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  • Using differentials, find the approximate value of each of the following: 33 ^- 1 / 5

1. Using differentials, find the approximate value of each of the following:
(b) ( 33) ^{-1/5 }

Answers (1)

best_answer

Let y = x^\frac{-1}{5}  and  x = 32 \ and \ \Delta x = 1
\Delta y = (x+\Delta x)^\frac{-1}{5}-x^\frac{-1}{5}
         = (32+1)^\frac{-1}{5}-(32)^\frac{-1}{5}
(33)^\frac{-1}{4} = \Delta y + \frac{1}{2}
Now, we know that \Delta y is approximately equals to dy
So,
dy = \frac{dy}{dx}.\Delta x \\ = \frac{-1}{5x^\frac{6}{5}}.1 \ \ \ \ \ \ \ (\because y = x^\frac{-1}{5} \ and \ \Delta x = 1)\\ = \frac{-1}{5(32)^\frac{6}{5}}.1 = \frac{-1}{5\times 64}.1= \frac{-1}{320}
Now,
(33)^\frac{-1}{5} = \Delta y + \frac{1}{2} = \frac{-1}{320}+\frac{1}{2} = \frac{159}{320} = 0.497
Hence, (33)^\frac{-1}{5} is approximately equals to 0.497

Posted by

Gautam harsolia

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