Improving Integration Accuracy with Simpson's Rule

[fluidistic]

The question is : Determine the degree of precision of the formula for \int_{-1}^{1} f(x)dx~\frac{4}{3}f(-0.5)-\frac{2}{3}f(0)+\frac{4}{3}f(0.5).
My guess is that I must answer like "the degree of precision is that this formula is exact for polynomials of grade \leqslant 2", for example.
My attempt are just thoughts... Can't start. Watching the coefficients in the right side of the "approximation", it is similar to the Simpson's rule.
Please help me telling me how can I start. Thanks!

 

[Dick]

It is exact for degree of polynomial<=2. To check this put f(x)=ax^2+bx+c. I'll give you a hint. It's also exact for cubics. Can you show this the same way? Is it exact for quartics?

 

[fluidistic]

Thanks for your help

Thank you! Before reading your message I tried a quadratic one and it worked, then a cubic one but didn't worked (now I found my calculus error!) and then I gave up because I had the sentiment I wasn't proving anything. I didn't had the idea to put the f(x) as a general form like a_0x^3+a_1x^2+a_2x+a_3. Now it worked till cubic ones, so the degree of precision of the formula is 3.
So to solve the problem, we have first to get a vague idea of the answer and then try testing the polynomials. There is no way to do it in one shot... (Maybe unless to see the coefficient in front of the variable of the higher degree and to see that integrated they are not equal to them in the formula given... hard).

 

[Dick]

Thinking about this, a somewhat simpler approach is just to check 1, x, x^2 and x^3 separately. Since the integral and the formula are both linear, it will then work for linear combinations of those.