Question

dertemine o valor aproximado da raiz quadrada nao exata de 28

Answer 1

$\mathrm{\sqrt{28}=\sqrt{2^2.7}\ \to\ \sqrt{28}=2\sqrt{7}}\\\\ \mathrm{Obteremos\ o\ valor\ aproximado\ de\ \sqrt{7}}\\ \mathrm{atrav\'es\ do\ m\'etodo\ de\ Newton-Raphson:}\\\\ \mathrm{x_{n+1}=x_n-\cfrac{f(x_n)}{f'(x_n)}}$

$\mathrm{Se\ f(x)=x^2-7,\ quando\ f(x)=0,\ x=\sqrt{7}.\ Logo:}\\\\ \mathrm{f(x)=x^2-7\ \ \| \ \ f'(x)=2x\ \ \| \ \ 2\ \textless \ \sqrt{7}\ \textless \ 3\ \to\ x_0=2}\\\\ \mathrm{x_1=2-\cfrac{f(2)}{f'(2)}=2-\bigg(\cfrac{-3}{4}\bigg)=\cfrac{8}{4}+\cfrac{3}{4}=\cfrac{11}{4}=2,75}\\\\\\ x_2=2,75-\cfrac{f(2,75)}{f'(2,75)}=2,75-\cfrac{0,5625}{5,5}}\\\\\ \mathrm{x_2=\cfrac{15,125-0,5625}{5,5}=\cfrac{14,5625}{5,5}\approx2,64773}$

$\mathrm{x_3=2,64773-\cfrac{f(2,64773)}{f'(2,64773)}=2,64773-\cfrac{0,0104741529}{5,29546}}\\\\ \mathrm{x_3=\cfrac{14,0209483058-0,0104741529}{5,29546}=\cfrac{14,0104741529}{5,29546}}\\\\ \mathrm{x_3\approx2,64575}\ \to\ \mathbf{\sqrt{7}\approx2,64575}\\\\\\ \textrm{Portanto:}\\\\ \mathrm{\sqrt{28}=2\sqrt{7}\approx2.2,64575}\ \to\ \mathbf{\sqrt{28}\approx5,2915}$