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Application of Derivatives

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Short Answer Type

381.

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

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382.

Use differentials, find the approximate value of each of the following upto 3 places of decimal:
$left parenthesis 31.9 right parenthesis to the power of 1 fifth end exponent$

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383.

Use differentials, find the approximate value of each of the following upto 3 places of decimal:
$left parenthesis 0.999 right parenthesis to the power of 1 over 10 end exponent$

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384.

Use differentials, find the approximate value of each of the following upto 3 places of decimal:
$left parenthesis 32.15 right parenthesis to the power of 1 fifth end exponent$

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385.

Use differentials, find the approximate value of each of the following upto 3 places of decimal:
$left parenthesis 33 right parenthesis to the power of negative 1 fifth end exponent$

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386.
If y = x⁴ – 10 and if x changes from 2 to 1.99, what is the approximate change in y?

Now, $δy$ is approximately equal to dy and
Here, $straight y space equals straight x to the power of 4 minus 10$ $dy space equals space dy over dx dx space equals space 4 straight x cubed dx$
$therefore space space space dy over dx space equals 4 straight x cubed$ $equals space 4 left parenthesis 2 right parenthesis cubed space left parenthesis negative 0.01 right parenthesis space equals space minus 32 space cross times space 0.01 space equals space minus 0.32$
$When space straight x space equals space 2 comma space space space straight y space equals left parenthesis 2 right parenthesis to the power of 4 space minus 10 space equals space 16 minus 10 space equals space 6$
$therefore space space space space straight x space changes space from space 2 space to space 1.99 space space space space space space space space space space space space space therefore space space space straight y space plus space δy space equals space 6 plus left parenthesis negative 0.32 right parenthesis space equals space 5.68 therefore space space space space straight x space equals space 2 comma space space space space space space δx space equals space dx space equals space minus 0.01 space space space space space space space space space therefore space space space straight y space changes space from space 6 space to space 5.68$