Answer
1. The curve ${{\bf{r}}_1}\left( t \right) = \left( {t,{t^2}} \right)$.
The tangent vector at $t=1$ is ${{\bf{r}}_1}'\left( 1 \right) = \left( {1,2} \right)$. The point corresponding to $t=1$ is ${{\bf{r}}_1}\left( 1 \right) = \left( {1,1} \right)$.
2. The curve ${{\bf{r}}_2}\left( t \right) = \left( {{t^3},{t^6}} \right)$.
The tangent vector at $t=1$ is ${{\bf{r}}_2}'\left( 1 \right) = \left( {3,6} \right)$. The point corresponding to $t=1$ is ${{\bf{r}}_2}\left( 1 \right) = \left( {1,1} \right)$.
Please see the figure attached. The red arrow represents the tangent vector.
Work Step by Step
1. The curve ${{\bf{r}}_1}\left( t \right) = \left( {t,{t^2}} \right)$.
We get ${{\bf{r}}_1}'\left( t \right) = \left( {1,2t} \right)$. Thus, the tangent vector at $t=1$ is ${{\bf{r}}_1}'\left( 1 \right) = \left( {1,2} \right)$. The point corresponding to $t=1$ is ${{\bf{r}}_1}\left( 1 \right) = \left( {1,1} \right)$.
2. The curve ${{\bf{r}}_2}\left( t \right) = \left( {{t^3},{t^6}} \right)$.
We get ${{\bf{r}}_2}'\left( t \right) = \left( {3{t^2},6{t^5}} \right)$. Thus, the tangent vector at $t=1$ is ${{\bf{r}}_2}'\left( 1 \right) = \left( {3,6} \right)$. The point corresponding to $t=1$ is ${{\bf{r}}_2}\left( 1 \right) = \left( {1,1} \right)$.
We evaluate several points for the interval $ - 3 \le t \le 3$ and list them in the following table. Then we plot the points and join them to obtain the curves.
$\begin{array}{*{20}{c}}
t&{{{\bf{r}}_1}\left( t \right) = \left( {x,y} \right)}\\
{ - 3}&{\left( { - 3,9} \right)}\\
{ - 2}&{\left( { - 2,4} \right)}\\
{ - 1}&{\left( { - 1,1} \right)}\\
0&{\left( {0,0} \right)}\\
1&{\left( {1,1} \right)}\\
2&{\left( {2,4} \right)}\\
3&{\left( {3,9} \right)}
\end{array}\begin{array}{*{20}{c}}
{}&{}
\end{array} \begin{array}{*{20}{c}}
t&{{{\bf{r}}_2}\left( t \right) = \left( {x,y} \right)}\\
{ - 3}&{\left( { - 27,729} \right)}\\
{ - 2}&{\left( { - 8,64} \right)}\\
{ - 1}&{\left( { - 1,1} \right)}\\
0&{\left( {0,0} \right)}\\
1&{\left( {1,1} \right)}\\
2&{\left( {8,64} \right)}\\
3&{\left( {27,729} \right)}
\end{array}$
