I only have a basic knowledge of calculus but I would like to know if it's possible to, given a set of points each with their own slopes, construct the simplest (or any) polynomial function that perfectly fits the given information.
In other words, a polynomial $P(x)$ would need to take on the value of each given ordinate at each given abscissa, and it's derivative, $P'(x)$ would need to take on the value of each given slope and each given abscissa.
I'm thinking that it might not be possible because if there's a series of points that lie on a straight line among many other, randomly distributed ones, the derivative $P'(x)$ would have a segment of constant value in the middle of the chaotic curvature surrounding it. I don't think that polynomial functions can have this kind of behavior, can they?
If it's indeed impossible to do this, is there any other way to construct a curve that fits the given data?
Best Answer
I think what you want is Hermite interpolation.
Look here:
https://en.wikipedia.org/wiki/Hermite_interpolation