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Application Of Derivatives

Question

Wired Faculty App

Use differentials to approximate fourth root of $17 over 81$ .

Solution

open parentheses 17 over 81 close parentheses to the power of 1 fourth end exponent space equals space fraction numerator left parenthesis 17 right parenthesis to the power of begin display style 1 fourth end style end exponent over denominator left parenthesis 81 right parenthesis to the power of begin display style 1 fourth end style end exponent end fraction space equals space fraction numerator left parenthesis 17 right parenthesis to the power of begin display style 1 fourth end style end exponent over denominator 3 end fraction

Let

straight f left parenthesis straight x right parenthesis space equals space straight x to the power of 1 fourth end exponent comma space space space straight x space equals space 16 comma space space dx space equals space 1 comma space space straight f left parenthesis straight x plus increment straight x right parenthesis space equals space 17 to the power of 1 fourth end exponent

increment straight y space equals space straight f left parenthesis straight x plus increment straight x right parenthesis space minus space straight f left parenthesis straight x right parenthesis

therefore space space space straight f left parenthesis straight x plus increment straight x right parenthesis space equals space straight f left parenthesis straight x right parenthesis plus increment straight y

δy

is approximately equal to dy

and space space space space space space dy space equals space dy over dx cross times dx space equals space 1 fourth straight x to the power of 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 space dx space equals space fraction numerator 1 over denominator 4 left parenthesis 16 right parenthesis to the power of begin display style 3 over 4 end style end exponent end fraction cross times 1 space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 1 over denominator 4 space cross times 8 end fraction space equals space 1 over 32 therefore space space space space space space 17 to the power of 1 fourth end exponent space equals space 2 plus 1 over 32 space equals space 2.03125 therefore space space space space space space open parentheses 17 over 81 close parentheses to the power of 1 fourth end exponent equals space fraction numerator left parenthesis 17 right parenthesis to the power of begin display style 1 fourth end style end exponent over denominator 3 end fraction space equals space fraction numerator 2.03125 over denominator 3 end fraction space equals space 0.677083 space equals space 0.677

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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