Question

Find the approximate value of : $(3.97{)}^4$

Answer 1

Let $f\left(x\right)=x^4$
Then $f^{\prime }\left(x\right)=\frac{d}{dx}\left(x^4\right)=4x^3$
Take $a=4$ and $h=-0.03$.
Then $f\left(a\right)=f\left(4\right)=(4{)}^4=256$ and
$f^{\prime }\left(a\right)=f^{\prime }\left(4\right)=4(4{)}^3=256$
The formula for approximation is
$f(a+h)\doteqdot f\left(a\right)+h.f^{\prime }\left(a\right)$
$\therefore (3.97{)}^4=f(3.97)=f(4-00.03)$
$\doteqdot f\left(4\right)-\left(0.03\right)f^{\prime }\left(4\right)$
$\doteqdot 256-0.03\times 256$
$\doteqdot 256-7.68$
$=248.32$
$\therefore (3.97{)}^4\doteqdot 248.32$.