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Haydo
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SOLVED
1. Homework Statement
Find polynomial of least degree satisfying:
p(1)=-1, p'(1)=2, p''(1)=0, p(2)=1, p'(2)=-2
In general, a Hermite Polynomial is defined by the following:
∑[f(xi)*hi(x)+f'(xi)*h2i(x)]
where:
hi(xj)=1 if i=j and 0 otherwise. Similarly with h'2. h'i(x)=0 and h2(x)=0. i.e., they are zero if they are integrated or derived.
Here is a page from wolfram with general information: http://mathworld.wolfram.com/HermitesInterpolatingPolynomial.html
First, I recognized x0=1 and x1=2. I tried to create some third coefficient term in order to satisfy p''(1)=0, but that seems to mean that I would have to make hi(xj) be zero for the second derivative, and I have no idea how to do this. I tried just setting this new h (call it h3) equal to (x)(x-2) when derived twice and integrating (so that h3''=1 for x1 and zero otherwise) but that was a total flop.
I'm starting to think that, despite the question's section (it was in the section regarding Hermite Interpolating Polynomials), there is a better way to approach it. Any help would be greatly appreciated.
1. Homework Statement
Find polynomial of least degree satisfying:
p(1)=-1, p'(1)=2, p''(1)=0, p(2)=1, p'(2)=-2
Homework Equations
In general, a Hermite Polynomial is defined by the following:
∑[f(xi)*hi(x)+f'(xi)*h2i(x)]
where:
hi(xj)=1 if i=j and 0 otherwise. Similarly with h'2. h'i(x)=0 and h2(x)=0. i.e., they are zero if they are integrated or derived.
Here is a page from wolfram with general information: http://mathworld.wolfram.com/HermitesInterpolatingPolynomial.html
The Attempt at a Solution
First, I recognized x0=1 and x1=2. I tried to create some third coefficient term in order to satisfy p''(1)=0, but that seems to mean that I would have to make hi(xj) be zero for the second derivative, and I have no idea how to do this. I tried just setting this new h (call it h3) equal to (x)(x-2) when derived twice and integrating (so that h3''=1 for x1 and zero otherwise) but that was a total flop.
I'm starting to think that, despite the question's section (it was in the section regarding Hermite Interpolating Polynomials), there is a better way to approach it. Any help would be greatly appreciated.
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