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Annales bac s etudes de fonctions

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Programme de TS/TES - Révision du BAC

Document de support à un cours réalisé sur www.learningshelter.com

Chapitre 1 : étude de fonction
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Annales bac s etudes de fonctions

  1. 1. Sujets de bac 2009-2012Chapitre 1 : Etude de fonctions
  2. 2. 2010 - LIBANPoint de cours abordés• Calcul de limite et de dérivées simples• Théorème des valeurs intermédiaires et de la bijection• Dérivées de fonctions mises à la puissance
  3. 3. Eléments de correction1. a. Sens de variation 1 ln ( x)  1 x U ( x)  2 x  U ( x)  0 U ( x)  x 2  2  ln( x) x x2  2x Limites ln(x)    U (x)    U ( x)  x  2  ln( x) 2 ln(x) 0   U (x) 0  1. b. Existence • La fonction est continue Ok 3 conditions • Elle est négative à un moment U (1)  1 à remplir • Elle est positive à un moment U (x)   
  4. 4. Eléments de correction 1. b. Unicité 1 condition supplémentaire La fonction est strictement monotone B B B B A A AA Théorème des valeurs intermédiaires Théorème de la bijection
  5. 5. Eléments de correction2. a. Dérivée x2  2x 1 1 f ( x)  2 x  2   (2  ln( x)) ((2  ln( x)) )  2   (2  ln( x)) 2 x x RAPPEL (1  x 3 )5 5  (3x 2 )  (1  x3 ) 4 ( g n ( x))  n  g ( x)  g n1 ( x) 5 (e 5 x  2  ) 8 8  (5e5 x  2  5 2 5 )  (e 5 x  2  ) 7 x x x 1 ( x 2  ln( x))3 3  (2 x  ln( x)  x 2  )  ( x 2  ln( x)) 2 x
  6. 6. 2010 – RATTRAPAGEPoint de cours abordés• Calcul de limite et de dérivées simples• Tableaux de variation• Croissance comparée, et notamment : x ln( x) 0 0 • Formule de la tangente
  7. 7. Eléments de correctionI. 1. Tableau de signe x 0 e ∞ x + + 1-ln(x) + 0 - f(x) + 0 -I. 2. Limites (1  ln( x))    U (x)    U ( x)  x  (1  ln( x)) (1  ln( x)) 0   U ( x) 0 ?  RAPPEL U ( x)  x  x ln( x)) x ln( x) 0 0 
  8. 8. Eléments de correctionI. 3. Dérivée 1 (1  ln( x))   1 x U ( x)  x  ( )  1 (1  ln( x)) U ( x)   ln( x) U ( x)  x(1  ln( x)) x x  1 Variations x 0 1 +∞ U’(x) + 0 - U(x) 0 1 -∞I. 4. Tangente y  f (a)  f (a)( x  a) RAPPELS La dérivée correspond au coefficient directeur de la tangente à la courbe en ce point
  9. 9. 2010 - ASIEPoints de cours abordés• Calcul de limite et de dérivées• Utilisation d’une fonction auxiliaire pour appliquer le théorème des valeurs intermédiaires et de la bijection• Tableau de variation
  10. 10. Eléments de correctionA. 1. Limites 1  e 1 x 1 f ( x)  0  1 f ( x)  2  e x 1 x f (x) 0   e 0  x  A. 2. Dérivée 1 1 1 1 (e )  2  e x x 1 1 1 x 2 1 1 f ( x)  ex f ( x)   ex  2  2 ex x2 x 3 x x 1 2 ( )  ( x 2 )  2  x 21  3 x2 x Etude de signe x 0 ∞ (2x+1) + -1/x4 - f’(x) -
  11. 11. Eléments de correction Variations x 0 ∞ f’(x) - f +∞ 0 FonctionA. 3. Existence U ( x)  f ( x )  2 auxiliaire • La fonction est continue Ok 3 conditions • Elle est négative à un moment à remplir U ( x) 0 2  • Elle est positive à un moment U (x) 0   Unicité 1 condition supplémentaire La fonction est strictement monotone
  12. 12. 2010 – POLYNESIEPoints de cours abordés• Calcul de limite et de dérivées simples• Tableaux de variation• Croissance comparée, et notamment :• Formule de la tangente
  13. 13. Eléments de correctionI. 1. Limites 1 1 xe x    f ( x)  e  xe  1 x x f ( x)  xe  (  1  x ) x x xe 1   0 xe x I. 2. Dérivée ( xe x )  xe x  e x  e x ( x  1) f ( x)  e x  xe x  1 f ( x)  e x (1  ( x  1))   xe x (e x )  e x Etude de signe x 0 ∞ f’(x) - f 2 -∞
  14. 14. Eléments de correctionII. 1. Dérivée 4x (e x  1)  e x 4  ( g ( x)) A( x)  x 4  (e x  1)  e x  4 x A ( x)  e 1 A ( x)  (4 x)  4 (e x  1) 2 (e x  1) 2III. 2. Tangente Etude de signe x 0 ∞ f’(x) - f 2 -∞
  15. 15. 2010 – AFRIQUEPoint de cours abordés• Calcul de limite et de dérivées simples• Tableaux de variation• Croissance comparée, et notamment :• Formule de la tangente
  16. 16. 2010 – AMERIQUE DU NORDPoint de cours abordés• Calcul de limite et de dérivées simples• Tableaux de variation• Croissance comparée, et notamment :• Formule de la tangente

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