Functions $f$, $g$, and $h$ are twice-differentiable functions with $g(2) = h(2) = 4$. The line $y = 4 + \dfrac{2}{3}(x - 2)$ is tangent to both the graph of $g$ at $x = 2$ and the graph of $h$ at $x = 2$.
(a) Find $h'(2)$.
(b) Let $a$ be the function given by $a(x) = 3x^3 h(x)$. Write an expression for $a'(x)$. Find $a'(2)$.
(c) The function $h$ satisfies $h(x) = \dfrac{x^2 - 4}{1 - (f(x))^3}$ for $x \neq 2$. It is known that $\lim_{x \to 2} h(x)$ can be evaluated using L'H\^{o}pital's Rule. Use $\lim_{x \to 2} h(x)$ to find $f(2)$ and $f'(2)$. Show the work that leads to your answers.
(d) It is known that $g(x) \leq h(x)$ for $1 < x < 3$. Let $k$ be a function satisfying $g(x) \leq k(x) \leq h(x)$ for $1 < x < 3$. Is $k$ continuous at $x = 2$? Justify your answer.

The functions $f$ and $g$ are twice differentiable. The table shown gives values of the functions and their first derivatives at selected values of $x$.

$x$	0	2	4	7
$f(x)$	10	7	4	5
$f'(x)$	$\frac{3}{2}$	$-8$	3	6
$g(x)$	1	2	$-3$	0
$g'(x)$	5	4	2	8

(a) Let $h$ be the function defined by $h(x) = f(g(x))$. Find $h'(7)$. Show the work that leads to your answer.
(b) Let $k$ be a differentiable function such that $k'(x) = (f(x))^{2} \cdot g(x)$. Is the graph of $k$ concave up or concave down at the point where $x = 4$? Give a reason for your answer.
(c) Let $m$ be the function defined by $m(x) = 5x^{3} + \int_{0}^{x} f'(t)\, dt$. Find $m(2)$. Show the work that leads to your answer.
(d) Is the function $m$ defined in part (c) increasing, decreasing, or neither at $x = 2$? Justify your answer.

6. The twice-differentiable function $f$ is defined for all real numbers and satisfies the following conditions:
$$f ( 0 ) = 2 , f ^ { \prime } ( 0 ) = - 4 , \text { and } f ^ { \prime \prime } ( 0 ) = 3$$
(a) The function $g$ is given by $g ( x ) = e ^ { a x } + f ( x )$ for all real numbers, where $a$ is a constant. Find $g ^ { \prime } ( 0 )$ and $g ^ { \prime \prime } ( 0 )$ in terms of $a$. Show the work that leads to your answers.
(b) The function $h$ is given by $h ( x ) = \cos ( k x ) f ( x )$ for all real numbers, where $k$ is a constant. Find $h ^ { \prime } ( x )$ and write an equation for the line tangent to the graph of $h$ at $x = 0$.

WRITE ALL WORK IN THE PINK EXAM BOOKLET.

END OF EXAM

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As a pot of tea cools, the temperature of the tea is modeled by a differentiable function $H$ for $0 \leq t \leq 10$, where time $t$ is measured in minutes and temperature $H ( t )$ is measured in degrees Celsius. Values of $H ( t )$ at selected values of time $t$ are shown in the table above. (a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time $t = 3.5$. Show the computations that lead to your answer. (b) Using correct units, explain the meaning of $\frac { 1 } { 10 } \int _ { 0 } ^ { 10 } H ( t ) d t$ in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate $\frac { 1 } { 10 } \int _ { 0 } ^ { 10 } H ( t ) d t$. (c) Evaluate $\int _ { 0 } ^ { 10 } H ^ { \prime } ( t ) d t$. Using correct units, explain the meaning of the expression in the context of this problem. (d) At time $t = 0$, biscuits with temperature $100 ^ { \circ } \mathrm { C }$ were removed from an oven. The temperature of the biscuits at time $t$ is modeled by a differentiable function $B$ for which it is known that $B ^ { \prime } ( t ) = - 13.84 e ^ { - 0.173 t }$. Using the given models, at time $t = 10$, how much cooler are the biscuits than the tea?

The function $f$ is defined by $f ( x ) = \sqrt { 25 - x ^ { 2 } }$ for $- 5 \leq x \leq 5$.
(a) Find $f ^ { \prime } ( x )$.
(b) Write an equation for the line tangent to the graph of $f$ at $x = - 3$.
(c) Let $g$ be the function defined by $g ( x ) = \begin{cases} f ( x ) & \text { for } - 5 \leq x \leq - 3 \\ x + 7 & \text { for } - 3 < x \leq 5 . \end{cases}$
Is $g$ continuous at $x = - 3$ ? Use the definition of continuity to explain your answer.
(d) Find the value of $\int _ { 0 } ^ { 5 } x \sqrt { 25 - x ^ { 2 } } d x$.

If $y = \sin ^ { 3 } x$, then $\frac { d y } { d x } =$
(A) $\cos ^ { 3 } x$
(B) $3 \cos ^ { 2 } x$
(C) $3 \sin ^ { 2 } x$
(D) $- 3 \sin ^ { 2 } x \cos x$
(E) $3 \sin ^ { 2 } x \cos x$

If $\arcsin x = \ln y$, then $\frac { d y } { d x } =$
(A) $\frac { y } { \sqrt { 1 - x ^ { 2 } } }$
(B) $\frac { x y } { \sqrt { 1 - x ^ { 2 } } }$
(C) $\frac { y } { 1 + x ^ { 2 } }$
(D) $e ^ { \arcsin x }$
(E) $\frac { e ^ { \arcsin x } } { 1 + x ^ { 2 } }$

Let $f$ and $g$ be the functions given by $f ( x ) = e ^ { x }$ and $g ( x ) = x ^ { 4 }$. On what intervals is the rate of change of $f ( x )$ greater than the rate of change of $g ( x )$ ?
(A) $( 0.831, 7.384 )$ only
(B) $( - \infty , 0.831 )$ and $( 7.384 , \infty )$
(C) $( - \infty , - 0.816 )$ and $( 1.430, 8.613 )$
(D) $( - 0.816, 1.430 )$ and $( 8.613 , \infty )$
(E) $( - \infty , \infty )$

On a certain workday, the rate, in tons per hour, at which unprocessed gravel arrives at a gravel processing plant is modeled by $G ( t ) = 90 + 45 \cos \left( \frac { t ^ { 2 } } { 18 } \right)$, where $t$ is measured in hours and $0 \leq t \leq 8$. At the beginning of the workday $( t = 0 )$, the plant has 500 tons of unprocessed gravel. During the hours of operation, $0 \leq t \leq 8$, the plant processes gravel at a constant rate of 100 tons per hour.
(a) Find $G ^ { \prime } ( 5 )$. Using correct units, interpret your answer in the context of the problem.
(b) Find the total amount of unprocessed gravel that arrives at the plant during the hours of operation on this workday.
(c) Is the amount of unprocessed gravel at the plant increasing or decreasing at time $t = 5$ hours? Show the work that leads to your answer.
(d) What is the maximum amount of unprocessed gravel at the plant during the hours of operation on this workday? Justify your answer.

Exercise 2 -- Common to all candidates
Consider the functions $f$ and $g$ defined for all real $x$ by: $$f(x) = \mathrm{e}^{x} \quad \text{and} \quad g(x) = 1 - \mathrm{e}^{-x}.$$ The representative curves of these functions in an orthogonal coordinate system of the plane, denoted respectively $\mathscr{C}_{f}$ and $\mathscr{C}_{g}$, are provided in the appendix.
Part A
These curves appear to admit two common tangent lines. Draw these tangent lines as accurately as possible on the figure in the appendix.
Part B
In this part, the existence of these common tangent lines is admitted. Let $\mathscr{D}$ denote one of them. This line is tangent to the curve $\mathscr{C}_{f}$ at point A with abscissa $a$ and tangent to the curve $\mathscr{C}_{g}$ at point B with abscissa $b$.

1. a. Express in terms of $a$ the slope of the tangent line to the curve $\mathscr{C}_{f}$ at point A. b. Express in terms of $b$ the slope of the tangent line to the curve $\mathscr{C}_{g}$ at point B. c. Deduce that $b = -a$.
2. Prove that the real number $a$ is a solution of the equation $$2(x - 1)\mathrm{e}^{x} + 1 = 0.$$

Part C
Consider the function $\varphi$ defined on $\mathbb{R}$ by $$\varphi(x) = 2(x - 1)\mathrm{e}^{x} + 1$$

1. a. Calculate the limits of the function $\varphi$ at $-\infty$ and $+\infty$. b. Calculate the derivative of the function $\varphi$, then study its sign. c. Draw the variation table of the function $\varphi$ on $\mathbb{R}$. Specify the value of $\varphi(0)$.
2. a. Prove that the equation $\varphi(x) = 0$ has exactly two solutions in $\mathbb{R}$. b. Let $\alpha$ denote the negative solution of the equation $\varphi(x) = 0$ and $\beta$ the positive solution of this equation. Using a calculator, give the values of $\alpha$ and $\beta$ rounded to the nearest hundredth.

Part D
In this part, we prove the existence of these common tangent lines, which was admitted in Part B. Let E be the point on the curve $\mathscr{C}_{f}$ with abscissa $\alpha$ and F the point on the curve $\mathscr{C}_{g}$ with abscissa $-\alpha$ ($\alpha$ is the real number defined in Part C).

1. Prove that the line $(EF)$ is tangent to the curve $\mathscr{C}_{f}$ at point E.
2. Prove that $(EF)$ is tangent to $\mathscr{C}_{g}$ at point F.

On the graph below, we have drawn, in an orthonormal coordinate system $(\mathrm{O}, \vec{\imath}, \vec{\jmath})$, a curve $\mathscr{C}$ and the line $(\mathrm{AB})$ where A and B are the points with coordinates $(0;1)$ and $(-1;3)$ respectively.
We denote by $f$ the function differentiable on $\mathbb{R}$ whose representative curve is $\mathscr{C}$. We further assume that there exists a real number $a$ such that for all real $x$, $$f(x) = x + 1 + ax\mathrm{e}^{-x^{2}}$$

1. a. Justify that the curve $\mathscr{C}$ passes through point A. b. Determine the slope of the line (AB). c. Prove that for all real $x$, $$f^{\prime}(x) = 1 - a\left(2x^{2} - 1\right)\mathrm{e}^{-x^{2}}$$ d. We assume that the line $(\mathrm{AB})$ is tangent to the curve $\mathscr{C}$ at point A. Determine the value of the real number $a$.
2. According to the previous question, for all real $x$, $$f(x) = x + 1 - 3x\mathrm{e}^{-x^{2}} \text{ and } f^{\prime}(x) = 1 + 3\left(2x^{2} - 1\right)\mathrm{e}^{-x^{2}}.$$ a. Prove that for all real $x$ in the interval $]-1;0]$, $f(x) > 0$. b. Prove that for all real $x$ less than or equal to $-1$, $f^{\prime}(x) > 0$. c. Prove that there exists a unique real number $c$ in the interval $\left[-\frac{3}{2};-1\right]$ such that $f(c) = 0$. Justify that $c < -\frac{3}{2} + 2 \cdot 10^{-2}$.
3. We denote by $\mathscr{A}$ the area, expressed in square units, of the region defined by: $$c \leqslant x \leqslant 0 \quad \text{and} \quad 0 \leqslant y \leqslant f(x)$$ a. Write $\mathscr{A}$ in the form of an integral. b. We admit that the integral $I = \int_{-\frac{3}{2}}^{0} f(x)\,\mathrm{d}x$ is an approximate value of $\mathscr{A}$ to within $10^{-3}$. Calculate the exact value of the integral $I$.

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 $\mathcal{C}_g$.

Consider the function $f$ defined and differentiable on the interval $[0 ; +\infty[$ by
$$f ( x ) = x \mathrm { e } ^ { - x } - 0,1$$

1. Determine the limit of $f$ as $x \to +\infty$.
2. Study the variations of $f$ on $[0 ; +\infty[$ and draw the variation table.
3. Prove that the equation $f ( x ) = 0$ has a unique solution denoted $\alpha$ on the interval $[0 ; 1]$.

We admit the existence of a strictly positive real number $\beta$ such that $\alpha < \beta$ and $f ( \beta ) = 0$. We denote by $\mathscr { C }$ the representative curve of the function $f$ on the interval $[\alpha ; \beta]$ in an orthogonal coordinate system and $\mathscr { C } ^ { \prime }$ the curve symmetric to $\mathscr { C }$ with respect to the $x$-axis.
The unit on each axis represents 5 meters. These curves are used to delimit a floral bed in the shape of a candle flame on which tulips will be planted.

1. Prove that the function $F$, defined on the interval $[\alpha ; \beta]$ by $$F ( x ) = - ( x + 1 ) \mathrm { e } ^ { - x } - 0,1 x$$ is an antiderivative of the function $f$ on the interval $[\alpha ; \beta]$.
2. Calculate, in square units, a value rounded to 0.01 of the area of the region between the curves $\mathscr { C }$ and $\mathscr { C } ^ { \prime }$. Use the following values rounded to 0.001: $\alpha \approx 0.112$ and $\beta \approx 3.577$.
3. Knowing that 36 tulip plants can be placed per square meter, calculate the number of tulip plants needed for this floral bed.

Exercise 1 (5 points)
The plane is equipped with an orthogonal coordinate system $(\mathrm{O}, \mathrm{I}, \mathrm{J})$.
Consider the function $f$ defined on the interval $]0; 1]$ by $$f(x) = x(1 - \ln x)^2.$$
a. Determine an expression for the derivative of $f$ and verify that for all $x \in ]0; 1]$, $f'(x) = (\ln x + 1)(\ln x - 1)$.
b. Study the variations of the function $f$ and draw its variation table on the interval $]0; 1]$ (it will be admitted that the limit of the function $f$ at 0 is zero).

This exercise is a multiple choice questionnaire (MCQ). For each question, three statements are proposed, only one of these statements is correct.

1. Consider the function $f$ defined on $\mathbb{R}$ by $$f(x) = \left(x^2 - 2x - 1\right)\mathrm{e}^x$$ A. The derivative function of $f$ is the function defined by $f^{\prime}(x) = (2x-2)\mathrm{e}^x$.
   B. The function $f$ is decreasing on the interval $]-\infty; 2]$.
   C. $\lim_{x \rightarrow -\infty} f(x) = 0$.
   
2. Consider the function $f$ defined on $\mathbb{R}$ by $f(x) = \frac{3}{5 + \mathrm{e}^x}$.
   Its representative curve in a coordinate system has:
   A. only one horizontal asymptote;
   B. one horizontal asymptote and one vertical asymptote;
   C. two horizontal asymptotes.
   
3. Below is the curve $\mathcal{C}_{f^{\prime\prime}}$ representing the second derivative function $f^{\prime\prime}$ of a function $f$ defined and twice differentiable on the interval $[-3.5; 6]$.
   A. The function $f$ is convex on the interval $[-3; 3]$.
   B. The function $f$ has three inflection points.
   C. The derivative function $f^{\prime}$ of $f$ is decreasing on the interval $[0; 2]$.
   
4. Consider the sequence $(u_n)$ defined for every natural integer $n$ by $u_n = n^2 - 17n + 20$.
   A. The sequence $(u_n)$ is bounded below.
   B. The sequence $(u_n)$ is decreasing.
   C. One of the terms of the sequence $(u_n)$ equals 2021.
   
5. Consider the sequence $(u_n)$ defined by $u_0 = 2$ and, for every natural integer $n$, $u_{n+1} = 0.75 u_n + 5$. Consider the following ``threshold'' function written in Python: \begin{verbatim} def seuil() : u = 2 n = 0 while u < 45 : u = 0,75*u + 5 n = n+1 return n \end{verbatim} This function returns:
   A. the smallest value of $n$ such that $u_n \geqslant 45$;
   B. the smallest value of $n$ such that $u_n < 45$;
   C. the largest value of $n$ such that $u_n \geqslant 45$.

Consider the function defined on $\mathbb{R}$ by $$f(x) = x\mathrm{e}^{-2x}$$ Let $f^{\prime\prime}$ denote the second derivative of the function $f$. For any real number $x$, $f^{\prime\prime}(x)$ is equal to: a. $(1-2x)\mathrm{e}^{-2x}$ b. $4(x-1)\mathrm{e}^{-2x}$ c. $4\mathrm{e}^{-2x}$ d. $(x+2)\mathrm{e}^{-2x}$

This exercise is a multiple choice questionnaire. For each of the six following questions, only one of the four proposed answers is correct.
A wrong answer, multiple answers, or absence of an answer to a question earns neither points nor deducts points.

1. Consider the function $g$ defined and differentiable on $]0;+\infty[$ by: $$g(x) = \ln\left(x^2 + x + 1\right).$$ For every strictly positive real number $x$: a. $g^{\prime}(x) = \frac{1}{2x+1}$ b. $g^{\prime}(x) = \frac{1}{x^2+x+1}$ c. $g^{\prime}(x) = \ln(2x+1)$ d. $g^{\prime}(x) = \frac{2x+1}{x^2+x+1}$
   
2. The function $x \longmapsto \ln(x)$ admits as an antiderivative on $]0;+\infty[$ the function: a. $x \longmapsto \ln(x)$ b. $x \longmapsto \frac{1}{x}$ c. $x \longmapsto x\ln(x) - x$ d. $x \longmapsto \frac{\ln(x)}{x}$
   
3. Consider the sequence $(a_n)$ defined for all $n$ in $\mathbb{N}$ by: $$a_n = \frac{1 - 3^n}{1 + 2^n}.$$ The limit of the sequence $(a_n)$ is equal to: a. $-\infty$ b. $-1$ c. $1$ d. $+\infty$
   
4. Consider a function $f$ defined and differentiable on $[-2;2]$. The variation table of the function $f^{\prime}$ derivative of the function $f$ on the interval $[-2;2]$ is given by:
   

   $x$	$-2$	$-1$	$0$	$2$
   variations of $f^{\prime}$	$1$			$>_{-2}^{-1}$

   
   The function $f$ is: a. convex on $[-2;-1]$ b. concave on $[0;1]$ c. convex on $[-1;2]$ d. concave on $[-2;0]$
   
5. The representative curve of the derivative $f^{\prime}$ of a function $f$ defined on the interval $[-2;4]$ is given above. By graphical reading of the curve of $f^{\prime}$, determine the correct statement for $f$: a. $f$ is decreasing on $[0;2]$ b. $f$ is decreasing on $[-1;0]$ c. $f$ admits a maximum at $1$ on $[0;2]$ d. $f$ admits a maximum at $3$ on $[2;4]$
   
6. A stock is quoted at $57\,€$. Its value increases by $3\%$ every month. The Python function \texttt{seuil()} which returns the number of months to wait for its value to exceed $200\,€$ is: a. \begin{verbatim} def seuil() : m=0 v=57 while v < 200 : m=m+1 v = v*1.03 return m \end{verbatim} b. \begin{verbatim} def seuil() : m=0 v=57 while v > 200 : m=m+1 v = v*1.03 return m \end{verbatim} c. \begin{verbatim} def seuil() : v=57 for i in range (200) : v = v*1.03 return v \end{verbatim} d. \begin{verbatim} def seuil() : m=0 v=57 if v<200: m=m+1 else : v = v*1.03 return m \end{verbatim}

This exercise is a multiple choice questionnaire. For each of the six following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers, or no answer to a question earns no points and loses no points.

1. Consider the function $f$ defined and differentiable on $] 0 ; + \infty [$ by: $$f ( x ) = x \ln ( x ) - x + 1 .$$ Among the four expressions below, which one is the derivative of $f$?
   

   a. $\ln ( x )$

   b. $\frac { 1 } { x } - 1$

   c. $\ln ( x ) - 2$

   d. $\ln ( x ) - 1$

   
   
2. Consider the function $g$ defined on $] 0$; $+ \infty \left[ \text{ by } g ( x ) = x ^ { 2 } [ 1 - \ln ( x ) ] \right.$. Among the four statements below, which one is correct?
   

   a. $\lim _ { x \rightarrow 0 } g ( x ) = + \infty$	b. $\lim _ { x \rightarrow 0 } g ( x ) = - \infty$	c. $\lim _ { x \rightarrow 0 } g ( x ) = 0$	\begin{tabular}{ l } d. The function $g$
   does not have a li-
   mit at 0.

   \hline \end{tabular}
   
3. Consider the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x ^ { 3 } - 0,9 x ^ { 2 } - 0,1 x$. The number of solutions to the equation $f ( x ) = 0$ on $\mathbb { R }$ is:
   

   a. 0

   b. 1

   c. 2

   d. 3

   
   
4. If $H$ is an antiderivative of a function $h$ defined and continuous on $\mathbb { R }$, and if $k$ is the function defined on $\mathbb { R }$ by $k ( x ) = h ( 2 x )$, then an antiderivative $K$ of $k$ is defined on $\mathbb { R }$ by:
   

   a. $K ( x ) = H ( 2 x )$

   b. $K ( x ) = 2 H ( 2 x )$

   c. $K ( x ) = \frac { 1 } { 2 } H ( 2 x )$

   d. $K ( x ) = 2 H ( x )$

   
   
5. The equation of the tangent line at the point with abscissa 1 to the curve of the function $f$ defined on $\mathbb { R }$ by $f ( x ) = x \mathrm { e } ^ { x }$ is:
   

   a. $y = \mathrm { e } x + \mathrm { e }$

   b. $y = 2 \mathrm { e } x - \mathrm { e }$

   c. $y = 2 \mathrm { e } x + \mathrm { e }$

   d. $y = \mathrm { e } x$

   
   
6. The integers $n$ that are solutions to the inequality $( 0,2 ) ^ { n } < 0,001$ are all integers $n$ such that:
   

   a. $n \leqslant 4$

   b. $n \leqslant 5$

   c. $n \geqslant 4$

   d. $n \geqslant 5$

   
   

Consider the function $f$ defined on $\mathbb { R }$ by
$$f ( x ) = x ^ { 3 } \mathrm { e } ^ { x }$$
It is admitted that the function $f$ is differentiable on $\mathbb { R }$ and we denote $f ^ { \prime }$ its derivative function.

1. The sequence $(u _ { n })$ is defined by $u _ { 0 } = - 1$ and, for all natural integer $n$, $u _ { n + 1 } = f \left( u _ { n } \right)$. a. Calculate $u _ { 1 }$ then $u _ { 2 }$.
   Exact values will be given, then approximate values to $10 ^ { - 3 }$. b. Consider the function fonc, written in Python language below.
   Recall that in Python language, ``i in range (n)'' means that i varies from 0 to n -1.
   \begin{verbatim} def fonc (n): u =- 1 for i in range(n) : u=u**3*exp(u) return u \end{verbatim}
   Determine, without justification, the value returned by fonc (2) rounded to $10 ^ { - 3 }$.
2. a. Prove that, for all real $x$, we have $f ^ { \prime } ( x ) = x ^ { 2 } \mathrm { e } ^ { x } ( x + 3 )$. b. Justify that the variation table of $f$ on $\mathbb { R }$ is the one represented below:
   

   $x$	$- \infty$	- 3	$+ \infty$
   	0		$+ \infty$
   $f$			$+ 27 \mathrm { e } ^ { - 3 }$

   
   c. Prove, by induction, that for all natural integer $n$, we have:
   $$- 1 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 0$$
   d. Deduce that the sequence $(u _ { n })$ is convergent. e. We denote $\ell$ the limit of the sequence $(u _ { n })$.
   Recall that $\ell$ is a solution of the equation $f ( x ) = x$. Determine $\ell$. (For this, it will be admitted that the equation $x ^ { 2 } \mathrm { e } ^ { x } - 1 = 0$ has only one solution in $\mathbb { R }$ and that this solution is strictly greater than $\frac { 1 } { 2 }$).

Exercise 3 — Theme: Functions; Sequences This exercise is a multiple choice questionnaire. For each of the following questions, only one of the four proposed answers is correct. A wrong answer, multiple answers, or no answer to a question earns no points and loses no points. To answer, indicate on your paper the question number and the letter of the chosen answer. No justification is required.

1. Let $g$ be the function defined on $\mathbb{R}$ by $g(x) = x^{1000} + x$. We can affirm that: a. the function $g$ is concave on $\mathbb{R}$. b. the function $g$ is convex on $\mathbb{R}$. c. the function $g$ has exactly one inflection point. d. the function $g$ has exactly two inflection points.
2. Consider a function $f$ defined and differentiable on $\mathbb{R}$. Let $f'$ denote its derivative function. Let $\mathscr{C}$ denote the representative curve of $f$. Let $\Gamma$ denote the representative curve of $f'$. The curve $\Gamma$ is plotted below. Let $T$ denote the tangent to the curve $\mathscr{C}$ at the point with abscissa 0. We can affirm that the tangent $T$ is parallel to the line with equation: a. $y = x$ b. $y = 0$ c. $y = 1$ d. $x = 0$
3. Consider the sequence $(u_n)$ defined for every natural number $n$ by $u_n = \frac{(-1)^n}{n+1}$. We can affirm that the sequence $(u_n)$ is: a. bounded above and not bounded below. b. bounded below and not bounded above. c. bounded. d. not bounded above and not bounded below.
4. Let $k$ be a non-zero real number. Let $(v_n)$ be a sequence defined for every natural number $n$. Suppose that $v_0 = k$ and that for all $n$, we have $v_n \times v_{n+1} < 0$. We can affirm that $v_{10}$ is: a. positive. b. negative. c. of the same sign as $k$. d. of the same sign as $-k$.
5. Consider the sequence $(w_n)$ defined for every natural number $n$ by: $$w_{n+1} = 2w_n - 4 \quad \text{and} \quad w_2 = 8.$$ We can affirm that: a. $w_0 = 0$ b. $w_0 = 5$. c. $w_0 = 10$. d. It is not possible to calculate $w_0$.
6. Consider the sequence $(a_n)$ defined for every natural number $n$ by: $$a_{n+1} = \frac{\mathrm{e}^n}{\mathrm{e}^n + 1} a_n \quad \text{and} \quad a_0 = 1.$$ We can affirm that: a. the sequence $(a_n)$ is strictly increasing. b. the sequence $(a_n)$ is strictly decreasing. c. the sequence $(a_n)$ is not monotone. d. the sequence $(a_n)$ is constant.
7. A cell reproduces by dividing into two identical cells, which divide in turn, and so on. The generation time is defined as the time required for a given cell to divide into two cells. 1 cell was placed in culture. After 4 hours, there are approximately 4000 cells. We can affirm that the generation time is approximately equal to: a. less than one minute. b. 12 minutes. c. 20 minutes. d. 1 hour.

The function $f$ is defined on the interval $[ 0 ; 1 ]$ by: $$f ( x ) = \frac { 0,96 x } { 0,93 x + 0,03 }$$

1. Prove that, for all $x$ belonging to the interval $[ 0 ; 1 ]$, $$f ^ { \prime } ( x ) = \frac { 0,0288 } { ( 0,93 x + 0,03 ) ^ { 2 } }$$
2. Determine the direction of variation of the function $f$ on the interval $[ 0 ; 1 ]$.

Consider the function $f$ defined on the interval $[ 0 ; 1 ]$ by $$f ( x ) = 2 x \mathrm { e } ^ { - x } .$$ It is admitted that the function $f$ is differentiable on the interval $[ 0 ; 1 ]$.

1. 1. [a.] Solve on the interval $[ 0 ; 1 ]$ the equation $f ( x ) = x$.
   2. [b.] Prove that, for all $x$ belonging to the interval $[ 0 ; 1 ]$, $$f ^ { \prime } ( x ) = 2 ( 1 - x ) \mathrm { e } ^ { - x } .$$
   3. [c.] Give the table of variations of the function $f$ on the interval $[ 0 ; 1 ]$.
   
   Consider the sequence $(u_n)$ defined by $u_0 = 0,1$ and for all natural integer $n$, $$u_{n+1} = f(u_n).$$
   
2. 1. [a.] Prove by induction that, for all natural integer $n$, $$0 \leqslant u_n < u_{n+1} \leqslant 1.$$
   2. [b.] Deduce that the sequence $(u_n)$ is convergent.
3. Prove that the limit of the sequence $(u_n)$ is $\ln(2)$.
4. 1. [a.] Justify that for all natural integer $n$, $\ln(2) - u_n$ is positive.
   2. [b.] It is desired to write a Python script that returns an approximate value of $\ln(2)$ by default to within $10^{-4}$, as well as the number of steps to achieve this. Copy and complete the script below so that it answers the problem posed. \begin{verbatim} def seuil() : n = 0 u=0.1 while ln (2) - u ...0.0001 : n=n+1 u=... return (u,n) \end{verbatim}
   3. [c.] Give the value of the variable $n$ returned by the function seuil().

We consider the function $f$ defined on the interval $]-\infty; 1[$ by $$f(x) = \frac{\mathrm{e}^x}{x-1}$$ We admit that the function $f$ is differentiable on the interval $]-\infty; 1[$. We call $\mathscr{C}$ its representative curve in a coordinate system.

1. 1. [a.] Determine the limit of the function $f$ at 1.
   2. [b.] Deduce from this a graphical interpretation.
2. Determine the limit of the function $f$ at $-\infty$.
3. 1. [a.] Show that for every real number $x$ in the interval $]-\infty; 1[$, we have $$f'(x) = \frac{(x-2)\mathrm{e}^x}{(x-1)^2}$$
   2. [b.] Draw up, by justifying, the table of variations of the function $f$ on the interval $]-\infty; 1[$.
4. We admit that for every real number $x$ in the interval $]-\infty; 1[$, we have $$f''(x) = \frac{\left(x^2 - 4x + 5\right)\mathrm{e}^x}{(x-1)^3}.$$
   1. [a.] Study the convexity of the function $f$ on the interval $]-\infty; 1[$.
   2. [b.] Determine the reduced equation of the tangent line $T$ to the curve $\mathscr{C}$ at the point with abscissa 0.
   3. [c.] Deduce from this that, for every real number $x$ in the interval $]-\infty; 1[$, we have: $$\mathrm{e}^x \geqslant (-2x-1)(x-1).$$
5. 1. [a.] Justify that the equation $f(x) = -2$ admits a unique solution $\alpha$ on the interval $]-\infty; 1[$.
   2. [b.] Using a calculator, determine an interval containing $\alpha$ with amplitude $10^{-2}$.