Write down the linear of system of equations

[medinaj2160]

Homework Statement

write down the linear system of equations that must be satisfied to find the Newton-Cotes quadratic rule for the interval [0,1] that is exact for polynomials of degree 3 and uses the sample points x1=0, x2=1/3, x3=2/3, and x4= 1

Homework Equations

none

The Attempt at a Solution

I know hot to do find the equations for the cubic polynomial in the standard bases given a equation, but I don't how to do it this way. Is not homework is a sample test, for the test that I have on monday.

Thanks

 

[nate808]

for posting this question on the forum! To find the linear system of equations for the Newton-Cotes quadratic rule, we can use the following steps:

1. Start by writing out the general form of the quadratic rule for the interval [0,1]:

∫0^1 f(x)dx = A1*f(0) + A2*f(1/3) + A3*f(2/3) + A4*f(1)

2. Since we want this rule to be exact for polynomials of degree 3, we can substitute in the basis polynomials for f(x) (1, x, x^2, x^3):

∫0^1 A1 + A2*x + A3*x^2 + A4*x^3 dx = A1*1 + A2*1/3 + A3*2/3 + A4*1

3. Integrate this equation from 0 to 1 to get:

A1 + A2/6 + A3/9 + A4/12 = A1 + A2/3 + 2*A3/3 + A4

4. Since this rule should be exact for polynomials of degree 3, we can set the coefficients of each basis polynomial equal to 0:

A1: 1 = 1

A2: 1/6 = 1/3

A3: 1/9 = 2/3

A4: 1/12 = 1

5. Solving this system of equations, we get the following values for A1, A2, A3, and A4:

A1 = 1, A2 = 2/3, A3 = -1/6, A4 = 1/4

6. Substituting these values back into the original equation, we get the final linear system of equations for the Newton-Cotes quadratic rule:

∫0^1 f(x)dx = 1*f(0) + (2/3)*f(1/3) + (-1/6)*f(2/3) + (1/4)*f(1)

I hope this helps and good luck on your test! Remember to always check your work and make sure your answers make sense in the context of the problem.