Question

Using differential, find the approximate values of the following:
$\left(xv\right)$. $(80{)}^{1/4}$

Answer 1

$2.9907$

Let $y=x^{1/4}\Rightarrow \frac{dy}{dx}=\frac{1}{4x^{3/4}}$
And $x=81$ and $\Delta x=-1$
then,
$\Delta y=(x+\Delta x{)}^{1/4}-x^{1/4}\Rightarrow \Delta y=(81-1{)}^{1/4}-(81{)}^{1/4}\Rightarrow \Delta y=(80{)}^{1/4}-3\Rightarrow (80{)}^{1/4}=3+\Delta y\dots (1)$
Now, Approximate change in value of $y$ is given by
$\Delta y\approx \left(\frac{dy}{dx}\right)\times \Delta x\Rightarrow \Delta y\approx \frac{1}{4x^{3/4}}\times \Delta x\Rightarrow \Delta y\approx \frac{1}{4(81{)}^{3/4}}\times (-1)$
$\Rightarrow \Delta y\approx -\frac{1}{4\times 27}\approx -0.0093$
From equation $\left(1\right)$
$(80{)}^{1/4}\approx 3-0.0093\therefore (80{)}^{1/4}\approx 2.9907$