Question

अवकल का प्रयोग करके निम्नलिखित में से सन्निकट मान दशमलव के तीन स्थानों तक ज्ञात कीजिए:

${\displaystyle {\left(32.15\right)}^{\frac{1}{5}}}$

Answer 1

माना ${\displaystyle \text{y}={\left(32\right)}^{\frac{1}{5}}=2}$              ...${\displaystyle [\because 32.15=32+0.15]}$

${\displaystyle \therefore \text{y}={\text{x}}^{\frac{1}{5}},}$ x = 32 y = 2 तथा ${\displaystyle \Delta \text{x}=0.15}$ 

y = ${\displaystyle {\text{x}}^{\frac{1}{5}} \text{से} \frac{\text{dy}}{\text{dx}}=\frac{1}{5}{\text{x}}^{\frac{1}{5}-1}=\frac{1}{5}{\text{x}}^{\frac{-4}{5}}=\frac{1}{5{\text{x}}^{\frac{4}{5}}}}$

${\displaystyle \therefore \text{dy}=\frac{\text{dy}}{\text{dx}}\times \Delta \text{x}}$

${\displaystyle =\frac{1}{5{\text{x}}^{\frac{4}{5}}}\times \Delta \text{x}}$

${\displaystyle =\frac{1}{5\times 16}\times 0.15}$

${\displaystyle =\frac{0.15}{80}}$

= 0.001875

${\displaystyle \therefore {\left(32.15\right)}^{\frac{1}{5}}}$ = y + dy

= 2 + 0.01875

= 2.001875

= 2.002