Application of Derivatives

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Use differentials to approximate: cube root of 0.009 - Wired Faculty

Question

Use differentials to approximate:
cube root of 0.009

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 0.008 comma space space space dx space equals space δx space equals space 0.001 space space space so space that space space straight x plus space δx space equals space 0.009 Now comma space space straight y space plus space δy space equals space left parenthesis straight x plus δx right parenthesis to the power of 1 third end exponent rightwards double arrow space space space space space space δy space equals space left parenthesis straight x plus δx right parenthesis to the power of 1 third end exponent space minus straight y space equals space left parenthesis 0.009 right parenthesis to the power of 1 third end exponent minus space left parenthesis 0.008 right parenthesis to the power of 1 third end exponent space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space δy space equals space left parenthesis 0.009 right parenthesis to the power of 1 third end exponent space minus 0.2 space space

rightwards double arrow space space space space left parenthesis 0.009 right parenthesis to the power of 1 third end exponent space equals space δy space plus space 0.2

...(1)
Now,

δy

is approximately equal to dy

and space dy space equals space dy over dx dx space equals 1 third straight x to the power of negative 2 over 3 end exponent dx space equals space fraction numerator 1 over denominator 3 straight x to the power of begin display style 2 over 3 end style end exponent end fraction dx space equals space fraction numerator 0.001 over denominator 3 left parenthesis 0.008 right parenthesis to the power of begin display style 2 over 3 end style end exponent end fraction space equals space fraction numerator 0.001 over denominator 3 left parenthesis 0.2 right parenthesis squared end fraction space equals space fraction numerator 0.001 over denominator 0.12 end fraction space equals space 0.008 therefore space space space space from space left parenthesis 1 right parenthesis comma space space left parenthesis 0.009 right parenthesis to the power of 1 third end exponent space equals space 0.2 plus 0.008 space equals space 0.208.