Question

Use differentials to approximate fourth root of 81.5.

Solution

   $straight y space equals space straight x to the power of 1 fourth end exponent comma space space space straight x space equals space 81 comma space space space dx space equals space 0.5$
   $δy space equals space left parenthesis straight x plus dx right parenthesis to the power of 1 fourth end exponent minus straight x to the power of 1 fourth end exponent space equals space left parenthesis 81.5 right parenthesis to the power of 1 fourth end exponent space minus left parenthesis 81 right parenthesis to the power of 1 fourth end exponent space equals space left parenthesis 81.5 right parenthesis to the power of 1 fourth end exponent space minus space 3$
$therefore space space space space left parenthesis 81.5 right parenthesis to the power of 1 fourth end exponent space equals space 3 space plus space straight delta space straight y$ ...(1)
Now $δy$ is approximately equal to dy.
$and space space space space space space space space dy space equals space dy over dx dx space equals space 1 fourth straight x to the power of negative 3 over 4 end exponent dx space equals space fraction numerator 1 over denominator 4 straight x to the power of begin display style 3 over 4 end style end exponent end fraction cross times dx$
   $equals space fraction numerator 1 over denominator 4 left parenthesis 81 right parenthesis to the power of begin display style 3 over 4 end style end exponent end fraction cross times space 0.5 space equals space fraction numerator 0.5 over denominator 4 cross times 27 end fraction space equals space fraction numerator 0.5 over denominator 108 end fraction space equals space 0.0046$
$therefore space space space from space left parenthesis 1 right parenthesis comma space space space space space left parenthesis 81.5 right parenthesis to the power of 1 fourth end exponent space equals space 3 plus 0.0046 space equals space 3.0046 space equals space 3.005$

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Application Of Derivatives

Use differentials to approximate fourth root of 81.5.

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The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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