Question

Using differential, find the approximate values of the following:
$\left(xviii\right)$. $\sqrt{26}$

Answer 1

Let $y=\sqrt{x}\Rightarrow \frac{dy}{dx}=\frac{1}{2\sqrt{x}}$
And $x=25$ and $\Delta x=1$
then,
$\Delta y=\sqrt{x+\Delta x}-\sqrt{x}\Rightarrow \Delta y=\sqrt{25+1}-\sqrt{25}\Rightarrow \Delta y=\sqrt{26}-5\Rightarrow \sqrt{26}=5+\Delta y\dots \left(1\right)$
Now, Approximate change in value of $y$ is given by
$\Delta y\approx \left(\frac{dy}{dx}\right)\times \Delta x\Rightarrow \Delta y\approx \frac{1}{2\sqrt{x}}\times \Delta x\Rightarrow \Delta y\approx \frac{1}{2\sqrt{25}}\times 1$
$\Rightarrow \Delta y\approx 0.1$
From equation $\left(1\right)$
$\sqrt{26}\approx 5+0.1\therefore \sqrt{26}\approx 5.1$