$\text{The approximate value of }(0.0037{)}^{\frac{1}{2}}$

$\text{ Consider }y=x^{\frac{1}{2}}\text{. Let }x=0.0036\text{ and }\mathrm{\Delta }x=0.0001\text{. Then }$

$\begin{array}{r}\begin{array}{rl} & \mathrm{\Delta }y=(x+\mathrm{\Delta }x{)}^{1/2}-(x{)}^{1/2}=(0.0037{)}^{1/2}-(0.0036{)}^{1/2}\\ & =(0.0037{)}^{1/2}-0.06\\ & \text{ Or }(0.0037{)}^{1/2}=0.06+\mathrm{\Delta }y\end{array}\\ \text{ Now, }dy\text{ is approximately equal to }\mathrm{\Delta }y\text{ and is given by }\end{array}$

$\begin{array}{r}dy=\left(\frac{dy}{dx}\right)\mathrm{\Delta }x=\frac{1}{2\sqrt{x}\left(\mathrm{\Delta }x\right)}\left[\text{ as }y=x^{1/2}\right]\\ =\frac{1}{2\times 0.06}\left(0.0001\right)\\ =\frac{0.0001}{0.12}=0.00083\end{array}$

$\text{ Thus, the approximation value of }(0.0037{)}^{\frac{1}{2}}\text{ is }0.06+0.00083=0.06083\text{. }$