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Résumé sur les mathématiques fonctions exponentielles 3

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Ceci un cours résumé de la leçon des fonctions exponentielles 3 .Il est assez court bref et va directement au vif du sujet donc vous retrouvez dessus toutes les notions fondamentales pour assimiler comme il se doit ce cours

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  • November 27, 2022
  • 1
  • 2022/2023
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  • mathematics
  • fonctions exponentielles
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LAGDEM Mohamed Fonctions exponentielles 2BACS-2020/2021


 La fonction réciproque de 𝒍𝒏 s’appelle la fonction exponentielle
népérienne notée 𝒆𝒙𝒑
Définition  Notation : ∀𝒙 ∈ ℝ 𝒆𝒙𝒑(𝒙) = 𝒆𝒙
 La fonction est 𝒆𝒙𝒑 définie sur ℝ et on a : ∀𝒙 ∈ ℝ 𝒆𝒙 > 𝟎
∀𝒙 ∈ ℝ ∀𝒚 ∈ ]𝟎, +∞[ 𝒆𝒙 = 𝒚 ⟺ 𝒙 = 𝒍𝒏(𝒚)
𝒆𝒙𝒑(𝟎) = 𝟏 𝒆𝒙𝒑(𝟏) = 𝒆
 𝒆𝒙𝒑 est une fonction continue et dérivable sur ℝ
propriétés  𝒆𝒙𝒑 est une fonction strictement croissante sur ℝ
 𝒆𝒙 > 𝒆𝒚 ⟺ 𝒙 > 𝑦
 𝒆𝒙 = 𝒆𝒚 ⟺ 𝒙 = 𝑦
∀𝒙 ∈ ℝ 𝒍𝒏(𝒆𝒙 ) = 𝒙 ∀𝒙 ∈ ]𝟎, +∞[ 𝒆𝒍𝒏(𝒙) = 𝒙
𝟏
 𝒆𝒙+𝒚 = 𝒆𝒙 × 𝒆𝒙 ; 𝒆−𝒙 =
Propriétés 𝒆𝒙
𝒆𝒙
algébriques  𝒆𝒙−𝒚 = ; 𝒆𝒓𝒙 = (𝒆 )
𝒙 𝒓
𝒓∈ℚ
𝒆𝒚


Les limites :
𝐥𝐢𝐦 𝒆𝒙 = 𝟎 𝐥𝐢𝐦 𝒆𝒙 = +∞
𝒙→−∞ 𝒙→+∞
𝐥𝐢𝐦 𝒙𝒆𝒙 = 𝟎 𝒆𝒙
𝒙→−∞ 𝐥𝐢𝐦 = +∞
𝒙→+∞ 𝒙
𝐥𝐢𝐦 𝒙𝒏 𝒆𝒙 = 𝟎 𝒏∈ℕ 𝒆𝒙
𝒙→−∞ 𝐥𝐢𝐦 = +∞ 𝒏∈ℕ
𝒙→+∞ 𝒙𝒏
𝒆𝒙 − 𝟏
𝐥𝐢𝐦 =𝟏
𝒙→𝟎 𝒙

La dérivation :
 ∀𝒙 ∈ ℝ (𝒆𝒙 )′ = 𝒆𝒙
 Si 𝒖 est dérivable sur un intervalle 𝑰 alors la fonction 𝒙 → 𝒆𝒖(𝒙) est dérivable sur 𝑰
et on a : (𝒆𝒖(𝒙) )′ = 𝒖′ (𝒙) × 𝒆𝒖(𝒙) ∀𝒙 ∈ 𝑰

La fonction exponentielle de base 𝐚 :
Définition 𝒆𝒙𝒑𝒂 (𝒙) = 𝒂𝒙 = 𝒆𝒙𝒍𝒏(𝒂)
avec 𝒂 un réel strictement positive et différent de 𝟏
La dérivée ∀𝒙 ∈ ℝ (𝒂𝒙 )′ = (𝒍𝒏 𝒂)𝒂𝒙
Cas particulière La fonction définie sur ℝ par : 𝒙 → 𝟏𝟎𝒙
𝒂 = 𝟏𝟎 ∀𝒙 ∈ ℝ ∀𝒚 ∈ ]𝟎, +∞[ 𝟏𝟎𝒙 = 𝒚 ⟺ 𝒙 = 𝒍𝒐𝒈(𝒚)

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