Question

By using differentiation, find the nearest value of $(0.007{)}^{1/3}$

Answer 1

Given,

${\left(0.007\right)}^{\frac{1}{3}}$

$Let,y=x^{\frac{1}{3}},x=0.008and\Delta x=-0.001$

$\frac{dy}{dx}=\frac{1}{3x^{\frac{2}{3}}}$

we know,

$\Delta y=\frac{dy}{dx}\Delta x$

$\Delta y=\frac{1}{3{\left(0.008\right)}^{\frac{2}{3}}}\times -0.001$

$\Delta y=\frac{-0.001}{3\times 0.04}$

$\Delta y=\frac{-0.1}{12}$

We know,

$\Delta y=f(x+\Delta x)-f\left(x\right)$

$\Delta y={(x+\Delta x)}^{\frac{1}{3}}-x^{\frac{1}{3}}$

$-\frac{0.1}{12}={\left(0.007\right)}^{\frac{1}{3}}-(0.008{)}^{\frac{1}{3}}$

${\left(0.007\right)}^{\frac{1}{3}}=(0.008{)}^{\frac{1}{3}}-\frac{0.1}{12}$

${\left(0.007\right)}^{\frac{1}{3}}=0.2-\frac{0.1}{12}$

${\left(0.007\right)}^{\frac{1}{3}}=\frac{2.4-0.1}{12}$

${\left(0.007\right)}^{\frac{1}{3}}=\frac{2.3}{12}$

${\left(0.007\right)}^{\frac{1}{3}}=0.19166$