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

Question

Using differentials, find the approximate values of the following:
$(29)^{\frac{1}{3}}$

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Answer

Consider the function $\text{y}=\text{f} (\text{x})=(\text{x})^{\frac{1}{3}}$ Let:
$\text{x} =27$ $\text{x}+\triangle \text{x}=29$Then,
$\text{x}= 2$ For $\text{x}=27$
$\text{y}=(27)^{\frac {1}{3}}=3$Let:
$\text{dx}=\triangle \text{x}=2$Now $\text{y}=(\text {x})^{\frac{1}{3}}$
$\Rightarrow\frac {\text{dy}}{\text{dx}}=\frac{1}{3(\text {x})^{\frac{2}{3}}}$ $\Rightarrow\Big(\frac {\text{dy}}{\text{dx}}\Big)_{\text{x}= 27}=\frac{1}{27}$ $\therefore\triangle \text{y}=\text{dy}=\frac{\text{dy}} {\text{dx}}\text{dx}=\frac{1} {27}\times2=0.074$ $\Rightarrow\triangle \text{y} =0.074$ $\therefore(29)^{\frac {1}{3}}=\text{y}+\triangle\text{y} =3.074$

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