S-exp-ii Math Devoir S2 3
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اﻟﻤﺴﺘﻮى :اﻟـﺜﺎﻧﻴﺔ ﺛﺎﻧﻮي اﻟﺸﻌﺒﺔ :اﻟﻌﻠﻮم اﻟﺘﺠﺮﻳﺒﻴﺔ اﻷﺳﺘﺎذ :ﻣــــــــﻮﻣــــــﺎد ﺳﻠﻢ اﻟﺘﻨﻘﻴﻂ:
اﻟﻤﺆﺳـــﺴــــﺔ :ﻣـــﻮﺳﻰ ﺑـﻦ ﻧـﺼـﻴـﺮ اﻟﻨﻴﺎﺑﺔ:ﻣﺮاآﺶ ﺳﻴﺪي ﻳﻮﺳﻒ ﺑﻦ ﻋﻠﻲ اﻟــﺴــﻨــﺔ اﻟــﺪراﺳـﻴـﺔ2006/2005:
اﻟﻔﺮض اﻟﻤﺤﺮوس اﻷول اﻟﺪورة اﻟـﺜـﺎﻧـﻴـــــــﺔ ﻣﺪة اﻹﻧﺠﺎز :ﺳﺎﻋﺘـــــﺎن ﻣﻼﺣﻈﺔ :ﻳﺮاﻋﻰ ﻓﻲ اﻟﺘﺼﺤﻴﺢ ﺳﻼﻣﺔ اﻟﺘﻌﺒﻴﺮ و ﺣﺴﻦ اﻟﺘﻘﺪﻳﻢ
اﻟﺘﻤﺮﻳﻦ اﻷول 8) :ن(
⎞⎛0 ⎞ ⎛ −1 ⎞⎛1 ⎞⎛0 JG JJG JG ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ﻓﻲ اﻟﻔﻀﺎء ξاﻟﻤﻨﺴﻮب إﻟﻰ ﻣﻌﻠﻢ ﻣﺘﻌﺎﻣﺪ ﻣﻤﻨﻈﻢ ﻣﺒﺎﺷﺮ ، ℜ = O , i , j , kﻧﻌﺘﺒﺮ اﻟﻨﻘﻂ ⎟ C ⎜1⎟ ، B ⎜ 0 ⎟ ، A ⎜ 1و ⎟ Ω ⎜ 1 ⎟⎜1 ⎟⎜0 ⎟⎜1 ⎟⎜0 ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝
)
1 0,5 1
(
JJJG JJJJG (1أﺣﺴﺐ A B ∧ ACواﺳﺘﻨﺘﺞ أن B ، Aو Cﺗﺤﺪد ﻣﺴﺘﻮى ﻣﻌﺎدﻟﺘﻪ هﻲ− x + y + z − 1 = 0 : 3 (2ﻧﻌﺘﺒﺮ اﻟﻔﻠﻜﺔ ) (Sاﻟﺘﻲ ﻣﺮآﺰهﺎ Ωوﺷﻌﺎﻋﻬﺎ 3 (aأﻋﻂ اﻟﻤﻌﺎدﻟﺔ اﻟﺪﻳﻜﺎرﺗﻴﺔ ﻟﻠﻔﻠﻜﺔ ) (S (bأﺣﺴﺐ ﻣﺴﺎﻓﺔ اﻟﻨﻘﻄﺔ Ωﻋﻦ اﻟﻤﺴﺘﻮى ) . ( A BCﻣﺎذا ﺗﺴﺘﻨﺘﺞ؟ (3ﻟﻴﻜﻦ ) (Qاﻟﻤﺴﺘﻮى اﻟﺬي ﻣﻌﺎدﻟﺘﻪ x − y + 2z − 2 = 0
(a (b (c (d (e
1 1 1 1,5 1
ﺑﻴﻦ أن اﻟﻤﺴﺘﻮﻳﻴﻦ ) ( A BCو ) (Qﻣﺘﻌﺎﻣﺪان ﺑﻴﻦ أن اﻟﻤﺴﺘﻮى ) (Qو اﻟﻔﻠﻜﺔ ) (Sﻳﺘﻘﺎﻃﻌﺎن وﻓﻖ داﺋﺮة ) ( Γ أﻋﻂ ﺗﻤﺜﻴﻼ ﺑﺎرا ﻣﺘﺮﻳﺎ ﻟﻠﻤﺴﺘﻘﻴﻢ ) ( Δاﻟﻤﺎر ﻣﻦ Ωو اﻟﻌﻤﻮدي ﻋﻠﻰ اﻟﻤﺴﺘﻮى ) (Q ﺣﺪد ﻣﺮآﺰ و ﺷﻌﺎع اﻟﺪاﺋﺮة ) ( Γ ﺣﺪد ﺗﻘﺎﻃﻊ اﻟﻤﺴﺘﻘﻴﻢ ) ( Δﻣﻊ اﻟﻔﻠﻜﺔ ) . (S
اﻟﺘﻤﺮﻳﻦ اﻟﺜﺎﻧﻲ12 ) :ن( ﻟﺘﻜﻦ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ fﻟﻠﻤﺘﻐﻴﺮ اﻟﺤﻘﻴﻘﻲ xﺑﺤﻴﺚ :
وﻟﻴﻜﻦ
)
⎧f (x ) = x ln(x ) − x ; x > 0 ⎪ ⎨ ⎪f (x ) = xe − x ; x ≤ 0 ⎩
JG JG (C fاﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ اﻟﻤﺴﺘﻮى Pاﻟﻤﻨﺴﻮب إﻟﻰ ﻣﻌﻠﻢ ﻣﺘﻌﺎﻣﺪ ﻣﻤﻨﻈﻢ . ℜ = O , i , j
)
2
(1أﺛﺒﺖ أن fﻣﺘﺼﻠﺔ ﻓﻲ .0
2
(2اﺣﺴﺐ ﻧﻬﺎﻳﺘﻲ اﻟﺪاﻟﺔ fﻋﻨﺪ ∞ +و ﻋﻨﺪ ∞. −
1
) f (x ) f (x lim+و أن = 1 –(a (3أﺛﺒﺖ أن ∞ = − x → 0 x x
1
-(bأﻋﻂ ﺗﺄوﻳﻼ ﻣﺒﻴﺎﻧﻴﺎ ﻟﻠﻨﻬﺎﻳﺘﻴﻦ اﻟﺴﺎﺑﻘﺘﻴﻦ).اﻟﺴﺆال ( a
1
-(a (4اﺣﺴﺐ ) f ' (xﻟﻜﻞ xﻣﻦ \ −و أﺛﺒﺖ أﻧﻬﺎ ﻣﻮﺟﺒﺔ.
1
-(bاﺣﺴﺐ ) f ' (xﻟﻜﻞ xﻣﻦ \ +و ﺣﺪد إﺷﺎرﺗﻬﺎ.
1
-(cأﻋﻂ ﺟﺪول ﺗﻐﻴﺮات اﻟﺪاﻟﺔ . f
1
)
2
(5ادرس اﻟﻔﺮوع اﻟﻼﻧﻬﺎﺋﻴﺔ ﻟﻠﻤﻨﺤﻨﻰ (6ارﺳﻢ اﻟﻤﻨﺤﻨﻰ
lim
x → 0−
. (C f
) (C f http://arabmaths.ift.fr
(
اﻟﻤﺆﺳـــﺴــــﺔ :ﻣـــﻮﺳﻰ ﺑـﻦ ﻧـﺼـﻴـﺮ اﻟﻨﻴﺎﺑﺔ:ﻣﺮاآﺶ ﺳﻴﺪي ﻳﻮﺳﻒ ﺑﻦ ﻋﻠﻲ اﻟــﺴــﻨــﺔ اﻟــﺪراﺳـﻴـﺔ2006/2005:
اﻟﻔﺮض اﻟﻤﺤﺮوس اﻷول اﻟﺪورة اﻟـﺜـﺎﻧـﻴـــــــﺔ ﻣﺪة اﻹﻧﺠﺎز :ﺳﺎﻋﺘـــــﺎن ﻣﻼﺣﻈﺔ :ﻳﺮاﻋﻰ ﻓﻲ اﻟﺘﺼﺤﻴﺢ ﺳﻼﻣﺔ اﻟﺘﻌﺒﻴﺮ و ﺣﺴﻦ اﻟﺘﻘﺪﻳﻢ
اﻟﺘﻤﺮﻳﻦ اﻷول 8) :ن(
⎞⎛0 ⎞ ⎛ −1 ⎞⎛1 ⎞⎛0 JG JJG JG ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ﻓﻲ اﻟﻔﻀﺎء ξاﻟﻤﻨﺴﻮب إﻟﻰ ﻣﻌﻠﻢ ﻣﺘﻌﺎﻣﺪ ﻣﻤﻨﻈﻢ ﻣﺒﺎﺷﺮ ، ℜ = O , i , j , kﻧﻌﺘﺒﺮ اﻟﻨﻘﻂ ⎟ C ⎜1⎟ ، B ⎜ 0 ⎟ ، A ⎜ 1و ⎟ Ω ⎜ 1 ⎟⎜1 ⎟⎜0 ⎟⎜1 ⎟⎜0 ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝
)
1 0,5 1
(
JJJG JJJJG (1أﺣﺴﺐ A B ∧ ACواﺳﺘﻨﺘﺞ أن B ، Aو Cﺗﺤﺪد ﻣﺴﺘﻮى ﻣﻌﺎدﻟﺘﻪ هﻲ− x + y + z − 1 = 0 : 3 (2ﻧﻌﺘﺒﺮ اﻟﻔﻠﻜﺔ ) (Sاﻟﺘﻲ ﻣﺮآﺰهﺎ Ωوﺷﻌﺎﻋﻬﺎ 3 (aأﻋﻂ اﻟﻤﻌﺎدﻟﺔ اﻟﺪﻳﻜﺎرﺗﻴﺔ ﻟﻠﻔﻠﻜﺔ ) (S (bأﺣﺴﺐ ﻣﺴﺎﻓﺔ اﻟﻨﻘﻄﺔ Ωﻋﻦ اﻟﻤﺴﺘﻮى ) . ( A BCﻣﺎذا ﺗﺴﺘﻨﺘﺞ؟ (3ﻟﻴﻜﻦ ) (Qاﻟﻤﺴﺘﻮى اﻟﺬي ﻣﻌﺎدﻟﺘﻪ x − y + 2z − 2 = 0
(a (b (c (d (e
1 1 1 1,5 1
ﺑﻴﻦ أن اﻟﻤﺴﺘﻮﻳﻴﻦ ) ( A BCو ) (Qﻣﺘﻌﺎﻣﺪان ﺑﻴﻦ أن اﻟﻤﺴﺘﻮى ) (Qو اﻟﻔﻠﻜﺔ ) (Sﻳﺘﻘﺎﻃﻌﺎن وﻓﻖ داﺋﺮة ) ( Γ أﻋﻂ ﺗﻤﺜﻴﻼ ﺑﺎرا ﻣﺘﺮﻳﺎ ﻟﻠﻤﺴﺘﻘﻴﻢ ) ( Δاﻟﻤﺎر ﻣﻦ Ωو اﻟﻌﻤﻮدي ﻋﻠﻰ اﻟﻤﺴﺘﻮى ) (Q ﺣﺪد ﻣﺮآﺰ و ﺷﻌﺎع اﻟﺪاﺋﺮة ) ( Γ ﺣﺪد ﺗﻘﺎﻃﻊ اﻟﻤﺴﺘﻘﻴﻢ ) ( Δﻣﻊ اﻟﻔﻠﻜﺔ ) . (S
اﻟﺘﻤﺮﻳﻦ اﻟﺜﺎﻧﻲ12 ) :ن( ﻟﺘﻜﻦ اﻟﺪاﻟﺔ اﻟﻌﺪدﻳﺔ fﻟﻠﻤﺘﻐﻴﺮ اﻟﺤﻘﻴﻘﻲ xﺑﺤﻴﺚ :
وﻟﻴﻜﻦ
)
⎧f (x ) = x ln(x ) − x ; x > 0 ⎪ ⎨ ⎪f (x ) = xe − x ; x ≤ 0 ⎩
JG JG (C fاﻟﻤﻨﺤﻨﻰ اﻟﻤﻤﺜﻞ ﻟﻠﺪاﻟﺔ fﻓﻲ اﻟﻤﺴﺘﻮى Pاﻟﻤﻨﺴﻮب إﻟﻰ ﻣﻌﻠﻢ ﻣﺘﻌﺎﻣﺪ ﻣﻤﻨﻈﻢ . ℜ = O , i , j
)
2
(1أﺛﺒﺖ أن fﻣﺘﺼﻠﺔ ﻓﻲ .0
2
(2اﺣﺴﺐ ﻧﻬﺎﻳﺘﻲ اﻟﺪاﻟﺔ fﻋﻨﺪ ∞ +و ﻋﻨﺪ ∞. −
1
) f (x ) f (x lim+و أن = 1 –(a (3أﺛﺒﺖ أن ∞ = − x → 0 x x
1
-(bأﻋﻂ ﺗﺄوﻳﻼ ﻣﺒﻴﺎﻧﻴﺎ ﻟﻠﻨﻬﺎﻳﺘﻴﻦ اﻟﺴﺎﺑﻘﺘﻴﻦ).اﻟﺴﺆال ( a
1
-(a (4اﺣﺴﺐ ) f ' (xﻟﻜﻞ xﻣﻦ \ −و أﺛﺒﺖ أﻧﻬﺎ ﻣﻮﺟﺒﺔ.
1
-(bاﺣﺴﺐ ) f ' (xﻟﻜﻞ xﻣﻦ \ +و ﺣﺪد إﺷﺎرﺗﻬﺎ.
1
-(cأﻋﻂ ﺟﺪول ﺗﻐﻴﺮات اﻟﺪاﻟﺔ . f
1
)
2
(5ادرس اﻟﻔﺮوع اﻟﻼﻧﻬﺎﺋﻴﺔ ﻟﻠﻤﻨﺤﻨﻰ (6ارﺳﻢ اﻟﻤﻨﺤﻨﻰ
lim
x → 0−
. (C f
) (C f http://arabmaths.ift.fr
(
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