The graph of the velocity $v(t)$, in $\mathrm{ft/sec}$, of a car traveling on a straight road, for $0 \leq t \leq 50$, is shown above. A table of values for $v(t)$, at 5 second intervals of time $t$, is shown to the right of the graph.

$t$ (seconds)	$v(t)$ (feet per second)
0	0
5	12
10	20
15	30
20	55
25	70
30	78
35	81
40	75
45	60
50	72

(a) During what intervals of time is the acceleration of the car positive? Give a reason for your answer.
(b) Find the average acceleration of the car, in $\mathrm{ft/sec^2}$, over the interval $0 \leq t \leq 50$.
(c) Find one approximation for the acceleration of the car, in $\mathrm{ft/sec^2}$, at $t = 40$. Show the computations you used to arrive at your answer.
(d) Approximate $\int_{0}^{50} v(t)\, dt$ with a Riemann sum, using the midpoints of five subintervals of equal length. Using correct units, explain the meaning of this integral.

The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function $R$ of time $t$. The table below shows the rate as measured every 3 hours for a 24-hour period.

\begin{tabular}{ c } $t$

(hours)

&

$R ( t )$

(gallons per hour)

\hline 0 & 9.6
3 & 10.4 6 & 10.8 9 & 11.2 12 & 11.4 15 & 11.3 18 & 10.7 21 & 10.2 24 & 9.6 \end{tabular}
(a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate $\int _ { 0 } ^ { 24 } R ( t ) d t$. Using correct units, explain the meaning of your answer in terms of water flow.
(b) Is there some time $t , 0 < t < 24$, such that $R ^ { \prime } ( t ) = 0$ ? Justify your answer.
(c) The rate of water flow $R ( t )$ can be approximated by $Q ( t ) = \frac { 1 } { 79 } \left( 768 + 23 t - t ^ { 2 } \right)$. Use $Q ( t )$ to approximate the average rate of water flow during the 24-hour time period. Indicate units of measure.

The temperature, in degrees Celsius (${}^{\circ}\mathrm{C}$), of the water in a pond is a differentiable function $W$ of time $t$. The table below shows the water temperature as recorded every 3 days over a 15-day period.

\begin{tabular}{ c } $t$

(days)

&

$W(t)$

$\left({}^{\circ}\mathrm{C}\right)$

\hline\hline 0 & 20
3 & 31 6 & 28 9 & 24 12 & 22 15 & 21 \hline \end{tabular}
(a) Use data from the table to find an approximation for $W^{\prime}(12)$. Show the computations that lead to your answer. Indicate units of measure.
(b) Approximate the average temperature, in degrees Celsius, of the water over the time interval $0 \leq t \leq 15$ days by using a trapezoidal approximation with subintervals of length $\Delta t = 3$ days.
(c) A student proposes the function $P$, given by $P(t) = 20 + 10te^{(-t/3)}$, as a model for the temperature of the water in the pond at time $t$, where $t$ is measured in days and $P(t)$ is measured in degrees Celsius. Find $P^{\prime}(12)$. Using appropriate units, explain the meaning of your answer in terms of water temperature.
(d) Use the function $P$ defined in part (c) to find the average value, in degrees Celsius, of $P(t)$ over the time interval $0 \leq t \leq 15$ days.

The graph of a differentiable function $f$ on the closed interval $[-3, 15]$ is shown in the figure above. The graph of $f$ has a horizontal tangent line at $x = 6$. Let $g(x) = 5 + \int_{6}^{x} f(t)\, dt$ for $-3 \leq x \leq 15$.
(a) Find $g(6)$, $g^{\prime}(6)$, and $g^{\prime\prime}(6)$.
(b) On what intervals is $g$ decreasing? Justify your answer.
(c) On what intervals is the graph of $g$ concave down? Justify your answer.
(d) Find a trapezoidal approximation of $\int_{-3}^{15} f(t)\, dt$ using six subintervals of length $\Delta t = 3$.

The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function $R$ of time $t$. The graph of $R$ and a table of selected values of $R(t)$, for the time interval $0 \leq t \leq 90$ minutes, are shown above.
(a) Use data from the table to find an approximation for $R'(45)$. Show the computations that lead to your answer. Indicate units of measure.
(b) The rate of fuel consumption is increasing fastest at time $t = 45$ minutes. What is the value of $R''(45)$? Explain your reasoning.
(c) Approximate the value of $\int_{0}^{90} R(t)\,dt$ using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation less than the value of $\int_{0}^{90} R(t)\,dt$? Explain your reasoning.
(d) For $0 < b \leq 90$ minutes, explain the meaning of $\int_{0}^{b} R(t)\,dt$ in terms of fuel consumption for the plane. Explain the meaning of $\frac{1}{b}\int_{0}^{b} R(t)\,dt$ in terms of fuel consumption for the plane. Indicate units of measure in both answers.

A test plane flies in a straight line with positive velocity $v(t)$, in miles per minute at time $t$ minutes, where $v$ is a differentiable function of $t$. Selected values of $v(t)$ for $0 \leq t \leq 40$ are shown in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 \hline

$v(t)$

(miles per minute)

& 7.0 & 9.2 & 9.5 & 7.0 & 4.5 & 2.4 & 2.4 & 4.3 & 7.3 \hline \end{tabular}
(a) Use a midpoint Riemann sum with four subintervals of equal length and values from the table to approximate $\int_{0}^{40} v(t)\,dt$. Show the computations that lead to your answer. Using correct units, explain the meaning of $\int_{0}^{40} v(t)\,dt$ in terms of the plane's flight.
(b) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval $0 < t < 40$? Justify your answer.
(c) The function $f$, defined by $f(t) = 6 + \cos\left(\frac{t}{10}\right) + 3\sin\left(\frac{7t}{40}\right)$, is used to model the velocity of the plane, in miles per minute, for $0 \leq t \leq 40$. According to this model, what is the acceleration of the plane at $t = 23$? Indicate units of measure.
(d) According to the model $f$, given in part (c), what is the average velocity of the plane, in miles per minute, over the time interval $0 \leq t \leq 40$?

A metal wire of length 8 centimeters (cm) is heated at one end. The table below gives selected values of the temperature $T ( x )$, in degrees Celsius $\left( { } ^ { \circ } \mathrm { C } \right)$, of the wire $x$ cm from the heated end. The function $T$ is decreasing and twice differentiable.

\begin{tabular}{ c } Distance

$x ( \mathrm {~cm} )$

& 0 & 1 & 5 & 6 & 8 \hline

Temperature

$T ( x ) \left( { } ^ { \circ } \mathrm { C } \right)$

& 100 & 93 & 70 & 62 & 55 \hline \end{tabular}
(a) Estimate $T ^ { \prime } ( 7 )$. Show the work that leads to your answer. Indicate units of measure.
(b) Write an integral expression in terms of $T ( x )$ for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure.
(c) Find $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) d x$, and indicate units of measure. Explain the meaning of $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) d x$ in terms of the temperature of the wire.
(d) Are the data in the table consistent with the assertion that $T ^ { \prime \prime } ( x ) > 0$ for every $x$ in the interval $0 < x < 8$ ? Explain your answer.

Concert tickets went on sale at noon $( t = 0 )$ and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time $t$ is modeled by a twice-differentiable function $L$ for $0 \leq t \leq 9$. Values of $L ( t )$ at various times $t$ are shown in the table below.

$t$ (hours)	0	1	3	4	7	8	9
$L ( t )$ (people)	120	156	176	126	150	80	0

(a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5:30 P.M. $( t = 5.5 )$. Show the computations that lead to your answer. Indicate units of measure.
(b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first 4 hours that tickets were on sale.
(c) For $0 \leq t \leq 9$, what is the fewest number of times at which $L ^ { \prime } ( t )$ must equal 0 ? Give a reason for your answer.
(d) The rate at which tickets were sold for $0 \leq t \leq 9$ is modeled by $r ( t ) = 550 t e ^ { - t / 2 }$ tickets per hour. Based on the model, how many tickets were sold by 3 P.M. ( $t = 3$ ), to the nearest whole number?

A tank contains 50 liters of oil at time $t = 4$ hours. Oil is being pumped into the tank at a rate $R ( t )$, where $R ( t )$ is measured in liters per hour, and $t$ is measured in hours. Selected values of $R ( t )$ are given in the table above.

$t$ (hours)	4	7	12	15
$R ( t )$ (liters/hour)	6.5	6.2	5.9	5.6

Using a right Riemann sum with three subintervals and data from the table, what is the approximation of the number of liters of oil that are in the tank at time $t = 15$ hours?
(A) 64.9
(B) 68.2
(C) 114.9
(D) 116.6
(E) 118.2

A student starts reading a book at time $t = 0$ minutes and continues reading for the next 10 minutes. The rate at which the student reads is modeled by the differentiable function $R$, where $R ( t )$ is measured in words per minute. Selected values of $R ( t )$ are given in the table shown.

$t$ (minutes)	0	2	8	10
$R ( t )$ (words per minute)	90	100	150	162

A. Approximate $R ^ { \prime } ( 1 )$ using the average rate of change of $R$ over the interval $0 \leq t \leq 2$. Show the work that leads to your answer. Indicate units of measure.
B. Must there be a value $c$, for $0 < c < 10$, such that $R ( c ) = 155$ ? Justify your answer.
C. Use a trapezoidal sum with the three subintervals indicated by the data in the table to approximate the value of $\int _ { 0 } ^ { 10 } R ( t ) d t$. Show the work that leads to your answer.
D. A teacher also starts reading at time $t = 0$ minutes and continues reading for the next 10 minutes. The rate at which the teacher reads is modeled by the function $W$ defined by $W ( t ) = - \frac { 3 } { 10 } t ^ { 2 } + 8 t + 100$, where $W ( t )$ is measured in words per minute. Based on the model, how many words has the teacher read by the end of the 10 minutes? Show the work that leads to your answer.

The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function $R$ of time $t$. The table below shows the rate as measured every 3 hours for a 24-hour period.

\begin{tabular}{ c } $t$

(hours)

&

$R(t)$

(gallons per hour)

\hline 0 & 9.6
3 & 10.4 6 & 10.8 9 & 11.2 12 & 11.4 15 & 11.3 18 & 10.7 21 & 10.2 24 & 9.6 \end{tabular}
(a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate $\int_{0}^{24} R(t)\, dt$. Using correct units, explain the meaning of your answer in terms of water flow.
(b) Is there some time $t$, $0 < t < 24$, such that $R'(t) = 0$? Justify your answer.
(c) The rate of water flow $R(t)$ can be approximated by $Q(t) = \frac{1}{79}\left(768 + 23t - t^2\right)$. Use $Q(t)$ to approximate the average rate of water flow during the 24-hour time period. Indicate units of measure.

The temperature, in degrees Celsius (${}^{\circ}\mathrm{C}$), of the water in a pond is a differentiable function $W$ of time $t$. The table below shows the water temperature as recorded every 3 days over a 15-day period.

\begin{tabular}{ c } $t$

(days)

&

$W(t)$

$\left({}^{\circ}\mathrm{C}\right)$

\hline\hline 0 & 20
3 & 31 6 & 28 9 & 24 12 & 22 15 & 21 \hline \end{tabular}
(a) Use data from the table to find an approximation for $W'(12)$. Show the computations that lead to your answer. Indicate units of measure.
(b) Approximate the average temperature, in degrees Celsius, of the water over the time interval $0 \leq t \leq 15$ days by using a trapezoidal approximation with subintervals of length $\Delta t = 3$ days.
(c) A student proposes the function $P$, given by $P(t) = 20 + 10te^{(-t/3)}$, as a model for the temperature of the water in the pond at time $t$, where $t$ is measured in days and $P(t)$ is measured in degrees Celsius. Find $P'(12)$. Using appropriate units, explain the meaning of your answer in terms of water temperature.
(d) Use the function $P$ defined in part (c) to find the average value, in degrees Celsius, of $P(t)$ over the time interval $0 \leq t \leq 15$ days.

A blood vessel is 360 millimeters (mm) long with circular cross sections of varying diameter. The table below gives the measurements of the diameter of the blood vessel at selected points along the length of the blood vessel, where $x$ represents the distance from one end of the blood vessel and $B(x)$ is a twice-differentiable function that represents the diameter at that point.

\begin{tabular}{ c } Distance

$x$

$(\mathrm{~mm})$

& 0 & 60 & 120 & 180 & 240 & 300 & 360 \hline

Diameter

$B(x)$

$(\mathrm{mm})$

& 24 & 30 & 28 & 30 & 26 & 24 & 26 \hline \end{tabular}
(a) Write an integral expression in terms of $B(x)$ that represents the average radius, in mm, of the blood vessel between $x = 0$ and $x = 360$.
(b) Approximate the value of your answer from part (a) using the data from the table and a midpoint Riemann sum with three subintervals of equal length. Show the computations that lead to your answer.
(c) Using correct units, explain the meaning of $\pi \int_{125}^{275} \left(\frac{B(x)}{2}\right)^2 dx$ in terms of the blood vessel.
(d) Explain why there must be at least one value $x$, for $0 < x < 360$, such that $B''(x) = 0$.

A test plane flies in a straight line with positive velocity $v ( t )$, in miles per minute at time $t$ minutes, where $v$ is a differentiable function of $t$. Selected values of $v ( t )$ for $0 \leq t \leq 40$ are shown in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 \hline

$v ( t )$

(miles per minute)

& 7.0 & 9.2 & 9.5 & 7.0 & 4.5 & 2.4 & 2.4 & 4.3 & 7.3 \hline \end{tabular}
(a) Use a midpoint Riemann sum with four subintervals of equal length and values from the table to approximate $\int _ { 0 } ^ { 40 } v ( t ) d t$. Show the computations that lead to your answer. Using correct units, explain the meaning of $\int _ { 0 } ^ { 40 } v ( t ) d t$ in terms of the plane's flight.
(b) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval $0 < t < 40$ ? Justify your answer.
(c) The function $f$, defined by $f ( t ) = 6 + \cos \left( \frac { t } { 10 } \right) + 3 \sin \left( \frac { 7 t } { 40 } \right)$, is used to model the velocity of the plane, in miles per minute, for $0 \leq t \leq 40$. According to this model, what is the acceleration of the plane at $t = 23$ ? Indicate units of measure.
(d) According to the model $f$, given in part (c), what is the average velocity of the plane, in miles per minute, over the time interval $0 \leq t \leq 40$ ?

A metal wire of length 8 centimeters (cm) is heated at one end. The table below gives selected values of the temperature $T ( x )$, in degrees Celsius $\left( {}^{\circ} \mathrm{C} \right)$, of the wire $x$ cm from the heated end. The function $T$ is decreasing and twice differentiable.

\begin{tabular}{ c } Distance

$x ( \mathrm{~cm} )$

& 0 & 1 & 5 & 6 & 8 \hline

Temperature

$T ( x ) \left( {}^{\circ} \mathrm{C} \right)$

& 100 & 93 & 70 & 62 & 55 \hline \end{tabular}
(a) Estimate $T ^ { \prime } ( 7 )$. Show the work that leads to your answer. Indicate units of measure.
(b) Write an integral expression in terms of $T ( x )$ for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure.
(c) Find $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) \, d x$, and indicate units of measure. Explain the meaning of $\int _ { 0 } ^ { 8 } T ^ { \prime } ( x ) \, d x$ in terms of the temperature of the wire.
(d) Are the data in the table consistent with the assertion that $T ^ { \prime \prime } ( x ) > 0$ for every $x$ in the interval $0 < x < 8$ ? Explain your answer.

The temperature of water in a tub at time $t$ is modeled by a strictly increasing, twice-differentiable function $W$, where $W(t)$ is measured in degrees Fahrenheit and $t$ is measured in minutes. At time $t = 0$, the temperature of the water is $55^{\circ}\mathrm{F}$. The water is heated for 30 minutes, beginning at time $t = 0$. Values of $W(t)$ at selected times $t$ for the first 20 minutes are given in the table below.

$t$ (minutes)	0	4	9	15	20
$W(t)$ (degrees Fahrenheit)	55.0	57.1	61.8	67.9	71.0

(a) Use the data in the table to estimate $W'(12)$. Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem.
(b) Use the data in the table to evaluate $\int_{0}^{20} W'(t)\, dt$. Using correct units, interpret the meaning of $\int_{0}^{20} W'(t)\, dt$ in the context of this problem.
(c) For $0 \leq t \leq 20$, the average temperature of the water in the tub is $\frac{1}{20} \int_{0}^{20} W(t)\, dt$. Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate $\frac{1}{20} \int_{0}^{20} W(t)\, dt$. Does this approximation overestimate or underestimate the average temperature of the water over these 20 minutes? Explain your reasoning.
(d) For $20 \leq t \leq 25$, the function $W$ that models the water temperature has first derivative given by $W'(t) = 0.4\sqrt{t}\cos(0.06t)$. Based on the model, what is the temperature of the water at time $t = 25$?

A tank has a height of 10 feet. The area of the horizontal cross section of the tank at height $h$ feet is given by the function $A$, where $A(h)$ is measured in square feet. The function $A$ is continuous and decreases as $h$ increases. Selected values for $A(h)$ are given in the table below.

\begin{tabular}{ c } $h$

(feet)

& 0 & 2 & 5 & 10 \hline

$A ( h )$

(square feet)

& 50.3 & 14.4 & 6.5 & 2.9 \hline \end{tabular}
(a) Use a left Riemann sum with the three subintervals indicated by the data in the table to approximate the volume of the tank. Indicate units of measure.
(b) Does the approximation in part (a) overestimate or underestimate the volume of the tank? Explain your reasoning.
(c) The area, in square feet, of the horizontal cross section at height $h$ feet is modeled by the function $f$ given by $f(h) = \frac{50.3}{e^{0.2h} + h}$. Based on this model, find the volume of the tank. Indicate units of measure.
(d) Water is pumped into the tank. When the height of the water is 5 feet, the height is increasing at the rate of 0.26 foot per minute. Using the model from part (c), find the rate at which the volume of water is changing with respect to time when the height of the water is 5 feet. Indicate units of measure.

The density of a bacteria population in a circular petri dish at a distance $r$ centimeters from the center of the dish is given by an increasing, differentiable function $f$, where $f ( r )$ is measured in milligrams per square centimeter. Values of $f ( r )$ for selected values of $r$ are given in the table below.

\begin{tabular}{ c } $r$

(centimeters)

& 0 & 1 & 2 & 2.5 & 4 \hline

$f ( r )$

(milligrams per square centimeter)

& 1 & 2 & 6 & 10 & 18 \hline \end{tabular}
(a) Use the data in the table to estimate $f ^ { \prime } ( 2.25 )$. Using correct units, interpret the meaning of your answer in the context of this problem.
(b) The total mass, in milligrams, of bacteria in the petri dish is given by the integral expression $2 \pi \int _ { 0 } ^ { 4 } r f ( r ) d r$. Approximate the value of $2 \pi \int _ { 0 } ^ { 4 } r f ( r ) d r$ using a right Riemann sum with the four subintervals indicated by the data in the table.
(c) Is the approximation found in part (b) an overestimate or underestimate of the total mass of bacteria in the petri dish? Explain your reasoning.
(d) The density of bacteria in the petri dish, for $1 \leq r \leq 4$, is modeled by the function $g$ defined by $g ( r ) = 2 - 16 ( \cos ( 1.57 \sqrt { r } ) ) ^ { 3 }$. For what value of $k , 1 < k < 4$, is $g ( k )$ equal to the average value of $g ( r )$ on the interval $1 \leq r \leq 4$ ?

The function $f$ is twice differentiable for all $x$ with $f(0) = 0$. Values of $f'$, the derivative of $f$, are given in the table for selected values of $x$.

$x$	0	$\pi$	$2\pi$
$f'(x)$	5	6	0

(a) For $x \geq 0$, the function $h$ is defined by $h(x) = \int_{0}^{x} \sqrt{1 + \left(f'(t)\right)^2}\, dt$. Find the value of $h'(\pi)$. Show the work that leads to your answer.
(b) What information does $\int_{0}^{\pi} \sqrt{1 + \left(f'(x)\right)^2}\, dx$ provide about the graph of $f$?
(c) Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f(2\pi)$. Show the computations that lead to your answer.
(d) Find $\int (t + 5)\cos\left(\frac{t}{4}\right)\, dt$. Show the work that leads to your answer.

Exercise 1 -- Part A

Consider the sequence $\left( u _ { n } \right)$ defined for every natural integer $n$ by: $$u _ { n } = \int _ { 0 } ^ { n } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x$$ We will not attempt to calculate $u _ { n }$ as a function of $n$.

1. a. Show that the sequence $(u_n)$ is increasing. b. Prove that for every real number $x \geqslant 0$, we have: $- x ^ { 2 } \leqslant - 2 x + 1$, then: $$\mathrm { e } ^ { - x ^ { 2 } } \leqslant \mathrm { e } ^ { - 2 x + 1 }$$ Deduce that for every natural integer $n$, we have: $u _ { n } < \frac { \mathrm { e } } { 2 }$. c. Prove that the sequence $(u_n)$ is convergent. We will not attempt to calculate its limit.
   
2. In this question, we propose to obtain an approximate value of $u _ { 2 }$.
   In the orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ below, we have drawn the curve $\mathscr{C}_f$ representing the function $f$ defined on the interval $[0;2]$ by $f(x) = \mathrm{e}^{-x^2}$, and the rectangle OABC where $\mathrm{A}(2;0)$, $\mathrm{B}(2;1)$ and $\mathrm{C}(0;1)$. We have shaded the region $\mathscr{D}$ between the curve $\mathscr{C}_f$, the horizontal axis, the vertical axis and the line with equation $x = 2$.
   Consider the random experiment consisting of choosing a point $M$ at random inside the rectangle OABC. We admit that the probability $p$ that this point belongs to the region is: $p = \frac{\text{area of } \mathscr{D}}{\text{area of } \mathrm{OABC}}$. a. Justify that $u_2 = 2p$. b. Consider the following algorithm:
   

   L1	Variables: $N, C$ integers; $X, Y, F$ real numbers
   L2	Input: Enter $N$
   L3	Initialization: $C$ takes the value 0
   L4	Processing:
   L5	For $k$ varying from 1 to $N$
   L6	$X$ takes the value of a random number between 0 and 2
   L7	$Y$ takes the value of a random number between 0 and 1
   L8	If $Y \leqslant \mathrm{e}^{-X^2}$ then
   L9	$C$ takes the value $C + 1$
   L10	End if
   L11	End for
   L12	Display $C$
   L13	$F$ takes the value $C/N$
   L14	Display $F$

   
   i. What does the condition on line $L8$ allow us to test regarding the position of point $M(X;Y)$? ii. Interpret the value $F$ displayed by this algorithm. iii. What can we conjecture about the value of $F$ when $N$ becomes very large? c. By running this algorithm for $N = 10^6$, we obtain $C = 441138$.
   We admit in this case that the value $F$ displayed by the algorithm is an approximate value of the probability $p$ to within $10^{-3}$. Deduce an approximate value of $u_2$ to within $10^{-2}$.

Part B

The sign, modeled by the region $\mathscr{D}$ defined in Part A, is cut from a rectangular sheet of 2 meters by 1 meter. It is represented in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$; the chosen unit is the meter.
For $x$ a real number belonging to the interval $[0;2]$, we denote:

• $M$ the point on the curve $\mathscr{C}_f$ with coordinates $(x; \mathrm{e}^{-x^2})$,
• $N$ the point with coordinates $(x; 0)$,
• $P$ the point with coordinates $\left(0; \mathrm{e}^{-x^2}\right)$,
• $A(x)$ the area of rectangle $ONMP$.

1. Justify that for every real number $x$ in the interval $[0;2]$, we have: $A(x) = x\mathrm{e}^{-x^2}$.
2. Determine the position of point $M$ on the curve $\mathscr{C}_f$ for which the area of rectangle $ONMP$ is maximum.
3. The rectangle $ONMP$ of maximum area obtained in question 2. must be painted blue, and the rest of the sign in white. Determine, in $\mathrm{m}^2$ and to within $10^{-2}$, the measure of the surface to be painted blue and that of the surface to be painted white.

Define $$x = \sum _ { i = 1 } ^ { 10 } \frac { 1 } { 10 \sqrt { 3 } } \frac { 1 } { 1 + \left( \frac { i } { 10 \sqrt { 3 } } \right) ^ { 2 } } \quad \text { and } \quad y = \sum _ { i = 0 } ^ { 9 } \frac { 1 } { 10 \sqrt { 3 } } \frac { 1 } { 1 + \left( \frac { i } { 10 \sqrt { 3 } } \right) ^ { 2 } } .$$ Show that a) $x < \frac { \pi } { 6 } < y$ and b) $\frac { x + y } { 2 } < \frac { \pi } { 6 }$. (Hint: Relate these sums to an integral.)

In the case $I = [0,1]$ and $\forall x \in I, w(x) = 1$, we seek to approximate $\int_0^1 f(x)\,\mathrm{d}x$ when $f$ is a continuous function from $[0,1]$ to $\mathbb{R}$.
Determine the order of the quadrature formula $I_0(f) = f(0)$ and represent graphically the associated error $e(f)$.

In the case $I = [0,1]$ and $\forall x \in I, w(x) = 1$, we seek to approximate $\int_0^1 f(x)\,\mathrm{d}x$ when $f$ is a continuous function from $[0,1]$ to $\mathbb{R}$.
Determine the order of the quadrature formula $I_0(f) = f(1/2)$ and represent graphically the associated error $e(f)$.

In the case $I = [0,1]$ and $\forall x \in I, w(x) = 1$, we seek to approximate $\int_0^1 f(x)\,\mathrm{d}x$ when $f$ is a continuous function from $[0,1]$ to $\mathbb{R}$.
Determine the coefficients $\lambda_0, \lambda_1, \lambda_2$ so that the formula $I_2(f) = \lambda_0 f(0) + \lambda_1 f(1/2) + \lambda_2 f(1)$ is exact on $\mathbb{R}_2[X]$. Is this quadrature formula of order 2?

Let $n \in \mathbb{N}$. We consider $n+1$ distinct points in $I$, denoted $x_0 < x_1 < \cdots < x_n$, and the Lagrange basis $(L_0, \ldots, L_n)$ associated with these points.
Suppose that, for all $k \in \mathbb{N}$, the map $x \mapsto x^k w(x)$ is integrable on $I$. Show that the quadrature formula $I_n(f) = \sum_{j=0}^n \lambda_j f(x_j)$ is exact on $\mathbb{R}_n[X]$ if and only if $$\forall j \in \llbracket 0, n \rrbracket, \quad \lambda_j = \int_I L_j(x) w(x)\,\mathrm{d}x.$$