Post on 07-Apr-2018
8/3/2019 ,MATHEMATICA.GR
1/30
:
2010 11
24
2011
2011: , , , ,
, , , , , , maths!, Kostas12345,Rania, , , , , .., .
:
www.mathematica.gr
8/3/2019 ,MATHEMATICA.GR
2/30
: |
2
1 (mathxl)
1 2 2009x ,x ,...,x f
f x x s x CV , s,CV
s 5
1
.
. N x 2
. : y x 1 f
3
i. s 4
ii. 1 1 2 2 2009 2009x ,y , x ,y ,..., x ,y ()
iii.
( )
) 1 :
1 2 2009x x ... x 1 1 ... 1 sx 1 s CV s2009 2009 x
f (x) 0 :
f (x) 0 x s x CV 0 x ( ,CV] [s, )
x=CV x=s
ss 2CV 2 x 2
CV
B)i : y x 1 f 3
:
2
3 sf (3) 1 3 s 3 CV 1 9 3CV 3s sCV 1 9 s 3s 1
2 2
s 5
2s 9s 20 0 s 4 s 5 0 s 4
8/3/2019 ,MATHEMATICA.GR
3/30
: |
3
ii),
1 2 2009 1 2 20091 2 2009 x 1 x 1 ... x 1 2009 x x ...xy y ... yy 1 x 1 2 12009 2009 2009
iii)
22009
i2009i 12 2 2 2 2 2
x i
i 1
x1
S x x (x) x 4 x 4 16 202009 2009
2 ( )
.
[ - ) ....... Fi%
.- 2 ................
.. -........ 30
- 6 ....... 60
-...............
1. 5 16/3.
2. F1%=10 8:
i. fi% .
ii. fi% xi.
iii. 2 6 ;
iv. 2 5 ;
v. 1 5 ;
vi. 5 ;
vii. 5;
viii. .
ix.
x. .
xi. 10%, , , .
xii. ( ) .
xiii. 23% ;
( ) ( )
1) .
t , : 2 + 2t = 6 t = 2
8/3/2019 ,MATHEMATICA.GR
4/30
8/3/2019 ,MATHEMATICA.GR
5/30
: |
5
:
iii. 2 6 20%+30%=50%
iv. 2 5 20%+15%=35%
v. 1 5 5%+20%+15%=40%
vi. 5 45%.
: x
5 60 x 6 5
2x 90 x 45%x 30 5 4
vii. 5 100%-45%=55%
viii. ( ) x 5 ,16
3 .
ix. R=8 ,
2 24 4
i i i i4 412 2 2 1
i i i i
1 1
x x1 1
s x xv
4 4 42 2 22 2 2i i
i i i i
1 1 1
f
x x x f x x x
%
=28.9-25=31 .900
s 3.9 =1.975
x. s s 1.975
CV 0.395x x 5
39.5%>10% .
xi. 10%=10/100=0.1 i i i iy =x +0.1x =1.1x
y x1.1 1.1 (16 / 3) 17.6 / 3 y 1.1 5 5.5
ys =1.1 1.975=2.1725
xii. c ,
ys 1.975
CV 0.1 0.1 0.1 c 14.75y 5 c
.
14.75
8/3/2019 ,MATHEMATICA.GR
6/30
8/3/2019 ,MATHEMATICA.GR
7/30
: |
7
: f(x) = x - 2 , :
x -3, :
x 3 x 3
f (x) f ( 3) x 2 ( 5))f ( 3) lim lim 1
x 3 x 3
. f B(0, 2) A(0, 2).
f
OA = OB =2, OAB 2 .
f xx, 1
4
, 0, x 0,
2
,
.
4 ( )
. : .h
h 0
e 1lim 1
h
.
h 1
ln hlim 1
h 1
.
h
h 1
e elim e
h 1
. : .ah
h 0
e 1lim
h
.
hh 1
ln xlim
e e
(kwstas12345)
A. ) , h h 0
0
h 0 h 0
e 1 e elim lim f 0 e 1
h h
xf x e xf x e
) h 1 h 1
ln h ln h ln1lim lim f 1 1
h 1 h 1
f x ln x 1
f x , x 0x
1f 1 1
1
) h h 1
1
h 1 h 1
e e e elim lim f 1 e e
h 1 h 1
xf x e xf x e
B. ) xf x e , x R f 0 1 R xf x e
f 0 1 (1), :
h 0 h 0
hf h f 0 e 1f 0
h 0 hlim lim
(2)
h 0
1 h
2
e 1f 0
hlim
) :hhh 1 h 1
lnx
ln x 1h 1lim lim
e ee e eh 1
8/3/2019 ,MATHEMATICA.GR
8/30
: |
8
5 ( )
3
x 1 , x 1f x x 1
3 , x 1
.
.
. f x =1
. f
C (1, f(1))
. f
( )
1) f R
2) H f x 1 ( :
).
f 1
23
x 1 x 1
x 1 x x 1x 1lim lim 3 f 1
x 1 x 1
.
f 3. f R.
3) f 1
h 0
f 1 h f 1lim
h
.
3
23 2
2 2h 0 h 0 h 0 h 0
h 1 13f 1 h f 1 h h 3h 3hhlim lim lim lim
h h h h
h 0lim h 3 3
.
f 1 f 1 3 .
4) (1,f(1)) y = x + f 1 3 (1,3)
.
3 3*1 0
y=3x.
5) f
C
8/3/2019 ,MATHEMATICA.GR
9/30
: |
9
6 ( )
f : 3
f x x x , , .
1) f [ -6, 6]
2) : 31
f x x x27
3) f.
4) f , , , .
5) , , 4 ,
.
6) , : P A f 1 P B f 4 .
:
) A B ) 27P A B 20 0
( )
1) f
[-6, -3] , [-3, 3] [3, 6].
x1 = -3 f(-3) x2 = 3 f(3) .
:
f x 0 6, 3 3, 6
f x 0 3,3
f 3 f 3 0
8/3/2019 ,MATHEMATICA.GR
10/30
: |
10
2) :
0
f (0) 0 0 0 1f ( 3) 2 27 3 2 54 6 4
27f ( 6) 2 216a 6 2 216a 6 2
1
31
f x x x,x27
3) H f
C .
4) f
C (6, 2): 11 : y f (6) f (6)(x 6) : y 3x 16
fC (-6, -2) : 22 : y f( 6) f ( 6)(x 6 : y 3x 16)
f
C (-3, 2) : 33 : y f ( 3) f ( 3)(x : y 23)
f
C (3, -2) : 44 : y f (3) f (3)(x 3) : y 2
5) :
1 2
43
1 2
3 4
3KNML
0
1 4, : 1 414
, 23
: 3 4d , M 14 128
K MN 2 KM 2 4 62 3 3
6) 26
P(A) f( 1)27
7
P(B)9
a) . :47
P(A B) P(A) P(B) 127
P(A B) 1 .
A B
b) ,47
P(A B) 1 P(A) P(B) P(A B) 1 P(A B) 1 27P(A B) 20 027
7 ( )
3
f x x 1 5 0x 2 ,
.
) f 2.
) [0, 10], .
8/3/2019 ,MATHEMATICA.GR
11/30
: |
11
) 5 ,
) ;
.
( KAKABAS BASILEIOS)
3
f x x 1 5 f (2) 6 , 2f (x) 3(x 1) f (2) 3
f
C 0x 2 f (2) f (2)(x 2) 6 3(x 2) 3x
[0-10] 5 ,
2.
,
1 2 3 4 5N 23 6, N 43 12 , N 63 18, N 83 24, N 103 30
) s 2 2
cv 0,1x 5
.
i iy x a y x a y 5 a y xs s 2 2
: yys
2 2cv 0,1 0,1 0,1 20 2 5 a 20 2 5 ay 5 a
a 24.
8 (maths-!!!)
140 .
98 56 .
:
: .
8/3/2019 ,MATHEMATICA.GR
12/30
: |
12
: .
: .
i. ,
, .
ii. , :
0.1P()0.4 .
0P()0.3 .
0.3P()0.6
( )
i. :
: : :
98 56 140
P K P A P A P P A 0,1140 140 140
P N P A 0
98 14 84
P M P A P A P A140 140 140
:
P A 0 1 P A 0 P A 1 .
II. ,
0 P A 1 154
P A P P A 1 P A 1 0,1 P A140
=()
K A 56
P K P A P 0,4140
, N A A A
980 P N P A 1 P A 1 0,3
140
, 0,1 P K 0,4 0,1 P A 0,4 0,4 P A 0,1
P A 0,4 P A P A P A 0,1 0,3 P M 0,6
9 ( )
,
8/3/2019 ,MATHEMATICA.GR
13/30
: |
13
3 21 1 3 5f (x) x x P(A B) x ,x R
12 2 2 6
P(A B) A B .
0,1,P(B ),P(A B),P(A B )
13x
25
2
5
i. P(A) P(B)
ii. 1 2
P(A B)5 5
iii. f
iv. f(1) 0 ,
( )
(1) A B B A A B A B
(2) A B A B
i. Venn :
A B B A
0 1 P B P A B P A B13 13x
25 5 25
13
1 1 P B P A P A B P B A5
13
2 P B P A P A B 1 P B A5
13
3 P B P A P A B P B P A B5
2
2P B P A5
(1)
B A B 0 P A B P B P A B 1
2 2 3
P B P B5 5 5
(2)
(1) 4
P A5
ii. : 2B P A B P B P A B 5 (3)
8/3/2019 ,MATHEMATICA.GR
14/30
: |
14
1 1 3
P A B P A P A B P A B5 5 5
(4)
A B B P A B P B
(3) (4) 1 2
P A B5 5
iii. 21 3
f x x x P A B4 2
x
: 3 1
1 P A B P A B2 2
2 1 2 1 1
P A B P A B 05 2 5 2 10
f x 0 x 1
0 04
f . R.
iv., 1 1 3 5 1
f 1 0 P A B 0 P A B 512 2 2 6 4
1 1 4 1 11
P A B P A P A B P A B P A B4 4 5 4 20
3 11 1
P B A P B P A B P B A P B A5 20 20
, : A B B A
:
1 1 6
P A B B A P A B P B A4 20 20
3
P A B B A10
10 ( )
{1,2,3,4} P(2) P(3) P(4)
P(1)2 3 4
:
={ , 2, 4, 5 , 10}
={ x 2ln(x x 1) 0 }
i.
ii. ={3,4} ={2,3,4}
8/3/2019 ,MATHEMATICA.GR
15/30
: |
15
iii. P(A B) P(A B)
iv. , P(X) A X B
( )
i. P 2 P 3 P 4
P 12 3 4
: P 1 , P 2 2 , P 3 3 , P 4 4
1
P 1 P 2 P 3 P 4 1 10 110
: 1 2 3 4
P 1 , P 2 ,P 3 , P 410 10 10 10
ii.
2v
i 2vi 12 2 2 2
i
i 1
x1 1 144
S x S 46v v 4 4
2 2 2 2 2
1 5S 46 36 S
4 2
2 2 25S 10 10 4 2
2
= 3 = 4, A 3,4
2x x 1 0 x 3 0
2 2 2ln x x 1 0 ln x x 1 ln1 x x 1 1 x x 1 0 , x < 0 x > 1 x
x =2 x =3 x = 4. , B 2,3,4
iii., A B P A B 0
A B 1,2 2,3,4 A B P A B 1
iv. P X X, A X B
.
A 3,4 B 2,3,4 X 2 , 2 1
P X P 2 P X10 5
( ):
iv .
P(A X) P(B) P(A) P(X) P(A X) P(B) (1)
8/3/2019 ,MATHEMATICA.GR
16/30
: |
16
3 4 7
P(A) P(3) P(4)10 10 10
2 3 4 9
P(B) P(2) P(3) P(4)10 10 10 10
(1) 7 9 2
P(X) P(A X) P(A X) P(X)10 10 10
2
0 P(A X) 1 0 P(X) 110
2 1
P(X)10 5
X 1
5 X {2}
11 ( )
.
)
) ;
)
)
)
( )
. , 1= 5
: 1 2 1 5v v
f fv v
, 1 5f f 0,2 1.
fi% Fi% ,
i i ix f % F %
1 20 20
2 16 36
3 14 50
4 30 80
5 20 100
,
50%50%
1,2,3,4,5
,
50% ( ) ,
. 0
i i360 f
0 0
1 5 360 0,2 72 ,0 0 0 0 0 0
2 3 4360 0,16 57,6 , 360 0,14 50,4 , 360 0,3 108
.
xi fi
1
2 0,16
3 0,14
4 0,3
5
8/3/2019 ,MATHEMATICA.GR
17/30
: |
17
. (3+4):2=3,5
4 5
. :
5
i i 5i 1
i i
i 1
x v
x x f 1,8v
. ,
52
i i 522 i 1
i i
i 1
x x v
s x x f v
52
i i 52 2 2 2i 1
i i
i 1
x v
s x x f xv
:2 356 81s 11
25 25 s 11 s 11 500 11CV% 100 100
x 1,8 9
( 100%!!).
12 ( )
2 4 , 3 15 +15 ,
5 . .
1) : A : .
2)N P(A ) . 0 .
3) 0 . 0A
: ,
: .
( )
1): 223 15 3 15
P(A )4 3 15 15
2) : 23x 15
f (x) ,x 5,x
. : P(A ) f ( ), 5,6,... .
x 5 : 2
4 3
0, x 103x 3x 15 2x x 10
f (x) 3 0, x=10x x
0, x 10
8/3/2019 ,MATHEMATICA.GR
18/30
: |
18
f x=10. : f (x) f (10) , x 5 . :
f( ) f(10) , 5 10P(A ) P(A ) , 5 . ,
=10 P(A ) . : 0 10 .
3) =10. , : 210 410 60 , :3.10-15=15
:10+15=25. : 1015 60 25
P(A ) , P(K) , P(M) .100 100 100
13 ( )
2 2 2 2 , 2 .
.
1) : A : .
2) : P(A ) P(A )
3) .
( )
1):
2 2
22 2
2 2P(A ) 22 2
.
2):
2
2
2P(A ) P(A ) P(A ) 1 P(A ) 2 1 4
2
.
3) : 2
2
x x 2f (x) ,x 2,
2x x
.
, : P(A ) f( ), 2,3,...
x 2 , :
8/3/2019 ,MATHEMATICA.GR
19/30
: |
19
2 2 2
2 22 2
0, x 2+ 3(2x 1)(2x x) (x x 2)(4x 1) x 4x 1
f (x) 0, x=2+ 3
2x x 2x x0, x 2+ 3
f 2,2 3
2 3, . , : 2 3 2 3 4 5 ... , : f(2)>f(3) f(4)
8/3/2019 ,MATHEMATICA.GR
20/30
: |
20
13,5%+14%=27,5%, : (27,5*800)/100=220
. ,
2 22 2 2 2 2i i
i
x x
s x 16 32 x 16 32 800v 800
: 22
2i16 32 800x
x 16 32 1040v 800
:4 8
2 2 2 2ix 8,653610
s x 1040 1081710 1040 100v 800
: s 10 10 1
CV1040 104
CV CV
.
15 ( )
1 2 v, , ..., .
1
P( ) x7v
* x 0 x 1 .
) x, .
) x ,
.
( )
) :v
k
k 1
1 v(v 1) 15 vP(a ) 1 xv 1 x
7v 2 14v
) x [0,1) x = 0, = 15.: kk 1
P(a ) , k 1, 2,...,15105
( ): x [0,1) ,15
0 x 1 0 1 0 15 1414
0 15 , , =1,2,3,...,15 15 v
x14v
. ,
=15 (
-) x = 0.
16 ( )
() ,
, : 225P (A) 21P(A) P(B) 5 0 (1)
8/3/2019 ,MATHEMATICA.GR
21/30
: |
21
1) .
2) : A B
( )
1) A B , : P(A B) P(A) P(B) P(A B) 1 ,
P(A) P(B) 1 : P(B) 1 P(A) . , (1), :
2 225P (A) 21P(A) 5 P(B) 1 P(A) 25P (A) 20P(A) 4 0
2 2
5P(A) 2 0 5P(A) 2 0 P A .5
(1), : 3P(B)5
.
2):2 3
P(A B) P(A) P(B) 1 P(A B) 15 5
.
, : A B . A B . N(A B) N( ) P(A B) 1 ,
1
8/3/2019 ,MATHEMATICA.GR
22/30
: |
22
( )
s 0 , 2
ii 1
1
s 0 (x x) 0
2
i 1 2(x x) 0 x x ... x x , .
) :
* P(A) 1 , f (x) x , .
* P(A) 1 , f (1 P(A))xf (x) x(1 P(A))e ,
- x 0 f (x) 0 , f , .
- x 0 f (x) 0 , f , .
) CV 0.1 s 0.1|x | (I)
(0, f(0)) 1
y xs
,
f (0) s x(1 P(A)) s (II)
* x 0 (I) s 0.1x
II x(1 P(A)) 0.1x 1 P(A) 0.1 P(A) 0.9
* x 0 (I) s 0.1x
II : x(1 P(A)) 0.1x 1 P(A) 0.1 P(A) 1.1 , .
CV=0.1 .
) M(0,x) f (0) x(1 0.9) 0.1x
() f M : y x 0.1xx (I)
() xx A 10,0 , () yy B 0,x .
() :1 1
E OAOB 10 x2 2
.
,9 9 1
P(A) 0.09E 10 x10 100 2
1
1 x x 22
: s 0.12 0.2 .
18 ( )
100 , 1 2 3 100x ,x ,x ,...,x ,
1002
i
i 1
x 646400
CV 10% .
8/3/2019 ,MATHEMATICA.GR
23/30
: |
23
1. x
2. R 160 .
) ( )
2
2 2 2
2
s sCV 0.1 CV 0.01 s 0.01x
x x .
2 2100 100 100 1002 2
i i i i10012 2 2 2 21 1 1
i1
x x x x1
s x s s x
1002
i2 2 2 21
x
0.01x x 1.01x 6464 x 6400 x 80
x 80
)( )
,
2 2 2
2 2 21 2 1002 2
1 2 100
x x x x x x
s x x x x x x 100s100
2 2 2
1 2 100x x x x x x 6400.
xi i = 1,2, , 100 :
22 2 2 2 2 2
i 1 2 100 i ix x x x x x x x x x 6400 x x 80
i ix x 80 80 x x 80.
* x 80 i i80 x 80 80 0 x 160 xi i = 1,2, , 100.
minx maxx , :
max0 x 160 1 min min min0 x 160 0 x 160 160 x 0 2
(1) (2) :
max min max min max min max min160 x x 160 x x 160 x x x x 160 R 160
* x 80 , i i80 x 80 80 160 x 0 xi i = 1,2, , 100, ,
.
8/3/2019 ,MATHEMATICA.GR
24/30
: |
24
19 ( )
, +5 , +1 , 2-3 , 7 > 0 .
) 2
2 2 20 64x 2, s5
.
) > 8 R , : 2 2R 28 10s .
) :
1) 5 : = +1 .
2) = 5 .
) > 0 : 1
ln s( )
2
.
( )
) ,5 1 2 3 7 5 10
x 25 5
,
2 2 2 2 2 2
22
2 5 2 1 2 3 2 7 2 2 5 14 2 20 64s
5 5 5
) 2- 3 ( ) 7 ( >8>7) , R 2 3 7 2 10 , 2 2 2R 28 4 40 128 10s
1) 0 4 :+1> >2-3 +1 +1
8/3/2019 ,MATHEMATICA.GR
25/30
: |
25
( ):
ln(s()) = 5
1 12 1 1 1
ln s 5 ln ln 2.8 lne2 5 2 2 2
20 ( )
{ , , , } . ,
0,1 , 0,1 , 0,1 , 0,1 s
. 0,1 , 0,1 , 0,1 , 0,1 221 s , s , CV .
( )
, 44
2 2 2 2 2 2 2 2
0,1 0,1 0,1 0,1M
0,1 0,1 0,1 0,
2 2 2 2 2 2 2 2 2 2 2 2
M M 214 4 4
2s 21
2
2 2 2 22 21 21 21 16
4 16
4 ,214 21
M16 4
2s 21 16 5 s 5 .
: 0 21 . 0 0 0
21 (xgastone)
. .
.
:
16
16 + 2x
16 ln(ex)
8/3/2019 ,MATHEMATICA.GR
26/30
: |
26
x 1/e e15
.
) ln ex
2x
)
ln exf x
2x ln ex
) x
) .
( )
) ix : iy : , :
i i i ix y
16 x y 16 (1)
i ix
16 2x x 16 2x (2)
i ix
16 2x x 16 2x (2)
(2), (3) (1) . ln ex2x
) ,
ln ex
ln ex2xP Aln ex 2x ln ex
12x
)
15
ln ex 1f (x) , x e
2x ln ex e
,
2
2 1 ln exf x ...
2x ln ex
x = 1 f
) , lne 1
P A2 lne 3
1 2
P B P A 1 P A 1 2P A3 3
22 (panathas13)
, ' 80% 60%
. ' ,
8/3/2019 ,MATHEMATICA.GR
27/30
: |
27
3/5.
) .
) ' ,
.
) 5, ' .
( )
.
:
* 0,8 0,4 ,
* 0,6 0,2 .
) (-) 3/5, :
0, 4a 0,8 3... a
a 5
) 0,6, :
0,6 0,60,3 30%
a 2
.
) : 0,2=5, = 25, 50 .
23 ( )
, , 17, . 2
16,
)
) , .
)( ) , 1 2 1w 3 4 2w , 1 2w ,w 0 , :
i) O 15 16
ii) ,
: 2 1w 3w
iii) ,
.
(sorfan)
) ( )
17 17 32 152
8/3/2019 ,MATHEMATICA.GR
28/30
: |
28
) R 2 2 2
, : ,15,17, 2 .
15 ( ), 3 4
2 17 15 ( + 2 17,
), = 15, : 15, 15, 17, 17
)( )
1 1 2 2 1 2.
1 1 2 2 1 2
15 w 15 w 17 w 17 w 15 w 17 wx
w w w w w w
i) : .15 x 17
, 1 2. 21 2
15 w 17 w15 x 15 0 w
w w
1 2. 11 2
15 w 17 wx 17 17 0 w
w w
.15 x 17
ii) , .15 15 17 17
x 164
,
. .x x 0,5 16 0,5 16,5
21 2
. 1 2 1 2 2 1 2 1
1 2
15 w 17 wx 16,5 16,5 15 w 17 w 16,5 w 16,5 w 0,5 w 1,5 w w 3w
w w
iii) ,
1 2w w
11 2 1 1. .
1 2 1 1 1
15 17 w15 w 17 w 15 w 17 wx 16 xw w w w 2w
: i iii , ,
.min maxx x x , .x : . .1 2 vw w ... w x x , 1 2 vw ,w ,...,w
8/3/2019 ,MATHEMATICA.GR
29/30
: |
29
24 ( ..)
={1,2,3,,10}
P 1 P 2 P 3 P 4 2P 5 , 1
P 9 P 10 P 12
P 5 P 6 P 7 3P 8 .
2A x / x 7x 10 0 , A B 3,4,5 , = {1, 8}
2
3 2
2x 1
4 2 x 3 / lim 9 20 11
2x 5x 3
.
)
) , , .
) A , B,A B
( , Rania )
) P(8)=m P(5)=P(6)=P(7)=3m P(1)=P(2)=P(3)=P(4)=6m P(9)=P(10)=3m
,10
i 1
1P(i) 1 m 15m 24m 1 m
40
:1
P(8)40 ,
3P(5) P(6) P(7) P(9) P(10) 40
6P(1) P(2) P(3) P(4)
40
) A x / (x 2)(x 7) 0 A 2,3,4,5,6,7
27
P(A) P(2) P(3) P(4) P(5) P(6) P(7)40
B A B A
B B (A A ) (B A ) (B A) 1,3, 4,5,8
22P(B) P(1) P(3) P(4) P(5) P(8)
40
2
2x 1
4x 4lim 1
(2x 3)(x 1)(4 2 x 3)
orner 1 vieta 1,2,6
) :
={2, 3, 4, 5, 6, 7} ' ={1, 8, 9}
={1, 3, 4, 5, 8} ' = {2, 6, 7, 9, 10}
8/3/2019 ,MATHEMATICA.GR
30/30
={1, 2, 6} ' = {3, 4, 5, 7, 8, 9, 10}
:
A 2,3,4,5,6,7,8,9,10
B 3,4,5,8
A B 9 .
6 3 1 34P(A ) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(10) 3 5
40 40 40 40
6 3 1 16
P( B) P(3) P(4) P(5) P(8) 2 40 40 40 40
3P(A B ) P(9)
40