A homeowner wants to have a water tank built. This water tank must comply with the following specifications:

• it must be located two metres from his house;
• the maximum depth must be two metres;
• it must measure five metres long;
• it must follow the natural slope of the land.

The curved part is modelled by the curve $\mathscr{C}_f$ of the function $f$ on the interval $[2; 2e]$ defined by: $$f(x) = x \ln\left(\frac{x}{2}\right) - x + 2$$
The curve $\mathscr{C}_f$ is represented in an orthonormal coordinate system with unit $1\mathrm{m}$ and constitutes a profile view of the tank. We consider the points $\mathrm{A}(2; 2)$, $\mathrm{I}(2; 0)$ and $\mathrm{B}(2\mathrm{e}; 2)$.
Part A
The objective of this part is to evaluate the volume of the tank.

1. Justify that the points B and I belong to the curve $\mathscr{C}_f$ and that the x-axis is tangent to the curve $\mathscr{C}_f$ at point I.
2. We denote by $\mathscr{T}$ the tangent to the curve $\mathscr{C}_f$ at point B, and D the point of intersection of the line $\mathscr{T}$ with the x-axis. a. Determine an equation of the line $\mathscr{T}$ and deduce the coordinates of D. b. We call $S$ the area of the region bounded by the curve $\mathscr{C}_f$, the lines with equations $y = 2$, $x = 2$ and $x = 2\mathrm{e}$. $S$ can be bounded by the area of triangle ABI and that of trapezoid AIDB. What bounds on the volume of the tank can we deduce?
3. a. Show that, on the interval $[2; 2\mathrm{e}]$, the function $G$ defined by $$G(x) = \frac{x^2}{2} \ln\left(\frac{x}{2}\right) - \frac{x^2}{4}$$ is an antiderivative of the function $g$ defined by $g(x) = x \ln\left(\frac{x}{2}\right)$. b. Deduce an antiderivative $F$ of the function $f$ on the interval $[2; 2\mathrm{e}]$. c. Determine the exact value of the area $S$ and deduce an approximate value of the volume $V$ of the tank to the nearest $\mathrm{m}^3$.

Part B
For any real number $x$ between 2 and $2\mathrm{e}$, we denote by $v(x)$ the volume of water, expressed in $\mathrm{m}^3$, in the tank when the water level in the tank is equal to $f(x)$. We admit that, for any real number $x$ in the interval $[2; 2\mathrm{e}]$, $$v(x) = 5\left[\frac{x^2}{2}\ln\left(\frac{x}{2}\right) - 2x\ln\left(\frac{x}{2}\right) - \frac{x^2}{4} + 2x - 3\right]$$

1. What volume of water, to the nearest $\mathrm{m}^3$, is in the tank when the water level in the tank is one metre?
2. We recall that $V$ is the total volume of the tank, $f$ is the function defined at the beginning of the exercise and $v$ the function defined in Part B. We consider the following algorithm:

   \begin{tabular}{l} Variables:

   Processing:

   &

   $a$ is a real number

   $b$ is a real number

   $a$ takes the value 2

   $b$ takes the value $2\mathrm{e}$

   While $v(b) - v(a) > 10^{-3}$ do:

   $c$ takes the value $(a + b)/2$

   If $v(c) < V/2$, then:

   $a$ takes the value $c$

   Otherwise

   $b$ takes the value $c$

   End If

   End While

   Display $f(c)$

   \hline \end{tabular} Interpret the result that this algorithm allows to display.

Let $a$ be a real number between 0 and 1. We denote by $f _ { a }$ the function defined on $\mathbf { R }$ by:
$$f _ { a } ( x ) = a \mathrm { e } ^ { a x } + a .$$
We denote by $I ( a )$ the integral of the function $f _ { a }$ between 0 and 1:
$$I ( a ) = \int _ { 0 } ^ { 1 } f _ { a } ( x ) \mathrm { d } x$$

1. In this question, we set $a = 0$. Determine $I ( 0 )$.
2. In this question, we set $a = 1$. We therefore study the function $f _ { 1 }$ defined on $\mathbf { R }$ by: $$f _ { 1 } ( x ) = \mathrm { e } ^ { x } + 1$$ a. Without detailed study, sketch the graph of the function $f _ { 1 }$ on your paper in an orthogonal coordinate system and show the number $I ( 1 )$. b. Calculate the exact value of $I ( 1 )$, then round to the nearest tenth.
3. Does there exist a value of $a$ for which $I ( a )$ equals 2? If so, give an interval of width $10 ^ { - 2 }$ containing this value.

The two parts of this exercise are independent.

Part A

Let the function $f$ defined on the set of real numbers by
$$f ( x ) = 2 \mathrm { e } ^ { x } - \mathrm { e } ^ { 2 x }$$
and $\mathscr { C }$ its representative curve in an orthonormal coordinate system. We admit that, for all $x$ belonging to $[ 0 ; \ln ( 2 ) ] , f ( x )$ is positive. Indicate whether the following proposition is true or false by justifying your answer.
Proposition A: The area of the region bounded by the lines with equations $x = 0$ and $x = \ln ( 2 )$, the $x$-axis and the curve $\mathscr { C }$ is equal to 1 unit of area.

Part B

Let $n$ be a strictly positive integer. Let the function $f _ { n }$ defined on the set of real numbers by
$$f _ { n } ( x ) = 2 n \mathrm { e } ^ { x } - \mathrm { e } ^ { 2 x }$$
and $\mathscr { C } _ { n }$ its representative curve in an orthonormal coordinate system. We admit that $f _ { n }$ is differentiable and that $\mathscr { C } _ { n }$ admits a horizontal tangent at a unique point $S _ { n }$. Indicate whether the following proposition is true or false by justifying your answer.
Proposition B: For all strictly positive integer $n$, the ordinate of the point $S _ { n }$ is $n ^ { 2 }$.

The purpose of the problem is the study of the integrals $I$ and $J$ defined by:
$$I = \int_{0}^{1} \frac{1}{1 + x} \mathrm{~d}x \quad \text{and} \quad J = \int_{0}^{1} \frac{1}{1 + x^{2}} \mathrm{~d}x$$
Part A: exact value of the integral $I$

1. Give a geometric interpretation of the integral $I$.
2. Calculate the exact value of $I$.

Part B: estimation of the value of $J$
Let $g$ be the function defined on the interval $[0; 1]$ by $g(x) = \frac{1}{1 + x^{2}}$. We denote $\mathscr{C}_{g}$ its representative curve in an orthonormal frame of the plane. We therefore have: $J = \int_{0}^{1} g(x) \mathrm{d}x$. The purpose of this part is to evaluate the integral $J$ using the probabilistic method described below. We choose at random a point $\mathrm{M}(x; y)$ by drawing independently its coordinates $x$ and $y$ at random according to the uniform distribution on $[0; 1]$. We admit that the probability $p$ that a point drawn in this manner is located below the curve $\mathscr{C}_{g}$ is equal to the integral $J$. In practice, we initialize a counter $c$ to 0, we fix a natural number $n$ and we repeat $n$ times the following process:

• we choose at random and independently two numbers $x$ and $y$, according to the uniform distribution on $[0; 1]$;
• if $\mathrm{M}(x; y)$ is below the curve $\mathscr{C}_{g}$ we increment the counter $c$ by 1.

We admit that $f = \frac{c}{n}$ is an approximate value of $J$. This is the principle of the so-called Monte-Carlo method.

The rate (as a percentage) of $\mathrm{CO}_2$ contained in a room after $t$ minutes of hood operation is modelled by the function $f$ defined for all real $t$ in the interval $[0;20]$ by: $$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$ Let $V_m$ denote the average rate (as a percentage) of $\mathrm{CO}_2$ present in the room during the first 11 minutes of operation of the extractor hood. a. Let $F$ be the function defined on the interval $[0;11]$ by: $$F(t) = (-1{,}6t - 3{,}6)\mathrm{e}^{-0{,}5t} + 0{,}03t.$$ Show that the function $F$ is an antiderivative of the function $f$ on the interval $[0;11]$. b. Deduce the average rate $V_m$, the average value of the function $f$ on the interval $[0;11]$. Round the result to the nearest thousandth, that is to $0.1\%$.

Consider the function $f$ defined on the interval $[0; +\infty[$ by $f(x) = k\mathrm{e}^{-kx}$ where $k$ is a strictly positive real number. We call $\mathcal{C}_f$ its graph in the orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. Consider point A on the curve $\mathcal{C}_f$ with x-coordinate 0 and point B on the curve $\mathcal{C}_f$ with x-coordinate 1. Point C has coordinates $(1; 0)$.

1. Determine an antiderivative of function $f$ on the interval $[0; +\infty[$.
2. Express, as a function of $k$, the area of triangle OCB and that of the region $\mathcal{D}$ bounded by the y-axis, the curve $\mathcal{C}_f$ and the segment $[OB]$.
3. Show that there exists a unique value of the strictly positive real number $k$ such that the area of region $\mathcal{D}$ is twice that of triangle OCB.

We consider, for every integer $n > 0$, the functions $f_{n}$ defined on the interval $[1; 5]$ by: $$f_{n}(x) = \frac{\ln x}{x^{n}}$$ For every integer $n > 0$, we denote by $\mathscr{C}_{n}$ the representative curve of the function $f_{n}$ in an orthogonal reference frame.

1. Show that, for every integer $n > 0$ and every real $x$ in the interval $[1; 5]$: $$f_{n}^{\prime}(x) = \frac{1 - n\ln(x)}{x^{n+1}}$$
2. For every integer $n > 0$, we admit that the function $f_{n}$ has a maximum on the interval $[1; 5]$. We denote by $A_{n}$ the point of the curve $\mathscr{C}_{n}$ having as ordinate this maximum. Show that all points $A_{n}$ belong to the same curve $\Gamma$ with equation $$y = \frac{1}{\mathrm{e}} \ln(x)$$
3. a. Show that, for every integer $n > 1$ and every real $x$ in the interval $[1; 5]$: $$0 \leqslant \frac{\ln(x)}{x^{n}} \leqslant \frac{\ln(5)}{x^{n}}$$ b. Show that for every integer $n > 1$: $$\int_{1}^{5} \frac{1}{x^{n}} \mathrm{~d}x = \frac{1}{n-1}\left(1 - \frac{1}{5^{n-1}}\right)$$ c. For every integer $n > 0$, we are interested in the area, expressed in square units, of the surface under the curve $f_{n}$, that is the area of the region of the plane bounded by the lines with equations $x = 1$, $x = 5$, $y = 0$ and the curve $\mathscr{C}_{n}$. Determine the limiting value of this area as $n$ tends to $+\infty$.

Below is the graphical representation $\mathscr { C } _ { g }$ in an orthogonal coordinate system of a function $g$ defined and continuous on $\mathbb { R }$. The curve $\mathscr { C } _ { g }$ is symmetric with respect to the $y$-axis and lies in the half-plane $y > 0$.
For all $t \in \mathbb { R }$ we define: $$G ( t ) = \int _ { 0 } ^ { t } g ( u ) \mathrm { d } u$$

Part A

The justifications of the answers to the following questions may be based on graphical considerations.

1. Is the function $G$ increasing on $[ 0 ; + \infty [$ ? Justify.
2. Justify graphically the inequality $G ( 1 ) \leqslant 0.9$.
3. Is the function $G$ positive on $\mathbb { R }$ ? Justify.

In the rest of the problem, the function $g$ is defined on $\mathbb { R }$ by $g ( u ) = \mathrm { e } ^ { - u ^ { 2 } }$.

Part B

1. Study of $g$ a. Determine the limits of the function $g$ at the boundaries of its domain. b. Calculate the derivative of $g$ and deduce the table of variations of $g$ on $\mathbb { R }$. c. Specify the maximum of $g$ on $\mathbb { R }$. Deduce that $g ( 1 ) \leqslant 1$.
2. We denote $E$ the set of points $M$ located between the curve $\mathscr { C } _ { g }$, the $x$-axis and the lines with equations $x = 0$ and $x = 1$. We call $I$ the area of this set. We recall that: $$I = G ( 1 ) = \int _ { 0 } ^ { 1 } g ( u ) \mathrm { d } u$$ We wish to estimate the area $I$ by the method called ``Monte-Carlo'' described below.
   • We choose a point $M ( x ; y )$ by randomly drawing its coordinates $x$ and $y$ independently according to the uniform distribution on the interval $[ 0 ; 1 ]$. It is admitted that the probability that the point $M$ belongs to the set $E$ is equal to $I$.
   • We repeat $n$ times the experiment of choosing a point $M$ at random. We count the number $c$ of points belonging to the set $E$ among the $n$ points obtained.
   • The frequency $f = \frac { c } { n }$ is an estimate of the value of $I$. a. The figure below illustrates the method presented for $n = 100$. Determine the value of $f$ corresponding to this graph. b. The execution of the algorithm below uses the Monte-Carlo method described previously to determine a value of the number $f$. Copy and complete this algorithm. $f , x$ and $y$ are real numbers, $n , c$ and $i$ are natural integers. ALEA is a function that randomly generates a number between 0 and 1. \begin{verbatim} $c \leftarrow 0$ For $i$ varying from 1 to $n$ do: $x \leftarrow$ ALEA $y \leftarrow$ ALEA If $y \leqslant \ldots$ then $c \leftarrow \ldots$ end If end For $f \leftarrow \ldots$ \end{verbatim} c. An execution of the algorithm for $n = 1000$ gives $f = 0.757$. Deduce a confidence interval, at the 95\% confidence level, for the exact value of $I$.

Part C

We recall that the function $g$ is defined on $\mathbb { R }$ by $g ( u ) = \mathrm { e } ^ { - u ^ { 2 } }$ and that the function $G$ is defined on $\mathbb { R }$ by: $$G ( t ) = \int _ { 0 } ^ { t } g ( u ) \mathrm { d } u$$ We propose to determine an upper bound for $G ( t )$ for $t \geqslant 1$.

1. A preliminary result. It is admitted that, for all real $u \geqslant 1$, we have $g ( u ) \leqslant \frac { 1 } { u ^ { 2 } }$. Deduce that, for all real $t \geqslant 1$, we have: $$\int _ { 1 } ^ { t } g ( u ) \mathrm { d } u \leqslant 1 - \frac { 1 } { t }$$
2. Show that, for all real $t \geqslant 1$, $$G ( t ) \leqslant 2 - \frac { 1 } { t }$$ What can we say about the possible limit of $G ( t )$ as $t$ tends to $+ \infty$ ?

Part C

In this part, we consider the function $h$ defined on $\mathbb { R }$ by $$h ( x ) = ( x - 1 ) \mathrm { e } ^ { - 2 x } + 1 .$$ We admit that the function $h$ is differentiable on $\mathbb { R }$. We place ourselves in an orthonormal coordinate system ( O ; I, J). We denote $\mathscr { C } _ { h }$ the representative curve of the function $h$ and $d$ the line with equation $y = x$. We admit that the curve $\mathscr { C } _ { h }$ is above the line $d$ on the interval $[ 0 ; 1 ]$. Let $\mathscr { D }$ be the region of the plane bounded by the curve $\mathscr { C } _ { h }$, the line $d$ and the vertical lines with equations $x = 0$ and $x = 1$. Let $\mathscr { A }$ be the area of $\mathscr { D }$ expressed in square units.

1. On ANNEX 1, shade the region $\mathscr { D }$ and justify that $$\mathscr { A } = \int _ { 0 } ^ { 1 } [ h ( x ) - x ] \mathrm { d } x$$
2. a. Prove that, for all real $x$, $$h ( x ) - x = ( 1 - x ) \left( 1 - \mathrm { e } ^ { - 2 x } \right) .$$ b. We admit that, for all real $x$, $\mathrm { e } ^ { - 2 x } \geqslant 1 - 2 x$. Prove that, for all real $x$ in the interval $[ 0 ; 1 ]$, $$h ( x ) - x \leqslant 2 x - 2 x ^ { 2 } .$$ c. Deduce that $\mathscr { A } \leqslant \frac { 1 } { 3 }$.
3. Let $H$ be the function defined on $[ 0 ; 1 ]$ by $$H ( x ) = \frac { 1 } { 4 } ( 1 - 2 x ) \mathrm { e } ^ { - 2 x } + x$$ We admit that the function $H$ is an antiderivative of the function $h$ on $[ 0 ; 1 ]$. Determine the exact value of $\mathscr { A }$.

Consider a function $h$ defined and continuous on $\mathbb{R}$ whose variation table is given below:

$x$	$-\infty$	1	$+\infty$
Variations of $h$		0
	$-\infty$

We denote $H$ the antiderivative of $h$ defined on $\mathbb{R}$ which vanishes at 0. It satisfies the property: a. $H$ is positive on $]-\infty; 0]$. b. $H$ is increasing on $]-\infty; 1]$. c. $H$ is negative on $]-\infty; 1]$. d. $H$ is increasing on $\mathbb{R}$.

Let $a$ be a strictly positive real number. We consider the function $f$ defined on the interval $]0; +\infty[$ by $$f(x) = a\ln(x)$$ We denote $\mathscr{C}$ its representative curve in an orthonormal coordinate system. Let $x_0$ be a real number strictly greater than 1.

1. Determine the abscissa of the point of intersection of the curve $\mathscr{C}$ and the x-axis.
2. Verify that the function $F$ defined by $F(x) = a[x\ln(x) - x]$ is a primitive of the function $f$ on the interval $]0; +\infty[$.
3. Deduce the area of the blue region as a function of $a$ and $x_0$.

We denote $T$ the tangent line to the curve $\mathscr{C}$ at the point $M$ with abscissa $x_0$. We call $A$ the point of intersection of the tangent line $T$ with the y-axis and $B$ the orthogonal projection of $M$ onto the y-axis.

1. Prove that the length AB is equal to a constant (that is, to a number that does not depend on $x_0$) which we will determine. The candidate will take care to make their approach explicit.

A integral $\int_0^2 (3x^2 + 2x)\,dx$ é igual a
(A) 8 (B) 10 (C) 12 (D) 14 (E) 16

QUESTION 141
A cylindrical tank has a base radius of 2 m and a height of 5 m. The volume of this tank, in cubic meters, is
(A) $10\pi$
(B) $15\pi$
(C) $20\pi$
(D) $25\pi$
(E) $30\pi$

QUESTION 159
The value of $\int_0^2 (3x^2 + 2x)\, dx$ is
(A) 8
(B) 10
(C) 12
(D) 14
(E) 16

A water tank in the form of a right rectangular parallelepiped, with 4 m in length, 3 m in width, and 2 m in height, needs to be sanitized. In this operation, the tank will need to be emptied in 20 minutes at most. The water will be removed with the help of a pump with constant flow rate, where flow rate is the volume of liquid that passes through the pump per unit of time.
The minimum flow rate, in liters per second, that this pump should have so that the tank is emptied in the stipulated time is
(A) 2.
(B) 3.
(C) 5.
(D) 12.
(E) 20.

A blood bank receives 450 mL of blood from each donor. After separating blood plasma from red blood cells, the former is stored in bags with 250 mL capacity. The blood bank rents refrigerators from a company for storage of plasma bags, according to its needs. Each refrigerator has a storage capacity of 50 bags. Over the course of a week, 100 people donated blood to that bank.
Assume that from every 60 mL of blood, 40 mL of plasma is extracted.
The minimum number of freezers that the bank needed to rent to store all the plasma bags from that week was
(A) 2.
(B) 3.
(C) 4.
(D) 6.
(E) 8.

The package of snacks preferred by a girl is sold in packages with different quantities. Each package is assigned a number of points in the promotion: ``When you total exactly 12 points in packages and add another R\$ 10.00 to the purchase value, you will win a stuffed animal''.
This snack is sold in three packages with the following masses, points, and prices:

\begin{tabular}{ c } Package

mass (g)

&

Package

points

& Price (R\$) \hline 50 & 2 & 2.00 \hline 100 & 4 & 3.60 \hline 200 & 6 & 6.40 \hline \end{tabular}
The smallest amount to be spent by this girl that allows her to take the stuffed animal in this promotion is
(A) R\$ 10.80.
(B) R\$ 12.80.
(C) R\$ 20.80.
(D) R\$ 22.00.
(E) R\$ 22.80.

The venue for Olympic swimming competitions uses the most advanced technology to provide swimmers with ideal conditions. This involves reducing the impact of undulation and currents caused by swimmers in their movement. To achieve this, the competition pool has a uniform depth of 3 m, which helps reduce the ``reflection'' of water (the movement against a surface and the return in the opposite direction, reaching the swimmers), in addition to the already traditional 50 m length and 25 m width. A club wishes to reform its pool of 50 m length, 20 m width and 2 m depth so that it has the same dimensions as Olympic pools.
After the reform, the capacity of this pool will exceed the capacity of the original pool by a value closest to
(A) $20\%$.
(B) $25\%$.
(C) $47\%$.
(D) $50\%$.
(E) $88\%$.

A couple is moving to a new home and needs to place a cubic object, with 80 cm edges, in a cardboard box, which cannot be disassembled. They have five boxes available, with different dimensions, as described:

• Box 1: $86 \mathrm{~cm} \times 86 \mathrm{~cm} \times 86 \mathrm{~cm}$
• Box 2: $75 \mathrm{~cm} \times 82 \mathrm{~cm} \times 90 \mathrm{~cm}$
• Box 3: $85 \mathrm{~cm} \times 82 \mathrm{~cm} \times 90 \mathrm{~cm}$
• Box 4: $82 \mathrm{~cm} \times 95 \mathrm{~cm} \times 82 \mathrm{~cm}$
• Box 5: $80 \mathrm{~cm} \times 95 \mathrm{~cm} \times 85 \mathrm{~cm}$

The couple needs to choose a box in which the object fits, so that the least free space remains in its interior.
The box chosen by the couple should be number
(A) 1.
(B) 2.
(C) 3.
(D) 4.
(E) 5.

The value of $\displaystyle\int_0^2 (3x^2 + 2x)\,dx$ is:
(A) 8
(B) 10
(C) 12
(D) 14
(E) 16

Prove that $\int _ { 1 } ^ { b } a ^ { \log _ { b } x } d x > \ln b$ where $a , b > 0 , b \neq 1$.

To find the volume of a cave, we fit $\mathrm { X } , \mathrm { Y }$ and Z axes such that the base of the cave is in the XY-plane and the vertical direction is parallel to the Z-axis. The base is the region in the XY-plane bounded by the parabola $y ^ { 2 } = 1 - x$ and the Y-axis. Each cross-section of the cave perpendicular to the X-axis is a square.
(a) Show how to write a definite integral that will calculate the volume of this cave.
(b) Evaluate this definite integral. Is it possible to evaluate it without using a formula for indefinite integrals?

Calculate the following integrals whenever possible. If a given integral does not exist, state so. Note that $[ x ]$ denotes the integer part of $x$, i.e., the unique integer $n$ such that $n \leq x < n + 1$. a) $\int _ { 1 } ^ { 4 } x ^ { 2 } d x$
Answer: $\_\_\_\_$ b) $\int _ { 1 } ^ { 3 } [ x ] ^ { 2 } d x$
Answer: $\_\_\_\_$ c) $\int _ { 1 } ^ { 2 } \left[ x ^ { 2 } \right] d x$
Answer: $\_\_\_\_$ d) $\int _ { - 1 } ^ { 1 } \frac { 1 } { x ^ { 2 } } d x$
Answer: $\_\_\_\_$

Consider the function $f(x) = x^{\cos(x) + \sin(x)}$ defined for $x \geq 0$.
(a) Prove that
$$0.4 \leq \int_{0}^{1} f(x)\, dx \leq 0.5$$
(b) Suppose the graph of $f(x)$ is being traced on a computer screen with the uniform speed of 1 cm per second (i.e., this is how fast the length of the curve is increasing). Show that at the moment the point corresponding to $x = 1$ is being drawn, the $x$ coordinate is increasing at the rate of
$$\frac{1}{\sqrt{2 + \sin(2)}} \text{ cm per second.}$$

Write the values of the following.
(a) $\int_{-3}^{3} \left| 3x^{2} - 3 \right| dx$.
(b) $f'(1)$ where $f(t) = \int_{0}^{t} \left| 3x^{2} - 3 \right| dx$.