Application of Derivatives

Use differentials to approximate . - Wired Faculty

Question

Use differentials to approximate $square root of 0.037 end root$ .

Solution

Take $straight y space equals space square root of straight x comma space space space space straight x space equals space 0.0361 comma space space space space dx space equals space δx space equals space 0.0009 space space so space that space space straight x plus δx space equals space 0.037$
Now $straight y plus δy space equals space square root of straight x plus δx end root$
$rightwards double arrow space space space space space δy space equals square root of straight x plus δx end root space minus space square root of straight x space equals space square root of 0.037 end root space minus space square root of 0.0361 end root space equals space square root of 0.037 end root space minus 0.19 rightwards double arrow space space space space square root of 0.037 end root space equals space δy space plus 0.19 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis Now space δy space is space approximately space equal space to space dy$
$and space space dy space equals dy over dx dx space equals space fraction numerator 1 over denominator 2 square root of straight x end fraction dx space equals space fraction numerator 1 over denominator 2 square root of 0.0361 end root end fraction left parenthesis 0.0009 right parenthesis space space space space space space equals space fraction numerator 0.0009 over denominator 2 left parenthesis 0.19 right parenthesis end fraction space equals space fraction numerator 0.0009 over denominator 0.38 end fraction space equals 0.023 therefore space space space space space from space left parenthesis 1 right parenthesis comma space space square root of 0.037 end root space equals space 0.0023 plus space 0.19 space equals space 0.1923.$

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Application Of Derivatives

Use differentials to approximate $square root of 0.037 end root$ .

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