## Calculus (3rd Edition)

$$8.33333\times 10^{-3},\ \ \ 3.445 \times 10^{-5}$$
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 $$|0.00829885-0.00833333|=3.445 \times 10^{-5}$$