( 2 .
i)
)()( xxf
xx
xx
x
xxxx
x
x22
22
2
1
)()(
.
x
x2
1)( .
ii)
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34
)3(332)3(32])3[()(2
xxxxxxxf xxxx 63)32(33323 ,
22 .
2. , ttttxx 96)(
23 , t x .
i) t.ii) 2 s 4 s;iii) () ;iv)
;
v) 5 s.
i)
9123)96()()(223 ttttttxt .
ii)
st 2 m/s3921223)2( 2
st 4 m/s9941243)4( 2 .
iii) , 0)( t ,
091232 tt
0342 tt
1t 3t .
, 1 s 3 s.
iv) , 0)( t ,
091232 tt
0342 tt
0)3)(1( tt
1t 3t .
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35
, 1t 3t ( )31 t .
:
x=x(t)
t=0
t=3
t=1
40
v) :
m4|04||)0()1(|1 xxS .
1t 3t m4|40||)1()3(|2 xxS
3t 5t m20|020||)3()5(|3 xxS
, S 5s
m282044321 SSSS .
( 1-18)
1. i) 5)( xf ii) 4)( xxf iii) 9)( xxf .
2. i)2/3
)( xxf ii) 3)( xxf iii) 5)( xxf .
3. i) 3)( xxf ii) 5 2)( xxf , 0x .
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36
4. i)x
xf 1)( ii) 31)( xxf iii)
5 21)(x
xf , 0x .
5. i)3
4)( xxf ii) 56)( xxf iii) 205
2)( xxf .
6. i)4
6)(
xxf
ii) xxxf 6)( .
7. i)24
3)( xxxf ii)x
xxf 35)(2 iii)
xxxxf 12)(
2
.
8. i) 58)( 3 xxxf ii) )(86)( 2 xxxxf .
9. i) )1)(1()(43 xxxf ii) )1()( xxxf .
10. i) )1)(1(3)( xxxxxf ii) xxxxxf 34)( 22 .
11. i)1
)(2
x
xxf ii)
x
xxf
)( iii)
x
xxxf
1
)(
.
12. i)x
xf1
1)(
ii)
2)1(
3)(
xxf .
13. i)5
)1()( xxf ii) 5)12()( xxf iii) 52 )32()( xxxf .
14. i) xxf3)( ii) 3)( xxf iii) xxxf 4)( iv) xxf 3)( .
15. i) xxxf 22)( ii) xxf 1)( .
16. i)xexf 3)( ii)
2
)(xexf iii) axexf )( iv)
xx
x
ee
exf
)( .
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37
17. i) xxf 2ln)( ii) 31ln)( xxf .
iii) )ln()( xxf iv) 1ln)( xxf .
18. i)x
xxf
ln)( ii) xexf x ln)( .
19. i)
1
)(2
3
t
ttf ))3(,3( fA .
ii)
f
)(
3,
3
f
.
20. t
2
)2(4
11)( ttB ,
8t . : i) t ii) 1, 2 8 .
21. )3,0(A )0,(xB x
10x .
22. )1()( xxxf ))0(,0( fO
xx 060 .
1. 1
3)(
x
xxf
53 xy ;
2.
O
x
y
B(x,0)
A(0,3)
-
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38
496)(23
xxxxf xx .
3.
1)(
x
xxf yx0 .
4. t
ttttx 23 2)( .
t .
;
5. xBxAxf )( , 0)()( 2 xfxf .
6. pxpx eexf )( , )()( 2 xfpxf .
7. xexf )( , 0)(4)(3)( xfxfxf .
8.
xxxf 2)( 3x .
9. P
I
IIP
)( , 0I , , . i)
)(IP , , ,
)0(P . ii) 2)(11
)( IP
IP .
10. tAty )( , t , .
i)
t.
ii)y.
iii), 0,
.
-
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39
1.4
. ,
, .
.
.
, ,
m/s200 . 2
m/s10g , , , h
t
2520)( ttth tt 205 2 . (1)
h(t) :
t 0t 4t .
22
t
20)2( h .
2t 2t .
2010)( tth ]4,0[ .
0)( th , 2
, ,
2 , .
O
h
t
20
2 4
16
-
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40
t 0 2 4
)(th 0
.
:
f
0)( xf , f
.
f
0)( xf , f
.
2t h, h , 2t h .
:
f 0)( 0 xf ),(0 x , 0)( xf ),( 0x 0)( xf ),( 0 x , f
),( 0xx .
f 0)( 0 xf ),(0 x , 0)( xf ),( 0x 0)( xf ),( 0 x , f
),( 0xx .
O
y
x
f(x0)=0
f(x)>0 f(x)
-
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41
1. 0,)( 2 xxxf .
:
xxxxf 2)()( 2 .
xxxf
2020)( .
xxxf
2020)( .
, 0 , 0)( xf
x
2 0)( xf
x
2 .
x
2
)(xf 0 0 , 0)( xf
x
2 0)( xf
x
2 .
x
2
)(xf 0 , 0,)( 2 xxxf
x
2
0 0 .
f
4
4
222
22
.
2. 1000
2100
10001000)(
t
ttp ,
t.
;
-
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42
22
2
22
2
)100(
)100(1000
)100(
21000)100(1000)(
t
t
t
ttttp
.
10100)100(10000)( 2 ttttp .
t, 10t .
010001000)100(
)100(10000)(
22
22
2
tt
t
ttp
10100)10)(10( ttt .
0)( tp 100 t .
0)10( p , 0)( tp (0, 10) 0)( tp , (10, ).
10
1050200
1010001000
10100
1010001000)10(
2
p .
f, 0xx .
, 0x ,
0xx 0xx .
f ,
. f ,
, f .
, 0)( 0
xf .
f, 0xx .
, x 0x ,
0xx 0xx .
f ,
O x0 x
y
y=f(x)
O
x0x
y
y=f(x)
18
19
-
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43
. , f , f ., 0)( 0 xf .
:
f 0)( 0 xf 0)( 0 xf , f () 0xx .
f 0)( 0 xf 0)( 0 xf , f () 0xx .
f 0)( 0 xf 0)( 0 xf ,
2
f.
1. . 30 .
.
x. x2 . y ,:
3022
12 xyxy
xxy 2302 (1)
,
.
,
222
2230
2
1)230(
2
12 x
xxxxxxxyE
,
BA
yy
2x
xx
-
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44
22
230)( xxxE
.
xxE )4(30)(
xxE
4
300)( .
0)4()( xE ,
x
4
30
. x (1)
y
4
60
4
30)2(302
4
30
y .
m2,44
30
yx , .
2. P t:
3,250
20020)(2
ttt
tP .
.
t
ttP 2
25020)(
2.
512502250
0)( 32
tttt
tP .
5t .
02500
202500
202250
20)(332
ttt
ttP , 3t .
5t
250012520)2550200(20)5( P .
-
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45
1.
i) xxxf 2)(2 ii) 63)( 2 xxf iii) 42)( 2 xxxf .
2.
i) 56)(23 xxxf ii) 13)( 3 xxxf .
3.
i)3
2)( xxf ii) 16)( 3 xxf iii) 1033)( 23 xxxxf
iv) 1153)(23 xxxxf .
4. 40.
.
5. 2m100
;
6.
32 3dm .
,
.
7.
12 2dm ,
;
8. 32 xy
.
9. c
c ,
c .
;
10. :
10
.
-
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46
1. qrrp 100)ln1(100 , pq,
q
pr .
2.
xx
1ln
2, ,
e
x1
.
3. 60 cm ,
,
.
,
.
4. 160002
m
.
900 ./m
600 ./m.
,
.
5.
.
6.
.
,
.
7. ,
ty
tktk eekk
Aty 21
12
)(
, , 1k 2k
12 kk . t
.
60
60
-
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47
8. km/h,
30001,06 .
i)
1000km .
ii)
,
1000km.
ii)
.
9. 450.
;
10. 20
. 40 km/h,
20 km/h. 10 km,
;
1. ,
m
(kgr/m).
x
)(xfm , x
h
xfhxf
h
)()(lim
0
, .
x+hx
xxfm )( , x
,
i) ]21,1,1[
ii) 1x .
-
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48
2. C x v :
32353)( vvxxxC .
v16 .
, .
3.
123)( 23 xxxxf
;
4. ),( A 1.
0x0y p q .
qp 2)( .
5. 20 cm ;
6. 1lt.
.
7. R .
.
R
R
O
8. )(xC x , C ,
x
xCxc
)()(
h
xChxC
h
)()(lim
0
.
-
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49
) x, :
=.
) () x
260021000
1)( 2 xxxC .
i) ,
1000 , 2000 3000 .
ii)
;
9. x , )(xp .
x ,
)()( xpxxR . R
R . x
)()()( xCxRxP . P
P .)x,
.
)
, 2
001,053800)( xxxC
xxp 01,050)( ;
10. 1 2 .
Fermat,
.
i)
2
21
2
1
21
2)(
)(
dxd
dxxt
ii) )(xt .
iii) 2
1
.
d2
d1
a
d
x d-x
-
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50
1.
)3)(2()( xxxxV .
:
. ),0[ . )2,0( . ]0,( . ]3,2[ .
2.
. x .2
x . 12 xx . 21 xx .
3.
63.
. 8 . 2 . -6 . 10.
4.
22
6)(
xxf
.
x
22
62
x
:
. 2x . 11 x . 22 x . 2x .
5.
xxf )( 2
1)( xg .
:
.2
.
3
.
3
2 .
6
x
3x
2x
O
B
A
x x
y
y=x2
y=x1
x
y
B(k,4)A(1,4)
(k,-3)(1,-3)
O x
y
3
2
1
321-1-2-3
1 2/ B
x
y
A
O
-
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51
6. f .
)()( 2 xfxf :
. 2 . 3 . 4 . 5 . 6
7. f g .
)2()2( gf :
. 5 . 4 . 3 . 2
8. y
f 4 .
:
. 1 . 1
. 2 . 12 .
9.
]3,0[/2)(2 Axxxf
f .
10.
.
x
y
5
O x
y=g(x)
y=f(x)
y
4321
1
2
3
4
y=f(x)
(1,-1) x
y
(-3,-2) (4,-2)
O x
y
3
2
1
-1
321-1
-
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52
i) ii)
x
y
y
x
x
2
x
x
x
2
0lim
O
yx
x
3
x
y
x
x
x
3
0lim
iii) iv)
x
y
y
x x
x
| |
2
x
xx
x
||lim
0
yx
1
1 2( )
x
y
O-1-2 321
21 )1(
1lim
xx
v) vi)
x
y
y
x
x
21
1
-2
-1
21
1
-1
1
1lim
2
1
x
x
x
yx
x
3 1
1
x
y
O-1-2
3
2
1
1
1
1lim
3
1
x
x
x
vii)
y x 2 1
x
y
O 2
3
2
1
1
12lim2
xx
-
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53
11. i) 2
3)( xxf 12)( f , ;
ii) x
xf1
)( 9
1)( f , ;
iii) xxf )( 2
3)( f , ;
12. f g 4)3( f , 2)3( g , 6)3( f
5)3( g 3x
) gf ) gf ) gf )g
f
13. ))(()( xgfxh 6)3( g , 4)3( g 7)6( f ,
)3(h .
14.
.
.
O x
y
O x
y
Ox
y
Ox
y
(1) (2) (3) (4)
O
x
y
O
x
y
O x
y
O
x
y
() () () ()
-
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2
status
(, ) ,
.
(,
, ).
, ,
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,
.
2238 ..
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, .
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().
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1348 400 ,
. 1620
Graunt
88 3 .
, 13.200 1620,
387.200 .
-
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56
, 11,
, .. 16
19,
, ,
.
,
:
,
.
, ,
.
R.A. Fisher (1890-1962),
:
:
.
(experimental design) ,
(descriptive statistics),
. ,
(inferential statistics)
,
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100
20
100 ,
. 20%
,
() .
.
.
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(, , ,
-
8/3/2019 Mathhmatika Kai Stoixeia Statistikhs-Biblio Mathiti
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57
, ), (, , , ), (, , , ),
(, , , )
(, , ,
Marketing, ),
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-
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58
-
,
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-
8/3/2019 Mathhmatika Kai Stoixeia Statistikhs-Biblio Mathiti
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59
2. , :
i) , .
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( 1,2,),
( 1,2,,6) .
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-
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60
.
,
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, , .
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,
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.
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, 2001
20 .
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1936. Literary Digest
2.400.000 Landon 57%.
, G. Gallup
50.000 Roosvelt
62%!
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(Sampling), . ,
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-
8/3/2019 Mathhmatika Kai Stoixeia Statistikhs-Biblio Mathiti
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61
1. ;;
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-
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62
,
.
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( )
-,
) , .
.
:
) ,
,
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,
) (),
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.
1
()
()
1971
1981
1991
1993
1994
0-14
15-64
65
2,22
5,58
0,96
2,31
6,19
1,24
1,97
6,88
1,40
1,85
6,99
1,54
1,81
7,04
1,58
: , 1996
2
, 1991, 10.000 .
-
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63
.. 1971 1981 1991
8.261,183
3.661,637
1.635,998
1.401,459
842,796
734,014
585,312
477,942
476,288
406,612
389,434
383,672301,106
287,611
110,581
91,503
84,069
77,014
456.471
162.986
97.008
66.606
52.487
31.787
89.578
32.664
17.367
30.180
14.201
13.31622.917
16.650
13.097
23.065
18.642
9.553
502.082
185.626
88.601
87.831
48.700
27.649
96.533
31.629
15.721
30.011
14.037
13.11119.947
20.350
14.295
28.574
19.668
11.127
539.938
205.502
87.151
98.181
51.060
29.392
104.781
33.032
17.645
32.556
14.838
13.52719.350
26.379
15.706
34.272
19.870
11.639
: , 1991
3
1990-94
1990 1991 1992 1993 1994
15
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-5960-64
65-69
18
731
3323
4277
3952
3589
3237
2839
2727
2304728
121
9
564
2785
3921
3700
3146
2803
2593
2564
2230720
150
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437
2755
4246
3388
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2185688
140
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735
2981
3881
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5
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2696
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1664523
96
27846 25185 25063 23959 22608
: ,
4
40 .
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64
. * (cm) (Kg) (cm) (cm)
1 K 4 1 15 170 60 172 168
2 A 1 0 17 180 68 185 165
3 K 4 2 12 178 62 181 160
4 K 5 1 18 165 47 180 162
5 K 5 0 15 170 54 180 168
6 K 4 3 16 168 56 185 168
7 K 4 2 15 175 58 193 162
8 A 4 1 15 175 72 174 174
9 A 2 3 13 173 67 182 160
10 K 3 1 15 162 50 176 170
11 K 4 1 16 160 51 176 164
12 A 2 1 11 170 58 182 16513 K 7 3 20 167 50 174 170
14 A 1 1 18 177 81 177 169
15 A 1 0 17 180 70 170 165
16 K 2 2 19 170 63 165 174
17 A 2 0 14 182 71 176 173
18 A 7 2 17 178 73 182 170
19 K 4 1 14 165 58 180 161
20 A 5 1 16 178 74 173 168
21 K 5 1 12 156 44 170 158
22 K 5 1 13 175 53 170 165
23 A 5 2 18 172 60 178 165
24 K 6 1 16 173 64 182 162
25 K 6 2 14 167 57 172 15726 A 5 0 14 187 85 185 170
27 K 6 1 17 170 62 180 165
28 A 3 1 12 180 80 180 167
29 A 3 0 15 178 73 173 170
30 A 2 1 10 191 86 180 170
31 A 2 0 16 176 65 180 172
32 K 4 1 12 169 57 170 167
33 K 4 2 14 167 61 179 158
34 K 4 1 19 166 62 178 165
35 A 3 1 19 179 76 178 160
36 A 3 1 16 178 68 180 160
37 A 5 1 19 180 85 170 163
38 K 5 1 19 164 64 184 17039 K 3 0 15 170 63 165 167
40 K 4 1 15 173 63 186 162
*1: , 2:, 3: - , 4:,
5: -, 6:, 7:
: 1 (. 98).
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65
xxx ,...,, 21 , v, . ix
() (frequency) i ,
ix
.
, :
v ...21 (1)
, : 4
01 x , 12 x , 23 x , 34 x , ,
81 , 222 , 73 , 34 404321 .
, 5.
|
.
5
:
4.
ix
i
if
%if
0
1
2
3
| | | | | | |
| | | | | | | | | | | | | | | | |
|
| | | | | |
| | |
8
22
7
3
0,200
0,550
0,175
0,075
20,0
55,0
17,5
7,5
: 40 1,000 100,0
i ,
(relative frequency) if ix ,
i
f ii ,...,2,1, . (2)
:
(i) 10 if i ,...,2,1 i 0 .
(ii) 1...21 fff ,
1...
...... 212121
fff .
, if ,
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66
%if , ii ff 100%
. , 01 x , 12 x , 23 x , 34 x
: :
20,040
81 f , 55,0
40
222 f , 175,0
40
73 f 075,0
40
34 f
1075,0175,055,020,04321 ffff .
%20%1 f , %55%2 f , %5,17%3 f %5,7%4 f
%100%%%% 4321 ffff .
iii fx ,,
,
. , ),( ii x
),( ii fx ,
%),( ii fx , . 5
: 4.
i if
(cumulative frequencies) iN
(cumulative relative frequencies) iF ,
ix . iF 100 ,
ii FF 100% , 6. xxx ,...,, 21
,
ix ii N ...21 . ,
ii fffF ...21 , i ,...,2,1 . ,
: 4 811 N , 30212 N ,
373213 N 4043214 N ,
20,011 fF , 75,0212 ffF , 925,03213 fffF
143214 ffffF , %20%1 F , %75%2 F , %5,92%3 F
%100%4 F . :
,11 N 1122 ,..., NNNN
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67
,11 Ff 1122 ,..., FFfFFf .
6
4.
ix
i
.
.
if
.
.
if %
.
.
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1. ( ) () ()
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, :
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1956
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: The World Almanac and Book of Facts, 1994.
8. 2
km 1960 1974 :
, 63 64 64 64 66 65 65 66 66 67 67 67 67 68 68
, 1960 61 62 63 64 65 66 67 68 69 1970 71 72 73 74
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129
(: 1960 1x ).
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262
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1995, 2000.
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