Hello everyone,

I have a question regarding my Spline interpolations. I am not an expert on the theory there, but the maple help tells me that the first derivative of an 3rd degree spline interpolation should exist at the knots. But a derivation returns "undefined" at some of the knots instead. Here is my example:

x(t):=Spline([[0, 0], [1, 1], [2, 2], [3, 2.2], [4, 1.8]],t,degree=3);

returns

x(t):=piecewise(t < 1, .953571428571429*t+0.464285714285714e-1*t^3, t < 2, -0.9286e-1+1.09285714285714*t+.139285714285714*(t-1)^2-.232142857142857*(t-1)^3, t < 3, .65000+.675000000000000*t-.557142857142857*(t-2)^2+0.821428571428571e-1*(t-2)^3, 2.77857-.192857142857143*t-.310714285714286*(t-3)^2+.103571428571429*(t-3)^3);

diff(x(t),t);

returns

piecewise(t < 1., .953571+.139286*t^2, t = 1., Float(undefined), t < 2., .814286+.278571*t-.696429*(t-1.)^2, t = 2., Float(undefined), t <= 3., 2.90357-1.11429*t+.246429*(t-2.)^2, 3. < t, 1.67143-.621429*t+.310714*(t-3.)^2);

Not defined at t=1 and t=2. Is it possible to get an interpolation of which the first derivative exists at every point? Thank you very much!