Application of Derivatives

Name: Wired Faculty - The Best Learning App
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Use differentials to approximate: cube root of 26 - Wired Faculty

Question

Use differentials to approximate:
cube root of 26

Solution

Take space straight y space equals space straight x to the power of 1 third end exponent comma space space space straight x space equals space 27 comma space space space space dx space equals space δx space equals space minus 1 space space space space space so space that space straight x space plus space δx space equals space 26 Now 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 δy space equals space left parenthesis straight x plus δx right parenthesis to the power of 1 third end exponent space minus space straight y space equals space left parenthesis 26 right parenthesis to the power of 1 third end exponent space minus space left parenthesis 27 right parenthesis to the power of 1 third end exponent space equals space left parenthesis 26 right parenthesis to the power of 1 third end exponent minus 3

rightwards double arrow space space space space left parenthesis 26 right parenthesis to the power of 1 third end exponent space equals space δy space plus space 3 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space.... left parenthesis 1 right parenthesis Now space space space δy space is space approximately space equal space to space dy and space dy space equals space dy over dx dx space equals space 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 1 over denominator 3 left parenthesis 27 right parenthesis to the power of begin display style 2 over 3 end style end exponent end fraction left parenthesis negative 1 right parenthesis space equals space minus 1 over 27 therefore space space space space space from space left parenthesis 1 right parenthesis comma space space space space left parenthesis 26 right parenthesis to the power of 1 third end exponent space equals space minus 1 over 27 plus 3 space equals space 80 over 27.