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 .

Please see the attached file for the complete solution.
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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 Summary

Problems relating to the differentiation of polynomials are solved. The interpolation polynomials are analyzed.

... An analysis of the nature of the roots of a given polynomial is presented using the properties of the derivative. An elliptic curve proof and real roots are ...

... Let p(x) be a polynomial in 1/x, ie p(x ... The n-th derivative of g(x) for x not zero is of ... zero (which is again easy) Combining these steps will proof the result. ...

... It helps provides proof for a number, identifies elements of a ... more precise let p , q , and r be polynomials. ... p ' = q ' , where p ' denotes the derivative of p ...

... constants and the H_n are the Hermite polynomials. ... positive to negative x, the first derivative can only ... continuous (I state this here without rigorous proof). ...

...Proof: See Appendix A ... of E/P with respect to MVE is more negative than the first derivative of Ei ... The roots (1 + u,) are the solution to the polynomial equation. ...

... Since in the estimation we only need the derivative, we specify directly the ... shape of f (ljt ) is remarkably similar for all higher-order polynomials. ...Proof. ...

... data, one that uses integration by parts in a proof. ... How would a typical polynomial of this type look? ... the partial fraction the degree of the derivative of the ...

... if and only if f(x) is not relatively prime to its derivative (which can ...Polynomials, fields and derivatives are investigated in the following posting. ...Proof: ...

... For example, Prove that for x>0, Proof. ... Definition of a Limit and Derivative, Product Rule ... Tangent Line Approximation, Taylor Polynomial Approximation, Newton's ...

...Proof. ... But most irreducible polynomials of degree 5 actually have Galois group S5 ; an example is ... The derivative f (x) = 5x4 + 3 is larger than 3 for all x, so ...