Consider the function $g$ defined on $\mathbb { R }$ by: $$g ( x ) = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 } .$$ The representative curve of the function $g$ admits as an asymptote at $+ \infty$ the line with equation: a. $x = 2$; b. $y = 2$; c. $y = 0$; d. $x = - 1$

Exercise 1 (7 points) Themes: exponential function, sequences In the context of a clinical trial, two treatment protocols for a disease are being considered. The objective of this exercise is to study, for these two protocols, the evolution of the quantity of medication present in a patient's blood as a function of time.
Parts $A$ and $B$ are independent

Part A: Study of the first protocol

The first protocol consists of having the patient take a medication in tablet form. The quantity of medication present in the patient's blood, expressed in mg, is modelled by the function $f$ defined on the interval $[0; 10]$ by $$f(t) = 3t \mathrm{e}^{-0.5t + 1},$$ where $t$ denotes the time, expressed in hours, elapsed since taking the tablet.

1. a. It is admitted that the function $f$ is differentiable on the interval $[0; 10]$ and we denote $f'$ its derivative function. Show that, for every real number $t$ in $[0; 10]$, we have: $f'(t) = 3(-0.5t + 1)\mathrm{e}^{-0.5t + 1}$. b. Deduce the table of variations of the function $f$ on the interval $[0; 10]$. c. According to this model, after how much time will the quantity of medication present in the patient's blood be maximum? What is this maximum quantity?
2. a. Show that the equation $f(t) = 5$ admits a unique solution on the interval $[0; 2]$ denoted $\alpha$, of which you will give an approximate value to $10^{-2}$ near. It is admitted that the equation $f(t) = 5$ admits a unique solution on the interval $[2; 10]$, denoted $\beta$, and that an approximate value of $\beta$ to $10^{-2}$ near is 3.46. b. It is considered that this treatment is effective when the quantity of medication present in the patient's blood is greater than or equal to 5 mg. Determine, to the nearest minute, the duration of effectiveness of the medication in the case of this protocol.

Part B: Study of the second protocol

The second protocol consists of initially injecting the patient, by intravenous injection, a dose of 2 mg of medication and then re-injecting every hour a dose of $1.8$ mg. It is assumed that the medication diffuses instantaneously into the blood and is then progressively eliminated. It is estimated that when one hour has elapsed after an injection, the quantity of medication in the blood has decreased by $30\%$ compared to the quantity present immediately after this injection. This situation is modelled using the sequence $(u_n)$ where, for every natural number $n$, $u_n$ denotes the quantity of medication, expressed in mg, present in the patient's blood immediately after the injection at the $n$-th hour. We therefore have $u_0 = 2$.

1. Calculate, according to this model, the quantity $u_1$, of medication (in mg) present in the patient's blood immediately after the injection at the first hour.
2. Justify that, for every natural number $n$, we have: $u_{n+1} = 0.7u_n + 1.8$.
3. a. Show by induction that, for every natural number $n$, we have: $u_n \leqslant u_{n+1} < 6$. b. Deduce that the sequence $(u_n)$ is convergent. We denote its limit by $\ell$. c. Determine the value of $\ell$. Interpret this value in the context of the exercise.
4. Consider the sequence $(v_n)$ defined, for every natural number $n$, by $v_n = 6 - u_n$. a. Show that the sequence $(v_n)$ is a geometric sequence with ratio 0.7 and specify its first term. b. Determine the expression of $v_n$ as a function of $n$, then of $u_n$ as a function of $n$. c. With this protocol, injections are stopped when the quantity of medication present in the patient's blood is greater than or equal to $5.5$ mg. Determine, by detailing the calculations, the number of injections carried out when applying this protocol.

Exercise 2 — 6 points

Theme: Exponential function Main topics covered: Sequences; Functions, Logarithm function. This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct.
A correct answer earns one point. An incorrect answer, a multiple answer, or no answer to a question earns or loses no points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.

1. A container initially containing 1 litre of water is left in the sun. Every hour, the volume of water decreases by $15 \%$. After how many whole hours does the volume of water become less than a quarter of a litre? a. 2 hours b. 8 hours. c. 9 hours d. 13 hours
   
2. We consider the function $f$ defined on the interval $] 0$; $+ \infty [ \operatorname { by } f ( x ) = 4 \ln ( 3 x )$. For every real $x$ in the interval $] 0$; $+ \infty [$, we have: a. $f ( 2 x ) = f ( x ) + \ln ( 24 )$ b. $f ( 2 x ) = f ( x ) + \ln ( 16 )$ c. $f ( 2 x ) = \ln ( 2 ) + f ( x )$ d. $f ( 2 x ) = 2 f ( x )$
   
3. We consider the function $g$ defined on the interval $] 1 ; + \infty [$ by: $$g ( x ) = \frac { \ln ( x ) } { x - 1 } .$$ We denote $\mathscr { C } _ { g }$ the representative curve of the function $g$ in an orthogonal coordinate system. The curve $\mathscr { C } _ { g }$ has: a. a vertical asymptote and a horizontal asymptote. b. a vertical asymptote and no horizontal asymptote. c. no vertical asymptote and a horizontal asymptote. d. no vertical asymptote and no horizontal asymptote.
   In the rest of the exercise, we consider the function $h$ defined on the interval ]0;2] by: $$h ( x ) = x ^ { 2 } [ 1 + 2 \ln ( x ) ] .$$ We denote $\mathscr { C } _ { h }$ the representative curve of $h$ in a coordinate system of the plane. We assume that $h$ is twice differentiable on the interval ]0; 2]. We denote $h ^ { \prime }$ its derivative and $h ^ { \prime \prime }$ its second derivative. We assume that, for every real $x$ in the interval ] 0 ; 2], we have: $$h ^ { \prime } ( x ) = 4 x ( 1 + \ln ( x ) ) .$$
   
4. On the interval $\left. ] \frac { 1 } { \mathrm { e } } ; 2 \right]$, the function $h$ equals zero: a. exactly 0 times. b. exactly 1 time. c. exactly 2 times. d. exactly 3 times.
   
5. An equation of the tangent line to $\mathscr { C } _ { h }$ at the point with abscissa $\sqrt { \mathrm { e } }$ is: a. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x$ b. $y = ( 6 \sqrt { \mathrm { e } } ) \cdot x + 2 \mathrm { e }$ c. $y = 6 \mathrm { e } ^ { \frac { x } { 2 } }$ d. $y = \left( 6 \mathrm { e } ^ { \frac { 1 } { 2 } } \right) \cdot x - 4 \mathrm { e }$.
   
6. On the interval $] 0 ; 2 ]$, the number of inflection points of the curve $\mathscr { C } _ { h }$ is equal to: a. 0 b. 1 c. 2 d. 3
   
7. We consider the sequence $\left( u _ { n } \right)$ defined for every natural number $n$ by $$u _ { n + 1 } = \frac { 1 } { 2 } u _ { n } + 3 \quad \text { and } \quad u _ { 0 } = 6 .$$ We can affirm that: a. the sequence $\left( u _ { n } \right)$ is strictly increasing. b. the sequence $( u _ { n } )$ is strictly decreasing. c. the sequence $( u _ { n } )$ is not monotonic. d. the sequence $( u _ { n } )$ is constant.

Exercise 3 — 7 points
Themes: Exponential function and sequence
Part A:
Let $h$ be the function defined on $\mathbb{R}$ by $$h(x) = \mathrm{e}^x - x$$

1. Determine the limits of $h$ at $-\infty$ and $+\infty$.
2. Study the variations of $h$ and draw up its variation table.
3. Deduce that: if $a$ and $b$ are two real numbers such that $0 < a < b$ then $h(a) - h(b) < 0$.

Part B:
Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = \mathrm{e}^x$$ We denote $\mathscr{C}_f$ its representative curve in a coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

1. Determine an equation of the tangent line $T$ to $\mathscr{C}_f$ at the point with abscissa 0.

In the rest of the exercise we are interested in the gap between $T$ and $\mathscr{C}_f$ in the neighbourhood of 0. This gap is defined as the difference of the ordinates of the points of $T$ and $\mathscr{C}_f$ with the same abscissa. We are interested in points with abscissa $\frac{1}{n}$, with $n$ a non-zero natural number. We then consider the sequence $(u_n)$ defined for all non-zero natural numbers $n$ by: $$u_n = \exp\left(\frac{1}{n}\right) - \frac{1}{n} - 1$$

1. Determine the limit of the sequence $(u_n)$.
2. a. Prove that, for all non-zero natural numbers $n$, $$u_{n+1} - u_n = h\left(\frac{1}{n+1}\right) - h\left(\frac{1}{n}\right)$$ where $h$ is the function defined in Part A. b. Deduce the direction of variation of the sequence $(u_n)$.
3. The table below gives approximate values to $10^{-9}$ of the first terms of the sequence $(u_n)$.
   

   $n$	$u_n$
   1	0.718281828
   2	0.148721271
   3	0.062279092
   4	0.034025417
   5	0.021402758
   6	0.014693746
   7	0.010707852
   8	0.008148453
   9	0.006407958
   10	0.005170918

   
   Give the smallest value of the natural number $n$ for which the gap between $T$ and $\mathscr{C}_f$ appears to be less than $10^{-2}$.

Exercise 3 — 6 points

Theme: Exponential function Main topics covered: Sequences; Functions, Exponential function.

Part A

We consider the function $f$ defined for every real $x$ by: $$f ( x ) = 1 + x - \mathrm { e } ^ { 0,5 x - 2 } .$$ We assume that the function $f$ is differentiable on $\mathbb { R }$. We denote $f ^ { \prime }$ its derivative.

1. a. Determine the limit of the function $f$ at $- \infty$. b. Prove that, for every non-zero real $x$, $f ( x ) = 1 + 0,5 x \left( 2 - \frac { \mathrm { e } ^ { 0,5 x } } { 0,5 x } \times \mathrm { e } ^ { - 2 } \right)$. Deduce the limit of the function $f$ at $+ \infty$.
2. a. Determine $f ^ { \prime } ( x )$ for every real $x$. b. Prove that the set of solutions of the inequality $f ^ { \prime } ( x ) < 0$ is the interval $] 4 + 2 \ln ( 2 ) ; + \infty [$.
3. Deduce from the previous questions the variation table of the function $f$ on $\mathbb { R }$. The exact value of the image of $4 + 2 \ln ( 2 )$ by $f$ should be shown.
4. Show that the equation $f ( x ) = 0$ has a unique solution on the interval $[ - 1 ; 0 ]$.

Part B

We consider the sequence $( u _ { n } )$ defined by $u _ { 0 } = 0$ and, for every natural number $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$ where $f$ is the function defined in Part A.

1. a. Prove by induction that, for every natural number $n$, we have: $$u _ { n } \leqslant u _ { n + 1 } \leqslant 4 .$$ b. Deduce that the sequence $( u _ { n } )$ converges. We denote its limit by $\ell$.
2. a. We recall that $f$ satisfies the relation $\ell = f ( \ell )$. Prove that $\ell = 4$. b. We consider the function value written below in the Python language: \begin{verbatim} def valeur (a) : u = 0 n = 0 while u <= a: u=1 + u - exp(0.5*u - 2) n = n+1 return n \end{verbatim} The instruction valeur(3.99) returns the value 12. Interpret this result in the context of the exercise.

For each of the following statements, indicate whether it is true or false. Justify each answer.

1. Statement 1: For all real $x : 1 - \frac { 1 - \mathrm { e } ^ { x } } { 1 + \mathrm { e } ^ { x } } = \frac { 2 } { 1 + \mathrm { e } ^ { - x } }$.
   
2. We consider the function $g$ defined on $\mathbb { R }$ by $g ( x ) = \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 }$. Statement 2: The equation $g ( x ) = \frac { 1 } { 2 }$ admits a unique solution in $\mathbb { R }$.
   
3. We consider the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x ^ { 2 } \mathrm { e } ^ { - x }$ and we denote $\mathscr { C }$ its curve in an orthonormal coordinate system. Statement 3: The $x$-axis is tangent to the curve $\mathscr { C }$ at only one point.
   
4. We consider the function $h$ defined on $\mathbb { R }$ by $h ( x ) = \mathrm { e } ^ { x } \left( 1 - x ^ { 2 } \right)$. Statement 4: In the plane equipped with an orthonormal coordinate system, the curve representing the function $h$ does not admit an inflection point.
   
5. Statement 5: $\lim _ { x \rightarrow + \infty } \frac { \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + x } = 0$.
   
6. Statement 6: For all real $x , 1 + \mathrm { e } ^ { 2 x } \geqslant 2 \mathrm { e } ^ { x }$.

Exercise 4 (7 points) Themes: numerical functions, exponential function
Part A: study of two functions Consider the two functions $f$ and $g$ defined on the interval $[0; +\infty[$ by: $$f(x) = 0.06\left(-x^2 + 13.7x\right) \quad \text{and} \quad g(x) = (-0.15x + 2.2)\mathrm{e}^{0.2x} - 2.2.$$ We admit that the functions $f$ and $g$ are differentiable and we denote $f'$ and $g'$ their respective derivative functions.

1. The complete table of variations of function $f$ on the interval $[0; +\infty[$ is given.
   

   $x$	0	6.85	$+\infty$
   \multirow{2}{*}{$f(x)$}		$\nearrow f(6.85)$
   	0		$\underline{-}_{\infty}$

   
   a. Justify the limit of $f$ at $+\infty$. b. Justify the variations of function $f$. c. Solve the equation $f(x) = 0$.
2. a. Determine the limit of $g$ at $+\infty$. b. Prove that, for every real $x$ belonging to $[0; +\infty[$ we have: $g'(x) = (-0.03x + 0.29)\mathrm{e}^{0.2x}$. c. Study the variations of function $g$ and draw its table of variations on $[0; +\infty[$. Specify an approximate value to $10^{-2}$ of the maximum of $g$. d. Show that the equation $g(x) = 0$ has a unique non-zero solution and determine, to $10^{-2}$ near, an approximate value of this solution.

Part B: trajectories of a golf ball We wish to use the functions $f$ and $g$ studied in Part A to model in two different ways the trajectory of a golf ball. We assume that the terrain is perfectly flat. We will admit here that 13.7 is the value that cancels the function $f$ and an approximation of the value that cancels the function $g$. For $x$ representing the horizontal distance traveled by the ball in tens of yards after the shot (with $0 < x < 13.7$), $f(x)$ (or $g(x)$ depending on the model) represents the corresponding height of the ball above the ground, in tens of yards. The ``takeoff angle'' of the ball is called the angle between the x-axis and the tangent to the curve ($\mathscr{C}_f$ or $\mathscr{C}_g$ depending on the model) at its point with abscissa 0. A measure of the takeoff angle of the ball is a real number $d$ such that $\tan(d)$ is equal to the slope of this tangent. Similarly, the ``landing angle'' of the ball is called the angle between the x-axis and the tangent to the curve ($\mathscr{C}_f$ or $\mathscr{C}_g$ depending on the model) at its point with abscissa 13.7. A measure of the landing angle of the ball is a real number $a$ such that $\tan(a)$ is equal to the opposite of the slope of this tangent. All angles are measured in degrees.

1. First model: recall that here, the unit being tens of yards, $x$ represents the horizontal distance traveled by the ball after the shot and $f(x)$ the corresponding height of the ball. According to this model: a. What is the maximum height, in yards, reached by the ball during its trajectory? b. Verify that $f'(0) = 0.822$. c. Give a measure in degrees of the takeoff angle of the ball, rounded to the nearest tenth. (You may use the table below). d. What graphical property of the curve $\mathscr{C}_f$ allows us to justify that the takeoff and landing angles of the ball are equal?
2. Second model: recall that here, the unit being tens of yards, $x$ represents the horizontal distance traveled by the ball after the shot and $g(x)$ the corresponding height of the ball. According to this model: a. What is the maximum height, in yards, reached by the ball during its trajectory? We specify that $g'(0) = 0.29$ and $g'(13.7) \approx -1.87$. b. Give a measure in degrees of the takeoff angle of the ball, rounded to the nearest tenth. (You may use the table below). c. Justify that 62 is an approximate value, rounded to the nearest unit, of a measure in degrees of the landing angle of the ball.

Table: excerpt from a spreadsheet giving a measure in degrees of an angle when its tangent is known:

	A	B	C	D	E	F	G	H	I	J	K	L	M
1	$\tan(\theta)$	0.815	0.816	0.817	0.818	0.819	0.82	0.821	0.822	0.823	0.824	0.825	0.826
2	$\theta$ in degrees	39.18	39.21	39.25	39.28	39.32	39.35	39.39	39.42	39.45	39.49	39.52	39.56
3
4	$\tan(\theta)$	0.285	0.286	0.287	0.288	0.289	0.29	0.291	0.292	0.293	0.294	0.295	0.296
5	$\theta$ in degrees	15.91	15.96	16.01	16.07	16.12	16.17	16.23	16.28	16.33	16.38	16.44	16.49

Part C: Interrogating the models Based on a large number of observations of professional players' performances, the following average results were obtained:

Launch angle in degrees	Maximum height in yards	Landing angle in degrees	Horizontal distance in yards at the point of impact
24	32	52	137

Which model, among the two previously studied, seems most suitable for describing the ball strike by a professional player? The answer will be justified.

We admit that the function $f$ from part $\mathbf{A}$ is defined on $\mathbb{R}$ by
$$f(x) = \left(x^{2} - 5x + 6\right)\mathrm{e}^{x}$$
We denote $\mathscr{C}$ the representative curve of the function $f$ in a coordinate system.

1. a. Determine the limit of the function $f$ at $+\infty$. b. Determine the limit of the function $f$ at $-\infty$.
2. Show that, for all real $x$, we have $f^{\prime}(x) = \left(x^{2} - 3x + 1\right)\mathrm{e}^{x}$.
3. Deduce the direction of variation of the function $f$.
4. Determine the reduced equation of the tangent line $(\mathscr{T})$ to the curve $\mathscr{C}$ at the point with abscissa 0.

We admit that the function $f$ is twice differentiable on $\mathbb{R}$. We denote $f^{\prime\prime}$ the second derivative function of $f$. We admit that, for all real $x$, we have $f^{\prime\prime}(x) = (x+1)(x-2)\mathrm{e}^{x}$.
5. a. Study the convexity of the function $f$ on $\mathbb{R}$. b. Show that, for all $x$ belonging to the interval $[-1; 2]$, we have $f(x) \leqslant x + 6$.

We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \mathrm{e}^{3x} - (2x+1)\mathrm{e}^{x}$$
The purpose of this exercise is to study the function $f$ on $\mathbb{R}$.
Part A - Study of an auxiliary function
We define the function $g$ on $\mathbb{R}$ by: $$g(x) = 3\mathrm{e}^{2x} - 2x - 3$$

1. a. Determine the limit of function $g$ at $-\infty$. b. Determine the limit of function $g$ at $+\infty$.
2. a. We admit that function $g$ is differentiable on $\mathbb{R}$, and we denote by $g'$ its derivative function. Prove that for every real number $x$, we have $g'(x) = 6\mathrm{e}^{2x} - 2$. b. Study the sign of the derivative function $g'$ on $\mathbb{R}$. c. Deduce the table of variations of function $g$ on $\mathbb{R}$. Verify that function $g$ has a minimum equal to $\ln(3) - 2$.
3. a. Show that $x = 0$ is a solution of the equation $g(x) = 0$. b. Show that the equation $g(x) = 0$ has a second non-zero solution, denoted $\alpha$, for which you will give an interval of amplitude $10^{-1}$.
4. Deduce from the previous questions the sign of function $g$ on $\mathbb{R}$.

Part B - Study of function $f$

1. Function $f$ is differentiable on $\mathbb{R}$, and we denote by $f'$ its derivative function. Prove that for every real number $x$, we have $f'(x) = \mathrm{e}^{x} g(x)$, where $g$ is the function defined in Part A.
2. Deduce the sign of the derivative function $f'$ and then the variations of function $f$ on $\mathbb{R}$.
3. Why is function $f$ not convex on $\mathbb{R}$? Explain.

We consider the function $f$ defined on $[ 0 ; + \infty [$ by: $$f ( x ) = x \mathrm { e } ^ { - x }$$ We denote $\mathscr { C } _ { f }$ its representative curve in an orthonormal coordinate system of the plane. We admit that $f$ is twice differentiable on $[ 0 ; + \infty [$. We denote $f ^ { \prime }$ its derivative and $f ^ { \prime \prime }$ its second derivative.

1. By noting that for all $x$ in $[ 0 ; + \infty [$, we have

$$f ( x ) = \frac { x } { \mathrm { e } ^ { x } }$$ prove that the curve $\mathscr { C } _ { f }$ has an asymptote at $+ \infty$ for which you will give an equation.
2. Prove that for all real $x$ belonging to $[ 0 ; + \infty [$ : $$f ^ { \prime } ( x ) = ( 1 - x ) \mathrm { e } ^ { - x }$$

1. Draw up the table of variations of $f$ on $[ 0 ; + \infty [$, on which you will show the values at the boundaries as well as the exact value of the extremum.
2. Determine, on the interval $[ 0 ; + \infty [$, the number of solutions of the equation

$$f ( x ) = \frac { 367 } { 1000 }$$

1. We admit that for all $x$ belonging to $[ 0 ; + \infty [$ :

$$f ^ { \prime \prime } ( x ) = \mathrm { e } ^ { - x } ( x - 2 )$$ Study the convexity of the function $f$ on the interval $[ 0 ; + \infty [$. 6. Let $a$ be a real number belonging to $[ 0 ; + \infty [$ and A the point of the curve $\mathscr { C } _ { f }$ with abscissa $a$. We denote $T _ { a }$ the tangent to $\mathscr { C } _ { f }$ at A. We denote $\mathrm { H } _ { a }$ the point of intersection of the line $T _ { a }$ and the ordinate axis. We denote $g ( a )$ the ordinate of $\mathrm { H } _ { a }$. a. Prove that a reduced equation of the tangent $T _ { a }$ is: $$y = \left[ ( 1 - a ) \mathrm { e } ^ { - a } \right] x + a ^ { 2 } \mathrm { e } ^ { - a }$$ b. Deduce the expression of $g ( a )$. c. Prove that $g ( a )$ is maximum when A is an inflection point of the curve $\mathscr { C } _ { f }$.

Biologists are studying the evolution of an insect population in a botanical garden. At the beginning of the study, the population is 100,000 insects. To preserve the balance of the natural environment, the number of insects must not exceed 400,000.

Part A: Study of a first model in the laboratory

Observation of the evolution of these insect populations in the laboratory, in the absence of any predator, shows that the number of insects increases by $60\%$ each month. Taking this observation into account, biologists model the evolution of the insect population using a sequence $(u_n)$ where, for every natural number $n$, $u_n$ models the number of insects, expressed in millions, after $n$ months. We therefore have $u_0 = 0.1$.

1. Justify that for every natural number $n$: $u_n = 0.1 \times 1.6^n$.
2. Determine the limit of the sequence $(u_n)$.
3. By solving an inequality, determine the smallest natural number $n$ from which $u_n > 0.4$.
4. According to this model, would the balance of the natural environment be preserved? Justify your answer.

Part B: Study of a second model

Taking into account the constraints of the natural environment in which the insects evolve, biologists choose a new model. They model the number of insects using the sequence $(v_n)$, defined by: $$v_0 = 0.1 \text{ and, for every natural number } n, v_{n+1} = 1.6v_n - 1.6v_n^2,$$ where, for every natural number $n$, $v_n$ is the number of insects, expressed in millions, after $n$ months.

1. Determine the number of insects after one month.
2. We consider the function $f$ defined on the interval $\left[0; \frac{1}{2}\right]$ by $$f(x) = 1.6x - 1.6x^2.$$ a. Solve the equation $f(x) = x$. b. Show that the function $f$ is increasing on the interval $\left[0; \frac{1}{2}\right]$.
3. a. Show by induction that, for every natural number $n$, $0 \leqslant v_n \leqslant v_{n+1} \leqslant \frac{1}{2}$. b. Show that the sequence $(v_n)$ is convergent. We denote by $\ell$ the value of its limit. We admit that $\ell$ is a solution of the equation $f(x) = x$. c. Determine the value of $\ell$. According to this model, will the balance of the natural environment be preserved? Justify your answer.
4. The threshold function is given below, written in Python language. a. What do we observe if we enter \texttt{seuil(0.4)}? b. Determine the value returned by entering \texttt{seuil(0.35)}. Interpret this value in the context of the exercise. \begin{verbatim} def seuil(a) : v=0.1 n=0 while v

Consider the function $g$ defined on $[0; +\infty[$ by $g(t) = \frac{a}{b + \mathrm{e}^{-t}}$ where $a$ and $b$ are two real numbers. We know that $g(0) = 2$ and $\lim_{t \rightarrow +\infty} g(t) = 3$. The values of $a$ and $b$ are:
A. $a = 2$ and $b = 3$
B. $a = 4$ and $b = \frac{4}{3}$
C. $a = 4$ and $b = 1$
D. $a = 6$ and $b = 2$

Exercise 3 — 5 points Theme: exponential function, algorithms 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. Statement: The function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^x - x$ is convex.
2. Statement: The equation $(2\mathrm{e}^x - 6)(\mathrm{e}^x + 2) = 0$ has $\ln(3)$ as its unique solution in $\mathbb{R}$.
3. Statement: $$\lim_{x \to +\infty} \frac{\mathrm{e}^{2x} - 1}{\mathrm{e}^x - x} = 0.$$
4. Let $f$ be the function defined on $\mathbb{R}$ by $f(x) = (6x + 5)\mathrm{e}^{3x}$ and $F$ the function defined on $\mathbb{R}$ by: $F(x) = (2x + 1)\mathrm{e}^{3x} + 4$. Statement: $F$ is the antiderivative of $f$ on $\mathbb{R}$ that takes the value 5 when $x = 0$.
5. We consider the function \texttt{mystere} defined below which takes a list $L$ of numbers as a parameter. We recall that \texttt{len(L)} represents the length of list $L$. \begin{verbatim} def mystere(L) : S = 0 for i in range(len(L)) : S = S + L[i] return S / len(L) \end{verbatim} Statement: The execution of \texttt{mystere([1, 9, 9, 5, 0, 3, 6, 12, 0, 5])} returns 50.

We consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \ln\left(1 + \mathrm{e}^{-x}\right),$$ where $\ln$ denotes the natural logarithm function. We denote by $\mathscr{C}$ its representative curve in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$.

1. a. Determine the limit of the function $f$ at $-\infty$. b. Determine the limit of the function $f$ at $+\infty$. Interpret this result graphically. c. We admit that the function $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative function. Calculate $f'(x)$ then show that, for every real number $x$, $f'(x) = \frac{-1}{1 + \mathrm{e}^x}$. d. Draw the complete table of variations of the function $f$ on $\mathbb{R}$.
2. We denote by $T_0$ the tangent line to the curve $\mathscr{C}$ at its point with abscissa 0. a. Determine an equation of the tangent line $T_0$. b. Show that the function $f$ is convex on $\mathbb{R}$. c. Deduce that, for every real number $x$, we have: $$f(x) \geqslant -\frac{1}{2}x + \ln(2)$$
3. For every real number $a$ different from 0, we denote by $M_a$ and $N_a$ the points of the curve $\mathscr{C}$ with abscissas $-a$ and $a$ respectively. We therefore have: $M_a(-a; f(-a))$ and $N_a(a; f(a))$. a. Show that, for every real number $x$, we have: $f(x) - f(-x) = -x$. b. Deduce that the lines $T_0$ and $(M_a N_a)$ are parallel.

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 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: $$\int_{0}^{1} x\mathrm{e}^{-x}\,\mathrm{d}x = 1 - 2\mathrm{e}^{-1}$$

Part A We consider the function $f$ defined on $\mathbb{R}$ by: $$f(x) = \frac{6}{1 + 5e^{-x}}$$ We have represented on the diagram below the representative curve $\mathscr{C}_f$ of the function $f$.

1. Show that point A with coordinates $(\ln 5 ; 3)$ belongs to the curve $\mathscr{C}_f$.
2. Show that the line with equation $y = 6$ is an asymptote to the curve $\mathscr{C}_f$.
3. a. We admit that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ its derivative function. Show that for every real $x$, we have: $$f'(x) = \frac{30\mathrm{e}^{-x}}{\left(1 + 5\mathrm{e}^{-x}\right)^2}.$$ b. Deduce the complete table of variations of $f$ on $\mathbb{R}$.
4. We admit that:
   • $f$ is twice differentiable on $\mathbb{R}$, we denote $f''$ its second derivative;
   • for every real $x$,
   $$f''(x) = \frac{30\mathrm{e}^{-x}\left(5\mathrm{e}^{-x} - 1\right)}{\left(1 + 5\mathrm{e}^{-x}\right)^3}.$$ a. Study the convexity of $f$ on $\mathbb{R}$. In particular, we will show that the curve $\mathscr{C}_f$ admits an inflection point. b. Justify that for every real $x$ belonging to $]-\infty ; \ln 5]$, we have: $f(x) \geqslant \frac{5}{6}x + 1$.
5. We consider a function $F_k$ defined on $\mathbb{R}$ by $F_k(x) = k\ln\left(\mathrm{e}^x + 5\right)$, where $k$ is a real constant. a. Determine the value of the real $k$ so that $F_k$ is a primitive of $f$ on $\mathbb{R}$. b. Deduce that the area, in square units, of the domain bounded by the curve $\mathscr{C}_f$, the $x$-axis, the $y$-axis and the line with equation $x = \ln 5$ is equal to $6\ln\left(\frac{5}{3}\right)$.

Part B The objective of this part is to study the following differential equation: $$(E) \quad y' = y - \frac{1}{6}y^2.$$ We recall that a solution of equation $(E)$ is a function $u$ defined and differentiable on $\mathbb{R}$ such that for every real $x$, we have: $$u'(x) = u(x) - \frac{1}{6}[u(x)]^2.$$

1. Show that the function $f$ defined in part A is a solution of the differential equation $(E)$.
2. Solve the differential equation $y' = -y + \frac{1}{6}$.
3. We denote by $g$ a function differentiable on $\mathbb{R}$ that does not vanish. We denote by $h$ the function defined on $\mathbb{R}$ by $h(x) = \frac{1}{g(x)}$. We admit that $h$ is differentiable on $\mathbb{R}$. We denote $g'$ and $h'$ the derivative functions of $g$ and $h$. a. Show that if $h$ is a solution of the differential equation $y' = -y + \frac{1}{6}$, then $g$ is a solution of the differential equation $y' = y - \frac{1}{6}y^2$. b. For every positive real $m$, we consider the functions $g_m$ defined on $\mathbb{R}$ by: $$g_m(x) = \frac{6}{1 + 6m\mathrm{e}^{-x}}.$$ Show that for every positive real $m$, the function $g_m$ is a solution of the differential equation $(E): \quad y' = y - \frac{1}{6}y^2$.

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.

2. Determine, by justifying, the limit of $f$ as $+ \infty$.
We will admit that the limit of $f$ at 0 is equal to 0.

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.

Question 151
Um capital de R\$ 5 000,00 é aplicado a juros simples de 2\% ao mês. Após 8 meses, o montante obtido será de
(A) R\$ 5 080,00 (B) R\$ 5 800,00 (C) R\$ 5 850,00 (D) R\$ 5 900,00 (E) R\$ 6 000,00

Question 164
Uma função $f$ é definida por $f(x) = 2^x$. O valor de $f(3) - f(1)$ é
(A) 2 (B) 4 (C) 6 (D) 8 (E) 16

Uma função exponencial é definida por $f(x) = 2^x$. O valor de $f(3) - f(1)$ é
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

In September 1987, Goiânia was the site of the largest radioactive accident that occurred in Brazil, when a sample of caesium-137, removed from an abandoned radiotherapy device, was inadvertently handled by part of the population. The half-life of a radioactive material is the time required for the mass of that material to be reduced to half. The half-life of caesium-137 is 30 years and the amount of remaining mass of a radioactive material, after $t$ years, is calculated by the expression $M(t) = A \cdot (2.7)^{kt}$, where $A$ is the initial mass and $k$ is a negative constant.
Consider 0.3 as an approximation for $\log_{10} 2$.
What is the time required, in years, for an amount of caesium-137 mass to be reduced to 10\% of the initial amount?
(A) 27 (B) 36 (C) 50 (D) 54 (E) 100

QUESTION 173
The function $f(x) = 3^x$ passes through the point
(A) $(0, 0)$
(B) $(0, 1)$
(C) $(1, 0)$
(D) $(1, 3)$
(E) $(3, 1)$