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Question Number 12612 by frank ntulah last updated on 26/Apr/17

$$\mathrm{Use}\mathrm{Newtown}\mathrm{Raphson}\mathrm{method}\mathrm{to}\mathrm{find}\mathrm{aproximate}$$$$\mathrm{value}\mathrm{of}\mathit{X}=\sqrt{\left(\frac{7}{\mathit{x}+1}\right)},\mathrm{starting}\mathrm{with}{\mathit{x}}_0=2.$$$$\mathit{p}\mathit{e}\mathit{r}\mathit{f}\mathit{o}\mathit{r}\mathit{m}4\mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{a}\mathit{n}\mathit{d}\mathit{a}\mathit{l}\mathit{l}\mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}$$$$\mathit{s}\mathit{h}\mathit{o}\mathit{u}\mathit{l}\mathit{d}\mathit{b}\mathit{e}\mathit{p}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{t}\mathit{e}\mathit{d}\mathit{i}\mathit{n}4\mathit{d}\mathit{e}\mathit{c}\mathit{i}\mathit{m}\mathit{a}\mathit{l}\mathit{p}\mathit{l}\mathit{a}\mathit{c}\mathit{e}\mathit{s}$$

Answered by mrW1 last updated on 26/Apr/17

$$f\left(x\right)=\sqrt{\frac{7}{x+1}}-x$$$$f^{\prime }\left(x\right)=\frac{1}{2\sqrt{\frac{7}{x+1}}}\times \frac{-7}{{(x+1)}^2}-1$$$$\frac{f\left(x\right)}{f^{\prime }\left(x\right)}=-\frac{\sqrt{\frac{7}{x+1}}-x}{\frac{1}{2\sqrt{\frac{7}{x+1}}}\times \frac{7}{{(x+1)}^2}+1}=-\frac{\frac{7}{x+1}-x\sqrt{\frac{7}{x+1}}}{\frac{7}{2{(x+1)}^2}+\sqrt{\frac{7}{x+1}}}$$$$=-\frac{14(x+1)-2x\sqrt{7{(x+1)}^3}}{7+2\sqrt{7{(x+1)}^3}}=g\left(x\right)$$$$x_{n+1}=x_n-\frac{f\left(x_n\right)}{f^{\prime }\left(x_n\right)}=x_n-g\left(x_n\right)$$$$$$$$x_0=2$$$$x_1=x_0-g\left(x_0\right)=2-g\left(2\right)=1.6234$$$$x_2=x_1-g\left(x_1\right)=1.6234-g(1.6234)=1.6310$$$$x_3=x_2-g\left(x_2\right)=1.6310-g(1.6310)=1.6311$$$$x_4=x_2-g\left(x_3\right)=1.6311-g(1.6311)=1.6311$$$$\Rightarrow x\approx 1.6311$$

Commented by frank ntulah last updated on 27/Apr/17

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

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