Question

Using differentials, find the approximate value of each of the following up to 3 places of decimal ?
$(26.57{)}^{\frac{1}{3}}$

Answer 1

$\left(26.57{)}^{\frac{1}{3}}Considerf\right(x)=x^{\frac{1}{3}}\Rightarrow f^{\prime }(x)=\frac{1}{3}{x}^{\frac{-2}{3}}=\frac{1}{3x^{\frac{2}{3}}}Letx=27and\Delta x=-0.43Now,f(x+\Delta x)\simeq f(x)+\Delta x{f}^{\prime }(x)\Rightarrow \sqrt{x+\Delta x}\simeq \sqrt{x}+\frac{1}{2\sqrt{x}}\times \Delta x\Rightarrow \sqrt{0.0036+0.0001}=\sqrt{0.0036}+\frac{0.0001}{2\sqrt{0.0036}}=0.06+\frac{0.0001}{2\times 0.06}=0.06+0.0008=0.0608\Rightarrow \sqrt{0.0037}\simeq 0.0608$