Question

Using differential, find the approximate value of the \[\left( 1 . 999 \right)^5\] ?

Answer 1

\[\text { Consider the function } y = f\left( x \right) = x^5 . \]

\[\text { Let }: \]

\[ x = 2 \]

\[ x + ∆ x = 1 . 999\]

\[\text { Then }, \]

\[ ∆ x = - 0 . 001\]

\[\text { For } x = 2, \]

\[ y = 2^5 = 32\]

\[\text { Let }: \]

\[ dx = ∆ x = - 0 . 001\]

\[\text { Now }, y = x^5 \]

\[ \Rightarrow \frac{dy}{dx} = 5 x^4 \]

\[ \Rightarrow \left( \frac{dy}{dx} \right)_{x = 2} = 80\]

\[ \therefore ∆ y = dy = \frac{dy}{dx}dx = 80 \times \left( - 0 . 001 \right) = - 0 . 08\]

\[ \Rightarrow ∆ y = - 0 . 08\]

\[ \therefore 1 . {999}^5 = y + ∆ y = 31 . 92\]