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Question Number 44291 by peter frank last updated on 25/Sep/18

$$withioutusingcalculatorfindapproximatefor$$$$\sqrt{9.01}$$

Commented by maxmathsup by imad last updated on 25/Sep/18

$$\sqrt{9,01}=\sqrt{9+0,01}=\sqrt{9+{10}^{-2}}=3\sqrt{1+\frac{{10}^{-2}}{9}}butforx\in v\left(0\right)$$$$\sqrt{1+x}\sim 1+\frac{x}{2}\Rightarrow \sqrt{1+\frac{{10}^{-2}}{9}}\sim 1+\frac{{10}^{-2}}{18}\Rightarrow \sqrt{9,01}\sim 3(1+\frac{{10}^{-2}}{18})\Rightarrow$$$$\sqrt{9,01}\sim 3+\frac{{10}^{-2}}{6}...$$

Commented by Joel578 last updated on 26/Sep/18

$$\mathrm{Sir}\mathrm{please}\mathrm{explain}\sqrt{1+x}\sim 1+\frac{x}{2}$$

Commented by Necxx last updated on 26/Sep/18

$$itslinearapproximation$$

Commented by abdo.msup.com last updated on 26/Sep/18

$$f\left(x\right)=f\left(0\right)+\frac{x}{1!}{f}^{{}^{\prime }}\left(0\right)+\frac{x^2}{2}\xi \left(x\right)with$$$${lim}_{x\to 0}\xi \left(x\right)=0\Rightarrow f\left(x\right)\sim f\left(0\right)+{xf}^{{}^{\prime }}\left(x\right)(x\to 0)$$$$lettakef\left(x\right)=\sqrt{1+x}\Rightarrow f\left(0\right)=1and$$$$f^{{}^{\prime }}\left(x\right)=\frac{1}{2\sqrt{x+1}}\Rightarrow f^{{}^{\prime }}\left(0\right)=\frac{1}{2}\Rightarrow$$$$f\left(x\right)\sim 1+\frac{x}{2}.$$

Commented by maxmathsup by imad last updated on 26/Sep/18

$$f\left(x\right)\sim f\left(0\right)+x{f}^{{}^{\prime }}\left(0\right)(x\to 0)$$

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Sep/18

$$y=\sqrt{x}$$$$\frac{dy}{dx}=\frac{1}{2\sqrt{x}}$$$$herex=9x+△x=9.01△x=0.01$$$$y=\sqrt{9}=3y+△y=tofind$$$$\frac{△y}{△x}\approx \frac{dy}{dx}$$$$\frac{dy}{dx}=\frac{△y}{△x}$$$$△y=\frac{dy}{dx}△x$$$$△y=\frac{1}{2\sqrt{9}}\times 0.01=\frac{0.01}{6}=\frac{1}{6}\times \frac{1}{100}\approx 0.1666\times \frac{1}{100}=0.0016666$$$$\sqrt{9.01}=3.0016666$$$$$$

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