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Application Of Derivatives

Question

Wired Faculty App

Use differentials to approximate:
$open parentheses 25 close parentheses to the power of 1 third end exponent$

Solution

Take space straight y space equals space straight x to the power of 1 third end exponent comma space space space space straight x space equals space 27 comma space space space dx space equals space minus 2 space space space space so space that space straight x plus dx space equals space 25 Now space space space space space space space space space space δy space equals space space left parenthesis straight x plus δx right parenthesis to the power of 1 third end exponent space minus space straight x to the power of 1 third end exponent space equals space left parenthesis 27 minus 2 right parenthesis to the power of 1 third end exponent space minus space 27 to the power of 1 third end exponent space equals space left parenthesis 25 right parenthesis to the power of 1 third end exponent minus 3

therefore space space space space space space space left parenthesis 25 right parenthesis to the power of 1 third end exponent space equals space 3 space plus space δy

...(1)
Now

δy

is approximately equal to dy and

dy space equals space open parentheses dy over dx close parentheses dx space equals space open parentheses 1 third straight x to the power of negative 2 over 3 end exponent close parentheses space dx space equals space 1 third left parenthesis 27 right parenthesis to the power of 2 over 3 end exponent left parenthesis negative 2 right parenthesis space space space space space space space space equals space 1 third left parenthesis 3 cubed right parenthesis to the power of negative 2 over 3 end exponent left parenthesis negative 2 right parenthesis space equals space 1 third left parenthesis 3 right parenthesis to the power of negative 2 end exponent left parenthesis negative 2 right parenthesis space equals space 1 third cross times 1 over 9 cross times left parenthesis negative 2 right parenthesis space equals space minus 2 over 27 therefore space space space from space left parenthesis 1 right parenthesis comma space left parenthesis 25 right parenthesis to the power of 1 third end exponent space equals space 3 minus 2 over 27 space equals fraction numerator 81 minus 2 over denominator 27 end fraction space equals 79 over 27 space equals 2.926

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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