Find the approximate value of (1.999)^5. - Vidyadip

Question

Find the approximate value of $(1.999)^5$.

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Answer

Let $x = 2$
And $\triangle\text{x}=-0.001$ $[\because2-0.001=1.999]$
Let $y = x^5$
On differentiating both sides w.r.t. x, we get
$\frac{\text{dy}}{\text{dx}}=5\text{x}^4$
Now, $\triangle\text{y}=\frac{\text{dy}}{\text{dx}}\triangle\text{x}=5\text{x}^4\times\triangle\text{x}=5\times2^4\times[-0.001]=-80\times0.001=-0.080$
$\therefore\ (1.999)^5-\text{y}+\triangle\text{y}$
$=2^5+(-0.080)$
$=32-0.080=31.920$

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