Find a Piecewise Smooth Parametrization of the Path C Given

Multiple Choice

Find a piecewise smooth parametrization of the path C given in the following graph.

A) $\mathbf { r } ( t ) = \left\{ \begin{array} { l l } t \mathbf { i } + t \mathbf { j } & 0 \leq t \leq 2 \\ \left( 4 + t ^ { 2 } \right) \mathbf { i } - 4 \mathbf { j } & 2 \leq t \leq 4 \\ \left( 8 - t ^ { 2 } \right) \mathbf { j } & 4 \leq t \leq 10 \end{array} \right.$

B) $\mathbf { r } ( t ) = \left\{ \begin{array} { l l } t \mathbf { i } - t ^ { 2 } \mathbf { j } & 0 \leq t \leq 2 \\ ( 4 + t ) \mathbf { i } + 4 \mathbf { j } & 2 \leq t \leq 4 \\ ( 8 + t ) \mathbf { j } & 4 \leq t \leq 8 \end{array} \right.$

C)
$\mathbf { r } ( t ) = \left\{ \begin{array} { l l } t \mathbf { i } + t ^ { 2 } \mathbf { j } & 0 \leq t \leq 2 \\( 4 + t ) \mathbf { i } - 4 \mathbf { j } & 2 \leq t \leq 4 \\\left( 8 - t ^ { 2 } \right) \mathbf { j } & 4 \leq t \leq 8\end{array} \right.$

D)
$\mathbf { r } ( t ) = \left\{ \begin{array} { l l } t ^ { 2 } + t ^ { 2 } \mathbf { j } & 0 \leq t \leq 2 \\( 4 - t ) \mathbf { i } + 4 \mathbf { j } & 2 \leq t \leq 4 \\( 8 - t ) \mathbf { j } & 4 \leq t \leq 8\end{array} \right.$
E)
$\mathbf { r } ( t ) = \left\{ \begin{array} { l l } t \mathbf { i } + t \mathbf { j } & 0 \leq t \leq 2 \\\left( 4 - t ^ { 2 } \right) \mathbf { i } + 4 \mathbf { j } & 2 \leq t \leq 4 \\\left( 8 - t ^ { 2 } \right) \mathbf { j } & 4 \leq t \leq 10\end{array} \right.$