Question

Using differential, find the approximate value of the following:  \[\left( 0 . 009 \right)^\frac{1}{3}\]

Answer 1

\[\text { Consider the function } y = f\left( x \right) = \sqrt[3]{x} . \]

\[\text { Let }: \]

\[ x = 0 . 008\]

\[x + ∆ x = 0 . 009\]

\[\text { Then }, ∆ x = 0 . 001\]

\[\text { For } x = 0 . 008, \]

\[ y = \sqrt{0 . 008} = 0 . 2\]

\[\text { Let }: \]

\[ dx = ∆ x = 0 . 001\]

\[\text { Now,} y = \sqrt[3]{x}\]

\[ \Rightarrow \frac{dy}{dx} = \frac{1}{3 \left( x \right)^\frac{2}{3}}\]

\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 0 . 008} = \frac{1}{3 \times 0 . 04} = \frac{1}{0 . 12}\]

\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = \frac{1}{0 . 12} \times 0 . 001 = \frac{1}{120}\]

\[ \Rightarrow ∆ y = \frac{1}{120} = 0 . 008333\]

\[ \therefore \left( 0 . 009 \right)^\frac{1}{3} = y + ∆ y = 0 . 208333\]