Using differentials, find the approximate value of each of the following up to 3 places of decimal.
$(i) \sqrt{25.3} (ii) \sqrt{49.5}$
$(i i i) \sqrt{0.6}$

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Solution

Let $y = \sqrt{x}$

$\frac{d y}{d x} = \frac{1}{2 \sqrt{x}} d x$

$i] y = \sqrt{25}$ and $d x = 0.3$

$y = 5$

$\frac{d y}{d x} = \frac{1}{2 \sqrt{25}} (0.3)$

$\frac{d y}{d x} = 0.03$

$y + \frac{d y}{d x} - = 5.03$

$i i] y = \sqrt{49} d x = 0.5$

$y = 7, x = 49$

$\frac{d y}{d x} - = \frac{1}{2 \sqrt{49}} (0.5)$

$\frac{d y}{d x} = 0.0357$

$y + \frac{d y}{d x} = 7.0357$

$i i i] y = \sqrt{0.64} d x = - 0.04$

$\frac{d y}{d x} = \frac{1}{2 \sqrt{- 0.064}} (- 0.04)$

$\frac{d y}{d x} - = - 0.025$

$y + \frac{d y}{d x} - = 0.8 - 0.025$

$= 0.775$

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