Introduction
I'm given this quadrature formula:
$$\int_a^b f(x)dx = h \sum_{i=0}^{n-1}\left[ f(x_i) + \alpha h f'(x_i) + \beta h^2 f''(x_i) \right] = Q_n(a,b;f)$$
where
$$h = \frac{b-a}{n}, \ x_i = a + ih, \ i=0,1,\dots,n$$
My goal is to find $\alpha$ and $\beta$ so that the quadrature formula has the highest degree of exactness.
My question
I noticed that $Q_n(a,b;f)$ can be interpreted as a composite quadrature formula. Since a composite and its simple quadrature formulas have the same degree of exactness, my idea is to work with the simple formula and impose the exactness conditions on it.
However, how can I know what simple formula this composite formula was generated from?
Can I say that $n=1$ and the simple formula is:
$$\int_a^b f(x)dx = h\left[ f(a) + \alpha h f'(a) + \beta h^2 f''(a) \right]$$
?