Answer
$$v(1)=\sqrt{74}.$$
Work Step by Step
Since $r(t)=\lt t^3, 1-t, 4t^2\gt$, then the velocity vector is given by
$$r'(t)=\lt 3 t^2, -1, 8t\gt$$
and the acceleration vector is given by
$$r''(t)=\lt 6 t, 0, 8\gt.$$
Moreover, the speed at $t=1$ is
$$v(1)=\|r'(1)\|=\sqrt{9+1+64}=\sqrt{74}.$$
