Let $y = f ( t )$ be a solution to the differential equation $\frac { d y } { d t } = k y$, where $k$ is a constant. Values of $f$ for selected values of $t$ are given in the table below:

$t$	0	2
$f ( t )$	4	12

Which of the following is an expression for $f ( t )$ ?
(A) $4 e ^ { \frac { t } { 2 } \ln 3 }$
(B) $e ^ { \frac { t } { 2 } \ln 9 } + 3$
(C) $2 t ^ { 2 } + 4$
(D) $4 t + 4$

Grass clippings are placed in a bin, where they decompose. For $0 \leq t \leq 30$, the amount of grass clippings remaining in the bin is modeled by $A ( t ) = 6.687 ( 0.931 ) ^ { t }$, where $A ( t )$ is measured in pounds and $t$ is measured in days.
(a) Find the average rate of change of $A ( t )$ over the interval $0 \leq t \leq 30$. Indicate units of measure.
(b) Find the value of $A ^ { \prime } ( 15 )$. Using correct units, interpret the meaning of the value in the context of the problem.
(c) Find the time $t$ for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval $0 \leq t \leq 30$.
(d) For $t > 30$, $L ( t )$, the linear approximation to $A$ at $t = 30$, is a better model for the amount of grass clippings remaining in the bin. Use $L ( t )$ to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer.

Exercise 1 -- Common to all candidates

Part A

Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{1-x}$$

1. Verify that for all real $x$, $f(x) = \mathrm{e} \times \frac{x}{\mathrm{e}^{x}}$.
2. Determine the limit of the function $f$ at $-\infty$.
3. Determine the limit of the function $f$ at $+\infty$. Interpret this limit graphically.
4. Determine the derivative of the function $f$.
5. Study the variations of the function $f$ on $\mathbb{R}$ then draw up the variation table.

Part B

For every non-zero natural number $n$, we consider the functions $g_n$ and $h_n$ defined on $\mathbb{R}$ by: $$g_n(x) = 1 + x + x^2 + \cdots + x^n \quad \text{and} \quad h_n(x) = 1 + 2x + \cdots + nx^{n-1}.$$

1. Verify that, for all real $x$: $(1-x)g_n(x) = 1 - x^{n+1}$.

We then obtain, for all real $x \neq 1$: $g_n(x) = \frac{1 - x^{n+1}}{1-x}$.
2. Compare the functions $h_n$ and $g_n'$, $g_n'$ being the derivative of the function $g_n$.
Deduce that, for all real $x \neq 1$: $h_n(x) = \frac{nx^{n+1} - (n+1)x^n + 1}{(1-x)^2}$.
3. Let $S_n = f(1) + f(2) + \ldots + f(n)$, $f$ being the function defined in Part A.
Using the results from Part B, determine an expression for $S_n$ then its limit as $n$ tends to $+\infty$.

Let $f$ be the differentiable function defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = \mathrm { e } ^ { x } + \frac { 1 } { x }$$
1. Study of an auxiliary function
a. Let the function $g$ be differentiable, defined on $[ 0 ; + \infty [$ by
$$g ( x ) = x ^ { 2 } \mathrm { e } ^ { x } - 1 .$$
Study the direction of variation of the function $g$.
b. Prove that there exists a unique real number $a$ belonging to $[ 0 ; + \infty [$ such that $g ( a ) = 0$.
Prove that $a$ belongs to the interval $[ 0{,}703 ; 0{,}704 [$.
c. Determine the sign of $g ( x )$ on $[ 0 ; + \infty [$.
2. Study of the function $f$
a. Determine the limits of the function $f$ at 0 and at $+ \infty$.
b. Let $f ^ { \prime }$ denote the derivative function of $f$ on the interval $] 0 ; + \infty [$.
Prove that for every strictly positive real number $x , f ^ { \prime } ( x ) = \frac { g ( x ) } { x ^ { 2 } }$.
c. Deduce the direction of variation of the function $f$ and draw its variation table on the interval $] 0 ; + \infty [$.
d. Prove that the function $f$ has as its minimum the real number
$$m = \frac { 1 } { a ^ { 2 } } + \frac { 1 } { a } .$$
e. Justify that $3{,}43 < m < 3{,}45$.

Part A

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

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

Part B

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

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

Exercise 2 (6 points)

We consider the function $f$ defined on $[ 0 ; + \infty [$ by
$$f ( x ) = 5 \mathrm { e } ^ { - x } - 3 \mathrm { e } ^ { - 2 x } + x - 3$$
We denote $\mathscr { C } _ { f }$ the graphical representation of the function $f$ and $\mathscr { D }$ the line with equation $y = x - 3$ in an orthogonal coordinate system of the plane.

Part A: Relative positions of $\mathscr { C } _ { f }$ and $\mathscr { D }$

Let $g$ be the function defined on the interval $[ 0 ; + \infty [$ by $g ( x ) = f ( x ) - ( x - 3 )$.

1. Justify that, for every real number $x$ in the interval $[ 0 ; + \infty [ , g ( x ) > 0$.
2. Do the curve $\mathscr { C } _ { f }$ and the line $\mathscr { D }$ have a common point? Justify.

Part B: Study of the function $\boldsymbol { g }$

We denote $M$ the point with abscissa $x$ of the curve $\mathscr { C } _ { f } , N$ the point with abscissa $x$ of the line $\mathscr { D }$ and we are interested in the evolution of the distance $M N$.

1. Justify that, for all $x$ in the interval $[ 0 ; + \infty [$, the distance $M N$ is equal to $g ( x )$.
2. We denote $g ^ { \prime }$ the derivative function of the function $g$ on the interval $[ 0 ; + \infty [$.

For all $x$ in the interval $\left[ 0 ; + \infty \left[ \right. \right.$, calculate $g ^ { \prime } ( x )$.
3. Show that the function $g$ has a maximum on the interval $[ 0 ; + \infty [$ which we will determine. Give a graphical interpretation of this.

Part C: Study of an area

We consider the function $\mathscr { A }$ defined on the interval $[ 0 ; + \infty [$ by
$$\mathscr { A } ( x ) = \int _ { 0 } ^ { x } [ f ( t ) - ( t - 3 ) ] \mathrm { d } t$$

1. Shade on the graph given in Appendix 1 (to be returned with your work) the region whose area is given by $\mathscr { A } ( 2 )$.
2. Justify that the function $\mathscr { A }$ is increasing on the interval $[ 0 ; + \infty [$.
3. For all positive real $x$, calculate $\mathscr { A } ( x )$.
4. Does there exist a value of $x$ such that $\mathscr { A } ( x ) = 2$ ?

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

Part A

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

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

Part B

Consider the following algorithm in which the variables are

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

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

\hline Processing &

For $i$ ranging from 1 to $K$

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

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

End For

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

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

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

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

Exercise 1 -- Common to all candidates

Part A

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

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

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

Part B

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

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

Pharmacokinetics studies the evolution of a drug after its administration in the body, by measuring its plasma concentration, that is to say its concentration in the plasma. In this exercise we study the evolution of plasma concentration in a patient of the same dose of drug, considering different modes of administration.
Part A: administration by intravenous route
We denote $f ( t )$ the plasma concentration, expressed in microgram per litre ( $\mu \mathrm { g } . \mathrm { L } ^ { - 1 }$ ), of the drug, after $t$ hours following administration by intravenous route. The mathematical model is: $f ( t ) = 20 \mathrm { e } ^ { - 0,1 t }$, with $t \in [ 0 ; + \infty [$. The initial plasma concentration of the drug is therefore $f ( 0 ) = 20 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$.

1. The half-life of the drug is the duration (in hours) after which the plasma concentration of the drug is equal to half the initial concentration. Determine this half-life, denoted $t _ { 0,5 }$.
2. It is estimated that the drug is eliminated as soon as the plasma concentration is less than $0.2 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. Determine the time from which the drug is eliminated. The result will be given rounded to the nearest tenth.
3. In pharmacokinetics, we call AUC (or ``area under the curve''), in $\mu \mathrm { g } . \mathrm { L } ^ { - 1 }$, the number $\lim _ { x \rightarrow + \infty } \int _ { 0 } ^ { x } f ( t ) \mathrm { d } t$. Verify that for this model, the AUC is equal to $200 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$.

Part B: administration by oral route
We denote $g ( t )$ the plasma concentration of the drug, expressed in microgram per litre ( $\mu g.L^{-1}$ ), after $t$ hours following ingestion by oral route. The mathematical model is: $g ( t ) = 20 \left( \mathrm { e } ^ { - 0,1 t } - \mathrm { e } ^ { - t } \right)$, with $t \in [ 0 ; + \infty [$. In this case, the effect of the drug is delayed, since the initial plasma concentration is equal to: $g ( 0 ) = 0 \mu g . \mathrm { L } ^ { - 1 }$.

1. Prove that, for all $t$ in the interval $[ 0 ; + \infty [$, we have: $g ^ { \prime } ( t ) = 20 \mathrm { e } ^ { - t } \left( 1 - 0,1 \mathrm { e } ^ { 0,9 t } \right)$.
2. Study the variations of the function $g$ on the interval $[ 0 ; + \infty [$. (The limit at $+ \infty$ is not required.) Deduce the duration after which the plasma concentration of the drug is maximum. The result will be given to the nearest minute.

Part C: repeated administration by intravenous route
We decide to inject at regular time intervals the same dose of drug by intravenous route. The time interval (in hours) between two injections is chosen equal to the half-life of the drug, that is to say the number $t _ { 0,5 }$ which was calculated in A - 1. Each new injection causes an increase in plasma concentration of $20 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. We denote $u _ { n }$ the plasma concentration of the drug immediately after the $n$-th injection. Thus, $u _ { 1 } = 20$ and, for all integer $n$ greater than or equal to 1, we have: $u _ { n + 1 } = 0,5 u _ { n } + 20$.

1. Prove by induction that, for all integer $n \geqslant 1 : u _ { n } = 40 - 40 \times 0,5 ^ { n }$.
2. Determine the limit of the sequence $( u _ { n } )$ as $n$ tends to $+ \infty$.
3. We consider that equilibrium is reached as soon as the plasma concentration exceeds $38 \mu \mathrm {~g} . \mathrm { L } ^ { - 1 }$. Determine the minimum number of injections necessary to reach this equilibrium.

A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:
$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$
The clearance $a$ of a certain patient is 7, and we choose an infusion rate $d$ equal to 84. In this part, the function $C$ is therefore defined on $[0; +\infty[$ by:
$$C ( t ) = 12 \left( 1 - \mathrm { e } ^ { - \frac { 7 } { 80 } t } \right)$$

1. Study the monotonicity of the function $C$ on $[0; +\infty[$.
2. For the treatment to be effective, the plateau must equal 15. Is the treatment of this patient effective?

A treatment protocol for a disease in children involves long-term infusion of an appropriate medication. The concentration of the medication in the blood over time is modeled by the function $C$ defined on the interval $[0; +\infty[$ by:
$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$
where $C$ denotes the concentration of the medication in the blood (in micromoles per liter), $t$ the time elapsed since the start of the infusion (in hours), $d$ the infusion rate (in micromoles per hour), $a$ a strictly positive real parameter called clearance (in liters per hour).
Part C: determination of appropriate treatment
The purpose of this part is to determine, for a given patient, the value of the infusion rate that allows the treatment to be effective, that is, the plateau to equal 15. The infusion rate $d$ is provisionally set to 105.

1. We seek to determine the clearance $a$ of a patient. The infusion rate is provisionally set to 105. a. Express as a function of $a$ the concentration of the medication 6 hours after the start of the infusion. b. After 6 hours, analyses allow us to know the concentration of the medication in the blood; it is equal to 5.9 micromoles per liter. Determine an approximate value, to the nearest tenth of a liter per hour, of the clearance of this patient.
2. Determine the value of the infusion rate $d$ guaranteeing the effectiveness of the treatment.

An aquaculture farm operates a shrimp population that evolves according to natural reproduction and harvesting. The initial mass of this shrimp population is estimated at 100 tonnes. Given the reproduction and harvesting conditions, the mass of the shrimp population, expressed in tonnes, as a function of time, expressed in weeks, is modelled by the function $f _ { p }$, defined on the interval $[ 0 ; + \infty [$ by :
$$f _ { p } ( t ) = \frac { 100 p } { 1 - ( 1 - p ) \mathrm { e } ^ { - p t } }$$
where $p$ is a parameter strictly between 0 and 1 and which depends on the various living and exploitation conditions of the shrimp.

1. Model consistency a. Calculate $f _ { p } ( 0 )$. b. Recall that $0 < p < 1$.

Prove that for all real number $t \geqslant 0,1 - ( 1 - p ) \mathrm { e } ^ { - p t } \geqslant p$. c. Deduce that for all real number $t \geqslant 0,0 < f _ { p } ( t ) \leqslant 100$.
2. Study of evolution when $p = 0.9$
In this question, we take $p = 0.9$ and study the function $f _ { 0.9 }$ defined on $[ 0 ; + \infty [$ by :
$$f _ { 0.9 } ( t ) = \frac { 90 } { 1 - 0.1 \mathrm { e } ^ { - 0.9 t } }$$
a. Determine the variations of the function $f _ { 0.9 }$. b. Prove that for all real number $t \geqslant 0 , f _ { 0.9 } ( t ) \geqslant 90$. c. Interpret the results of questions 2. a. and 2. b. in context.
3. Return to the general case
Recall that $0 < p < 1$. Express as a function of $p$ the limit of $f _ { p }$ as $t$ tends to $+ \infty$.
4. In this question, we take $p = \frac { 1 } { 2 }$. a. Show that the function $H$ defined on the interval $[ 0 ; + \infty [$ by :
$$H ( t ) = 100 \ln \left( 2 - \mathrm { e } ^ { - \frac { t } { 2 } } \right) + 50 t$$
is an antiderivative of the function $f _ { 1/2 }$ on this interval. b. Deduce the average mass of shrimp during the first 5 weeks of exploitation, that is the average value of the function $f _ { 1/2 }$ on the interval $[ 0 ; 5 ]$. Give an approximate value rounded to the nearest tonne.

In a factory, a kiln bakes ceramics at a temperature of $1000^{\circ}\mathrm{C}$. At the end of baking, it is turned off and cools down. The temperature of the kiln is expressed in degrees Celsius (${}^{\circ}\mathrm{C}$). The kiln door can be opened safely for the ceramics as soon as its temperature is below $70^{\circ}\mathrm{C}$.
In this part, we denote $t$ the time (in hours) elapsed since the moment the kiln was turned off. The temperature of the kiln (in degrees Celsius) at time $t$ is given by the function $f$ defined, for every positive real number $t$, by: $$f(t) = a\mathrm{e}^{-\frac{t}{5}} + b,$$ where $a$ and $b$ are two real numbers. We admit that $f$ satisfies the following relation: $f'(t) + \frac{1}{5}f(t) = 4$.

1. Determine the values of $a$ and $b$ knowing that initially, the temperature of the kiln is $1000^{\circ}\mathrm{C}$, that is $f(0) = 1000$.
2. For the following, we admit that, for every positive real number $t$: $$f(t) = 980\mathrm{e}^{-\frac{t}{5}} + 20.$$ a. Determine the limit of $f$ as $t$ tends to $+\infty$. b. Study the variations of $f$ on $[0; +\infty[$. Deduce its complete table of variations. c. With this model, after how many minutes can the kiln be opened safely for the ceramics?
3. The average temperature (in degrees Celsius) of the kiln between two times $t_1$ and $t_2$ is given by: $\frac{1}{t_2 - t_1}\int_{t_1}^{t_2} f(t)\,\mathrm{d}t$. a. Using the graphical representation of $f$, give an estimate of the average temperature $\theta$ of the kiln over the first 15 hours of cooling. Explain your approach. b. Calculate the exact value of this average temperature $\theta$ and give its value rounded to the nearest degree Celsius.
4. In this question, we are interested in the temperature drop (in degrees Celsius) of the kiln over the course of one hour, that is between two times $t$ and $(t+1)$. This drop is given by the function $d$ defined, for every positive real number $t$, by: $d(t) = f(t) - f(t+1)$. a. Verify that, for every positive real number $t$: $d(t) = 980\left(1 - \mathrm{e}^{-\frac{1}{5}}\right)\mathrm{e}^{-\frac{t}{5}}$. b. Determine the limit of $d(t)$ as $t$ tends to $+\infty$. What interpretation can be given to this?

Exercise 2

4 points

Common to all candidates

When the tail of a wall lizard breaks, it regrows on its own in about sixty days. During regrowth, the length in centimeters of the lizard's tail is modeled as a function of the number of days. This length is modeled by the function $f$ defined on $[ 0 ; + \infty [$ by:
$$f ( x ) = 10 \mathrm { e } ^ { u ( x ) }$$
where $u$ is the function defined on $[ 0 ; + \infty [$ by:
$$u ( x ) = - \mathrm { e } ^ { 2 - \frac { x } { 10 } }$$
It is admitted that the function $f$ is differentiable on $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $f ^ { \prime }$ its derivative function.

1. Verify that for all positive $x$ we have $f ^ { \prime } ( x ) = - u ( x ) \mathrm { e } ^ { u ( x ) }$.

Deduce the direction of variation of the function $f$ on $[ 0 ; + \infty [$.
2. a. Calculate $f ( 20 )$.
Deduce an estimate, rounded to the nearest millimeter, of the length of the lizard's tail after twenty days of regrowth. b. According to this model, can the lizard's tail measure 11 cm?
3. We wish to determine after how many days the growth rate is maximum.
It is admitted that the growth rate after $x$ days is given by $f ^ { \prime } ( x )$. It is admitted that the derivative function $f ^ { \prime }$ is differentiable on $\left[ 0 ; + \infty \left[ \right. \right.$, we denote $f ^ { \prime \prime }$ the derivative function of $f ^ { \prime }$ and it is admitted that:
$$f ^ { \prime \prime } ( x ) = \frac { 1 } { 10 } u ( x ) \mathrm { e } ^ { u ( x ) } ( 1 + u ( x ) )$$
a. Determine the variations of $f ^ { \prime }$ on $[ 0 ; + \infty [$. b. Deduce after how many days the growth rate of the length of the lizard's tail is maximum.

Exercise 1 (6 points) -- Part A
Let $a$ and $b$ be real numbers. We consider a function $f$ defined on $[ 0 ; + \infty [$ by $$f ( x ) = \frac { a } { 1 + \mathrm { e } ^ { - b x } }$$ The curve $\mathscr { C } _ { f }$ representing the function $f$ in an orthogonal coordinate system is given. The curve $\mathscr { C } _ { f }$ passes through the point $\mathrm { A } ( 0 ; 0.5 )$. The tangent line to the curve $\mathscr { C } _ { f }$ at point A passes through the point B(10; 1).

1. Justify that $a = 1$.

We then obtain, for all real $x \geqslant 0$, $$f ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - b x } }$$

1. It is admitted that the function $f$ is differentiable on $\left[ 0 ; + \infty \left[ \right. \right.$ and we denote $f ^ { \prime }$ its derivative function. Verify that, for all real $x \geqslant 0$ $$f ^ { \prime } ( x ) = \frac { b \mathrm { e } ^ { - b x } } { \left( 1 + \mathrm { e } ^ { - b x } \right) ^ { 2 } }$$
2. Using the data from the problem statement, determine $b$.

Exercise 1 -- Part B
The proportion of individuals who possess a certain type of equipment in a population is modelled by the function $p$ defined on $[ 0 ; + \infty [$ by $$p ( x ) = \frac { 1 } { 1 + \mathrm { e } ^ { - 0,2 x } } .$$ The real number $x$ represents the time elapsed, in years, since January 1st, 2000. The number $p ( x )$ models the proportion of equipped individuals after $x$ years. Thus, for this model, $p ( 0 )$ is the proportion of equipped individuals on January 1st, 2000 and $p ( 3.5 )$ is the proportion of equipped individuals in the middle of 2003.

1. What is, for this model, the proportion of equipped individuals on January 1st, 2010? Give a value rounded to the nearest hundredth.
2. 1. [a.] Determine the direction of variation of the function $p$ on $[ 0 ; + \infty [$.
   2. [b.] Calculate the limit of the function $p$ as $x \to + \infty$.
   3. [c.] Interpret this limit in the context of the exercise.
3. It is considered that, when the proportion of equipped individuals exceeds $95\%$, the market is saturated. Determine, by explaining the approach, the year in which this occurs.
4. The average proportion of equipped individuals between 2008 and 2010 is defined by $$m = \frac { 1 } { 2 } \int _ { 8 } ^ { 10 } p ( x ) \mathrm { d } x$$
   1. [a.] Verify that, for all real $x \geqslant 0$, $$p ( x ) = \frac { \mathrm { e } ^ { 0,2 x } } { 1 + \mathrm { e } ^ { 0,2 x } }$$
   2. [b.] Deduce an antiderivative of the function $p$ on $[ 0 ; + \infty [$.
   3. [c.] Determine the exact value of $m$ and its approximation to the nearest hundredth.

Exercise 1 -- Part A

We consider the function $f$ defined on the set $\mathbb{R}$ of real numbers by: $$f(x) = \frac{7}{2} - \frac{1}{2}\left(\mathrm{e}^{x} + \mathrm{e}^{-x}\right)$$

1. a. Determine the limit of the function $f$ at $+\infty$. b. Show that the function $f$ is strictly decreasing on the interval $[0; +\infty[$. c. Show that the equation $f(x) = 0$ admits, on the interval $[0; +\infty[$, a unique solution, which we denote $\alpha$.
2. By noting that, for all real $x$, $f(-x) = f(x)$, justify that the equation $f(x) = 0$ admits exactly two solutions in $\mathbb{R}$ and that they are opposite.

Part B

The plane is given an orthonormal coordinate system with unit 1 meter. The function $f$ and the real number $\alpha$ are defined in Part A. In the rest of the exercise, we model a greenhouse arch by the curve $\mathscr{C}$ of the function $f$ on the interval $[-\alpha; +\alpha]$.

1. Calculate the height of an arch.
2. a. In this question, we propose to calculate the exact value of the length of the curve $\mathscr{C}$ on the interval $[0; \alpha]$. It is admitted that this length is given, in meters, by the integral: $$I = \int_{0}^{\alpha} \sqrt{1 + \left(f'(x)\right)^{2}} \, dx$$ Show that, for all real $x$, we have: $1 + \left(f'(x)\right)^{2} = \frac{1}{4}\left(\mathrm{e}^{x} + \mathrm{e}^{-x}\right)^{2}$ b. Deduce the value of the integral $I$ as a function of $\alpha$. Justify that the length of an arch, in meters, is equal to: $\mathrm{e}^{\alpha} - \mathrm{e}^{-\alpha}$.

Part C

We wish to build a garden greenhouse in the shape of a tunnel. We fix four metal arches to the ground, whose shape is that described in the previous part, spaced 1.5 meters apart. On the south facade, we plan an opening modeled by the rectangle $ABCD$ with width 1 meter and length 2 meters.

1. Show that the quantity of sheet necessary to cover the south and north facades is given, in $m^2$, by: $$\mathscr{A} = 4\int_{0}^{\alpha} f(x)\,dx - 2$$
2. We take 1.92 as an approximate value of $\alpha$. Determine, to the nearest $\mathrm{m}^2$, the total area of plastic sheet necessary to build this greenhouse.

We consider the function $f$ defined on $\mathbb { R }$ by:
$$f ( x ) = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { x } + 1 }$$
The representative curve $\mathscr { C }$ of the function $f$ in an orthonormal coordinate system is given.

1. Calculate the limit of the function $f$ at negative infinity and interpret the result graphically.
2. Show that the line with equation $y = 2$ is a horizontal asymptote to the curve $\mathscr { C }$.
3. Calculate $f ^ { \prime } ( x )$, where $f ^ { \prime }$ is the derivative function of $f$, and verify that for all real numbers $x$ we have: $$f ^ { \prime } ( x ) = \frac { f ( x ) } { \mathrm { e } ^ { x } + 1 } .$$
4. Show that the function $f$ is increasing on $\mathbb { R }$.
5. Show that the curve $\mathscr { C }$ passes through the point $\mathrm { I } ( 0 ; 1 )$ and that its tangent at this point has slope 0.5.

Part A

The function $g$ is defined on $[ 0 ; + \infty [$ by $$g ( x ) = 1 - \mathrm { e } ^ { - x } .$$ We admit that the function $g$ is differentiable on $[ 0 ; + \infty [$.

1. Determine the limit of the function $g$ at $+ \infty$.
2. Study the variations of the function $g$ on $[ 0 ; + \infty [$ and draw its variation table.

Part A
Let $g$ be the function defined on the set of real numbers $\mathbf { R }$, by $$g ( x ) = x ^ { 2 } + x + \frac { 1 } { 4 } + \frac { 4 } { \left( 1 + \mathrm { e } ^ { x } \right) ^ { 2 } }$$
It is admitted that the function $g$ is differentiable on $\mathbf { R }$ and we denote by $g ^ { \prime }$ its derivative function.
1. Determine the limits of $g$ at $+ \infty$ and at $- \infty$.
2. It is admitted that the function $g ^ { \prime }$ is strictly increasing on $\mathbf { R }$ and that $g ^ { \prime } ( 0 ) = 0$.
Determine the sign of the function $g ^ { \prime }$ on $\mathbf { R }$.
3. Draw up the table of variations of the function $g$ and calculate the minimum of the function $g$ on $\mathbf { R }$.
Part B
Let $f$ be the function defined on $\mathbf { R }$ by: $$f ( x ) = 3 - \frac { 2 } { 1 + \mathrm { e } ^ { x } }$$
We denote by $\mathscr { C } _ { f }$ the representative curve of $f$ in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ).
Let A be the point with coordinates $\left( - \frac { 1 } { 2 } ; 3 \right)$.
1. Prove that point $\mathrm { B } ( 0 ; 2 )$ belongs to $\mathscr { C } _ { f }$.
2. Let $x$ be any real number. We denote by $M$ the point on the curve $\mathscr { C } _ { f }$ with coordinates $( x ; f ( x ) )$.
Prove that $\mathrm { A } M ^ { 2 } = g ( x )$.
3. It is admitted that the distance $\mathrm { A } M$ is minimal if and only if $\mathrm { A } M ^ { 2 }$ is minimal.
Determine the coordinates of the point on the curve $\mathscr { C } _ { f }$ such that the distance AM is minimal.
4. It is admitted that the function $f$ is differentiable on $\mathbf { R }$ and we denote by $f ^ { \prime }$ its derivative function.
a. Calculate $f ^ { \prime } ( x )$ for all real $x$.
b. Let $T$ be the tangent to the curve $\mathscr { C } _ { f }$ at point B.
Prove that the reduced equation of $T$ is $y = \frac { x } { 2 } + 2$.
5. Prove that the line $T$ is perpendicular to the line (AB).

For each of the following statements, indicate whether it is true or false. You will justify each answer.
Statement 1: For all real numbers $a$ and $b$, $\left( \mathrm{e}^{a+b} \right)^{2} = \mathrm{e}^{2a} + \mathrm{e}^{2b}$.
Statement 2: In the plane with a coordinate system, the tangent line at point A with abscissa 0 to the representative curve of the function $f$ defined on $\mathbb{R}$ by $f(x) = -2 + (3-x)\mathrm{e}^{x}$ has the reduced equation $y = 2x + 1$.
Statement 3: $\lim_{x \rightarrow +\infty} \left( \mathrm{e}^{2x} - \mathrm{e}^{x} + \frac{3}{x} \right) = 0$.
Statement 4: The equation $1 - x + \mathrm{e}^{-x} = 0$ has a unique solution belonging to the interval $[0 ; 2]$.
Statement 5: The function $g$ defined on $\mathbb{R}$ by $g(x) = x^{2} - 5x + \mathrm{e}^{x}$ is convex.

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 $-2$.

Main topics covered: Exponential function; differentiation.
The graph below represents, in an orthogonal coordinate system, the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ defined on $\mathbb{R}$ by:
$$f(x) = x^{2}\mathrm{e}^{-x} \text{ and } g(x) = \mathrm{e}^{-x}.$$
Question 3 is independent of questions 1 and 2.

1. a. Determine the coordinates of the intersection points of $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$. b. Study the relative position of the curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$.
2. For every real number $x$ in the interval $[-1; 1]$, we consider the points $M$ with coordinates $(x; f(x))$ and $N$ with coordinates $(x; g(x))$, and we denote by $d(x)$ the distance $MN$. We assume that: $d(x) = \mathrm{e}^{-x} - x^{2}\mathrm{e}^{-x}$. We assume that the function $d$ is differentiable on the interval $[-1; 1]$ and we denote by $d^{\prime}$ its derivative function. a. Show that $d^{\prime}(x) = \mathrm{e}^{-x}\left(x^{2} - 2x - 1\right)$. b. Deduce the variations of the function $d$ on the interval $[-1; 1]$. c. Determine the common abscissa $x_{0}$ of the points $M_{0}$ and $N_{0}$ allowing to obtain a maximum distance $d(x_{0})$, and give an approximate value to 0.1 of the distance $M_{0}N_{0}$.
3. Let $\Delta$ be the line with equation $y = x + 2$. We consider the function $h$ differentiable on $\mathbb{R}$ and defined by: $h(x) = \mathrm{e}^{-x} - x - 2$. By studying the number of solutions of the equation $h(x) = 0$, determine the number of intersection points of the line $\Delta$ and the curve $\mathscr{C}_{g}$.

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 $[ 0 ; + \infty [$?

Question 4: Consider the function $f$ defined on $\mathbb{R}$ by $f(x) = 3\mathrm{e}^x - x$.

a. $\lim_{x\rightarrow+\infty} f(x) = 3$	b. $\lim_{x\rightarrow+\infty} f(x) = +\infty$	c. $\lim_{x\rightarrow+\infty} f(x) = -\infty$	\begin{tabular}{l} d. We cannot
determine the limit
of the function $f$
as $x$ tends to
$+\infty$

\hline \end{tabular}