Chapter 12 - Vector-Valued Functions - Review Exercises - Page 864: 38

\[\begin{align} & \left( \text{a} \right) \\ & \mathbf{v}\left( t \right)=\left\langle 1,-{{\sec }^{2}}t,{{e}^{t}} \right\rangle \\ & \text{speed}=\sqrt{1+{{\sec }^{4}}t+{{e}^{2t}}} \\ & \mathbf{a}\left( t \right)=\left\langle 0,-2{{\sec }^{2}}t\tan t,{{e}^{t}} \right\rangle \\ & \left( \text{b} \right) \\ & \mathbf{v}\left( 0 \right)=\left\langle 1,-1,1 \right\rangle \\ & \mathbf{a}\left( 0 \right)=\left\langle 0,0,1 \right\rangle \\ \end{align}\]

\[\begin{align} & \mathbf{r}\left( t \right)=\left\langle t,-\tan t,{{e}^{t}} \right\rangle ,\text{ }t=0 \\ & \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[ \left\langle t,-\tan t,{{e}^{t}} \right\rangle \right] \\ & \mathbf{v}\left( t \right)=\left\langle 1,-{{\sec }^{2}}t,{{e}^{t}} \right\rangle \\ & \text{speed}=\left\| \mathbf{v}\left( t \right) \right\|=\left\| \left\langle 1,-{{\sec }^{2}}t,{{e}^{t}} \right\rangle \right\| \\ & \text{speed}=\sqrt{{{\left( 1 \right)}^{2}}+{{\left( -{{\sec }^{2}}t \right)}^{2}}+{{\left( {{e}^{t}} \right)}^{2}}} \\ & \text{speed}=\sqrt{1+{{\sec }^{4}}t+{{e}^{2t}}} \\ & \\ & \mathbf{a}\left( t \right)=\mathbf{v}'\left( t \right) \\ & \mathbf{a}\left( t \right)=\frac{d}{dt}\left[ \left\langle 1,-{{\sec }^{2}}t,{{e}^{t}} \right\rangle \right] \\ & \mathbf{a}\left( t \right)=\left\langle 0,-2\sec t\sec t\tan t,{{e}^{t}} \right\rangle \\ & \mathbf{a}\left( t \right)=\left\langle 0,-2{{\sec }^{2}}t\tan t,{{e}^{t}} \right\rangle \\ & \\ & \left( \mathbf{b} \right)\text{Evaluating }\mathbf{v}\left( t \right)\text{ and }\mathbf{a}\left( t \right)\text{ at }t=0 \\ & \mathbf{v}\left( 0 \right)=\left\langle 1,-{{\sec }^{2}}0,{{e}^{0}} \right\rangle \\ & \mathbf{v}\left( 0 \right)=\left\langle 1,-1,1 \right\rangle \\ & \mathbf{a}\left( 0 \right)=\left\langle 0,-2{{\sec }^{2}}0\tan 0,{{e}^{0}} \right\rangle \\ & \mathbf{a}\left( 0 \right)=\left\langle 0,0,1 \right\rangle \\ & \\ & \text{Summary} \\ & \left( \text{a} \right) \\ & \mathbf{v}\left( t \right)=\left\langle 1,-{{\sec }^{2}}t,{{e}^{t}} \right\rangle \\ & \text{speed}=\sqrt{1+{{\sec }^{4}}t+{{e}^{2t}}} \\ & \mathbf{a}\left( t \right)=\left\langle 0,-2{{\sec }^{2}}t\tan t,{{e}^{t}} \right\rangle \\ & \left( \text{b} \right) \\ & \mathbf{v}\left( 0 \right)=\left\langle 1,-1,1 \right\rangle \\ & \mathbf{a}\left( 0 \right)=\left\langle 0,0,1 \right\rangle \\ \end{align}\]