Showing a parametric equation is smoothly parametrized?
A parametric equation, say r(t), is smoothly parametrized if:
1. its derivative is continuous, and
2. its derivative does not equal zero for all t in the domain of r.
Now that sounds simple enough. Now lets say we have the tractrix:
r(t) = (t-tanht)i + sechtj,
then r'(t) = [ 1-1/(1+t^2) ]i + [ tantsect ]j, right?
FIRSTLY, do I have the derivative correct? --and
SECONDLY, without reverting to MatLab or Maple to view the graph, how do we deduce that r'(t) is continuous (or not)?
Do I just state that is is/isn't -by inspection, or ...?