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‫ﺳﻠﺴﻠﺔ رﻗﻢ ‪:5‬اﻟﺪوال اﻷﺻﻠﯿﺔ‬
‫أﻛﺎدﻳﻤﯿﺔ‬
‫اﻟﺠﮫﺔ‬
‫اﻟﺸﺮﻗﯿﺔ‬
‫اﻟﻤﺴﺘﻮى ‪ :‬اﻟﺜﺎﻧﯿﺔ ﺑﺎك ﻋﻠﻮم ﻓﯿﺰﯾﺎﺋﯿﺔ وﻋﻠﻮم اﻟﺤﯿﺎة‬
‫واﻷرض واﻟﻌﻠﻮم اﻟﺰراﻋﯿﺔ‬
‫ﺗﻤﺮﯾﻦ‪ :1‬ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ ‪ f‬اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ ¡ ﻛﺎﻟﺘﺎﻟﻲ‪:‬‬
‫‪f ( x) = x2 + 2x + 3‬‬
‫ﺣﺪد داﻟﺔ ‪ F‬ﻗﺎﺑﻠﺔ ﻟﻼﺷﺘﻘﺎق‬
‫‪.1‬‬
‫ﻋﻠﻰ ¡ ﺑﺤﯿﺚ ) ‪( "x Î ¡ ) ; F ¢ ( x ) = f ( x‬‬
‫‪ .2‬ھﻞ ﺗﻮﺟﺪ داﻟﺔ أﺧﺮى ‪ G‬ﺑﺤﯿﺚ ) ‪( "x Î ¡ ) ; G ¢ ( x ) = f ( x‬‬
‫‪ .3‬ﻛﻢ ﺗﻮﺟﺪ ﻣﻦ داﻟﺔ ‪ F‬ﺑﺤﯿﺚ ) ‪ ( "x Î ¡ ) ; F ¢ ( x ) = f ( x‬؟‬
‫ﺗﻤﺮﯾﻦ‪ :2‬ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ ‪ f‬اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [‪ ]0; +¥‬ﻛﺎﻟﺘﺎﻟﻲ‪:‬‬
‫‪1‬‬
‫‪x2‬‬
‫‪f ( x ) = 2x 2 + x + 1 +‬‬
‫‪ .1‬ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ ‪ f‬ﻋﻠﻰ [‪]0; +¥‬‬
‫‪ .2‬ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ ‪ F‬ﻟﻠﺪاﻟﺔ ‪ f‬ﺑﺤﯿﺚ ‪F (1) = 3‬‬
‫ﺗﻤﺮﯾﻦ‪ :3‬ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ ‪:‬‬
‫‪+ cos x + sin x -1 (2 f ( x ) = 5x + 3x +1 (1‬‬
‫‪4‬‬
‫‪1‬‬
‫‪x‬‬
‫= )‪f ( x‬‬
‫‪f ( x ) = ( 2 x - 1) (4 f ( x) = sin x + x cos x (3‬‬
‫‪3‬‬
‫‪(5‬‬
‫‪x‬‬
‫‪2‬‬
‫)‪( x - 1‬‬
‫‪2‬‬
‫‪(5‬‬
‫‪2‬‬
‫) ‪( x3 + 2‬‬
‫= )‪f ( x‬‬
‫= )‪f ( x‬‬
‫ﺗﻤﺮﯾﻦ‪ :5‬ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ ‪:‬‬
‫‪x (2 f x = 2 2 x + 1 (1‬‬
‫) (‬
‫= )‪f ( x‬‬
‫‪x2 + 1‬‬
‫ﺗﻤﺮﯾﻦ‪ :6‬ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ ‪:‬‬
‫‪x2 (2 f x = x x 2 + 1 (1‬‬
‫) (‬
‫= )‪f ( x‬‬
‫‪8 + x3‬‬
‫‪(3‬‬
‫ﺗﻤﺮﯾﻦ‪ :8‬ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ ‪f‬‬
‫اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [ ‪ [1; +¥‬ﻛﺎﻟﺘﺎﻟﻲ ‪f ( x ) = x x - 1 :‬‬
‫‪ .1‬ﺑﯿﻦ أن ‪"x Î [1; +¥ [ f ( x ) = ( x - 1)3 + x - 1 :‬‬
‫‪ .2‬ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ ‪ F‬ﻟﻠﺪاﻟﺔ ‪ f‬ﺑﺤﯿﺚ ‪F ( 2 ) = 1‬‬
‫ﺗﻤﺮﯾﻦ‪:9‬ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ ‪ f‬اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ ¡ ﻛﺎﻟﺘﺎﻟﻲ‪:‬‬
‫‪5 x 4 + 40 x 2 + 20 x + 80‬‬
‫‪2‬‬
‫)‪f ( x ) = sin ( 4 x - 1‬‬
‫‪f ( x ) = ( sin x ) cos x (5 f ( x ) = cos ( 2 x + 8 ) (4‬‬
‫ﺗﻤﺮﯾﻦ‪ :7‬ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ ‪ f‬اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ [‪[0; +¥‬‬
‫ﻛﺎﻟﺘﺎﻟﻲ ‪:‬‬
‫‪2‬‬
‫)‪( x + 1‬‬
‫‪ .1‬ﺣﺪد اﻟﻌﺪدﯾﻦ اﻟﺤﻘﯿﻘﯿﯿﻦ ‪ a‬و ‪ b‬ﺑﺤﯿﺚ‪:‬‬
‫‪b‬‬
‫‪2‬‬
‫)‪( x + 1‬‬
‫‪2‬‬
‫)‪+ 4‬‬
‫‪2‬‬
‫‪(x‬‬
‫‪ .2‬ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺔ ‪f‬‬
‫‪ .3‬ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ ‪ F‬ﻟﻠﺪاﻟﺔ ‪ f‬ﺑﺤﯿﺚ ‪F ( 0 ) = c‬‬
‫ﺗﻤﺮﯾﻦ‪:10‬ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺔ ‪ f‬اﻟﻤﻌﺮﻓﺔ ﻋﻠﻰ ] ‪ [0; p‬ﻛﺎﻟﺘﺎﻟﻲ ‪:‬‬
‫‪f ( x ) = x cos x - sin x‬‬
‫اﻟﻤﻌﺮﻓﺔ ﻛﺎﻟﺘﺎﻟﻲ ‪g ( x ) = x sin x - cos x :‬‬
‫‪ .3‬ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ ‪ G‬ﻟﻠﺪاﻟﺔ ‪ g‬ﺑﺤﯿﺚ ‪G (p ) = 0‬‬
‫ﺗﻤﺮﯾﻦ‪:11‬ﻧﻌﺘﺒﺮ اﻟﺪاﻟﺘﯿﻦ ‪ f‬و ‪ g‬اﻟﻤﻌﺮﻓﺘﯿﻦ‬
‫‪x‬‬
‫‪ù pé‬‬
‫ﻋﻠﻰ ‪ ú 0; ê‬ﻛﺎﻟﺘﺎﻟﻲ ‪ f ( x ) = tan 2 x :‬و ‪g ( x ) = cos 2‬‬
‫‪2‬‬
‫‪û 2ë‬‬
‫ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪاﻟﺘﯿﻦ ‪ f‬و ‪g‬‬
‫ﺗﻤﺮﯾﻦ‪ :12‬ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ ‪:‬‬
‫‪x (2 f x = 2 2 x + 1 (1‬‬
‫) (‬
‫= )‪f ( x‬‬
‫‪x2 + 1‬‬
‫ﺗﻤﺮﯾﻦ‪ :13‬ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ ‪:‬‬
‫‪x5 (2 f x = x 2 x 3 + 2 (1‬‬
‫) (‬
‫= )‪f ( x‬‬
‫‪3 + x6‬‬
‫‪(3‬‬
‫) ‪f ( x ) = 3sin ( x + 2‬‬
‫‪4‬‬
‫‪f ( x ) = ( cos x ) ´ sin x (5 f ( x ) = 2 cos ( 5 x + 1) (4‬‬
‫= )‪f ( x‬‬
‫‪« c’est en forgeant que l’on devient‬‬
‫‪forgeron » dit un proverbe.‬‬
‫‪c’est en s’entraînant régulièrement‬‬
‫‪"x Î [ 0; +¥ [ f ( x ) = a +‬‬
‫‪aux calculs et exercices que l’on‬‬
‫‪ .2‬ﺣﺪد اﻟﺪاﻟﺔ اﻷﺻﻠﯿﺔ ‪ F‬ﻟﻠﺪاﻟﺔ ‪f‬‬
‫‪1‬‬
‫)‪+ 4‬‬
‫‪2‬‬
‫‪(x‬‬
‫= )‪f ( x‬‬
‫‪ .1‬ﺣﺪد اﻷﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ ‪ a‬و ‪ b‬و ‪c‬‬
‫ﺑﺤﯿﺚ ‪"x Î [ 0; +¥[ f ( x ) = ax + b + c :‬‬
‫‪2‬‬
‫‪x2 + 2x‬‬
‫ﻧﺠﯿﺐ‬
‫ﻋﺜﻤﺎﻧﻲ‬
‫‪ .1‬أﺣﺴﺐ ) ‪f ¢ ( x‬‬
‫‪ .2‬اﺳﺘﻨﺘﺞ ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ‪ G‬ﻟﻠﺪاﻟﺔ ‪g‬‬
‫ﺗﻤﺮﯾﻦ‪ :4‬ﺣﺪد ﻣﺠﻤﻮﻋﺔ اﻟﺪوال اﻷﺻﻠﯿﺔ ﻟﻠﺪوال اﻟﺘﺎﻟﯿﺔ ‪:‬‬
‫‪3‬‬
‫‪2‬‬
‫‪f ( x) = 2cos x -sin x -3 (2 f ( x) = 8x + 4x + x + 6 (1‬‬
‫‪2‬‬
‫‪2‬‬
‫‪f ( x ) = ( 4 x + 5 ) (4 f ( x) = 2xsinx+ x cosx (3‬‬
‫‪x2‬‬
‫اﻷﺳﺘﺎذ‪:‬‬
‫ﺑﺤﯿﺚ ‪5‬‬
‫‪2‬‬
‫اﻷﺳﺘﺎذ‪ :‬ﻧﺠﯿﺐ ﻋﺜﻤﺎﻧﻲ‬
‫= )‪F (1‬‬
‫‪devient un mathématicien‬‬
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