Chapter 1.1, Problem 13ES

a.

To determine

The third polynomial $P_3\left(x\right)$ for the function $f\left(x\right)=\left(x-1\right)\ln x$ about $x_0=1$ and then to use $P_3\left(0.5\right)$ to approximate $f\left(0.5\right)$. Also, find an upper bound for error $\left|f\left(0.5\right)-P_3\left(0.5\right)\right|$ using the error formula and compare it to the actual error.

b.

To determine

The upper bound for the error $\left|f\left(x\right)-P_3\left(x\right)\right|$ in using $P_3\left(x\right)$ to approximate $f\left(x\right)$ on the interval $\left[0.5,15\right]$.

c.

To determine

The value of ${\displaystyle \underset{0.5}{\overset{1.5}{\int }}f\left(x\right)}dx$, using ${\displaystyle \underset{0.5}{\overset{1.5}{\int }}{P}_3\left(x\right)}dx$.

d.

To determine

The upper bound for the error in (c) using ${\displaystyle \underset{0.5}{\overset{1.5}{\int }}\left|R\left(x\right)\right|dx}$.