Tajribi Math Sx (34)

‫ﻮع‬ ‫ﻮﺿ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬

‫ﺔ‬ ‫ﯿ‬ ‫ﻠ‬ ‫ﯿ‬ ‫ﻫ‬ ‫ﺄ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫ﺎت‬ ‫ﻨ‬ ‫ﻣ‬ ‫ﺔد‬ ‫ﯾ‬ ‫ﻮ‬ ‫ﻧ‬ ‫ﺎ‬ ‫ﺛ‬ ‫ل‬ ‫ﯾﻼ‬ ‫ز‬ ‫ﺎت ‪-‬أ‬ ‫ﻨ‬ ‫ﻣ‬ ‫د‬ ‫ﺎ‬ ‫ﯾ‬ ‫ر‬ ‫ﻮ‬ ‫ﻟ‬ ‫ﺎ‬ ‫ﻜ‬ ‫ﺒ‬ ‫ﻟ‬ ‫ا‬ ‫ن‬ ‫ﺎ‬ ‫ﺘﺤ‬ ‫ﻣ‬ ‫ا‬ ‫ﺪ‬ ‫ﺣ‬ ‫ﻮ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﺒﻲ‬ ‫ﯾ‬ ‫ﺮ‬ ‫ﺠ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫ن‬ ‫ﺎ‬ ‫ﺤ‬ ‫ﺘ‬ ‫ﻣ‬ ‫ﻻ‬ ‫ا‬ ‫ﻞ ‪2007‬‬ ‫ﯾ‬ ‫ﺮ‬ ‫ﺑ‬ ‫أ‬ ‫ة‬ ‫ر‬ ‫و‬ ‫د‬

‫ﺎت‬ ‫ﯿ‬ ‫ﺎﺿ‬ ‫ﯾ‬ ‫ﺮ‬ ‫ﻟ‬ ‫ا‬ ‫ة‪:‬‬ ‫د‬ ‫ﺎ‬ ‫ﻣ‬ ‫ﺎت‬ ‫ﻋ‬ ‫ﺎ‬ ‫ز ‪3 :‬ﺳ‬ ‫ﺎ‬ ‫ﺠ‬ ‫ﻧ‬ ‫ﻹ‬ ‫ا‬ ‫ة‬ ‫ﺪ‬ ‫ﻣ‬

‫ﻢ‬ ‫ﻠ‬ ‫ﺳ‬ ‫ﯿﻂ‬ ‫ﻘ‬ ‫ﻨ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬

‫ﻠﻚ‬ ‫ﻦﺳ‬ ‫ﺔﻣ‬ ‫ﯿ‬ ‫ﻧ‬ ‫ﺎ‬ ‫ﺜ‬ ‫ﻟ‬ ‫ﻮى ‪:‬ا‬ ‫ﺘ‬ ‫ﻤﺴ‬ ‫ﻟ‬ ‫ا‬ ‫ﺎ‬ ‫ﯾ‬ ‫ر‬ ‫ﻮ‬ ‫ﻟ‬ ‫ﺎ‬ ‫ﻜ‬ ‫ﺒ‬ ‫ﻟ‬ ‫ا‬ ‫ﺔ‬ ‫ﯿ‬ ‫ﺒ‬ ‫ﯾ‬ ‫ﺮ‬ ‫ﺘﺠ‬ ‫ﻟ‬ ‫ا‬ ‫م‬ ‫ﻮ‬ ‫ﻠ‬ ‫ﻌ‬ ‫ﻟ‬ ‫ا‬ ‫ﺔ‪:‬‬ ‫ﺒ‬ ‫ﻌ‬ ‫ﻟﺸ‬ ‫ا‬

‫‪1/2‬‬

‫ﺔ‬ ‫ﺠ‬ ‫ﻣ‬ ‫ﺮ‬ ‫ﺒ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﺮ‬ ‫ﯿ‬ ‫ﺔﻏ‬ ‫ﺒ‬ ‫ﺎﺳ‬ ‫ﺤ‬ ‫ﻟ‬ ‫ا‬ ‫ﺔ‬ ‫ﻟ‬ ‫ﻵ‬ ‫ا‬ ‫ل‬ ‫ﺎ‬ ‫ﻤ‬ ‫ﻌ‬ ‫ﺘ‬ ‫ﺎﺳ‬ ‫ﺑ‬ ‫ﻤﺢ‬ ‫ﯾﺴ‬

‫ل‪2.5 ) :‬ن(‬ ‫و‬ ‫ﻷ‬ ‫ا‬ ‫ﯾﻦ‬ ‫ﺮ‬ ‫ﻤ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫ﺔ )‪A(1,1,1‬‬ ‫ﻘﻄ‬ ‫ﻨ‬ ‫ﻟ‬ ‫ا‬ ‫ﺮ ) ‪(O, i , j , k‬‬ ‫ﺷ‬ ‫ﺎ‬ ‫ﺒ‬ ‫ﻢﻣ‬ ‫ﻨﻈ‬ ‫ﻤ‬ ‫ﺪﻣ‬ ‫ﻣ‬ ‫ﺎ‬ ‫ﻌ‬ ‫ﺘ‬ ‫ﻢﻣ‬ ‫ﻠ‬ ‫ﻌ‬ ‫ﻟﻰﻣ‬ ‫إ‬ ‫ﻮب‬ ‫ﺴ‬ ‫ﻨ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ء‪‬ا‬ ‫ﺎ‬ ‫ﻔﻀ‬ ‫ﻟ‬ ‫ا‬ ‫ﻓﻲ‬ ‫ﺮ‬ ‫ﺒ‬ ‫ﺘ‬ ‫ﻌ‬ ‫ﻧ‬ ‫ﻪ ‪. x y z 2 0‬‬ ‫ﺘ‬ ‫ﻟ‬ ‫د‬ ‫ﺎ‬ ‫ﻌ‬ ‫ﺬيﻣ‬ ‫ﻟ‬ ‫ﻮى )‪(P‬ا‬ ‫ﺘ‬ ‫ﻤﺴ‬ ‫ﻟ‬ ‫ا‬ ‫و‬ ‫‪0.25‬‬ ‫ﻮى ) ‪. (P‬‬ ‫ﺘ‬ ‫ﺴ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﻠﻰ‬ ‫ديﻋ‬ ‫ﻮ‬ ‫ﻤ‬ ‫ﻌ‬ ‫ﻟ‬ ‫ا‬ ‫ﺔ ‪A‬و‬ ‫ﻘﻄ‬ ‫ﻨ‬ ‫ﻟ‬ ‫ا‬ ‫رﻣﻦ‬ ‫ﺎ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ﻢ )‪(D‬ا‬ ‫ﯿ‬ ‫ﻘ‬ ‫ﺘ‬ ‫ﻤﺴ‬ ‫ﻠ‬ ‫ﻟ‬ ‫ﺎ‬ ‫ﯾ‬ ‫ﺮ‬ ‫ﺘ‬ ‫اﻣ‬ ‫ر‬ ‫ﺎ‬ ‫ﺑ‬ ‫ﯿﻼ‬ ‫ﺜ‬ ‫ﻤ‬ ‫ﺗ‬ ‫د‬ ‫ﺪ‬ ‫‪ -1‬ﺣ‬ ‫‪0.5‬‬ ‫ﻮى ) ‪. (P‬‬ ‫ﺘ‬ ‫ﺴ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﻢ )‪ (D‬و‬ ‫ﯿ‬ ‫ﻘ‬ ‫ﺘ‬ ‫ﻤﺴ‬ ‫ﻟ‬ ‫ا‬ ‫ﻊ‬ ‫ﺎﻃ‬ ‫ﻘ‬ ‫ﺗ‬ ‫ﺔ‬ ‫ﻘﻄ‬ ‫ﺎت ‪B‬ﻧ‬ ‫ﯿ‬ ‫ﺛ‬ ‫ا‬ ‫ﺪ‬ ‫إﺣ‬ ‫د‬ ‫ﺪ‬ ‫‪ -2‬ﺣ‬ ‫ﺎ ‪.r = 7‬‬ ‫ﻬ‬ ‫ﺎﻋ‬ ‫ﻌ‬ ‫ﺎ ‪A‬وﺷ‬ ‫ﻫ‬ ‫ﺰ‬ ‫ﻛ‬ ‫ﺮ‬ ‫ﺘﻲﻣ‬ ‫ﻟ‬ ‫ﺔ ) ‪(S‬ا‬ ‫ﻜ‬ ‫ﻠ‬ ‫ﻔ‬ ‫ﻟ‬ ‫ا‬ ‫ﺮ‬ ‫ﺒ‬ ‫ﺘ‬ ‫ﻌ‬ ‫‪-3‬ﻧ‬ ‫ﺔ ) ‪. (S‬‬ ‫ﻜ‬ ‫ﻠ‬ ‫ﻔ‬ ‫ﻠ‬ ‫ﻟ‬ ‫ﺔ‬ ‫ﯿ‬ ‫ﺗ‬ ‫ر‬ ‫ﺎ‬ ‫ﻜ‬ ‫ﯾ‬ ‫ﺔد‬ ‫ﻟ‬ ‫د‬ ‫ﺎ‬ ‫ﻌ‬ ‫ﻋﻂﻣ‬ ‫أ‪-‬أ‬ ‫‪0.25‬‬ ‫ﺎ‪.‬‬ ‫ﻬ‬ ‫ﻋ‬ ‫ﺎ‬ ‫ﻌ‬ ‫ﺎوﺷ‬ ‫ﻫ‬ ‫ﺰ‬ ‫ﻛ‬ ‫ﺮ‬ ‫اﻣ‬ ‫د‬ ‫ﺪ‬ ‫ة ) ‪(C‬ﻣﺤ‬ ‫ﺮ‬ ‫ﺋ‬ ‫ا‬ ‫ﻓﻖد‬ ‫ﺔ ) ‪(S‬و‬ ‫ﻜ‬ ‫ﻠ‬ ‫ﻔ‬ ‫ﻟ‬ ‫ا‬ ‫ﻊ‬ ‫ﻘﻄ‬ ‫ﻮى ) ‪(P‬ﯾ‬ ‫ﺘ‬ ‫ﺴ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ن‬ ‫أ‬ ‫ﻦ‬ ‫ﯿ‬ ‫ب‪-‬ﺑ‬ ‫‪0.5‬‬ ‫ﺔ ) ‪. (S‬‬ ‫ﻜ‬ ‫ﻠ‬ ‫ﻔ‬ ‫ﻠ‬ ‫ﻟ‬ ‫ﻦ‬ ‫ﯿ‬ ‫ﺳ‬ ‫ﺎ‬ ‫ﻤ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﻮى )‪(P‬و‬ ‫ﺘ‬ ‫ﻤﺴ‬ ‫ﻠ‬ ‫ﻟ‬ ‫ﻦ‬ ‫ﯿ‬ ‫ﯾ‬ ‫ز‬ ‫ا‬ ‫ﻮ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﯿﻦ‬ ‫ﯾ‬ ‫ﻮ‬ ‫ﺘ‬ ‫ﻤﺴ‬ ‫ﻟ‬ ‫ا‬ ‫ﺘﻲ‬ ‫ﻟ‬ ‫د‬ ‫ﺎ‬ ‫ﻌ‬ ‫دﻣ‬ ‫ﺪ‬ ‫‪-4‬ﺣ‬ ‫‪1‬‬ ‫ﻧﻲ‪3.5) :‬ن(‬ ‫ﺎ‬ ‫ﺜ‬ ‫ﻟ‬ ‫ا‬ ‫ﯾﻦ‬ ‫ﺮ‬ ‫ﻤ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫‪3‬‬ ‫‪2‬‬ ‫ﯿﺚ ‪P( z ) z 2(2 3i ) z 4(1 5i )z 16(1 i) :‬‬ ‫ﺔ )‪P (z‬ﺣ‬ ‫ﯾ‬ ‫د‬ ‫و‬ ‫ﺪ‬ ‫ﻟﺤ‬ ‫ﻓﻲ ‪C‬ا‬ ‫ﺮ‬ ‫ﺒ‬ ‫ﺘ‬ ‫ﻌ‬ ‫ﻧ‬ ‫‪0.25‬‬ ‫ﺔ ‪.P‬‬ ‫ﯾ‬ ‫د‬ ‫و‬ ‫ﺪ‬ ‫ﻠﺤ‬ ‫ﻟ‬ ‫ر‬ ‫ﺬ‬ ‫أن ‪z0 2‬ﺟ‬ ‫ﻘﻖ‬ ‫ﺤ‬ ‫أ‪-‬ﺗ‬ ‫‪-1‬‬ ‫‪2‬‬ ‫‪0.5‬‬ ‫ﯿﺚ ‪P( z ) ( z z 0 )( z az b) :‬‬ ‫ﻦ ‪ a‬و ‪b‬ﺑﺤ‬ ‫ﯿ‬ ‫ﯾ‬ ‫د‬ ‫ﺪ‬ ‫ﻘ‬ ‫ﻌ‬ ‫ﻟ‬ ‫ا‬ ‫ﻦ‬ ‫ﯾ‬ ‫د‬ ‫ﺪ‬ ‫ﻌ‬ ‫ﻟ‬ ‫ا‬ ‫د‬ ‫ﺪ‬ ‫ب‪ -‬ﺣ‬ ‫‪1‬‬ ‫ﺔ‪. (E ) : P ( z) 0 :‬‬ ‫ﻟ‬ ‫د‬ ‫ﺎ‬ ‫ﻌ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﻓﻲ ‪C‬‬ ‫ﻞ‬ ‫ج‪-‬ﺣ‬ ‫‪1‬‬ ‫ﯿﺚ ‪. e (z 2 ) 0‬‬ ‫ﺔ ) ‪(E‬ﺣ‬ ‫ﻟ‬ ‫د‬ ‫ﺎ‬ ‫ﻌ‬ ‫ﻤ‬ ‫ﻠ‬ ‫ﻟ‬ ‫ﯾﻦ‬ ‫ﺮ‬ ‫ﺧ‬ ‫اﻵ‬ ‫ﯿﻦ‬ ‫ﻠ‬ ‫ﻟﺤ‬ ‫ﻜﻦ ‪z1‬و ‪z 2‬ا‬ ‫ﯿ‬ ‫‪-2‬ﻟ‬ ‫ﻢ‬ ‫ل ‪u 0 z0‬ﺛ‬ ‫و‬ ‫ﻷ‬ ‫ا‬ ‫ﺎ‬ ‫ﻫ‬ ‫ﺪ‬ ‫ﺔﺣ‬ ‫ﯿ‬ ‫ﺪﺳ‬ ‫ﻨ‬ ‫ﺔﻫ‬ ‫ﯿ‬ ‫ﻟ‬ ‫ﺎ‬ ‫ﺘ‬ ‫ﺘ‬ ‫ﺔﻣﻦﻣ‬ ‫ﻌ‬ ‫ﺑ‬ ‫ﺎ‬ ‫ﺘ‬ ‫ﺘ‬ ‫دﻣ‬ ‫و‬ ‫ﺪ‬ ‫ﺔ ) ‪(E‬ﻫﻲﺣ‬ ‫ﻟ‬ ‫د‬ ‫ﺎ‬ ‫ﻌ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﻮل‬ ‫ﻠ‬ ‫أنﺣ‬ ‫ﯿﻦ‬ ‫ﺑ‬ ‫ﺪ ‪. u16‬‬ ‫ﻟﺤ‬ ‫ا‬ ‫ﺎ ‪q‬و‬ ‫ﻬ‬ ‫ﺳ‬ ‫ﺎ‬ ‫أﺳ‬ ‫د‬ ‫ﺪ‬ ‫ﺣ‬ ‫‪0.25‬‬ ‫ﻘﻂ )‪ A( 2‬و ) ‪ B( 2 2i‬و ) ‪. C (4i‬‬ ‫ﻨ‬ ‫ﻟ‬ ‫ا‬ ‫ﺪي‬ ‫ﻘ‬ ‫ﻌ‬ ‫ﻟ‬ ‫ا‬ ‫ﻮى‬ ‫ﺘ‬ ‫ﺴ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﻞﻓﻲ‬ ‫ﺜ‬ ‫أ‪-‬ﻣ‬ ‫‪-3‬‬ ‫‪0.5‬‬ ‫ﺔ )‪ (A,1‬و )‪( B, 1‬و )‪. (C,1‬‬ ‫ﻧ‬ ‫ﺰ‬ ‫ﺘ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﻘﻂ‬ ‫ﻨ‬ ‫ﻟ‬ ‫ا‬ ‫ﺢ‬ ‫ﺮﺟ‬ ‫ﺔ ‪G‬ﻣ‬ ‫ﻘﻄ‬ ‫ﻨ‬ ‫ﻟ‬ ‫ا‬ ‫ﻟﺤﻖ‬ ‫د‬ ‫ﺪ‬ ‫ب‪ -‬ﺣ‬ ‫ﻟﺚ ‪3.5) :‬ن(‬ ‫ﺎ‬ ‫ﺜ‬ ‫ﻟ‬ ‫ا‬ ‫ﯾﻦ‬ ‫ﺮ‬ ‫ﻤ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫‪1‬‬ ‫‪n x‬‬ ‫*‬ ‫‪. I n ‬‬ ‫ﻊ ‪x e dx‬‬ ‫ﻞ ‪n IN‬ﻧﻀ‬ ‫أﺟ‬ ‫ﻣﻦ‬ ‫‪0‬‬ ‫‪0.5‬‬ ‫‪0.5‬‬ ‫‪0.5‬‬ ‫‪0.5‬‬ ‫‪0.75‬‬ ‫‪0.25‬‬ ‫‪0.5‬‬

‫ﺴﺐ ‪. I 1‬‬ ‫اﺣ‬ ‫ء‬ ‫ا‬ ‫ﺰ‬ ‫ﻷﺟ‬ ‫ﺎ‬ ‫ﺑ‬ ‫ﺔ‬ ‫ﻠ‬ ‫ﻣ‬ ‫ﺎ‬ ‫ﻜ‬ ‫لﻣ‬ ‫ﺎ‬ ‫ﻤ‬ ‫ﻌ‬ ‫ﺘ‬ ‫ﺳ‬ ‫ﺎ‬ ‫ﺑ‬ ‫‪-1‬‬ ‫‪IN * ‬‬ ‫ﻦ‪1‬‬ ‫ﻞ ‪n‬ﻣ‬ ‫ﻜ‬ ‫ﻟ‬ ‫ن‬ ‫أ‬ ‫ﯿﻦ‬ ‫أ‪-‬ﺑ‬ ‫‪-2‬‬

‫‪e‬‬

‫ﺣﺴﺐ ‪ I 2‬و ‪. I 3‬‬ ‫ب‪-‬ا‬

‫‪1‬‬

‫‪.‬‬ ‫ﻞ ‪(2 x 3 4 x 2 )e x dx‬‬ ‫ﻣ‬ ‫ﺎ‬ ‫ﻜ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫ﺴﺐ‬ ‫ج‪-‬اﺣ‬ ‫‪0‬‬ ‫د ‪.0‬‬ ‫ﺪ‬ ‫ﻌ‬ ‫ﻟ‬ ‫ﺎ‬ ‫ﺑ‬ ‫ة‬ ‫ر‬ ‫ﻮ‬ ‫ﻐ‬ ‫ﺔوﻣﺼ‬ ‫ﯿ‬ ‫ﻗﺼ‬ ‫ﺎ‬ ‫ﻨ‬ ‫ﺔ ‪(I n ) nIN‬ﺗ‬ ‫ﯿ‬ ‫ﻟ‬ ‫ﺎ‬ ‫ﺘ‬ ‫ﺘ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ن‬ ‫أ‬ ‫ﯿﻦ‬ ‫أ‪-‬ﺑ‬ ‫‪-3‬‬ ‫ﺔ‪.‬‬ ‫ﺑ‬ ‫ر‬ ‫ﺎ‬ ‫ﻘ‬ ‫ﺘ‬ ‫أن ‪(I n ) nIN‬ﻣ‬ ‫ﺘﺞ‬ ‫ﻨ‬ ‫ﺘ‬ ‫ﺳ‬ ‫ب‪-‬ا‬ ‫*‬

‫*‬

‫ﻦ * ‪IN‬‬ ‫ﻞ ‪n‬ﻣ‬ ‫ﻜ‬ ‫ﻟ‬ ‫ن‬ ‫أ‬ ‫ﻦ‬ ‫ﯿ‬ ‫ج‪-‬ﺑ‬

‫ﺔ‪10.5 ) :‬ن(‬ ‫ﻟ‬ ‫ﺄ‬ ‫ﻣﺴ‬

‫‪0.75‬‬

‫‪. I n nI n 1 1‬‬

‫‪1‬‬ ‫‪. nlim‬‬ ‫‪‬‬ ‫ﺘ‬ ‫ﻨ‬ ‫ﺘ‬ ‫ﺳ‬ ‫ا‬ ‫ﻢ‬ ‫ﺛ‬ ‫‪In ‬‬ ‫ﺞ ‪n‬‬ ‫‪‬‬ ‫‪‬‬

‫‪n 1‬‬

‫ل‪:‬‬ ‫و‬ ‫اﻷ‬ ‫ء‬ ‫ﺰ‬ ‫ﺠ‬ ‫ﻟ‬ ‫ا‬ ‫ﻠﻲ ‪:‬‬ ‫ﯾ‬ ‫ﺎ‬ ‫ﻤ‬ ‫ﺑ‬ ‫ﺔ‬ ‫ﻓ‬ ‫ﺮ‬ ‫ﻌ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﺔ‬ ‫ﯾ‬ ‫د‬ ‫ﺪ‬ ‫ﻌ‬ ‫ﻟ‬ ‫ا‬ ‫ﺔ‬ ‫ﻟ‬ ‫ا‬ ‫ﺪ‬ ‫ﻟ‬ ‫ﻜﻦ ‪g‬ا‬ ‫ﺘ‬ ‫ﻟ‬ ‫‪. xlim‬‬ ‫ﺴﺐ ) ‪ lim g ( x‬و )‪g ( x‬‬ ‫‪-1‬اﺣ‬ ‫‪‬‬ ‫‪‬‬

‫‪x 0‬‬

‫)‪. g ( x) x 1 ln( x‬‬

‫ﺔ‬ ‫ﯿ‬ ‫ﻠ‬ ‫ﯿ‬ ‫ﻫ‬ ‫ﺄ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫ﺎت‬ ‫ﻨ‬ ‫ﻣ‬ ‫ﺔد‬ ‫ﯾ‬ ‫ﻮ‬ ‫ﻧ‬ ‫ﺎ‬ ‫ﺛ‬ ‫ﻞ‪2007‬‬ ‫ﯾ‬ ‫ﺮ‬ ‫ﺑ‬ ‫أ‬ ‫ة‬ ‫ر‬ ‫و‬ ‫د‬

‫‪0.5‬‬ ‫‪0.25‬‬

‫ﺪ‬ ‫ﻮﺣ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﺒﻲ‬ ‫ﯾ‬ ‫ﺮ‬ ‫ﺠ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫ﺎن‬ ‫ﺘﺤ‬ ‫ﻣ‬ ‫ﻻ‬ ‫ا‬ ‫****************‬ ‫ﺎت‬ ‫ﺎﻋ‬ ‫ز‪3 :‬ﺳ‬ ‫ﺎ‬ ‫ﺠ‬ ‫ﻧ‬ ‫ﻹ‬ ‫ا‬ ‫ة‬ ‫ﺪ‬ ‫ﻣ‬

‫ﺎ‬ ‫ﯾ‬ ‫ر‬ ‫ﻮ‬ ‫ﻟ‬ ‫ﺎ‬ ‫ﻜ‬ ‫ﺔﺑ‬ ‫ﯿ‬ ‫ﻧ‬ ‫ﺎ‬ ‫ﺜ‬ ‫ﻟ‬ ‫ا‬ ‫ﻮى‪:‬‬ ‫ﺘ‬ ‫ﻤﺴ‬ ‫ﻟ‬ ‫ا‬ ‫ﺔ‬ ‫ﯿ‬ ‫ﺒ‬ ‫ﯾ‬ ‫ﺮ‬ ‫ﺠ‬ ‫مﺗ‬ ‫ﻮ‬ ‫ﻠ‬ ‫ﺔ‪:‬ﻋ‬ ‫ﺒ‬ ‫ﻌ‬ ‫ﻟﺸ‬ ‫ا‬

‫‪2/2‬‬

‫ات ‪. g‬‬ ‫ﺮ‬ ‫ﯿ‬ ‫ﻐ‬ ‫ﺗ‬ ‫ل‬ ‫و‬ ‫ﺪ‬ ‫ﻋﻂﺟ‬ ‫أ‬ ‫ﻢ‬ ‫ﻦ ‪IR *‬ﺛ‬ ‫ﻞ ‪x‬ﻣ‬ ‫ﻜ‬ ‫ﺴﺐ ) ‪g ' ( x‬ﻟ‬ ‫ﺣ‬ ‫ا‬ ‫‪-2‬‬ ‫‪*‬‬ ‫ﻜﻞ ‪x‬ﻣﻦ ‪. g ( x) 0 IR‬‬ ‫ﻟ‬ ‫أن‬ ‫ﺘﺞ‬ ‫ﻨ‬ ‫ﺘ‬ ‫ﺳ‬ ‫ا‬ ‫‪-3‬‬ ‫ﻧﻲ‪:‬‬ ‫ﺎ‬ ‫ﺜ‬ ‫ﻟ‬ ‫ا‬ ‫ء‬ ‫ﺰ‬ ‫ﻟﺠ‬ ‫ا‬

‫ﻠﻲ‬ ‫ﯾ‬ ‫ﺎ‬ ‫ﻤ‬ ‫‪‬ﺑ‬ ‫‪0, ‬‬ ‫ﻠﻰ‪‬‬ ‫ﺔﻋ‬ ‫ﻓ‬ ‫ﺮ‬ ‫ﻌ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ﻘﻲ ‪x‬ا‬ ‫ﯿ‬ ‫ﻘ‬ ‫ﺤ‬ ‫ﻟ‬ ‫ا‬ ‫ﺮ‬ ‫ﯿ‬ ‫ﻐ‬ ‫ﺘ‬ ‫ﻤ‬ ‫ﻠ‬ ‫ﺔ ‪f‬ﻟ‬ ‫ﯾ‬ ‫د‬ ‫ﺪ‬ ‫ﻌ‬ ‫ﻟ‬ ‫ا‬ ‫ﺔ‬ ‫ﻟ‬ ‫ا‬ ‫ﺪ‬ ‫ﻟ‬ ‫ا‬ ‫ﺮ‬ ‫ﺒ‬ ‫ﺘ‬ ‫ﻌ‬ ‫ﻧ‬

‫‪:‬‬

‫‪x 1‬‬ ‫‪‬‬ ‫‪f ( x ) e x ln( x) , x 0‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪f (0) 0‬‬

‫ﻢ‪.‬‬ ‫ﻨﻈ‬ ‫ﻤ‬ ‫ﺪﻣ‬ ‫ﻣ‬ ‫ﺎ‬ ‫ﻌ‬ ‫ﺘ‬ ‫ﻢﻣ‬ ‫ﻠ‬ ‫ﻌ‬ ‫ﻓﻲﻣ‬ ‫ﺎ‬ ‫ﻫ‬ ‫ﺎ‬ ‫ﻨ‬ ‫ﻨﺤ‬ ‫ﻦ ) ‪(c f‬ﻣ‬ ‫ﻜ‬ ‫ﯿ‬ ‫ﻟ‬ ‫و‬ ‫‪0.5‬‬

‫‪. xlim‬‬ ‫ﺴﺐ )‪f ( x‬‬ ‫اﺣ‬ ‫ﺔ ‪0‬و‬ ‫ﻘﻄ‬ ‫ﻨ‬ ‫ﻟ‬ ‫ا‬ ‫ﻓﻲ‬ ‫ﯿﻦ‬ ‫ﻤ‬ ‫ﯿ‬ ‫ﻟ‬ ‫ا‬ ‫ﻠﻰ‬ ‫ﺔ ‪f‬ﻋ‬ ‫ﻟ‬ ‫ا‬ ‫ﺪ‬ ‫ﻟ‬ ‫ا‬ ‫ل‬ ‫ﺎ‬ ‫ﺗﺼ‬ ‫ا‬ ‫رس‬ ‫د‬ ‫‪-1‬ا‬ ‫‪‬‬ ‫‪1‬‬

‫‪0.5‬‬ ‫‪0.5‬‬ ‫‪0.25‬‬ ‫‪1.5‬‬

‫)‪ln( x‬‬ ‫)‪f ( x‬‬ ‫‪.‬‬ ‫ل‪0,‬‬ ‫ﺎ‬ ‫ﻤﺠ‬ ‫ﻟ‬ ‫ا‬ ‫ﻦ‬ ‫ﻞ ‪x‬ﻣ‬ ‫ﻜ‬ ‫ﻟ‬ ‫‪e x‬‬ ‫ن‪:‬‬ ‫أ‬ ‫ﻘﻖ‬ ‫أ‪-‬ﺗﺤ‬ ‫‪-2‬‬ ‫‪x‬‬ ‫ﺎ‪.‬‬ ‫ﯿ‬ ‫ﺳ‬ ‫ﺪ‬ ‫ﻨ‬ ‫ﺔﻫ‬ ‫ﯿﺠ‬ ‫ﺘ‬ ‫ﻨ‬ ‫ﻟ‬ ‫ا‬ ‫ل‬ ‫و‬ ‫أ‬ ‫ﻢ‬ ‫ﺔ ‪0‬ﺛ‬ ‫ﻘﻄ‬ ‫ﻨ‬ ‫ﻟ‬ ‫ا‬ ‫ﻓﻲ‬ ‫ﯿﻦ‬ ‫ﻤ‬ ‫ﯿ‬ ‫ﻟ‬ ‫ا‬ ‫ﻠﻰ‬ ‫ﺔ ‪f‬ﻋ‬ ‫ﻟ‬ ‫ا‬ ‫ﺪ‬ ‫ﻟ‬ ‫ا‬ ‫ﺎق‬ ‫ﻘ‬ ‫ﺘ‬ ‫ﺷ‬ ‫ا‬ ‫رس‬ ‫د‬ ‫ب‪-‬ا‬ ‫)‪f ( x‬‬ ‫‪. xlim‬‬ ‫ﺔ‬ ‫ﯾ‬ ‫ﺎ‬ ‫ﻬ‬ ‫ﻨ‬ ‫ﻟ‬ ‫ا‬ ‫ﺴﺐ‬ ‫ج‪-‬اﺣ‬ ‫‪ x‬‬ ‫‪e t 1‬‬ ‫‪f ( x) x‬‬ ‫‪( lim‬‬ ‫ﺔ ‪1‬‬ ‫ﺠ‬ ‫ﯿ‬ ‫ﺘ‬ ‫ﻨ‬ ‫ﻟ‬ ‫ا‬ ‫ل‬ ‫ﺎ‬ ‫ﻤ‬ ‫ﻌ‬ ‫ﺘ‬ ‫ﺳ‬ ‫ا‬ ‫ﻚ‬ ‫ﻨ‬ ‫ﻜ‬ ‫ﻤ‬ ‫ﯾ‬ ‫)‬ ‫‪lim‬‬ ‫ن ‪1‬‬ ‫أ‬ ‫ﻦ‬ ‫ﯿ‬ ‫د‪-‬ﺑ‬ ‫‪t 0‬‬ ‫)‪x  ln( x‬‬ ‫‪t‬‬

‫ﺔ ‪.f‬‬ ‫ﻟ‬ ‫ا‬ ‫ﺪ‬ ‫ﻟ‬ ‫ا‬ ‫ﻨﻰ‬ ‫ﺤ‬ ‫ﻨ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ﺋﻲ‬ ‫ﺎ‬ ‫ﻬ‬ ‫ﻧ‬ ‫ﻟﻼ‬ ‫ا‬ ‫ﺮع‬ ‫ﻔ‬ ‫ﻟ‬ ‫ا‬ ‫د‬ ‫ﺪ‬ ‫ﻢﺣ‬ ‫‪xlim‬ﺛ‬ ‫ن ‪f ( x ) x ‬‬ ‫أ‬ ‫ﺞ‬ ‫ﺘ‬ ‫ﻨ‬ ‫ﺘ‬ ‫ﺳ‬ ‫ا‬ ‫‪‬‬

‫‪1‬‬ ‫‪0.25‬‬ ‫‪1‬‬

‫ﺔ ‪. f‬‬ ‫ﻟ‬ ‫ا‬ ‫ﺪ‬ ‫ﻟ‬ ‫ا‬ ‫ات‬ ‫ﺮ‬ ‫ﯿ‬ ‫ﻐ‬ ‫ﺗ‬ ‫ول‬ ‫ﺪ‬ ‫أﻋﻂﺟ‬ ‫ﺎو‬ ‫ﻬ‬ ‫ﺗ‬ ‫ر‬ ‫ﺎ‬ ‫ﺷ‬ ‫إ‬ ‫رس‬ ‫د‬ ‫ا‬ ‫ﻢ‬ ‫‪‬ﺛ‬ ‫ل‪0,‬‬ ‫ﺎ‬ ‫ﺠ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﻦ‬ ‫ﻞ ‪x‬ﻣ‬ ‫ﻜ‬ ‫ﺴﺐ ) ‪f ' ( x‬ﻟ‬ ‫‪-3‬اﺣ‬ ‫ﺴﺐ )‪f (1‬و )‪f (2‬و )‪. f (3‬‬ ‫ﺣ‬ ‫أ‪-‬ا‬ ‫‪-4‬‬ ‫ﻨﻰ ) ‪. (c f‬‬ ‫ﺤ‬ ‫ﻨ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﻧﺸﺊ‬ ‫ب‪-‬أ‬ ‫‪4‬‬

‫ﻘﻂ )‪.( A(1,1‬‬ ‫ﻨ‬ ‫ﻟ‬ ‫ا‬ ‫ﻓﻲ‬ ‫ﺎف‬ ‫ﻌﻄ‬ ‫ﻧ‬ ‫ا‬ ‫ﺔ‬ ‫ﻘﻄ‬ ‫ﻨﻰ ) ‪(c f‬ﻧ‬ ‫ﻨﺤ‬ ‫ﻤ‬ ‫ﻠ‬ ‫ﻟ‬ ‫ن‬ ‫أ‬ ‫ﻞ‬ ‫ﺒ‬ ‫ﻘ‬ ‫ﻧ‬ ‫ﻌﻄﻲ ‪3 3 4,3‬و‬ ‫ﻧ‬ ‫)‬ ‫‪1‬‬ ‫‪1‬‬

‫ﻞ‬ ‫ﯿ‬ ‫ﺎﺻ‬ ‫ﻓ‬ ‫ﻷ‬ ‫ا‬ ‫ر‬ ‫ﻮ‬ ‫ﺤ‬ ‫ﺔ ‪g‬وﻣ‬ ‫ﻟ‬ ‫ا‬ ‫ﺪ‬ ‫ﻟ‬ ‫ا‬ ‫ﻨﻰ‬ ‫ﺤ‬ ‫ﻨ‬ ‫ﻦﻣ‬ ‫ﯿ‬ ‫ﺑ‬ ‫ر‬ ‫ﻮ‬ ‫ﻤﺤﺼ‬ ‫ﻟ‬ ‫ا‬ ‫ﻮى‬ ‫ﺘ‬ ‫ﺴ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﺰ‬ ‫ﯿ‬ ‫ﺔﺣ‬ ‫ﺎﺣ‬ ‫ﺴﺐ )‪ A(‬ﻣﺴ‬ ‫ﺣ‬ ‫أ‪-‬ا‬ ‫‪-5‬‬ ‫ﯿﺚ ‪. 1‬‬ ‫ﻟﻲ ‪ x 1‬و ‪ x ‬ﺣ‬ ‫ا‬ ‫ﻮ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫ﻠﻰ‬ ‫ﺎﻋ‬ ‫ﻤ‬ ‫ﻫ‬ ‫ﺎ‬ ‫ﺘ‬ ‫ﻟ‬ ‫د‬ ‫ﺎ‬ ‫ﻌ‬ ‫ﯾﻦﻣ‬ ‫ﺬ‬ ‫ﻠ‬ ‫ﻟ‬ ‫ا‬ ‫ﯿﻦ‬ ‫ﻤ‬ ‫ﯿ‬ ‫ﻘ‬ ‫ﺘ‬ ‫ﺴ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫و‬ ‫‪. lim‬‬ ‫ﺴﺐ )‪(‬‬ ‫ﺣ‬ ‫ب‪-‬ا‬ ‫‪‬‬ ‫ﻟﺚ‪:‬‬ ‫ﺎ‬ ‫ﺜ‬ ‫ﻟ‬ ‫ا‬ ‫ء‬ ‫ﺰ‬ ‫ﻟﺠ‬ ‫ا‬ ‫ﻠﻲ‪:‬‬ ‫ﯾ‬ ‫ﺎ‬ ‫ﻤ‬ ‫ﺑ‬ ‫ﺔ‬ ‫ﻓ‬ ‫ﺮ‬ ‫ﻌ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ﺔ ) ‪(u n‬ا‬ ‫ﯾ‬ ‫د‬ ‫ﺪ‬ ‫ﻌ‬ ‫ﻟ‬ ‫ا‬ ‫ﺔ‬ ‫ﯿ‬ ‫ﻟ‬ ‫ﺎ‬ ‫ﺘ‬ ‫ﺘ‬ ‫ﻤ‬ ‫ﻟ‬ ‫ا‬ ‫ﺮ‬ ‫ﺒ‬ ‫ﺘ‬ ‫ﻌ‬ ‫ﻧ‬ ‫‪u 0 1‬‬ ‫‪‬‬ ‫‪‬‬ ‫‪u n 1 g (u n ), n IN‬‬ ‫‪‬‬

‫‪0.5‬‬ ‫‪0.5‬‬ ‫‪0.5‬‬

‫أن‬ ‫ﯿﻦ‬ ‫‪-1‬ﺑ‬ ‫ﺔ‪.‬‬ ‫ﯾ‬ ‫ﺪ‬ ‫ﯾ‬ ‫ا‬ ‫ﺰ‬ ‫أن ) ‪(un‬ﺗ‬ ‫ﯿﻦ‬ ‫‪-2‬ﺑ‬ ‫ﺴﺐ‬ ‫اﺣ‬ ‫ﺔو‬ ‫ﺑ‬ ‫ر‬ ‫ﺎ‬ ‫ﻘ‬ ‫ﺘ‬ ‫ن ) ‪(un‬ﻣ‬ ‫أ‬ ‫ﺘﺞ‬ ‫ﻨ‬ ‫ﺘ‬ ‫‪-3‬اﺳ‬ ‫‪1 u n e‬‬

‫‪. n IN‬‬ ‫‪. nlim‬‬ ‫‪un‬‬ ‫‪‬‬ ‫ﯿﻖ‬ ‫ﻓ‬ ‫ﻮ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫ﻟﻲ‬ ‫اﷲو‬ ‫و‬

‫ﻢ‬ ‫ﯾ‬ ‫ﺪ‬ ‫ﻘ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫ﻦ‬ ‫ﺮوﺣﺴ‬ ‫ﯿ‬ ‫ﺒ‬ ‫ﻌ‬ ‫ﺘ‬ ‫ﻟ‬ ‫ا‬ ‫ﺔ‬ ‫ﻣ‬ ‫ﺢﺳﻼ‬ ‫ﯿ‬ ‫ﺤ‬ ‫ﺘﺼ‬ ‫ﻟ‬ ‫ا‬ ‫ﻓﻲ‬ ‫ﻋﻰ‬ ‫ا‬ ‫ﺮ‬ ‫ﺔ‪:‬ﯾ‬ ‫ﻈ‬ ‫ﺣ‬ ‫ﻣﻼ‬ ‫ﻊ‬ ‫ﯿ‬ ‫ﻤ‬ ‫ﻠﺠ‬ ‫ﻟ‬ ‫ﺪ‬ ‫ﯿ‬ ‫ﻌ‬ ‫ﺣﻆﺳ‬