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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:
$left parenthesis 31.9 right parenthesis to the power of 1 fifth end exponent$

76 Views

383.
Use differentials, find the approximate value of each of the following upto 3 places of decimal:
$left parenthesis 0.999 right parenthesis to the power of 1 over 10 end exponent$

Take space straight y space equals space straight x to the power of 1 over 10 end exponent comma space space space space straight x space equals space 1 comma space space space dx space equals space δx space equals space minus 0.001 space space space so space that space straight x space plus space δx space equals space 0.999 Now space straight y space plus space δy space equals space left parenthesis straight x plus δx right parenthesis to the power of 1 over 10 end exponent space space space rightwards double arrow space space space space space δy space equals space left parenthesis straight x plus δx right parenthesis to the power of 1 over 10 end exponent space minus space straight y space equals space left parenthesis 0.999 right parenthesis to the power of 1 over 10 end exponent minus 1 rightwards double arrow space space space space space left parenthesis 0.999 right parenthesis to the power of 1 over 10 end exponent space equals space δy plus 1 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 space.... left parenthesis 1 right parenthesis 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 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

Now 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 over 10 straight x to the power of negative 9 over 10 end exponent space equals fraction numerator 1 over denominator 10 straight x to the power of begin display style 9 over 10 end style end exponent end fraction space equals space fraction numerator 1 over denominator 10 left parenthesis 1 right parenthesis to the power of begin display style 9 over 10 end style end exponent end fraction left parenthesis negative 0.001 right parenthesis space equals space fraction numerator negative 0.001 over denominator 10 end fraction space equals space minus 0.0001

therefore space space space from space left parenthesis 1 right parenthesis comma space space space left parenthesis 0.999 right parenthesis to the power of 1 over 10 end exponent space equals space minus 0.0001 space plus 1 space equals space 0.9999

therefore space space space space space left parenthesis 0.999 right parenthesis to the power of 1 over 10 end exponent space equals space 0.9999

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384.

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

78 Views

385.

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