Chapter 12 - Vector-Valued Functions - 12.6 Motion Along A Curve - Exercises Set 12.6 - Page 891: 8

$$\eqalign{ & {\bf{v}}\left( t \right) = \left( {{e^t}\cos t + {e^t}\sin t} \right){\bf{i}} + \left( {{e^t}\cos t - {e^t}\sin t} \right){\bf{j}} + {\bf{k}} \cr & {\bf{a}}\left( t \right) = 2{e^t}\cos t{\bf{i}} - 2{e^t}\sin t{\bf{j}} \cr & {\bf{v}}\left( {\frac{\pi }{2}} \right) = {e^{\pi /2}}{\bf{i}} - {e^{\pi /2}}{\bf{j}} + {\bf{k}} \cr & \left\| {{\bf{v}}\left( {\frac{\pi }{2}} \right)} \right\| = \sqrt {2{e^\pi } + 1} \cr & {\bf{a}}\left( {\frac{\pi }{2}} \right) = - 2{e^{\pi /2}}{\bf{j}} \cr} $$

$$\eqalign{ & {\bf{r}}\left( t \right) = {e^t}\sin t{\bf{i}} + {e^t}\cos t{\bf{j}} + t{\bf{k}};\,\,\,\,t = \pi /2 \cr & {\text{Calculate the velocity at }}t = \pi /2 \cr & {\bf{v}}\left( t \right) = \frac{{d{\bf{r}}}}{{dt}} \cr & {\bf{v}}\left( t \right) = \frac{d}{{dt}}\left[ {{e^t}\sin t{\bf{i}} + {e^t}\cos t{\bf{j}} + t{\bf{k}}} \right] \cr & {\bf{v}}\left( t \right) = \left( {{e^t}\cos t + {e^t}\sin t} \right){\bf{i}} + \left( {{e^t}\cos t - {e^t}\sin t} \right){\bf{j}} + {\bf{k}} \cr & {\bf{v}}\left( {\frac{\pi }{2}} \right) = \left( {{e^{\pi /2}}\cos \frac{\pi }{2} + {e^{\pi /2}}\sin \frac{\pi }{2}} \right){\bf{i}} + \left( {{e^{\pi /2}}\cos \frac{\pi }{2} - {e^{\pi /2}}\sin \frac{\pi }{2}} \right){\bf{j}} + {\bf{k}} \cr & {\bf{v}}\left( {\frac{\pi }{2}} \right) = {e^{\pi /2}}{\bf{i}} - {e^{\pi /2}}{\bf{j}} + {\bf{k}} \cr & \cr & {\text{Calculate the speed at }}t = 1 \cr & \left\| {{\bf{v}}\left( t \right)} \right\| = \sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2} + {{\left( {\frac{{dz}}{{dt}}} \right)}^2}} \cr & \left\| {{\bf{v}}\left( t \right)} \right\| = \sqrt {{{\left( {{e^t}\cos t + {e^t}\sin t} \right)}^2} + {{\left( {{e^t}\cos t - {e^t}\sin t} \right)}^2} + {{\left( 1 \right)}^2}} \cr & \left\| {{\bf{v}}\left( t \right)} \right\| = \sqrt {{e^{2t}} + 2{e^{2t}}\sin t + {e^{2t}} - 2{e^{2t}}\sin t + 1} \cr & \left\| {{\bf{v}}\left( t \right)} \right\| = \sqrt {2{e^{2t}} + 1} \cr & \left\| {{\bf{v}}\left( {\frac{\pi }{2}} \right)} \right\| = \sqrt {2{e^{2\left( {\pi /2} \right)}} + 1} \cr & \left\| {{\bf{v}}\left( {\frac{\pi }{2}} \right)} \right\| = \sqrt {2{e^\pi } + 1} \cr & \cr & {\text{Calculate the acceleration at }}t = 1 \cr & {\bf{a}}\left( t \right) = \frac{{d{\bf{v}}}}{{dt}} \cr & {\bf{a}}\left( t \right) = \frac{d}{{dt}}\left[ {\left( {{e^t}\cos t + {e^t}\sin t} \right){\bf{i}} + \left( {{e^t}\cos t - {e^t}\sin t} \right){\bf{j}} + {\bf{k}}} \right] \cr & {\bf{a}}\left( t \right) = \left( {{e^t}\cos t - {e^t}\sin t + {e^t}\cos t + {e^t}\sin t} \right){\bf{i}} \cr & \,\,\,\,\,\,\,\,\,\,\, + \left( {{e^t}\cos t - {e^t}\sin t - {e^t}\cos t - {e^t}\sin t} \right){\bf{j}} + 0 \cr & {\bf{a}}\left( t \right) = 2{e^t}\cos t{\bf{i}} - 2{e^t}\sin t{\bf{j}} \cr & {\bf{a}}\left( {\frac{\pi }{2}} \right) = 2{e^{\pi /2}}\cos \frac{\pi }{2}{\bf{i}} - 2{e^{\pi /2}}\sin \frac{\pi }{2}{\bf{j}} \cr & {\bf{a}}\left( {\frac{\pi }{2}} \right) = - 2{e^{\pi /2}}{\bf{j}} \cr} $$