We consider the case $I = [0,1]$ and $\forall x \in I, w(x) = 1$. Determine the Lagrange basis associated with the points $(0, 1/2, 1)$ and thus recover the coefficients of the quadrature formula $I_2(f)$ from question 3.

We consider the case of an arbitrary segment $I = [a,b]$ (with $a < b$), subdivided into $n+1$ equidistant points $a_0, \ldots, a_n$: $$\forall i \in \llbracket 0, n \rrbracket, \quad a_i = a + ih,$$ where $h = \frac{b-a}{n}$ is the step of the subdivision. The trapezoidal rule is $$T_n(f) = \frac{b-a}{n} \sum_{i=0}^{n-1} \frac{f(a_i) + f(a_{i+1})}{2}.$$
Represent graphically $T_n(f)$.

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 $I = [a,b]$ with associated error $e_n(f) = \int_a^b f(x)\,\mathrm{d}x - T_n(f)$, where $f$ is of class $\mathcal{C}^2$.
Deduce the error bound $$\left|e_n(f)\right| \leqslant \frac{(b-a)^3}{12n^2} \sup_{x \in [a,b]} |f''(x)|.$$

We consider a function $f : [a,b] \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$ and the trapezoidal method $$T_n(f) = h \sum_{i=0}^{n-1} \frac{f(a_i) + f(a_{i+1})}{2},$$ where $h = \frac{b-a}{n}$ and $\forall i \in \llbracket 0, n-1 \rrbracket, a_i = a + ih$.
Using the result of question 38, show that, for all integer $m \geqslant 1$, $$\int_a^b f(x)\,\mathrm{d}x = T_n(f) - \sum_{p=1}^m \frac{\gamma_{2p}}{n^{2p}} + \rho_{2m}(n)$$ where the coefficients $\gamma_{2p}$ are given by $$\gamma_{2p} = \frac{(b-a)^{2p} b_{2p}}{(2p)!}\left(f^{(2p-1)}(b) - f^{(2p-1)}(a)\right)$$ and $\rho_{2m}(n)$ is a remainder integral satisfying the bound $$|\rho_{2m}(n)| \leqslant \frac{C_{2m}}{n^{2m}}$$ where $C_{2m}$ is a constant to be determined depending only on $m$, $a$ and $b$.

$\lim_{n \rightarrow \infty} \frac{3}{n}\left\{4 + \left(2 + \frac{1}{n}\right)^{2} + \left(2 + \frac{2}{n}\right)^{2} + \ldots + \left(3 - \frac{1}{n}\right)^{2}\right\}$ is equal to
(1) 12
(2) $\frac{19}{3}$
(3) 0
(4) 19

$$\begin{aligned} & f : [ 1,3 ] \rightarrow [ 2,10 ] \\ & f ( x ) = 1 + x ^ { 2 } \end{aligned}$$
The interval $[ 1,3 ]$ is divided into two subintervals of equal length, and the right endpoints of these subintervals are marked as $x _ { 1 }$ and $x _ { 2 }$. Then, two rectangles are drawn with each subinterval as the base and heights $f \left( x _ { 1 } \right), f \left( x _ { 2 } \right)$ respectively.
If the sum of the areas of these rectangles is A and the area of the region between the function f and the x-axis is B, what is the difference A - B in square units?
A) $\frac { 11 } { 2 }$
B) $\frac { 13 } { 3 }$
C) $\frac { 15 } { 4 }$
D) $\frac { 19 } { 6 }$
E) $\frac { 23 } { 6 }$