Chapter 1.2, Problem 8E

a.

To determine

To find: The third Taylor polynomial $P_3\left(x\right)$ for the function $f\left(x\right)=\left(x-1\right)\ln x$ about $x_0=1$ , use $P_3\left(0.5\right)$ to approximate $f\left(0.5\right)$ , 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

To find: A 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 $\left[0.5,1.5\right]$ .

c.

To determine

To find: The approximate value of ${\displaystyle {\int }_{0.5}^{1.5}f\left(x\right)dx}$ using ${\displaystyle {\int }_{0.5}^{1.5}{P}_3\left(x\right)dx}$ .

d.

To determine

To find: An upper boundfor the error of approximation of ${\displaystyle {\int }_{0.5}^{1.5}f\left(x\right)dx}$ using ${\displaystyle {\int }_{0.5}^{1.5}{P}_3\left(x\right)dx}$ , using ${\displaystyle {\int }_{0.5}^{1.5}\left|R_3\left(x\right)dx\right|}$ and compare the bound to the actual error.