Question

Using differential, find the approximate value of the \[\left( 3 . 968 \right)^\frac{3}{2}\] ?

Answer 1

\[\text { Consider the function } y = f\left( x \right) = \left( x \right)^\frac{3}{2} . \]

\[\text { Let }: \]

\[ x = 4 \]

\[ x + ∆ x = 3 . 968\]

\[\text { Then }, \]

\[ ∆ x = - 0 . 032\]

\[\text { For } x = 4, \]

\[ y = \left( 4 \right)^\frac{3}{2} = 8\]

\[\text { Let }: \]

\[ dx = ∆ x = - 0 . 032\]

\[\text { Now }, y = \left( x \right)^\frac{3}{2} \]

\[ \Rightarrow \frac{dy}{dx} = \frac{3\sqrt{x}}{2}\]

\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 4} = 3\]

\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = 3 \times \left( - 0 . 032 \right) = - 0 . 096\]

\[ \Rightarrow ∆ y = - 0 . 096\]

\[ \therefore \left( 3 . 968 \right)^\frac{3}{2} = y + ∆ y = 7 . 904\]