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 .

... 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). ...

... 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 ...

... f(x) has a multiple root if and only if f(x) is not relatively prime to its derivative (which can ...Polynomials, fields and derivatives are investigated. ...Proof: ...

...Proof. ... 5. What is the Taylor polynomial approximation to a function? Explain how to use tangent line approximation of the derivative of a function to get a ...

...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 ...

... where Pn(x) is the Legendre polynomial of degree n ... equation proof and a Legendre's differential equation proof. ... 1 − J p xx Taking the derivative of equation 1 ...

... 2 Then the fourth Taylor polynomial for f (t) = et /2 is. ... Here's the outline of the proof. ... Since by hypothesis G has a derivative in the second variable, we see ...