The graph of $f ^ { \prime }$, the derivative of a function $f$, consists of two line segments and a semicircle, as shown in the figure above. If $f ( 2 ) = 1$, then $f ( - 5 ) =$
(A) $2 \pi - 2$
(B) $2 \pi - 3$
(C) $2 \pi - 5$
(D) $6 - 2 \pi$
(E) $4 - 2 \pi$

$\int _ { 1 } ^ { \infty } x e ^ { - x ^ { 2 } } d x$ is
(A) $- \frac { 1 } { e }$
(B) $\frac { 1 } { 2 e }$
(C) $\frac { 1 } { e }$
(D) $\frac { 2 } { e }$
(E) divergent

Let $g$ be a continuously differentiable function with $g ( 1 ) = 6$ and $g ^ { \prime } ( 1 ) = 3$. What is $\lim _ { x \rightarrow 1 } \frac { \int _ { 1 } ^ { x } g ( t ) d t } { g ( x ) - 6 }$ ?
(A) 0
(B) $\frac { 1 } { 2 }$
(C) 1
(D) 2
(E) The limit does not exist.

The graph of the piecewise linear function $f$ is shown above. What is the value of $\int _ { - 1 } ^ { 9 } ( 3 f ( x ) + 2 ) d x$ ?
(A) 7.5
(B) 9.5
(C) 27.5
(D) 47
(E) 48.5

What is the average value of $y = \sqrt { \cos x }$ on the interval $0 \leq x \leq \frac { \pi } { 2 }$ ?
(A) - 0.637
(B) 0.500
(C) 0.763
(D) 1.198
(E) 1.882

If $f ^ { \prime } ( x ) > 0$ for all real numbers $x$ and $\int _ { 4 } ^ { 7 } f ( t ) d t = 0$, which of the following could be a table of values for the function $f$ ?
(A)

$x$	$f ( x )$
4	- 4
5	- 3
7	0

(B)

$x$	$f ( x )$
4	- 4
5	- 2
7	5

(C)

$x$	$f ( x )$
4	- 4
5	6
7	3

(D)

$x$	$f ( x )$
4	0
5	0
7	0

(E)

$x$	$f ( x )$
4	0
5	4
7	6

The function $h$ is differentiable, and for all values of $x$, $h ( x ) = h ( 2 - x )$. Which of the following statements must be true?
I. $\int _ { 0 } ^ { 2 } h ( x ) d x > 0$
II. $h ^ { \prime } ( 1 ) = 0$
III. $h ^ { \prime } ( 0 ) = h ^ { \prime } ( 2 ) = 1$
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III

The function $f$ is defined on the closed interval $[ - 5, 4 ]$. The graph of $f$ consists of three line segments and is shown in the figure above. Let $g$ be the function defined by $g ( x ) = \int _ { - 3 } ^ { x } f ( t ) \, dt$.
(a) Find $g ( 3 )$.
(b) On what open intervals contained in $- 5 < x < 4$ is the graph of $g$ both increasing and concave down? Give a reason for your answer.
(c) The function $h$ is defined by $h ( x ) = \frac { g ( x ) } { 5 x }$. Find $h ^ { \prime } ( 3 )$.
(d) The function $p$ is defined by $p ( x ) = f \left( x ^ { 2 } - x \right)$. Find the slope of the line tangent to the graph of $p$ at the point where $x = - 1$.

The figure above shows the graph of the piecewise-linear function $f$. For $- 4 \leq x \leq 12$, the function $g$ is defined by $g ( x ) = \int _ { 2 } ^ { x } f ( t ) \, d t$.
(a) Does $g$ have a relative minimum, a relative maximum, or neither at $x = 10$ ? Justify your answer.
(b) Does the graph of $g$ have a point of inflection at $x = 4$ ? Justify your answer.
(c) Find the absolute minimum value and the absolute maximum value of $g$ on the interval $- 4 \leq x \leq 12$. Justify your answers.
(d) For $- 4 \leq x \leq 12$, find all intervals for which $g ( x ) \leq 0$.

People enter a line for an escalator at a rate modeled by the function $r$ given by
$$r ( t ) = \begin{cases} 44 \left( \frac { t } { 100 } \right) ^ { 3 } \left( 1 - \frac { t } { 300 } \right) ^ { 7 } & \text { for } 0 \leq t \leq 300 \\ 0 & \text { for } t > 300 \end{cases}$$
where $r ( t )$ is measured in people per second and $t$ is measured in seconds. As people get on the escalator, they exit the line at a constant rate of 0.7 person per second. There are 20 people in line at time $t = 0$.
(a) How many people enter the line for the escalator during the time interval $0 \leq t \leq 300$ ?
(b) During the time interval $0 \leq t \leq 300$, there are always people in line for the escalator. How many people are in line at time $t = 300$ ?
(c) For $t > 300$, what is the first time $t$ that there are no people in line for the escalator?
(d) For $0 \leq t \leq 300$, at what time $t$ is the number of people in line a minimum? To the nearest whole number, find the number of people in line at this time. Justify your answer.

Fish enter a lake at a rate modeled by the function $E$ given by $E ( t ) = 20 + 15 \sin \left( \frac { \pi t } { 6 } \right)$. Fish leave the lake at a rate modeled by the function $L$ given by $L ( t ) = 4 + 2 ^ { 0.1 t ^ { 2 } }$. Both $E ( t )$ and $L ( t )$ are measured in fish per hour, and $t$ is measured in hours since midnight $( t = 0 )$.
(a) How many fish enter the lake over the 5-hour period from midnight $( t = 0 )$ to 5 A.M. $( t = 5 )$? Give your answer to the nearest whole number.
(b) What is the average number of fish that leave the lake per hour over the 5-hour period from midnight $( t = 0 )$ to 5 A.M. $( t = 5 )$?
(c) At what time $t$, for $0 \leq t \leq 8$, is the greatest number of fish in the lake? Justify your answer.
(d) Is the rate of change in the number of fish in the lake increasing or decreasing at 5 A.M. ($t = 5$)? Explain your reasoning.

The continuous function $f$ is defined on the closed interval $- 6 \leq x \leq 5$. The figure above shows a portion of the graph of $f$, consisting of two line segments and a quarter of a circle centered at the point $( 5, 3 )$. It is known that the point $( 3, 3 - \sqrt { 5 } )$ is on the graph of $f$.
(a) If $\int _ { - 6 } ^ { 5 } f ( x ) \, dx = 7$, find the value of $\int _ { - 6 } ^ { - 2 } f ( x ) \, dx$. Show the work that leads to your answer.
(b) Evaluate $\int _ { 3 } ^ { 5 } \left( 2 f ^ { \prime } ( x ) + 4 \right) dx$.
(c) The function $g$ is given by $g ( x ) = \int _ { - 2 } ^ { x } f ( t ) \, dt$. Find the absolute maximum value of $g$ on the interval $- 2 \leq x \leq 5$. Justify your answer.
(d) Find $\lim _ { x \rightarrow 1 } \frac { 10 ^ { x } - 3 f ^ { \prime } ( x ) } { f ( x ) - \arctan x }$.

A customer at a gas station is pumping gasoline into a gas tank. The rate of flow of gasoline is modeled by a differentiable function $f$, where $f(t)$ is measured in gallons per second and $t$ is measured in seconds since pumping began. Selected values of $f(t)$ are given in the table.

\begin{tabular}{ c } $t$

(seconds)

& 0 & 60 & 90 & 120 & 135 & 150 \hline

$f ( t )$

(gallons per second)

& 0 & 0.1 & 0.15 & 0.1 & 0.05 & 0 \hline \end{tabular}
(a) Using correct units, interpret the meaning of $\int_{60}^{135} f(t)\, dt$ in the context of the problem. Use a right Riemann sum with the three subintervals $[60,90]$, $[90, 120]$, and $[120, 135]$ to approximate the value of $\int_{60}^{135} f(t)\, dt$.
(b) Must there exist a value of $c$, for $60 < c < 120$, such that $f'(c) = 0$? Justify your answer.
(c) The rate of flow of gasoline, in gallons per second, can also be modeled by $g(t) = \left(\frac{t}{500}\right)\cos\left(\left(\frac{t}{120}\right)^{2}\right)$ for $0 \leq t \leq 150$. Using this model, find the average rate of flow of gasoline over the time interval $0 \leq t \leq 150$. Show the setup for your calculations.
(d) Using the model $g$ defined in part (c), find the value of $g'(140)$. Interpret the meaning of your answer in the context of the problem.

The graph of the differentiable function $f$, shown for $-6 \leq x \leq 7$, has a horizontal tangent at $x = -2$ and is linear for $0 \leq x \leq 7$. Let $R$ be the region in the second quadrant bounded by the graph of $f$, the vertical line $x = -6$, and the $x$- and $y$-axes. Region $R$ has area 12.
(a) The function $g$ is defined by $g(x) = \int_{0}^{x} f(t)\, dt$. Find the values of $g(-6)$, $g(4)$, and $g(6)$.
(b) For the function $g$ defined in part (a), find all values of $x$ in the interval $0 \leq x \leq 6$ at which the graph of $g$ has a critical point. Give a reason for your answer.
(c) The function $h$ is defined by $h(x) = \int_{-6}^{x} f'(t)\, dt$. Find the values of $h(6)$, $h'(6)$, and $h''(6)$. Show the work that leads to your answers.

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 continuous function $f$ is defined on the closed interval $- 6 \leq x \leq 12$. The graph of $f$, consisting of two semicircles and one line segment, is shown in the figure.
Let $g$ be the function defined by $g ( x ) = \int _ { 6 } ^ { x } f ( t ) d t$.
A. Find $g ^ { \prime } ( 8 )$. Give a reason for your answer.
B. Find all values of $x$ in the open interval $- 6 < x < 12$ at which the graph of $g$ has a point of inflection. Give a reason for your answer.
C. Find $g ( 12 )$ and $g ( 0 )$. Label your answers.
D. Find the value of $x$ at which $g$ attains an absolute minimum on the closed interval $- 6 \leq x \leq 12$. Justify your answer.

Let $f$ be a function defined and differentiable on $\mathbb{R}$. We denote by $\mathscr{C}$ its representative curve in the plane equipped with a coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$.
Part A
In the graphs below, we have represented the curve $\mathscr{C}$ and three other curves $\mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_3$ with the tangent line at their point with abscissa 0.

1. Determine by reading the graph, the sign of $f(x)$ according to the values of $x$.
2. We denote by $F$ a primitive of the function $f$ on $\mathbb{R}$. a. Using the curve $\mathscr{C}$, determine $F'(0)$ and $F'(-2)$. b. One of the curves $\mathscr{C}_1, \mathscr{C}_2, \mathscr{C}_3$ is the representative curve of the function $F$. Determine which one by justifying the elimination of the other two.

Part B
In this part, we assume that the function $f$ mentioned in Part A is the function defined on $\mathbb{R}$ by $$f(x) = (x+2)\mathrm{e}^{\frac{1}{2}x}$$

1. Observation of the curve $\mathscr{C}$ allows us to conjecture that the function $f$ admits a minimum. a. Prove that for all real $x$, $f'(x) = \frac{1}{2}(x+4)\mathrm{e}^{\frac{1}{2}x}$. b. Deduce a validation of the previous conjecture.
2. Let $I = \int_0^1 f(x)\,\mathrm{d}x$. a. Give a geometric interpretation of the real number $I$. b. Let $u$ and $v$ be the functions defined on $\mathbb{R}$ by $u(x) = x$ and $v(x) = \mathrm{e}^{\frac{1}{2}x}$. Verify that $f = 2(u'v + uv')$. c. Deduce the exact value of the integral $I$.
3. The algorithm below is given.
   

   Variables :	$k$ and $n$ are natural integers. $s$ is a real number.
   Input :	Assign to $s$ the value 0.
   Processing :	For $k$ ranging from 0 to $n-1$
   	End of loop.
   Output :	Display $s$.

   
   We denote by $s_n$ the number displayed by this algorithm when the user enters a strictly positive natural integer as the value of $n$. a. Justify that $s_3$ represents the area, expressed in square units, of the shaded region in the graph below where the three rectangles have the same width. b. What can be said about the value of $s_n$ provided by the proposed algorithm when $n$ becomes large?

We consider the function $g$ defined for all real $x$ in the interval $[0;1]$ by: $$g(x) = 1 + \mathrm{e}^{-x}$$ We admit that, for all real $x$ in the interval $[0;1], g(x) > 0$.
We denote $\mathscr{C}$ the representative curve of function $g$ in an orthogonal coordinate system, and $\mathscr{D}$ the plane region bounded on one hand between the $x$-axis and curve $\mathscr{C}$, on the other hand between the lines with equations $x = 0$ and $x = 1$.
The purpose of this exercise is to divide region $\mathscr{D}$ into two regions of equal area, first by a line parallel to the $y$-axis (part A), then by a line parallel to the $x$-axis (part B).

Part A

Let $a$ be a real number such that $0 \leqslant a \leqslant 1$. We denote $\mathscr{A}_1$ the area of the region between curve $\mathscr{C}$, the $x$-axis, the lines with equations $x = 0$ and $x = a$, and $\mathscr{A}_2$ that of the region between curve $\mathscr{C}$, the $x$-axis and the lines with equations $x = a$ and $x = 1$. $\mathscr{A}_1$ and $\mathscr{A}_2$ are expressed in square units.

1. a. Prove that $\mathscr{A}_1 = a - \mathrm{e}^{-a} + 1$. b. Express $\mathscr{A}_2$ as a function of $a$.
2. Let $f$ be the function defined for all real $x$ in the interval $[0;1]$ by: $$f(x) = 2x - 2\mathrm{e}^{-x} + \frac{1}{\mathrm{e}}$$ a. Draw the variation table of function $f$ on the interval $[0;1]$. The exact values of $f(0)$ and $f(1)$ will be specified. b. Prove that function $f$ vanishes once and only once on the interval $[0;1]$, at a real number $\alpha$. Give the value of $\alpha$ rounded to the nearest hundredth.
3. Using the previous questions, determine an approximate value of the real $a$ for which the areas $\mathscr{A}_1$ and $\mathscr{A}_2$ are equal.

Part B

Let $b$ be a positive real number. In this part, we propose to divide region $\mathscr{D}$ into two regions of equal area by the line with equation $y = b$. We admit that there exists a unique positive real $b$ that is a solution.

1. Justify the inequality $b < 1 + \frac{1}{\mathrm{e}}$. You may use a graphical argument.
2. Determine the exact value of the real $b$.

4. We define the number $I = \int _ { 0 } ^ { 1 } f _ { 1 } ( x ) \mathrm { d } x$.
Show that $I = \ln \left( \frac { 1 + \mathrm { e } } { 2 } \right)$. Give a graphical interpretation of $I$.

Part B

In this part, we choose $k = - 1$ and we wish to sketch the curve $\mathscr { C } _ { - 1 }$ representing the function $f _ { - 1 }$. For all real $x$, we call $P$ the point on $\mathscr { C } _ { 1 }$ with abscissa $x$ and $M$ the point on $\mathscr { C } _ { - 1 }$ with abscissa $x$. We denote by $K$ the midpoint of segment [ $M P$ ].

1. Show that, for all real $x , f _ { 1 } ( x ) + f _ { - 1 } ( x ) = 1$.
2. Deduce that point $K$ belongs to the line with equation $y = \frac { 1 } { 2 }$.
3. Sketch the curve $\mathscr { C } _ { - 1 }$ on the APPENDIX, to be returned with your answer sheet.
4. Deduce the area, in square units, of the region bounded by the curves $\mathscr { C } _ { 1 } , \mathscr { C } _ { - 1 }$, the $y$-axis and the line with equation $x = 1$.

Part C

In this part, we do not privilege any particular value of the parameter $k$. For each of the following statements, say whether it is true or false and justify your answer.

1. Whatever the value of the real number $k$, the graph of the function $f _ { k }$ is strictly between the lines with equations $y = 0$ and $y = 1$.
2. Whatever the value of the real $k$, the function $f _ { k }$ is strictly increasing.
3. For all real $u _ { n }$ & 4502 & 13378 & 39878 & 119122 & 356342 & 1066978 & 3196838 & 9582322 & 28730582 \hline \end{tabular}
   b. What conjecture can be made concerning the monotonicity of the sequence $\left( u _ { n } \right)$ ?

Part A

In the plane with an orthonormal coordinate system, we denote by $\mathscr { C } _ { 1 }$ the curve representing the function $f _ { 1 }$ defined on $\mathbb { R }$ by: $$f _ { 1 } ( x ) = x + \mathrm { e } ^ { - x } .$$

1. Justify that $\mathscr { C } _ { 1 }$ passes through point A with coordinates $( 0 ; 1 )$.
2. Determine the variation table of the function $f _ { 1 }$. Specify the limits of $f _ { 1 }$ at $+ \infty$ and at $- \infty$.

Part B

The purpose of this part is to study the sequence $\left( I _ { n } \right)$ defined on $\mathbb { N }$ by: $$I _ { n } = \int _ { 0 } ^ { 1 } \left( x + \mathrm { e } ^ { - n x } \right) \mathrm { d } x .$$

1. In the plane with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ), for every natural integer $n$, we denote by $\mathscr { C } _ { n }$ the curve representing the function $f _ { n }$ defined on $\mathbb { R }$ by $$f _ { n } ( x ) = x + \mathrm { e } ^ { - n x } .$$ a. Give a geometric interpretation of the integral $I _ { n }$. b. Using this interpretation, formulate a conjecture about the direction of variation of the sequence ( $I _ { n }$ ) and its possible limit. Specify the elements on which you base your conjecture.
2. Prove that for every natural integer $n$ greater than or equal to 1, $$I _ { n + 1 } - I _ { n } = \int _ { 0 } ^ { 1 } \mathrm { e } ^ { - ( n + 1 ) x } \left( 1 - \mathrm { e } ^ { x } \right) \mathrm { d } x$$ Deduce the sign of $I _ { n + 1 } - I _ { n }$ and then prove that the sequence ( $I _ { n }$ ) is convergent.
3. Determine the expression of $I _ { n }$ as a function of $n$ and determine the limit of the sequence $\left( I _ { n } \right)$.

Let $f$ be the function defined and differentiable on the interval $[ 0 ; + \infty [$ such that:
$$f ( x ) = \frac { x } { \mathrm { e } ^ { x } - x }$$
It is admitted that the function $f$ is positive on the interval $[ 0 ; + \infty [$. We denote by $\mathscr { C }$ the representative curve of the function $f$ in an orthogonal coordinate system of the plane. The curve $\mathscr { C }$ is represented in the appendix, to be returned with the answer sheet.

Part A

Let the sequence $\left( I _ { n } \right)$ be defined for every natural integer $n$ by $I _ { n } = \int _ { 0 } ^ { n } f ( x ) \mathrm { d } x$. We will not seek to calculate the exact value of $I _ { n }$ as a function of $n$.

1. Show that the sequence ( $I _ { n }$ ) is increasing.
2. It is admitted that for every real $x$ in the interval $\left[ 0 ; + \infty \left[ , \mathrm { e } ^ { x } - x \geqslant \frac { \mathrm { e } ^ { x } } { 2 } \right. \right.$. a. Show that, for every natural integer $n , I _ { n } \leqslant \int _ { 0 } ^ { n } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x$. b. Let $H$ be the function defined and differentiable on the interval $[ 0 ; + \infty [$ such that: $$H ( x ) = ( - x - 1 ) \mathrm { e } ^ { - x }$$ Determine the derivative function $H ^ { \prime }$ of the function $H$. c. Deduce that, for every natural integer $n , I _ { n } \leqslant 2$.
3. Show that the sequence ( $I _ { n }$ ) is convergent. The value of its limit is not required.

Part B

Consider the following algorithm in which the variables are

• $K$ and $i$ natural integers, $K$ being non-zero;
• $A , x$ and $h$ real numbers.

Input:	Enter $K$ non-zero natural integer
Initialization	\begin{tabular}{l} Assign to $A$ the value 0
Assign to $x$ the value 0
Assign to $h$ the value $\frac { 1 } { K }$

\hline Processing &

For $i$ ranging from 1 to $K$

Assign to $A$ the value $A + h \times f ( x )$

Assign to $x$ the value $x + h$

End For

\hline Output & Display $A$ \hline \end{tabular}

1. Reproduce and complete the following table by running this algorithm for $K = 4$. The successive values of $A$ will be rounded to the nearest thousandth.
   

   $i$	$A$	$x$
   1
   2
   3
   4

   
   
2. By illustrating it on the appendix to be returned with the answer sheet, give a graphical interpretation of the result displayed by this algorithm for $K = 8$.
3. What does the algorithm give when $K$ becomes large?

A factory produces mineral water in bottles. The shape of the bottle labels is bounded by the x-axis and the curve $\mathscr { C }$ with equation $y = a \cos x$ with $x \in \left[ - \frac { \pi } { 2 } ; \frac { \pi } { 2 } \right]$ and $a$ a strictly positive real number.
A disk located inside is intended to receive information given to buyers. We consider the disk with centre at point A with coordinates $\left( 0 ; \frac { a } { 2 } \right)$ and radius $\frac { a } { 2 }$. It is admitted that this disk is entirely below the curve $\mathscr { C }$ for values of $a$ less than 1.4.

1. Justify that the area of the region between the x-axis, the lines with equations $x = - \frac { \pi } { 2 }$ and $x = \frac { \pi } { 2 }$, and the curve $\mathscr { C }$ equals $2 a$ square units.
2. For aesthetic reasons, it is desired that the area of the disk equals the area of the shaded surface. What value should be given to the real number $a$ to satisfy this constraint?

Exercise 1 -- Common to all candidates

Part A

Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = \frac{3}{1 + \mathrm{e}^{-2x}}$$ In the graph below, we have drawn, in an orthogonal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, the representative curve $\mathscr{C}$ of the function $f$ and the line $\Delta$ with equation $y = 3$.

1. Prove that the function $f$ is strictly increasing on $\mathbb{R}$.
2. Justify that the line $\Delta$ is an asymptote to the curve $\mathscr{C}$.
3. Prove that the equation $f(x) = 2.999$ has a unique solution $\alpha$ on $\mathbb{R}$.

Determine an interval containing $\alpha$ with amplitude $10^{-2}$.

Part B

Let $h$ be the function defined on $\mathbb{R}$ by $h(x) = 3 - f(x)$.

1. Justify that the function $h$ is positive on $\mathbb{R}$.
2. We denote by $H$ the function defined on $\mathbb{R}$ by $H(x) = -\frac{3}{2}\ln\left(1 + \mathrm{e}^{-2x}\right)$.
   Prove that $H$ is an antiderivative of $h$ on $\mathbb{R}$.
3. Let $a$ be a strictly positive real number. a. Give a graphical interpretation of the integral $\int_{0}^{a} h(x)\,\mathrm{d}x$. b. Prove that $\int_{0}^{a} h(x)\,\mathrm{d}x = \frac{3}{2}\ln\left(\frac{2}{1 + \mathrm{e}^{-2a}}\right)$. c. We denote by $\mathscr{D}$ the set of points $M(x\,;\,y)$ in the plane defined by $$\left\{\begin{array}{l} x \geqslant 0 \\ f(x) \leqslant y \leqslant 3 \end{array}\right.$$ Determine the area, in square units, of the region $\mathscr{D}$.

The manufacturer of padlocks of the brand ``K'' wishes to print a logo for his company. This logo has the shape of a stylized capital letter K, inscribed in a square ABCD, with side length one unit of length. We place ourselves in the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD})$.
Part B: study of Proposal B
This proposal is characterized by the following two conditions:

• the line with endpoints A and E is a portion of the graph of the function $f$ defined for all real $x \geqslant 0$ by: $f(x) = \ln(2x + 1)$;
• the line with endpoints B and G is a portion of the graph of the function $g$ defined for all real $x > 0$ by: $g(x) = k\left(\frac{1 - x}{x}\right)$, where $k$ is a positive real number to be determined.

1. a) Determine the abscissa of point E. b) Determine the value of the real number $k$, knowing that the abscissa of point G is equal to 0.5.
   
2. a) Prove that the function $f$ has as a primitive the function $F$ defined for all real $x \geqslant 0$ by: $$F(x) = (x + 0.5) \times \ln(2x + 1) - x.$$ b) Prove that $r = \frac{\mathrm{e}}{2} - 1$.
   
3. Determine a primitive $G$ of the function $g$ on the interval $]0; +\infty[$.
   
4. It is admitted that the previous results allow us to establish that $s = [\ln(2)]^2 + \frac{\ln(2) - 1}{2}$. Does Proposal B satisfy the conditions imposed by the manufacturer?

Let $f$ be a function defined on the interval $[0;1]$, continuous and positive on this interval, and $a$ a real number such that $0 < a < 1$.
We denote:

• $\mathscr{C}$ the representative curve of the function $f$ in an orthogonal coordinate system;
• $\mathscr{A}_1$ the area of the plane region bounded by the $x$-axis and the curve $\mathscr{C}$ on one hand, the lines with equations $x = 0$ and $x = a$ on the other hand.
• $\mathscr{A}_2$ the area of the plane region bounded by the $x$-axis and the curve $\mathscr{C}$ on one hand, the lines with equations $x = a$ and $x = 1$ on the other hand.

The purpose of this exercise is to determine, for different functions $f$, a value of the real number $a$ satisfying condition (E): ``the areas $\mathscr{A}_1$ and $\mathscr{A}_2$ are equal''. We admit the existence of such a real number $a$ for each of the functions considered.
Part A: Study of some examples

1. Verify that in the following cases, condition (E) is satisfied for a unique real number $a$ and determine its value. a. $f$ is a strictly positive constant function. b. $f$ is defined on $[0;1]$ by $f(x) = x$.
   
2. a. Using integrals, express, in units of area, the areas $\mathscr{A}_1$ and $\mathscr{A}_2$. b. Let $F$ be a primitive of the function $f$ on the interval $[0;1]$. Prove that if the real number $a$ satisfies condition (E), then $F(a) = \dfrac{F(0) + F(1)}{2}$. Is the converse true?
   
3. In this question, we consider two other particular functions. a. The function $f$ is defined for all real $x$ in $[0;1]$ by $f(x) = \mathrm{e}^x$. Verify that condition (E) is satisfied for a unique real number $a$ and give its value. b. The function $f$ defined for all real $x$ in $[0;1]$ by $f(x) = \dfrac{1}{(x+2)^2}$. Verify that the value $a = \dfrac{2}{5}$ works.

Part B: Using a sequence to determine an approximate value of $a$
In this part, we consider the function $f$ defined for all real $x$ in $[0;1]$ by $f(x) = 4 - 3x^2$.

1. Prove that if $a$ is a real number satisfying condition (E), then $a$ is a solution of the equation: $$x = \frac{x^3}{4} + \frac{3}{8}$$ In the rest of the exercise, we will admit that this equation has a unique solution in the interval $[0;1]$. We denote this solution by $a$.
   
2. We consider the function $g$ defined for all real $x$ in $[0;1]$ by $g(x) = \dfrac{x^3}{4} + \dfrac{3}{8}$ and the sequence $(u_n)$ defined by: $u_0 = 0$ and, for all natural number $n$, $u_{n+1} = g(u_n)$. a. Calculate $u_1$. b. Prove that the function $g$ is increasing on the interval $[0;1]$. c. Prove by induction that, for all natural number $n$, we have $0 \leqslant u_n \leqslant u_{n+1} \leqslant 1$. d. Prove that the sequence $(u_n)$ is convergent. Using operations on limits, prove that the limit is $a$. e. We admit that the real number $a$ satisfies the inequality $0 < a - u_{10} < 10^{-9}$. Calculate $u_{10}$ to $10^{-8}$ precision.