Answer
$$8.33333\times 10^{-3},\ \ \ 3.445 \times 10^{-5}$$
Work Step by Step
Given $$8.1^{1 / 3}-2 $$
Consider $f(x)=x^{1 / 3}, a=8,$ and $\Delta x=0.1$, since
\begin{align*}
f^{\prime}(x)&=\frac{1}{3}x^{-2/3}\\
f^{\prime}(8)&=\frac{1}{12}
\end{align*}
Then
\begin{align*}
\Delta f&=f (a+\Delta x)-f(a)\\
&\approx f'(a)\Delta x\\
&\approx \frac{0.1}{12}\\
&\approx 8.33333\times 10^{-3}
\end{align*}
By using a calculator, $8.1^{1 / 3}-2=0.00829885$ and the error in the linear approximation is given by
$$
|0.00829885-0.00833333|=3.445 \times 10^{-5}
$$
