Question

Using differential, find the approximate values of the following:
$\left(xvii\right)$. $(66{)}^{1/3}$

Answer 1

$4.0416$

Let $y=x^{1/3}\Rightarrow \frac{dy}{dx}=\frac{1}{3x^{2/3}}$
And $x=64$ and $\Delta x=2$
then,
$\Delta y=(x+\Delta x{)}^{1/3}-x^{1/3}\Rightarrow \Delta y=(64+2{)}^{1/3}-(64{)}^{1/3}\Rightarrow \Delta y=(66{)}^{1/3}-4\Rightarrow (66{)}^{1/3}=4+\Delta y\dots (1)$
Now, Approximate change in value of $y$ is given by
$\Delta y\approx \left(\frac{dy}{dx}\right)\times \Delta x\Rightarrow \Delta y\approx \frac{1}{3x^{2/3}}\times \Delta x\Rightarrow \Delta y\approx \frac{1}{3(64{)}^{\frac{2}{3}}}\times \left(2\right)$
$\Rightarrow \Delta y\approx -\frac{1}{3\times 16}\approx 0.0416$
From equation $\left(1\right)$
$(66{)}^{1/3}\approx 4+0.0416\therefore (66{)}^{1/3}\approx 4.0416$