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  .

Attached file(s):

19Nov2002 - ex1.doc  View File