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Solution for Using Differential, Find the Approximate Value of the ( 255 ) 1 4 . - CBSE (Science) Class 12 - Mathematics

Question

Using differential, find the approximate value of the $\left( 255 \right)^\frac{1}{4}$ ?

Solution

$\text { Consider the function y } = f\left( x \right) = \left( x \right)^\frac{1}{4} .$

$\text { Let }:$

$x = 256$

$x + ∆ x = 255$

$\text { Then},$

$∆ x = - 1$

$\text { For } x = 256,$

$y = \left( 256 \right)^\frac{1}{4} = 4$

$\text { Let }:$

$dx = ∆ x = - 1$

$\text { Now,} y = \left( x \right)^\frac{1}{4}$

$\Rightarrow \frac{dy}{dx} = \frac{1}{4 \left( x \right)^\frac{3}{4}}$

$\Rightarrow \left( \frac{dy}{dx} \right)_{x = 256} = \frac{1}{256}$

$\therefore ∆ y = dy = \frac{dy}{dx}dx = \frac{1}{256} \times - 1 = \frac{- 1}{256}$

$\Rightarrow ∆ y = \frac{- 1}{256} = - 0 . 003906$

$\therefore \left( 255 \right)^\frac{1}{4} = y + ∆ y = 3 . 99609 \approx 3 . 9961$

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Solution Using Differential, Find the Approximate Value of the ( 255 ) 1 4 . Concept: Approximations.
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