Chapter 8.6, Problem 18E

(a)

To determine

To justify: Each equal sign $3\left({\text{r}}_3-{\text{r}}_2\right)={\text{z}}^{\prime }\left(1\right)$ $=0.5{\text{x}}^{\prime }\left(1\right)$ $=\frac{3}{2}\left({\text{p}}_3-{\text{p}}_2\right)$.

(b)

To determine

To show that: The point ${\text{r}}_2$ is the midpoint of the segment from ${\text{p}}_2$ to ${\text{p}}_3$.

(c)

To determine

To justify: Each equal sign $3\left({\text{r}}_1-{\text{r}}_0\right)={\text{z}}^{\prime }\left(0\right)$ $=0.5{\text{x}}^{\prime }\left(0.5\right)$.

(d)

To determine

To show that: The point $8{\text{r}}_1=-{\text{p}}_0-{\text{p}}_1+{\text{p}}_2+{\text{p}}_3+8{\text{r}}_0$.

(e)

To determine

To show that: The point ${\text{r}}_1=\frac{1}{2}\left({\text{r}}_2+\frac{1}{2}\left({\text{p}}_1+{\text{p}}_2\right)\right)$.