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 $M(0) = 5$.

The depth of seawater at a location can be modeled by the function $H$ that satisfies the differential equation $\frac{dH}{dt} = \frac{1}{2}(H - 1)\cos\left(\frac{t}{2}\right)$, where $H(t)$ is measured in feet and $t$ is measured in hours after noon $(t = 0)$. It is known that $H(0) = 4$.
(a) A portion of the slope field for the differential equation is provided. Sketch the solution curve, $y = H(t)$, through the point $(0, 4)$.
(b) For $0 < t < 5$, it can be shown that $H(t) > 1$. Find the value of $t$, for $0 < t < 5$, at which $H$ has a critical point. Determine whether the critical point corresponds to a relative minimum, a relative maximum, or neither a relative minimum nor a relative maximum of the depth of seawater at the location. Justify your answer.
(c) Use separation of variables to find $y = H(t)$, the particular solution to the differential equation $\frac{dH}{dt} = \frac{1}{2}(H - 1)\cos\left(\frac{t}{2}\right)$ with initial condition $H(0) = 4$.

Let $y = f ( x )$ be the particular solution to the differential equation $\frac { d y } { d x } = ( 3 - x ) y ^ { 2 }$ with initial condition $f ( 1 ) = - 1$.
A. Find $f ^ { \prime \prime } ( 1 )$, the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 1 , - 1 )$. Show the work that leads to your answer.
B. Write the second-degree Taylor polynomial for $f$ about $x = 1$.
C. The second-degree Taylor polynomial for $f$ about $x = 1$ is used to approximate $f ( 1.1 )$. Given that $\left| f ^ { \prime \prime \prime } ( x ) \right| \leq 60$ for all $x$ in the interval $1 \leq x \leq 1.1$, use the Lagrange error bound to show that this approximation differs from $f ( 1.1 )$ by at most 0.01.
D. Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 1.4 )$. Show the work that leads to your answer.

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 non-negative real $t$, we denote $\theta(t)$ the temperature of the coffee at instant $t$, with $\theta(t)$ expressed in degrees Celsius and $t$ in minutes. Thus $\theta(0) = 80$.
In this model, more precise than that of part A, we assume that $\theta$ is a function differentiable on the interval $[0; +\infty[$ and that, for any real $t$ in this interval, Newton's law is modeled by the equality: $$\theta'(t) = -0{,}2(\theta(t) - M).$$

1. In this question, we choose $M = 0$. We then seek a function $\theta$ differentiable on the interval $[0; +\infty[$ satisfying $\theta(0) = 80$ and, for any real $t$ in this interval: $\theta'(t) = -0{,}2\theta(t)$. a. If $\theta$ is such a function, we set for any $t$ in the interval $[0; +\infty[$, $f(t) = \frac{\theta(t)}{\mathrm{e}^{-0{,}2t}}$. Show that the function $f$ is differentiable on $[0; +\infty[$ and that, for any real $t$ in this interval, $f'(t) = 0$. b. Keeping the hypothesis from a., calculate $f(0)$. Deduce, for any $t$ in the interval $[0; +\infty[$, an expression for $f(t)$, then for $\theta(t)$. c. Verify that the function $\theta$ found in b. is a solution to the problem.
2. In this question, we choose $M = 10$. We admit that there exists a unique function $g$ differentiable on $[0; +\infty[$, modeling the temperature of the coffee at any non-negative instant $t$, and that, for any $t$ in the interval $[0; +\infty[$: $$g(t) = 10 + 70\mathrm{e}^{-0{,}2t},$$ where $t$ is expressed in minutes and $g(t)$ in degrees Celsius.
   A person likes to drink their coffee at $40^{\circ}\mathrm{C}$. Show that there exists a unique real $t_0$ in $[0; +\infty[$ such that $g(t_0) = 40$. Give the value of $t_0$ rounded to the nearest second.

We consider the function $f$ defined on $[0; +\infty[$ by $$f(x) = \ln\left(\frac{3x+1}{x+1}\right).$$ We admit that the function $f$ is differentiable on $[0; +\infty[$ and we denote by $f'$ its derivative function. We denote by $\mathscr{C}_f$ the representative curve of the function $f$ in an orthogonal coordinate system.

Part A

1. Determine $\lim_{x \rightarrow +\infty} f(x)$ and give a graphical interpretation.
2. a. Prove that, for every non-negative real number $x$, $$f'(x) = \frac{2}{(x+1)(3x+1)}$$ b. Deduce that the function $f$ is strictly increasing on $[0; +\infty[$.

Part B

Let $(u_n)$ be the sequence defined by $$u_0 = 3 \text{ and, for every natural number } n,\ u_{n+1} = f(u_n).$$

1. Prove by induction that, for every natural number $n$, $\frac{1}{2} \leqslant u_{n+1} \leqslant u_n$.
2. Prove that the sequence $(u_n)$ converges to a strictly positive limit.

Part C

We denote by $\ell$ the limit of the sequence $(u_n)$. We admit that $f(\ell) = \ell$. The objective of this part is to determine an approximate value of $\ell$. We introduce for this purpose the function $g$ defined on $[0; +\infty[$ by $g(x) = f(x) - x$. We give below the table of variations of the function $g$ on $[0; +\infty[$ where $x_0 = \frac{-2+\sqrt{7}}{3} \approx 0.215$ and $g(x_0) \approx 0.088$, rounded to $10^{-3}$.

$x$	0	$x_0$	$+\infty$
Variations		$g(x_0)$
of the
function $g$	0		$-\infty$

1. Prove that the equation $g(x) = 0$ has a unique strictly positive solution. We denote it by $\alpha$.
2. a. Copy and complete the algorithm below so that the last value taken by the variable $x$ is an approximate value of $\alpha$ by excess to 0.01 near. b. Give then the last value taken by the variable $x$ during the execution of the algorithm. $$x \leftarrow 0.22$$ While $\_\_\_\_$ do $$x \leftarrow x + 0.01$$ End While
3. Deduce an approximate value to 0.01 near of the limit $\ell$ of the sequence $(u_n)$.

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 A (Main topics: Sequences, Differential equations)
In this exercise, we are interested in the growth of Moso bamboo with maximum height 20 meters. Ludwig von Bertalanffy's growth model assumes that the growth rate for such bamboo is proportional to the difference between its height and the maximum height.

Part I: discrete model

In this part, we observe a bamboo with initial height 1 meter. For every natural integer $n$, we denote $u_n$ the height, in meters, of the bamboo $n$ days after the start of observation. Thus $u_0 = 1$. Von Bertalanffy's model for bamboo growth between two consecutive days is expressed by the equality: $$u_{n+1} = u_n + 0.05\left(20 - u_n\right) \text{ for every natural integer } n.$$

1. Verify that $u_1 = 1.95$.
2. a. Show that for every natural integer $n$, $u_{n+1} = 0.95 u_n + 1$. b. We set for every natural integer $n$, $v_n = 20 - u_n$. Prove that the sequence $(v_n)$ is a geometric sequence and specify its initial term $v_0$ and its common ratio. c. Deduce that, for every natural integer $n$, $u_n = 20 - 19 \times 0.95^n$.
3. Determine the limit of the sequence $(u_n)$.

Part II: continuous model

In this part, we wish to model the height of the same Moso bamboo by a function giving its height, in meters, as a function of time $t$ expressed in days. According to von Bertalanffy's model, this function is a solution of the differential equation $$(E) \quad y^{\prime} = 0.05(20 - y)$$ where $y$ denotes a function of the variable $t$, defined and differentiable on $[0; +\infty[$ and $y^{\prime}$ denotes its derivative function. Let the function $L$ defined on the interval $[0; +\infty[$ by $$L(t) = 20 - 19\mathrm{e}^{-0.05t}$$

1. Verify that the function $L$ is a solution of $(E)$ and that we also have $L(0) = 1$.
2. We take this function $L$ as our model and we admit that, if we denote $L^{\prime}$ its derivative function, $L^{\prime}(t)$ represents the growth rate of the bamboo at time $t$. a. Compare $L^{\prime}(0)$ and $L^{\prime}(5)$. b. Calculate the limit of the derivative function $L^{\prime}$ at $+\infty$. Is this result consistent with the description of the growth model presented at the beginning of the exercise?

Part A: Let $g$ be the function defined on $\mathbb{R}$ by: $$g(x) = 2\mathrm{e}^{\frac{-1}{3}x} + \frac{2}{3}x - 2$$

1. We admit that the function $g$ is differentiable on $\mathbb{R}$ and we denote $g^{\prime}$ its derivative function. Show that, for every real number $x$: $$g^{\prime}(x) = \frac{-2}{3}e^{-\frac{1}{3}x} + \frac{2}{3}.$$
2. Deduce the direction of variation of the function $g$ on $\mathbb{R}$.
3. Determine the sign of $g(x)$, for every real $x$.

Part B:

1. Consider the differential equation $$(E):\quad 3y^{\prime} + y = 0.$$ Solve the differential equation (E).
2. Determine the particular solution whose representative curve, in a coordinate system of the plane, passes through the point $\mathrm{M}(0;2)$.
3. Let $f$ be the function defined on $\mathbb{R}$ by: $$f(x) = 2\mathrm{e}^{-\frac{1}{3}x}$$ and $\mathscr{C}_f$ its representative curve. a. Show that the tangent line $(\Delta_0)$ to the curve $\mathscr{C}_f$ at the point $\mathrm{M}(0;2)$ has an equation of the form: $$y = -\frac{2}{3}x + 2$$ b. Study, on $\mathbb{R}$, the position of this curve $\mathscr{C}_f$ relative to the tangent line $(\Delta_0)$.

Part C:

1. Let A be the point on the curve $\mathscr{C}_f$ with abscissa $a$, where $a$ is any real number. Show that the tangent line $(\Delta_a)$ to the curve $\mathscr{C}_f$ at point A intersects the $x$-axis at a point P with abscissa $a+3$.
2. Explain the construction of the tangent line $(\Delta_{-2})$ to the curve $\mathscr{C}_f$ at point B with abscissa $-2$.

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 $[1;+\infty[$.

Main topics covered: Differential equations; exponential function.
We consider the differential equation
$$\text { (E) } y ^ { \prime } = y + 2 x \mathrm { e } ^ { x }$$
We seek the set of functions defined and differentiable on the set $\mathbb { R }$ of real numbers that are solutions to this equation.

1. Let $u$ be the function defined on $\mathbb { R }$ by $u ( x ) = x ^ { 2 } \mathrm { e } ^ { x }$. We admit that $u$ is differentiable and we denote $u ^ { \prime }$ its derivative function. Prove that $u$ is a particular solution of $( E )$.
2. Let $f$ be a function defined and differentiable on $\mathbb { R }$. We denote $g$ the function defined on $\mathbb { R }$ by: $$g ( x ) = f ( x ) - u ( x )$$ a. Prove that if the function $f$ is a solution of the differential equation $( E )$ then the function $g$ is a solution of the differential equation: $y ^ { \prime } = y$. We admit that the converse of this property is also true. b. Using the solution of the differential equation $y ^ { \prime } = y$, solve the differential equation (E).
3. Study of the function $u$ a. Study the sign of $u ^ { \prime } ( x )$ for $x$ varying in $\mathbb { R }$. b. Draw the table of variations of the function $u$ on $\mathbb { R }$ (limits are not required). c. Determine the largest interval on which the function $u$ is concave.

EXERCISE A Main topics covered: Exponential function, convexity, differentiation, differential equations
This exercise consists of three independent parts. Below is represented, in an orthonormal coordinate system, a portion of the representative curve $\mathscr { C }$ of a function $f$ defined on $\mathbb { R }$.
Consider the points $\mathrm { A } ( 0 ; 2 )$ and $\mathrm { B } ( 2 ; 0 )$.

Part 1

Knowing that the curve $\mathscr { C }$ passes through A and that the line (AB) is tangent to the curve $\mathscr { C }$ at point A, give by reading the graph:

1. The value of $f ( 0 )$ and that of $f ^ { \prime } ( 0 )$.
2. An interval on which the function $f$ appears to be convex.

Part 2

We denote $(E)$ the differential equation $$y ^ { \prime } = - y + \mathrm { e } ^ { - x }$$ It is admitted that $g : x \longmapsto x \mathrm { e } ^ { - x }$ is a particular solution of $(E)$.

1. Give all solutions on $\mathbb { R }$ of the differential equation $( H ) : y ^ { \prime } = - y$.
2. Deduce all solutions on $\mathbb { R }$ of the differential equation $(E)$.
3. Knowing that the function $f$ is the particular solution of $(E)$ which satisfies $f ( 0 ) = 2$, determine an expression of $f ( x )$ as a function of $x$.

Part 3

It is admitted that for every real number $x , f ( x ) = ( x + 2 ) \mathrm { e } ^ { - x }$.

1. Recall that $f ^ { \prime }$ denotes the derivative function of the function $f$. a. Show that for all $x \in \mathbb { R } , f ^ { \prime } ( x ) = ( - x - 1 ) \mathrm { e } ^ { - x }$. b. Study the sign of $f ^ { \prime } ( x )$ for all $x \in \mathbb { R }$ and draw up the table of variations of $f$ on $\mathbb { R }$. Neither the limit of $f$ at $- \infty$ nor the limit of $f$ at $+ \infty$ will be specified. Calculate the exact value of the extremum of $f$ on $\mathbb { R }$.
2. Recall that $f ^ { \prime \prime }$ denotes the second derivative function of the function $f$. a. Calculate for all $x \in \mathbb { R } , f ^ { \prime \prime } ( x )$. b. Can we assert that $f$ is convex on the interval $[ 0 ; + \infty [$?

Exercise 1

PART A Consider the differential equation $$( E ) : \quad y ^ { \prime } + \frac { 1 } { 4 } y = 20 \mathrm { e } ^ { - \frac { 1 } { 4 } x } ,$$ with unknown $y$, a function defined and differentiable on the interval $[ 0 ; + \infty [$.

1. Determine the value of the real number $a$ such that the function $g$ defined on the interval $[ 0 ; + \infty [$ by $g ( x ) = a x \mathrm { e } ^ { - \frac { 1 } { 4 } x }$ is a particular solution of the differential equation $( E )$.
2. Consider the differential equation $$\left( E ^ { \prime } \right) : \quad y ^ { \prime } + \frac { 1 } { 4 } y = 0 ,$$ with unknown $y$, a function defined and differentiable on the interval $[ 0 ; + \infty [$. Determine the solutions of the differential equation ( $E ^ { \prime }$ ).
3. Deduce the solutions of the differential equation ( $E$ ).
4. Determine the solution $f$ of the differential equation ( $E$ ) such that $f ( 0 ) = 8$.

PART B Consider the function $f$ defined on the interval $[ 0 ; + \infty [$ by $$f ( x ) = ( 20 x + 8 ) \mathrm { e } ^ { - \frac { 1 } { 4 } x }$$ It is admitted that the function $f$ is differentiable on the interval $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $f ^ { \prime }$ its derivative function on the interval $\left[ 0 ; + \infty \left[ \right. \right.$. Moreover, it is admitted that $\lim _ { x \rightarrow + \infty } f ( x ) = 0$.

1. a. Justify that, for every positive real number $x$, $$f ^ { \prime } ( x ) = ( 18 - 5 x ) \mathrm { e } ^ { - \frac { 1 } { 4 } x }$$ b. Deduce the table of variations of the function $f$. The exact value of the maximum of the function $f$ on the interval $[ 0 ; + \infty [$ will be specified.
2. In this question we are interested in the equation $f ( x ) = 8$. a. Justify that the equation $f ( x ) = 8$ admits a unique solution, denoted $\alpha$, in the interval [14; 15]. b. Copy and complete the table below by running step by step the solution\_equation function opposite, written in Python language
   

   $a$	14
   $b$	15
   $b - a$	1
   $m$	14,5
   \begin{tabular}{ l } Condition
   $f ( m ) > 8$

   & FALSE & & & & \hline \end{tabular}
   \begin{verbatim} from math import exp def f(x) : return (20* x +8)*exp(-1/4* x) def solution_equation() : a,b = 14,15 while b-a>0.1: m = (a+b)/2 if f (m) > 8 : |a=m else : | b=m return a,b \end{verbatim}
   c. What is the objective of the solution\_equation function in the context of the question?

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. Consider the function $f$ defined on $\mathbb { R }$ by: $f ( x ) = 5 x \mathrm { e } ^ { - x }$.

We denote by $C _ { f }$ the representative curve of $f$ in an orthonormal coordinate system.
Statement 1: The $x$-axis is a horizontal asymptote to the curve $C _ { f }$.
Statement 2: The function $f$ is a solution on $\mathbb { R }$ of the differential equation $( E ) : y ^ { \prime } + y = 5 \mathrm { e } ^ { - x }$.
2. Consider the sequences $\left( u _ { n } \right) , \left( v _ { n } \right)$ and $\left( w _ { n } \right)$, such that, for every natural integer $n$ :
$$u _ { n } \leqslant v _ { n } \leqslant w _ { n } .$$
Moreover, the sequence $( u _ { n } )$ converges to $- 1$ and the sequence $( w _ { n } )$ converges to $1$.
Statement 3: The sequence $\left( \nu _ { n } \right)$ converges to a real number $\ell$ belonging to the interval $[ - 1 ; 1 ]$.
We further assume that the sequence $( u _ { n } )$ is increasing and that the sequence $( w _ { n } )$ is decreasing.
Statement 4: For every natural integer $n$, we then have: $\quad u _ { 0 } \leqslant v _ { n } \leqslant w _ { 0 }$.

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.

This exercise contains 5 statements. For each statement, answer TRUE or FALSE by justifying the answer. Any lack of justification or incorrect justification will not be taken into account in the grading.
Part 1
We consider the sequence $(u_n)$ defined by: $$u_0 = 10 \text{ and for all natural integer } n,\ u_{n+1} = \frac{1}{3}u_n + 2.$$

1. Statement 1: The sequence $(u_n)$ is decreasing and bounded below by 0.
2. Statement 2: $\lim_{n \rightarrow +\infty} u_n = 0$.
3. Statement 3: The sequence $(v_n)$ defined for all natural integer $n$ by $v_n = u_n - 3$ is geometric.

Part 2
We consider the differential equation $(E): y' = \frac{3}{2}y + 2$ with unknown $y$, a function defined and differentiable on $\mathbb{R}$.

1. Statement 4: There exists a constant function that is a solution to the differential equation $(E)$.
2. In an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$ we denote $\mathscr{C}_f$ the representative curve of the function $f$ solution of $(E)$ such that $f(0) = 0$. Statement 5: The tangent line at the point with abscissa 1 of $\mathscr{C}_f$ has slope $2\mathrm{e}^{\frac{3}{2}}$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer receives no points.

1. Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \mathrm{e}^{x} + x.$$ Statement A: The function $f$ has the following variation table:
   

   $x$	$-\infty$
   variations of $f$	$+\infty$

   
   Statement B: The equation $f(x) = -2$ has two solutions in $\mathbb{R}$.
2. Statement C: $$\lim_{x \rightarrow +\infty} \frac{\ln(x) - x^{2} + 2}{3x^{2}} = -\frac{1}{3}.$$
3. Consider the function $k$ defined and continuous on $\mathbb{R}$ by $$k(x) = 1 + 2\mathrm{e}^{-x^{2}+1}$$ Statement D: There exists a primitive of the function $k$ that is decreasing on $\mathbb{R}$.
4. Consider the differential equation $$(E): \quad 3y' + y = 1.$$ Statement E: The function $g$ defined on $\mathbb{R}$ by $$g(x) = 4\mathrm{e}^{-\frac{1}{3}x} + 1$$ is a solution of the differential equation $(E)$ with $g(0) = 5$.
5. Statement F: Integration by parts allows us to obtain: $$\int_{0}^{1} x\mathrm{e}^{-x}\,\mathrm{d}x = 1 - 2\mathrm{e}^{-1}$$

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: $$\int_{0}^{\frac{\pi}{2}} \left[ -2\mathrm{e}^{x} + \sin(x) + 2\cos(x) \right] \mathrm{d}x$$

Exercise 2

Part A

We consider the function $f$ defined on the interval $[0; +\infty[$ by:
$$f(x) = \frac{1}{a + \mathrm{e}^{-bx}}$$
where $a$ and $b$ are two strictly positive real constants. We admit that the function $f$ is differentiable on the interval $[0; +\infty[$. The function $f$ has for graphical representation the curve $\mathscr{C}_f$.
We consider the points $\mathrm{A}(0; 0.5)$ and $\mathrm{B}(10; 1)$. We admit that the line (AB) is tangent to the curve $\mathscr{C}_f$ at point A.

1. By graphical reading, give an approximate value of $f(10)$.
2. We admit that $\lim_{x \rightarrow +\infty} f(x) = 1$. Give a graphical interpretation of this result.
3. Justify that $a = 1$.
4. Determine the slope of the line (AB).
5. a. Determine the expression of $f'(x)$ as a function of $x$ and the constant $b$. b. Deduce the value of $b$.

Part B

We admit, in the rest of the exercise, that the function $f$ is defined on the interval $[0; +\infty[$ by:
$$f(t) = \frac{1}{1 + \mathrm{e}^{-0.2x}}$$

1. Determine $\lim_{x \rightarrow +\infty} f(x)$.
2. Study the variations of the function $f$ on the interval $[0; +\infty[$.
3. Show that there exists a unique positive real number $\alpha$ such that $f(\alpha) = 0.97$.
4. Using a calculator, give a bound for the real number $\alpha$ by two consecutive integers. Interpret this result in the context of the statement.

Part C

1. Show that, for all $x$ belonging to the interval $[0; +\infty[$, $f(x) = \dfrac{\mathrm{e}^{0.2x}}{1 + \mathrm{e}^{0.2x}}$.
2. Deduce an antiderivative of the function $f$ on the interval $[0; +\infty[$.
3. Calculate the average value of the function $f$ on the interval $[0; 40]$, that is: $$I = \frac{1}{40} \int_0^{40} \frac{1}{1 + \mathrm{e}^{-0.2x}} \,\mathrm{d}x$$ The exact value and an approximate value to the nearest thousandth will be given.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. In a class of 24 students, there are 14 girls and 10 boys.
   Statement 1: It is possible to form 272 different groups of four students composed of two girls and two boys.
   
2. Let $f$ be the function defined on $\mathbb { R }$ by $f ( x ) = 3 \sin ( 2 x + \pi )$ and $C$ its representative curve in a given coordinate system.
   Statement 2: An equation of the tangent line to $C$ at the point with abscissa $\frac { \pi } { 2 }$ is $y = 6 x - 3 \pi$.
   
3. We consider the function $F$ defined on $] 0$; $+ \infty [$ by $F ( x ) = ( 2 x + 1 ) \ln ( x )$.
   Statement 3: The function $F$ is an antiderivative of the function $f$ defined on $] 0 ; + \infty [$ by $f ( x ) = \frac { 2 } { x }$.
   
4. We consider the function $g$ defined on $\mathbb { R }$ by $g ( t ) = 45 \mathrm { e } ^ { 0.06 t } + 20$.
   Statement 4: The function $g$ is the unique solution of the differential equation $\left( E _ { 1 } \right) y ^ { \prime } + 0.06 y = 1.2$ satisfying $g ( 0 ) = 65$.
   
5. We consider the differential equation: $$\left( E _ { 2 } \right) : \quad y ^ { \prime } - y = 3 \mathrm { e } ^ { 0.4 x }$$ where $y$ is a positive function of the real variable $x$, defined and differentiable on $\mathbb { R }$ and $y ^ { \prime }$ the derivative function of the function $y$.
   Statement 5: The solutions of the equation $\left( E _ { 2 } \right)$ are convex functions on $\mathbb { R }$.

We study the evolution of the population of an animal species within a nature reserve. The numbers of this population were recorded in different years. The collected data are presented in the following table:

Year	2000	2005	2010	2015
Number of individuals	50	64	80	100

To anticipate the evolution of this population, the reserve management chose to model the number of individuals as a function of time. For this, it uses a function, defined on the interval $[0; +\infty[$, where the variable $x$ represents the elapsed time, in years, from the year 2000. In its model, the image of 0 by this function equals 50, which corresponds to the number of individuals in the year 2000.
Part A. Model 1
In this part, the reserve management makes the hypothesis that the function sought satisfies the following differential equation: $$y' = 0.05y - 0.5 \quad (E_1)$$

1. Solve the differential equation $(E_1)$ with the initial condition $y(0) = 50$.
2. Compare the results in the table with those that would be obtained with this model.

Part B. Model 2
In this part, the reserve management makes the hypothesis that the function sought satisfies the following differential equation: $$y' = 0.05y(1 - 0.00125y)$$
Let $f$ be the function defined on $[0; +\infty[$ by: $$f(x) = \frac{800}{1 + 15\mathrm{e}^{-0.05x}}$$ and $C$ its representative curve in an orthonormal coordinate system.
Using computer algebra software, the following results were obtained. For the rest of the exercise, these results may be used without proof, except for question 5.

	Instruction	Result
1	$f(x) := \frac{800}{1 + 15\mathrm{e}^{-0.05x}}$	$f(x) = \frac{800}{1 + 15\mathrm{e}^{-0.05x}}$
2	$f'(x) :=$ Derivative$(f(x))$	$f'(x) = \frac{600\mathrm{e}^{-0.05x}}{\left(1 + 15\mathrm{e}^{-0.05x}\right)^2}$
3	$f''(x) :=$ Derivative$(f'(x))$	$f''(x) = \frac{30\mathrm{e}^{-0.05x}}{\left(1 + 15\mathrm{e}^{-0.05x}\right)^3}\left(15\mathrm{e}^{-0.05x} - 1\right)$
4	Solve$(15\mathrm{e}^{-0.05x} - 1 \geqslant 0)$	$x \leqslant 20\ln(15)$

1. Prove that the function $f$ satisfies $f(0) = 50$ and that for all $x \in \mathbb{R}$: $$f'(x) = 0.05f(x)(1 - 0.00125f(x))$$ We admit that this function $f$ is the unique solution of $(E_2)$ taking the initial value of 50 at 0.
2. With this new model $f$, estimate the population size in 2050. Round the result to the nearest integer.
3. Calculate the limit of $f$ as $x \to +\infty$. What can be deduced about the curve $C$? Interpret this limit in the context of this concrete problem.
4. Justify that the function $f$ is increasing on $[0; +\infty[$.
5. Prove the result obtained in line 4 of the software.
6. We admit that the growth rate of the population of this species, expressed in number of individuals per year, is modeled by the function $f'$. a. Study the convexity of the function $f$ on the interval $[0; +\infty[$ and determine the coordinates of any inflection points of the curve $C$. b. The reserve management claims: ``According to this model, the growth rate of the population of this species will increase for a little more than fifty years, then will decrease''. Is the management correct? Justify.

We propose to study the concentration in the blood of a medication ingested by a person for the first time. Let $t$ be the time (in hours) elapsed since the ingestion of this medication. We admit that the concentration of this medication in the blood, in grams per litre of blood, is modelled by a function $f$ of the variable $t$ defined on the interval $[ 0 ; + \infty [$.

Part A: graphical readings

The graph above shows the representative curve of the function $f$. With the precision allowed by the graph, give without justification:

1. The time elapsed from the moment of ingestion of this medication to the moment when the concentration of medication in the blood is maximum according to this model.
2. The set of solutions to the inequality $f ( t ) \geqslant 1$.
3. The convexity of the function $f$ on the interval $[ 0 ; 8 ]$.

Part B: determination of the function $\boldsymbol { f }$

We consider the differential equation
$$( E ) : \quad y ^ { \prime } + y = 5 \mathrm { e } ^ { - t }$$
of unknown $y$, where $y$ is a function defined and differentiable on the interval $[ 0 ; + \infty [$. We admit that the function $f$ is a solution of the differential equation $( E )$.

1. Solve the differential equation $\left( E ^ { \prime } \right) : y ^ { \prime } + y = 0$.
2. Let $u$ be the function defined on the interval $\left[ 0 ; + \infty \left[ \operatorname { by } u ( t ) = a t \mathrm { e } ^ { - t } \right. \right.$ with $a \in \mathbb { R }$.

Determine the value of the real number $a$ such that the function $u$ is a solution of equation $( E )$.
3. Deduce the set of solutions of the differential equation $( E )$.
4. Since the person has not taken this medication before, we admit that $f ( 0 ) = 0$.
Determine the expression of the function $f$.

Part C: study of the function $\boldsymbol { f }$

In this part, we admit that $f$ is defined on the interval $\left[ 0 ; + \infty \left[ \operatorname { by } f ( t ) = 5 t \mathrm { e } ^ { - t } \right. \right.$.

1. Determine the limit of $f$ at $+ \infty$.

Interpret this result in the context of the exercise.
2. Study the variations of $f$ on the interval $[ 0 ; + \infty [$ then draw up its complete variation table.
3. Prove that there exist two real numbers $t _ { 1 }$ and $t _ { 2 }$ such that $f \left( t _ { 1 } \right) = f \left( t _ { 2 } \right) = 1$.
Give an approximate value to $10 ^ { - 2 }$ of the real numbers $t _ { 1 }$ and $t _ { 2 }$.
4. For a medication concentration greater than or equal to 1 gram per litre of blood, there is a risk of drowsiness. What is the duration in hours and minutes of the drowsiness risk when taking this medication?

Part D: average concentration

The average concentration of the medication (in grams per litre of blood) during the first hour is given by:
$$T _ { m } = \int _ { 0 } ^ { 1 } f ( t ) \mathrm { d } t$$
where $f$ is the function defined on $\left[ 0 ; + \infty \left[ \operatorname { by } f ( t ) = 5 t \mathrm { e } ^ { - t } \right. \right.$. Calculate this average concentration. Give the exact value then an approximate value to 0.01.

Part A

We consider the function $f$ defined on the interval $]-1; +\infty[$ by $$f(x) = 4\ln(x+1) - \frac{x^2}{25}$$ We admit that the function $f$ is differentiable on the interval $]-1; +\infty[$.

1. Determine the limit of the function $f$ at $-1$.
2. Show that, for all $x$ belonging to the interval $]-1; +\infty[$, we have: $$f'(x) = \frac{100 - 2x - 2x^2}{25(x+1)}$$
3. Study the variations of the function $f$ on the interval $]-1; +\infty[$ and then deduce that the function $f$ is strictly increasing on the interval $[2; 6.5]$.
4. We consider $h$ the function defined on the interval $[2; 6.5]$ by $h(x) = f(x) - x$.
   The table of variations of the function $h$ is given (showing $h$ increases then decreases on $[2;6.5]$ with $h(2) < 0$ and $h(6.5) < 0$ and a positive maximum in between).
   Show that the equation $h(x) = 0$ admits a unique solution $\alpha \in [2; 6.5]$.
5. Consider the following script, written in Python language: \begin{verbatim} from math import * def f(x): return 4*log(1+x)-(x**2)/25 def bornes(n) : p = 1/10**n x = 6 while f(x)-x > 0 : x = x + p return (x-p,x) \end{verbatim} We recall that in Python language:
   • the command $\log(x)$ returns the value $\ln x$;
   • the command $\mathrm{c}**\mathrm{d}$ returns the value of $c^d$.
   a. Give the values returned by the command \texttt{bornes(2)}. The values will be given rounded to the nearest hundredth. b. Interpret these values in the context of the exercise.

Part B

In this part, we may use the results obtained in Part A. We consider the sequence $(u_n)$ defined by $u_0 = 2$, and, for all natural integer $n$, $u_{n+1} = f(u_n)$.

1. Show by induction that for all natural integer $n$, $$2 \leqslant u_n \leqslant u_{n+1} < 6.5.$$
2. Deduce that the sequence $(u_n)$ converges to a limit $\ell$.
3. We recall that the real number $\alpha$, defined in Part A, is the solution of the equation $h(x) = 0$ on the interval $[2; 6.5]$. Justify that $\ell = \alpha$.

We consider the sequence $(u_n)$ defined by $u_0 = 5$ and, for all natural integers $n$: $$u_{n+1} = 2 + \ln\left(u_n^2 - 3\right)$$ We admit that this sequence is well defined.
Part A: Exploitation of Python programs

1. Copy and complete the Python script below so that \texttt{suite(k)} which takes a natural integer $k$ as parameter returns the list of the first $k$ values of the sequence $(u_n)$.
   Remark: We specify that, for any strictly positive real number $a$, $\log(a)$ returns the value of the natural logarithm of $a$.
   \begin{verbatim} def suite(k): L = [] u = 5 for i in range(......): L.append(u) u=............ return(......) \end{verbatim}
   
2. We executed \texttt{suite(9)} below. Make two conjectures: one on the direction of variation of the sequence $(u_n)$ and another on its possible convergence.
   \begin{verbatim} >>> suite(9) [ 5, 5.091042453358316, 5.131953749864703, 5.150037910978289, 5.157974010229213, 5.1614456706362954, 5.162962248594583, 5.163624356938671, 5.163913344065642] \end{verbatim}
   
3. We then created the function \texttt{mystere(n)} given below and executed \texttt{mystere(10000)}, which returned 1. Does this output contradict the conjecture made about the direction of variation of the sequence $(u_n)$? Justify.
   \begin{verbatim} def mystere(n): L = suite(n) c = 1 for i in range(n - 1): if L[i] > L[i + 1]: c = 0 return c
   >>> mystere(10000) 1 \end{verbatim}

Part B: Study of the convergence of the sequence $(u_n)$
We consider the function $g$ defined on $[2; +\infty[$ by: $$g(x) = 2 + \ln\left(x^2 - 3\right)$$ We admit that $g$ is differentiable on $[2; +\infty[$ and we denote $g'$ its derivative function.

1. Prove that the function $g$ is increasing on $[2; +\infty[$.
2. a. Prove by induction that, for all natural integers $n$: $$4 \leqslant u_n \leqslant u_{n+1} \leqslant 6$$ b. Deduce that the sequence $(u_n)$ converges.

Part C: Study of the limit value
We consider the function $f$ defined on $[2; +\infty[$ by: $$f(x) = 2 + \ln\left(x^2 - 3\right) - x$$ We admit that $f$ is differentiable on $[2; +\infty[$ and we denote $f'$ its derivative function. We give the following variation table of $f$. No justification is requested.

$x$	2	3	$+\infty$
		$\ln(6) - 1$
$f(x)$
	0		$-\infty$

1. a. Show that the equation $f(x) = 0$ has exactly two solutions on $[2; +\infty[$ which we will denote $\alpha$ and $\beta$ with $\alpha < \beta$. b. Give the exact value of $\alpha$ and an approximate value to $10^{-3}$ of $\beta$.
2. Let $\ell$ be the limit of the sequence $(u_n)$. Justify that $f(\ell) = 0$ and determine $\ell$.