ا ا ا
ا نا تا ةد
ا ارا :
2008 –
2009
2009/05/24 : را ر ا:ا
ة$%ا :
4
&' ت )*و
لو#ا ع%%&ا
لو#ا '&ا :
ﺔﺌﻁﺎﺨ ﺕﻨﺎﻜ ﺍﺫﺇ ﺍﺩﺎﻀﻤ ﻻﺎﺜﻤﻭ ﺔﺤﻴﺤﺼ ﺕﻨﺎﻜ ﺍﺫﺇ ﺎﻨﺎﻫﺭﺒ ﻡﺩﻗ ﺔﻴﺘﻵﺍ لﻤﺠﻝﺍ ﻥﻤ ﺔﻠﻤﺠ لﻜﻝ
(1 ﻲﻌﻴﺒﻁﻝﺍ ﺩﺩﻌﻝﺍ ﻲﻝﻭﺃ 2009
.
(2 ﻥﺍﺩﺩﻌﻝﺍ ﻭ 2009
ﺎﻤﻬﻨﻴﺒ ﺎﻤﻴﻓ ﻥﺎﻴﻝﻭﺃ 1430 .
(3 ﺔﻝﺩﺎﻌﻤﻝﺍ 2009 x + 21 y = 7
ﻲﻓ ﻼﺤ لﻗﻷﺍ ﻰﻠﻋ لﺒﻘﺘ
Z2
(4 ﺔﻝﺩﺎﻌﻤﻝﺍ لﻭﻠﺤ 24 x + 35 y = 9
ﻲﻓ
Z2
ﺕﺎﻴﺌﺎﻨﺜﻝﺍ ﻲﻫ (70k-144 ; 99 -24k)
ﺙﻴﺤ ﺢﻴﺤﺼ ﺩﺩﻋ k
.
(5 ﺩﺩﻌﻝﺍ ﻪﻴﻓ ﺏﺘﻜﻴ ﺩﺍﺩﻌﺘ ﻡﺎﻅﻨ ﺩﺠﻭﻴ لﻜﺸﻝﺍ ﻰﻠﻋ 2009
. 809
'&ا ()ا : ABCDEFGH ﺙﻴﺤ ﺕﻼﻴﻁﺘﺴﻤ ﻱﺯﺍﻭﺘﻤ
: AB = AE = 2 ﻭ
AD = 4 .
ﻲﻤﺴﻨ ﺯﻜﺭﻤ I
ﻊﺒﺭﻤﻝﺍ ﻭ ABFE
ﺔﻌﻁﻘﻝﺍ ﻑﺼﺘﻨﻤ J . [EH]
ﺱﻨﺎﺠﺘﻤﻝﺍ ﺩﻤﺎﻌﺘﻤﻝﺍ ﻡﻠﻌﻤﻝﺍ ﻰﻝﺇ ﺀﺎﻀﻔﻝﺍ ﺏﺴﻨﻴ
1 1 1
; ; ;
2 4 2
A AB AD AE
uuur uuur uuur
* (1 ﻁﻘﻨﻝﺍ ﻥﻤ ﺔﻁﻘﻨ لﻜ ﺕﺎﻴﺜﺍﺩﺤﺇ ﻥﻴﻋ ، B
، C ، E ، F ﻡﺜ H ﻭ I . J
* ﻥﻴﻋﺎﻌﺸﻝﺍ ﻥﻤ ﻉﺎﻌﺸ لﻜ ﺕﺎﺒﻜﺭﻤ ﻥﻴﻋ
IJ
ﻭ uur JC
uuur
* ﻉﺎﻌﺸﻝﺍ ﻥﺃ ﻥﻴﺒ
AF uuur
ﻱﻭﺘﺴﻤﻠﻝ ﻡﻅﺎﻨ ﻉﺎﻌﺸ (IJC)
.
* ﻱﻭﺘﺴﻤﻠﻝ ﺔﻴﺘﺭﺎﻜﻴﺩ ﺔﻝﺩﺎﻌﻤ ﻥﻴﻋ
(IJC) ﻁﻘﻨﻝﺍ ﻥﺃ ﻕﻘﺤﺘ ﻡﺜ ، B
، C
، E
ﻪﻴﻝﺇ ﻲﻤﺘﻨﺘ H
.
(2 ﻲﻤﺴﻨ (Γ) ﻁﻘﻨﻝﺍ ﺔﻋﻭﻤﺠﻤ ﺙﻴﺤ ﺀﺎﻀﻔﻝﺍ ﻥﻤ M
:
2 2 2 2
48 MB +MC +ME +MH =
* ﻥﺃ ﻥﻴﺒ
(Γ) ﺎﻫﺯﻜﺭﻤ ﺕﺎﻴﺜﺍﺩﺤﺇ ﺩﻴﺩﺤﺘ ﺏﻠﻁﻴ ﺓﺭﻜ ﺢﻁﺴ ω
ﺎﻫﺭﻁﻗ ﻑﺼﻨﻭ .
* ﺔﻁﻘﻨﻝﺍ ﻥﺃ ﻕﻘﺤﺘ
ω ﺙﻠﺜﻤﻝﺍ لﻘﺜ ﺯﻜﺭﻤ . IJC
* ﺓﺭﺌﺍﺩﻝﺍ ﺯﻜﺭﻤ ﺕﺎﻴﺜﺍﺩﺤﺇ ﻭ ﺭﻁﻘﻝﺍ ﻑﺼﻨ ﻥﻴﻋ
(γ) لﻴﻁﺘﺴﻤﻝﺎﺒ ﺔﻁﻴﺤﻤﻝﺍ . EBCH
* ﺓﺭﺌﺍﺩﻠﻝ ﺎﻴﺘﺭﺎﻜﻴﺩ ﻼﻴﺜﻤﺘ ﺞﺘﻨﺘﺴﺍ
(γ) .
ﺙﻝﺎﺜﻝﺍ ﻥﻴﺭﻤﺘﻝﺍ :
(1
ﺔﺒﻜﺭﻤﻝﺍ ﺩﺍﺩﻌﻝﺍ ﺔﻋﻭﻤﺠﻤ ﻲﻓ لﺤ لﻭﻬﺠﻤﻝﺍ ﺕﺍﺫ ﺔﻝﺩﺎﻌﻤﻝﺍ C
: z
2 1 0
z + + =z
.
ﻲﻤﺴﻨ ﺏﺠﻭﻤ ﻲﻠﻴﺨﺘﻝﺍ ﻩﺅﺯﺠ ﻱﺫﻝﺍ لﺤﻝﺍ j
.
(2
ﻥﻴﺩﺩﻌﻝﺍ ﺏﺘﻜﺍ ﻭ j
1
ﻲﺴﻷﺍ لﻜﺸﻝﺍ ﻰﻠﻋ j
.
(3
ﺱﻨﺎﺠﺘﻤﻝﺍ ﺩﻤﺎﻌﺘﻤﻝﺍ ﻡﻠﻌﻤﻝﺍ ﻰﻝﺇ ﺏﻭﺴﻨﻤ ﺏﻜﺭﻤﻝﺍ ﻱﻭﺘﺴﻤﻝﺍ
(
O ; ; ur vr)
. ﻥﻴﺘﻁﻘﻨﻝﺍ ﺭﺒﺘﻌﻨ ﺎﻬﺘﻘﺤﻻ A
α = +2 i
ﻭ ﺎﻬﺘﻘﺤﻻ M
. z ﻲﻤﺴﻨ ﺔﻘﺤﻼﻝﺍ ﺕﺍﺫ ﺔﻁﻘﻨﻝﺍ B
β α= j
ﻭ ﺓﺭﻭﺼ M’
ﻩﺯﻜﺭﻤ ﻱﺫﻝﺍ ﻥﺍﺭﻭﺩﻝﺎﺒ M O
ﻪﺘﻴﻭﺍﺯﻭ
2 3
− π
/1
4
• ﻥﻋ ﺭﺒﻋ ﺔﻘﺤﻻ z’
ﺔﻝﻻﺩﺒ M’
ﻭ z . j
ﺩﺩﻌﻝﺍ ﻱﺭﺒﺠﻝﺍ لﻜﺸﻝﺍ ﻰﻠﻋ ﺏﺘﻜﺍ •
' z z
β α
−
. −
• ﺩﺩﻌﻝﺍ ﺓﺩﻤﻋﻭ ﺔﻠﻴﻭﻁ ﻥﻴﻋ
' z z
β α
−
. −
ﺎﻴﺴﺩﻨﻫ ﻥﻴﺘﺠﻴﺘﻨﻝﺍ ﺭﺴﻓ .
• ﻥﻴﺘﻁﻘﻨﻝﺍ ﻡﺴﺭﻝﺍ ﻰﻠﻋ ﻥﻴﻋ ﻭ B
ﺎﻤﻝ M’
1 3 z= + i
.
ﻊﺒﺍﺭﻝﺍ ﻥﻴﺭﻤﺘﻝﺍ :
لﻭﻷﺍ ﺀﺯﺠﻝﺍ
ϕ : ﻰﻠﻋ ﺔﻓﺭﻌﻤﻝﺍ ﺔﻴﺩﺩﻌﻝﺍ ﺔﻝﺍﺩﻝﺍ ﻜ R
ﻲﻠﻴﺎﻤ
2
2(x +1).e−x 1−
= ϕ (x)
(1 * ﺔﻴﺎﻬﻨ ﺏﺴﺤﺍ ﺩﻨﻋ ϕ
ﺩﻨﻋ ﻭ -∞
. + ∞
* ﺔﻝﺍﺩﻝﺍ ﺕﺍﺭﻴﻐﺘ ﻩﺎﺠﺘﺍ ﺱﺭﺩﺍ ﺎﻬﺘﺍﺭﻴﻐﺘ لﻭﺩﺠ ﺯﺠﻨﺃ ﻡﺜ ϕ
(2
* ﺔﻝﺩﺎﻌﻤﻝﺍ ﻥﺃ ﻥﻴﺒ
(x) = 0
ﺍﺩﻴﺤﻭ ﻼﺤ لﺒﻘﺘ ϕ
لﺎﺠﻤﻝﺍ ﻰﻝﺇ ﻲﻤﺘﻨﻴ α [2 ;3]
ﻥﻴﻋ ﻡﺜ
ﺩﻌﻠﻝ ﺭﺼﺤ
ﺩ
α ﻪﺘﻌﺴ
10−1
.
* ﺓﺭﺎﺸﺇ لﻭﺩﺠ ﺯﺠﻨﺃ
ϕ(x)
.
ﻲﻨﺎﺜﻝﺍ ﺀﺯﺠﻝﺍ ) :
ﺔﺤﺎﺴﻤ ﺏﺎﺴﺤ ﻭ ﻥﻴﻴﻨﺤﻨﻤ ﺔﻴﻌﻀﻭ ﺔﺴﺍﺭﺩ (
ﺔﻝﺍﺩﻠﻝ لﻭﻷﺍ ، ﻥﻴﻴﻨﺎﻴﺒﻝﺍ ﻥﻴﻠﻴﺜﻤﺘﻝﺍ ﻉﻭﻀﻭﻤﻝﺍ ﺭﺨﺁ ﻲﻓ ﻰﻁﻌﺘ ﺔﻝﺍﺩﻠﻝ ﻲﻨﺎﺜﻝﺍ ﻭ f
ﻥﻴﺘﻓﺭﻌﻤﻝﺍ g
ﻰﻠﻋ R
ﻜ ﻲﻠﻴ ﺎﻤ
:
( ) 4 . x f x = x e−
ﻭ
2
( ) 2
1 g x x
= x
+
ﻲﻤﺴﻨ ( )Cf
ﻰﻨﺤﻨﻤ ﻭ f
( )
Cgﻰﻨﺤﻨﻤ . g
ﺱﻨﺎﺠﺘﻤ ﺩﻤﺎﻌﺘﻤ ﻡﻠﻌﻤ ﻲﻓ ( , , )o i j
r uur )
ﺓﺩﺤﻭﻝﺍ : 2cm
(
(1
* ﻴﻨﺤﻨﻤﻝﺍ ﻥﺃ ﻥﻴﺒ ﺔﻁﻘﻨﻝﺍ ﻥﻼﻤﺸﻴ ﻥﻴ
ﻡﻠﻌﻤﻝﺍ ﺃﺩﺒﻤ o
.
* ﻥﻤ لﻜﻝ ﺱﺎﻤﻤﻝﺍ ﺔﻝﺩﺎﻌﻤ ﺏﺘﻜﺍ ( )Cf
( )
Cg ﻭ ﺔﻁﻘﻨﻝﺍ ﺩﻨﻋ .o
(2
* ﻲﻘﻴﻘﺤ ﺩﺩﻋ لﻜ لﺠﺃ ﻥﻤ ﻪﻨﺍ ﻥﻴﺒ : x
2
2 ( ) ( ) ( )
1 x x g x f x
x ϕ
− = −
ﺙﻴﺤ ، +
ﺔﺴﻭﺭﺩﻤﻝﺍ ﺔﻝﺍﺩﻝﺍ ϕ
لﻭﻷﺍ ﺀﺯﺠﻝﺍ ﻲﻓ .
* ﺓﺭﺎﺸﺇ ﺱﺭﺩﺍ
g(x) – f(x)
*
ﻥﻴﻴﻨﺤﻨﻤﻠﻝ ﺔﻴﺒﺴﻨﻝﺍ ﺔﻴﻌﻀﻭﻝﺍ ﺞﺘﻨﺘﺴﺍ ( )Cf
( )
Cg ﻭ .* (3
ﺔﻝﺍﺩﻝﺍ ﻥﺃ ﻥﻴﺒ ﻰﻠﻋ ﺔﻓﺭﻌﻤﻝﺍ h
ﻲﻠﻴﺎﻤﻜ R
2 -x :
ln(x +1) + (4x+4).e
h(x) =
ﺔﻴﻠﺼﺃ ﺔﻝﺍﺩ
ﺔﻝﺍﺩﻠﻝ
g(x) – f(x)
ﻰﻠﻋ .R
* ﻡﺴﺭﻝﺍ ﻲﻓ لﻠﻀﻤﻝﺍ ﻱﻭﺘﺴﻤﻝﺍ ﺯﻴﺤﻝﺍ ﺔﺤﺎﺴﻤﻝ ﺓﺩﺤﻭﻝﺍ ﻰﻝﺇ ﺔﻴﺒﻴﺭﻘﺘﻝﺍ ﻡﺜ ﺔﻁﻭﺒﻀﻤﻝﺍ ﺔﻤﻴﻘﻝﺍ ﺞﺘﻨﺘﺴﺍ
/2
4
ﻝﺍ
ﻲﻨﺎﺜﻝﺍ ﻉﻭﻀﻭﻤ :
لﻭﻷﺍ ﻥﻴﺭﻤﺘﻝﺍ :
(1 ﻲﻌﻴﺒﻁ ﺩﺩﻋ لﻜ لﺠﺃ ﻥﻤ ﻪﻨﺃ ﻥﻴﺒ ﺍ n
ﺩﺩﻌﻝ
3n3−11n+48
ﻰﻠﻋ ﺔﻤﺴﻘﻝﺍ لﺒﻘﻴ
3 n+
(2 ﻲﻌﻴﺒﻁ ﺩﺩﻋ لﻜ لﺠﺃ ﻥﻤ ﻪﻨﺃ ﻥﻴﺒ ﺩﺩﻌﻝﺍ n
3n2−9n+16
ﻡﻭﺩﻌﻤ ﺭﻴﻏ ﻲﻌﻴﺒﻁ ﺩﺩﻋ .
* (3 ﻲﻌﻴﺒﻁ ﺩﺩﻋ لﻜ لﺠﺃ ﻥﻤ ﻪﻨﺃ ﻥﻴﺒ ﻱﻭﺎﺴﻴ ﻭﺃ ﺭﺒﻜﺍ n
: 2
(
3 3 11 , n + 3)
(48 , n + 3)PGCD n − n =PGCD
) ( , b)
PGCD a
ﻥﻴﻴﻌﻴﺒﻁﻝﺍ ﻥﻴﺩﺩﻌﻠﻝ ﺭﺒﻜﻷﺍ ﻙﺭﺘﺸﻤﻝﺍ ﻡﺴﺎﻘﻠﻝ ﺯﻤﺭﻴ ﻭ a
( b
* ﺩﺩﻌﻠﻝ ﺔﻴﻌﻴﺒﻁﻝﺍ ﻡﺴﺍﻭﻘﻝﺍ ﺔﻋﻭﻤﺠﻤ ﻥﻴﻋ
. 48
* ﺔﻴﻌﻴﺒﻁﻝﺍ ﺩﺍﺩﻋﻷﺍ ﺔﻋﻭﻤﺠﻤ ﺞﺘﻨﺘﺴﺍ
ﺩﺩﻌﻝﺍ ﻥﻭﻜﻴ ﺎﻬﻠﺠﺃ ﻥﻤ ﻲﺘﻝﺍ n
3 3 11 3
n n
n
−
ﺎﻴﻌﻴﺒﻁ +
.
ﻲﻨﺎﺜﻝﺍ ﻥﻴﺭﻤﺘﻝﺍ :
ﺎﻋﻭ
1 ﺀ ﻡﻀﻴ A
ﻊﺒﺭﺍ ﺔﻤﻗﺭﻤ ﺕﺎﺼﻴﺭﻗ ، 9
، 9 ، 0 ﺀﺎﻋﻭ ﻭ 0
A2
ﺔﻤﻗﺭﻤ ﺕﺎﺼﻴﺭﻗ ﻊﺒﺭﺃ ﻡﻀﻴ
، 9
، 0 ، 0 . 1 ﻴﺌﺍﻭﺸﻋ ﺭﺎﺘﺨﻨ ﺎ
ﺔﺼﻴﺭﻗ ﻪﻨﻤ ﺏﺤﺴﻨ ﻡﺜ ﻥﻴﺌﺎﻋﻭﻝﺍ ﺩﺤﺃ .
ﺏﺤﺴ ﺔﺠﻴﺘﻨ لﻜﺒ ﻕﻓﺭﻨ
ﺩﺩﻌﻝﺍ ﺙﻴﺤ 200a
ﺔﺒﻭﺤﺴﻤﻝﺍ ﺔﺼﻴﺭﻘﻝﺍ ﻪﻠﻤﺤﺘ ﻱﺫﻝﺍ ﻡﻗﺭﻝﺍ ﻭﻫ a .
* (1
ﺏﺴﺎﻨﻤﻝﺍ لﺎﻤﺘﺤﻻﺍ ﻡﺎﻬﻔﺘﺴﻻﺍ ﺔﻤﻼﻋ ﻥﺎﻜﻤ ﻲﻓ ﻊﻀﻭﺒ ﺔﻴﺘﻵﺍ ﺕﻻﺎﻤﺘﺤﻻﺍ ﺓﺭﺠﺸ لﻤﻜﺃ
0
1
2
1 A 1
2
؟ 9
؟ 0
؟
؟ 1
A2
9
1 4
ﺔﺜﺩﺎﺤﻝﺍ لﺎﻤﺘﺤﺍ ﺏﺴﺤﺍ •
" A ﺩﺩﻌﻝﺍ ﻱﻭﺎﺴﻴ N
" 2009
ﺩﺩﻌﻝﺍ ﻥﺃ ﺎﻤﻠﻋ • ﻱﻭﺎﺴﻴ N
ﺀﺎﻋﻭﻝﺍ ﻥﻤ ﺔﺒﻭﺤﺴﻤﻝﺍ ﺔﺼﻴﺭﻘﻝﺍ ﻥﻭﻜﺘ ﻥﺃ لﺎﻤﺘﺤﺍ ﻭﻫﺎﻤ ، 2009
A1
.
(2 ـﺒ ﺯﻤﺭﻨ ﺩﺩﻌﻝﺍ ﻡﺎﻗﺭﺃ ﻉﻭﻤﺠﻤ ﺏﺤﺴ ﺔﺠﻴﺘﻨ لﻜﺒ ﻕﻓﺭﻴ ﻱﺫﻝﺍ ﻲﺌﺍﻭﺸﻌﻝﺍ ﺭﻴﻐﺘﻤﻠﻝ X
N
* ﺭﻴﻐﺘﻤﻠﻝ ﺔﻨﻜﻤﻤﻝﺍ ﻡﻴﻘﻝﺍ ﻥﻴﻋ
. X
* ﺭﻴﻐﺘﻤﻝﺍ لﺎﻤﺘﺤﺍ ﻥﻭﻨﺎﻗ ﻥﻴﻋ
ﻲﻀﺎﻴﺭﻝﺍ ﻪﻠﻤﺃ ﺏﺴﺤﺃﻭ X .
ﺙﻝﺎﺜﻝﺍ ﻥﻴﺭﻤﺘﻝﺍ :
ﺎﻤﺎﻤﺘ ﺏﺠﻭﻤ ﻲﻘﻴﻘﺤ ﺩﺩﻋ a .
(1 ﻷﺍ ﺔﻋﻭﻤﺠﻤ ﻲﻓ لﺤ ﺔﺒﻜﺭﻤﻝﺍ ﺩﺍﺩﻋ
لﻭﻬﺠﻤﻝﺍ ﺕﺍﺫ ﺔﻝﺩﺎﻌﻤﻝﺍ C
: z
(
4z2−a2) (2z− 3ai)
=0
(2 ﻝﺇ ﺏﻭﺴﻨﻤﻝﺍ ﺏﻜﺭﻤﻝﺍ ﻱﻭﺘﺴﻤﻝﺍ ﻲﻓ ﺱﻨﺎﺠﺘﻤﻝﺍ ﺩﻤﺎﻌﺘﻤﻝﺍ ﻡﻠﻌﻤﻝﺍ ﻰ
(
O ; ; ur vr)
ﻁﻘﻨﻝﺍ ﺭﺒﺘﻌﻨ ، ، A
، B C
ﺏﻴﺘﺭﺘﻝﺍ ﻰﻠﻋ ﺎﻬﻘﺤﺍﻭﻝ ﻲﺘﻝﺍ
3 2 ai 2 ،
−a 2 ﻭ a
/3 4
• ﺩﺩﻌﻝﺍ ﻲﺴﻷﺍ لﻜﺸﻝﺍ ﻰﻠﻋ ﺏﺘﻜﺍ :
A B
C A
z z z z
−
−
• ﻥﺍﺭﻭﺩﻝﺍ ﺔﻴﻭﺍﺯ ﺞﺘﻨﺘﺴﺍ لﻭﺤﻴ ﻱﺫﻝﺍ R
ﻰﻝﺇ A لﻭﺤﻴﻭ B ﻰﻝﺇ C
. A
ﻕﻘﺤﺘ • ﻥﺃ ﻥﺍﺭﻭﺩﻝﺍ ﺯﻜﺭﻤ ﺙﻠﺜﻤﻝﺍ لﻘﺜ ﺯﻜﺭﻤ ﻊﻤ ﻕﺒﻁﻨﻴ R
. ABC
(3 ﻲﻤﺴﻨ (Γ) ﻁﻘﻨﻝﺍ ﺔﻋﻭﻤﺠﻤ ﺙﻴﺤ ﻱﻭﺘﺴﻤﻝﺍ ﻥﻤ M
2 2 2 2 :
2 MA +MB +MC = a
* ﻁﻘﻨﻝﺍ ﻥﺃ ﻕﻘﺤﺘ
، A ﻭ B ﺔﻋﻭﻤﺠﻤﻝﺍ ﻰﻝﺇ ﻲﻤﺘﻨﺘ C (Γ)
.
* ﺔﻌﻴﺒﻁ ﻥﻴﻋ
(Γ) ﺎﻫﺭﺼﺎﻨﻋﻭ .
* لﺠﺃ ﻥﻤ لﻜﺸﻝﺍ ﻡﺴﺭﺍ
a = 4 ) ﺓﺩﺤﻭﻝﺍ : .( 2 cm
ﻊﺒﺍﺭﻝﺍ ﻥﻴﺭﻤﺘﻝﺍ :
ﻰﻠﻋ ﺔﻓﺭﻌﻤﻝﺍ ﺔﻴﺩﺩﻌﻝﺍ ﺔﻝﺍﺩﻝﺍ f
[
0 ; + ∞[
ﻲﺘﺄﻴ ﺎﻤﻜ :
f(0)=0
لﺠﺃ ﻥﻤﻭ ﻰﻝﺇ ﻲﻤﺘﻨﻴ x
]
0 ; + ∞[
( ) :( ) 2 1 2 ln( ) f x =x − x
ﻲﻤﺴﻨ
( )
Cfﻰﻨﺤﻨﻤ ﺱﻨﺎﺠﺘﻤﻝﺍ ﺩﻤﺎﻌﺘﻤﻝﺍ ﻡﻠﻌﻤﻝﺍ ﻰﻝﺇ ﺏﻭﺴﻨﻤﻝﺍ ﻱﻭﺘﺴﻤﻝﺍ ﻲﻓ f
(
O ; ; ri rj)
.
* (1 ﺔﻴﺎﻬﻨ ﺏﺴﺤﺍ ﺩﻨﻋ f
+ ∞
* ﻥﺃ ﻥﻴﺒ
0
lim ( ) 0
x
f x
→ x =
ﺎﻴﺴﺩﻨﻫ ﺔﺠﻴﺘﻨﻝﺍ ﺭﺴﻓ ،
* ﻥﺃ ﻥﻴﺒ
ﻰﻠﻋ ﻕﺎﻘﺘﺸﻼﻝ ﺔﻠﺒﺎﻗ f
]
0 ; + ∞[
ﻥﺃﻭ
'( ) 4 ln( ) f x = − x x
)
'
ـﻝ ﺔﻘﺘﺸﻤﻝﺍ ﺔﻝﺍﺩﻠﻝ ﺯﻤﺭﺘ f
( f
* ﺭﻴﻐﺘ ﻩﺎﺠﺘﺍ ﺱﺭﺩﺍ
ﺎﻬﺘﺍﺭﻴﻐﺘ لﻭﺩﺠ لﻜﺸ ﻡﺜ f .
* (2 ﻊﻁﺎﻘﺘ ﻲﺘﻁﻘﻨ ﺕﺎﻴﺜﺍﺩﺤﺇ ﻥﻴﻋ
( )
Cfلﺼﺍﻭﻔﻝﺍ ﺭﻭﺤﻤ ﻊﻤ
* ﺱﺎﻤﻤﻠﻝ ﺔﻝﺩﺎﻌﻤ ﺏﺘﻜﺍ
ـﻝ (d)
( )
Cfﺔﻠﺼﺎﻔﻝﺍ ﺕﺍﺫ ﺔﻁﻘﻨﻝﺍ ﺩﻨﻋ e
* ﻰﻨﺤﻨﻤﻝﺍ ﺊﺸﻨﺃ
( )
Cfﺱﺎﻤﻤﻝﺍﻭ . (d)
) ﺓﺩﺤﻭﻝﺍ : ( 3 cm
* (3 ﺔﻝﺍﺩﻝﺍ ﻥﺃ ﻥﻴﺒ ﺔﻓﺭﻌﻤﻝﺍ g
( ) ـﺒ
3
( ) 5 6 ln( ) 9
g x = x − x
ﺔﻝﺍﺩﻠﻝ ﺔﻴﻠﺼﺃ ﺔﻝﺍﺩ ﻰﻠﻋ f
]
0 ; + ∞[
.
* لﺎﺠﻤﻝﺍ ﻥﻤ ﻲﻘﻴﻘﺤ ﺩﺩﻋ λ
0 ; e
. ﺔﻝﻻﺩﺒ ﺏﺴﺤﺍ ﺔﺤﺎﺴﻤﻝﺍ λ
( ) S λ
ﻰﻨﺤﻨﻤﻝﺎﺒ ﺩﺩﺤﻤﻝﺍ ﻱﻭﺘﺴﻤﻝﺍ ﺯﻴﺤﻠﻝ
( )
Cfﺎﻤﻬﻴﺘﻝﺩﺎﻌﻤ ﻥﻴﺫﻠﻝﺍ ﻥﻴﻤﻴﻘﺘﺴﻤﻝﺍﻭ لﺼﺍﻭﻔﻝﺍ ﺭﻭﺤﻤ ﻭ
x=λ
ﻭ
x= e
.
• ﺔﻴﺎﻬﻨ ﺏﺴﺤﺍ
( ) S λ
ﺎﻤﻝ λ ﻰﻝﺇ لﻭﺅﻴ . 0
/ـــــــــا
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