S-exp-ii Math Devoir S1 8

‫ﻓﺮض آﺘﺎﺑﻲ ﻓﻲ اﻟﺮﻳﺎﺿﻴﺎت ‪1‬‬

‫ﺛﺎﻧﻮﻳﺔ اﻻﻣﻴﺮ ﻣﻮﻻي اﻟﺮﺷﻴﺪ‪ -‬ﻣﻴﺪﻟﺖ‬ ‫اﻟﺘﻤﺮﻳﻦ اﻻول‪:‬‬ ‫ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ ‪ f‬اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ‬

‫[∞‪I = [1, +‬‬

‫‪2‬ﺑﺎك ‪4+3‬‬

‫اﻟﻤﺪة‪ 2:‬س‬

‫ﺑﻤﺎ ﻳﻠﻲ‪f ( x) = x − 2 x :‬‬

‫‪ (1‬ﺑﻴﻦ ان ‪ f‬داﻟﺔ ﺕﻘﺎﺑﻠﻴﺔ ﻣﻦ ‪ I‬ﻧﺤﻮ ﻣﺠﺎل ‪ J‬ﻳﺘﻢ ﺕﺤﺪﻳﺪﻩ‬

‫‪ (2‬ﺡﺪد )‪f −1 (0‬‬ ‫‪ (3‬اﺡﺴﺐ ﻟﻜﻞ ‪f −1 ( x) : x ∈ J‬‬ ‫اﻟﺘﻤﺮﻳﻦ اﻟﺜﺎﻧﻲ‪:‬‬

‫‪1‬‬ ‫= ‪u0 = 0, un +1‬‬ ‫ﻟﻜﻞ ﻋﺪد ﻃﺒﻴﻌﻲ ‪n‬‬ ‫ﻟﺘﻜﻦ اﻟﻤﺘﺘﺎﻟﻴﺔ اﻟﻤﻌﺮﻓﺔ ب‪:‬‬ ‫‪2 − un‬‬ ‫واآﺘﺒﻬﺎ ﻋﻠﻰ ﺷﻜﻞ آﺴﻮر ﻣﺨﺘﺰﻟﺔ‬ ‫‪ (1‬اﺡﺴﺐ اﻻﻋﺪاد ‪u1 , u2 , u3‬‬ ‫‪n‬‬ ‫‪ (2‬ﻗﺎرن ﺕﺒﺎﻋﺎ اﻟﺤﺪود اﻻرﺑﻌﺔ اﻻوﻟﻰ ﻟﻠﻤﺘﺘﺎﻟﻴﺔ ) ‪ (un‬ﻣﻊ اﻟﺤﺪود اﻻرﺑﻌﺔ اﻻوﻟﻰ ﻟﻠﻤﺘﺘﺎﻟﻴﺔ ) ‪ ( wn‬اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ ` ب‪:‬‬ ‫‪n +1‬‬ ‫‪ (3 (3‬ﺑﺎﺱﺘﻌﻤﺎل اﺱﺘﺪﻻل ﺑﺎﻟﺘﺮﺝﻊ اﺛﺒﺖ ان‪∀n ∈ `, wn = un :‬‬

‫= ‪wn‬‬

‫اﻟﺘﻤﺮﻳﻦ اﻟﺘﺎﻟﺖ‪:‬‬

‫ﻧﻌﺘﺒﺮ اﻟﻤﺘﺘﺎﻟﻴﺔ ) ‪ (un‬اﻟﻤﻌﺮﻓﺔ آﻤﺎ ﻳﻠﻲ‪u0 = 1, un +1 = un + 2n + 3 :‬‬ ‫‪ (1‬ادرس رﺕﺎﺑﺔ اﻟﻤﺘﺘﺎﻟﻴﺔ ) ‪(un‬‬ ‫‪ (2‬ﺑﺮهﻦ ان‪ ’ ∀n ∈ `, un 〉 n :‬ﻣﺎ هﻲ ﻧﻬﺎﻳﺔ اﻟﻤﺘﺘﺎﻟﻴﺔ ) ‪ (un‬؟‬ ‫‪2‬‬

‫‪ (3‬اﺡﺴﺐ اﻟﻤﺠﻤﻮع ‪:‬‬

‫)‪ s = 3 + 5 + 7 + ........ + (2n + 3‬ﺛﻢ اﺡﺴﺐ ‪ un‬ﺑﺪﻻﻟﺔ ‪n‬‬

‫اﻟﺘﻤﺮﻳﻦ اﻟﺮاﺑﻊ‪:‬‬

‫ﻟﺘﻜﻦ اﻟﺪاﻟﺔ ‪f‬‬

‫اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ اﻟﻤﺠﺎل‬

‫‪ (1‬ادرس ﺕﻐﻴﺮات اﻟﺪاﻟﺔ ‪ f‬ﻋﻠﻰ‬

‫‪(2‬‬

‫) ‪ (un‬و‬

‫) ‪(vn‬‬

‫]‪[0, 2‬‬ ‫]‪[0, 2‬‬

‫‪2x +1‬‬ ‫ب‪:‬‬ ‫‪x +1‬‬

‫= )‪f ( x‬‬

‫ﺛﻢ ﺑﻴﻦ ان ]‪f ([ 0, 2]) ⊂ [ 0, 2‬‬

‫ﻣﺘﺘﺎﻟﻴﺘﻴﻦ ﻋﺪدﻳﺘﻴﻦ ﻣﻌﺮﻓﺘﻴﻦ ﻋﻠﻰ ` ` ب‪ u0 = 1, un +1 = f (un ) :‬و ) ‪v0 = 2, vn +1 = f (vn‬‬

‫ﺑﻴﻦ ان ﻟﻜﻞ ` ∈ ‪ 1 ≤ vn ≤ 2 : n‬و ‪vn +1 ≤ vn‬‬ ‫ﺑﻴﻦ ان ﻟﻜﻞ ` ∈ ‪ 1 ≤ un ≤ 2 : n‬و ‪un ≤ un +1‬‬ ‫‪vn − un‬‬ ‫‪ (3‬اﺛﺒﺚ ان ﻟﻜﻞ ` ∈ ‪: n‬‬ ‫= ‪vn +1 − un +1‬‬ ‫)‪(vn + 1)(un + 1‬‬ ‫واﺱﺘﻨﺘﺞ ان ﻟﻜﻞ ` ∈ ‪: n‬‬

‫‪vn − un ≥ 0‬‬

‫و‬

‫‪1‬‬ ‫) ‪vn +1 − un +1 ≤ (vn − un‬‬ ‫‪4‬‬

‫‪n‬‬

‫‪1‬‬ ‫‪ , vn − un ≤  ‬ﻟﻜﻞ ` ∈ ‪n‬‬ ‫‪ (4‬اﺱﺘﻨﺘﺞ ان‪:‬‬ ‫‪4‬‬ ‫ﺑﻴﻦ ان اﻟﻤﺘﺘﺎﻟﻴﺘﻴﻦ ) ‪ (un‬و ) ‪ (vn‬ﻣﺘﻘﺎرﺑﺘﻴﻦ وﻟﻬﻤﺎ ﻧﻔﺲ اﻟﻨﻬﺎﻳﺔ ‪ α‬ﺛﻢ ﺡﺪد ‪α‬‬ ‫اﻟﺘﻤﺮﻳﻦ اﻟﺨﺎﻣﺲ‪:‬‬

‫‪u +v‬‬ ‫‪v0 = 3, vn +1 = n +1 n‬‬ ‫‪2‬‬

‫‪u + vn‬‬ ‫‪ u0 = 3, un +1 = n‬و‬ ‫) ‪ (un‬و ) ‪ (vn‬ﻣﺘﺘﺎﻟﻴﺘﻴﻦ ﻣﻌﺮﻓﺘﻴﻦ ب‪:‬‬ ‫‪2‬‬ ‫‪ (1‬اﺡﺴﺐ ‪u1 , v1 , u2 , v2‬‬ ‫‪1‬‬ ‫ﺛﻢ اﺡﺴﺐ ) ‪ ( wn‬ﺑﺪﻻﻟﺔ ‪ n‬ﺛﻢ اﺱﺘﻨﺘﺞ ان‪ vn ≥ un :‬ﺛﻢ ﺡﺪد ‪lim wn‬‬ ‫ﺑﻴﻦ ان اﻟﻤﺘﺘﺎﻟﻴﺔ ) ‪ ( wn‬هﻨﺪﺱﻴﺔ اﺱﺎﺱﻬﺎ‬ ‫‪ (2‬ﻧﻀﻊ ‪wn = vn − un‬‬ ‫∞→ ‪n‬‬ ‫‪4‬‬ ‫‪ (3‬ﺑﻴﻦ ان ) ‪ (un‬ﺕﺰاﻳﺪﻳﺔ وان ) ‪ (vn‬ﺕﻨﺎﻗﺼﻴﺔ ﺛﻢ اﺱﺘﻨﺘﺞ ان اﻟﻤﺘﺘﺎﻟﻴﺘﻴﻦ ﻣﺘﻘﺎرﺑﺘﻴﻦ وﻟﻬﻤﺎ ﻧﻔﺲ اﻟﻨﻬﺎﻳﺔ‬ ‫‪u + 2vn‬‬ ‫اﻋﺪاد‪:‬ﺑﻮﻏﺎﺑﻲ ﺥﻠﻴﻞ‬ ‫‪ tn = n‬ﺑﻴﻦ ان اﻟﻤﺘﺘﺎﻟﻴﺔ ) ‪ (tn‬ﺛﺎﺑﺘﺔ واﺱﺘﻨﺘﺞ ﻧﻬﺎﻳﺔ ) ‪ (un‬و ) ‪(vn‬‬ ‫ﻧﻀﻊ‬ ‫‪3‬‬ ‫اﷲ وﻟﻲ اﻟﺘﻮﻓﻴﻖ‬ ‫ﺱﻴﺎﺥﺪ ﺑﻌﻴﻦ اﻻﻋﺘﺒﺎر اﻟﻜﺘﺎﺑﺔ اﻟﻮاﺿﺤﺔ و اﻟﺘﻨﻈﻴﻢ اﻟﺠﻴﺪ ﻟﻠﻮرﻗﺔ‬

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