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.

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.

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.

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.

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

In the Pyrenees National Park, a researcher is working on the decline of a protected species in high-mountain lakes: the ``midwife toad''. Parts I and II can be approached independently.

Part I: Effect of the introduction of a new species

In certain lakes in the Pyrenees, trout have been introduced by humans to enable fishing activities in the mountains. The researcher studied the impact of this introduction on the midwife toad population in a lake. His previous studies lead him to model the evolution of this population as a function of time by the following function $f$: $$f ( t ) = \left( 0.04 t ^ { 2 } - 8 t + 400 \right) \mathrm { e } ^ { \frac { t } { 50 } } + 40 \text { for } t \in [ 0 ; 120 ]$$ The variable $t$ represents the elapsed time, in days, from the introduction at time $t = 0$ of trout into the lake, and $f ( t )$ models the number of toads at time $t$.

1. Determine the number of toads present in the lake when the trout are introduced.
2. We admit that the function $f$ is differentiable on the interval $[0 ; 120]$ and we denote $f ^ { \prime }$ its derivative function. Show, by displaying the calculation steps, that for every real number $t$ belonging to the interval $[0 ; 120]$ we have: $$f ^ { \prime } ( t ) = t ( t - 100 ) \mathrm { e } ^ { \frac { t } { 50 } } \times 8 \times 10 ^ { - 4 }$$
3. Study the variations of the function $f$ on the interval $[0 ; 120]$, then draw up the variation table of $f$ on this interval (approximate values to the nearest hundredth will be given).
4. According to this model: a. Determine the number of days $J$ necessary for the number of toads to reach its minimum. What is this minimum number? b. Justify that, after reaching its minimum, the number of toads will one day exceed 140 individuals. c. Using a calculator, determine the duration in days from which the number of toads will exceed 140 individuals.

Part II: Effect of Chytridiomycosis on a tadpole population

One of the main causes of the decline of this toad species in high mountains is a disease, ``Chytridiomycosis'', caused by a fungus. The researcher considers that:

• Three quarters of the mountain lakes in the Pyrenees are not infected by the fungus, that is, they contain no contaminated tadpoles (toad larvae).
• In the remaining lakes, the probability that a tadpole is contaminated is 0.74.

The researcher randomly chooses a lake in the Pyrenees and takes samples from it. For the rest of the exercise, results will be rounded to the nearest thousandth when necessary. The researcher randomly takes a tadpole from the chosen lake to perform a test before releasing it. We denote $T$ the event ``The tadpole is contaminated by the disease'' and $L$ the event ``The lake is infected by the fungus''. We denote $\bar { L }$ the opposite event of $L$ and $\bar { T }$ the opposite event of $T$.

1. Copy and complete the following probability tree using the data from the problem statement.
2. Show that the probability $P ( T )$ that the sampled tadpole is contaminated is 0.185.
3. The tadpole is not contaminated. What is the probability that the lake is infected?

Exercise 4 (7 points) Themes: numerical functions, exponential function
Part A: study of two functions Consider the two functions $f$ and $g$ defined on the interval $[0; +\infty[$ by: $$f(x) = 0.06\left(-x^2 + 13.7x\right) \quad \text{and} \quad g(x) = (-0.15x + 2.2)\mathrm{e}^{0.2x} - 2.2.$$ We admit that the functions $f$ and $g$ are differentiable and we denote $f'$ and $g'$ their respective derivative functions.

1. The complete table of variations of function $f$ on the interval $[0; +\infty[$ is given.
   

   $x$	0	6.85	$+\infty$
   \multirow{2}{*}{$f(x)$}		$\nearrow f(6.85)$
   	0		$\underline{-}_{\infty}$

   
   a. Justify the limit of $f$ at $+\infty$. b. Justify the variations of function $f$. c. Solve the equation $f(x) = 0$.
2. a. Determine the limit of $g$ at $+\infty$. b. Prove that, for every real $x$ belonging to $[0; +\infty[$ we have: $g'(x) = (-0.03x + 0.29)\mathrm{e}^{0.2x}$. c. Study the variations of function $g$ and draw its table of variations on $[0; +\infty[$. Specify an approximate value to $10^{-2}$ of the maximum of $g$. d. Show that the equation $g(x) = 0$ has a unique non-zero solution and determine, to $10^{-2}$ near, an approximate value of this solution.

Part B: trajectories of a golf ball We wish to use the functions $f$ and $g$ studied in Part A to model in two different ways the trajectory of a golf ball. We assume that the terrain is perfectly flat. We will admit here that 13.7 is the value that cancels the function $f$ and an approximation of the value that cancels the function $g$. For $x$ representing the horizontal distance traveled by the ball in tens of yards after the shot (with $0 < x < 13.7$), $f(x)$ (or $g(x)$ depending on the model) represents the corresponding height of the ball above the ground, in tens of yards. The ``takeoff angle'' of the ball is called the angle between the x-axis and the tangent to the curve ($\mathscr{C}_f$ or $\mathscr{C}_g$ depending on the model) at its point with abscissa 0. A measure of the takeoff angle of the ball is a real number $d$ such that $\tan(d)$ is equal to the slope of this tangent. Similarly, the ``landing angle'' of the ball is called the angle between the x-axis and the tangent to the curve ($\mathscr{C}_f$ or $\mathscr{C}_g$ depending on the model) at its point with abscissa 13.7. A measure of the landing angle of the ball is a real number $a$ such that $\tan(a)$ is equal to the opposite of the slope of this tangent. All angles are measured in degrees.

1. First model: recall that here, the unit being tens of yards, $x$ represents the horizontal distance traveled by the ball after the shot and $f(x)$ the corresponding height of the ball. According to this model: a. What is the maximum height, in yards, reached by the ball during its trajectory? b. Verify that $f'(0) = 0.822$. c. Give a measure in degrees of the takeoff angle of the ball, rounded to the nearest tenth. (You may use the table below). d. What graphical property of the curve $\mathscr{C}_f$ allows us to justify that the takeoff and landing angles of the ball are equal?
2. Second model: recall that here, the unit being tens of yards, $x$ represents the horizontal distance traveled by the ball after the shot and $g(x)$ the corresponding height of the ball. According to this model: a. What is the maximum height, in yards, reached by the ball during its trajectory? We specify that $g'(0) = 0.29$ and $g'(13.7) \approx -1.87$. b. Give a measure in degrees of the takeoff angle of the ball, rounded to the nearest tenth. (You may use the table below). c. Justify that 62 is an approximate value, rounded to the nearest unit, of a measure in degrees of the landing angle of the ball.

Table: excerpt from a spreadsheet giving a measure in degrees of an angle when its tangent is known:

	A	B	C	D	E	F	G	H	I	J	K	L	M
1	$\tan(\theta)$	0.815	0.816	0.817	0.818	0.819	0.82	0.821	0.822	0.823	0.824	0.825	0.826
2	$\theta$ in degrees	39.18	39.21	39.25	39.28	39.32	39.35	39.39	39.42	39.45	39.49	39.52	39.56
3
4	$\tan(\theta)$	0.285	0.286	0.287	0.288	0.289	0.29	0.291	0.292	0.293	0.294	0.295	0.296
5	$\theta$ in degrees	15.91	15.96	16.01	16.07	16.12	16.17	16.23	16.28	16.33	16.38	16.44	16.49

Part C: Interrogating the models Based on a large number of observations of professional players' performances, the following average results were obtained:

Launch angle in degrees	Maximum height in yards	Landing angle in degrees	Horizontal distance in yards at the point of impact
24	32	52	137

Which model, among the two previously studied, seems most suitable for describing the ball strike by a professional player? The answer will be justified.

A biologist has modeled the evolution of a bacterial population (in thousands of entities) by the function $f$ defined on $[0; +\infty[$ by
$$f(t) = e^3 - e^{-0.5t^2 + t + 2}$$
where $t$ denotes the time in hours since the beginning of the experiment. Based on this modeling, he proposes the three statements below. For each of them, indicate, by justifying, whether it is true or false.

• Statement 1: ``The population increases permanently''.
• Statement 2: ``In the long term, the population will exceed 21000 bacteria''.
• Statement 3: ``The bacterial population will have a count of 10000 on two occasions over time''.

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $V \mathrm {~mm} ^ { 3 }$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is of radius equal to the outer radius of the container. If the volume of the material used to make the container is minimum when the inner radius of the container is 10 mm, then the value of $\frac { V } { 250 \pi }$ is

Consider all rectangles lying in the region
$$\left\{ ( x , y ) \in \mathbb { R } \times \mathbb { R } : 0 \leq x \leq \frac { \pi } { 2 } \text { and } 0 \leq y \leq 2 \sin ( 2 x ) \right\}$$
and having one side on the $x$-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
(A) $\frac { 3 \pi } { 2 }$
(B) $\pi$
(C) $\frac { \pi } { 2 \sqrt { 3 } }$
(D) $\frac { \pi \sqrt { 3 } } { 2 }$

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx} = 100 - 12\sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is
(1) 3500
(2) 4500
(3) 2500
(4) 3000

If the volume of a spherical ball is increasing at the rate of $4 \pi \mathrm { cc } / \mathrm { sec }$ then the rate of increase of its radius (in $\mathrm { cm } / \mathrm { sec }$), when the volume is $288 \pi \mathrm { cc }$ is
(1) $\frac { 1 } { 9 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 24 }$
(4) $\frac { 1 } { 36 }$

If a right circular cone, having maximum volume, is inscribed in a sphere of radius $3$ cm, then the curved surface area (in $\mathrm { cm } ^ { 2 }$) of this cone is :
(1) $8 \sqrt { 2 } \pi$
(2) $6 \sqrt { 2 } \pi$
(3) $8 \sqrt { 3 } \pi$
(4) $6 \sqrt { 3 } \pi$

The shortest distance between the line $x - y = 1$ and the curve $x ^ { 2 } = 2 y$ is:
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { \sqrt { 2 } }$
(3) $\frac { 1 } { 2 \sqrt { 2 } }$
(4) 0

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is $\tan ^ { - 1 } \frac { 3 } { 4 }$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is $\_\_\_\_$ .

An internet company can serve at most 1000 customers and can reach this number when it sets the monthly internet fee at 40 TL. The company has observed that after each 5 TL increase in the monthly internet fee, the number of customers decreases by 50.
At what monthly internet fee should this company set its rate to maximize the total revenue from the monthly internet fee?
A) 55 B) 60 C) 65 D) 70 E) 75

The amount of fuel consumed in 1 hour by a rocket moving at a constant speed of $V$ kilometers per hour is calculated by the function
$$f ( V ) = \frac { V ^ { 3 } } { 20 } - 7 \cdot V ^ { 2 } + 265 \cdot V$$
in units.
Accordingly, what is the minimum amount of fuel in units that this rocket must consume to travel 100 kilometers at a constant speed?
A) 1000
B) 2000
C) 3000
D) 4000
E) 5000

In the rectangular coordinate plane, the shaded region between the lines $y = 2 + 3x$ and $y = 12 - x$ and the positive $x$ and $y$-axes is given below.
Rectangles are constructed in this region with one side on the $x$-axis and one vertex each on the lines $y = 2 + 3x$ and $y = 12 - x$.
What is the length of the side on the $x$-axis of the rectangle with the largest area?
A) 6 B) $\dfrac{19}{3}$ C) $\dfrac{20}{3}$ D) 7 E) $\dfrac{22}{3}$