Chapter 12 - Vector-Valued Functions - 12.3 Exercises - Page 838: 7

$${\bf{v}}\left( \pi \right) = \left\langle {2,0} \right\rangle $$$${\bf{a}}\left( \pi \right) = \left\langle {0, - 1} \right\rangle $$

$$\eqalign{ & {\bf{r}}\left( t \right) = \left\langle {t - \sin t,1 - \cos t} \right\rangle ,{\text{ }}\left( {\pi ,2} \right) \cr & {\text{Let }}t = \pi \cr & {\bf{r}}\left( \pi \right) = \left\langle {\pi - \sin \pi ,1 - \cos \pi } \right\rangle \cr & {\bf{r}}\left( \pi \right) = \left\langle {\pi ,2} \right\rangle \cr & {\text{Then, at }}\left( {\pi ,2} \right){\text{ }}t = \pi \cr & \left( {\bf{a}} \right){\text{Find the vectors: }}{\bf{v}}\left( t \right),{\text{ }}{\bf{a}}\left( t \right){\text{ and speed}}{\text{.}} \cr & {\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) \cr & {\bf{v}}\left( t \right) = \frac{d}{{dt}}\left[ {\left\langle {t - \sin t,1 - \cos t} \right\rangle } \right] \cr & {\bf{v}}\left( t \right) = \left\langle {1 - \cos t,\sin t} \right\rangle \cr & {\text{speed}} = \left\| {{\bf{v}}\left( t \right)} \right\| = \left\| {\left\langle {1 - \cos t,\sin t} \right\rangle } \right\| \cr & {\text{speed}} = \sqrt {{{\left( {1 - \cos t} \right)}^2} + {{\left( {\sin t} \right)}^2}} = \sqrt {1 - 2\cos t + {{\cos }^2}t + {{\sin }^2}t} \cr & {\text{speed}} = \sqrt {1 - 2\cos t + {{\cos }^2}t + {{\sin }^2}t} \cr & {\text{speed}} = \sqrt {2 - 2\cos t} \cr & {\bf{a}}\left( t \right) = {\bf{v}}'\left( t \right) \cr & {\bf{a}}\left( t \right) = \frac{d}{{dt}}\left[ {\left\langle {1 - \cos t,\sin t} \right\rangle } \right] \cr & {\bf{a}}\left( t \right) = \left\langle {\sin t,\cos t} \right\rangle \cr & \cr & \left( {\bf{b}} \right){\text{Evaluating }}{\bf{v}}\left( t \right),{\text{ }}{\bf{a}}\left( t \right){\text{ and speed at the given point}}{\text{.}} \cr & {\text{At }}\left( {\pi ,2} \right){\text{ }}t = \pi \cr & {\bf{v}}\left( \pi \right) = \left\langle {1 - \cos \pi ,\sin \pi } \right\rangle \cr & {\bf{v}}\left( \pi \right) = \left\langle {2,0} \right\rangle \cr & {\text{speed}} = \sqrt {2 - 2\cos \left( \pi \right)} \cr & {\text{speed}} = 2 \cr & {\bf{a}}\left( \pi \right) = \left\langle {\sin \pi ,\cos \pi } \right\rangle \cr & {\bf{a}}\left( \pi \right) = \left\langle {0, - 1} \right\rangle \cr & \cr & \left( {\bf{c}} \right){\text{ Sketching}} \cr & {\bf{r}}\left( t \right) = \left\langle {t - \sin t,1 - \cos t} \right\rangle \cr & x = t - \sin t,{\text{ }}y = 1 - \cos t \cr & {\text{Graph}} \cr} $$