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Q.
Use differentials to approximate the cube root of $127$.
Application of Derivatives
Solution:
Since, we have to find the approximate value of the cube root of $127$. So, we consider the function
$y=f(x) = x^{1/3}$
Let $x = 125$ and $x + \Delta x = 127$.
Then, $\Delta x = 127 - 125 = 2$
For $x = 125$, we have
$y - (125)^{1/3} = 5$.
[Putting $x = 125$ in $y = x^{1/3}$]
Let $dx = \Delta x = 2$
Now, $y = x^{1/3}$
$\Rightarrow \frac{dy}{dx} = \frac{1}{3x^{2/3}}$
$\Rightarrow \left(\frac{dy}{dx}\right)_{x = 125} = \frac{1}{3\left(125\right)^{^{2/3}}}$
$ = \frac{1}{3\left(5^{3}\right)^{^{2/3}} } = \frac{1}{75}$
$\therefore dy = \frac{dy}{dx} dx$
$\Rightarrow dy = \frac{1}{75}\left(2\right) = \frac{2}{75}$
$\Rightarrow \Delta y = \frac{2}{75}$
$\left[\because \Delta y \cong dy\right]$
Hence, $\left(127\right)^{1/3} = y + \Delta y = 5 + \frac{2}{75}$
$= 5.026$.