Chapter 12, Problem 12RE

Suppose that ${\text{r}}_1\left(t\right){\text{and r}}_{\text{2}}\left(t\right)$ are smooth vector-valued functions such that ${\text{r}}_1\left(0\right)=〈-1,1,2〉,{\text{r}}_2\left(0\right)=〈1,2,1〉,{\text{<msup> r <mo> ′ </mo> </msup>}}_{\text{1}}\left(0\right)=〈1,0,1〉\text{,}{\text{and <msup> r <mo> ′ </mo> </msup>}}_2\left(0\right)=〈4,0,2〉.$ Use this information to evaluate the derivative at $t=0$ of each function.

$\begin{array}{l}\left(\text{a}\right)\text{r}\left(t\right)=3{\text{r}}_1\left(t\right)+2{\text{r}}_2\left(t\right)\hfill \\ \left(\text{b}\right)\text{r}\left(t\right)=\left[\ln \left(t+1\right)\right]{\text{r}}_1\left(t\right)\hfill \\ \left(\text{c}\right)\text{r}\left(t\right)={\text{r}}_1\left(t\right)\times {\text{r}}_2\left(t\right)\hfill \\ \left(\text{d}\right)f\left(t\right)={\text{r}}_1\left(t\right)\cdot {\text{r}}_2\left(t\right)\hfill \end{array}$