Question

Using differential, find the approximate values of the following:
$\left(xxiv\right)$. $\sqrt{36.6}$

Answer 1

$6.05$

Let $y=\sqrt{x}\Rightarrow \frac{dy}{dx}=\frac{1}{2\sqrt{x}}$
And $x=36$ and $\Delta x=0.6$
then,
$\Delta y=\sqrt{x+\Delta x}-\sqrt{x}\Rightarrow \Delta y=\sqrt{36+0.6}-\sqrt{36}\Rightarrow \Delta y=\sqrt{36.6}-6\Rightarrow \sqrt{36.6}=6+\Delta y\dots \left(1\right)$
Now, Approximate change in value of $y$ is given by
$\Delta y\approx \left(\frac{dy}{dx}\right)\times \Delta x\Rightarrow \Delta y\approx \frac{1}{2\sqrt{x}}\times \Delta x\Rightarrow \Delta y\approx \frac{1}{2\sqrt{36}}\times 0.6$
$\Rightarrow \Delta y\approx 0.05$
From equation $\left(1\right)$
$\sqrt{36.6}\approx 6+0.05\therefore \sqrt{36.6}\approx 6.05$