Math Devoir 10

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‫‪ò‹Ñïå‚@óibïä‬‬ ‫‪öa‹èÜa@óá€bÐ@óîíäbq‬‬

‫‪‘ì‹«@‹Ð‬‬

‫‪pbïšbî‹€€€€€€€€€€€€€€€€€ÜaZò†b€€€€€€€€€€€€€€€€€€€€€€€€€€€¾a‬‬ ‫‪p@Ë@l@‘2@ó€åÜaZõín€€€€€€€€€€€€€€€€€€€€€€¾a‬‬ ‫‪2HZŒb−fia@ò‡€€€€€€€€€€€€€€€€€€à‬‬

‫‪Zßìÿa@æî‹ánÜa‬‬ ‫‪3‬‬ ‫‪x +1− 3 x +1‬‬ ‫‪x +5 −2‬‬ ‫‪ 1‬‬ ‫‪ 1 ‬‬ ‫‪lim n  cos   − cos ‬‬ ‫‪، lim‬‬ ‫‪، lim‬‬ ‫أ ا  ت ا‪:‬‬ ‫‪‬‬ ‫‪‬‬ ‫∞‪n →+‬‬ ‫‪x →3‬‬ ‫‪x‬‬ ‫‪x −3‬‬ ‫‪ n + 1   x →0‬‬ ‫‪ n ‬‬ ‫‪  2 n ‬‬ ‫‪n‬‬ ‫‪. lim 3 Arc tan    ‬‬ ‫‪ 3  ‬‬ ‫∞‪n →+‬‬ ‫‪‬‬ ‫‪‬‬

‫‪ZðäbrÜa@æî‹ánÜa‬‬

‫‪1‬‬ ‫‪‬‬ ‫‪a n +1 − a n = n +1‬‬ ‫‪ ‬ﺃﺤﺴﺏ ‪ a n‬ﺒﺩﻻﻝﺔ ‪. n‬‬ ‫ﻝﺘﻜﻥ ) ‪ ( a n‬ﻤﺘﺘﺎﻝﻴﺔ ﻋﺩﺩﻴﺔ ﺒﺤﻴﺙ‪3 :‬‬ ‫‪‬‬ ‫‪a0 = 1‬‬ ‫‪ZsÜbrÜa@æî‹ánÜa‬‬ ‫ﺍﻝﺠﺯﺀ‪:A‬‬

‫‪x +1‬‬ ‫‪‬‬ ‫‪f‬‬ ‫‪x‬‬ ‫=‬ ‫‪− 1; x ≥ 1‬‬ ‫(‬ ‫)‬ ‫‪‬‬ ‫‪x‬‬ ‫ﻝﺘﻜﻥ ‪ f‬ﺩﺍﻝﺔ ﻋﺩﺩﻴﺔ ﺒﺤﻴﺙ‪:‬‬ ‫‪‬‬ ‫‪f ( x ) = x + 1 ;0 ≤ x ≺ 1‬‬ ‫‪‬‬ ‫‪x‬‬ ‫‪ -1‬ﺃﺤﺴﺏ ) ‪ lim f ( x‬ﻭ ) ‪. lim f ( x‬‬ ‫∞‪x →+‬‬

‫‪x →0‬‬

‫ﻓﻲ ‪. x 0 = 1‬‬

‫‪ -2‬ﺃﺩﺭﺱ ﺍﺘﺼﺎل ﺍﻝﺩﺍﻝﺔ ‪f‬‬ ‫‪ -3‬ﺒﻴﻥ ﺃﻥ ﺍﻝﺩﺍﻝﺔ ‪ f‬ﺘﺯﺍﻴﺩﻴﺔ ﻗﻁﻌﺎ ﻋﻠﻰ ﺍﻝﻤﺠﺎل [∞‪. I = [1, +‬‬ ‫‪−1‬‬ ‫‪ -4‬ﺒﻴﻥ ﺃﻥ ‪ f‬ﺘﻘﺎﺒل ﻤﻥ ﺍﻝﻤﺠﺎل ‪ I‬ﻨﺤﻭ ﻤﺠﺎل ‪ J‬ﻴﻨﺒﻐﻲ ﺘﺤﺩﻴﺩﻩ ﺜﻡ ﺤﺩﺩ ) ‪ f ( x‬ﻝﻜل ‪ x‬ﻤﻥ ‪. J‬‬ ‫‪x +1‬‬ ‫= ‪ f ( x ) − x‬ﺜﻡ ﺍﺴﺘﻨﺘﺞ ﺇﺸﺎﺭﺓ ‪ f ( x ) − x‬ﻋﻠﻰ ‪. I‬‬ ‫‪ -5‬ﺒﻴﻥ ﺃﻥ‪1 − x ; ∀x ∈ I :‬‬ ‫‪x‬‬

‫)‬

‫(‬

‫ﺍﻝﺠﺯﺀ‪:B‬‬

‫‪1+ un‬‬ ‫‪‬‬ ‫‪− 1; ∀n ∈ ℕ‬‬ ‫= ‪u n +1‬‬ ‫‪u‬‬ ‫ﻝﺘﻜﻥ ) ‪ (u n‬ﺩﺍﻝﺔ ﻋﺩﺩﻴﺔ ﺒﺤﻴﺙ‪:‬‬ ‫‪‬‬ ‫‪n‬‬ ‫‪‬‬ ‫‪u0 = 2‬‬ ‫‪‬‬ ‫‪ -1‬ﺒﻴﻥ ﺒﺎﻝﺘﺭﺠﻊ ﺃﻥ‪. ∀n ∈ ℕ : u n ≥ 1:‬‬ ‫‪ -2‬ﺃﺩﺭﺱ ﺭﺘﺎﺒﺔ ﺍﻝﻤﺘﺘﺎﻝﻴﺔ ) ‪ (u n‬ﻭﺍﺴﺘﻨﺘﺞ ﺃﻥ‪. ∀n ∈ ℕ : u n ≤ 2 :‬‬

‫‪ -3‬ﻫل ﺍﻝﻤﺘﺘﺎﻝﻴﺔ ) ‪ (u n‬ﻤﺘﻘﺎﺭﺒﺔ؟ﺇﺫﺍ ﻜﺎﻥ ﺍﻝﺠﻭﺍﺏ ﺒﻨﻌﻡ ﺃﺤﺴﺏ ) ‪(u n‬‬

‫‪. lim‬‬

‫∞‪x →+‬‬

‫ﺍﻝﺠﺯﺀ‪:C‬‬

‫‪‬‬ ‫‪1 +v n‬‬ ‫‪; ∀n ∈ ℕ‬‬ ‫= ‪v n +1‬‬ ‫‪2‬‬ ‫ﻝﺘﻜﻥ ) ‪ (v n‬ﺩﺍﻝﺔ ﻋﺩﺩﻴﺔ ﺒﺤﻴﺙ‪:‬‬ ‫‪‬‬ ‫‪1‬‬ ‫‪‬‬ ‫= ‪v0‬‬ ‫‪‬‬ ‫‪2‬‬ ‫‪ π ‬‬ ‫‪v n = cos ‬ﻝﻜل ‪ n‬ﻤﻥ ‪ ℕ‬ﺜﻡ ﺃﺤﺴﺏ ) ‪. lim (v n‬‬ ‫‪ -1‬ﺒﻴﻥ ﺃﻥ‪ -2. ∀n ∈ ℕ : 0 ≤ v n ≤ 1 :‬ﺒﻴﻥ ﺒﺎﻝﺘﺭﺠﻊ ﺃﻥ‪:‬‬ ‫‪n ‬‬ ‫∞‪x →+‬‬ ‫‪ 3.2 ‬‬

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