A particle moves along the $x$-axis with its position at time $t$ given by $x ( t ) = ( t - a ) ( t - b )$, where $a$ and $b$ are constants and $a \neq b$. For which of the following values of $t$ is the particle at rest?
(A) $t = a b$
(B) $t = \frac { a + b } { 2 }$
(C) $t = a + b$
(D) $t = 2 ( a + b )$
(E) $t = a$ and $t = b$

For $t \geq 0$, the position of a particle moving along the $x$-axis is given by $x ( t ) = \sin t - \cos t$. What is the acceleration of the particle at the point where the velocity is first equal to 0 ?
(A) $- \sqrt { 2 }$
(B) $-1$
(C) 0
(D) 1
(E) $\sqrt { 2 }$

A particle moves along a line so that its acceleration for $t \geq 0$ is given by $a ( t ) = \frac { t + 3 } { \sqrt { t ^ { 3 } + 1 } }$. If the particle's velocity at $t = 0$ is 5, what is the velocity of the particle at $t = 3$ ?
(A) 0.713
(B) 1.134
(C) 6.134
(D) 6.710
(E) 11.710

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.

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.

The function $f$ is differentiable on the closed interval $[-6, 5]$ and satisfies $f(-2) = 7$. The graph of $f'$, the derivative of $f$, consists of a semicircle and three line segments, as shown in the figure.
(a) Find the values of $f(-6)$ and $f(5)$.
(b) On what intervals is $f$ increasing? Justify your answer.
(c) Find the absolute minimum value of $f$ on the closed interval $[-6, 5]$. Justify your answer.
(d) For each of $f''(-5)$ and $f''(3)$, find the value or explain why it does not exist.

The graph of the continuous function $g$, the derivative of the function $f$, is shown above. The function $g$ is piecewise linear for $- 5 \leq x < 3$, and $g ( x ) = 2 ( x - 4 ) ^ { 2 }$ for $3 \leq x \leq 6$.
(a) If $f ( 1 ) = 3$, what is the value of $f ( - 5 )$ ?
(b) Evaluate $\int _ { 1 } ^ { 6 } g ( x ) d x$.
(c) For $- 5 < x < 6$, on what open intervals, if any, is the graph of $f$ both increasing and concave up? Give a reason for your answer.
(d) Find the $x$-coordinate of each point of inflection of the graph of $f$. Give a reason for your answer.

Let $f$ be a continuous function defined on the closed interval $- 4 \leq x \leq 6$. The graph of $f$, consisting of four line segments, is shown above. Let $G$ be the function defined by $G ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.
(a) On what open intervals is the graph of $G$ concave up? Give a reason for your answer.
(b) Let $P$ be the function defined by $P ( x ) = G ( x ) \cdot f ( x )$. Find $P ^ { \prime } ( 3 )$.
(c) Find $\lim _ { x \rightarrow 2 } \frac { G ( x ) } { x ^ { 2 } - 2 x }$.
(d) Find the average rate of change of $G$ on the interval $[ - 4,2 ]$. Does the Mean Value Theorem guarantee a value $c , - 4 < c < 2$, for which $G ^ { \prime } ( c )$ is equal to this average rate of change? Justify your answer.

Let $f$ be a differentiable function with $f ( 4 ) = 3$. On the interval $0 \leq x \leq 7$, the graph of $f ^ { \prime }$, the derivative of $f$, consists of a semicircle and two line segments, as shown in the figure above.
(a) Find $f ( 0 )$ and $f ( 5 )$.
(b) Find the $x$-coordinates of all points of inflection of the graph of $f$ for $0 < x < 7$. Justify your answer.
(c) Let $g$ be the function defined by $g ( x ) = f ( x ) - x$. On what intervals, if any, is $g$ decreasing for $0 \leq x \leq 7$ ? Show the analysis that leads to your answer.
(d) For the function $g$ defined in part (c), find the absolute minimum value on the interval $0 \leq x \leq 7$. Justify your answer.

Throughout what follows, $m$ denotes any real number.
Part A
Let $f$ be the function defined and differentiable on the set of real numbers $\mathbb{R}$ such that: $$f(x) = (x+1)\mathrm{e}^x$$

1. Calculate the limit of $f$ at $+\infty$ and $-\infty$.
2. We denote by $f'$ the derivative function of $f$ on $\mathbb{R}$. Prove that for all real $x$, $f'(x) = (x+2)\mathrm{e}^x$.
3. Draw the variation table of $f$ on $\mathbb{R}$.

Part B
We define the function $g_m$ on $\mathbb{R}$ by: $$g_m(x) = x + 1 - m\mathrm{e}^{-x}$$ and we denote by $\mathscr{C}_m$ the curve of function $g_m$ in a frame $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$ of the plane.

1. a. Prove that $g_m(x) = 0$ if and only if $f(x) = m$. b. Deduce from Part $A$, without justification, the number of intersection points of curve $\mathscr{C}_m$ with the $x$-axis as a function of the real number $m$.
2. We have represented in appendix 2 the curves $\mathscr{C}_0$, $\mathscr{C}_{\mathrm{e}}$, and $\mathscr{C}_{-\mathrm{e}}$ (obtained by taking respectively for $m$ the values 0, e and $-$e). Identify each of these curves in the figure of appendix 2 by justifying.
3. Study the position of curve $\mathscr{C}_m$ relative to the line $\mathscr{D}$ with equation $y = x + 1$ according to the values of the real number $m$.
4. a. We call $D_2$ the part of the plane between curves $\mathscr{C}_{\mathrm{e}}$, $\mathscr{C}_{-\mathrm{e}}$, the axis $(Oy)$ and the line $x = 2$. Shade $D_2$ on appendix 2. b. In this question, $a$ denotes a positive real number, $D_a$ the part of the plane between $\mathscr{C}_{\mathrm{e}}$, $\mathscr{C}_{-\mathrm{e}}$, the axis $(Oy)$ and the line $\Delta_a$ with equation $x = a$. We denote by $\mathscr{A}(a)$ the area of this part of the plane, expressed in square units. Prove that for all positive real $a$: $\mathscr{A}(a) = 2\mathrm{e} - 2\mathrm{e}^{1-a}$. Deduce the limit of $\mathscr{A}(a)$ as $a$ tends to $+\infty$.

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 $+\infty$.

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 $\mathscr { A } ( x ) = 2$ ?

Exercise 2

We consider the function $f$ defined and differentiable on the set $\mathbb { R }$ of real numbers by
$$f ( x ) = x + 1 + \frac { x } { \mathrm { e } ^ { x } }$$
We denote by $\mathscr { C }$ its representative curve in an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath }$ ).

Part A

1. Let $g$ be the function defined and differentiable on the set $\mathbb { R }$ by $$g ( x ) = 1 - x + \mathrm { e } ^ { x }$$ Draw up, by justifying, the table giving the variations of the function $g$ on $\mathbb { R }$ (the limits of $g$ at the boundaries of its domain are not required). Deduce the sign of $g ( x )$.
2. Determine the limit of $f$ at $- \infty$ then the limit of $f$ at $+ \infty$.
3. We call $f ^ { \prime }$ the derivative of the function $f$ on $\mathbb { R }$. Prove that, for all real $x$, $$f ^ { \prime } ( x ) = \mathrm { e } ^ { - x } g ( x )$$
4. Deduce the variation table of the function $f$ on $\mathbb { R }$.
5. Prove that the equation $f ( x ) = 0$ admits a unique real solution $\alpha$ on $\mathbb { R }$. Prove that $- 1 < \alpha < 0$.
6. a. Prove that the line T with equation $y = 2 x + 1$ is tangent to the curve $\mathscr { C }$ at the point with abscissa 0. b. Study the relative position of the curve $\mathscr { C }$ and the line T.

Part B

1. Let H be the function defined and differentiable on $\mathbb { R }$ by $$\mathrm { H } ( x ) = ( - x - 1 ) \mathrm { e } ^ { - x }.$$ Prove that H is a primitive on $\mathbb { R }$ of the function $h$ defined by $h ( x ) = x \mathrm { e } ^ { - x }$.
2. We denote by $\mathscr { D }$ the domain bounded by the curve $\mathscr { C }$, the line T and the lines with equations $x = 1$ and $x = 3$. Calculate, in square units, the area of the domain $\mathscr { D }$.

For each of the following propositions, indicate whether it is true or false and justify your chosen answer. One point is awarded for each correct answer that is properly justified. An answer without justification is not taken into account. An absence of an answer is not penalized.
Proposition 1 Every positive increasing sequence tends to $+ \infty$.
Proposition 2 $g$ is the function defined on $] - \frac { 1 } { 2 } ; + \infty [$ by $$g ( x ) = 2 x \ln ( 2 x + 1 ) .$$ On $] - \frac { 1 } { 2 } ; + \infty$ [, the equation $g ( x ) = 2 x$ has a unique solution: $\frac { \mathrm { e } - 1 } { 2 }$.
Proposition 3 The slope of the tangent line to the curve representing the function $g$ at the point with abscissa $\frac { 1 } { 2 }$ is: $1 + \ln 4$.
Proposition 4 Space is equipped with an orthonormal coordinate system ( $\mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k }$ ). $\mathscr { P }$ and $\mathscr { R }$ are the planes with equations respectively: $2 x + 3 y - z - 11 = 0$ and $x + y + 5 z - 11 = 0$. The planes $\mathscr { P }$ and $\mathscr { R }$ intersect perpendicularly.

Let $f$ be the function defined on the interval $[0; +\infty[$ by
$$f(x) = x\mathrm{e}^{-x}$$
We denote by $\mathscr{C}$ the representative curve of $f$ in an orthogonal coordinate system.

Part A

1. We denote by $f'$ the derivative function of the function $f$ on the interval $[0; +\infty[$. For every real number $x$ in the interval $[0; +\infty[$, calculate $f'(x)$. Deduce the variations of the function $f$ on the interval $[0; +\infty[$.
2. Determine the limit of the function $f$ at $+\infty$. What graphical interpretation can be made of this result?

Part B

Let $\mathscr{A}$ be the function defined on the interval $[0; +\infty[$ as follows: for every real number $t$ in the interval $[0; +\infty[$, $\mathscr{A}(t)$ is the area, in square units, of the region bounded by the $x$-axis, the curve $\mathscr{C}$ and the lines with equations $x = 0$ and $x = t$.

1. Determine the direction of variation of the function $\mathscr{A}$.
2. We admit that the area of the region bounded by the curve $\mathscr{C}$ and the $x$-axis is equal to 1 square unit. What can we deduce about the function $\mathscr{A}$?
3. We seek to prove the existence of a real number $\alpha$ such that the line with equation $x = \alpha$ divides the region between the $x$-axis and the curve $\mathscr{C}$ into two parts of equal area, and to find an approximate value of this real number. a) Prove that the equation $\mathscr{A}(t) = \frac{1}{2}$ has a unique solution on the interval $[0; +\infty[$ b) On the graph provided in the appendix (to be returned with the answer sheet) are drawn the curve $\mathscr{C}$, as well as the curve $\Gamma$ representing the function $\mathscr{A}$. On the graph in the appendix, identify the curves $\mathscr{C}$ and $\Gamma$, then draw the line with equation $y = \frac{1}{2}$. Deduce an approximate value of the real number $\alpha$. Shade the region corresponding to $\mathscr{A}(\alpha)$.
4. We define the function $g$ on the interval $[0; +\infty[$ by $$g(x) = (x + 1)\mathrm{e}^{-x}$$ a) We denote by $g'$ the derivative function of the function $g$ on the interval $[0; +\infty[$. For every real number $x$ in the interval $[0; +\infty[$, calculate $g'(x)$. b) Deduce, for every real number $t$ in the interval $[0; +\infty[$, an expression for $\mathscr{A}(t)$. c) Calculate an approximate value to $10^{-2}$ of $\mathscr{A}(6)$.

It is desired to create a gate. Each leaf measures 2 metres wide.

Part A: modelling the upper part of the gate

The upper edge of the right leaf of the gate is modelled with a function $f$ defined on the interval [0;2] by
$$f ( x ) = \left( x + \frac { 1 } { 4 } \right) \mathrm { e } ^ { - 4 x } + b$$
where $b$ is a real number. Let $f ^ { \prime }$ denote the derivative function of $f$ on the interval $[ 0 ; 2 ]$.

1. a. Calculate $f ^ { \prime } ( x )$, for every real $x$ belonging to the interval $[ 0 ; 2 ]$. b. Deduce the direction of variation of the function $f$ on the interval $[ 0 ; 2 ]$.
2. Determine the number $b$ so that the maximum height of the gate is equal to $1{,}5 \mathrm{~m}$.

In the following, the function $f$ is defined on the interval $[ 0 ; 2 ]$ by
$$f ( x ) = \left( x + \frac { 1 } { 4 } \right) \mathrm { e } ^ { - 4 x } + \frac { 5 } { 4 }$$

Part B: determination of an area

Each leaf is made using a metal plate. We want to calculate the area of each plate, knowing that the lower edge of the leaf is at $0{,}05 \mathrm{~m}$ height from the ground.

1. Show that the function $F$ defined on the interval $[ 0 ; 2 ]$ by $$F ( x ) = \left( - \frac { 1 } { 4 } x - \frac { 5 } { 16 } \right) \mathrm { e } ^ { - 4 x } + \frac { 5 } { 4 } x$$ is an antiderivative of $f$ on the interval $[ 0 ; 2 ]$.
2. Calculate the area, in square metres, of each metal plate.

Let $f$ and $g$ be the functions defined on $\mathbb{R}$ by $$f(x) = \mathrm{e}^{x} \quad \text{and} \quad g(x) = 2\mathrm{e}^{\frac{x}{2}} - 1.$$ We denote $\mathcal{C}_f$ and $\mathcal{C}_g$ the representative curves of functions $f$ and $g$ in an orthogonal coordinate system.

1. Prove that the curves $\mathcal{C}_f$ and $\mathcal{C}_g$ have a common point with abscissa 0 and that at this point, they have the same tangent line $\Delta$ whose equation we will determine.
2. Study of the relative position of curve $\mathcal{C}_g$ and line $\Delta$
   Let $h$ be the function defined on $\mathbb{R}$ by $h(x) = 2\mathrm{e}^{\frac{x}{2}} - x - 2$. a. Determine the limit of function $h$ at $-\infty$. b. Justify that, for every real $x \neq 0, h(x) = x\left(\frac{\mathrm{e}^{\frac{x}{2}}}{\frac{x}{2}} - 1 - \frac{2}{x}\right)$. Deduce the limit of function $h$ at $+\infty$. c. We denote $h'$ the derivative function of function $h$ on $\mathbb{R}$. For every real $x$, calculate $h'(x)$ and study the sign of $h'(x)$ according to the values of $x$. d. Draw the variation table of function $h$ on $\mathbb{R}$. e. Deduce that, for every real $x, 2\mathrm{e}^{\frac{x}{2}} - 1 \geqslant x + 1$. f. What can we deduce about the relative position of curve $\mathcal{C}_g$ and line $\Delta$?
3. Study of the relative position of curves $\mathcal{C}_f$ and $\mathcal{C}_g$. a. For every real $x$, expand the expression $\left(\mathrm{e}^{\frac{x}{2}} - 1\right)^2$. b. Determine the relative position of curves $\mathcal{C}_f$ and $\mathcal{C}_g$.

We consider the function $g$ defined for all real $x$ by $g(x) = \mathrm{e}^{1-x}$. The representative curves $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ of the functions $f$ and $g$ respectively are drawn in a coordinate system. The purpose of this part is to study the relative position of these two curves.

1. After observing the graph, what conjecture can be made?
2. Justify that, for all real $x$ belonging to $]-\infty; 0]$, $f(x) < g(x)$.
3. In this question, we consider the interval $]0; +\infty[$. We set, for all strictly positive real $x$, $\Phi(x) = \ln x - x^{2} + x$. a. Show that, for all strictly positive real $x$, $$f(x) \leqslant g(x) \text{ is equivalent to } \Phi(x) \leqslant 0$$ It is admitted for the rest that $f(x) = g(x)$ is equivalent to $\Phi(x) = 0$. b. It is admitted that the function $\Phi$ is differentiable on $]0; +\infty[$. Draw the table of variations of the function $\Phi$. (The limits at 0 and $+\infty$ are not required.) c. Deduce that, for all strictly positive real $x$, $\Phi(x) \leqslant 0$.
4. a. Is the conjecture made in question 1 of Part B valid? b. Show that $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$ have a unique common point, denoted $A$. c. Show that at this point $A$, these two curves have the same tangent line.

Let $f$ be the function defined on $\mathbb{R}$ by $$f(x) = x - \ln\left(x^{2} + 1\right).$$

1. Solve in $\mathbb{R}$ the equation: $f(x) = x$.
2. Justify all elements of the variation table below except for the limit of the function $f$ at $+\infty$ which is admitted.
   

   $x$	$-\infty$		1		$+\infty$
   $f^{\prime}(x)$		+	0	+
   						$+\infty$
   $f(x)$
   	$-\infty$

   
   
3. Show that, for every real $x$ belonging to $[0;1]$, $f(x)$ belongs to $[0;1]$.
4. Consider the following algorithm:
   

   Variables	$N$ and $A$ natural integers;
   Input	Enter the value of $A$
   Processing	\begin{tabular}{ l } $N$ takes the value 0
   While $N - \ln\left(N^{2} + 1\right) < A$
   $N$ takes the value $N + 1$
   End while

   \hline Output & Display $N$ \hline \end{tabular}
   a. What does this algorithm do? b. Determine the value $N$ provided by the algorithm when the value entered for $A$ is 100.

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.

Let $f$ and $g$ be the functions defined on $] 0 ; + \infty [$ by
$$f ( x ) = \mathrm { e } ^ { - x } \quad \text { and } \quad g ( x ) = \frac { 1 } { x ^ { 2 } } \mathrm { e } ^ { - \frac { 1 } { x } } .$$
We admit that $f$ and $g$ are differentiable on $] 0 ; + \infty [$. We denote $f ^ { \prime }$ and $g ^ { \prime }$ their respective derivative functions.

Part A - Graphical conjectures

In each of the questions in this part, no explanation is required.

1. Conjecture graphically a solution to the equation $f ( x ) = g ( x )$ on $] 0 ; + \infty [$.
2. Conjecture graphically a solution to the equation $g ^ { \prime } ( x ) = 0$ on $] 0 ; + \infty [$.

Part B - Study of the function $g$

1. Calculate the limit of $g ( x )$ as $x$ tends to $+ \infty$.
2. We admit that the function $g$ is strictly positive on $] 0 ; + \infty [$.

Let $h$ be the function defined on $] 0 ; + \infty [$ by $h ( x ) = \ln ( g ( x ) )$. a. Prove that, for every strictly positive real number $x$,
$$h ( x ) = \frac { - 1 - 2 x \ln x } { x } .$$
b. Calculate the limit of $h ( x )$ as $x$ tends to 0. c. Deduce the limit of $g ( x )$ as $x$ tends to 0.
3. Prove that, for every strictly positive real number $x$,
$$g ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { - \frac { 1 } { x } } ( 1 - 2 x ) } { x ^ { 4 } } .$$

1. Deduce the variations of the function $g$ on $] 0 ; + \infty [$.

Part C - Area of the two regions between the curves $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$

1. Prove that point A with coordinates $( 1 ; \mathrm { e } ^ { - 1 } )$ is an intersection point of $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$. We admit that this point is the unique intersection point of $\mathscr { C } _ { f }$ and $\mathscr { C } _ { g }$, and that $\mathscr { C } _ { f }$ is above $\mathscr { C } _ { g }$ on the interval $] 0 ; 1 [$ and below on the interval $] 1 ; + \infty [$.
2. Let $a$ and $b$ be two strictly positive real numbers. Prove that

$$\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) \mathrm { d } x = \mathrm { e } ^ { - a } + \mathrm { e } ^ { - \frac { 1 } { a } } - \mathrm { e } ^ { - b } - \mathrm { e } ^ { - \frac { 1 } { b } } .$$

1. Prove that

$$\lim _ { a \rightarrow 0 } \int _ { a } ^ { 1 } ( f ( x ) - g ( x ) ) \mathrm { d } x = 1 - 2 \mathrm { e } ^ { - 1 } .$$

1. We admit that

$$\lim _ { a \rightarrow 0 } \int _ { a } ^ { 1 } ( f ( x ) - g ( x ) ) \mathrm { d } x = \lim _ { b \rightarrow + \infty } \int _ { 1 } ^ { b } ( g ( x ) - f ( x ) ) \mathrm { d } x .$$
Give a graphical interpretation of this equality.

In this exercise, the plane is equipped with an orthonormal coordinate system.
The curve with equation is represented below: $$y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } - 2 \right) .$$ This curve is called a ``catenary''.
We are interested here in ``catenary arcs'' delimited by two points of this curve that are symmetric with respect to the $y$-axis. Such an arc is represented on the graph below in solid line. We define the ``width'' and ``height'' of the catenary arc delimited by the points $M$ and $M^{\prime}$ as indicated on the graph.
The purpose of the exercise is to study the possible positions on the curve of the point $M$ with strictly positive abscissa so that the width of the catenary arc is equal to its height.

1. Justify that the problem studied reduces to finding the strictly positive solutions of the equation $$( E ) : \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } - 4 x - 2 = 0$$
   
2. Let $f$ be the function defined on the interval $[ 0 ; + \infty [$ by: $$f ( x ) = \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } - 4 x - 2 .$$ a. Verify that for all $x > 0 , f ( x ) = x \left( \frac { \mathrm { e } ^ { x } } { x } - 4 \right) + \mathrm { e } ^ { - x } - 2$. b. Determine $\lim _ { x \rightarrow + \infty } f ( x )$.
   
3. a. Let $f ^ { \prime }$ denote the derivative function of $f$. Calculate $f ^ { \prime } ( x )$, where $x$ belongs to the interval $[ 0 ; + \infty [$. b. Show that the equation $f ^ { \prime } ( x ) = 0$ is equivalent to the equation: $\left( \mathrm { e } ^ { x } \right) ^ { 2 } - 4 \mathrm { e } ^ { x } - 1 = 0$. c. By setting $X = \mathrm { e } ^ { x }$, show that the equation $f ^ { \prime } ( x ) = 0$ has as its unique real solution the number $\ln ( 2 + \sqrt { 5 } )$.
   
4. The sign table of the derivative function $f ^ { \prime }$ of $f$ is given below:
   

   $x$	0		$\ln ( 2 + \sqrt { 5 } )$	$+ \infty$
   $f ^ { \prime } ( x )$	-	0	+

   
   a. Draw up the variation table of the function $f$. b. Prove that the equation $f ( x ) = 0$ has a unique strictly positive solution which we denote by $\alpha$.
   
5. Consider the following algorithm where the variables $a$, $b$ and $m$ are real numbers: \begin{verbatim} While $b - a > 0.1$ do: $m \leftarrow \frac { a + b } { 2 }$ If $\mathrm { e } ^ { m } + \mathrm { e } ^ { - m } - 4 m - 2 > 0$, then: $b \leftarrow m$ Else: $a \leftarrow m$ End If End While \end{verbatim} a. Before execution of this algorithm, the variables $a$ and $b$ contain respectively the values 2 and 3. What do they contain at the end of the algorithm execution? Justify the answer by reproducing and completing the table opposite with the different values taken by the variables at each step of the algorithm.
   

   $m$	$a$	$b$	$b - a$
   	2	3	1
   2.5
   $\ldots$	$\ldots$	$\ldots$

   
   b. How can we use the values obtained at the end of the algorithm in the previous question?
   
6. The width of the Gateway Arch arc, expressed in metres, is equal to twice the strictly positive solution of the equation: $$\left( E ^ { \prime } \right) : \mathrm { e } ^ { \frac { t } { 39 } } + \mathrm { e } ^ { - \frac { t } { 39 } } - 4 \frac { t } { 39 } - 2 = 0$$ Give a bound for the height of the Gateway Arch.

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.

Vasopressin is a hormone that promotes the reabsorption of water by the body. The level of vasopressin in the blood is considered normal if it is less than $2.5 \mu\mathrm{g}/\mathrm{mL}$. This hormone is secreted as soon as blood volume decreases. In particular, vasopressin is produced following a hemorrhage.
The following model will be used:
$$f(t) = 3t\mathrm{e}^{-\frac{1}{4}t} + 2 \text{ with } t \geqslant 0$$
where $f(t)$ represents the level of vasopressin (in $\mu\mathrm{g}/\mathrm{mL}$) in the blood as a function of time $t$ (in minutes) elapsed after the start of a hemorrhage.

1. a. What is the level of vasopressin in the blood at time $t = 0$? b. Justify that twelve seconds after a hemorrhage, the level of vasopressin in the blood is not normal. c. Determine the limit of the function $f$ as $t \to +\infty$. Interpret this result.
2. We admit that the function $f$ is differentiable on $[0; +\infty[$.
   Verify that for every positive real number $t$,
   $$f^{\prime}(t) = \frac{3}{4}(4 - t)\mathrm{e}^{-\frac{1}{4}t}$$
   
3. a. Study the direction of variation of $f$ on the interval $[0; +\infty[$ and draw the variation table of the function $f$ (including the limit as $t \to +\infty$). b. At what time is the level of vasopressin maximal? What is this level then? Give an approximate value to $10^{-2}$ near.
4. a. Prove that there exists a unique value $t_0$ belonging to $[0; 4]$ such that $f(t_0) = 2.5$. Give an approximate value to $10^{-3}$ near. We admit that there exists a unique value $t_1$ belonging to $[4; +\infty[$ satisfying $f(t_1) = 2.5$. An approximate value of $t_1$ to $10^{-3}$ near is given: $t_1 \approx 18.930$. b. Determine for how long, in a person who has suffered a hemorrhage, the level of vasopressin remains above $2.5 \mu\mathrm{g}/\mathrm{mL}$ in the blood.
5. Let $F$ be the function defined on $[0; +\infty[$ by $F(t) = -12(t + 4)\mathrm{e}^{-\frac{1}{4}t} + 2t$. a. Prove that the function $F$ is an antiderivative of the function $f$ and deduce an approximate value of $\int_{t_0}^{t_1} f(t)\,\mathrm{d}t$ to the nearest unit. b. Deduce an approximate value to 0.1 near of the average level of vasopressin, during a hemorrhagic accident during the period when this level is above $2.5 \mu\mathrm{g}/\mathrm{mL}$.

Consider the function $f$ defined on $\mathbb { R }$ by $$f ( x ) = x \mathrm { e } ^ { - x ^ { 2 } + 1 }$$ Let ( $\mathscr { C }$ ) denote the representative curve of $f$ in an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath }$ ).

1. a. Show that for every real $x$, $f ( x ) = \frac { \mathrm { e } } { x } \times \frac { x ^ { 2 } } { \mathrm { e } ^ { x ^ { 2 } } }$. b. Deduce the limit of $f ( x )$ as $x$ tends to $+ \infty$.
2. For every real $x$, consider the points $M$ and $N$ on the curve $( \mathscr { C } )$ with abscissae $x$ and $- x$ respectively. a. Show that the point O is the midpoint of the segment $[ M N ]$. b. What can be deduced about the curve $( \mathscr { C } )$ ?
3. Study the variations of the function $f$ on the interval $[ 0 ; + \infty [$.
4. a. Show that the equation $f ( x ) = 0.5$ admits on $[ 0 ; + \infty [$ exactly two solutions denoted $\alpha$ and $\beta$ (with $\alpha < \beta$ ). b. Deduce the solutions on $[ 0 ; + \infty [$ of the inequality $f ( x ) \geqslant 0.5$. c. Give an approximate value to $10 ^ { - 2 }$ of $\alpha$ and $\beta$.
5. Let $A$ be a strictly positive real number. Set $I _ { A } = \int _ { 0 } ^ { A } f ( x ) \mathrm { d } x$. a. Justify that $I _ { A } = \frac { 1 } { 2 } \left( \mathrm { e } - \mathrm { e } ^ { - A ^ { 2 } + 1 } \right)$. b. Calculate the limit of $I _ { A }$ as $A$ tends to $+ \infty$.
6. As illustrated in the graph below, we are interested in the shaded part of the plane which is bounded by:
   • the curve $( \mathscr { C } )$ on $\mathbb { R }$ and the curve $\left( \mathscr { C } ^ { \prime } \right)$ symmetric to $( \mathscr { C } )$ with respect to the x-axis;
   • the circle with centre $\Omega \left( \frac { \sqrt { 2 } } { 2 } ; 0 \right)$ and radius 0.5 and its symmetric with respect to the y-axis.
   It is admitted that the disk with centre $\Omega \left( \frac { \sqrt { 2 } } { 2 } ; 0 \right)$ and radius 0.5 and its symmetric with respect to the y-axis are situated entirely between the curve ( $\mathscr { C }$ ) and the curve ( $\mathscr { C } ^ { \prime }$ ). Determine an approximate value in square units to the nearest hundredth of the area of this shaded part of the plane.