Answer
$$0.083333,\ \ \ \ 0.080084,\ \ 3.25 \times 10^{-3} $$
Work Step by Step
Given $$9^{1 / 3}-2$$
Consider $f(x)= x^{1/3}$, $a= 8 $, $\Delta x=1$, since
\begin{align*}
f'(x) &= \frac{1}{3}x^{-2/3}\\
f'(8)&=0.0833333
\end{align*}
Then the linear approximation is given by
\begin{align*}
\Delta &f \approx f^{\prime}(a) \Delta x\\
&= (0.0833333)(1)\\
&= 0.0833333
\end{align*}
and the actual change is given by
\begin{align*}
\Delta f&=f(a+\Delta x)-f(a)\\
&=f(9)-f(8)\\
& =0.080084
\end{align*}
Hence the error is
$$ |0.080084-0.083333| \approx 3.25 \times 10^{-3} $$
