Use differentials to approximate the cube root of 28. from Math

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Application of Derivatives

Short Answer Type

361.

Use differentials to approximate:
$square root of 3.968 end root$

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362.

Use differentials to approximate:
$square root of 25.2 end root$

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363.

Use differentials to approximate:
$left parenthesis 0.0037 right parenthesis to the power of 1 half end exponent$

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364.

Use differentials to approximate:
$square root of 36.3 end root$

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365.

Use differentials to approximate:
$square root of 49.7 end root$

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366.

Use differentials to approximate:
$square root of 401$

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367.

Use differentials to approximate:
$square root of 402$

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368.

Use differentials to approximate:
$square root of 51$

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369.

Use differentials to approximate:
$square root of 26$

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370. Use differentials to approximate the cube root of 28.

Take space straight y space equals straight x to the power of 1 third end exponent comma space space space straight x space equals space 27 comma space space space dx space equals space 1 space space space space space so space that space space straight x plus space dx space equals space 28 Now 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 x to the power of 1 third end exponent space equals space left parenthesis 27 plus 1 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 28 right parenthesis to the power of 1 third end exponent space minus space 3

therefore space space space left parenthesis 28 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 space space space space space space equals 1 third left parenthesis 27 right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 1 right parenthesis space equals space 1 third left parenthesis 3 cubed right parenthesis to the power of negative 2 over 3 end exponent left parenthesis 1 right parenthesis space equals space 1 third left parenthesis 3 right parenthesis to the power of negative 2 end exponent left parenthesis 1 right parenthesis space equals space 1 third cross times 1 over 9 cross times left parenthesis 1 right parenthesis space equals 1 over 27 therefore space space from space left parenthesis 1 right parenthesis comma space left parenthesis 28 right parenthesis to the power of 1 third end exponent space is space approximately space equal space to space 3 plus 1 over 27 space equals 82 over 27.

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