Question 2

Durée : 5 mn

Note maximale : 6

Question

Donner les expressions approchées, quand \(\Delta x << x ,\) de :

  • \(a.~~\sqrt{x+\Delta x}\)

  • \(b.~~\ln(x+\Delta x)\)

  • \(c.~~\tan(x+\Delta x)\)

Solution

Nous avons : \(\Delta f = f(x+\Delta x)-f(x)\cong df=f'(x)\Delta x\)

si \(\Delta x << x\) d'où :

  • \(a.~~\color{blue}\sqrt{x+\Delta x}\cong\sqrt x+\frac{\Delta x}{2\sqrt x}~~\color{red}\textrm{(2 points)}\)

  • \(b.~~\color{blue}\ln(x+\Delta x)\cong\ln x+\frac{\Delta x}{x}~~\color{red}\textrm{(2 points)}\)

  • \(c.~~\color{blue}\tan(x+\Delta x)\cong\tan x+(1+\tan^2x)\Delta x~~\color{red}\textrm{(2 points)}\)