If $\displaystyle x>0$ show that $\displaystyle |(1+x)^{1/3}-(1+\frac{1}{3}x-\frac{1}{9}x^2)|\leq (5/81)x^3$ .
Use this inequality to approximate $\displaystyle \sqrt[3]{1.2}$ and $\displaystyle \sqrt[3]{2}$ .
If $\displaystyle x>0$ show that $\displaystyle |(1+x)^{1/3}-(1+\frac{1}{3}x-\frac{1}{9}x^2)|\leq (5/81)x^3$ .
Use this inequality to approximate $\displaystyle \sqrt[3]{1.2}$ and $\displaystyle \sqrt[3]{2}$ .
Apply Taylor's formula and look at the remainder term. Report back if you have any issues.