Chapter 12 - Vector-Valued Functions - 12.6 Motion Along A Curve - Exercises Set 12.6 - Page 891: 5

$$\eqalign{ & {\bf{v}}\left( t \right) = {\bf{i}} + t{\bf{j}} + {t^2}{\bf{k}} \cr & {\bf{v}}\left( 1 \right) = {\bf{i}} + {\bf{j}} + {\bf{k}} \cr & \left\| {{\bf{v}}\left( 1 \right)} \right\| = \sqrt 3 \cr & {\bf{a}}\left( 1 \right) = {\bf{j}} + 2{\bf{k}} \cr & {\bf{a}}\left( t \right) = {\bf{j}} + 2t{\bf{k}} \cr} $$

$$\eqalign{ & {\bf{r}}\left( t \right) = t{\bf{i}} + \frac{1}{2}{t^2}{\bf{j}} + \frac{1}{3}{t^3}{\bf{k}};\,\,\,\,t = 1 \cr & {\text{Calculate the velocity at }}t = 1 \cr & {\bf{v}}\left( t \right) = \frac{{d{\bf{r}}}}{{dt}} \cr & {\bf{v}}\left( t \right) = \frac{d}{{dt}}\left[ {t{\bf{i}} + \frac{1}{2}{t^2}{\bf{j}} + \frac{1}{3}{t^3}{\bf{k}}} \right] \cr & {\bf{v}}\left( t \right) = {\bf{i}} + t{\bf{j}} + {t^2}{\bf{k}} \cr & {\bf{v}}\left( 1 \right) = {\bf{i}} + \left( 1 \right){\bf{j}} + {\left( 1 \right)^2}{\bf{k}} \cr & {\bf{v}}\left( 1 \right) = {\bf{i}} + {\bf{j}} + {\bf{k}} \cr & \cr & {\text{Calculate the speed at }}t = 1 \cr & \left\| {{\bf{v}}\left( t \right)} \right\| = \sqrt {{{\left( {\frac{{dx}}{{dt}}} \right)}^2} + {{\left( {\frac{{dy}}{{dt}}} \right)}^2} + {{\left( {\frac{{dz}}{{dt}}} \right)}^2}} \cr & \left\| {{\bf{v}}\left( t \right)} \right\| = \sqrt {{{\left( t \right)}^2} + {{\left( {{t^2}} \right)}^2} + {{\left( {{t^3}} \right)}^2}} \cr & \left\| {{\bf{v}}\left( t \right)} \right\| = \sqrt {{t^2} + {t^4} + {t^6}} \cr & \left\| {{\bf{v}}\left( 1 \right)} \right\| = \sqrt 3 \cr & \cr & {\text{Calculate the acceleration at }}t = 1 \cr & {\bf{a}}\left( t \right) = \frac{{d{\bf{v}}}}{{dt}} \cr & {\bf{a}}\left( t \right) = \frac{d}{{dt}}\left[ {{\bf{i}} + t{\bf{j}} + {t^2}{\bf{k}}} \right] \cr & {\bf{a}}\left( t \right) = {\bf{j}} + 2t{\bf{k}} \cr & {\bf{a}}\left( 1 \right) = {\bf{j}} + 2\left( 1 \right){\bf{k}} \cr & {\bf{a}}\left( 1 \right) = {\bf{j}} + 2{\bf{k}} \cr} $$