Question

Using differential, find the approximate values of the following:
$\left(xxix\right)$. $\sqrt{0.082}$

Answer 1

$0.2867$

Let $y=\sqrt{x}\Rightarrow \frac{dy}{dx}=\frac{1}{2\sqrt{x}}$
And $x=0.09$ and $\Delta x=-0.008$
then,
$\Delta y=\sqrt{x+\Delta x}-\sqrt{x}\Rightarrow \Delta y=\sqrt{0.09-0.008}-\sqrt{0.09}\Rightarrow \Delta y=\sqrt{0.082}-0.3\Rightarrow \sqrt{0.082}=0.3+\Delta y\dots \left(1\right)$
Now, Approximate change in value of $y$ is given by
$\Delta y\approx \left(\frac{dy}{dx}\right)\times \Delta x\Rightarrow \Delta y\approx \frac{1}{2\sqrt{x}}\times \Delta x\Rightarrow \Delta y\approx \frac{1}{2\sqrt{0.09}}\times (-0.008)$
$\Rightarrow \Delta y\approx -0.0133$
From equation $\left(1\right)$
$\sqrt{0.082}\approx 0.3-0.0133\therefore \sqrt{0.082}\approx 0.2867$