Chapter 12 - Vector-Valued Functions - 12.3 Exercises - Page 838: 5

$${\bf{v}}\left( {\frac{\pi }{4}} \right) = - \sqrt 2 {\bf{i}} + \sqrt 2 {\bf{j}}$$$${\bf{a}}\left( {\frac{\pi }{4}} \right) = - \sqrt 2 {\bf{i}} - \sqrt 2 {\bf{j}}$$

$$\eqalign{ & {\bf{r}}\left( t \right) = 2\cos t{\bf{i}} + 2\sin t{\bf{j}},{\text{ }}\left( {\sqrt 2 ,\sqrt 2 } \right) \cr & {\text{Let }}t = \frac{\pi }{4} \cr & {\bf{r}}\left( {\frac{\pi }{4}} \right) = 2\cos \left( {\frac{\pi }{4}} \right){\bf{i}} + 2\sin \left( {\frac{\pi }{4}} \right){\bf{j}} \cr & {\bf{r}}\left( {\frac{\pi }{4}} \right) = \sqrt 2 {\bf{i}} + \sqrt 2 {\bf{j}} \cr & {\text{Then, at }}\left( {\sqrt 2 ,\sqrt 2 } \right){\text{ }}t = \frac{\pi }{4} \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[ {2\cos t{\bf{i}} + 2\sin t{\bf{j}}} \right] \cr & {\bf{v}}\left( t \right) = - 2\sin t{\bf{i}} + 2\cos t{\bf{j}} \cr & {\text{speed}} = \left\| {{\bf{v}}\left( t \right)} \right\| = \left\| { - 2\sin t{\bf{i}} + 2\cos t{\bf{j}}} \right\| \cr & {\text{speed}} = \sqrt {{{\left( { - 2\sin t} \right)}^2} + {{\left( {2\cos t} \right)}^2}} = \sqrt {4{{\sin }^2}t + 4{{\cos }^2}t} \cr & {\text{speed}} = 2 \cr & {\bf{a}}\left( t \right) = {\bf{v}}'\left( t \right) \cr & {\bf{a}}\left( t \right) = \frac{d}{{dt}}\left[ { - 2\sin t{\bf{i}} + 2\cos t{\bf{j}}} \right] \cr & {\bf{a}}\left( t \right) = - 2\cos t{\bf{i}} - 2\sin t{\bf{j}} \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( {\sqrt 2 ,\sqrt 2 } \right){\text{ }}t = \frac{\pi }{4} \cr & {\bf{v}}\left( {\frac{\pi }{4}} \right) = - 2\sin \left( {\frac{\pi }{4}} \right){\bf{i}} + 2\cos \left( {\frac{\pi }{4}} \right){\bf{j}} \cr & {\bf{v}}\left( {\frac{\pi }{4}} \right) = - \sqrt 2 {\bf{i}} + \sqrt 2 {\bf{j}} \cr & {\text{speed}} = 2 \cr & {\bf{a}}\left( {\frac{\pi }{4}} \right) = - 2\cos \left( {\frac{\pi }{4}} \right){\bf{i}} - 2\sin \left( {\frac{\pi }{4}} \right){\bf{j}} \cr & {\bf{a}}\left( {\frac{\pi }{4}} \right) = - \sqrt 2 {\bf{i}} - \sqrt 2 {\bf{j}} \cr & \cr & \left( {\bf{c}} \right){\text{ Sketching}} \cr & {\bf{r}}\left( t \right) = 2\cos t{\bf{i}} + 2\sin t{\bf{j}} \cr & x = 2\cos t,{\text{ }}y = 2\sin t \cr & {x^2} + {y^2} = {\left( {2\cos t} \right)^2} + {\left( {2\sin t} \right)^2} = 4 \cr & {x^2} + {y^2} = 4,{\text{ circle}} \cr & {\text{Graph}} \cr} $$