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November 12th, 2008, 07:32 PM   #1
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finding smooth parameterization

Background:

Let A(t) = (x(t),y(t)) describe a curve in the plane (parameterized by time t).
Call a parameterization smooth if functions x(t) and y(t) are smooth and A'(t) (velocity) is not 0.

The curve y = |x| for -1 <= x <= 1 is not smooth at x-0 but has a parameterization A(t) = (x(t),y(t)) with x(t),y(t) smooth everywhere
but A'(t) is 0 at the corner.

Question:

Define A: R-->R... by A(t) = e^(-1/x) for x > 0, and 0 elsewhere.

Use A to find smooth functions x(t),y(t) that parameterize y = |x| for -1 <= x <= 1

I'm not sure what to do but it seems like I should define y = -x for -1<= x < 0 and y = x for 0 <= x <= 1

Thanks in advance.
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