Application of Derivatives

Use differentials to approximate - Wired Faculty

Question

Use differentials to approximate $square root of 36.6 end root$

Solution

Take $straight y space equals space square root of straight x comma space space space space straight x space equals space 36 comma space space space dx space equals space dδ space equals space 0.6 comma space so space that space straight x space plus space δx space equals space 36.6$
Now, $straight y plus δy space equals space square root of straight x plus δx end root$
$therefore space space space space space space space space space space space space space δy space equals space square root of straight x plus δx end root space minus space straight y space equals space square root of 36.6 end root space minus space square root of 36 rightwards double arrow space space space space space space space space space space space space space δy space equals space square root of 36.6 end root space minus space 6 rightwards double arrow space space space space space space space space square root of 36.6 end root space equals space δy minus 6 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis Now space δy space is space approximately space equal space to space dy$
and $dy space equals space dy over dx dx space equals space fraction numerator 1 over denominator 2 square root of straight x end fraction dx$
$equals space fraction numerator 1 over denominator 2 square root of 36 end fraction left parenthesis 0.6 right parenthesis space equals space fraction numerator 0.3 over denominator 6 end fraction space equals space 0.05$
$therefore space space space from space left parenthesis 1 right parenthesis comma space space space square root of 36.6 end root space equals space 0.05 plus 6 space equals space 6.05.$

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Application Of Derivatives

Use differentials to approximate $square root of 36.6 end root$

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