We consider the trapezoidal rule on $I = [a,b]$: $$T_n(f) = \frac{b-a}{n} \sum_{i=0}^{n-1} \frac{f(a_i) + f(a_{i+1})}{2},$$ where $a_i = a + ih$ and $h = \frac{b-a}{n}$, with associated error $e_n(f) = \int_a^b f(x)\,\mathrm{d}x - T_n(f)$. Suppose that $f$ is a function of class $\mathcal{C}^2$ from $[a,b]$ to $\mathbb{R}$. Show that $$e_n(f) = \frac{b-a}{n} \sum_{i=0}^{n-1} e(g_i)$$ where $e$ is the error associated with the quadrature formula $I_1$ studied in question 11 and the $g_i : [0,1] \rightarrow \mathbb{R}$ are functions to be specified.
We consider the trapezoidal rule on $I = [a,b]$:
$$T_n(f) = \frac{b-a}{n} \sum_{i=0}^{n-1} \frac{f(a_i) + f(a_{i+1})}{2},$$
where $a_i = a + ih$ and $h = \frac{b-a}{n}$, with associated error $e_n(f) = \int_a^b f(x)\,\mathrm{d}x - T_n(f)$.
Suppose that $f$ is a function of class $\mathcal{C}^2$ from $[a,b]$ to $\mathbb{R}$. Show that
$$e_n(f) = \frac{b-a}{n} \sum_{i=0}^{n-1} e(g_i)$$
where $e$ is the error associated with the quadrature formula $I_1$ studied in question 11 and the $g_i : [0,1] \rightarrow \mathbb{R}$ are functions to be specified.