Chapter 12 - Vector-Valued Functions - Review Exercises - Page 863: 21

\[\begin{align} & \left( \mathbf{a} \right)6{{t}^{2}}\mathbf{i}+4\mathbf{j}-2t\mathbf{k} \\ & \left( \mathbf{b} \right)12t\mathbf{i}-2\mathbf{k} \\ & \left( \mathbf{c} \right)72{{t}^{3}}+4t \\ & \left( \mathbf{d} \right)-8\mathbf{i}-12{{t}^{2}}\mathbf{j}-48t\mathbf{k} \\ \end{align}\]

\[\begin{align} & \mathbf{r}\left( t \right)=2{{t}^{3}}\mathbf{i}+4t\mathbf{j}-{{t}^{2}}\mathbf{k} \\ & \left( \mathbf{a} \right)\text{Find }\mathbf{r}'\left( t \right) \\ & \mathbf{r}'\left( t \right)=\frac{d}{dt}\left[ 2{{t}^{3}}\mathbf{i}+4t\mathbf{j}-{{t}^{2}}\mathbf{k} \right] \\ & \mathbf{r}'\left( t \right)=6{{t}^{2}}\mathbf{i}+4\mathbf{j}-2t\mathbf{k} \\ & \\ & \left( \mathbf{b} \right)\text{Find }\mathbf{r}''\left( t \right) \\ & \mathbf{r}''\left( t \right)=\frac{d}{dt}\left[ 6{{t}^{2}}\mathbf{i}+4\mathbf{j}-2t\mathbf{k} \right] \\ & \mathbf{r}''\left( t \right)=12t\mathbf{i}+0\mathbf{j}-2\mathbf{k} \\ & \\ & \left( \mathbf{c} \right)\text{Find }\mathbf{r}'\left( t \right)\cdot \mathbf{r}''\left( t \right) \\ & \mathbf{r}'\left( t \right)\cdot \mathbf{r}''\left( t \right)=\left( 6{{t}^{2}}\mathbf{i}+4\mathbf{j}-2t\mathbf{k} \right)\cdot \left( 12t\mathbf{i}+0\mathbf{j}-2\mathbf{k} \right) \\ & \mathbf{r}'\left( t \right)\cdot \mathbf{r}''\left( t \right)=72{{t}^{3}}+0+4t \\ & \mathbf{r}'\left( t \right)\cdot \mathbf{r}''\left( t \right)=72{{t}^{3}}+4t \\ & \\ & \left( \mathbf{d} \right)\text{Find }\mathbf{r}'\left( t \right)\times \mathbf{r}''\left( t \right) \\ & \mathbf{r}'\left( t \right)\times \mathbf{r}''\left( t \right)=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 6{{t}^{2}} & 4 & -2t \\ 12t & 0 & -2 \\ \end{matrix} \right| \\ & \mathbf{r}'\left( t \right)\times \mathbf{r}''\left( t \right)=\left| \begin{matrix} 4 & -2t \\ 0 & -2 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 6{{t}^{2}} & -2t \\ 12t & -2 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 6{{t}^{2}} & 4 \\ 12t & 0 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{r}'\left( t \right)\times \mathbf{r}''\left( t \right)=-8\mathbf{i}-12{{t}^{2}}\mathbf{j}-48t\mathbf{k} \\ & \\ & \text{summary} \\ & \left( \mathbf{a} \right)6{{t}^{2}}\mathbf{i}+4\mathbf{j}-2t\mathbf{k} \\ & \left( \mathbf{b} \right)12t\mathbf{i}-2\mathbf{k} \\ & \left( \mathbf{c} \right)72{{t}^{3}}+4t \\ & \left( \mathbf{d} \right)-8\mathbf{i}-12{{t}^{2}}\mathbf{j}-48t\mathbf{k} \\ \end{align}\]