Answer
\begin{align}
r(t)&= \left\langle 1, t, 8 t^2\right\rangle .
\end{align}
Work Step by Step
By integration, we have
\begin{align}
r'(t)&= \left\langle c_1, c_2, 16 t +c_3 \right\rangle .
\end{align}
By the condition $r'(0)=\lt 0,1,0\gt$, we get
$$0=c_1, \quad 1=c_2, \quad 0=c_3$$
Hence we have
\begin{align}
r'(t)&= \left\langle 0, 1, 16 t \right\rangle .
\end{align}
Again, by integration we have
\begin{align}
r(t)&= \left\langle a, t+b, 8 t^2+c \right\rangle .
\end{align}
By the condition $r(0)=\lt1,0,0\gt$, we get
$$1=a, \quad 0=c_2, \quad 0=c$$
Hence we have
\begin{align}
r(t)&= \left\langle 1, t, 8 t^2\right\rangle .
\end{align}
