Question

Using differentials, find the approximate values of the following:
(vi)$(255{)}^{\frac{1}{4}}$

Answer 1

Let $y=x^{\frac{1}{4}}\Rightarrow \frac{dy}{dx}=\frac{1}{4x^{\frac{3}{4}}}$
And x = 256 and $\Delta x=-1$
Then,
$\Delta y=(x+\Delta x{)}^{\frac{1}{4}}-x^{\frac{1}{4}}$
$\Rightarrow \Delta y=(256-1{)}^{\frac{1}{4}}-(256{)}^{\frac{1}{4}}$
$\Rightarrow \Delta y=(255{)}^{\frac{1}{4}}-4$
$\Rightarrow (255{)}^{\frac{1}{4}}=4+\Delta y$ ...(1)
Now, approximate change in value of y is given by,
$\because \Delta y\approx \left(\frac{dy}{dx}\right)\times \Delta x$
$\Rightarrow \Delta y\approx \frac{1}{4x^{\frac{3}{4}}}\times \Delta x$
$\Rightarrow \Delta y\approx \frac{1}{(256{)}^{\frac{3}{4}}}\times (-1)$
$\Rightarrow \Delta y\approx -\frac{1}{4\times 64}\approx -0.0039$
From equation(1)
$(255{)}^{\frac{1}{4}}\approx 4-0.0039$
$\therefore (255{)}^{\frac{1}{4}}\approx 3.9961$