Chapter 12 - Vector-Valued Functions - Review Exercises - Page 864: 35

\[\begin{align} & \left( \text{a} \right) \\ & \mathbf{v}\left( t \right)=4\mathbf{i}+3{{t}^{2}}\mathbf{j}-\mathbf{k} \\ & \left\| \mathbf{v}\left( t \right) \right\|=\text{speed}=\sqrt{17+9{{t}^{4}}} \\ & \mathbf{a}\left( t \right)=6t\mathbf{j} \\ & \left( \text{b} \right) \\ & \mathbf{v}\left( 1 \right)=4\mathbf{i}+3\mathbf{j}-\mathbf{k} \\ & \mathbf{a}\left( 1 \right)=6\mathbf{j} \\ \end{align}\]

\[\begin{align} & \mathbf{r}\left( t \right)=4t\mathbf{i}+{{t}^{3}}\mathbf{j}-t\mathbf{k},\text{ }t=1 \\ & \left( \mathbf{a} \right)\text{Find the vectors: }\mathbf{v}\left( t \right),\text{ }\mathbf{a}\left( t \right)\text{ and speed}\text{.} \\ & \mathbf{v}\left( t \right)=\mathbf{r}'\left( t \right) \\ & \mathbf{v}\left( t \right)=\frac{d}{dt}\left[ 4t\mathbf{i}+{{t}^{3}}\mathbf{j}-t\mathbf{k} \right] \\ & \mathbf{v}\left( t \right)=4\mathbf{i}+3{{t}^{2}}\mathbf{j}-\mathbf{k} \\ & \text{speed}=\left\| \mathbf{v}\left( t \right) \right\|=\left\| 4\mathbf{i}+3{{t}^{2}}\mathbf{j}-\mathbf{k} \right\| \\ & \text{speed}=\sqrt{{{\left( 4 \right)}^{2}}+{{\left( 3{{t}^{2}} \right)}^{2}}+{{\left( -1 \right)}^{2}}} \\ & \text{speed}=\sqrt{17+9{{t}^{4}}} \\ & \mathbf{a}\left( t \right)=\mathbf{v}'\left( t \right) \\ & \mathbf{a}\left( t \right)=\frac{d}{dt}\left[ 4\mathbf{i}+3{{t}^{2}}\mathbf{j}-\mathbf{k} \right] \\ & \mathbf{a}\left( t \right)=6t\mathbf{j} \\ & \\ & \left( \mathbf{b} \right)\text{Evaluating }\mathbf{v}\left( t \right)\text{ and }\mathbf{a}\left( t \right)\text{ at }t=1 \\ & \mathbf{v}\left( 1 \right)=4\mathbf{i}+3{{\left( 1 \right)}^{2}}\mathbf{j}-\mathbf{k} \\ & \mathbf{v}\left( 1 \right)=4\mathbf{i}+3\mathbf{j}-\mathbf{k} \\ & \mathbf{a}\left( 1 \right)=6\left( 1 \right)\mathbf{j} \\ & \mathbf{a}\left( 1 \right)=6\mathbf{j} \\ & \\ & \text{Summary} \\ & \left( \text{a} \right) \\ & \mathbf{v}\left( t \right)=4\mathbf{i}+3{{t}^{2}}\mathbf{j}-\mathbf{k} \\ & \left\| \mathbf{v}\left( t \right) \right\|=\text{speed}=\sqrt{17+9{{t}^{4}}} \\ & \mathbf{a}\left( t \right)=6t\mathbf{j} \\ & \left( \text{b} \right) \\ & \mathbf{v}\left( 1 \right)=4\mathbf{i}+3\mathbf{j}-\mathbf{k} \\ & \mathbf{a}\left( 1 \right)=6\mathbf{j} \\ \end{align}\]