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

Question

Using differentials, find the approximate values of the following:
$(1.999)^5$

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Answer

Consider the function $\text{y}=\text{f} (\text{x})=\text{x}^5$Let:
$\text{x}=5$
$\text{x}+\triangle \text{x}=1.999$Then,
$\triangle\text{x}=0.001$For $\text{x}=2$
$\text{y}=2^5=32$Let:
$\text{dx}=\triangle \text{x}=-0.001$Now, $\text{y}=\text{x}^5$
$\Rightarrow\frac {\text{dy}}{\text{dx}}=5\text{x}^4$ $\Rightarrow\Big(\frac {\text{dy}}{\text{dx}}\Big)_{\text{x}=2}=80$ $\therefore\triangle \text{y}=\text{dy}=\frac{\text{dy}} {\text{dx}}\text{dx}=80\times(-0.001)=-0.08$ $\Rightarrow\triangle\text{y} =-0.08$ $\therefore1.999^5=\text{y}+\triangle\text{y}=31.92$

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