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Differentiation of Polynomials : Proofs

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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 differentiation of polynomials are solved. The interpolation polynomials are analyzed.

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

1) Show that when g is a polynomial of degree less than or equal to 3.
Proof. We consider four cases.

Case 1. g is a polynomial of degree 0, that is, , C is a constant.
In this case,
Case 2. g is a polynomial of degree 1, that is, , C and D are constants.
In this case, Case 3. g is a polynomial of degree 2, that is, , C , D and E ...

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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