A medication is administered to a patient. The amount, in milligrams, of the medication in the patient at time $t$ hours is modeled by a function $y = A(t)$ that satisfies the differential equation $\frac{dy}{dt} = \frac{12 - y}{3}$. At time $t = 0$ hours, there are 0 milligrams of the medication in the patient.
(a) A portion of the slope field for the differential equation $\frac{dy}{dt} = \frac{12 - y}{3}$ is given. Sketch the solution curve through the point $(0, 0)$.
(b) Using correct units, interpret the statement $\lim_{t \rightarrow \infty} A(t) = 12$ in the context of this problem.
(c) Use separation of variables to find $y = A(t)$, the particular solution to the differential equation $\frac{dy}{dt} = \frac{12 - y}{3}$ with initial condition $A(0) = 0$.
(d) A different procedure is used to administer the medication to a second patient. The amount, in milligrams, of the medication in the second patient at time $t$ hours is modeled by a function $y = B(t)$ that satisfies the differential equation $\frac{dy}{dt} = 3 - \frac{y}{t + 2}$. At time $t = 1$ hour, there are 2.5 milligrams of the medication in the second patient. Is the rate of change of the amount of medication in the second patient increasing or decreasing at time $t = 1$? Give a reason for your answer.

Consider the differential equation $\dfrac{dy}{dx} = \dfrac{1}{2}\sin\!\left(\dfrac{\pi}{2}x\right)\sqrt{y+7}$. Let $y = f(x)$ be the particular solution to the differential equation with the initial condition $f(1) = 2$. The function $f$ is defined for all real numbers.
(a) A portion of the slope field for the differential equation is given. Sketch the solution curve through the point $(1, 2)$.
(b) Write an equation for the line tangent to the solution curve in part (a) at the point $(1, 2)$. Use the equation to approximate $f(0.8)$.
(c) It is known that $f''(x) > 0$ for $-1 \leq x \leq 1$. Is the approximation found in part (b) an overestimate or an underestimate for $f(0.8)$? Give a reason for your answer.
(d) Use separation of variables to find $y = f(x)$, the particular solution to the differential equation $\dfrac{dy}{dx} = \dfrac{1}{2}\sin\!\left(\dfrac{\pi}{2}x\right)\sqrt{y+7}$ with the initial condition $f(1) = 2$.

A bottle of milk is taken out of a refrigerator and placed in a pan of hot water to be warmed. The increasing function $M$ models the temperature of the milk at time $t$, where $M(t)$ is measured in degrees Celsius (${}^{\circ}\mathrm{C}$) and $t$ is the number of minutes since the bottle was placed in the pan. $M$ satisfies the differential equation $\frac{dM}{dt} = \frac{1}{4}(40 - M)$. At time $t = 0$, the temperature of the milk is $5^{\circ}\mathrm{C}$. It can be shown that $M(t) < 40$ for all values of $t$.
(a) A slope field for the differential equation $\frac{dM}{dt} = \frac{1}{4}(40 - M)$ is shown. Sketch the solution curve through the point $(0, 5)$.
(b) Use the line tangent to the graph of $M$ at $t = 0$ to approximate $M(2)$, the temperature of the milk at time $t = 2$ minutes.
(c) Write an expression for $\frac{d^{2}M}{dt^{2}}$ in terms of $M$. Use $\frac{d^{2}M}{dt^{2}}$ to determine whether the approximation from part (b) is an underestimate or an overestimate for the actual value of $M(2)$. Give a reason for your answer.
(d) Use separation of variables to find an expression for $M(t)$, the particular solution to the differential equation $\frac{dM}{dt} = \frac{1}{4}(40 - M)$ with initial condition $M(0) = 5$.

Consider the differential equation given by $\dfrac{dy}{dx} = \dfrac{xy}{2}$.
(a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.
(b) Let $y = f(x)$ be the particular solution to the given differential equation with the initial condition $f(0) = 3$. Use Euler's method starting at $x = 0$, with a step size of 0.1, to approximate $f(0.2)$. Show the work that leads to your answer.
(c) Find the particular solution $y = f(x)$ to the given differential equation with the initial condition $f(0) = 3$. Use your solution to find $f(0.2)$.

6. Consider the differential equation given by $\frac { d y } { d x } = x ( y - 1 ) ^ { 2 }$.
(a) On the axes provided, sketch a slope field for the given differential equation at the eleven points indicated. (Note: Use the axes provided in the pink test booklet.) [Figure]
(b) Use the slope field for the given differential equation to explain why a solution could not have the graph shown below. [Figure]
(c) Find the particular solution $y = f ( x )$ to the given differential equation with the initial condition $f ( 0 ) = - 1$.
(d) Find the range of the solution found in part (c).

END OF EXAMINATION

Copyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved. AP is a registered trademark of the College Entrance Examination Board.

Consider the differential equation $\dfrac{dy}{dx} = \dfrac{3 - x}{y}$.
(a) Let $y = f(x)$ be the particular solution to the given differential equation for $1 < x < 5$ such that the line $y = -2$ is tangent to the graph of $f$. Find the $x$-coordinate of the point of tangency, and determine whether $f$ has a local maximum, local minimum, or neither at this point. Justify your answer.
(b) Let $y = g(x)$ be the particular solution to the given differential equation for $-2 < x < 8$, with the initial condition $g(6) = -4$. Find $y = g(x)$.

Consider the differential equation $\frac { d y } { d x } = 2 x - y$.
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point $( 0, 1 )$.
(b) The solution curve that passes through the point $( 0, 1 )$ has a local minimum at $x = \ln \left( \frac { 3 } { 2 } \right)$. What is the $y$-coordinate of this local minimum?
(c) Let $y = f ( x )$ be the particular solution to the given differential equation with the initial condition $f ( 0 ) = 1$. Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f ( - 0.4 )$. Show the work that leads to your answer.
(d) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. Determine whether the approximation found in part (c) is less than or greater than $f ( - 0.4 )$. Explain your reasoning.

Consider the differential equation $\frac{dy}{dx} = 5x^{2} - \frac{6}{y-2}$ for $y \neq 2$. Let $y = f(x)$ be the particular solution to this differential equation with the initial condition $f(-1) = -4$.
(a) Evaluate $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$ at $(-1, -4)$.
(b) Is it possible for the $x$-axis to be tangent to the graph of $f$ at some point? Explain why or why not.
(c) Find the second-degree Taylor polynomial for $f$ about $x = -1$.
(d) Use Euler's method, starting at $x = -1$ with two steps of equal size, to approximate $f(0)$. Show the work that leads to your answer.

The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time $t = 0$, when the bird is first weighed, its weight is 20 grams. If $B ( t )$ is the weight of the bird, in grams, at time $t$ days after it is first weighed, then
$$\frac { d B } { d t } = \frac { 1 } { 5 } ( 100 - B ) .$$
Let $y = B ( t )$ be the solution to the differential equation above with initial condition $B ( 0 ) = 20$.
(a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain your reasoning.
(b) Find $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ in terms of $B$. Use $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ to explain why the graph of $B$ cannot resemble the following graph.
(c) Use separation of variables to find $y = B ( t )$, the particular solution to the differential equation with initial condition $B ( 0 ) = 20$.

The points $( - 1 , - 1 )$ and $( 1 , - 5 )$ are on the graph of a function $y = f ( x )$ that satisfies the differential equation $\frac { d y } { d x } = x ^ { 2 } + y$. Which of the following must be true?
(A) $( 1 , - 5 )$ is a local maximum of $f$.
(B) $( 1 , - 5 )$ is a point of inflection of the graph of $f$.
(C) $( - 1 , - 1 )$ is a local maximum of $f$.
(D) $( - 1 , - 1 )$ is a local minimum of $f$.
(E) $( - 1 , - 1 )$ is a point of inflection of the graph of $f$.

Let $k$ be a positive constant. Which of the following is a logistic differential equation?
(A) $\frac { d y } { d t } = k t$
(B) $\frac { d y } { d t } = k y$
(C) $\frac { d y } { d t } = k t ( 1 - t )$
(D) $\frac { d y } { d t } = k y ( 1 - t )$
(E) $\frac { d y } { d t } = k y ( 1 - y )$

If $P ( t )$ is the size of a population at time $t$, which of the following differential equations describes linear growth in the size of the population?
(A) $\frac { d P } { d t } = 200$
(B) $\frac { d P } { d t } = 200 t$
(C) $\frac { d P } { d t } = 100 t ^ { 2 }$
(D) $\frac { d P } { d t } = 200 P$
(E) $\frac { d P } { d t } = 100 P ^ { 2 }$

Consider the differential equation $\frac { d y } { d x } = 2 x - y$.
(a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated.
(b) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. Determine the concavity of all solution curves for the given differential equation in Quadrant II. Give a reason for your answer.
(c) Let $y = f ( x )$ be the particular solution to the differential equation with the initial condition $f ( 2 ) = 3$. Does $f$ have a relative minimum, a relative maximum, or neither at $x = 2$? Justify your answer.
(d) Find the values of the constants $m$ and $b$ for which $y = m x + b$ is a solution to the differential equation.

A bottle of milk is taken out of a refrigerator and placed in a pan of hot water to be warmed. The increasing function $M$ models the temperature of the milk at time $t$, where $M(t)$ is measured in degrees Celsius (${}^{\circ}\mathrm{C}$) and $t$ is the number of minutes since the bottle was placed in the pan. $M$ satisfies the differential equation $\frac{dM}{dt} = \frac{1}{4}(40 - M)$. At time $t = 0$, the temperature of the milk is $5^{\circ}\mathrm{C}$. It can be shown that $M(t) < 40$ for all values of $t$.
(a) A slope field for the differential equation $\frac{dM}{dt} = \frac{1}{4}(40 - M)$ is shown. Sketch the solution curve through the point $(0, 5)$.
(b) Use the line tangent to the graph of $M$ at $t = 0$ to approximate $M(2)$, the temperature of the milk at time $t = 2$ minutes.
(c) Write an expression for $\frac{d^{2}M}{dt^{2}}$ in terms of $M$. Use $\frac{d^{2}M}{dt^{2}}$ to determine whether the approximation from part (b) is an underestimate or an overestimate for the actual value of $M(2)$. Give a reason for your answer.
(d) Use separation of variables to find an expression for $M(t)$, the particular solution to the differential equation $\frac{dM}{dt} = \frac{1}{4}(40 - M)$ with initial condition $M(0) = 5$.

Let $a$ be a fixed non-zero real number. The purpose of this exercise is to study the sequence $(u_n)$ defined by: $$u_0 = a \quad \text{and, for all } n \text{ in } \mathbb{N}, \quad u_{n+1} = \mathrm{e}^{2u_n} - \mathrm{e}^{u_n}.$$ Note that this equality can also be written: $u_{n+1} = e^{u_n}(\mathrm{e}^{u_n} - 1)$.

1. Let $g$ be the function defined for all real $x$ by $$g(x) = \mathrm{e}^{2x} - \mathrm{e}^x - x.$$ a) Calculate $g'(x)$ and prove that, for all real $x$: $g'(x) = (\mathrm{e}^x - 1)(2\mathrm{e}^x + 1)$. b) Determine the variations of the function $g$ and give the value of its minimum. c) By noting that $u_{n+1} - u_n = g(u_n)$, study the direction of variation of the sequence $(u_n)$.
   
2. In this question, we assume that $a \leqslant 0$. a) Prove by induction that, for all natural integer $n$, $u_n \leqslant 0$. b) Deduce from the previous questions that the sequence $(u_n)$ is convergent. c) In the case where $a$ equals 0, give the limit of the sequence $(u_n)$.
   
3. In this question, we assume that $a > 0$.
   Since the sequence $(u_n)$ is increasing, question 1 allows us to assert that, for all natural integer $n$, $u_n \geqslant a$. a) Prove that, for all natural integer $n$, we have: $u_{n+1} - u_n \geqslant g(a)$. b) Prove by induction that, for all natural integer $n$, we have: $$u_n \geqslant a + n \times g(a).$$ c) Determine the limit of the sequence $(u_n)$.
   
4. In this question, we take $a = 0.02$.
   The following algorithm is intended to determine the smallest integer $n$ such that $u_n > M$, where $M$ denotes a positive real number. This algorithm is incomplete.
   

   Variables	$n$ is an integer, $u$ and $M$ are two real numbers
   Initialization	\begin{tabular}{l} $u$ takes the value 0.02
   $n$ takes the value 0
   Enter the value of $M$

   \hline Processing & While $\cdots$ & $\ldots$ & $\ldots$ End while & \end{tabular}
   a) On your paper, rewrite the ``Processing'' part by completing it. b) Using a calculator, determine the value that this algorithm will display if $M = 60$.

The director of a marine reserve counted 3000 cetaceans in this reserve on June 1st, 2017. He is concerned because he knows that the classification of the area as a ``marine reserve'' will not be renewed if the number of cetaceans in this reserve falls below 2000.
A study allows him to develop a model according to which, each year:

• between June 1st and October 31st, 80 cetaceans arrive in the marine reserve;
• between November 1st and May 31st, the reserve experiences a decline of $5\%$ of its population compared to that of October 31st of the preceding year.

The evolution of the number of cetaceans is modelled by a sequence $(u_n)$. According to this model, for any natural number $n$, $u_n$ denotes the number of cetaceans on June 1st of the year $2017 + n$. We have $u_0 = 3000$.

1. Justify that $u_1 = 2926$.
2. Justify that, for any natural number $n$, $u_{n+1} = 0.95u_n + 76$.
3. Using a spreadsheet, the first 8 terms of the sequence $(u_n)$ were calculated. The director configured the cell format so that only numbers rounded to the nearest integer are displayed.
   

   	A	B	C	D	E	F	G	H	I
   1	$n$	0	1	2	3	4	5	6	7
   2	$u_n$	3000	2926	2856	2789	2725	2665	2608	2553

   
   What formula can be entered in cell C2 to obtain, by copying to the right, the terms of the sequence $(u_n)$?
4. a. Prove that, for any natural number $n$, $u_n \geqslant 1520$. b. Prove that the sequence $(u_n)$ is decreasing. c. Justify that the sequence $(u_n)$ is convergent. We will not seek to find the value of the limit here.
5. We denote by $(v_n)$ the sequence defined by, for any natural number $n$, $v_n = u_n - 1520$. a. Prove that the sequence $(v_n)$ is a geometric sequence with ratio 0.95 and specify its first term. b. Deduce that, for any natural number $n$, $u_n = 1480 \times 0.95^n + 1520$. c. Determine the limit of the sequence $(u_n)$.
6. Copy and complete the following algorithm to determine the year from which the number of cetaceans present in the marine reserve will be less than 2000. $$\begin{array}{|l|} \hline n \leftarrow 0 \\ u \leftarrow 3000 \\ \text{While } \ldots \\ \quad n \leftarrow \ldots \\ u \leftarrow \ldots \\ \text{End While} \end{array}$$

Exercise 4 — For candidates who have NOT followed the speciality course We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \frac{1}{2}x^{2} - x + \frac{3}{2}$$ Let $a$ be a positive real number. We define the sequence $(u_{n})$ by $u_{0} = a$ and, for every natural number $n$: $u_{n+1} = f(u_{n})$. The purpose of this exercise is to study the behaviour of the sequence $(u_{n})$ as $n$ tends to $+\infty$, depending on different values of its first term $u_{0} = a$.

1. Using a calculator, conjecture the behaviour of the sequence $(u_{n})$ as $n$ tends to $+\infty$, for $a = 2.9$ then for $a = 3.1$.
2. In this question, we assume that the sequence $(u_{n})$ converges to a real number $\ell$. a. By noting that $u_{n+1} = \frac{1}{2}u_{n}^{2} - u_{n} + \frac{3}{2}$, show that $\ell = \frac{1}{2}\ell^{2} - \ell + \frac{3}{2}$. b. Show that the possible values of $\ell$ are 1 and 3.
3. In this question, we take $a = 2.9$. a. Show that $f$ is increasing on the interval $[1; +\infty[$. b. Show by induction that, for every natural number $n$, we have: $1 \leqslant u_{n+1} \leqslant u_{n}$. c. Show that $(u_{n})$ converges and determine its limit.
4. In this question, we take $a = 3.1$ and we admit that the sequence $(u_{n})$ is increasing. a. Using the previous questions show that the sequence $(u_{n})$ is not bounded above. b. Deduce the behaviour of the sequence $(u_{n})$ as $n$ tends to $+\infty$. c. The following algorithm calculates the smallest rank $p$ for which $u_{p} > 10^{6}$. Copy and complete this algorithm. $P$ is a natural number and $U$ is a real number. \begin{verbatim} P <- 0 U..... Tant que... P ...... U ...... Fin Tant que \end{verbatim}

Newton's law of cooling states that the rate of change of the temperature of a body is proportional to the difference between the temperature of this body and that of the surrounding environment.
A cup of coffee is served at an initial temperature of $80^{\circ}\mathrm{C}$ in an environment whose temperature, expressed in degrees Celsius, assumed to be constant, is denoted $M$.
In this part, for any natural number $n$, we denote $T_n$ the temperature of the coffee at instant $n$, with $T_n$ expressed in degrees Celsius and $n$ in minutes. Thus $T_0 = 80$.
We model Newton's law between any two consecutive minutes $n$ and $n+1$ by the equality: $$T_{n+1} - T_n = k(T_n - M)$$ where $k$ is a real constant.
In the rest of part A, we choose $M = 10$ and $k = -0{,}2$. Thus, for any natural number $n$, we have: $T_{n+1} - T_n = -0{,}2(T_n - 10)$.

1. Based on the context, can we conjecture the direction of variation of the sequence $(T_n)$?
2. Show that for any natural number $n$: $T_{n+1} = 0{,}8T_n + 2$.
3. We set, for any natural number $n$: $u_n = T_n - 10$. a. Show that $(u_n)$ is a geometric sequence. Specify its common ratio and its first term $u_0$. b. Show that, for any natural number $n$, we have: $T_n = 70 \times 0{,}8^n + 10$. c. Determine the limit of the sequence $(T_n)$.
4. Consider the following algorithm: \begin{verbatim} While $T \geqslant 40$ $T \leftarrow 0,8T + 2$ $n \leftarrow n + 1$ End While \end{verbatim} a. Initially, we assign the value 80 to the variable $T$ and the value 0 to the variable $n$. What numerical value does the variable $n$ contain at the end of the algorithm's execution? b. Interpret this value in the context of the exercise.

EXERCISE - B

Main topics covered: Function study, exponential function; Differential equations

Part I

Let us consider the differential equation $$y' = -0.4y + 0.4$$ where $y$ denotes a function of the variable $t$, defined and differentiable on $[0; +\infty[$.

1. a. Determine a particular constant solution of this differential equation. b. Deduce the set of solutions of this differential equation. c. Determine the function $g$, solution of this differential equation, which satisfies $g(0) = 10$.

Part II

Let $p$ be the function defined and differentiable on the interval $[0; +\infty[$ by $$p(t) = \frac{1}{g(t)} = \frac{1}{1 + 9\mathrm{e}^{-0.4t}}$$

1. Determine the limit of $p$ at $+\infty$.
2. Show that $p'(t) = \frac{3.6\mathrm{e}^{-0.4t}}{\left(1 + 9\mathrm{e}^{-0.4t}\right)^{2}}$ for all $t \in [0; +\infty[$.
3. a. Show that the equation $p(t) = \frac{1}{2}$ has a unique solution $\alpha$ on $[0; +\infty[$. b. Determine an approximate value of $\alpha$ to $10^{-1}$ near using a calculator.

Part III

1. $p$ denotes the function from Part II. Verify that $p$ is a solution of the differential equation $y' = 0.4y(1 - y)$ with the initial condition $y(0) = \frac{1}{10}$ where $y$ denotes a function defined and differentiable on $[0; +\infty[$.
2. In a developing country, in the year 2020, 10\% of schools have access to the internet.
   A voluntary equipment policy is implemented and we are interested in the evolution of the proportion of schools with access to the internet. We denote $t$ the time elapsed, expressed in years, since the year 2020. The proportion of schools with access to the internet at time $t$ is modelled by $p(t)$. Interpret in this context the limit from question II 1 then the approximate value of $\alpha$ from question II 3. b. as well as the value $p(0)$.

EXERCISE B - Differential equation
Part A: Determination of a function $f$ and resolution of a differential equation
Consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \mathrm{e}^x + ax + b\mathrm{e}^{-x}$$ where $a$ and $b$ are real numbers that we propose to determine in this part. In the plane with a coordinate system with origin O, the curve $\mathscr{C}$, representing the function $f$, and the tangent line $(T)$ to the curve $\mathscr{C}$ at the point with abscissa 0 are shown.

1. By reading the graph, give the values of $f(0)$ and $f'(0)$.
2. Using the expression of the function $f$, express $f(0)$ as a function of $b$ and deduce the value of $b$.
3. We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative function. a. Give, for every real $x$, the expression of $f'(x)$. b. Express $f'(0)$ as a function of $a$. c. Using the previous questions, determine $a$, then deduce the expression of $f(x)$.
4. Consider the differential equation: $$( E ) : \quad y' + y = 2\mathrm{e}^x - x - 1$$ a. Verify that the function $g$ defined on $\mathbb{R}$ by: $$g(x) = \mathrm{e}^x - x + 2\mathrm{e}^{-x}$$ is a solution of equation $(E)$. b. Solve the differential equation $y' + y = 0$. c. Deduce all solutions of equation $(E)$.

Part B: Study of the function $g$ on $[1;+\infty[$

1. Verify that for every real $x$, we have: $$\mathrm{e}^{2x} - \mathrm{e}^x - 2 = (\mathrm{e}^x - 2)(\mathrm{e}^x + 1)$$
2. Deduce a factored expression of $g'(x)$, for every real $x$.
3. We will admit that, for all $x \in [1;+\infty[$, $\mathrm{e}^x - 2 > 0$. Study the direction of variation of the function $g$ on $[1;+\infty[$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.
Statement 1: Let (E) be the differential equation: $y ^ { \prime } - 2 y = - 6 x + 1$. The function $f$ defined on $\mathbb { R }$ by: $f ( x ) = \mathrm { e } ^ { 2 x } - 6 x + 1$ is a solution of the differential equation (E).
Statement 2: Consider the sequence $\left( u _ { n } \right)$ defined on $\mathbb { N }$ by $$u _ { n } = 1 + \frac { 3 } { 4 } + \left( \frac { 3 } { 4 } \right) ^ { 2 } + \cdots + \left( \frac { 3 } { 4 } \right) ^ { n }$$ The sequence $(u _ { n })$ has limit $+ \infty$.
Statement 3: Consider the sequence $(u _ { n })$ defined in Statement 2. The instruction suite(50) below, written in Python language, returns $u _ { 50 }$. \begin{verbatim} def suite(k): S=0 for i in range(k): S=S+(3/4)**k return S \end{verbatim}
Statement 4: Let $a$ be a real number and $f$ the function defined on $] 0 ; + \infty [$ by: $$f ( x ) = a \ln ( x ) - 2 x$$ Let $C$ be the representative curve of the function $f$ in a coordinate system $(O ; \vec { \imath } , \vec { \jmath })$. There exists a value of $a$ for which the tangent to $C$ at the point with abscissa 1 is parallel to the horizontal axis.

Alain owns a swimming pool that contains $50\mathrm{~m}^3$ of water. Recall that $1\mathrm{~m}^3 = 1000\mathrm{~L}$. To disinfect the water, he must add chlorine. The chlorine level in the water, expressed in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, is defined as the mass of chlorine per unit volume of water. Pool specialists recommend a chlorine level between 1 and $3\mathrm{~mg}\cdot\mathrm{L}^{-1}$. Under the action of the ambient environment, particularly ultraviolet rays, chlorine decomposes and gradually disappears. On Wednesday, June 19, he measures a chlorine level of $0.70\mathrm{~mg}\cdot\mathrm{L}^{-1}$.
Part A: study of a discrete model.
To maintain the chlorine level in his pool, Alain decides, starting from Thursday, June 20, to add 15 g of chlorine each day. It is admitted that this chlorine mixes uniformly in the pool water.

1. Justify that this addition of chlorine increases the level by $0.3\mathrm{~mg}\cdot\mathrm{L}^{-1}$.
2. For any natural number $n$, we denote by $v_n$ the chlorine level, in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, obtained with this new protocol $n$ days after Wednesday, June 19. Thus $v_0 = 0.7$. It is admitted that for any natural number $n$, $$v_{n+1} = 0.92 v_n + 0.3.$$ a. Show by induction that for any natural number $n$, $\quad v_n \leqslant v_{n+1} \leqslant 4$. b. Show that the sequence $(v_n)$ is convergent and calculate its limit.
3. In the long term, will the chlorine level comply with the pool specialists' recommendation? Justify your answer.
4. Reproduce and complete the algorithm below written in Python language so that the function \texttt{alerte\_chlore} returns, when it exists, the smallest integer $n$ such that $v_n > s$. \begin{verbatim} def alerte_chlore(s) : n = 0 u=0.7 while...: n = ... u=... return n \end{verbatim}
5. What value is obtained by entering the instruction \texttt{alerte\_chlore(3)}? Interpret this result in the context of the exercise.

Part B: study of a continuous model.
Alain decides to call on a specialized engineering firm. This firm uses a continuous model to describe the chlorine level in the pool. In this model, for a duration $x$ (in days elapsed from Wednesday, June 19), $f(x)$ represents the chlorine level, in $\mathrm{mg}\cdot\mathrm{L}^{-1}$, in the pool. It is admitted that the function $f$ is a solution of the differential equation $(E): y' = -0.08y + \frac{q}{50}$, where $q$ is the quantity of chlorine, in grams, added to the pool each day.

1. Justify that the function $f$ is of the form $f(x) = C\mathrm{e}^{-0.08x} + \frac{q}{4}$ where $C$ is a real constant.
2. a. Express as a function of $q$ the limit of $f$ at $+\infty$. b. Recall that the chlorine level observed on Wednesday, June 19 is equal to $0.7\mathrm{~mg}\cdot\mathrm{L}^{-1}$. It is desired that the chlorine level stabilizes in the long term around $2\mathrm{~mg}\cdot\mathrm{L}^{-1}$. Determine the values of $C$ and $q$ so that these two conditions are satisfied.

A company manufactures plastic objects by injecting molten material at $210 ^ { \circ } \mathrm { C }$ into a mold. We seek to model the cooling of the material using a function $f$ giving the temperature of the injected material as a function of time $t$. Time is expressed in seconds and temperature is expressed in degrees Celsius. We assume that the function $f$ sought is a solution to a differential equation of the following form, where $m$ is a real constant that we seek to determine:
$$( E ) : \quad y ^ { \prime } + 0,02 y = m$$
Part A

1. Justify the following output from a computer algebra system:

Input:	SolveDifferentialEquation $\left( y ^ { \prime } + 0,02 y = m \right)$
Output:	$\rightarrow y = k * \exp ( - 0.02 * t ) + 50 * m$

1. The workshop temperature is $30 ^ { \circ } \mathrm { C }$. We assume that the temperature $f ( t )$ tends toward $30 ^ { \circ } \mathrm { C }$ as $t$ tends toward infinity. Prove that $m = 0,6$.
2. Determine the expression of the function $f$ sought, taking into account the initial condition $f ( 0 ) = 210$.

Part B We assume here that the temperature (expressed in degrees Celsius) of the injected material as a function of time (expressed in seconds) is given by the function whose expression and graphical representation are given below:
$$f ( t ) = 180 \mathrm { e } ^ { - 0,02 t } + 30$$

1. The object can be removed from the mold when its temperature becomes less than $50 ^ { \circ } \mathrm { C }$. a. By graphical reading, give an approximate value of the number $T$ of seconds to wait before removing the object from the mold. b. Determine by calculation the exact value of this time $T$.
2. Using an integral, calculate the average value of the temperature over the first 100 seconds.

This exercise is a multiple choice questionnaire (MCQ) comprising five questions. The five questions are independent. For each question, only one of the four answers is correct.

1. The solution $f$ of the differential equation $y' = -3y + 7$ such that $f(0) = 1$ is the function defined on $\mathbb{R}$ by:
   A. $f(x) = \mathrm{e}^{-3x}$
   B. $f(x) = -\frac{4}{3}\mathrm{e}^{-3x} + \frac{7}{3}$
   C. $f(x) = \mathrm{e}^{-3x} + \frac{7}{3}$
   D. $f(x) = -\frac{10}{3}\mathrm{e}^{-3x} - \frac{7}{3}$
   
2. The curve of a function $f$ defined on $[0; +\infty[$ is given below.
   A bound for the integral $I = \int_{1}^{5} f(x)\,\mathrm{d}x$ is:
   A. $0 \leqslant I \leqslant 4$
   B. $1 \leqslant I \leqslant 5$
   C. $5 \leqslant I \leqslant 10$
   D. $10 \leqslant I \leqslant 15$
   
3. Consider the function $g$ defined on $\mathbb{R}$ by $g(x) = x^2 \ln\left(x^2 + 4\right)$.
   Then $\int_{0}^{2} g'(x)\,\mathrm{d}x$ equals, to $10^{-1}$ near:
   A. 4.9
   B. 8.3
   C. 1.7
   D. 7.5
   
4. A teacher teaches the mathematics specialization in a class of 31 final year students. She wants to form a group of 5 students. In how many different ways can she form such a group of 5 students?
   A. $31^5$
   B. $31 \times 30 \times 29 \times 28 \times 27$
   C. $31 + 30 + 29 + 28 + 27$
   D. $\binom{31}{5}$
   
5. The teacher is now interested in the other specialization of the 31 students in her group:
   • 10 students chose the physics-chemistry specialization;
   • 20 students chose the SES specialization;
   • 1 student chose the Spanish LLCE specialization.
   She wants to form a group of 5 students containing exactly 3 students who chose the SES specialization. In how many different ways can she form such a group?
   A. $\binom{20}{3} \times \binom{11}{2}$
   B. $\binom{20}{3} + \binom{11}{2}$
   C. $\binom{20}{3}$
   D. $20^3 \times 11^2$

Consider the differential equation $$\left( E_0 \right) : \quad y^{\prime} = y$$ where $y$ is a differentiable function of the real variable $x$.

1. Prove that the unique constant function solution of the differential equation $\left( E_0 \right)$ is the zero function.
2. Determine all solutions of the differential equation $(E_0)$.

Consider the differential equation $$(E) : \quad y^{\prime} = y - \cos(x) - 3\sin(x)$$ where $y$ is a differentiable function of the real variable $x$.

1. The function $h$ is defined on $\mathbb{R}$ by $h(x) = 2\cos(x) + \sin(x)$. It is admitted that it is differentiable on $\mathbb{R}$. Prove that the function $h$ is a solution of the differential equation $(E)$.
2. Consider a function $f$ defined and differentiable on $\mathbb{R}$. Prove that: ``$f$ is a solution of $(E)$'' is equivalent to ``$f - h$ is a solution of $\left(E_0\right)$''.
3. Deduce all solutions of the differential equation $(E)$.
4. Determine the unique solution $g$ of the differential equation $(E)$ such that $g(0) = 0$.
5. Calculate: $$\int_{0}^{\frac{\pi}{2}} \left[ -2\mathrm{e}^{x} + \sin(x) + 2\cos(x) \right] \mathrm{d}x$$