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Application Of Derivatives

Question

Wired Faculty App

Use differentials, find the approximate value of each of the following upto 3 places of decimal:
$left parenthesis 33 right parenthesis to the power of negative 1 fifth end exponent$

Solution

left parenthesis 33 right parenthesis to the power of negative 1 fifth end exponent space equals space fraction numerator 1 over denominator left parenthesis 33 right parenthesis to the power of begin display style 1 fifth end style end exponent end fraction

Let space space straight y space equals space straight f left parenthesis straight x right parenthesis space equals space straight x to the power of 1 fifth end exponent comma space space space space straight x space equals space 32 comma space space space dx space equals space 1

straight f left parenthesis straight x plus increment straight x right parenthesis space equals space left parenthesis 33 right parenthesis to the power of 1 fifth end exponent space equals space straight f left parenthesis straight x right parenthesis space plus space increment straight y space equals space 2 space plus space dy over dx cross times dy space space space space space space space space space space space space space space space space space space equals space 2 plus fraction numerator 1 over denominator 5 straight x to the power of begin display style 4 over 5 end style end exponent end fraction cross times 1 space equals space 2 plus fraction numerator 1 over denominator 5 space cross times space left parenthesis 32 right parenthesis to the power of begin display style 4 over 5 end style end exponent end fraction space space space space space space space space space space space space space space space space equals 2 plus fraction numerator 1 over denominator 5 space cross times space 16 end fraction space equals space 2 plus 1 over 80 space equals space 161 over 80 space space space space space space space space space space space space space space space space open parentheses 1 over 33 close parentheses to the power of 1 fifth end exponent space equals fraction numerator 1 over denominator left parenthesis 33 right parenthesis to the power of begin display style 1 fifth end style end exponent end fraction space equals fraction numerator 1 over denominator begin display style 161 over 80 end style end fraction space equals space 80 over 161 space equals space 0.4968 space equals space 0.497

Some More Questions From Application of Derivatives Chapter

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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