A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let $h$ be the depth of the coffee in the pot, measured in inches, where $h$ is a function of time $t$, measured in seconds. The volume $V$ of coffee in the pot is changing at the rate of $-5\pi\sqrt{h}$ cubic inches per second. (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r^2 h$.)
(a) Show that $\dfrac{dh}{dt} = -\dfrac{\sqrt{h}}{5}$.
(b) Given that $h = 17$ at time $t = 0$, solve the differential equation $\dfrac{dh}{dt} = -\dfrac{\sqrt{h}}{5}$ for $h$ as a function of $t$.
(c) At what time $t$ is the coffeepot empty?

If $P ( t )$ is the size of a population at time $t$, which of the following differential equations describes linear growth in the size of the population?
(A) $\frac { d P } { d t } = 200$
(B) $\frac { d P } { d t } = 200 t$
(C) $\frac { d P } { d t } = 100 t ^ { 2 }$
(D) $\frac { d P } { d t } = 200 P$
(E) $\frac { d P } { d t } = 100 P ^ { 2 }$

At time $t = 0$, a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is 91 degrees Celsius $\left({}^{\circ}\mathrm{C}\right)$ at time $t = 0$, and the internal temperature of the potato is greater than $27^{\circ}\mathrm{C}$ for all times $t > 0$. The internal temperature of the potato at time $t$ minutes can be modeled by the function $H$ that satisfies the differential equation $\frac{dH}{dt} = -\frac{1}{4}(H - 27)$, where $H(t)$ is measured in degrees Celsius and $H(0) = 91$.
(a) Write an equation for the line tangent to the graph of $H$ at $t = 0$. Use this equation to approximate the internal temperature of the potato at time $t = 3$.
(b) Use $\frac{d^2H}{dt^2}$ to determine whether your answer in part (a) is an underestimate or an overestimate of the internal temperature of the potato at time $t = 3$.
(c) For $t < 10$, an alternate model for the internal temperature of the potato at time $t$ minutes is the function $G$ that satisfies the differential equation $\frac{dG}{dt} = -(G - 27)^{2/3}$, where $G(t)$ is measured in degrees Celsius and $G(0) = 91$. Find an expression for $G(t)$. Based on this model, what is the internal temperature of the potato at time $t = 3$?

A cylindrical barrel with a diameter of 2 feet contains collected rainwater. The water drains out through a valve (not shown) at the bottom of the barrel. The rate of change of the height $h$ of the water in the barrel with respect to time $t$ is modeled by $\dfrac{dh}{dt} = -\dfrac{1}{10}\sqrt{h}$, where $h$ is measured in feet and $t$ is measured in seconds. (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r^2 h$.)
(a) Find the rate of change of the volume of water in the barrel with respect to time when the height of the water is 4 feet. Indicate units of measure.
(b) When the height of the water is 3 feet, is the rate of change of the height of the water with respect to time increasing or decreasing? Explain your reasoning.
(c) At time $t = 0$ seconds, the height of the water is 5 feet. Use separation of variables to find an expression for $h$ in terms of $t$.

The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its current weight. At time $t = 0$, when the bird is first weighed, its weight is 20 grams. If $B(t)$ is the weight of the bird, in grams, at time $t$ days after it is first weighed, then $$\frac{dB}{dt} = \frac{1}{5}(100 - B).$$ Let $y = B(t)$ be the solution to the differential equation above with initial condition $B(0) = 20$.
(a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain your reasoning.
(b) Find $\frac{d^{2}B}{dt^{2}}$ in terms of $B$. Use $\frac{d^{2}B}{dt^{2}}$ to explain why the graph of $B$ cannot resemble the following graph.
(c) Use separation of variables to find $y = B(t)$, the particular solution to the differential equation with initial condition $B(0) = 20$.

Let $k$ be a positive constant. Which of the following is a logistic differential equation?
(A) $\frac { d y } { d t } = k t$
(B) $\frac { d y } { d t } = k y$
(C) $\frac { d y } { d t } = k t ( 1 - t )$
(D) $\frac { d y } { d t } = k y ( 1 - t )$
(E) $\frac { d y } { d t } = k y ( 1 - y )$

If $P ( t )$ is the size of a population at time $t$, which of the following differential equations describes linear growth in the size of the population?
(A) $\frac { d P } { d t } = 200$
(B) $\frac { d P } { d t } = 200 t$
(C) $\frac { d P } { d t } = 100 t ^ { 2 }$
(D) $\frac { d P } { d t } = 200 P$
(E) $\frac { d P } { d t } = 100 P ^ { 2 }$

At time $t = 0$, a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal temperature of the potato is 91 degrees Celsius (${}^{\circ}\mathrm{C}$) at time $t = 0$, and the internal temperature of the potato is greater than $27^{\circ}\mathrm{C}$ for all times $t > 0$. The internal temperature of the potato at time $t$ minutes can be modeled by the function $H$ that satisfies the differential equation $\frac{dH}{dt} = -\frac{1}{4}(H - 27)$, where $H(t)$ is measured in degrees Celsius and $H(0) = 91$.
(a) Write an equation for the line tangent to the graph of $H$ at $t = 0$. Use this equation to approximate the internal temperature of the potato at time $t = 3$.
(b) Use $\frac{d^2H}{dt^2}$ to determine whether your answer in part (a) is an underestimate or an overestimate of the internal temperature of the potato at time $t = 3$.
(c) For $t < 10$, an alternate model for the internal temperature of the potato at time $t$ minutes is the function $G$ that satisfies the differential equation $\frac{dG}{dt} = -(G - 27)^{2/3}$, where $G(t)$ is measured in degrees Celsius and $G(0) = 91$. Find an expression for $G(t)$. Based on this model, what is the internal temperature of the potato at time $t = 3$?

A cylindrical barrel with a diameter of 2 feet contains collected rainwater. The water drains out through a valve at the bottom of the barrel. The rate of change of the height $h$ of the water in the barrel with respect to time $t$ is modeled by $\frac { d h } { d t } = - \frac { 1 } { 10 } \sqrt { h }$, where $h$ is measured in feet and $t$ is measured in seconds. (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r ^ { 2 } h$.)
(a) Find the rate of change of the volume of water in the barrel with respect to time when the height of the water is 4 feet. Indicate units of measure.
(b) When the height of the water is 3 feet, is the rate of change of the height of the water with respect to time increasing or decreasing? Explain your reasoning.
(c) At time $t = 0$ seconds, the height of the water is 5 feet. Use separation of variables to find an expression for $h$ in terms of $t$.

Newton's law of cooling states that the rate of change of the temperature of a body is proportional to the difference between the temperature of this body and that of the surrounding environment.
A cup of coffee is served at an initial temperature of $80^{\circ}\mathrm{C}$ in an environment whose temperature, expressed in degrees Celsius, assumed to be constant, is denoted $M$.
In this part, for any non-negative real $t$, we denote $\theta(t)$ the temperature of the coffee at instant $t$, with $\theta(t)$ expressed in degrees Celsius and $t$ in minutes. Thus $\theta(0) = 80$.
In this model, more precise than that of part A, we assume that $\theta$ is a function differentiable on the interval $[0; +\infty[$ and that, for any real $t$ in this interval, Newton's law is modeled by the equality: $$\theta'(t) = -0{,}2(\theta(t) - M).$$

1. In this question, we choose $M = 0$. We then seek a function $\theta$ differentiable on the interval $[0; +\infty[$ satisfying $\theta(0) = 80$ and, for any real $t$ in this interval: $\theta'(t) = -0{,}2\theta(t)$. a. If $\theta$ is such a function, we set for any $t$ in the interval $[0; +\infty[$, $f(t) = \frac{\theta(t)}{\mathrm{e}^{-0{,}2t}}$. Show that the function $f$ is differentiable on $[0; +\infty[$ and that, for any real $t$ in this interval, $f'(t) = 0$. b. Keeping the hypothesis from a., calculate $f(0)$. Deduce, for any $t$ in the interval $[0; +\infty[$, an expression for $f(t)$, then for $\theta(t)$. c. Verify that the function $\theta$ found in b. is a solution to the problem.
2. In this question, we choose $M = 10$. We admit that there exists a unique function $g$ differentiable on $[0; +\infty[$, modeling the temperature of the coffee at any non-negative instant $t$, and that, for any $t$ in the interval $[0; +\infty[$: $$g(t) = 10 + 70\mathrm{e}^{-0{,}2t},$$ where $t$ is expressed in minutes and $g(t)$ in degrees Celsius.
   A person likes to drink their coffee at $40^{\circ}\mathrm{C}$. Show that there exists a unique real $t_0$ in $[0; +\infty[$ such that $g(t_0) = 40$. Give the value of $t_0$ rounded to the nearest second.

Exercise A (Main topics: Sequences, Differential equations)
In this exercise, we are interested in the growth of Moso bamboo with maximum height 20 meters. Ludwig von Bertalanffy's growth model assumes that the growth rate for such bamboo is proportional to the difference between its height and the maximum height.

Part I: discrete model

In this part, we observe a bamboo with initial height 1 meter. For every natural integer $n$, we denote $u_n$ the height, in meters, of the bamboo $n$ days after the start of observation. Thus $u_0 = 1$. Von Bertalanffy's model for bamboo growth between two consecutive days is expressed by the equality: $$u_{n+1} = u_n + 0.05\left(20 - u_n\right) \text{ for every natural integer } n.$$

1. Verify that $u_1 = 1.95$.
2. a. Show that for every natural integer $n$, $u_{n+1} = 0.95 u_n + 1$. b. We set for every natural integer $n$, $v_n = 20 - u_n$. Prove that the sequence $(v_n)$ is a geometric sequence and specify its initial term $v_0$ and its common ratio. c. Deduce that, for every natural integer $n$, $u_n = 20 - 19 \times 0.95^n$.
3. Determine the limit of the sequence $(u_n)$.

Part II: continuous model

In this part, we wish to model the height of the same Moso bamboo by a function giving its height, in meters, as a function of time $t$ expressed in days. According to von Bertalanffy's model, this function is a solution of the differential equation $$(E) \quad y^{\prime} = 0.05(20 - y)$$ where $y$ denotes a function of the variable $t$, defined and differentiable on $[0; +\infty[$ and $y^{\prime}$ denotes its derivative function. Let the function $L$ defined on the interval $[0; +\infty[$ by $$L(t) = 20 - 19\mathrm{e}^{-0.05t}$$

1. Verify that the function $L$ is a solution of $(E)$ and that we also have $L(0) = 1$.
2. We take this function $L$ as our model and we admit that, if we denote $L^{\prime}$ its derivative function, $L^{\prime}(t)$ represents the growth rate of the bamboo at time $t$. a. Compare $L^{\prime}(0)$ and $L^{\prime}(5)$. b. Calculate the limit of the derivative function $L^{\prime}$ at $+\infty$. Is this result consistent with the description of the growth model presented at the beginning of the exercise?

A company manufactures plastic objects by injecting molten material at $210 ^ { \circ } \mathrm { C }$ into a mold. We seek to model the cooling of the material using a function $f$ giving the temperature of the injected material as a function of time $t$. Time is expressed in seconds and temperature is expressed in degrees Celsius. We assume that the function $f$ sought is a solution to a differential equation of the following form, where $m$ is a real constant that we seek to determine:
$$( E ) : \quad y ^ { \prime } + 0,02 y = m$$
Part A

1. Justify the following output from a computer algebra system:

Input:	SolveDifferentialEquation $\left( y ^ { \prime } + 0,02 y = m \right)$
Output:	$\rightarrow y = k * \exp ( - 0.02 * t ) + 50 * m$

1. The workshop temperature is $30 ^ { \circ } \mathrm { C }$. We assume that the temperature $f ( t )$ tends toward $30 ^ { \circ } \mathrm { C }$ as $t$ tends toward infinity. Prove that $m = 0,6$.
2. Determine the expression of the function $f$ sought, taking into account the initial condition $f ( 0 ) = 210$.

Part B We assume here that the temperature (expressed in degrees Celsius) of the injected material as a function of time (expressed in seconds) is given by the function whose expression and graphical representation are given below:
$$f ( t ) = 180 \mathrm { e } ^ { - 0,02 t } + 30$$

1. The object can be removed from the mold when its temperature becomes less than $50 ^ { \circ } \mathrm { C }$. a. By graphical reading, give an approximate value of the number $T$ of seconds to wait before removing the object from the mold. b. Determine by calculation the exact value of this time $T$.
2. Using an integral, calculate the average value of the temperature over the first 100 seconds.

We propose to study the concentration in the blood of a medication ingested by a person for the first time. Let $t$ be the time (in hours) elapsed since the ingestion of this medication. We admit that the concentration of this medication in the blood, in grams per litre of blood, is modelled by a function $f$ of the variable $t$ defined on the interval $[ 0 ; + \infty [$.

Part A: graphical readings

The graph above shows the representative curve of the function $f$. With the precision allowed by the graph, give without justification:

1. The time elapsed from the moment of ingestion of this medication to the moment when the concentration of medication in the blood is maximum according to this model.
2. The set of solutions to the inequality $f ( t ) \geqslant 1$.
3. The convexity of the function $f$ on the interval $[ 0 ; 8 ]$.

Part B: determination of the function $\boldsymbol { f }$

We consider the differential equation
$$( E ) : \quad y ^ { \prime } + y = 5 \mathrm { e } ^ { - t }$$
of unknown $y$, where $y$ is a function defined and differentiable on the interval $[ 0 ; + \infty [$. We admit that the function $f$ is a solution of the differential equation $( E )$.

1. Solve the differential equation $\left( E ^ { \prime } \right) : y ^ { \prime } + y = 0$.
2. Let $u$ be the function defined on the interval $\left[ 0 ; + \infty \left[ \operatorname { by } u ( t ) = a t \mathrm { e } ^ { - t } \right. \right.$ with $a \in \mathbb { R }$.

Determine the value of the real number $a$ such that the function $u$ is a solution of equation $( E )$.
3. Deduce the set of solutions of the differential equation $( E )$.
4. Since the person has not taken this medication before, we admit that $f ( 0 ) = 0$.
Determine the expression of the function $f$.

Part C: study of the function $\boldsymbol { f }$

In this part, we admit that $f$ is defined on the interval $\left[ 0 ; + \infty \left[ \operatorname { by } f ( t ) = 5 t \mathrm { e } ^ { - t } \right. \right.$.

1. Determine the limit of $f$ at $+ \infty$.

Interpret this result in the context of the exercise.
2. Study the variations of $f$ on the interval $[ 0 ; + \infty [$ then draw up its complete variation table.
3. Prove that there exist two real numbers $t _ { 1 }$ and $t _ { 2 }$ such that $f \left( t _ { 1 } \right) = f \left( t _ { 2 } \right) = 1$.
Give an approximate value to $10 ^ { - 2 }$ of the real numbers $t _ { 1 }$ and $t _ { 2 }$.
4. For a medication concentration greater than or equal to 1 gram per litre of blood, there is a risk of drowsiness. What is the duration in hours and minutes of the drowsiness risk when taking this medication?

Part D: average concentration

The average concentration of the medication (in grams per litre of blood) during the first hour is given by:
$$T _ { m } = \int _ { 0 } ^ { 1 } f ( t ) \mathrm { d } t$$
where $f$ is the function defined on $\left[ 0 ; + \infty \left[ \operatorname { by } f ( t ) = 5 t \mathrm { e } ^ { - t } \right. \right.$. Calculate this average concentration. Give the exact value then an approximate value to 0.01.

Exercise 2

Part A

We consider the function $f$ defined on the interval $[0; +\infty[$ by:
$$f(x) = \frac{1}{a + \mathrm{e}^{-bx}}$$
where $a$ and $b$ are two strictly positive real constants. We admit that the function $f$ is differentiable on the interval $[0; +\infty[$. The function $f$ has for graphical representation the curve $\mathscr{C}_f$.
We consider the points $\mathrm{A}(0; 0.5)$ and $\mathrm{B}(10; 1)$. We admit that the line (AB) is tangent to the curve $\mathscr{C}_f$ at point A.

1. By graphical reading, give an approximate value of $f(10)$.
2. We admit that $\lim_{x \rightarrow +\infty} f(x) = 1$. Give a graphical interpretation of this result.
3. Justify that $a = 1$.
4. Determine the slope of the line (AB).
5. a. Determine the expression of $f'(x)$ as a function of $x$ and the constant $b$. b. Deduce the value of $b$.

Part B

We admit, in the rest of the exercise, that the function $f$ is defined on the interval $[0; +\infty[$ by:
$$f(t) = \frac{1}{1 + \mathrm{e}^{-0.2x}}$$

1. Determine $\lim_{x \rightarrow +\infty} f(x)$.
2. Study the variations of the function $f$ on the interval $[0; +\infty[$.
3. Show that there exists a unique positive real number $\alpha$ such that $f(\alpha) = 0.97$.
4. Using a calculator, give a bound for the real number $\alpha$ by two consecutive integers. Interpret this result in the context of the statement.

Part C

1. Show that, for all $x$ belonging to the interval $[0; +\infty[$, $f(x) = \dfrac{\mathrm{e}^{0.2x}}{1 + \mathrm{e}^{0.2x}}$.
2. Deduce an antiderivative of the function $f$ on the interval $[0; +\infty[$.
3. Calculate the average value of the function $f$ on the interval $[0; 40]$, that is: $$I = \frac{1}{40} \int_0^{40} \frac{1}{1 + \mathrm{e}^{-0.2x}} \,\mathrm{d}x$$ The exact value and an approximate value to the nearest thousandth will be given.

A team of biologists is studying the evolution of the area covered by a marine algae called seagrass, on the bottom of Alycastre Bay, near the island of Porquerolles. The studied area has a total area of 20 hectares (ha), and on July 1, 2024, the seagrass covered 1 ha of this area.
Part A: Study of a discrete model
For any natural integer $n$, we denote by $u _ { n }$ the area of the zone, in hectares, covered by seagrass on July 1 of the year $2024 + n$. Thus, $u _ { 0 } = 1$. A study conducted on this area made it possible to establish that for any natural integer $n$: $$u _ { n + 1 } = - 0{,}02 u _ { n } ^ { 2 } + 1{,}3 u _ { n }$$

1. Calculate the area that seagrass should cover on July 1, 2025 according to this model.
2. We denote by $h$ the function defined on $[0;20]$ by $$h ( x ) = - 0{,}02 x ^ { 2 } + 1{,}3 x$$ We admit that $h$ is increasing on $[0;20]$. a. Prove that for any natural integer $n$, $1 \leqslant u _ { n } \leqslant u _ { n + 1 } \leqslant 20$. b. Deduce that the sequence $(u _ { n })$ converges. We denote by $L$ its limit. c. Justify that $L = 15$.
3. The biologists wish to know after how long the area covered by seagrass will exceed 14 hectares. a. Without any calculation, justify that, according to this model, this will occur. b. Copy and complete the following algorithm so that at the end of execution, it displays the answer to the biologists' question. \begin{verbatim} def seuil(): n=0 u= 1 while ...... : n=...... u=...... return n \end{verbatim}

Part B: Study of a continuous model
We wish to describe the area of the studied zone covered by seagrass over time with a continuous model. In this model, for a duration $t$, in years, elapsed from July 1, 2024, the area of the studied zone covered by seagrass is given by $f ( t )$, where $f$ is a function defined on $[ 0 ; + \infty [$ satisfying:

• $f ( 0 ) = 1$;
• $f$ does not vanish on $[ 0 ; + \infty [$;
• $f$ is differentiable on $[ 0 ; + \infty [$;
• $f$ is a solution on $[ 0 ; + \infty [$ of the differential equation $$\left( E _ { 1 } \right) : \quad y ^ { \prime } = 0{,}02 y ( 15 - y ) .$$

We admit that such a function $f$ exists; the purpose of this part is to determine an expression for it. We denote by $f ^ { \prime }$ the derivative function of $f$.

1. Let $g$ be the function defined on $\left[ 0 ; + \infty \left[ \text{ by } g ( t ) = \frac { 1 } { f ( t ) } \right. \right.$. Show that $g$ is a solution of the differential equation $$\left( E _ { 2 } \right) : \quad y ^ { \prime } = - 0{,}3 y + 0{,}02 .$$
2. Give the solutions of the differential equation $( E _ { 2 } )$.
3. Deduce that for all $t \in [ 0 ; + \infty [$: $$f ( t ) = \frac { 15 } { 14 \mathrm { e } ^ { - 0{,}3 t } + 1 }$$
4. Determine the limit of $f$ as $+ \infty$.
5. Solve in the interval $[ 0 ; + \infty [$ the inequality $f ( t ) > 14$. Interpret the result in the context of the exercise.

We study the evolution of the population of an animal species within a nature reserve. The numbers of this population were recorded in different years. The collected data are presented in the following table:

Year	2000	2005	2010	2015
Number of individuals	50	64	80	100

To anticipate the evolution of this population, the reserve management chose to model the number of individuals as a function of time. For this, it uses a function, defined on the interval $[0; +\infty[$, where the variable $x$ represents the elapsed time, in years, from the year 2000. In its model, the image of 0 by this function equals 50, which corresponds to the number of individuals in the year 2000.
Part A. Model 1
In this part, the reserve management makes the hypothesis that the function sought satisfies the following differential equation: $$y' = 0.05y - 0.5 \quad (E_1)$$

1. Solve the differential equation $(E_1)$ with the initial condition $y(0) = 50$.
2. Compare the results in the table with those that would be obtained with this model.

Part B. Model 2
In this part, the reserve management makes the hypothesis that the function sought satisfies the following differential equation: $$y' = 0.05y(1 - 0.00125y)$$
Let $f$ be the function defined on $[0; +\infty[$ by: $$f(x) = \frac{800}{1 + 15\mathrm{e}^{-0.05x}}$$ and $C$ its representative curve in an orthonormal coordinate system.
Using computer algebra software, the following results were obtained. For the rest of the exercise, these results may be used without proof, except for question 5.

	Instruction	Result
1	$f(x) := \frac{800}{1 + 15\mathrm{e}^{-0.05x}}$	$f(x) = \frac{800}{1 + 15\mathrm{e}^{-0.05x}}$
2	$f'(x) :=$ Derivative$(f(x))$	$f'(x) = \frac{600\mathrm{e}^{-0.05x}}{\left(1 + 15\mathrm{e}^{-0.05x}\right)^2}$
3	$f''(x) :=$ Derivative$(f'(x))$	$f''(x) = \frac{30\mathrm{e}^{-0.05x}}{\left(1 + 15\mathrm{e}^{-0.05x}\right)^3}\left(15\mathrm{e}^{-0.05x} - 1\right)$
4	Solve$(15\mathrm{e}^{-0.05x} - 1 \geqslant 0)$	$x \leqslant 20\ln(15)$

1. Prove that the function $f$ satisfies $f(0) = 50$ and that for all $x \in \mathbb{R}$: $$f'(x) = 0.05f(x)(1 - 0.00125f(x))$$ We admit that this function $f$ is the unique solution of $(E_2)$ taking the initial value of 50 at 0.
2. With this new model $f$, estimate the population size in 2050. Round the result to the nearest integer.
3. Calculate the limit of $f$ as $x \to +\infty$. What can be deduced about the curve $C$? Interpret this limit in the context of this concrete problem.
4. Justify that the function $f$ is increasing on $[0; +\infty[$.
5. Prove the result obtained in line 4 of the software.
6. We admit that the growth rate of the population of this species, expressed in number of individuals per year, is modeled by the function $f'$. a. Study the convexity of the function $f$ on the interval $[0; +\infty[$ and determine the coordinates of any inflection points of the curve $C$. b. The reserve management claims: ``According to this model, the growth rate of the population of this species will increase for a little more than fifty years, then will decrease''. Is the management correct? Justify.

The population $p(t)$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{dp(t)}{dt} = 0.5\,p(t) - 450$. If $p(0) = 850$, then the time at which the population becomes zero is
(1) $\ln 18$
(2) $\ln 9$
(3) $\frac{1}{2}\ln 18$
(4) $2\ln 18$

Let the population of rabbits surviving at a time $t$ be governed by the differential equation $\frac { d p ( t ) } { d t } = \frac { 1 } { 2 } \{ p ( t ) - 400 \}$. If $p ( 0 ) = 100$, then $p ( t )$ equals
(1) $600 - 500 e ^ { \frac { t } { 2 } }$
(2) $400 - 300 e ^ { \frac { - t } { 2 } }$
(3) $400 - 300 e ^ { t / 2 }$
(4) $300 - 200 e ^ { \frac { - t } { 2 } }$

The Laplace transform of the function $f ( t )$, defined for $t \geq 0$, is denoted by $F ( s ) = \mathcal { L } [ f ( t ) ]$ and its definition is given by
$$F ( s ) = \mathcal { L } [ f ( t ) ] = \int _ { 0 } ^ { \infty } f ( t ) \exp ( - s t ) d t$$
where $s$ is a complex number. In the following, the set of all complex numbers is denoted by $\mathbb { C }$, and the set of the complex numbers with positive real parts is denoted by $\mathbb { C } ^ { + }$.
I. Consider the following function $g ( t )$ defined for $t \geq 0$ :
$$g ( t ) = \int _ { 0 } ^ { \infty } \frac { \sin ^ { 2 } ( t x ) } { x ^ { 2 } } d x$$

1. Find the Laplace transform $G ( s ) = \mathcal { L } [ g ( t ) ] \left( s \in \mathbb { C } ^ { + } \right)$ of the function $g ( t )$.
2. Obtain the value of the following integral using the result of Question I.1: $$\int _ { - \infty } ^ { \infty } \frac { \sin ^ { 2 } ( x ) } { x ^ { 2 } } d x$$

II. Consider the function $u ( x , t )$ that satisfies the following partial differential equation:
$$\frac { \partial u ( x , t ) } { \partial t } = \frac { \partial ^ { 2 } u ( x , t ) } { \partial x ^ { 2 } } ( 0 < x < 1 , t > 0 )$$
under the boundary conditions:
$$\left\{ \begin{array} { l } \left. \frac { \partial u ( x , t ) } { \partial x } \right| _ { x = 0 } = 0 \quad ( t \geq 0 ) \\ u ( 1 , t ) = 1 \quad ( t \geq 0 ) \\ u ( x , 0 ) = \frac { \cosh ( x ) } { \cosh ( 1 ) } \quad ( 0 < x < 1 ) \end{array} \right.$$

1. The Laplace transform of $u ( x , t )$ is denoted by $U ( x , s ) = \mathcal { L } [ u ( x , t ) ]$ ( $s \in \mathbb { C } ^ { + }$). Derive the ordinary differential equation and boundary conditions for $U ( x , s )$ with respect to the independent variable $x$. Here, the function $u ( x , t )$ can be assumed to be bounded. The following relations can also be used: $$\begin{aligned} & \mathcal { L } \left[ \frac { \partial u ( x , t ) } { \partial x } \right] = \frac { \partial U ( x , s ) } { \partial x } \\ & \mathcal { L } \left[ \frac { \partial ^ { 2 } u ( x , t ) } { \partial x ^ { 2 } } \right] = \frac { \partial ^ { 2 } U ( x , s ) } { \partial x ^ { 2 } } \end{aligned}$$
2. Using an analytic function $Q ( s ) ( s \in \mathbb { C } )$, the function $U _ { \mathrm { c } } ( x , s )$ is defined as follows: $$U _ { c } ( x , s ) = \frac { \cosh ( x ) } { ( s - 1 ) \cosh ( 1 ) } - \frac { \cosh ( x \sqrt { s } ) } { Q ( s ) } \quad ( 0 \leq x \leq 1 )$$ When the function $U ( x , s ) = U _ { \mathrm { c } } ( x , s )$ satisfies the differential equation and the boundary conditions derived in Question II.1 for $s \in \mathbb { C } ^ { + }$, find the function $Q ( s )$.
3. Using the function $Q ( s )$ derived in Question II.2, the sequence of complex numbers $\left\{ a _ { r } \right\} ( r = 1,2 , \cdots )$ is defined by arranging all of the roots of $Q ( s ) = 0 ( s \in \mathbb { C } )$ in ascending order of their absolute values. In this case, the following limits $R _ { r } ( x , t )$ are finite for $t \geq 0,0 \leq x \leq 1$, and $r \geq 1$ : $$R _ { r } ( x , t ) = \lim _ { s \rightarrow a _ { r } } \left( s - a _ { r } \right) U _ { \mathrm { c } } ( x , s ) \exp ( s t )$$ and the solution of the partial differential equation is given by $$u ( x , t ) = \sum _ { r = 1 } ^ { \infty } R _ { r } ( x , t )$$ Determine $R _ { 1 } ( x , t ) , R _ { 2 } ( x , t )$, and $R _ { r } ( x , t )$ for $r \geq 3$.