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Consider the functions f(t) = t^(7/3) and g(t) = t^(5/3). Let each function
represent the position at time t of a cart moving along a straight track.
Both these functions vanish at the origin, have first derivatives which
vanish at the origin, but have second derivatives which behave much
differently at the origin -- f" vanishes, while g" goes belly up.
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There is also something that bothers me about the f-cart. Even though f" is
zero at t = 0, f''' becomes infinite there. So the force on the f-cart is
zero at t = 0, but it is changing infinitely fast. What would that feel
like?
Paul Goodman