Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.4 - Curvature and Normal Vectors of a Curve - Exercises 13.4 - Page 765: 13

$${\bf{T}} = \frac{t}{{\sqrt {{t^2} + 1} }}{\bf{i}} + \frac{1}{{\sqrt {{t^2} + 1} }}{\bf{j}},\,\,\,\,\,\,{\bf{N}} = \frac{1}{{\sqrt {{t^2} + 1} }}{\bf{i}} - \frac{t}{{\sqrt {{t^2} + 1} }}{\bf{j}},\,\,\,\,\,\,\,\kappa = \frac{1}{{t{{\left( {{t^2} + 1} \right)}^{3/2}}}}$$

$$\eqalign{ & {\bf{r}}\left( t \right) = \left( {{t^3}/3} \right){\bf{i}} + \left( {{t^2}/2} \right){\bf{j}},\,\,\,\,\,\,\,\,t > 0 \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( {{t^3}/3} \right){\bf{i}} + \left( {{t^2}/2} \right){\bf{j}}} \right] \cr & {\bf{v}}\left( t \right) = \frac{{3{t^2}}}{3}{\bf{i}} + \frac{{2t}}{2}{\bf{j}} \cr & {\bf{v}}\left( t \right) = {t^2}{\bf{i}} + t{\bf{j}} \cr & \cr & {\text{We calculate }}{\bf{T}}{\text{ from the velocity vector}} \cr & {\bf{v}}\left( t \right) = {t^2}{\bf{i}} + t{\bf{j}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{{\left( {{t^2}} \right)}^2} + {{\left( t \right)}^2}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{t^4} + {t^2}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{t^2}\left( {{t^2} + 1} \right)} \cr & \left| {{\bf{v}}\left( t \right)} \right| = t\sqrt {{t^2} + 1} \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{{{t^2}{\bf{i}} + t{\bf{j}}}}{{t\sqrt {{t^2} + 1} }} \cr & {\bf{T}} = \frac{{{t^2}}}{{t\sqrt {{t^2} + 1} }}{\bf{i}} + \frac{t}{{t\sqrt {{t^2} + 1} }}{\bf{j}} \cr & {\bf{T}} = \frac{t}{{\sqrt {{t^2} + 1} }}{\bf{i}} + \frac{1}{{\sqrt {{t^2} + 1} }}{\bf{j}} \cr & {\bf{T}} = t{\left( {{t^2} + 1} \right)^{ - 1/2}}{\bf{i}} + {\left( {{t^2} + 1} \right)^{ - 1/2}}{\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( {t{{\left( {{t^2} + 1} \right)}^{ - 1/2}}{\bf{i}} + {{\left( {{t^2} + 1} \right)}^{ - 1/2}}{\bf{j}}} \right) \cr & \frac{{d{\bf{T}}}}{{dt}} = \left[ {t\left( { - \frac{1}{2}} \right){{\left( {{t^2} + 1} \right)}^{ - 3/2}}\left( {2t} \right) + {{\left( {{t^2} + 1} \right)}^{ - 1/2}}} \right]{\bf{i}} + \left( { - \frac{1}{2}{{\left( {{t^2} + 1} \right)}^{ - 3/2}}\left( {2t} \right)} \right){\bf{j}} \cr & \frac{{d{\bf{T}}}}{{dt}} = \left[ { - {t^2}{{\left( {{t^2} + 1} \right)}^{ - 3/2}} + {{\left( {{t^2} + 1} \right)}^{ - 1/2}}} \right]{\bf{i}} - t{\left( {{t^2} + 1} \right)^{ - 3/2}}{\bf{j}} \cr & \frac{{d{\bf{T}}}}{{dt}} = \left[ {{{\left( {{t^2} + 1} \right)}^{ - 3/2}}\left( { - {t^2} + {t^2} + 1} \right)} \right]{\bf{i}} - t{\left( {{t^2} + 1} \right)^{ - 3/2}}{\bf{j}} \cr & \frac{{d{\bf{T}}}}{{dt}} = {\left( {{t^2} + 1} \right)^{ - 3/2}}{\bf{i}} - t{\left( {{t^2} + 1} \right)^{ - 3/2}}{\bf{j}} \cr & \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {{{\left( {{{\left( {{t^2} + 1} \right)}^{ - 3/2}}} \right)}^2} + {{\left( { - t{{\left( {{t^2} + 1} \right)}^{ - 3/2}}} \right)}^2}} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {{{\left( {{t^2} + 1} \right)}^{ - 3}} + {t^2}{{\left( {{t^2} + 1} \right)}^{ - 3}}} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {{{\left( {{t^2} + 1} \right)}^{ - 3}}\left( {1 + {t^2}} \right)} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \frac{1}{{{t^2} + 1}} \cr & \cr & {\bf{N}} = \frac{{d{\bf{T}}/dt}}{{\left| {d{\bf{T}}/dt} \right|}} = \frac{{{{\left( {{t^2} + 1} \right)}^{ - 3/2}}{\bf{i}} - t{{\left( {{t^2} + 1} \right)}^{ - 3/2}}{\bf{j}}}}{{1/\left( {{t^2} + 1} \right)}} \cr & {\bf{N}} = {\left( {{t^2} + 1} \right)^{ - 1/2}}{\bf{i}} - t{\left( {{t^2} + 1} \right)^{ - 1/2}}{\bf{j}} \cr & {\bf{N}} = \frac{1}{{\sqrt {{t^2} + 1} }}{\bf{i}} - \frac{t}{{\sqrt {{t^2} + 1} }}{\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}{{t\sqrt {{t^2} + 1} }}\left( {\frac{1}{{{t^2} + 1}}} \right) \cr & \kappa = \frac{1}{{t{{\left( {{t^2} + 1} \right)}^{3/2}}}} \cr} $$