Question: "Determine constants a, b, c, d, e that will produce a quadrature formula

$\int_{-1}^{1}f(x)dx=af(-1)+bf(0)+cf(1)+df'(-1)+ef'(1)$

that has degree of precision 4"

Some background information: degree of precision of a quadrature formula is the largest positive integer n such that the formula is exact for x^k for each k = 0,1, ... , n.

I'm not sure about this problem... I think d and e are 0 because a quadrature formula is given by constants times certain values f(x) at different x's. ie. the quadrature formula is:

$\int_{a}^{b} f(x) dx \approx \sum_{i=0}^{n} a_{i} f(x_{i})$

Any suggestions?

Originally Posted by limddavid
Question: "Determine constants a, b, c, d, e that will produce a quadrature formula

$\int_{-1}^{1}f(x)dx=af(-1)+bf(0)+cf(1)+df'(-1)+ef'(1)$

that has degree of precision 4"

Some background information: degree of precision of a quadrature formula is the largest positive integer n such that the formula is exact for x^k for each k = 0,1, ... , n.

I'm not sure about this problem... I think d and e are 0 because a quadrature formula is given by constants times certain values f(x) at different x's. ie. the quadrature formula is:

$\int_{a}^{b} f(x) dx \approx \sum_{i=0}^{n} a_{i} f(x_{i})$

Any suggestions?
You have a set of linear equations:

$\int_{-1}^{1}dx=a+b+c$

$\int_{-1}^{1}x dx=a(-1)+b(0)+c(1)+d+e$

$\int_{-1}^{1}x^2dx=a(-1)^2+b(0)^2+c(1)^2+2d(-1)+2e(1)$

etc

which you can then solve for $a,b,c,d,e$

CB