Chapter 12 - Vector-Valued Functions - Review Exercises - Page 864: 36

$$\eqalign{ & \left( {\text{a}} \right) \cr & {\bf{v}}\left( t \right) = \frac{1}{{2\sqrt t }}{\bf{i}} + 5{\bf{j}} + 4t{\bf{k}} \cr & {\text{speed}} = \sqrt {\frac{1}{{4t}} + 25 + 16{t^2}} \cr & {\bf{a}}\left( t \right) = - \frac{1}{{4t\sqrt t }}{\bf{i}} + 4{\bf{k}} \cr & \left( {\text{b}} \right) \cr & {\bf{v}}\left( 4 \right) = \frac{1}{4}{\bf{i}} + 5{\bf{j}} + 16{\bf{k}} \cr & {\bf{a}}\left( 4 \right) = - \frac{1}{{32}}{\bf{i}} + 4{\bf{k}} \cr} $$

$$\eqalign{ & {\bf{r}}\left( t \right) = \sqrt t {\bf{i}} + 5t{\bf{j}} + 2{t^2}{\bf{k}},{\text{ }}t = 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[ {\sqrt t {\bf{i}} + 5t{\bf{j}} + 2{t^2}{\bf{k}}} \right] \cr & {\bf{v}}\left( t \right) = \frac{1}{{2\sqrt t }}{\bf{i}} + 5{\bf{j}} + 4t{\bf{k}} \cr & {\text{speed}} = \left\| {{\bf{v}}\left( t \right)} \right\| = \left\| {\frac{1}{{2\sqrt t }}{\bf{i}} + 5{\bf{j}} + 4t{\bf{k}}} \right\| \cr & {\text{speed}} = \sqrt {{{\left( {\frac{1}{{2\sqrt t }}} \right)}^2} + {{\left( 5 \right)}^2} + {{\left( {4t} \right)}^2}} \cr & {\text{speed}} = \sqrt {\frac{1}{{4t}} + 25 + 16{t^2}} \cr & {\bf{a}}\left( t \right) = {\bf{v}}'\left( t \right) \cr & {\bf{a}}\left( t \right) = \frac{d}{{dt}}\left[ {\frac{1}{{2\sqrt t }}{\bf{i}} + 5{\bf{j}} + 4t{\bf{k}}} \right] \cr & {\bf{a}}\left( t \right) = - \frac{1}{{4t\sqrt t }}{\bf{i}} + 4{\bf{k}} \cr & \cr & \left( {\bf{b}} \right){\text{Evaluating }}{\bf{v}}\left( t \right){\text{ and }}{\bf{a}}\left( t \right){\text{ at }}t = 4 \cr & {\bf{v}}\left( 4 \right) = \frac{1}{{2\sqrt 4 }}{\bf{i}} + 5{\bf{j}} + 4\left( 4 \right){\bf{k}} \cr & {\bf{v}}\left( 4 \right) = \frac{1}{4}{\bf{i}} + 5{\bf{j}} + 16{\bf{k}} \cr & {\bf{a}}\left( 4 \right) = - \frac{1}{{4\left( 4 \right)\sqrt 4 }}{\bf{i}} + 4{\bf{k}} \cr & {\bf{a}}\left( 4 \right) = - \frac{1}{{32}}{\bf{i}} + 4{\bf{k}} \cr & \cr & {\text{Summary}} \cr & \left( {\text{a}} \right) \cr & {\bf{v}}\left( t \right) = \frac{1}{{2\sqrt t }}{\bf{i}} + 5{\bf{j}} + 4t{\bf{k}} \cr & {\text{speed}} = \sqrt {\frac{1}{{4t}} + 25 + 16{t^2}} \cr & {\bf{a}}\left( t \right) = - \frac{1}{{4t\sqrt t }}{\bf{i}} + 4{\bf{k}} \cr & \left( {\text{b}} \right) \cr & {\bf{v}}\left( 4 \right) = \frac{1}{4}{\bf{i}} + 5{\bf{j}} + 16{\bf{k}} \cr & {\bf{a}}\left( 4 \right) = - \frac{1}{{32}}{\bf{i}} + 4{\bf{k}} \cr} $$