Question

Using differential, find the approximate value of the \[\left( 66 \right)^\frac{1}{3}\] ?

Answer 1

\[\text { Consider  the  function }  y = f\left( x \right) =  \left( x \right)^\frac{1}{3}  . \] 

\[\text { Let }: \] 

\[  x   = 64  \] 

\[x +  ∆ x = 66\] 

\[\text { Then },   \] 

 

\[ ∆ x = 2\] 

\[\text { For }  x = 64, \] 

\[  y =  \left( 64 \right)^\frac{1}{3}  = 4\] 

\[\text{ Let }: \] 

\[  dx =  ∆ x = 2\] 

\[\text { Now },   y =  \left( x \right)^\frac{1}{3} \] 

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

\[ \Rightarrow  \left( \frac{dy}{dx} \right)_{x = 64}  = \frac{1}{48}\] 

\[ \therefore    ∆ y = dy = \frac{dy}{dx}dx = \frac{1}{48} \times 2 = 0 . 042\] 

\[ \Rightarrow  ∆ y = 0 . 042\] 

\[ \therefore    \left( 66 \right)^\frac{1}{3}  = y +  ∆ y = 4 . 042\]