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 :
\(a.~~\color{blue}\sqrt{x+\Delta x}\cong\sqrt x+\frac{\Delta x}{2\sqrt x}~~\color{red}\textrm{(2 points)}\)
\(b.~~\color{blue}\ln(x+\Delta x)\cong\ln x+\frac{\Delta x}{x}~~\color{red}\textrm{(2 points)}\)
\(c.~~\color{blue}\tan(x+\Delta x)\cong\tan x+(1+\tan^2x)\Delta x~~\color{red}\textrm{(2 points)}\)