Answer
$$0.002,\ \ \ ,0.0020012\ \ \ ,1.20\times 10^{-6}$$
Work Step by Step
Given $$625^{1 / 4}-624^{1 / 4} $$
Consider $f(x)=x^{1 /4}, a=625,$ and $\Delta x=-1$, since
\begin{align*}
f^{\prime}(x)&=\frac{1}{4}x^{-3/4}\\
f^{\prime}(625)&=\frac{-1}{500}= -0.002
\end{align*}
Then
\begin{align*}
\Delta f&=f (a+\Delta x)-f(a)\\
&\approx f'(a)\Delta x\\
&\approx (-0.002 )(-1) \\
&\approx 0.002
\end{align*}
By using a calculator, $625^{1 / 4}-624^{1 / 4}= 0.0020012 $ and the error in the linear approximation is
$$
|0.002 - 0.0020012 |=1.20\times 10^{-6}
$$
