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Q.
Find the approximate change in the volume $V$ of a cube of side $x$ meters caused by increasing the side by $2\%$.
Application of Derivatives
Solution:
It is given that $\frac{\Delta x}{x} \times 100 =2$
We have, $V = x^{3}$
$\Rightarrow \frac{dV}{dx} = 3x^{2}$
$\therefore \Delta V = \frac{dV}{dx} \times\Delta x$
$\Rightarrow \Delta V = 3x^{2}\,\Delta x$
$\Rightarrow \Delta V = 3x^{2} \times \frac{2x}{100}$
$\Rightarrow \Delta V = 0.06x^{2}$
Thus, the approximate change in volume is $0.06x^3\, m^3$.