1. a) Consider the problem of cubic polynomial interpolation

p(xi) = yi, I = 0,1,23

with deg(p) ≤ 3 and x0, x1, x2, x3 distinct. Convert the problem of finding p(x) to another problem involving the solution of a system of linear equations.

b) Express the system from (a) in the form Ax = b, identify the matrix A and the vectors b and x.

(See attached file for full problem description with equations)
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I need some guidance on a Matlab programming project. The project concerns Hermite polynomialinterpolation. We are given values and and their derivatives and at and respectively, and we look for coefficients such that if

(i) Use the tabulated data and Newton's method of divided differences to (a) calculate the degree three interpolating polynomial based on x_0, x_1, x_2, x_3 and (b) calculate the degree four interpolating polynomial based on x_0, x_1, x_2, x_3, x_4.
x_k || 0 | -1 | 1 | 3 | 2
y_k || 5 | 15 | 3 | 47 | 9
(Use exact arithmeti

Hi,
Your assistance with the following interpolation problem would be appreciated. I do have an existing Maple program that might be of assistance. Please let me know if you would like it.
Approximate the following function using the set of nodes
{1.0, 1.25, 1.75, 2.0, 2.5, 3.0}
Use the Newton divided - difference inter

1.Expand the following function into Maclaurin Series (see attached file) using properties of the power series.
2. The Lagrange interpolationpolynomial may be compactly written as is a shape function. Sketch the shape function in a graphic form.
3. Write a forward and backward difference Newton's interpolation formulas b

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 interpolationpolynomial of f at the points -1, , , 1.
a) Show that .
b

a) Given a polynomial P(x) and a point xo, what 2 things does Horner's method give us? How is the result useful for polynomial root-finding P(x)=0?
b) One use of polynomials is interpolation of given data points {(xk, fk)} , k=0,....,n
1. Write down the Lagrange building block Ln,k (x)- that is the nth degree polynomial wh

The problem looks at a general cubicpolynomial, and calculates the conditions needed for exactly two stationary points to exist. It also finds the inflexion point.
Consider the cubicpolynomial (degree 3) given by, y=ax^3+bx^2+cx+d, where a is not equal to 0.
(a) Find the condition on the constants a,b,c so that this f

Determine if possible whether
f(x)=3x^3-2x^2-7x+5 has a zero between a=1 and b=2. (e answer will be yes or cannot say
Determine if possible whether f(x)=x^3+3x^2-9x-13 has a zero between a=1 and b=2. (The answer will be either "yes", or "cannot".
Classify the polynomial as linear, quadratic, cubic, or quartic. Determine t

7. This problem generalizes the factorial function, as in n!=n(n-1)(n-2)...(2)(1), to more general arguments than just the positive integers.
(a) Use integration by parts to show that for any positive integer n, the integral with respect to x from 0 to infinity of xne-x is n!
(b) Make a clear case that the integral exists