Chapter 1.1, Problem 14ES

a.

To determine

The third polynomial $P_3\left(x\right)$ for the function $f\left(x\right)=2x\cos 2x-{\left(x-2\right)}^2$ about $x_0=0$ and then to use $P_3\left(x\right)$ to approximate $f\left(0.4\right)$.

b.

To determine

The upper bound for the error $\left|f\left(0.4\right)-P_3\left(0.4\right)\right|$ and obtain the actual error.

c.

To determine

The fourth Taylor polynomial $P_4\left(x\right)$ and use it to approximate $f\left(0.4\right)$.

d.

To determine

The upper bound for the error $\left|f\left(0.4\right)-P_4\left(0.4\right)\right|$ and obtain the actual error.