How to model a growing root?

I am trying to create a "growing root" animation. Just one pipe, no branching necessary. To do that I was planning to use a circle extruded along the path (partial extrude) and animate one of the ends.

It all works fine except for one "fine detail" - I cannot manage to get the tube to be thinner on the "growing" end.

I remember long ago seeing a tutorial explaining how to make a profile for an extruded surface that would allow one to control the changing thickness of the tube - but I am no longer sure if the tutorial was even dealing with maya.

Could anyone suggest a way to solve this newbie problem?