cours

1. ‫اﻟﻌﺪدﻳﺔ‬ ‫اﻟﺪوال‬ ‫ﺣﻮل‬ ‫ﻋﻤﻮﻣﻴﺎت‬
I-‫اﻟﺪاﻟﺔ‬–‫داﻟﺘﻴﻦ‬ ‫ﺗﺴﺎوي‬–‫ﻟﺪاﻟﺔ‬ ‫اﻟﻤﺒﻴﺎﻧﻲ‬ ‫اﻟﺘﻤﺜﻴﻞ‬
1/‫داﻟﺔ‬ ‫ﺗﻌﺮﻳﻒ‬–‫داﻟﺔ‬ ‫ﺗﻌﺮﻳﻒ‬ ‫ﻣﺠﻤﻮﻋﺔ‬
‫ﻧﺸﺎط‬
‫اﻟﻌﺪدﻳﺔ‬ ‫اﻟﺪاﻟﺔ‬ ‫ﺗﻌﺮﻳﻒ‬ ‫ﻣﺠﻤﻮﻋﺔ‬ ‫ﺣﺪد‬f‫اﻟﺤﻘﻴﻘﻲ‬ ‫ﻟﻠﻤﺘﻐﻴﺮ‬x‫اﻟﺘﺎﻟﻴ‬ ‫اﻟﺤﺎﻻت‬ ‫ﻓﻲ‬‫ﺔ‬
( ) ( )
( ) ( )
2
2
5
2 1 ( ; (
4
3 3 2
( ; (
2 33
f x x b f x a
x
x x
f x d f x c
x xx
= + =
−
−
= =
+ −−
‫اﻟﺤﻞ‬
( )
5
(
4
f x a
x
=
−
‫ﻟﺘﻜﻦ‬x ∈
fx D∈‫ﺗﻜﺎﻓﺊ‬4 0x− ≠
‫ﺗﻜﺎﻓﺊ‬4x ≠
‫اذن‬{ }4fD = −
( ) 2 1 (f x x b= +
fx D∈‫ﺗﻜﺎﻓﺊ‬2 1 0x +
‫ﺗﻜﺎﻓﺊ‬
1
2
x ≥ −
‫اذن‬
1
;
2
fD
 
= − +∞  
( )
2
2
3 2
; (
2 3
x
f x c
x x
−
=
+ −
fx D∈‫ﺗﻜﺎﻓﺊ‬2
2 3 0x x+ − ≠
‫ﻟﻴﻜﻦ‬∆‫اﻟﺤﺪود‬ ‫ﺛﻼﺛﻴﺔ‬ ‫ﻣﻤﻴﺰ‬2
2 3x x+ −
4 12 16∆ = + =
‫ﻟـ‬ ‫ﻣﻨﻪ‬ ‫و‬2
2 3x x+ −‫هﻤﺎ‬ ‫ﺟﺮﻳﻦ‬1
2 16
1
2
x
− +
= =‫و‬1
2 16
3
2
x
− −
= = −
‫إذن‬{ }3;1fD = − −
‫ﺗﻌﺮﻳﻒ‬
‫ﻋﺮﻓﻨﺎ‬ ‫اﻧﻨﺎ‬ ‫ﻧﻘﻮل‬‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬f‫ﻣﻦ‬ ‫ﻋﺪد‬ ‫آﻞ‬ ‫رﺑﻄﻨﺎ‬ ‫اذا‬‫ﻧﺮﻣﺰ‬ ‫ﺣﻘﻴﻲ‬ ‫ﺑﻌﺪد‬ ‫اﻻآﺜﺮ‬ ‫ﻋﻠﻰ‬
‫ﺑـ‬ ‫ﻟﻪ‬( )f x.
( )f x‫ﺻﻮرة‬ ‫ﺗﻘﺮأ‬x‫ﺑﺎﻟﺪاﻟﺔ‬f‫ﺑﺎﺧﺘﺼﺎر‬ ‫أو‬f‫ﻟـ‬x
‫ﺗﻌﺮﻳﻒ‬
‫ﻟﺘﻜﻦ‬f‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬.
‫اﻟﺪاﻟﺔ‬ ‫ﺗﻌﺮﻳﻒ‬ ‫ﻣﺠﻤﻮﻋﺔ‬f‫ﺑﺎﻟﺪاﻟﺔ‬ ‫ﺻﻮرة‬ ‫ﺗﻘﺒﻞ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺤﻘﻴﻘﻴﺔ‬ ‫اﻷﻋﺪاد‬ ‫ﺟﻤﻴﻊ‬ ‫ﻣﻦ‬ ‫اﻟﻤﻜﻮﻧﺔ‬ ‫اﻟﻤﺠﻤﻮﻋﺔ‬ ‫هﻲ‬f
‫ﺑـ‬ ‫ﻟﻬﺎ‬ ‫ﻧﺮﻣﺰ‬fD
2-‫داﻟﺘﻴﻦ‬ ‫ﺗﺴﺎوي‬
‫ﻧﺸﺎط‬
‫اﻟﻌﺪدﻳﺘﻴﻦ‬ ‫اﻟﺪاﻟﺘﻴﻦ‬ ‫ﻗﺎرن‬f‫و‬g‫اﻟﺘﺎﻟﻴﺘﻴﻦ‬ ‫اﻟﺤﺎﻟﺘﻴﻦ‬ ‫ﻓﻲ‬ ‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬
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2. ( ) ( )
( ) ( )
( )
2
1
1 ; (
1
1 1 2
; (
2 2
x
f x x g x a
x
f x g x b
x x x x
−
= − =
+
= − =
+ +
a/‫ﻟﺪﻳﻨﺎ‬fD =‫و‬{ }1gD = −
‫وﻣﻨﻪ‬f gD D≠‫اذن‬f g≠
b/‫ﻟﺪﻳﻨﺎ‬{ }*
2g fD D= = −
‫ﻟﺘﻜﻦ‬{ }*
2x ∈ −
( )
( ) ( )
( )
1 1 2 2
2 2 2
x x
f x g
x x x x x x
+ −
= − = = =
+ + +
‫إذن‬f g=
‫ﺗﻌﺮﻳﻒ‬
‫ﻟﺘﻜﻦ‬f‫و‬g‫ﺣﻘﻴﻘ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺘﻴﻦ‬ ‫داﻟﺘﻴﻦ‬‫ﻲ‬
‫ﺗﻜﻮن‬f‫و‬g‫اﻟﺘﻌﺮﻳﻒ‬ ‫ﻣﺠﻤﻮﻋﺔ‬ ‫ﻧﻔﺲ‬ ‫ﻟﻬﻤﺎ‬ ‫آﺎن‬ ‫اذا‬ ‫وﻓﻘﻂ‬ ‫اذا‬ ‫ﻣﺘﺴﺎوﻳﺘﻴﻦ‬D‫ﻟﻜﻞ‬ ‫و‬x‫ﻣﻦ‬D
( ) ( )f x g x=
3-‫اﻟﻤﺒ‬ ‫اﻟﺘﻤﺜﻴﻞ‬‫ﻟﺪاﻟﺔ‬ ‫ﻴﺎﻧﻲ‬
‫ﻧﺸﺎط‬
‫اﻟﻌﺪدﻳﺔ‬ ‫اﻟﺪاﻟﺔ‬ ‫ﻧﻌﺘﺒﺮ‬f‫اﻟﺤﻘﻴﻘﻲ‬ ‫ﻟﻠﻤﺘﻐﻴﺮ‬x‫ﺣﻴﺚ‬( )
2
4
2
x
f x
x
−
=
−
‫أ‬-‫ﺣﺪد‬fD
‫ب‬-‫أرﺗﻮﺑﻲ‬ ‫ﺣﺪد‬A‫و‬B‫اﻟﻤﻨﺤﻨﻰ‬ ‫ﻣﻦ‬ ‫ﻧﻘﻄﺘﻴﻦ‬fC‫اﻟﺘﻮاﻟﻲ‬ ‫ﻋﻠﻰ‬ ‫أﻓﺼﻮﻟﻴﻬﻤﺎ‬0‫و‬3
‫ج‬-‫اﻟﻨﻘﻂ‬ ‫هﻞ‬( ) ( ) ( )4; 6 ; 4;6 ; 2;0E D C− −‫إﻟﻰ‬ ‫ﺗﻨﺘﻤﻲ‬fC
‫د‬–‫أآﺘﺐ‬( )f x‫ﺛﻢ‬ ‫اﻟﻤﻄﻠﻘﺔ‬ ‫ﻟﻠﻘﻴﻤﺔ‬ ‫رﻣﺰ‬ ‫ﺑﺪون‬‫اﻟﻤﻨﺤﻨﻰ‬ ‫أﻧﺸﺊ‬fC‫ﻣﻌﻠﻢ‬ ‫اﻟﻰ‬ ‫ﻣﻨﺴﻮب‬ ‫ﻣﺴﺘﻮى‬ ‫ﻓﻲ‬
‫ﻣﻤﻨﻈﻢ‬ ‫ﻣﺘﻌﺎﻣﺪ‬( ); ;O i j
‫اﻟﺤﻞ‬
‫أ‬-‫ﻧ‬‫ﺤﺪد‬fD
fx D∈‫ﺗﻜﺎﻓﺊ‬2 0x − ≠
‫ﺗﻜﺎﻓﺊ‬2x ≠
‫ﺗﻜﺎﻓﺊ‬2x ≠‫أو‬2x = −
‫إذن‬{ }2;2fD = − −
‫ب‬-‫ﻧ‬‫أرﺗﻮﺑﻲ‬ ‫ﺤﺪد‬A‫و‬B‫اﻟﻤﻨﺤﻨﻰ‬ ‫ﻣﻦ‬ ‫ﻧﻘﻄﺘﻴﻦ‬fC‫ﻋﻠﻰ‬ ‫أﻓﺼﻮﻟﻴﻬﻤﺎ‬‫اﻟﺘﻮاﻟﻲ‬0‫و‬3
‫ﻟﺪﻳﻨﺎ‬( )
4
0 2
2
f
−
= =
−
‫ﻣﻨﻪ‬ ‫و‬( )0;2 fA C∈
‫ﻟﺪﻳﻨﺎ‬( )
9 4
3 5
3 2
f
−
= =
−
‫ﻣﻨﻪ‬ ‫و‬( )3;5 fB C∈
‫ح‬-‫اﻟﻨﻘﻂ‬ ‫هﻞ‬( ) ( ) ( )4; 6 ; 4;6 ; 2;0E D C− −‫إﻟﻰ‬ ‫ﺗﻨﺘﻤﻲ‬fC
‫ﻟﺪﻳﻨﺎ‬{ }2 2;2∉ − −‫وﻣﻨﻪ‬( )2;0 fC C∈
‫ﻟﺪﻳﻨﺎ‬( )
16 4
4 6
4 2
f
−
− = =
−
‫ﻣﻨﻪ‬ ‫و‬( )4;6 fD C− ∈
‫ﻟﺪﻳﻨﺎ‬( )
16 4
4 6
4 2
f
−
= =
−
‫ﻣﻨﻪ‬ ‫و‬( )4; 6 fE C− ∉
‫د‬–‫ﻧ‬‫ﻜﺘﺐ‬( )f x‫اﻟﻤﻄﻠﻘﺔ‬ ‫ﻟﻠﻘﻴﻤﺔ‬ ‫رﻣﺰ‬ ‫ﺑﺪون‬
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3. ‫ﻟﺪﻳﻨﺎ‬‫آﺎن‬ ‫إذا‬[ [ ] [0;2 2;x ∈ ∪ +∞‫ﻓﺎن‬( )
( )( )2 2 24
2
2 2
x xx
f x x
x x
− +−
= = = +
− −
‫آﺎن‬ ‫إذا‬] [ ] ]; 2 2;0x ∈ −∞ − ∪ −‫ﻓﺎن‬( )
( )( )2 2 24
2
2 2
x xx
f x x
x x
− +−
= = = − +
− − −
‫ﻧ‬‫اﻟﻤﻨﺤﻨﻰ‬ ‫ﻨﺸﺊ‬fC
‫ﻣﻌﺎدﻟﺔ‬‫ﺟﺰء‬fC‫ﻋﻠﻰ‬[ [ ] [0;2 2;∪ +∞‫هﻲ‬2y x= +‫ﻣﻨﻪ‬ ‫و‬fC‫اﻟﻨﻘﻄﺔ‬ ‫أﺻﻠﻪ‬ ‫ﻣﺴﺘﻘﻴﻢ‬ ‫ﻧﺼﻊ‬( )0;2A
‫اﻷﻓﺼﻮل‬ ‫ذات‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ﻣﻦ‬ ‫ﻣﺤﺮوم‬2
‫ﺟﺰء‬ ‫ﻣﻌﺎدﻟﺔ‬fC‫ﻋﻠﻰ‬] [ ] ]; 2 2;0−∞ − ∪ −‫هﻲ‬2y x= − +‫ﻣﻨﻪ‬ ‫و‬fC‫اﻟﻨﻘﻄﺔ‬ ‫أﺻﻠﻪ‬ ‫ﻣﺴﺘﻘﻴﻢ‬ ‫ﻧﺼﻊ‬
( )0;2A‫ﻣﺤﺮوم‬‫اﻷﻓﺼﻮل‬ ‫ذات‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ﻣﻦ‬2-
‫ﺗﻌﺮﻳﻒ‬
‫ﻟﺘﻜﻦ‬f‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬.
‫ﻟﻠﺪاﻟﺔ‬ ‫اﻟﻤﺒﻴﺎﻧﻲ‬ ‫اﻟﺘﻤﺜﻴﻞ‬f)‫اﻟﺪاﻟﺔ‬ ‫ﻣﻨﺤﻨﻰ‬ ‫أو‬f(‫اﻟﻨﻘﻂ‬ ‫ﻣﺠﻤﻮﻋﺔ‬ ‫هﻮ‬( )( );M x f x‫ﺣﻴﺚ‬fx D∈‫ﻧﺮﻣﺰ‬
‫ﺑﺎﻟﺮﻣﺰ‬ ‫ﻟﻬﺎ‬fC( )( ){ }; /f fC M x f x x D= ∈
‫ﻣﻼﺣﻈﺔ‬
( ); fM x y C∈‫ﺗﻜﺎﻓﺊ‬( )y f x=‫و‬fx D∈
‫اﻟﻌﻼﻗﺔ‬( )y f x=‫ﻟﻠﻤﻨﺤﻨﻰ‬ ‫دﻳﻜﺎرﺗﻴﺔ‬ ‫ﻣﻌﺎدﻟﺔ‬ ‫ﺗﺴﻤﻰ‬fC
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4. II-‫داﻟﺔ‬ ‫زوﺟﻴﺔ‬
‫اﻟﺰوﺟﻴﺔ‬ ‫اﻟﺪاﻟﺔ‬ -1
‫أ‬-‫ﺗﻌﺮﻳﻒ‬
‫ﻟﺘﻜﻦ‬f‫و‬ ‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬fD‫ﺗﻌﺮﻳﻔﻬﺎ‬ ‫ﺣﻴﺰ‬
‫ان‬ ‫ﻧﻘﻮل‬f‫اﻟﺘﺎﻟﻴﺎن‬ ‫اﻟﺸﺮﻃﺎن‬ ‫ﺗﺤﻘﻖ‬ ‫اذا‬ ‫زوﺟﻴﺔ‬ ‫داﻟﺔ‬:
*‫ﻟﻜﻞ‬x‫ﻣﻦ‬fDfx D− ∈
*‫ﻟﻜﻞ‬x‫ﻣﻦ‬fD( ) ( )f x f x− =
‫ﺗﻤﺮﻳﻦ‬
‫اﻟﻌﺪدﻳﺔ‬ ‫اﻟﺪاﻟﺔ‬ ‫هﻞ‬f‫اﻟﺘﺎﻟﻴﺔ‬ ‫اﻟﺤﺎﻻت‬ ‫ﻓﻲ‬ ‫زوﺟﻴﺔ‬
( ) ( )
( )
( )
3
2
1
1 ( ; (
2 0 4
(
2 0
f x x b f x x a
x
f x x x
c
f x x x
= + = −
 = ≤

= −
≺
≺
a/( ) 2
1
f x x
x
= −
*
fD =
‫ﻟﻜﻞ‬ ‫ﻟﺪﻳﻨﺎ‬*
x ∈*
x− ∈
‫ﻟﺘﻜﻦ‬*
x ∈
( )
( )
( )2 2
1 1
f x x x f x
xx
− = − − = − =
−
‫إذن‬f‫داﻟﺔ‬‫زوﺟﻴﺔ‬
b/( ) 3
1f x x= +
( ) ( ) ( )3 3
1 1 1 1 1 0 1 1 1 1 1 2f f− = − + = − + = = + = + =
‫وﻣﻨﻪ‬( ) ( )1 1f f− ≠
f‫داﻟﺔ‬‫ﻏﻴﺮ‬‫زوﺟﻴﺔ‬
c/
( )
( )
2 0 4
2 0
f x x x
f x x x
 = ≤

= −
≺
≺
] [ [ [ ] [;0 0;4 ;4fD = −∞ ∪ = −∞
‫أن‬ ‫ﻧﻼﺣﻆ‬6 fD− ∈‫و‬6 fD∉‫اذن‬f‫داﻟﺔ‬‫ﻏﻴﺮ‬‫زوﺟﻴﺔ‬
‫زوﺟﻴﺔ‬ ‫ﻟﺪاﻟﺔ‬ ‫اﻟﻤﺒﻴﺎﻧﻲ‬ ‫اﻟﺘﻤﺜﻴﻞ‬ ‫ب‬
f‫و‬ ‫زوﺟﻴﺔ‬ ‫داﻟﺔ‬fC‫ﻣﻤﻨﻈﻢ‬ ‫ﻣﺘﻌﺎﻣﺪ‬ ‫ﻣﻌﻠﻢ‬ ‫اﻟﻰ‬ ‫ﻣﻨﺴﻮب‬ ‫ﻣﺴﺘﻮى‬ ‫ﻓﻲ‬ ‫ﻣﻨﺤﻨﺎهﺎ‬( ); ;O i j
‫ﻟﺘﻜﻦ‬( )( );M x f x‫ﻣﻦ‬fC‫و‬'M‫اﻷراﺗﻴﺐ‬ ‫ﻟﻤﺤﻮر‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫ﻣﻤﺎﺛﻠﺘﻬﺎ‬.
‫وﻣﻨﻪ‬( )( )' ;M x f x−
‫أن‬ ‫ﺣﻴﺚ‬ ‫و‬f‫ﻓﺎن‬ ‫زوﺟﻴﺔ‬fx D− ∈‫و‬( ) ( )f x f x− =
‫وﻣﻨﻪ‬( )( )' ;M x f x− −‫ﺑﺎﻟﺘﺎﻟﻲ‬ ‫و‬' fM C∈
‫اذن‬fC‫اﻷراﺗﻴﺐ‬ ‫ﻟﻤﺤﻮر‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫ﻣﺘﻤﺎﺛﻞ‬
‫اﻟﻌﻜﺲ‬
‫آﺎن‬ ‫إذا‬ ‫أﻧﻪ‬ ‫ﺑﻴﻦ‬fC‫ﻓﺎن‬ ‫اﻷراﺗﻴﺐ‬ ‫ﻟﻤﺤﻮر‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫ﻣﺘﻤﺎﺛﻞ‬f‫زوﺟﻴﺔ‬ ‫داﻟﺔ‬
‫ﺧﺎﺻﻴﺔ‬
‫ﻟﺘﻜﻦ‬f‫و‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬fC‫ﻣﻤﻨﻈﻢ‬ ‫ﻣﺘﻌﺎﻣﺪ‬ ‫ﻣﻌﻠﻢ‬ ‫اﻟﻰ‬ ‫ﻣﻨﺴﻮب‬ ‫ﻣﺴﺘﻮى‬ ‫ﻓﻲ‬ ‫ﻣﻨﺤﻨﺎهﺎ‬( ); ;O i j
‫ﺗﻜﻮن‬f‫ﻟﻠﻤﻨﺤﻨﻰ‬ ‫ﺗﻤﺎﺛﻞ‬ ‫ﻣﺤﻮر‬ ‫اﻷراﺗﻴﺐ‬ ‫ﻣﺤﻮر‬ ‫آﺎن‬ ‫إذا‬ ‫وﻓﻘﻂ‬ ‫إذا‬ ‫زوﺟﻴﺔ‬ ‫داﻟﺔ‬fC
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5. ‫ﺗﻤﺮﻳﻦ‬
1-f‫اﻟﻤﻨﺤﻨﻰ‬ ‫أﺗﻤﻢ‬ ‫زوﺟﻴﺔ‬ ‫داﻟﺔ‬fC
2-f‫ﻳﻠﻲ‬ ‫آﻤﺎ‬ ‫ﻣﻨﺤﻨﺎهﺎ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬
‫هﻞ‬f‫زوﺟﻴﺔ‬
2-‫ﻓﺮدﻳﺔ‬ ‫داﻟﺔ‬
‫أ‬-‫ﺗﻌﺮﻳﻒ‬
‫ﻟﺘﻜﻦ‬f‫و‬ ‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬fD‫ﺗﻌﺮﻳﻔﻬﺎ‬ ‫ﺣﻴﺰ‬
‫ان‬ ‫ﻧﻘﻮل‬f‫اﻟﺘﺎﻟﻴﺎن‬ ‫اﻟﺸﺮﻃﺎن‬ ‫ﺗﺤﻘﻖ‬ ‫إذا‬ ‫ﻓﺮدﻳﺔ‬ ‫داﻟﺔ‬:
*‫ﻟﻜﻞ‬x‫ﻣﻦ‬fDfx D− ∈
*‫ﻟﻜﻞ‬x‫ﻣﻦ‬fD( ) ( )f x f x− = −
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6. ‫ﺗﻤﺮﻳﻦ‬
‫اﻟﻌﺪدﻳﺔ‬ ‫اﻟﺪاﻟﺔ‬ ‫هﻞ‬f‫اﻟﺤﺎ‬ ‫ﻓﻲ‬ ‫ﻓﺮدﻳﺔ‬‫اﻟﺘﺎﻟﻴﺔ‬ ‫ﻻت‬
( ) ( )
( )
( )
3
3
1
1 ( ; (
2 1 0 2
(
2 1 2 0
f x x b f x a
x
f x x x
c
f x x x
= + =
 = − + ≤ ≤

= − − − ≤ ≺
a/( ) 3
1
f x
x
=
*
fD =
‫ﻟﻜﻞ‬ ‫ﻟﺪﻳﻨﺎ‬*
x ∈*
x− ∈
‫ﻟﺘﻜﻦ‬*
x ∈
( )
( )
( )3 3
1 1
f x f x
xx
− = = − = −
−
‫إذن‬f‫داﻟﺔ‬‫ﻓﺮدﻳﺔ‬
b/( ) 3
1f x x= +
( ) ( ) ( )3 3
1 1 1 1 1 0 1 1 1 1 1 2f f− = − + = − + = = + = + =
‫وﻣﻨﻪ‬( ) ( )1 1f f− ≠ −
f‫داﻟﺔ‬‫ﻏﻴﺮ‬‫ﻓﺮدﻳﺔ‬
c/
( )
( )
2 1 0 2
2 1 2 0
f x x x
f x x x
 = − + ≤ ≤

= − − − ≤ ≺
[ [ [ ] [ ]2;0 0;2 2;2fD = − ∪ = −
‫ﻟﻜﻞ‬ ‫ﻟﺪﻳﻨﺎ‬[ ]2;2x ∈ −‫و‬[ ]2;2x− ∈ −
‫إذا‬‫آﺎن‬] ]0;2x ∈‫ﻓﺎن‬[ [2;0x− ∈ −
‫ﺑﺎﻟﺘﺎﻟﻲ‬ ‫و‬( ) 2 1f x x= − +‫و‬( ) ( )2 1 2 1f x x x− = − − − = −‫و‬‫ﻣﻨﻪ‬( ) ( )f x f x− = −
‫آﺎن‬ ‫إذا‬[ [2;0x ∈ −‫ﻓﺎن‬] ]0;2x− ∈
‫ﺑﺎﻟﺘﺎﻟﻲ‬ ‫و‬( ) 2 1f x x= − −‫و‬( ) ( )2 1 2 1f x x x− = − − + = +‫ﻣﻨﻪ‬ ‫و‬( ) ( )f x f x− = −
‫إذن‬‫ﻟﻜﻞ‬[ ]2;2x ∈ −( ) ( )f x f x− = −
‫إذن‬f‫داﻟﺔ‬‫ﻓﺮدﻳﺔ‬
‫ب‬-‫ﻓﺮدﻳﺔ‬ ‫ﻟﺪاﻟﺔ‬ ‫اﻟﻤﺒﻴﺎﻧﻲ‬ ‫اﻟﺜﻤﺜﻴﻞ‬
‫ﺧﺎﺻﻴﺔ‬
‫ﻟﺘﻜﻦ‬f‫و‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬fC‫ﻣﻤﻨﻈﻢ‬ ‫ﻣﺘﻌﺎﻣﺪ‬ ‫ﻣﻌﻠﻢ‬ ‫اﻟﻰ‬ ‫ﻣﻨﺴﻮب‬ ‫ﻣﺴﺘﻮى‬ ‫ﻓﻲ‬ ‫ﻣﻨﺤﻨﺎهﺎ‬( ); ;O i j
‫ﺗﻜﻮن‬f‫اﻟﻤﻨﺤﻨﻰ‬ ‫آﺎن‬ ‫إذا‬ ‫وﻓﻘﻂ‬ ‫إذا‬ ‫ﻓﺮدﻳﺔ‬ ‫داﻟﺔ‬fC‫ﺑﺎﻟﻨﺴﺒﺔ‬ ‫ﻣﺘﻤﺎﺛﻼ‬‫اﻟﻤﻌﻠﻢ‬ ‫ﻷﺻﻞ‬
‫ﺗﻤﺮﻳﻦ‬
f‫اﻟﻤﻨﺤﻨﻰ‬ ‫أﺗﻤﻢ‬ ‫ﻓﺮدﻳﺔ‬ ‫داﻟﺔ‬fC
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7. ‫ﺗﻤﺮﻳﻦ‬
‫اﻟ‬ ‫ﻧﻌﺘﺒﺮ‬‫اﻟﻌﺪدﻳﺔ‬ ‫ﺪاﻟﺔ‬f‫اﻟﺤﻘﻴﻘﻲ‬ ‫ﻟﻠﻤﺘﻐﻴﺮ‬x‫ﺣﻴﺚ‬( )
2
x x
f x
x
+
=
‫ﺣﺪد‬fD‫أن‬ ‫وﺑﻴﻦ‬f‫أﻧﺸﺊ‬ ‫ﺛﻢ‬ ‫ﻓﺮدﻳﺔ‬fC
‫ﻣﻼﺣﻈﺔ‬‫ﻟﻠ‬ ‫ﻳﻤﻜﻦ‬‫زوﺟﻴﺔ‬ ‫ﻏﻴﺮ‬ ‫و‬ ‫ﻓﺮدﻳﺔ‬ ‫ﻏﻴﺮ‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﺪاﻟﺔ‬
III-‫داﻟﺔ‬ ‫ﺗﻐﻴﺮات‬
1-‫داﻟﺔ‬ ‫ﺗﻐﻴﺮات‬ ‫ﻣﻨﺤﻰ‬
‫ﺗﻌﺮﻳﻒ‬
‫ﻟﺘﻜﻦ‬f‫و‬ ‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬I‫ﺿﻤﻦ‬ ‫ﻣﺠﺎل‬fD
-‫ﺗﻜﻮن‬f‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬I‫ﻟﻜﻞ‬ ‫آﺎن‬ ‫إذا‬ ‫ﻓﻘﻂ‬ ‫و‬ ‫إذا‬1x‫و‬2x‫ﻣﻦ‬I‫آﺎن‬ ‫إذا‬1 2x x≺‫ﻓﺎن‬
( ) ( )1 2f x f x≤
-‫ﺗﻜﻮن‬f‫ﻋﻠﻰ‬ ‫ﻗﻄﻌﺎ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬I‫ﻟﻜﻞ‬ ‫آﺎن‬ ‫إذا‬ ‫ﻓﻘﻂ‬ ‫و‬ ‫إذا‬1x‫و‬2x‫ﻣﻦ‬I‫آﺎن‬ ‫إذا‬1 2x x≺
‫ﻓﺎن‬( ) ( )1 2f x f x≺
-‫ﺗﻜﻮن‬f‫ﻋﻠﻰ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬I‫ﻟﻜﻞ‬ ‫آﺎن‬ ‫إذا‬ ‫ﻓﻘﻂ‬ ‫و‬ ‫إذا‬1x‫و‬2x‫ﻣﻦ‬I‫آﺎن‬ ‫إذا‬1 2x x≺‫ﻓﺎن‬
( ) ( )1 2f x f x≥
-‫ﺗﻜﻮن‬f‫ﻋﻠﻰ‬ ‫ﻗﻄﻌﺎ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬I‫ﻟﻜﻞ‬ ‫آﺎن‬ ‫إذا‬ ‫ﻓﻘﻂ‬ ‫و‬ ‫إذا‬1x‫و‬2x‫ﻣﻦ‬I‫آﺎن‬ ‫إذا‬1 2x x≺
‫ﻓﺎن‬( ) ( )1 2f x f x
‫ﻣﺜﺎل‬‫اﻟﺪاﻟﺔ‬ ‫ﺗﻐﻴﺮات‬ ‫أدرس‬f‫ﺣﻴﺚ‬( ) 2 1f x x= − +
‫ﻟﻴﻜﻦ‬a‫و‬b‫ﻣﻦ‬‫ﺣﻴﺚ‬a b≺
‫وﻣﻨﻪ‬2 2a b− −‫ﺑﺎﻟﺘﺎﻟﻲ‬ ‫و‬2 1 2 1a b− + − +( ) ( )f a f b
‫إذن‬f‫ﻗﻄﻌﺎ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬
‫ﺗﻤﺮﻳﻦ‬‫ﻧﻌﺘﺒﺮ‬( ) 2f x x= −
‫ﺗﻐﻴﺮات‬ ‫ﻣﻨﺤﻰ‬ ‫أدرس‬f‫ﻣﻦ‬ ‫آﻞ‬ ‫ﻋﻠﻰ‬] ];2−∞‫و‬] [2;+∞
‫أﻧﺸﺊ‬fC
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8. ‫ﺗﻤﺮﻳﻦ‬‫ﻟﻠﺪاﻟﺔ‬ ‫اﻟﻤﺒﻴﺎﻧﻲ‬ ‫اﻟﺘﻤﺜﻴﻞ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬f
‫اﻟﻤﺠﺎل‬ ‫ﻋﻠﻰ‬[ ]4;5−‫ﺗﻐﻴﺮات‬ ‫ﺣﺪد‬f
‫اﻟﺮﺗﻴﺒﺔ‬ ‫اﻟﺪاﻟﺔ‬ -2
‫ﺗﻌﺮﻳﻒ‬
‫ﻟﺘﻜﻦ‬f‫و‬ ‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬I‫ﺿﻤﻦ‬ ‫ﻣﺠﺎل‬fD
‫ان‬ ‫ﻧﻘﻮل‬f‫ﻋﻠﻰ‬ ‫رﺗﻴﺒﺔ‬I‫آﺎن‬ ‫إذا‬ ‫ﻓﻘﻂ‬ ‫و‬ ‫إذا‬f‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬ ‫إﻣﺎ‬I‫ﻋﻠﻰ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬ ‫إﻣﺎ‬ ‫و‬I.
‫ﻣﻼﺣﻈﺎت‬
-‫ﻣﺠﺎل‬ ‫ﻋﻠﻰ‬ ‫رﺗﻴﺒﺔ‬ ‫ﻏﻴﺮ‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﻟﺪاﻟﺔ‬ ‫ﻳﻤﻜﻦ‬I
-‫در‬‫رﺗﺎﺑﺔ‬ ‫اﺳﺔ‬f‫ﻣﺠﺎل‬ ‫ﻋﻠﻰ‬I‫ﺗﺠﺰيء‬ ‫ﻳﻌﻨﻲ‬I‫ﻣﺠﺎﻻت‬ ‫إﻟﻰ‬
‫ﻓﻴﻬﺎ‬ ‫ﺗﻜﻮن‬f‫رﺗﻴﺒﺔ‬.‫اﻟﺘﻐﻴﺮات‬ ‫ﺟﺪول‬ ‫ﻳﺴﻤﻰ‬ ‫ﺟﺪول‬ ‫ﻓﻲ‬ ‫اﻟﺪراﺳﺔ‬ ‫وﻧﻠﺨﺺ‬
3-‫اﻟﺘﻐﻴﺮ‬ ‫ﻣﻌﺪل‬
‫أ‬-‫ﺗﻌﺮﻳﻒ‬
‫ﻟﺘﻜﻦ‬f‫و‬ ‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬1x‫و‬2x‫ﻣﺨﺘﻠﻔﻴﻨﻤﻦ‬ ‫ﻋﻨﺼﺮﻳﻦ‬fD
‫اﻟﻌﺪد‬
( ) ( )2 1
2 1
f x f x
x x
−
−
‫اﻟﺪاﻟﺔ‬ ‫ﺗﻐﻴﺮ‬ ‫ﻣﻌﺪل‬ ‫ﻳﺴﻤﻰ‬f‫ﺑﻴﻦ‬1x‫و‬2x.
‫ﻣﺜﺎل‬‫ﻧﻌﺘﺒﺮ‬( ) 2
3f x x x= −
‫أﺣﺴﺐ‬‫ﻣﻌ‬‫ﺪ‬‫ل‬‫ﺗﻐﻴﺮات‬f‫ﺑﻴﻦ‬2‫و‬1-
‫ب‬-‫اﻟﺮﺗﺎﺑﺔ‬ ‫و‬ ‫اﻟﺘﻐﻴﺮ‬ ‫ﻣﻌﺪل‬
‫ﻧﺤﺼﻞ‬ ‫اﻟﺘﻌﺮﻳﻒ‬ ‫ﺑﺘﻮﻇﻴﻒ‬‫ﻋﻠﻰ‬
‫ﺧﺎﺻﻴﺔ‬
‫ﻟﺘﻜﻦ‬f‫و‬ ‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬I‫ﺿﻤﻦ‬ ‫ﻣﺠﺎل‬fD
-‫ﺗﻜﻮن‬f‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬I‫ﻟﻜﻞ‬ ‫آﺎن‬ ‫إذا‬ ‫ﻓﻘﻂ‬ ‫و‬ ‫إذا‬1x‫و‬2x‫ﻣﻦ‬ ‫ﻣﺨﺘﻠﻔﻴﻦ‬I
( ) ( )2 1
2 1
0
f x f x
x x
−
≥
−
-‫ﺗﻜﻮن‬f‫ﻋﻠﻰ‬ ‫ﻗﻄﻌﺎ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬I‫ﻟﻜﻞ‬ ‫آﺎن‬ ‫إذا‬ ‫ﻓﻘﻂ‬ ‫و‬ ‫إذا‬1x‫و‬2x‫ﻣﻦ‬ ‫ﻣﺨﺘﻠﻔﻴﻦ‬I
( ) ( )2 1
2 1
0
f x f x
x x
−
−
-‫ﺗﻜﻮن‬f‫ﻋﻠﻰ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬I‫ﻟﻜﻞ‬ ‫آﺎن‬ ‫إذا‬ ‫ﻓﻘﻂ‬ ‫و‬ ‫إذا‬1x‫و‬2x‫ﻣﻦ‬ ‫ﻣﺨﺘﻠﻔﻴﻦ‬I
( ) ( )2 1
2 1
0
f x f x
x x
−
≤
−
-‫ﺗﻜﻮن‬f‫ﻋﻠﻰ‬ ‫ﻗﻄﻌﺎ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬I‫ﻟﻜﻞ‬ ‫آﺎن‬ ‫إذا‬ ‫ﻓﻘﻂ‬ ‫و‬ ‫إذا‬1x‫و‬2x‫ﻣﻦ‬ ‫ﻣﺨﺘﻠﻔﻴﻦ‬I
( ) ( )2 1
2 1
0
f x f x
x x
−
−
≺
‫ﺗﻤﺮﻳﻦ‬
‫ﻧﻌﺘﺒﺮ‬( ) 2
4 1f x x x= − −
‫رﺗﺎﺑﺔ‬ ‫أدرس‬f‫ﻋﻠ‬‫اﻟﻤﺠﺎﻟﻴﻦ‬ ‫ﻣﻦ‬ ‫آﻞ‬ ‫ﻰ‬] ] ] [;2 ; 2;−∞ +∞
‫ﺗﻐﻴﺮات‬ ‫ﺟﺪول‬ ‫أﻋﻂ‬ ‫و‬f
‫اﻟﺠﻮاب‬
‫ﻟﻴﻜﻦ‬a‫و‬b‫ﻣﻦ‬‫ﺣﻴﺚ‬a b≠
( ) ( ) ( )( ) ( ) ( )( )2 2 4 44 1 4 1
4
f a f b a b a b a b a b a ba a b b
a b
a b a b a b a b
− − + − + − + −− − − + +
= = = = + −
− − − −
‫آﺎن‬ ‫إذا‬a‫و‬b‫ﻣﻦ‬] [2;+∞‫ﻓﺎن‬2a‫و‬2b‫وﻣﻨﻪ‬4a b+‫أي‬4 0a b+ −
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9. ‫إذن‬f‫ﻗﻄﻌﺎ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬‫ﻋﻠﻰ‬] [2;+∞
‫آﺎن‬ ‫إذا‬a‫و‬b‫ﻣﻦ‬] ];2−∞‫ﻓﺎن‬2a ≤‫و‬2b ≤‫وﻣﻨﻪ‬4a b+ ≤‫أي‬4 0a b+ − ≤
‫إذن‬f‫ﺗﻨﺎﻗﺼﻴﺔ‬‫ﻋﻠﻰ‬] ];2−∞
‫اﻟﺘﻐﻴﺮات‬ ‫ﺟﺪول‬
+∞2−∞x
1-
f
‫ﺗﻤﺮﻳﻦ‬
‫ﻧﻌﺘﺒﺮ‬( )
2 1
2
x
f x
x
−
=
+
‫رﺗﺎﺑﺔ‬ ‫أدرس‬f
‫داﻟﺔ‬ ‫وزوﺟﻴﺔ‬ ‫اﻟﺮﺗﺎﺑﺔ‬ -4
‫ﺧﺎﺻﻴﺔ‬ -‫أ‬
‫ﻟﺘﻜﻦ‬f‫و‬ ‫زوﺟﻴﺔ‬ ‫داﻟﺔ‬I‫ﺿﻤﻦ‬ ‫ﻣﺠﺎل‬fD +
∩‫و‬J‫ﻟـ‬ ‫ﻣﻤﺎﺛﻞ‬ ‫ﻣﺠﺎل‬I‫ﻟـ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬0{ }( )/J x x I= − ∈
-‫آﺎﻧﺖ‬ ‫إذا‬f‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬I‫ﻓﺎن‬f‫ﻋﻠﻰ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬J.
-‫آﺎﻧﺖ‬ ‫إذا‬f‫ﺗﻨ‬‫ﺎﻗﺼﻴﺔ‬‫ﻋﻠﻰ‬I‫ﻓﺎن‬f‫ﺗﺰاﻳﺪﻳﺔ‬‫ﻋﻠﻰ‬J.
‫اﻟﺒﺮهﺎن‬
‫ﻟﺘﻜﻦ‬f‫و‬ ‫زوﺟﻴﺔ‬ ‫داﻟﺔ‬1x‫و‬2x‫ﻣﻦ‬ ‫ﻣﺨﺘﻠﻔﻴﻦ‬ ‫ﻋﻨﺼﺮﻳﻦ‬J
‫ﻳﻮﺟﺪ‬ ‫وﻣﻨﻪ‬1 'x‫و‬2 'x‫ﻣﻦ‬I‫ﺣﻴﺚ‬1 1'x x= −‫و‬2 2'x x= −
( ) ( ) ( ) ( ) ( ) ( )2 1 2 1 2 1
2 1 2 1 2 1
' '
' '
f x f x f x f x f x f x
x x x x x x
− − − − −
= = −
− − + −
‫إذ‬‫ن‬‫ﺗﻐﻴﺮات‬f‫ﻋﻠﻰ‬I‫ﺗﻐﻴﺮات‬ ‫ﻋﻜﺲ‬f‫ﻋﻠﻰ‬J
‫ﺧﺎﺻﻴﺔ‬ -‫أ‬
‫ﻟﺘﻜﻦ‬f‫داﻟ‬‫و‬ ‫ﻓﺮدﻳﺔ‬ ‫ﺔ‬I‫ﺿﻤﻦ‬ ‫ﻣﺠﺎل‬fD +
∩‫و‬J‫ﻟـ‬ ‫ﻣﻤﺎﺛﻞ‬ ‫ﻣﺠﺎل‬I‫ﻟـ‬ ‫ﺑﺎﻟﻨﺴﺒﺔ‬0{ }( )/J x x I= − ∈
-‫آﺎﻧﺖ‬ ‫إذا‬f‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬I‫ﻓﺎن‬f‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬J.
-‫آﺎﻧﺖ‬ ‫إذا‬f‫ﻋ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬‫ﻠﻰ‬I‫ﻓﺎن‬f‫ﻋﻠﻰ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬J.
‫ﻣﻼﺣﻈﺔ‬
‫ﻋﻠﻰ‬ ‫ﺗﻐﻴﺮاﺗﻬﺎ‬ ‫دراﺳﺔ‬ ‫ﻳﻜﻔﻲ‬ ‫زوﺟﻴﺔ‬ ‫أو‬ ‫ﻓﺮدﻳﺔ‬ ‫داﻟﺔ‬ ‫ﺗﻐﻴﺮات‬ ‫ﻟﺪراﺳﺔ‬fD +
∩‫ﺗﻐﻴﺮاﺗﻬﺎ‬ ‫اﺳﺘﻨﺘﺎج‬ ‫ﺛﻢ‬
‫ﻋﻠﻰ‬fD −
∩
‫ﺗﻤﺮﻳﻦ‬
‫ﻧﻌﺘﺒﺮ‬( )
2
1x
f x
x
+
=
1-‫ﺣﺪد‬fD‫زوﺟﻴﺔ‬ ‫أدرس‬ ‫و‬f
2-‫ﺗﻐﻴﺮات‬ ‫أدرس‬f‫ﺗﻐﻴﺮاﺗﻬﺎ‬ ‫ﺟﺪول‬ ‫أﻋﻂ‬ ‫و‬
VI-‫اﻟﻘﺼﻮى‬ ‫اﻟﻘﻴﻤﺔ‬–‫اﻟﺪﻧﻴﺎ‬ ‫اﻟﻘﻴﻤﺔ‬
‫ﺗﻌﺮﻳﻒ‬
‫ﻟﺘﻜﻦ‬f‫ﻟﻤﺘ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬‫ﺣﻘﻴﻘﻲ‬ ‫ﻐﻴﺮ‬
-‫ان‬ ‫ﻧﻘﻮل‬f‫ﻋﻨﺪ‬ ‫ﻗﺼﻮى‬ ‫ﻗﻴﻤﺔ‬ ‫ﺗﻘﺒﻞ‬a‫ﻣﺠﺎل‬ ‫وﺟﺪ‬ ‫إذا‬I‫ﺿﻤﻦ‬fD‫و‬a I∈‫ﻟﻜﻞ‬ ‫ﺣﻴﺚ‬{ }x I a∈ −
( ) ( )f x f a≺
-‫ان‬ ‫ﻧﻘﻮل‬f‫ﻋﻨﺪ‬ ‫دﻧﻴﺎ‬ ‫ﻗﻴﻤﺔ‬ ‫ﺗﻘﺒﻞ‬a‫ﻣﺠﺎل‬ ‫وﺟﺪ‬ ‫إذا‬I‫ﺿﻤﻦ‬fD‫و‬a I∈‫ﻟﻜﻞ‬ ‫ﺣﻴﺚ‬{ }x I a∈ −
( ) ( )f x f a
‫اﺻﻄﻼح‬
‫ﺗﺴ‬ ‫اﻟﺪﻧﻴﺎ‬ ‫ﻗﻴﻢ‬ ‫و‬ ‫اﻟﻘﺼﻮى‬ ‫ﻗﻴﻢ‬ ‫ﻣﻦ‬ ‫آﻞ‬‫ﻟﺪاﻟﺔ‬ ‫ﻣﻄﺎرﻳﻒ‬ ‫ﻤﻰ‬f
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10. ‫ﺗﻤﺮﻳﻦ‬‫ﻧﻌﺘﺒﺮ‬( )
1
f x x
x
= +
1-‫زوﺟﻴﺔ‬ ‫أدرس‬f‫أﺣﺴﺐ‬( )1f
2-‫أن‬ ‫ﺑﻴﻦ‬‫ﻟﻜﻞ‬x‫ﻣﻦ‬] [0;+∞( ) 2f x ≥
3-‫ﻟـ‬ ‫ﻗﺼﻮى‬ ‫ﻗﻴﻤﺔ‬ ‫و‬ ‫دﻧﻴﺎ‬ ‫ﻗﻴﻤﺔ‬ ‫ﺣﺪد‬f‫وﺟﺪ‬ ‫إذا‬
‫اﻟﺠﻮاب‬
1-‫ﻧ‬‫زوﺟﻴﺔ‬ ‫ﺪرس‬f
*
fD =
‫ﻟﻜﻞ‬x ∈x− ∈
( ) ( )
1 1
f x x x f x
x x
 
− = − + = − + = − 
−  
‫إذن‬f‫ﻓﺮدﻳﺔ‬
‫ﺣﺴ‬‫ﺎ‬‫ب‬( )
1
1 1 2
1
f = + =
2-‫أن‬ ‫ﻧﺒﻴﻦ‬‫ﻟﻜﻞ‬x‫ﻣﻦ‬] [0;+∞( ) 2f x ≥
‫ﻟﻴﻜﻦ‬x‫ﻣﻦ‬] [0;+∞
( )
( )22 11 2 1
2 2
xx x
f x x
x x x
−− +
− = + − = =
‫أن‬ ‫ﺑﻤﺎ‬0x‫و‬( )2
1 0x − ≥‫ﻓﺎن‬( ) 2f x ≥
3-‫ﻧ‬‫ﻟـ‬ ‫ﻗﺼﻮى‬ ‫ﻗﻴﻤﺔ‬ ‫و‬ ‫دﻧﻴﺎ‬ ‫ﻗﻴﻤﺔ‬ ‫ﺤﺪد‬f
‫ﻣﻦ‬1/‫و‬2/‫أن‬ ‫ﻧﺴﺘﻨﺘﺞ‬‫ﻟﻜﻞ‬x‫ﻣﻦ‬] [0;+∞( ) ( )1f x f≥
‫اذن‬f‫ﻋﻨﺪ‬ ‫دﻧﻴﺎ‬ ‫ﻗﻴﻤﺔ‬ ‫ﺗﻘﺒﻞ‬1
‫ﻟﻴﻜﻦ‬] [;0x ∈ −∞‫ﻣﻨﻪ‬ ‫و‬] [0;x− ∈ +∞‫ﻧﺴ‬ ‫ﺳﺒﺚ‬ ‫ﻣﻤﺎ‬‫أن‬ ‫ﺘﻨﺘﺞ‬( ) ( )1f x f− ≥
‫ﺣﻴﺚ‬ ‫و‬f‫ﻓﺎن‬ ‫ﻓﺮدﻳﺔ‬( ) ( )1f x f− ≥‫ﺑﺎﻟﺘﺎﻟﻲ‬ ‫و‬( ) ( )1f x f≤ −‫أي‬( ) ( )1f x f≤ −
‫اذن‬f‫ﻋﻨﺪ‬ ‫ﻗﺼﻮى‬ ‫ﻗﻴﻤﺔ‬ ‫ﺗﻘﺒﻞ‬1-
‫ﺧﺎﺻﻴﺔ‬
‫ﻟﻴﻜﻦ‬a‫و‬b‫و‬c‫ﺣﻴﺚ‬ ‫ﺣﻘﻴﻘﻴﺔ‬ ‫أﻋﺪاد‬a b c≺ ≺‫و‬f‫داﻟﺔ‬
‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺔ‬
‫آﺎﻧﺖ‬ ‫إذا‬f‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬[ ];a b‫ﻋﻠﻰ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬ ‫و‬[ ];b c‫ﻓﺎن‬f
‫ﻋﻨﺪ‬ ‫ﻗﺼﻮى‬ ‫ﻗﻴﻤﺔ‬ ‫ﺗﻘﺒﻞ‬b
‫آﺎﻧﺖ‬ ‫إذا‬f‫ﻋﻠﻰ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬[ ];a b‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬ ‫و‬[ ];b c‫ﻓﺎن‬f
‫ﻋﻨﺪ‬ ‫دﻧﻴﺎ‬ ‫ﻗﻴﻤﺔ‬ ‫ﺗﻘﺒﻞ‬b
V-‫داﻟﺔ‬ ‫ﺗﻐﺒﺮات‬ ‫دراﺳﺔ‬–‫ﻣﻨﺤﻨﻴﻴﻦ‬ ‫وﺿﻌﻴﺔ‬ ‫دراﺳﺔ‬
‫داﻟﺔ‬ ‫ﺗﻐﻴﺮات‬ ‫دراﺳﺔ‬f‫ﻳﻌﻨﻲ‬
–‫ﺗﺤﺪﻳﺪ‬fD
-‫رﺗﺎﺑﺔ‬ ‫دراﺳﺔ‬f‫وﺗﻠ‬‫اﻟﺘﻐﻴﺮات‬ ‫ﺟﺪول‬ ‫ﻓﻲ‬ ‫ﺨﻴﺼﻬﺎ‬
------------------------------------------
‫ﻣﺒﻴﺎﻧﻴﺎ‬ ‫ﻣﻨﺤﻨﻴﻴﻦ‬ ‫وﺿﻊ‬ ‫دراﺳﺔ‬
‫ﻟﻴﻜﻦ‬fC‫و‬Cg‫ﻟﻠﺪاﻟﺘﻴﻦ‬ ‫ﻣﻨﺤﻨﻴﻴﻦ‬f‫و‬g‫اﻟﺘﻮاﻟﻲ‬ ‫ﻋﻠﻰ‬
‫ﻳﻜﻮن‬( ) ( )f x g x‫اﻟﻤﺠﺎل‬ ‫ﻋﻠﻰ‬I‫آﺎن‬ ‫ﻓﻘﻂ‬ ‫و‬ ‫اذا‬fC‫ق‬ ‫ﻓﻮ‬Cg‫اﻟﻤﺠﺎل‬ ‫ﻓﻲ‬I
‫ﻳﻜﻮن‬( ) ( )f x g x≺‫اﻟﻤﺠﺎل‬ ‫ﻋﻠﻰ‬I‫ﻓﻘﻂ‬ ‫و‬ ‫اذا‬‫آﺎن‬fC‫ﺗﺤﺖ‬Cg‫اﻟﻤﺠﺎل‬ ‫ﻓﻲ‬I
‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﺣﻠﻮل‬( ) ( )f x g x=‫اﻟﻤﺠﺎل‬ ‫ﻋﻠﻰ‬I‫اﻟﻤﻨﺤﻨﻴﻴﻦ‬ ‫ﺗﻘﺎﻃﻊ‬ ‫ﻧﻘﻂ‬ ‫أﻓﺎﺻﻴﻞ‬ ‫هﻲ‬fC‫ﺗﺤﺖ‬Cg‫اﻟﻤﺠﺎل‬ ‫ﻓﻲ‬I
‫ﺗﻤﺮﻳﻦ‬
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11. ‫ﺗﻐﻴﺮات‬ ‫أدرس‬f‫ﺣﻴﺚ‬( )
2 3
1
x
f x
x
− +
=
−
‫ﺗﻤﺮﻳﻦ‬
‫ﺗﻐﻴﺮات‬ ‫أدرس‬f‫ﺣﻴﺚ‬( ) 3
3f x x x= −
‫اﻟﺪاﻟﺔ‬ ‫ﻣﻄﺎرﻳﻒ‬ ‫ﺣﺪد‬f
‫ﺣﻠﻮل‬ ‫و‬ ‫ﺗﻤﺎرﻳﻦ‬
‫ﺗﻤﺮﻳﻦ‬1
‫ﻧﻌﺘﺒﺮ‬f‫ﺑـ‬ ‫ﻣﻌﺮﻓﺔ‬ ‫ﺣﻘﻴﻘﻲ‬ ‫ﻟﻤﺘﻐﻴﺮ‬ ‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬:( ) 4f x x x x= −
1–‫اﻟﺪاﻟﺔ‬ ‫زوﺟﻴﺔ‬ ‫أدرس‬f
2–‫أ‬(‫ﻣﺨﺘﻠﻔﻴﻦ‬ ‫ﻋﻨﺼﺮﻳﻦ‬ ‫ﻟﻜﻞ‬ ‫أن‬ ‫ﺑﻴﻦ‬x‫و‬y‫ﻣﻦ‬[ [0;+∞
( ) ( )
4
f x f y
x y
x y
−
= + −
−
‫ب‬(‫رﺗﺎﺑﺔ‬ ‫ﺣﺪد‬f‫ﻣﻦ‬ ‫آﻞ‬ ‫ﻋﻠﻰ‬[ [0;2‫و‬] [2;+∞‫رﺗﺎﺑﺔ‬ ‫اﺳﺘﻨﺘﺞ‬ ‫و‬f‫ﻣﻦ‬ ‫آﻞ‬ ‫ﻋﻠﻰ‬] ]2;0−‫و‬] [; 2−∞ −
‫ج‬(‫اﻟﺪاﻟﺔ‬ ‫ﺗﻐﻴﺮات‬ ‫ﺟﺪول‬ ‫اﻋﻂ‬f
3-‫اﻟﺪاﻟﺔ‬ ‫ﻣﻄﺎرﻳﻒ‬ ‫ﺣﺪد‬f‫وﺟﺪت‬ ‫إن‬
4-‫ﺗ‬ ‫ﺣﺪد‬‫اﻟﻤﻨﺤﻨﻰ‬ ‫ﻘﺎﻃﻊ‬( )fC‫اﻟﻤﺴﺘﻘﻴﻢ‬ ‫و‬( )D‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ذا‬2y x= −
( ) 4f x x x x= −
1–‫ﻧ‬‫اﻟﺪاﻟﺔ‬ ‫زوﺟﻴﺔ‬ ‫ﺪرس‬f
‫ﻟﺪﻳﻨﺎ‬fD =
‫ﻟﻜﻞ‬x‫ﻣﻦ‬:x− ∈
( ) ( ) ( )4 4f x x x x x x x f x− = − − + = − − = −
‫إذن‬f‫ﻓﺮدﻳﺔ‬ ‫داﻟﺔ‬
2–‫أ‬(‫ﻧ‬‫ﻣﺨﺘﻠﻔﻴﻦ‬ ‫ﻋﻨﺼﺮﻳﻦ‬ ‫ﻟﻜﻞ‬ ‫أن‬ ‫ﺒﻴﻦ‬x‫و‬y‫ﻣﻦ‬[ [0;+∞:
( ) ( )
4
f x f y
x y
x y
−
= + −
−
‫ﻟﻜﻞ‬ ‫ﻟﺪﻳﻨﺎ‬x‫ﻣﻦ‬[ [0;+∞:( ) 2
4f x x x= −
‫ﻟﻴﻜﻦ‬x‫و‬y‫ﻣﻦ‬[ [0;+∞‫ﺣﻴﺚ‬x y≠:
( ) ( )
( )( ) ( )
( )( )
2 2
4 4
4
4
4
f x f y x x y y
x y x y
x y x y x y
x y
x y x y
x y
x y
− − − +
=
− −
− + − −
=
−
− + −
=
−
= + −
‫ب‬(‫ﻧ‬‫رﺗﺎﺑﺔ‬ ‫ﺤﺪد‬f‫ﻣﻦ‬ ‫آﻞ‬ ‫ﻋﻠﻰ‬[ [0;2‫و‬] [2;+∞‫و‬‫ﻧ‬‫رﺗﺎﺑﺔ‬ ‫ﺴﺘﻨﺘﺞ‬f‫ﻣﻦ‬ ‫آﻞ‬ ‫ﻋﻠﻰ‬] ]2;0−‫و‬] [; 2−∞ −
*‫ﻟﻴﻜﻦ‬x‫و‬y‫ﻣﻦ‬[ [0;2‫ﺣﻴﺚ‬x y≠‫وﻣﻨﻪ‬0 2x≤ ≺‫و‬0 2y≤ ≺
‫ﺑﺎﻟﺘﺎﻟﻲ‬ ‫و‬0 4x y≤ + ≺‫أي‬4 4 0x y− ≤ + − ≺
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12. ‫وﻣﻨﻪ‬
( ) ( )
0
f x f y
x y
−
−
≺
‫إذن‬f‫ﻋﻠﻰ‬ ‫ﻗﻄﻌﺎ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬[ [0;2‫أن‬ ‫ﺣﻴﺚ‬ ‫و‬f‫ﻓﺎن‬ ‫ﻓﺮدﻳﺔ‬f‫ﻋﻠﻰ‬ ‫ﻗﻄﻌﺎ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬] ]2;0−
*‫ﻟﻴﻜﻦ‬x‫و‬y‫ﻣﻦ‬] [2;+∞‫ﺣﻴﺚ‬x y≠‫وﻣﻨﻪ‬2x‫و‬2y
‫وﺑﺎﻟﺘﺎﻟﻲ‬4 0x y+ −‫أي‬
( ) ( )
0
f x f y
x y
−
−
‫إذن‬f‫ﻋﻠﻰ‬ ‫ﻗﻄﻌﺎ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬] [2;+∞‫وﻣﻨﻪ‬f‫ﻋﻠﻰ‬ ‫ﻗﻄﻌﺎ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬] [; 2−∞ −
‫ج‬(‫اﻟﺪاﻟﺔ‬ ‫ﺗﻐﻴﺮات‬ ‫ﺟﺪول‬f
+∞22-−∞x
4
4-
f
3-‫ﻧ‬‫اﻟﺪاﻟﺔ‬ ‫ﻣﻄﺎرﻳﻒ‬ ‫ﺤﺪد‬f
‫أن‬ ‫ﺑﻤﺎ‬f‫ﻣﻦ‬ ‫آﻞ‬ ‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬] [2;+∞‫و‬] [; 2−∞ −‫ﻋﻠﻰ‬ ‫ﺗﻨﺎﻗﺼﻴﺔ‬ ‫و‬[ ]2;2−‫ﻓﺎن‬f‫ﻗﺼﻮى‬ ‫ﻗﻴﻤﺔ‬ ‫ﺗﻘﺒﻞ‬
‫ﻋﻨﺪ‬2-‫هﻲ‬4‫ﻋﻨﺪ‬ ‫دﻧﻴﺎ‬ ‫ﻗﻴﻤﺔ‬ ‫و‬2‫هﻲ‬4-
4-‫ﻧ‬‫اﻟﻤﻨﺤﻨﻰ‬ ‫ﺗﻘﺎﻃﻊ‬ ‫ﺤﺪد‬( )fC‫اﻟﻤﺴﺘﻘﻴﻢ‬ ‫و‬( )D‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ذا‬2y x= −
‫ﺗﺤﺪﻳﺪ‬‫اﻟﻤﻨﺤﻨﻰ‬ ‫ﺗﻘﺎﻃﻊ‬( )fC‫اﻟﻤﺴﺘﻘﻴﻢ‬ ‫و‬( )D‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﺣﻞ‬ ‫إﻟﻰ‬ ‫ﻳﺮﺟﻊ‬4 2x x x x− = −
4 2x x x x− = −‫ﺗﻜﺎﻓﺊ‬2 0x x x− =
‫ﺗﻜﺎﻓﺊ‬( )2 0x x − =
‫ﺗﻜﺎﻓﺊ‬0x =‫أو‬2x =
‫ﺗﻜﺎﻓﺊ‬0x =‫أو‬2x =‫أو‬2x = −
‫إذن‬‫اﻟﻤﻨﺤﻨﻰ‬( )fC‫اﻟﻤﺴﺘﻘﻴﻢ‬ ‫و‬( )D‫اﻷﻓﺎﺻﻴﻞ‬ ‫ذات‬ ‫اﻟﻨﻘﻂ‬ ‫ﻓﻲ‬ ‫ﻳﺘﻘﺎﻃﻌﺎن‬0‫و‬2‫و‬2-
‫ﺗﻤﺮﻳﻦ‬2
‫ﻧﻌﺘﺒﺮ‬f‫ﻋﺪدﻳﺔ‬ ‫داﻟﺔ‬‫ﺑـ‬ ‫ﻣﻌﺮﻓﺔ‬( ) 2
1
x
f x
x
−
=
−
1-‫ﺣﺪد‬fD‫أن‬ ‫ﺑﻴﻦ‬ ‫و‬f‫ﻓﺮدﻳﺔ‬ ‫داﻟﺔ‬
2-‫ﻣﺨﺘﻠﻔﻴﻦ‬ ‫ﻋﻨﺼﺮﻳﻦ‬ ‫ﻟﻜﻞ‬ ‫أن‬ ‫ﻳﺒﻦ‬a‫و‬b‫ﻣﻦ‬fD
( ) ( )
( )( )2 2
1
1 1
f a f b ab
a b a b
− +
=
− − −
3-‫ﺗﻐﻴﺮات‬ ‫ﻣﻨﺤﻰ‬ ‫ﺣﺪد‬f‫ﻋﻠﻰ‬[ [0;1‫و‬] [1;+∞‫ﻋﻠﻰ‬ ‫ﺗﻐﻴﺮاﺗﻬﺎ‬ ‫ﻣﻨﺤﻰ‬ ‫اﺳﺘﻨﺘﺞ‬ ‫و‬] ]1;0−‫و‬] [; 1−∞ −
4-‫ﺗﻐﻴﺮا‬ ‫ﺟﺪول‬ ‫أﻋﻂ‬‫ت‬f
‫اﻟﺤﻞ‬
( ) 2
1
x
f x
x
−
=
−
1-‫ﻧﺤﺪد‬fD
*-‫ﻟﻴﻜﻦ‬x∈
fx D∈‫ﻳﻜﺎﻓﺊ‬
2
1 0x − ≠
‫ﺗ‬‫ﻜﺎﻓﺊ‬
2
1x ≠
‫ﺗﻜﺎﻓﺊ‬1x ≠‫و‬1x ≠ −
‫إذن‬{ }1;1fD = − −
*-‫أن‬ ‫ﻧﺒﻴﻦ‬f‫ﻓﺮدﻳﺔ‬ ‫داﻟﺔ‬
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13. ‫ﻟﻜﻞ‬x‫ﻣﻦ‬{ }1;1− −:{ }1;1x− ∈ − −
‫ﻟﺘﻜﻦ‬{ }1;1x∈ − −
( ) ( )2 2
( )
( ) 1 1
x x
f x f x
x x
− − −
− = = − = −
− − −
‫إذن‬f‫ﻓﺮدﻳﺔ‬ ‫داﻟﺔ‬
2-‫ﻣﺨﺘﻠﻔﻴﻦ‬ ‫ﻋﻨﺼﺮﻳﻦ‬ ‫ﻟﻜﻞ‬ ‫أن‬ ‫ﻧﺒﺒﻦ‬a‫و‬b‫ﻣﻦ‬fD
( ) ( )
( )( )2 2
1
1 1
f a f b ab
a b a b
− +
=
− − −
‫ﻟﻴﻜﻦ‬a‫و‬b‫ﻣﻦ‬{ }1;1− −‫ﺣﻴﺚ‬a b≠
( ) ( ) ( ) ( )
( )( )
( ) ( )
( )( )( )
( )
( )( )( )
( ) ( ) ( )( )
( )( )( ) ( )( )
2 2
2 2
2 2
2 2 2 2
2 2 2
2 2
2
1 1 11 1
1 1
1 1 1 1
1 1
1 1 1 1
a b
a b b af a f b a b
a b a b a ba b
f a f b a ba b
a b a b a b a b a b
f a f b a b ab ab
a b a b a b a
ab a ba ba
b
b
− −
− − − + −− − −= = ×
− − −− −
− + −+ + −
= =
− − − − − − −
− − + +
= =
−
−
− − − − −
−
3-‫ﺗﻐﻴﺮا‬ ‫ﻣﻨﺤﻰ‬ ‫ﻧﺤﺪد‬‫ت‬f‫ﻋﻠﻰ‬[ [0;1‫و‬] [1;+∞‫ﻋﻠﻰ‬ ‫ﺗﻐﻴﺮاﺗﻬﺎ‬ ‫ﻣﻨﺤﻰ‬ ‫ﻧﺴﺘﻨﺘﺞ‬ ‫و‬] ]1;0−‫و‬] [; 1−∞ −
‫ﻣﺨﺘﻠﻔﻴﻦ‬ ‫ﻋﻨﺼﺮﻳﻦ‬ ‫ﻟﻜﻞ‬ ‫ﻟﺪﻳﻨﺎ‬a‫و‬b‫ﻣﻦ‬{ }1;1− −
( ) ( )
( )( )2 2
1
1 1
f a f b ab
a b a b
− +
=
− − −
‫ﻟﻴﻜﻦ‬a‫و‬b‫ﻣﻦ‬[ [0;1
‫وﻣﻨﻪ‬0 1 ; 0 1a b≤ ≤≺ ≺‫ﺑﺎﻟﺘﺎﻟﻲ‬ ‫و‬
2 2
0 1 0 1 0 1ab et a et b≤ ≤ ≤≺ ≺ ≺
‫وﻣﻨﻪ‬
2 2
1 1 2 1 1 0 1 1 0ab et a et b≤ + − ≤ − − ≤ −≺ ≺ ≺
‫إذن‬
( )( )2 2
1
0
1 1
ab
a b
+
− −
‫وﻣﻨﻪ‬f‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬[ [0;1
‫أن‬ ‫ﺣﻴﺚ‬ ‫و‬f‫ﻓﺎن‬ ‫ﻓﺮدﻳﺔ‬f‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬] ]1;0−
‫ﻟﻴﻜﻦ‬a‫و‬b‫ﻣﻦ‬] [1;+∞
‫وﻣﻨﻪ‬1 ; 1a b‫ﺑﺎﻟﺘﺎﻟﻲ‬ ‫و‬
2 2
1 0 1 1ab et a et b≤
‫وﻣﻨﻪ‬
2 2
1 2 1 0 1 0ab et a et b+ − −
‫إذن‬
( )( )2 2
1
0
1 1
ab
a b
+
− −
‫وﻣﻨﻪ‬f‫ﺗﺰاﻳﺪ‬‫ﻋﻠﻰ‬ ‫ﻳﺔ‬] [1;+∞
‫أن‬ ‫ﺣﻴﺚ‬ ‫و‬f‫ﻓﺎن‬ ‫ﻓﺮدﻳﺔ‬f‫ﻋﻠﻰ‬ ‫ﺗﺰاﻳﺪﻳﺔ‬] [; 1−∞ −
4-‫ﺗﻐﻴﺮات‬ ‫ﺟﺪول‬f
+∞101-−∞x
f
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