S-exp-ii Math Devoir S2 2

‫ﻤﺩﺓ ﺍﻻﻨﺠﺎﺯ ‪ :‬ﺴﺎﻋﺘﺎﻥ‬ ‫ﺍﻝﺴﻨﺔ ‪ 05-06‬ﺍﻝﺩﻭﺭﺓ ‪2‬‬

‫ﺜﺎﻨﻭﻴﺔ ﻤﻴﺩﺍﻥ ﺍﻝﻔﺭﻭﺴﻴﺔ ﻓﺎﺱ‬ ‫ﺍﻝﺜﺎﻨﻴﺔ ﺒﺎﻙ ﻋﻠﻭﻡ ﺘﺞ ‪3 .2 .1‬‬ ‫ﺍﻷﺴﺘﺎﺫ ﻤﺤﻤﺩ ﺸﺎﺸﻴل‬ ‫‪----------------------------------------------------------------------------------‬‬‫)‪(10-03-2006‬‬ ‫ﻓﺭﺽ ﻤﺤﺭﻭﺱ ‪1‬‬

‫ﺍﻝﺘﻤﺭﻴﻥ ﺍﻷﻭل ‪ 3,5 ):‬ﻨﻘﻁﺔ(‬ ‫ﺤل ﻓﻲ ‪ IR‬ﺍﻝﻤﻌﺎﺩﻝﺘﻴﻥ ﺍﻝﺘﺎﻝﻴﺘﻴﻥ‬ ‫‪(1‬‬ ‫‪1+1‬‬ ‫ﺃ ‪ln x + ln( x − 3) = ln(2 x − 4) -‬‬ ‫‪(2‬‬

‫‪1.5‬‬

‫أ ات ا ‪:‬‬

‫‪e2 x − e x‬‬ ‫‪x→0‬‬ ‫‪x‬‬

‫‪lim‬‬

‫ب ‪-‬‬ ‫‪,‬‬

‫‪3e x − 2e − x + 1 = 0‬‬

‫‪lim x − ln 2 x‬‬

‫‪,‬‬

‫∞‪x →+‬‬

‫‪ x +1‬‬ ‫‪lim x ln ‬‬ ‫‪‬‬ ‫∞‪x →−‬‬ ‫‪ x ‬‬

‫‪--------------------------------------------------------------------‬‬‫


‬ ‫ا  ا ‪ 6 ):‬ﻨﻘﻁ(  ا ء )‪ (E‬ا'‪-‬ب إ‪ !#)0 ( O, i, j , k ) !"#$ %&'$ ($)$ %*)$ +‬ا‪./‬‬ ‫)‪ A(1,4,3‬و )‪ B(0,2,2‬و ) ‪ C(1,-1,1‬وا'‪-‬ى)‪ (P‬ا‪6‬ي ‪)$‬د‪x-3y-2z+3=0 3‬‬ ‫‪ (1‬ﺍﻋﻁ ﺘﻤﺜﻴﻼ ﺒﺎﺭﻤﺘﺭﻴﺎ ﻝﻠﻤﺴﺘﻘﻴﻡ )‪(AB‬‬ ‫‪0.5‬‬ ‫ا‪-‬ى )‪((=> ? @/0  (P‬ه‬ ‫‪ 89 (2‬أن ا'‪!; (AB) %/‬ق‬ ‫‪1‬‬ ‫

‬ ‫‪ (3‬أ ‪ -‬أ ا'? ‪ AB ∧ AC‬و ا‪(C DA‬م ا‪ $/A‬ا‪ A ./‬و ‪ B‬و ‪C‬‬ ‫‪0.5‬‬ ‫ب ‪( -‬د ‪)$‬د د‪F‬ر> *'‪-‬ى )‪(ABC‬‬ ‫‪0.5‬‬ ‫‪ 89 (4‬أن ا'‪ (ABC) 8-‬و )‪)H/ (P‬ن و‪ 3*I'> ((=> ? %/$ J‬ا‪#‬ر‪!$‬ي‪.‬‬ ‫‪1.5‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪2‬‬ ‫‪ 8F (4‬ا *‪ (S) F‬ذات ا')د ‪x + y + z − 2 x + 2 y − 2 z − 1 = 0‬‬ ‫أ – (د ‪!$‬آ‪ N‬و ")ع ا *‪.(S) F‬‬ ‫‪0.5‬‬ ‫ب – ‪ 89‬ان ا'‪-‬ى )‪ P@/ (P‬ا *‪ (S) F‬و‪ J‬دا‪!R‬ة (د ‪!$‬آ‪N‬ه و ")‪C‬‬ ‫‪1.5‬‬ ‫‪-------------------------------------------------------------------‬‬‫ا  ا ‪ 10,5 ) :‬ﻨﻘﻁﺔ(‬ ‫*‬ ‫اء اول‪:‬‬ ‫‪9‬ـ ‪g ( x) = 1 − x + ln x :‬‬ ‫‪ g 8F‬ا(ا ا')! ‪IR+ +*C‬‬ ‫)‪lim g ( x‬‬ ‫‪,‬‬ ‫)‪lim g ( x‬‬ ‫‪ (1‬أ ا‪8‬‬ ‫‪1‬‬ ‫∞‪x →+‬‬ ‫‪x → 0+‬‬

‫‪1‬‬

‫‪ (2‬أدرس >‪!U‬ات ‪IR+* +*C g‬‬

‫‪0.5‬‬

‫‪ (3‬ا‪ DA‬أن ‪g ( x) ≤ 0‬‬

‫‪∀x ∈ IR‬‬

‫*‬ ‫‪+‬‬

‫‪----------------------------------------------------------------‬‬‫‪ !#)0‬ا(ا ا)(د ‪ f‬ا')! ‪.‬‬ ‫اءا‪:‬‬

‫‪(1‬‬ ‫‪(2‬‬ ‫‪(3‬‬

‫‪0.5‬‬ ‫‪1‬‬ ‫‪1‬‬

‫‪ 89‬أن ا(ا ‪0  *X$ f‬‬ ‫أ – أ ) ‪f ( x‬‬ ‫‪ xlim‬و‪ 89‬أن ∞‪lim f ( x) = +‬‬ ‫∞‪→−‬‬ ‫∞‪x →+‬‬

‫) ‪(C‬‬ ‫‪f‬‬

‫ج ‪ 89 -‬أن ) ‪  '?*" C! [#/ ( C f‬ا>?‪ Z‬ا'‪( ∆ ) %/‬‬ ‫ﺩ – ﺍﺩﺭﺱ ﺍﻝﻭﻀﻊ ﺍﻝﻨﺴﺒﻲ ﻝﻠﻤﻨﺤﻨﻰ ) ‪ ( C f‬و ا'‪ +*C ( ∆ ) %/‬ا'?ل [∞‪]0, +‬‬

‫‪0.5‬‬

‫‪(4‬‬ ‫‪(5‬‬

‫‪1‬‬

‫‪0,75+ 0.25‬‬

‫

‬

‫‪-?9‬ار ∞‪−‬‬

‫ا‪6‬ي ‪)$‬د‪-?9 y = x 3‬ار ∞‪+‬‬

‫‪0.5‬‬

‫‪1.5‬‬

‫‪f‬‬

‫ب ‪ 89 -‬أن ا'‪ %/‬ا*‪6‬ي ‪)$‬د‪/$ y = −2 x − 1 3‬رب ـ‬

‫‪0.5‬‬

‫‪0.5‬‬

‫‪ f ( x) = x − x .ln x‬‬ ‫‪x>0‬‬ ‫‪9‬ـ‪:‬‬ ‫‪‬‬ ‫‪x‬‬ ‫‪x≤0‬‬ ‫‪ f ( x) = e − 2 x − 1‬‬ ‫ﺤﺩﺩ ‪ W!)> C-'?$ D f‬ا(ا ‪f‬‬

‫و‬

‫) ‪(C‬‬

‫‪%&'$ ($)$ %*)$  f +=$‬‬

‫) ‪( O, i , j‬‬

‫‪.‬‬

‫أدرس ]‪ *9‬ا"‪ /‬ق ‪  f‬ا‪ ! X‬وا‪_> .C‬و^ ه(‪A‬‬ ‫‪ f ' 8F‬ا(ا ا'`‪(* /‬ا ‪IR* +*C f‬‬ ‫أ ‪ -‬أ )‪f '( x‬‬

‫‪ x [F‬ﻤﻥ [‪]−∞, 0‬‬

‫‪−1‬‬ ‫‪9‬ـ ‪ 89 -‬أن ) ‪g ( x‬‬ ‫‪x‬‬

‫= )‪f '( x‬‬

‫[∞‪∀x ∈ ]0, +‬‬

‫ﺝ ‪ -‬ﻀﻊ ﺠﺩﻭل ﺘﻐﻴﺭﺍﺕ ‪. f‬‬ ‫‪ (6‬أ‪`0‬ء ‪ )9‬ا'=‪( C f ) +‬‬ ‫‪-------------------------------------------------------------‬ﺤﻅ ﺴﻌﻴﺩ‬‫ﺴﻴﺅﺨﺫ ﺒﻌﻴﻥ ﺍﻹﻋﺘﺒﺎﺭ ﺍﻝﺩﻗﺔ ﻭ ﺍﻝﻌﻨﺎﻴﺔ ﺃﺜﻨﺎﺀ ﺍﻝﺘﺤﺭﻴﺭ ‪.‬‬ ‫‪http://arabmaths.ift.fr‬‬