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Differentiation of Polynomials : Proofs

Please see the attached file for the fully formatted problems.

Let g be a function which can be differentiated four times on the interval [-1,1].
Denote .

1) Show that when g is a polynomial of degree less than or equal to 3.
2) Let P be the interpolation polynomial of f at the points -1, , , 1.
a) Show that .
b) Show that , where
and is a constant which you will evaluate.
c) Deduce a number which is greater than or equal to the error .

3) Let f be a function which can be differentiated four times on an interval [a,b].
Let . Using x, show that the integral f can be calculated on [a,b] with the help of an integral on [-1,1].
4) Deduce an approximation of .
5) Using this method, calculate an approximation of .

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Solution Summary

Problems relating to the diferentiation of polynomials are solved.

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