Chapter 12 - Vector-Valued Functions - 12.6 Motion Along A Curve - Exercises Set 12.6 - Page 891: 7

$$\eqalign{ & {\bf{v}}\left( t \right) = - 2\sin t{\bf{i}} + 2\cos t{\bf{j}} + {\bf{k}} \cr & {\bf{v}}\left( {\frac{\pi }{4}} \right) = - \sqrt 2 {\bf{i}} + \sqrt 2 {\bf{j}} + {\bf{k}} \cr & \left\| {{\bf{v}}\left( {\frac{\pi }{4}} \right)} \right\| = \sqrt 5 \cr & {\bf{a}}\left( t \right) = - 2\cos t{\bf{i}} - 2\sin t{\bf{j}} \cr & {\bf{a}}\left( {\frac{\pi }{4}} \right) = - \sqrt 2 {\bf{i}} - \sqrt 2 {\bf{j}} \cr} $$

$$\eqalign{ & x = 2\cos t,\,\,\,y = 2\sin t,\,\,\,z = t,\,\,\,\,t = \pi /4 \cr & {\text{Calculate the velocity at }}t = \frac{\pi }{4} \cr & {\bf{v}}\left( t \right) = \frac{{dx}}{{dt}}{\bf{i}} + \frac{{dy}}{{dt}}{\bf{j}} + \frac{{dz}}{{dt}}{\bf{k}} \cr & {\bf{v}}\left( t \right) = \frac{d}{{dt}}\left[ {2\cos t} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {2\sin t} \right]{\bf{j}} + \frac{{dz}}{{dt}}\left[ t \right]{\bf{k}} \cr & {\bf{v}}\left( t \right) = - 2\sin t{\bf{i}} + 2\cos t{\bf{j}} + {\bf{k}} \cr & {\bf{v}}\left( {\frac{\pi }{4}} \right) = - 2\sin \left( {\frac{\pi }{4}} \right){\bf{i}} + 2\cos \left( {\frac{\pi }{4}} \right){\bf{j}} + {\bf{k}} \cr & {\bf{v}}\left( {\frac{\pi }{4}} \right) = - \sqrt 2 {\bf{i}} + \sqrt 2 {\bf{j}} + {\bf{k}} \cr & \cr & {\text{Calculate the speed at }}t = 2 \cr & \left\| {{\bf{v}}\left( t \right)} \right\| = \sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2} + {{\left( {\frac{{dz}}{{dt}}} \right)}^2}} \cr & \left\| {{\bf{v}}\left( t \right)} \right\| = \sqrt {{{\left( { - 2\sin t} \right)}^2} + {{\left( {2\cos t} \right)}^2} + {{\left( 1 \right)}^2}} \cr & \left\| {{\bf{v}}\left( t \right)} \right\| = \sqrt {4{{\sin }^2}t + 4{{\cos }^2}t + 1} \cr & \left\| {{\bf{v}}\left( t \right)} \right\| = \sqrt 5 \cr & \left\| {{\bf{v}}\left( {\frac{\pi }{4}} \right)} \right\| = \sqrt 5 \cr & \cr & {\text{Calculate the acceleration at }}t = 2 \cr & {\bf{a}}\left( t \right) = \frac{{d{\bf{v}}}}{{dt}} \cr & {\bf{a}}\left( t \right) = \frac{d}{{dt}}\left[ { - 2\sin t{\bf{i}} + 2\cos t{\bf{j}} + {\bf{k}}} \right] \cr & {\bf{a}}\left( t \right) = - 2\cos t{\bf{i}} - 2\sin t{\bf{j}} \cr & {\bf{a}}\left( {\frac{\pi }{4}} \right) = - 2\cos \left( {\frac{\pi }{4}} \right){\bf{i}} - 2\sin \left( {\frac{\pi }{4}} \right){\bf{j}} \cr & {\bf{a}}\left( {\frac{\pi }{4}} \right) = - \sqrt 2 {\bf{i}} - \sqrt 2 {\bf{j}} \cr} $$