Answer
$ a.\quad$ see image
$ b.\quad \mathrm{r}^{\prime}(t) =\langle 2t,3t^{2}\rangle$
$ c.\quad$ see image
Work Step by Step
$a.$
Parametric equations: $\mathrm{r}(t):\quad\left\{\begin{array}{llll}
x=t^{2} & \Rightarrow & t=x^{1/2} & *\\
y=t^{3} & & &
\end{array}\right.$
Substitute $(*)$ into the equation for y:
$y=(x^{1/2})^{3}$
$y=x^{3/2}$
see image for graph
$b.$
Differentiate $(\displaystyle \frac{d}{dt})$ each component function
$\mathrm{r}^{\prime}(t):\quad\left\{\begin{array}{l}
x=2t\\
y=3t^{2}
\end{array}\right.$
$\mathrm{r}^{\prime}(t) =\langle 2t,3t^{2}\rangle$
$c. $
position vector : black$\quad \mathrm{r}(1)$=$\langle 1,1\rangle$
tangent vector: red$\quad \mathrm{r}^{\prime}(1)$=$\langle 2,3\rangle$
