MATH307 December 2010
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Question 03 (a)
Suppose we are given 4 points (x1, y1), (x2, y2), (x3, y3) and (x4, y4) in the plane and we want to find a function ƒ(x), defined for , whose graph interpolates these points. Assume that
where each pi(x) is a polynomial.
(a) What equations, written in terms of pi(x) and possibly their derivatives, express the condition that ƒ(x) goes through the given points? Do the equations imply that ƒ(x) is continuous?
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The condition that goes through all the points means that . Rewrite this in terms of .
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Since the piece-wise function above has inclusive restrictions, we can see that both polynomials, and , have the same value . The same goes for polynomials and .
Thus, we need to satisfy
Hence these equations imply that is continuous as they show that both polynomials surrounding the given point equal each other at .
As well, to show that the function goes through the given points, we need to write the following six equations:
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