p 181 n°2

règle du cours : \(\exp(x+y) = \exp(x) \times \exp(y)\)

\(\exp(2) = \exp(1 + 1)\)
• \(\exp(2) = \exp(1) \times \exp(1)\)
• \(\exp(2) = (\exp(1))^2\)

\(\exp(3) = \exp(1 + 2)\)
• \(\exp(3) = \exp(1) \times \exp(2)\)
• \(\exp(3) = \exp(1) \times (\exp(1))^2\)
• \(\exp(3) = (\exp(1))^1 \times (\exp(1))^2\)
• \(\exp(3) = (\exp(1))^{1+2} = (\exp(1))^{3}\)

Remarque : \(\exp(3) = \exp(1,5 + 1,5) = \exp(1,5) \times \exp(1,5) = (\exp(1,5))^2\)

donc \((\exp(1))^{3} = (\exp(1,5))^2\)

\(\exp(10) = \exp( 5 + 5 )\)
• \(\exp(10) = \exp( 5 ) \times \exp(5)\)
• \(\exp(10) = (\exp( 5 ))^2\)
• \(\exp(10) = (\exp( 2+3 ))^2\)
• \(\exp(10) = \left(\exp( 2) \times \exp(3 )\right)^2\)
• \(\exp(10) = \left(\exp( 1)^2 \times \exp(1 )^3\right)^2\)
• \(\exp(10) = \left(\exp( 1)^{2+3}\right)^2\)
• \(\exp(10) = \left(\exp( 1)^{5}\right)^2\)
• \(\exp(10) = \exp( 1)^{5 \times 2} = \exp( 1)^{10}\)

Conjecture : \(\exp(n) = \exp(1)^n\)

Exercice du cours

Rappels sur les puissances

• \(2^3 \times 2^5 =2^8\)
• \(10^5 \times 10^{-5} =10^0 = 1\)
• \((2^3)^4 = 2^{12}\)
• \(\dfrac{8^9}{8^8} =8^1 = 8\)
• \(\dfrac{3^{10}}{3^6} =3^4\)
  
• \(\dfrac{4^2}{4^8} =4^{-6}\)

histoire de la racine carrée

Supposons qu'on puisse écrire la racine carrée sous forme d'une puissance : \(\sqrt{x} = x^p\) (avec \(x \geqslant 0\))

on sait \(\left(\sqrt{x}\right)^2 = x\)
donc il faut \(\left(x^p\right)^2 = x\)
c'est à dire \(x^{2p} = x\)

comparons des éléments de même nature : \(x^{2p} = x^1 \Leftrightarrow 2p = 1 \Leftrightarrow p = \frac{1}{2}\)

notation e

\(\exp(-x) = \dfrac{1}{\exp(x)}\) et \(e^{-x} = \dfrac{1}{e^x}\)

\(\exp(x + y) = \exp(x) \times \exp(y)\) et \(e^{x + y} = e^x \times e^y\)

au lieu d'écrire \(f(x)= \exp(x)\) , on écrira \(f(x) = e^x\)

p 191 n° 42

\(A = 3 \times \exp(x)^2 \times \exp(x)\)
• \(A = 3 \times \exp(x)^3\)
• \(A = 3 \times (e^x)^3\)

  (règle : \((a^x)^y = a^{x \times y}\))

• \(A = 3 \times e^{x \times 3}\)
• \(A = 3 e^{3x}\)

\(B = \dfrac{\exp{(3x+1)}}{\exp{(x^2)}}\)
• \(B = \dfrac{e^{3x+1}}{e^{x^2}}\)

  (règle : \(\dfrac{a^x}{a^y} = a^{x-y}\))

• \(B = e^{(3x+1) - x^2} = e^{-x^2 + 3x+1}\)

\(C= \exp(-x) \times \exp(x)\)
• \(C= e^{-x} \times e^{x} = e^{-x +x} = e^0 = 1\)