Which of the following curve is smooth and why?

Question

1. r(t) =
2. r(t) =
3. r(t) =
4. r(t) =
5. r(t) =

Answer 1

1,2 are smooth, 3,4,5 are not. For smoothness, the tangent vector cannot be the zero vector.

Answer 2

1 - smooth
Because if you take the derivative of this position function, you get r'(t) = <3t^2, 1, 9t^8>

It is never 0 for any t, therefore we can never get a location where the particle stops in space, which could give rise to a sharp corner in the graph. Since the particle is always moving, the path must be smoothly curved.

In all the other cases, look for where the derivative is completely the 0 vector. At that spot, you may have a place where the curve is not smooth.