Answer
a) 1.032222222;
b) Absoute error: $5.8\cdot 10^{-5}$
Work Step by Step
We are given the function $f$ and the Taylor polynomial $p_2$:
$f(x)=\sqrt[3]{1+x}$
$p_2(x)=1+\dfrac{x}{3}-\dfrac{x^2}{9}$
a) Let $x=0.1$
Compute $p_2(x)$ for $x=0.1$:
$p_2(0.1)=1+\dfrac{0.1}{3}-\dfrac{0.1^2}{9}=1.032222222$
b) Use a computer to determine the exact value of $\sqrt[3]{1.1}$:
$\sqrt[3]{1.1}=1.032280115$
Determine the absolute error:
$|\text{exact value}-\text{approximating value}|=|1.032280115-1.032222222|=0.000057893=5.8\cdot 10^{-5}$
