Chapter 11 - Vectors and Vector-Valued Functions - Review Exercises - Page 856: 45

$$\eqalign{ & \left. a \right)\left\langle {0,3,0} \right\rangle ,\,{\text{The limit does not exist}},\,\,\, \cr & \left. b \right){\bf{r}}'\left( 0 \right) = \left\langle {2\cos 2t, - 12\sin 4t,1} \right\rangle ,\,\,\,\left\langle {2,0,1} \right\rangle \cr & \left. c \right)\left\langle { - 4\sin 2t, - 48\cos 4t,0} \right\rangle \cr & \left. d \right)\left\langle { - \frac{1}{2}\cos 2t,\frac{3}{4}\sin 4t,\frac{{{t^2}}}{2}} \right\rangle + {\bf{C}} \cr} $$

$$\eqalign{ & {\text{Let }}{\bf{r}}\left( t \right) = \left\langle {\sin 2t,3\cos 4t,t} \right\rangle \cr & \left. a \right){\text{ Find }}\mathop {\lim }\limits_{t \to 0} {\bf{r}}\left( t \right){\text{ and }}\mathop {\lim }\limits_{t \to \infty } {\bf{r}}\left( t \right) \cr & \mathop {\lim }\limits_{t \to 0} {\bf{r}}\left( t \right) = \mathop {\lim }\limits_{t \to 0} \left\langle {\sin 2t,3\cos 4t,t} \right\rangle \cr & \mathop {\lim }\limits_{t \to 0} {\bf{r}}\left( t \right) = \left\langle {\sin 2\left( 0 \right),3\cos 4\left( 0 \right),0} \right\rangle \cr & \mathop {\lim }\limits_{t \to 0} {\bf{r}}\left( t \right) = \left\langle {0,3,0} \right\rangle \cr & and \cr & \mathop {\lim }\limits_{t \to \infty } {\bf{r}}\left( t \right) = \mathop {\lim }\limits_{t \to \infty } \left\langle {\sin 2t,3\cos 4t,t} \right\rangle \cr & \mathop {\lim }\limits_{t \to \infty } {\bf{r}}\left( t \right) = \mathop {\lim }\limits_{t \to \infty } \left\langle {\sin 2\left( \infty \right),3\cos 4\left( \infty \right),\infty } \right\rangle \cr & {\text{The limit does not exist}} \cr & \cr & \left. b \right){\text{ Find }}{\bf{r}}'\left( t \right){\text{ and evaluate }}{\bf{r}}'\left( 0 \right) \cr & {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left\langle {\sin 2t,3\cos 4t,t} \right\rangle \cr & {\bf{r}}'\left( t \right) = \left\langle {2\cos 2t, - 12\sin 4t,1} \right\rangle \cr & and \cr & {\bf{r}}'\left( 0 \right) = \left\langle {2\cos 2\left( 0 \right), - 12\sin 4\left( 0 \right),1} \right\rangle \cr & {\bf{r}}'\left( 0 \right) = \left\langle {2,0,1} \right\rangle \cr & \cr & \left. c \right){\text{ Find }}{\bf{r}}''\left( t \right) \cr & {\bf{r}}''\left( t \right) = \frac{d}{{dt}}\left\langle {2\cos 2t, - 12\sin 4t,1} \right\rangle \cr & {\bf{r}}''\left( t \right) = \left\langle { - 4\sin 2t, - 48\cos 4t,0} \right\rangle \cr & \cr & \left. d \right){\text{ Evaluate }}\int {{\bf{r}}\left( t \right)dt} \cr & \int {{\bf{r}}\left( t \right)dt} = \int {\left\langle {\sin 2t,3\cos 4t,t} \right\rangle dt} \cr & \int {{\bf{r}}\left( t \right)dt} = \left\langle { - \frac{1}{2}\cos 2t,\frac{3}{4}\sin 4t,\frac{{{t^2}}}{2}} \right\rangle + {\bf{C}} \cr} $$