## Thomas' Calculus 13th Edition

$${\bf{T}} = - \cos t{\bf{i}} + \sin t{\bf{j}},\,\,\,\,\,\,{\bf{N}} = \sin t{\bf{i}} + \cos t{\bf{j}},\,\,\,\,\,\,\,\kappa = \frac{1}{{3\sin t\cos t}}$$
\eqalign{ & {\bf{r}}\left( t \right) = \left( {{{\cos }^3}t} \right){\bf{i}} + \left( {{{\sin }^3}t} \right){\bf{j}},\,\,\,\,\,\,\,\,0 < t < \pi /2 \cr & {\text{Calculate }}{\bf{v}}\left( t \right).{\text{ Use }}{\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) \cr & {\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {\left( {{{\cos }^3}t} \right){\bf{i}} + \left( {{{\sin }^3}t} \right){\bf{j}}} \right] \cr & {\bf{v}}\left( t \right) = 3{\cos ^2}t\left( { - \sin t} \right){\bf{i}} + 3{\sin ^2}t\left( {\cos t} \right){\bf{j}} \cr & {\bf{v}}\left( t \right) = - 3{\cos ^2}t\sin t{\bf{i}} + 3{\sin ^2}t\cos t{\bf{j}} \cr & \cr & {\text{We calculate }}{\bf{T}}{\text{ from the velocity vector}} \cr & {\bf{v}}\left( t \right) = - 3{\cos ^2}t\sin t{\bf{i}} + 3{\sin ^2}t\cos t{\bf{j}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{{\left( { - 3{{\cos }^2}t\sin t} \right)}^2} + {{\left( {3{{\sin }^2}t\cos t} \right)}^2}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {9{{\cos }^4}t{{\sin }^2}t + 9{{\sin }^4}t{{\cos }^2}t} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {9{{\sin }^2}t{{\cos }^2}t\left( {{{\cos }^2}t + {{\sin }^2}t} \right)} \cr & \left| {{\bf{v}}\left( t \right)} \right| = 3\sin t\cos t \cr & \cr & {\text{use }}{\bf{T}}\left( t \right) = \frac{{{\bf{v}}\left( t \right)}}{{\left| {{\bf{v}}\left( t \right)} \right|}} \cr & {\bf{T}} = \frac{{ - 3{{\cos }^2}t\sin t{\bf{i}} + 3{{\sin }^2}t\cos t{\bf{j}}}}{{3\sin t\cos t}} \cr & {\bf{T}} = \frac{{ - 3{{\cos }^2}t\sin t{\bf{i}}}}{{3\sin t\cos t}} + \frac{{3{{\sin }^2}t\cos t{\bf{j}}}}{{3\sin t\cos t}} \cr & {\bf{T}} = - \cos t{\bf{i}} + \sin t{\bf{j}} \cr & \cr & {\text{Calculate }}{\bf{N}}\left( t \right){\text{ using the equation }}{\bf{N}} = \frac{{d{\bf{T}}/dt}}{{\left| {d{\bf{T}}/dt} \right|}}{\text{ }}\left( {{\text{see page 764}}} \right).{\text{ Then}} \cr & \frac{{d{\bf{T}}}}{{dt}} = \frac{d}{{dt}}\left( { - \cos t{\bf{i}} + \sin t{\bf{j}}} \right) \cr & \frac{{d{\bf{T}}}}{{dt}} = \sin t{\bf{i}} + \cos t{\bf{j}} \cr & \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {{{\left( {\sin t} \right)}^2} + {{\left( {\cos t} \right)}^2}} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = 1 \cr & \cr & {\bf{N}} = \frac{{d{\bf{T}}/dt}}{{\left| {d{\bf{T}}/dt} \right|}} = \frac{{\sin t{\bf{i}} + \cos t{\bf{j}}}}{1} \cr & {\bf{N}} = \sin t{\bf{i}} + \cos t{\bf{j}} \cr & \cr & {\text{Calculate }}\kappa {\text{ using the equation }}\kappa = \frac{1}{{\left| {\bf{v}} \right|}}\left| {\frac{{d{\bf{T}}}}{{dt}}} \right|{\text{ }}\left( {{\text{see page 764}}} \right).{\text{ Then}} \cr & \kappa = \frac{1}{{\left| {\bf{v}} \right|}}\left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \frac{1}{{3\sin t\cos t}}\left( 1 \right) \cr & \kappa = \frac{1}{{3\sin t\cos t}} \cr & \cr & {\bf{T}} = - \cos t{\bf{i}} + \sin t{\bf{j}},\,\,\,\,\,\,{\bf{N}} = \sin t{\bf{i}} + \cos t{\bf{j}},\,\,\,\,\,\,\,\kappa = \frac{1}{{3\sin t\cos t}} \cr}