Using differentials, find the approximate values of the following… - Vidyadip

Question

Using differentials, find the approximate values of the following:
$(66)^{\frac{1}{3}}$

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Answer

Consider the function $\text{y}=\text{f} (\text{x})=(\text{x})^{\frac{1}{3}}$ Let:
$\text{x}=64$
$\text{x}+\triangle \text{x}=66$Then,
$\triangle\text{x}=2$ For $\text{x}=64$
$\text{y}=(64)^{\frac {1}{3}}=4$Let:
$\text{dx}=\triangle \text{x}=2$Now, $\text{y}=(\text{x})^ {\frac{1}{3}}$
$\Rightarrow\frac {\text{dy}}{\text{dx}}=\frac{1}{3(\text {x})^{\frac{2}{3}}}$ $\Rightarrow\Big(\frac {\text{dy}}{\text{dx}}\Big)_{\text{x}} =64=\frac{1}{48}$ $\therefore\triangle \text{y}=\text{dy}=\frac{\text{dy}} {\text{dx}}\text{dx}=\frac{1} {48}\times2=0.042$ $\Rightarrow\triangle \text{y} =0.042$ $\therefore(66)^{\frac {1}{3}}=\text{y}+\triangle\text{y} =4.042$

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