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ﺔﻴﺒﻌﺸﻟﺍ ﺔﻴﻁﺍﺭﻘﻤﻴﺩﻟﺍ ﺔﻴﺭﺌﺍﺯﺠﻟﺍ ﺔﻴﺭﻭﻬﻤﺠﻟﺍ
ﺩﻌﺒ ﻥﻋ ﻥﻴﻭﻜﺘﻟﺍﻭ ﻡﻴﻠﻌﺘﻠﻟ ﻲﻨﻁﻭﻟﺍ ﻥﺍﻭﻴﺩﻟﺍ ﺔﻴﻨﻁﻭﻟﺍ ﺔﻴﺒﺭﺘﻟﺍ ﺓﺭﺍﺯﻭ ﻯﻭﺘﺴﻤﻟﺍ ﻥﺎﺤﺘﻤﺍ ﺏﺍﻭﺠ ﻡﻴﻤﺼﺘ
– ﻱﺎﻤ ﺓﺭﻭﺩ 2011
ﺔﺒﻌﺸﻟﺍﻭ ﻯﻭﺘﺴﻤﻟﺍ :
ﺔﻴﺒﻴﺭﺠﺘ ﻡﻭﻠﻋ ﻱﻭﻨﺎﺜ 3 ﺓﺩﺎﻤﻟﺍ
: ﺕﺎﻴﻀﺎﻴﺭ ﺔـــﻤﻼﻌﻟﺍ
ﺔﻠﻤﺎﻜ
ةأﺰﺠﻣ ﺔﺒﺎﺠﻹﺍ ﺭﺼﺎﻨﻋ ﺭﻭﺎﺤﻤ
ﻉﻭﻀﻭﻤﻟﺍ
ﻥ 04.5
ﻥ 01
ﻥ 01.5
ﻥ 0.5
ﻥ 01
ﻥ 0.5
1
2 4
3 3
n n
u + = u + ﻭ
0 1
u =
(1 ﻟﺍ ﻡﺴﺭ ﻰﻨﺤﻨﻤ
( )
Cfﺔﻟﺍﺩﻠﻟ لﺜﻤﻤﻟﺍ ﺙﻴﺤ f
3 : 4 3 ) 2
(x = x+ ﻡﻴﻘﺘﺴﻤﻟﺍ ﻭ f
( )
∆ ﺔﻟﺩﺎﻌﻤﻟﺍ ﻭﺫ xy=
(2 لﻴﺜﻤﺘ ﺍ ﺩﻭﺩﺤﻟ u0
، u1
ﻭ u2
:
u0 u1 u2
0 1
1
x y
(3 ﻥﻴﻤﺨﺘﻟﺍ :
ﺔﻴﻟﺎﺘﺘﻤﻟﺍ ﻥﺃ ﺭﻬﻅﻴ
( )
unﺩﺩﻌﻟﺍ ﻭﺤﻨ ﺔﺒﺭﺎﻘﺘﻤﻭ ﺓﺩﻴﺍﺯﺘﻤ .4
(4 ﻫﺭﺒ ﺎ ﻲﻌﻴﺒﻁ ﺩﺩﻋ لﻜ لﺠﺍ ﻥﻤ ﻪﻨﺃ ﻊﺠﺍﺭﺘﻟﺎﺒ ﻥ :n
1≤un ≺4 .
لﺠﺃ ﻥﻤ 0
n= ﺎﻨﻴﺩﻟ 1≤u0 ≺4 ﻥﻷ
0 1
u = .
ﻥﺃ ﺽﺭﻔﻨ 1≤un ≺4
ﻥﺃ ﻥﻫﺭﺒﻨﻭ 1≤un+1≺4
ﺎﻨﻴﺩﻟ 1≤un ≺4 ﻪﻨﻤﻭ
2 4
2 4
3un 3
≤ + ≺
ﻱﺃ 2≤un+1≺4 ﻪﻨﻤﻭ
1≤un+1≺4 لﻜﻟ ﻥﺫﺇ
n∈N : 1≤un ≺4
(5 ﺔﻴﻟﺎﺘﺘﻤﻟﺍ ﺭﻴﻐﺘ ﻩﺎﺠﺘﺍ
( )
un:
1
1 4
3 3 0
n n n
u + −u = − u + ≥
ﻪﻨﻤﻭ
( )
unﺓﺩﻴﺍﺯﺘﻤ .
ﻥﻴﺭﻤﺘﻟﺍ لﻭﻷﺍ
2 / 3 ﻥ 05
ﻥ 03
ﻥ 07.5
ﻥ 0.75 ﻥ 01
ﻥ 0.5
ﻥ 0.5
ﻥ 0.75
ﻥ 0.5
ﻥ 01
ﻥ 0.75 ﻥ 0.75 ﻥ 0.75 ﻥ 0.75 ﻥ 0.5
ﻥ 0.5
ﻥ 0.5
ﻥ 0.5
ﻥ 0.5
ﻥ 0.5
ﻥ 0.5
ﻥ 0.5
( 1 ﻠﺤ لﻭ ﺔﻟﺩﺎﻌﻤﻟﺍ :
0 4 3
2−2 Z+ =
. Z
( )
2 4 2i∆ = − = ،
A 3
Z = +i ﻭ
B 3
Z = −i
( 2 ﺃ ( ﺔﺒﺎﺘﻜ ﻥﻤ لﻜ ZA
ﻭ ZB
ﻲﺜﻠﺜﻤﻟﺍ لﻜﺸﻟﺍ ﻰﻠﻋ .
2 cos sin
6 6
ZA = ⎛ π +i π ⎞
⎜ ⎟
⎝ ⎠
ﻭ
2 cos sin
6 6
ZB = ⎛⎜⎝ ⎛⎜⎝−π ⎞⎟⎠+i ⎛⎜⎝−π ⎞⎟⎠⎞⎟⎠
ﺏ ( ﻱﺭﺒﺠﻟﺍ لﻜﺸﻟﺍ ﻟ
ﺩﺩﻌﻠ
2010
2 ⎟
⎠
⎜ ⎞
⎝
⎛ZA
:
( )
2010
cos sin 1
2 ZA
π i π
⎛ ⎞ = + = −
⎜ ⎟
⎝ ⎠
(3 لﻴﻭﺤﺘﻟﺍ ﺭﺒﺘﻌﻨ ﺔﻁﻘﻨ لﻜﺒ ﻕﻓﺭﻴ ﻱﺫﻟﺍ T
ﺎﻬﺘﻘﺤﻻ M ﺔﻁﻘﻨﻟﺍ Z
' ﺎﻬﺘﻘﺤﻻ M
' ﺙﻴﺤ Z :
Z e Z
i 3 2
'
π
. =
ﺃ ( لﻴﻭﺤﺘﻟﺍ ﺔﻌﻴﺒﻁ ﺓﺯﻴﻤﻤﻟﺍ ﻩﺭﺼﺎﻨﻋﻭ T
:
ﻩﺯﻜﺭﻤ ﻥﺍﺭﻭﺩ T ﻪﺘﻴﻭﺍﺯﻭ O
2 3 . π
ﺏ ( ﺔﻁﻘﻨﻟﺍ ﺓﺭﻭﺼ C
لﻴﻭﺤﺘﻟﺎﺒ A :T
C 3
Z = − +i
(ـﺟ
C A 3
B A
Z Z
Z Z i
− =
ﻨ − ﺞﺘﻨﺘﺴ ﻥﺃ ﺙﻠﺜﻤﻟﺍ ﻲﻓ ﻡﺌﺎﻗABC
.A
(1 ﺢﻴﺤﺼ .
( )
ﻥﻷ A∈ P( )
ﻭ B∈ P .(2 ﺄﻁﺨ .
( )
ﻥﻷ(
,)
2d D P = ≠ R
(3 ﺄﻁﺨ . ﻥﻷ ﻭ AB
ﺎﻴﻁﺨ ﻥﺎﻁﺒﺘﺭﻤ ﺭﻴﻏ AC .
(4 ﺄﻁﺨ . ﻥﻷ ﻱﺯﺍﻭﻴ ﻻAD
( )
P
e x
x x x
f( )= −( +1) −
(1
( )
lim ( ) lim x x
x f x x x xe− e−
→+∞ = →+∞ − − = +∞
(2
( ) ( )
lim ( ) lim x x 0
x f x x x xe− e−
→+∞ − = →+∞ − − =
ﻲﻨﺎﻴﺒﻟﺍ ﺭﻴﺴﻔﺘﻟﺍ
( )
C : ﺩﻨﻋ لﺌﺎﻤ ﺏﺭﺎﻘﻤ ﻡﻴﻘﺘﺴﻤ لﺒﻘﻴ ﻪﺘﻟﺩﺎﻌﻤ +∞y= x .
(3 ﺃ(
( )
' 1 x 0
f x = +xe−
ﺔﻟﺍﺩﻟﺍ ﺕﺍﺭﻴﻐﺘ f′
لﺎﺠﻤﻟﺍ ﻰﻠﻋ
[
− +∞1;[
:
( )
lim ( ) lim 1 x 1
x f x x xe−
→+∞ ′ = →+∞ + =
( ) (
1)
xf′′ x = −x e−
ﺓﺭﺎﺸﺇ
( )
f′′ x
ﻥﻴﺭﻤﺘﻟﺍ ﻲﻨﺎﺜﻟﺍ
ﻥﻴﺭﻤﺘﻟﺍ ﺎﺜﻟﺍ ﺙﻟ
ﻥﻴﺭﻤﺘﻟﺍ ﻊﺒﺍﺭﻟﺍ
3 / 3 ﻥ 0.5
ﻥ 0.5
ﻥ 0.5
ﻥ 0.5
ﻥ 01.5
ﺔﻟﺍﺩﻟﺍ ﺕﺍﺭﻴﻐﺘ لﻭﺩﺠ f′
:
ﺏ ( ﺔﻟﺩﺎﻌﻤﻟﺍ 0
) ( ' x = ﺩﻴﺤ ﻭ لﺤ لﺒﻘﺘ f
ﺤ α ﺙﻴ 56 , 0 57
,
0 −
− ≺α ≺
)
ﺔﻁﺴﻭﺘﻤﻟﺍ ﻡﻴﻘﻟﺍ ﺔﻨﻫﺭﺒﻤ .(
ﺟ (ـ ﺓﺭﺎﺸﺇ ) ( ' x ﻰﻠﻋ f لﺎﺠﻤﻟﺍ
[
− +∞1;[
.
( 4 ﺃ ( ﺔﻟﺍﺩﻟﺍ ﺕﺍﺭﻴﻐﺘ لﻭﺩﺠ : f
ﺏ ( ﻰﻨﺤﻨﻤﻟﺍ ﻡﺴﺭ
( )
C
(C)
0 1
1
x y
+∞
1
−1 x
- + 0
( )
f′′ x1+e−1
1 1−e
( )
f′ x
+∞
α
−1 x
+ 0
-
( )
f′ x+∞
α
−1 x
+ 0
-
( )
f′ x+∞
−1
( )
f α
( )
f x