8. If $f ( x ) = e ^ { x }$, then $\ln \left( f ^ { \prime } ( 2 ) \right) =$
(A) 2
(B) 0
(C) $\frac { 1 } { e ^ { 2 } }$
(D) $2 e$
(E) $e ^ { 2 }$

22. If $f ( x ) = \left( x ^ { 2 } + 1 \right) ^ { x }$, then $f ^ { \prime } ( x ) =$
(A) $\quad x \left( x ^ { 2 } + 1 \right) ^ { x - 1 }$
(B) $\quad 2 x ^ { 2 } \left( x ^ { 2 } + 1 \right) ^ { x - 1 }$
(C) $\quad x \ln \left( x ^ { 2 } + 1 \right)$
(D) $\quad \ln \left( x ^ { 2 } + 1 \right) + \frac { 2 x ^ { 2 } } { x ^ { 2 } + 1 }$
(E) $\left( x ^ { 2 } + 1 \right) ^ { x } \left[ \ln \left( x ^ { 2 } + 1 \right) + \frac { 2 x ^ { 2 } } { x ^ { 2 } + 1 } \right]$ [Figure]

The temperature outside a house during a 24-hour period is given by $$F(t) = 80 - 10\cos\left(\frac{\pi t}{12}\right), \quad 0 \leq t \leq 24,$$ where $F(t)$ is measured in degrees Fahrenheit and $t$ is measured in hours.
(a) Sketch the graph of $F$ on the grid provided.
(b) Find the average temperature, to the nearest degree Fahrenheit, between $t = 6$ and $t = 14$.
(c) An air conditioner cooled the house whenever the outside temperature was at or above 78 degrees Fahrenheit. For what values of $t$ was the air conditioner cooling the house?
(d) The cost of cooling the house accumulates at the rate of $\$0.05$ per hour for each degree the outside temperature exceeds 78 degrees Fahrenheit. What was the total cost, to the nearest cent, to cool the house for this 24-hour period?

Let $f$ be a function defined by $$f(x) = \begin{cases} 1 - 2\sin x & \text{for } x \leq 0 \\ e^{-4x} & \text{for } x > 0. \end{cases}$$
(a) Show that $f$ is continuous at $x = 0$.
(b) For $x \neq 0$, express $f'(x)$ as a piecewise-defined function. Find the value of $x$ for which $f'(x) = -3$.
(c) Find the average value of $f$ on the interval $[-1, 1]$.

If $f ( x ) = 7 x - 3 + \ln x$, then $f ^ { \prime } ( 1 ) =$
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Researchers on a boat are investigating plankton cells in a sea. At a depth of $h$ meters, the density of plankton cells, in millions of cells per cubic meter, is modeled by $p ( h ) = 0.2 h ^ { 2 } e ^ { - 0.0025 h ^ { 2 } }$ for $0 \leq h \leq 30$ and is modeled by $f ( h )$ for $h \geq 30$. The continuous function $f$ is not explicitly given.
(a) Find $p ^ { \prime } ( 25 )$. Using correct units, interpret the meaning of $p ^ { \prime } ( 25 )$ in the context of the problem.
(b) Consider a vertical column of water in this sea with horizontal cross sections of constant area 3 square meters. To the nearest million, how many plankton cells are in this column of water between $h = 0$ and $h = 30$ meters?
(c) There is a function $u$ such that $0 \leq f ( h ) \leq u ( h )$ for all $h \geq 30$ and $\int _ { 30 } ^ { \infty } u ( h ) d h = 105$. The column of water in part (b) is $K$ meters deep, where $K > 30$. Write an expression involving one or more integrals that gives the number of plankton cells, in millions, in the entire column. Explain why the number of plankton cells in the column is less than or equal to 2000 million.
(d) The boat is moving on the surface of the sea. At time $t \geq 0$, the position of the boat is $( x ( t ) , y ( t ) )$, where $x ^ { \prime } ( t ) = 662 \sin ( 5 t )$ and $y ^ { \prime } ( t ) = 880 \cos ( 6 t )$. Time $t$ is measured in hours, and $x ( t )$ and $y ( t )$ are measured in meters. Find the total distance traveled by the boat over the time interval $0 \leq t \leq 1$.

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 $+\infty$.

Consider the function $f$ defined on $\mathbb { R }$ by
$$f ( x ) = ( x + 2 ) \mathrm { e } ^ { - x }.$$
We denote by $\mathscr { C }$ the representative curve of the function $f$ in an orthogonal coordinate system.

1. Study of the function $f$. a. Determine the coordinates of the intersection points of the curve $\mathscr { C }$ with the axes of the coordinate system. b. Study the limits of the function $f$ at $- \infty$ and at $+ \infty$. Deduce any possible asymptotes of the curve $\mathscr { C }$. c. Study the variations of $f$ on $\mathbb { R }$.
2. Calculation of an approximate value of the area under a curve.

We denote by $\mathscr { D }$ the region between the $x$-axis, the curve $\mathscr { C }$ and the lines with equations $x = 0$ and $x = 1$. We approximate the area of the region $\mathscr { D }$ by calculating a sum of areas of rectangles. a. In this question, we divide the interval $[ 0 ; 1 ]$ into four intervals of equal length:

• On the interval $\left[ 0 ; \frac { 1 } { 4 } \right]$, we construct a rectangle of height $f ( 0 )$
• On the interval $\left[ \frac { 1 } { 4 } ; \frac { 1 } { 2 } \right]$, we construct a rectangle of height $f \left( \frac { 1 } { 4 } \right)$
• On the interval $\left[ \frac { 1 } { 2 } ; \frac { 3 } { 4 } \right]$, we construct a rectangle of height $f \left( \frac { 1 } { 2 } \right)$
• On the interval $\left[ \frac { 3 } { 4 } ; 1 \right]$, we construct a rectangle of height $f \left( \frac { 3 } { 4 } \right)$

The algorithm below allows us to obtain an approximate value of the area of the region $\mathscr { D }$ by adding the areas of the four preceding rectangles:

Variables :	$k$ is an integer
	$S$ is a real number
Initialization :	Assign to $S$ the value 0
Processing:	For $k$ varying from 0 to 3
	$\mid$ Assign to $S$ the value $S + \frac { 1 } { 4 } f \left( \frac { k } { 4 } \right)$
	End For
Output :	Display $S$

Give an approximate value to $10 ^ { - 3 }$ of the result displayed by this algorithm. b. In this question, $N$ is an integer strictly greater than 1. We divide the interval $[ 0 ; 1 ]$ into $N$ intervals of equal length. On each of these intervals, we construct a rectangle by proceeding in the same manner as in question 2.a. Modify the preceding algorithm so that it displays as output the sum of the areas of the $N$ rectangles thus constructed.
3. Calculation of the exact value of the area under a curve.
Let $g$ be the function defined on $\mathbb { R }$ by
$$g ( x ) = ( - x - 3 ) \mathrm { e } ^ { - x }$$
We admit that $g$ is an antiderivative of the function $f$ on $\mathbb { R }$. a. Calculate the area $\mathscr { A }$ of the region $\mathscr { D }$, expressed in square units. b. Give an approximate value to $10 ^ { - 3 }$ of the error made by replacing $\mathscr { A }$ by the approximate value found using the algorithm of question 2.a, that is the difference between these two values.

Let $f$ be the function defined on the interval $] 0 ; + \infty [$ by
$$f ( x ) = \frac { 1 + \ln ( x ) } { x ^ { 2 } }$$
and let $\mathscr { C }$ be the representative curve of the function $f$ in a coordinate system of the plane.

1. a. Study the limit of $f$ at 0. b. What is $\lim _ { x \rightarrow + \infty } \frac { \ln ( x ) } { x }$ ? Deduce the limit of the function $f$ at $+ \infty$. c. Deduce the possible asymptotes to the curve $\mathscr { C }$.
2. a. Let $f ^ { \prime }$ denote the derivative function of the function $f$ on the interval $] 0 ; + \infty [$. Prove that, for every real $x$ belonging to the interval $] 0 ; + \infty [$, $$f ^ { \prime } ( x ) = \frac { - 1 - 2 \ln ( x ) } { x ^ { 3 } }$$ b. Solve on the interval $] 0 ; + \infty [$ the inequality $- 1 - 2 \ln ( x ) > 0$. Deduce the sign of $f ^ { \prime } ( x )$ on the interval $] 0 ; + \infty [$. c. Draw up the table of variations of the function $f$.
3. a. Prove that the curve $\mathscr { C }$ has a unique point of intersection with the $x$-axis, whose coordinates you will specify. b. Deduce the sign of $f ( x )$ on the interval $] 0 ; + \infty [$.
4. For every integer $n \geqslant 1$, we denote by $I _ { n }$ the area, expressed in square units, of the region bounded by the $x$-axis, the curve $\mathscr { C }$ and the lines $x = 1$ and $x = n$.

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 $\mathscr { D }$.

Let $f$ be the function defined on the interval $]0; +\infty[$ by: $$f(x) = x - 5\ln x - \frac{4}{x}$$

1. Determine the limit of $f(x)$ as $x$ tends to 0. You may use without proof the fact that $\lim_{x \rightarrow 0} x \ln x = 0$.
2. Determine the limit of $f(x)$ as $x$ tends to $+\infty$.
3. Prove that, for all strictly positive real $x$, $f'(x) = u(x)$, where $u(x) = \frac{x^2 - 5x + 4}{x^2}$.

Deduce the table of variations of the function $f$ by specifying the limits and particular values.

For every natural number $n$, the function $f_{n}$ is defined for all real $x$ in the interval $[0; 1]$ by:
$$f_{n}(x) = x + \mathrm{e}^{n(x-1)}$$
Let $\mathscr{C}_{n}$ denote the graph of the function $f_{n}$ in an orthogonal coordinate system.

Part A: generalities on the functions $\boldsymbol{f}_{\boldsymbol{n}}$

1. Prove that, for every natural number $n$, the function $f_{n}$ is increasing and positive on the interval $[0; 1]$.
2. Show that the curves $\mathscr{C}_{n}$ all have a common point A, and specify its coordinates.
3. Using the graphical representations, can one conjecture the behavior of the slopes of the tangent lines at A to the curves $\mathscr{C}_{n}$ for large values of $n$? Prove this conjecture.

Part B: evolution of $\boldsymbol{f}_{\boldsymbol{n}}(\boldsymbol{x})$ when $x$ is fixed

Let $x$ be a fixed real number in the interval $[0; 1]$. For every natural number $n$, we set $u_{n} = f_{n}(x)$.

1. In this question, assume that $x = 1$. Study the possible limit of the sequence $(u_{n})$.
2. In this question, assume that $0 \leq x < 1$. Study the possible limit of the sequence $(u_{n})$.

Part C: area under the curves $\mathscr{C}_{\boldsymbol{n}}$

For every natural number $n$, let $A_{n}$ denote the area, expressed in square units, of the region located between the $x$-axis, the curve $\mathscr{C}_{n}$ and the lines with equations $x = 0$ and $x = 1$ respectively. Based on the graphical representations, conjecture the limit of the sequence $(A_{n})$ as the integer $n$ tends to $+\infty$, then prove this conjecture.

We consider the function $f$ defined for all real $x$ by $f(x) = x\mathrm{e}^{1-x^{2}}$.

1. Calculate the limit of the function $f$ at $+\infty$. Hint: you may use the fact that for all real $x$ different from 0, $f(x) = \frac{\mathrm{e}}{x} \times \frac{x^{2}}{\mathrm{e}^{x^{2}}}$. It is admitted that the limit of the function $f$ at $-\infty$ is equal to 0.
2. a. It is admitted that $f$ is differentiable on $\mathbb{R}$ and we denote by $f'$ its derivative. Prove that for all real $x$, $$f'(x) = \left(1 - 2x^{2}\right)\mathrm{e}^{1-x^{2}}$$ b. Deduce the table of variations of the function $f$.

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.

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:
$$C ( t ) = \frac { d } { a } \left( 1 - \mathrm { e } ^ { - \frac { a } { 80 } t } \right)$$
Part B: study of functions

1. Let $f$ be the function defined on $]0; +\infty[$ by:
   $$f ( x ) = \frac { 105 } { x } \left( 1 - \mathrm { e } ^ { - \frac { 3 } { 40 } x } \right)$$
   Prove that, for every real $x$ in $]0; +\infty[$, $f ^ { \prime } ( x ) = \frac { 105 g ( x ) } { x ^ { 2 } }$, where $g$ is the function defined on $[0; +\infty[$ by:
   $$g ( x ) = \frac { 3 x } { 40 } \mathrm { e } ^ { - \frac { 3 } { 40 } x } + \mathrm { e } ^ { - \frac { 3 } { 40 } x } - 1$$
   
2. The variation table of the function $g$ is given:
   

   $x$	0	$+\infty$
   	0
   $g ( x )$		- 1

   
   Deduce the monotonicity of the function $f$. The limits of the function $f$ are not required.
   
3. Show that the equation $f ( x ) = 5.9$ has a unique solution on the interval $[1; 80]$. Deduce that this equation has a unique solution on the interval $]0; +\infty[$. Give an approximate value of this solution to the nearest tenth.

Exercise 1 -- Common to all candidates

Part A

We consider the function $h$ defined on the interval $[ 0 ; + \infty [$ by: $$h ( x ) = x \mathrm { e } ^ { - x }$$

1. Determine the limit of the function $h$ at $+ \infty$.
2. Study the variations of the function $h$ on the interval $[ 0 ; + \infty [$ and draw up its table of variations.
3. The objective of this question is to determine a primitive of the function $h$. a. Verify that for every real number $x$ belonging to the interval $[ 0 ; + \infty [$, we have: $$h ( x ) = \mathrm { e } ^ { - x } - h ^ { \prime } ( x )$$ where $h ^ { \prime }$ denotes the derivative function of $h$. b. Determine a primitive on the interval $[ 0 ; + \infty [$ of the function $x \longmapsto \mathrm { e } ^ { - x }$. c. Deduce from the two previous questions a primitive of the function $h$ on the interval $[ 0 ; + \infty [$.

Part B

We define the functions $f$ and $g$ on the interval $[ 0 ; + \infty [$ by: $$f ( x ) = x \mathrm { e } ^ { - x } + \ln ( x + 1 ) \quad \text { and } \quad g ( x ) = \ln ( x + 1 )$$ We denote $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$ the respective graphical representations of the functions $f$ and $g$ in an orthonormal coordinate system.

1. For a real number $x$ belonging to the interval $[ 0 ; + \infty [$, we call $M$ the point with coordinates $( x ; f ( x ) )$ and $N$ the point with coordinates $( x ; g ( x ) )$: $M$ and $N$ are therefore the points with abscissa $x$ belonging respectively to the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$. a. Determine the value of $x$ for which the distance $MN$ is maximum and give this maximum distance. b. Place on the graph provided in the appendix the points $M$ and $N$ corresponding to the maximum value of $MN$.
2. Let $\lambda$ be a real number belonging to the interval $[ 0 ; + \infty [$. We denote $D _ { \lambda }$ the region of the plane bounded by the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { g }$ and by the lines with equations $x = 0$ and $x = \lambda$. a. Shade the region $D _ { \lambda }$ corresponding to the value $\lambda$ proposed on the graph in the appendix. b. We denote $A _ { \lambda }$ the area of the region $D _ { \lambda }$, expressed in square units. Prove that: $$A _ { \lambda } = 1 - \frac { \lambda + 1 } { \mathrm { e } ^ { \lambda } } .$$ c. Calculate the limit of $A _ { \lambda }$ as $\lambda$ tends to $+ \infty$ and interpret the result.
3. We consider the following algorithm: \begin{verbatim} Variables: $\lambda$ is a positive real number $S$ is a real number strictly between 0 and 1. Initialization: Input $S$ $\lambda$ takes the value 0 Processing: While $1 - \frac { \lambda + 1 } { \mathrm { e } ^ { \lambda } } < S$ do $\lambda$ takes the value $\lambda + 1$ End While Output: Display $\lambda$ \end{verbatim} a. What value does this algorithm display if we input the value $S = 0.8$? b. What is the role of this algorithm?

We consider the function $f$ defined on $] 0 ; + \infty [$ by
$$f ( x ) = \frac { ( \ln x ) ^ { 2 } } { x }$$
We denote $\mathscr { C }$ the representative curve of $f$ in an orthonormal coordinate system.

1. Determine the limit of the function $f$ at 0 and interpret the result graphically.
2. a. Prove that, for all $x$ belonging to $] 0 ; + \infty [$, $$f ( x ) = 4 \left( \frac { \ln ( \sqrt { x } ) } { \sqrt { x } } \right) ^ { 2 }$$ b. Deduce that the $x$-axis is an asymptote to the representative curve of the function $f$ in the neighbourhood of $+ \infty$.
3. We admit that $f$ is differentiable on $] 0 ; + \infty [$ and we denote $f ^ { \prime }$ its derivative function. a. Prove that, for all $x$ belonging to $] 0 ; + \infty [$, $$f ^ { \prime } ( x ) = \frac { \ln ( x ) ( 2 - \ln ( x ) ) } { x ^ { 2 } } .$$ b. Study the sign of $f ^ { \prime } ( x )$ according to the values of the strictly positive real number $x$. c. Calculate $f ( 1 )$ and $f \left( \mathrm { e } ^ { 2 } \right)$.
4. Prove that the equation $f ( x ) = 1$ admits a unique solution $\alpha$ on $] 0 ; + \infty [$ and give a bound for $\alpha$ with amplitude $10 ^ { - 2 }$.

Throughout the exercise, $n$ denotes a strictly positive natural number. The purpose of the exercise is to study the equation
$$\left( E _ { n } \right) : \quad \frac { \ln ( x ) } { x } = \frac { 1 } { n }$$
with unknown strictly positive real number $x$.

Part A

Let $f$ be the function defined on the interval $] 0$; $+ \infty [$ by
$$f ( x ) = \frac { \ln ( x ) } { x }$$
It is admitted that the function $f$ is differentiable on the interval $] 0 ; + \infty [$.

1. Study the variations of function $f$.
2. Determine its maximum.

Part B

1. Show that, for $n \geqslant 3$, the equation $f ( x ) = \frac { 1 } { n }$ has a unique solution on $[ 1 ; e]$ denoted $\alpha _ { n }$.
2. From the above, for every integer $n \geqslant 3$, the real number $\alpha _ { n }$ is a solution of equation $\left( E _ { n } \right)$. a. On the graph are drawn the lines $D _ { 3 } , D _ { 4 }$ and $D _ { 5 }$ with equations respectively $y = \frac { 1 } { 3 } , y = \frac { 1 } { 4 }$, $y = \frac { 1 } { 5 }$. Conjecture the direction of variation of the sequence ( $\alpha _ { n }$ ). b. Compare, for every integer $n \geqslant 3 , f \left( \alpha _ { n } \right)$ and $f \left( \alpha _ { n + 1 } \right)$. Determine the direction of variation of the sequence $\left( \alpha _ { n } \right)$. c. Deduce that the sequence ( $\alpha _ { n }$ ) converges. It is not asked to calculate its limit.
3. It is admitted that, for every integer $n \geqslant 3$, equation $\left( E _ { n } \right)$ has another solution $\beta _ { n }$ such that $$1 \leqslant \alpha _ { n } \leqslant \mathrm { e } \leqslant \beta _ { n }$$ a. It is admitted that the sequence ( $\beta _ { n }$ ) is increasing. Establish that, for every natural number $n$ greater than or equal to 3, $$\beta _ { n } \geqslant n \frac { \beta _ { 3 } } { 3 } .$$ b. Deduce the limit of the sequence ( $\beta _ { n }$ ).

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.

An advertiser wishes to print a logo on a T-shirt. He draws this logo using the curves of two functions $f$ and $g$ defined on $\mathbb{R}$ by: $$f(x) = \mathrm{e}^{-x}(-\cos x + \sin x + 1) \text{ and } g(x) = -\mathrm{e}^{-x}\cos x$$ It is admitted that the functions $f$ and $g$ are differentiable on $\mathbb{R}$.

Part A — Study of function $f$

1. Justify that, for all $x \in \mathbb{R}$: $$-\mathrm{e}^{-x} \leqslant f(x) \leqslant 3\mathrm{e}^{-x}$$
2. Deduce the limit of $f$ as $x \to +\infty$.
3. Prove that, for all $x \in \mathbb{R}$, $f'(x) = \mathrm{e}^{-x}(2\cos x - 1)$ where $f'$ is the derivative of $f$.
4. In this question, we study function $f$ on the interval $[-\pi; \pi]$. a. Determine the sign of $f'(x)$ for $x$ in the interval $[-\pi; \pi]$. b. Deduce the variations of $f$ on $[-\pi; \pi]$.

Part B — Area of the logo

We denote by $\mathscr{C}_f$ and $\mathscr{C}_g$ the graphs of functions $f$ and $g$ in an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. The graphical unit is 2 centimetres.

1. Study the relative position of curve $\mathscr{C}_f$ with respect to curve $\mathscr{C}_g$ on $\mathbb{R}$.
2. Let $H$ be the function defined on $\mathbb{R}$ by: $$H(x) = \left(-\frac{\cos x}{2} - \frac{\sin x}{2} - 1\right)\mathrm{e}^{-x}$$ It is admitted that $H$ is an antiderivative of the function $x \mapsto (\sin x + 1)\mathrm{e}^{-x}$ on $\mathbb{R}$. We denote by $\mathscr{D}$ the region bounded by curve $\mathscr{C}_f$, curve $\mathscr{C}_g$ and the lines with equations $x = -\frac{\pi}{2}$ and $x = \frac{3\pi}{2}$. a. Shade the region $\mathscr{D}$ on the graph in the appendix to be returned with your work. b. Calculate, in square units, the area of region $\mathscr{D}$, then give an approximate value to $10^{-2}$ in $\mathrm{cm}^2$.

We consider, for every integer $n > 0$, the functions $f_{n}$ defined on the interval $[1; 5]$ by: $$f_{n}(x) = \frac{\ln x}{x^{n}}$$ For every integer $n > 0$, we denote by $\mathscr{C}_{n}$ the representative curve of the function $f_{n}$ in an orthogonal reference frame.

1. Show that, for every integer $n > 0$ and every real $x$ in the interval $[1; 5]$: $$f_{n}^{\prime}(x) = \frac{1 - n\ln(x)}{x^{n+1}}$$
2. For every integer $n > 0$, we admit that the function $f_{n}$ has a maximum on the interval $[1; 5]$. We denote by $A_{n}$ the point of the curve $\mathscr{C}_{n}$ having as ordinate this maximum. Show that all points $A_{n}$ belong to the same curve $\Gamma$ with equation $$y = \frac{1}{\mathrm{e}} \ln(x)$$
3. a. Show that, for every integer $n > 1$ and every real $x$ in the interval $[1; 5]$: $$0 \leqslant \frac{\ln(x)}{x^{n}} \leqslant \frac{\ln(5)}{x^{n}}$$ b. Show that for every integer $n > 1$: $$\int_{1}^{5} \frac{1}{x^{n}} \mathrm{~d}x = \frac{1}{n-1}\left(1 - \frac{1}{5^{n-1}}\right)$$ c. For every integer $n > 0$, we are interested in the area, expressed in square units, of the surface under the curve $f_{n}$, that is the area of the region of the plane bounded by the lines with equations $x = 1$, $x = 5$, $y = 0$ and the curve $\mathscr{C}_{n}$. Determine the limiting value of this area as $n$ tends to $+\infty$.

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 $2.5 \mu\mathrm{g}/\mathrm{mL}$.

The flow of water from a tap has a constant and moderate flow rate.
We are particularly interested in a part of the flow profile represented in appendix 1 by the curve $C$ in an orthonormal coordinate system.

Part A

We consider that the curve $C$ given in appendix 1 is the graphical representation of a function $f$ differentiable on the interval $] 0 ; 1 ]$ which respects the following three conditions:
$$( H ) : f ( 1 ) = 0 \quad f ^ { \prime } ( 1 ) = 0.25 \quad \text { and } \lim _ { \substack { x \rightarrow 0 \\ x > 0 } } f ( x ) = - \infty .$$

1. Can the function $f$ be a polynomial function of degree two? Why?
2. Let $g$ be the function defined on the interval $]0 ; 1]$ by $g ( x ) = k \ln x$. a. Determine the real number $k$ so that the function $g$ respects the three conditions $( H )$. b. Does the representative curve of the function $g$ coincide with the curve $C$ ? Why?
3. Let $h$ be the function defined on the interval $]0; 1]$ by $h ( x ) = \frac { a } { x ^ { 4 } } + b x$ where $a$ and $b$ are real numbers. Determine $a$ and $b$ so that the function $h$ respects the three conditions ( $H$ ).

Part B

We admit in this part that the curve $C$ is the graphical representation of a function $f$ continuous, strictly increasing, defined and differentiable on the interval $] 0 ; 1 ]$ with expression:
$$f ( x ) = \frac { 1 } { 20 } \left( x - \frac { 1 } { x ^ { 4 } } \right)$$

1. Justify that the equation $f ( x ) = - 5$ admits on the interval $] 0 ; 1 ]$ a unique solution which will be denoted $\alpha$. Determine an approximate value of $\alpha$ to $10 ^ { - 2 }$ near.
2. It is admitted that the volume of water in $\mathrm { cm } ^ { 3 }$, contained in the first 5 centimetres of the flow, is given by the formula: $V = \int _ { \alpha } ^ { 1 } \pi x ^ { 2 } f ^ { \prime } ( x ) \mathrm { d } x$. a. Let $u$ be the function differentiable on $] 0; 1]$ defined by $u ( x ) = \frac { 1 } { 2 x ^ { 2 } }$. Determine its derivative function. b. Determine the exact value of $V$. Using the approximate value of $\alpha$ obtained in question 1, give an approximate value of $V$.

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.

In the plane with a coordinate system, we consider the curve $\mathscr{C}_{f}$ representative of a function $f$, twice differentiable on the interval $]0 ; +\infty[$. The curve $\mathscr{C}_{f}$ admits a horizontal tangent line $T$ at point A(1;4).

1. Specify the values $f(1)$ and $f^{\prime}(1)$.

We admit that the function $f$ is defined for every real number $x$ in the interval $]0 ; +\infty[$ by:
$$f(x) = \frac{a + b \ln x}{x} \text{ where } a \text{ and } b \text{ are two real numbers.}$$

1. Prove that, for every strictly positive real number $x$, we have: $$f^{\prime}(x) = \frac{b - a - b \ln x}{x^{2}}$$
2. Deduce the values of the real numbers $a$ and $b$.

In the rest of the exercise, we admit that the function $f$ is defined for every real number $x$ in the interval $]0; +\infty[$ by:
$$f(x) = \frac{4 + 4 \ln x}{x}$$

1. Determine the limits of $f$ at 0 and at $+\infty$.
2. Determine the variation table of $f$ on the interval $]0 ; +\infty[$.
3. Prove that, for every strictly positive real number $x$, we have: $$f^{\prime\prime}(x) = \frac{-4 + 8 \ln x}{x^{3}}$$
4. Show that the curve $\mathscr{C}_{f}$ has a unique inflection point B whose coordinates you will specify.