## Calculus (3rd Edition)

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}