Undetermined coefficients - deriving formula

[muso07]

Homework Statement

Derive a formula of the form $$\int\stackrel{b}{a}f(x)dx \approx c_{0}f(a)+c_{1}f(b)+c_{2}f'(a)+c_{3}f'(b)$$ that is exact for polynomials of the highest degree possible.

Apply a change of variable: y=x-a

Homework Equations

The Attempt at a Solution

I don't get the "highest degree possible" thing. Like, if it's exact for quadratics, I know it has to be exact for f(x)=1,x,x².

But applying the change of variable, I got $$\int\stackrel{b-a}{0}f(y)dy \approx c_{0}f(0)+c_{1}f(b-a)+c_{2}f'(0)+c_{3}f'(b-a)$$

Any help much appreciated. :)

 

[muso07]

Whoops, that's meant to be integral from a to b for the first one and 0 to b-a for the second one.

 

[muso07]

Can anyone even point me in the right direction?