We are interested in the evolution of the height of a corn plant as a function of time.
We decide to model this growth by a logistic function of the type: $$h ( t ) = \frac { a } { 1 + b \mathrm { e } ^ { - 0,04 t } }$$ where $a$ and $b$ are positive real constants, $t$ is the time variable expressed in days and $h ( t )$ denotes the height of the plant, expressed in metres.
We know that initially, for $t = 0$, the plant measures $0,1 \mathrm{~m}$ and that its height tends towards a limiting height of 2 m.
Part 1. Determine the constants $a$ and $b$ so that the function $h$ corresponds to the growth of the corn plant studied.
Part 2. We now consider that the growth of the corn plant is given by the function $f$ defined on $[0;250]$ by $$f ( t ) = \frac { 2 } { 1 + 19 \mathrm { e } ^ { - 0,04 t } }$$

1. Determine $f ^ { \prime } ( t )$ as a function of $t$ ($f ^ { \prime }$ denoting the derivative function of the function $f$). Deduce the variations of the function $f$ on the interval $[ 0 ; 250 ]$.
2. Calculate the time required for the corn plant to reach a height greater than $1,5 \mathrm{~m}$.
3. a. Verify that for all real $t$ belonging to the interval $[ 0 ; 250 ]$ we have $f ( t ) = \frac { 2 \mathrm { e } ^ { 0,04 t } } { \mathrm { e } ^ { 0,04 t } + 19 }$.
   Show that the function $F$ defined on the interval $[ 0 ; 250]$ by $F ( t ) = 50 \ln \left( \mathrm { e } ^ { 0,04 t } + 19 \right)$ is an antiderivative of the function $f$.
   b. Determine the average value of $f$ on the interval $[ 50 ; 100 ]$. Give an approximate value to $10 ^ { - 2 }$ and interpret this result.
4. We are interested in the growth rate of the corn plant; it is given by the derivative function of the function $f$. The growth rate is maximum for a value of $t$. Using the graph given in the appendix, determine an approximate value of this. Then estimate the height of the plant.

Exercise 4 (Candidates who have not followed the specialization course)
The common spruce is a species of coniferous tree that can measure up to 40 meters in height and live more than 150 years. The objective of this exercise is to estimate the age and height of a spruce based on the diameter of its trunk measured at $1.30 \mathrm {~m}$ from the ground.

Part A - Modeling the age of a spruce

For a spruce whose age is between 20 and 120 years, the relationship between its age (in years) and the diameter of its trunk (in meters) measured at $1.30 \mathrm {~m}$ from the ground is modeled by the function $f$ defined on the interval $] 0 ; 1 [$ by: $$f ( x ) = 30 \ln \left( \frac { 20 x } { 1 - x } \right)$$ where $x$ denotes the diameter expressed in meters and $f ( x )$ the age in years.

1. Prove that the function $f$ is strictly increasing on the interval $] 0 ; 1 [$.
2. Determine the values of the trunk diameter $x$ such that the age calculated in this model remains consistent with its validity conditions, that is, between 20 and 120 years.

Part B

The average height of spruces in representative samples of trees aged 50 to 150 years was measured. The following table, created using a spreadsheet, groups these results and allows calculation of the average growth rate of a spruce.

	A	B	C	D	E	F	G	H	I	J	K	L	M
1	Ages (in years)	50	70	80	85	90	95	100	105	110	120	130	150
2	Heights (in meters)	11.2	15.6	18.05	19.3	20.55	21.8	23	24.2	25.4	27.6	29.65	33
3	Growth rate (in meters per year)		0.22	0.245	0.25

1. a. Interpret the number 0.245 in cell D3. b. What formula should be entered in cell C3 to complete line 3 by copying cell C3 to the right?
2. Determine the expected height of a spruce whose trunk diameter measured at $1.30 \mathrm {~m}$ from the ground is 27 cm.
3. The quality of the wood is better when the growth rate is maximal. a. Determine an age interval during which the wood quality is best by explaining the approach. b. Is it consistent to ask loggers to cut trees when their diameter measures approximately 70 cm?

Exercise 4 — Candidates who have not followed the specialization course
We are interested in the fall of a water droplet that detaches from a cloud without initial velocity. A very simplified model makes it possible to establish that the instantaneous vertical velocity, expressed in $\mathrm{m.s^{-1}}$, of the droplet's fall as a function of the fall duration $t$ is given by the function $v$ defined as follows:
For every non-negative real number $t$, $v(t) = 9.81\dfrac{m}{k}\left(1 - \mathrm{e}^{-\frac{k}{m}t}\right)$; the constant $m$ is the mass of the droplet in milligrams and the constant $k$ is a strictly positive coefficient related to air friction.
We recall that instantaneous velocity is the derivative of position. Parts $A$ and $B$ are independent.
Part A - General case

1. Determine the variations of the velocity of the water droplet.
2. Does the droplet slow down during its fall?
3. Show that $\lim_{t \rightarrow +\infty} v(t) = 9.81\dfrac{m}{k}$. This limit is called the terminal velocity of the droplet.
4. A scientist claims that after a fall duration equal to $\dfrac{5m}{k}$, the velocity of the droplet exceeds $99\%$ of its terminal velocity. Is this claim correct?

Part B
In this part, we take $m = 6$ and $k = 3.9$. At a given instant, the instantaneous velocity of this droplet is $15\mathrm{~m.s^{-1}}$.

1. How long ago did the droplet detach from its cloud? Round the answer to the nearest tenth of a second.
2. Deduce the average velocity of this droplet between the moment it detached from the cloud and the instant when its velocity was measured. Round the answer to the nearest tenth of $\mathrm{m.s^{-1}}$.

A statistical study was conducted in a large city in France between January $1^{\text{st}}$ 2000 and January $1^{\text{st}}$ 2010 to assess the proportion of households with a fixed internet connection. On January $1^{\text{st}}$ 2000, one household in eight had a fixed internet connection and, on January $1^{\text{st}}$ 2010, $64\%$ of households did. Following this study, this proportion was modelled by the function $g$ defined on the interval $[0; +\infty[$ by: $$g(t) = \frac{1}{1 + k\mathrm{e}^{-at}},$$ where $k$ and $a$ are two positive real constants and the variable $t$ denotes the time, measured in years, elapsed since January $1^{\text{st}}$ 2000.

1. Determine the exact values of $k$ and $a$ so that $g(0) = \frac{1}{8}$ and $g(10) = \frac{64}{100}$.
2. In the following, we take $k = 7$ and $a = 0.25$. The function $g$ is therefore defined by: $$g(t) = \frac{1}{1 + 7\mathrm{e}^{-\left(\frac{t}{4}\right)}}$$ a. Show that the function $g$ is increasing on the interval $[0; +\infty[$. b. According to this model, can we assert that one day, at least $99\%$ of households in this city will have a fixed internet connection? Justify your answer.
3. a. Give, to the nearest hundredth, the proportion of households, predicted by the model, equipped with a fixed internet connection on January $1^{\text{st}}$ 2018. b. Given the development of mobile telephony, some statisticians believe that the modelling by the function $g$ of the evolution of the proportion of households with a fixed internet connection should be reconsidered. At the beginning of 2018 a survey was conducted. Out of 1000 households, 880 had a fixed connection. Does this survey support these sceptical statisticians? (You may use an asymptotic confidence interval at the $95\%$ level.)

Cécile has invited friends to lunch on her terrace. For dessert, she has planned an assortment of individual cakes that she bought frozen. She takes the cakes out of the freezer at $- 19 ^ { \circ } \mathrm { C }$ and brings them to the terrace where the temperature is $25 ^ { \circ } \mathrm { C }$. After 10 minutes, the temperature of the cakes is $1.3 ^ { \circ } \mathrm { C }$.

I- First model

We assume that the thawing rate is constant, that is, the temperature increase is the same minute after minute. According to this model, determine what the temperature of the cakes would be 25 minutes after they are taken out of the freezer. Does this model seem relevant?

II - Second model

We denote $T _ { n }$ the temperature of the cakes in degrees Celsius, after $n$ minutes following their removal from the freezer; thus $T _ { 0 } = - 19$. We assume that to model the evolution of temperature, we must have the following relation
$$\text { For all natural integers } n , T _ { n + 1 } - T _ { n } = - 0.06 \times \left( T _ { n } - 25 \right) \text {. }$$

1. Justify that, for all integers $n$, we have $T _ { n + 1 } = 0.94 T _ { n } + 1.5$
2. Calculate $T _ { 1 }$ and $T _ { 2 }$. Give values rounded to the nearest tenth.
3. Prove by induction that, for all natural integers $n$, we have $T _ { n } \leqslant 25$.

Returning to the situation studied, was this result foreseeable?
4. Study the direction of variation of the sequence $( T _ { n } )$.
5. Prove that the sequence $( T _ { n } )$ is convergent. 6. We set for all natural integers $n$, $U _ { n } = T _ { n } - 25$. a. Show that the sequence $( U _ { n } )$ is a geometric sequence and specify its common ratio and first term $U _ { 0 }$. b. Deduce that for all natural integers $n$, $T _ { n } = - 44 \times 0.94 ^ { n } + 25$. c. Deduce the limit of the sequence $( T _ { n } )$. Interpret this result in the context of the situation studied. 7. a. The manufacturer recommends consuming the cakes after half an hour at room temperature following their removal from the freezer. What is then the temperature reached by the cakes? Give a value rounded to the nearest integer. b. Cécile is a regular customer of these cakes, which she likes to enjoy while still fresh, at a temperature of $10 ^ { \circ } \mathrm { C }$. Give a range between two consecutive integers of the time in minutes after which Cécile should enjoy her cake. c. The following program, written in Python language, must return after its execution the smallest value of the integer $n$ for which $T _ { n } \geqslant 10$.
\begin{verbatim} def seuil() : n=0 T= while T T= n=n+1 return \end{verbatim}
Copy this program onto your paper and complete the incomplete lines so that the program returns the expected value.

Question 176
Um banco oferece uma aplicação com juros compostos de 1\% ao mês. Um cliente aplica R\$ 10 000,00. Após 2 meses, o montante obtido será de
(A) R\$ 10 100,00 (B) R\$ 10 200,00 (C) R\$ 10 201,00 (D) R\$ 10 210,00 (E) R\$ 10 220,00

Um capital é aplicado a juros compostos de 10\% ao ano. Após 2 anos, o montante é R\$\,12\,100,00. O capital inicial é
(A) R\$\,9\,000,00 (B) R\$\,9\,500,00 (C) R\$\,10\,000,00 (D) R\$\,10\,500,00 (E) R\$\,11\,000,00

Even when the surroundings suddenly become dark, the human eye perceives the change gradually. After the light intensity suddenly changes from 1000 to 10 and $t$ seconds have elapsed, the light intensity $I ( t )$ perceived by a person is $$I ( t ) = 10 + 990 \times a ^ { - 5 t } ( \text { where } a \text { is a constant with } a > 1 )$$ After the light intensity suddenly changes from 1000 to 10, let $s$ seconds elapse until the person perceives the light intensity as 21. What is the value of $s$? (Note: The unit of light intensity is Td (troland).) [3 points]
(1) $\frac { 1 + 2 \log 3 } { 5 \log a }$
(2) $\frac { 1 + 3 \log 3 } { 5 \log a }$
(3) $\frac { 2 + \log 3 } { 5 \log a }$
(4) $\frac { 2 + 2 \log 3 } { 5 \log a }$
(5) $\frac { 2 + 3 \log 3 } { 5 \log a }$

What is the sum of all real roots of the exponential equation $2 ^ { x } + 2 ^ { 2 - x } = 5$? [3 points]
(1) - 2
(2) - 1
(3) 0
(4) 1
(5) 2

What is the sum of all natural numbers $x$ that satisfy the exponential inequality $\left( 3 ^ { x } - 5 \right) \left( 3 ^ { x } - 100 \right) < 0$? [3 points]
(1) 5
(2) 7
(3) 9
(4) 11
(5) 13

For a certain financial product, when an initial asset $W _ { 0 }$ is invested, the expected asset $W$ after $t$ years is given as follows. $$\begin{aligned} & W = \frac { W _ { 0 } } { 2 } 10 ^ { a t } \left( 1 + 10 ^ { a t } \right) \\ & \text { (where } W _ { 0 } > 0 , t \geq 0 \text {, and } a \text { is a constant.) } \end{aligned}$$ When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 15 years is 3 times the initial asset. When an initial asset $w _ { 0 }$ is invested in this financial product, the expected asset after 30 years is $k$ times the initial asset. What is the value of the real number $k$? (where $w _ { 0 } > 0$) [3 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13

The line $y = 2 x + k$ meets the graphs of the two functions $$y = \left( \frac { 2 } { 3 } \right) ^ { x + 3 } + 1 , \quad y = \left( \frac { 2 } { 3 } \right) ^ { x + 1 } + \frac { 8 } { 3 }$$ at points $\mathrm { P }$ and $\mathrm { Q }$ respectively. When $\overline { \mathrm { PQ } } = \sqrt { 5 }$, what is the value of the constant $k$? [4 points]
(1) $\frac { 31 } { 6 }$
(2) $\frac { 16 } { 3 }$
(3) $\frac { 11 } { 2 }$
(4) $\frac { 17 } { 3 }$
(5) $\frac { 35 } { 6 }$

8. The shelf life $y$ (in hours) of a certain food and storage temperature $x$ (in ${ } ^ { \circ } \mathrm { C }$ ) satisfy the functional relationship $y = e ^ { k x + b }$ (where $e = 2.718 \ldots$ is the base of natural logarithm, and $k , b$ are constants). If the shelf life of this food at $0 { } ^ { \circ } \mathrm { C }$ is 192 hours and at $22 { } ^ { \circ } \mathrm { C }$ is 48 hours, then the shelf life at $33 { } ^ { \circ } \mathrm { C }$ is
(A) 16 hours
(B) 20 hours
(C) 24 hours
(D) 28 hours

13. The shelf life $y$ (in hours) of a certain food and the storage temperature $x$ (in ${}^{\circ} \mathrm { C }$) satisfy the functional relationship $y = e ^ { k x + b }$ ($e = 2.718 \cdots$ is the base of the natural logarithm, $k$ and $b$ are constants). If the shelf life of this food at $0 ^ { \circ } \mathrm{C}$ is designed to be $192$ hours, and the shelf life at $22 ^ { \circ } \mathrm{C}$ is $45$ hours, then the shelf life of this food at $33 ^ { \circ } \mathrm{C}$ is $\_\_\_\_$ hours.

The cooling process in the first 10 minutes is to be modeled for $0 \leq t \leq 10$ by a function $u _ { 1 }$ defined on $\mathbb { R }$ with $u _ { 1 } ( t ) = a + b \cdot \mathrm { e } ^ { - c \cdot t } , a , b , c \in \mathbb { R } , c > 0$.
Here, $t$ denotes the time in minutes since the beginning of the investigation and $u _ { 1 } ( t )$ denotes the temperature of the coffee in ${ } ^ { \circ } \mathrm { C }$.
Explain why $a = 18$ is chosen in the modeling, and then calculate the values of $b$ and $c$ based on the information in the table. [Check solution with rounded values: $u _ { 1 } ( t ) = 18 + 55,86 \cdot \mathrm { e } ^ { - 0,053 \cdot t }$.]

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Give a simple asymptotic equivalent of $f(x)$ as $x$ tends to $-1$.

108. The graph of $f(x) = -2 + \left(\dfrac{1}{2}\right)^{Ax+B}$ intersects the graph of $y = x^2 - x$ at two points with $x$-coordinates 1 and 2. What is $f(3)$?
(1) $3$ (2) $4$ (3) $5$ (4) $6$
\rule{\textwidth}{0.4pt} Calculation Space
%% Page 4

114. There are 24 grams of a radioactive element. If $\dfrac{1}{10}$ of the element decays every 30 days, after how many days will 8 grams remain? $(\log 3 = 0.48)$
$$360 \quad (1) \qquad 300 \quad (2) \qquad 270 \quad (3) \qquad 240 \quad (4)$$

5. Determine the value of the real parameter $k$ so that the two curves $y = e ^ { x }$, $y = 6 - k e ^ { - x }$ are tangent to each other, finding the coordinates of the point of tangency.

Q61. The sum of all the solutions of the equation $( 8 ) ^ { 2 x } - 16 \cdot ( 8 ) ^ { x } + 48 = 0$ is :
(1) $1 + \log _ { 8 } ( 6 )$
(2) $1 + \log _ { 6 } ( 8 )$
(3) $\log _ { 8 } ( 6 )$
(4) $\log _ { 8 } ( 4 )$

According to experimental statistics, a certain type of bacteria reproduces such that its quantity increases by a factor of 2.4 on average every 3.5 hours. Suppose a test tube in the laboratory initially contains 1000 of this type of bacteria. According to an exponential function model, approximately how many hours later will the quantity of this bacteria reach about $4 \times 10^{10}$? (Note: $\log 2 \approx 0.3010$, $\log 3 \approx 0.4771$)
(1) 63 hours
(2) 70 hours
(3) 77 hours
(4) 84 hours
(5) 91 hours

The half-life $T$ of a radioactive substance is defined as ``every time period $T$ passes, the mass of the substance decays to half of its original amount''. A lead container contains two radioactive substances $A$ and $B$ with half-lives $T _ { A }$ and $T _ { B }$ respectively. At the start of recording, the masses of these two substances are equal. After 112 days, measurement shows that the mass of substance $B$ is one-quarter of the mass of substance $A$. Based on the above, which of the following is the relationship between $T _ { A }$ and $T _ { B }$?
(1) $- 2 + \frac { 112 } { T _ { A } } = \frac { 112 } { T _ { B } }$
(2) $2 + \frac { 112 } { T _ { A } } = \frac { 112 } { T _ { B } }$
(3) $- 2 + \log _ { 2 } \frac { 112 } { T _ { A } } = \log _ { 2 } \frac { 112 } { T _ { B } }$
(4) $2 + \log _ { 2 } \frac { 112 } { T _ { A } } = \log _ { 2 } \frac { 112 } { T _ { B } }$
(5) $2 \log _ { 2 } \frac { 112 } { T _ { A } } = \log _ { 2 } \frac { 112 } { T _ { B } }$

14. $a , b , x$, and $y$ are real and positive. $a$ and $b$ are constants. $x$ and $y$ are related.
A graph of $\log y$ against $\log x$ is drawn.
For which one of the following relationships will this graph be a straight line?
A $y ^ { b } = a ^ { x }$
B $y = a b ^ { x }$
C $y ^ { 2 } = a + x ^ { b }$
D $y = a x ^ { b }$
E $y ^ { x } = a ^ { b }$

Find the sum of the real solutions of the equation:
$$3 ^ { x } - ( \sqrt { 3 } ) ^ { x + 4 } + 20 = 0$$
A 1
B 4
C 9
D $\quad \log _ { 3 } 20$
E $\quad 2 \log _ { 3 } 20$
F $\quad 4 \log _ { 3 } 20$

$$4 ^ { x } \cdot 6 ^ { x } \cdot 9 ^ { x } = 36$$
Given this, what is x?
A) $\frac { 2 } { 3 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 3 } { 8 }$
E) $\frac { 4 } { 9 }$