Question

Using differentials, find the approximate value of $(3.968{)}^{\frac{3}{2}}$

Answer 1

Let $f\left(x\right)=x^{\frac{3}{2}}$
$f^{\prime }\left(x\right)=\frac{3}{2}{x}^{\frac{1}{2}}$
Take $a=4,h=-0.032$
$\therefore f\left(a\right)=f\left(4\right)=4^{\frac{3}{2}}=8$ and $f^{\prime }\left(a\right)=f^{\prime }\left(4\right)=\frac{3}{2}.4^{\frac{1}{2}}=3$
Therefore by using formula for approximation $f(a+h)=f\left(a\right)+h.f^{\prime }\left(a\right)$
$\therefore (3.968{)}^{\frac{3}{2}}=f(4)+h{f}^{\prime }(a)$
$=8-0.032\times 3$
$=8-0.096$
$=7.904$