Solution

Find the velocity, acceleration, speed, and direction of motion of $\mathbf{r}(t) = e^{t}\sin t\hat{i} + (t^3 - 2t)\hat{j} + \arctan(2t)\hat{k}$ at $t = 0$. Interpret the results by describing the particle's motion at that instant, including how fast and in what direction it is moving, whether it is speeding up or slowing down, and how its path is curving.