The number of fish in a lake is modeled by the function $F$ that satisfies the logistic differential equation $\frac { d F } { d t } = 0.04 F \left( 1 - \frac { F } { 5000 } \right)$, where $t$ is the time in months and $F ( 0 ) = 2000$. What is $\lim _ { t \rightarrow \infty } F ( t )$?
(A) 10,000
(B) 5000
(C) 2500
(D) 2000

44. If $f$ is the solution of $x f ^ { \prime } ( x ) - f ( x ) = x$ such that $f ( - 1 ) = 1$, then $f \left( e ^ { - 1 } \right) =$
(A) $- 2 e ^ { - 1 }$
(B) 0
C) $e ^ { - 1 }$
(D) $- e ^ { - 1 }$
(E) $2 e ^ { - 2 }$

Consider the differential equation $\frac { d y } { d x } = \frac { y - 1 } { x ^ { 2 } }$, where $x \neq 0$.
(a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated. (Note: Use the axes provided in the exam booklet.)
(b) Find the particular solution $y = f ( x )$ to the differential equation with the initial condition $f ( 2 ) = 0$.
(c) For the particular solution $y = f ( x )$ described in part (b), find $\lim _ { x \rightarrow \infty } f ( x )$.

At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function $W$ models the total amount of solid waste stored at the landfill. Planners estimate that $W$ will satisfy the differential equation $\frac{dW}{dt} = \frac{1}{25}(W - 300)$ for the next 20 years. $W$ is measured in tons, and $t$ is measured in years from the start of 2010.
(a) Use the line tangent to the graph of $W$ at $t = 0$ to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of 2010 (time $t = \frac{1}{4}$).
(b) Find $\frac{d^2W}{dt^2}$ in terms of $W$. Use $\frac{d^2W}{dt^2}$ to determine whether your answer in part (a) is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time $t = \frac{1}{4}$.
(c) Find the particular solution $W = W(t)$ to the differential equation $\frac{dW}{dt} = \frac{1}{25}(W - 300)$ with initial condition $W(0) = 1400$.

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
$$\frac { d B } { d t } = \frac { 1 } { 5 } ( 100 - B ) .$$
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 $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ in terms of $B$. Use $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ 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 $B ( 0 ) = 20$.

Consider the differential equation $\frac { d y } { d x } = \frac { 1 } { 3 } x ( y - 2 ) ^ { 2 }$.
(a) A slope field for the given differential equation is shown below. Sketch the solution curve that passes through the point $( 0,2 )$, and sketch the solution curve that passes through the point $( 1,0 )$.
(b) Let $y = f ( x )$ be the particular solution to the given differential equation with initial condition $f ( 1 ) = 0$. Write an equation for the line tangent to the graph of $y = f ( x )$ at $x = 1$. Use your equation to approximate $f ( 0.7 )$.
(c) Find the particular solution $y = f ( x )$ to the given differential equation with initial condition $f ( 1 ) = 0$.

The depth of seawater at a location can be modeled by the function $H$ that satisfies the differential equation $\frac{dH}{dt} = \frac{1}{2}(H - 1)\cos\left(\frac{t}{2}\right)$, where $H(t)$ is measured in feet and $t$ is measured in hours after noon $(t = 0)$. It is known that $H(0) = 4$.
(a) A portion of the slope field for the differential equation is provided. Sketch the solution curve, $y = H(t)$, through the point $(0, 4)$.
(b) For $0 < t < 5$, it can be shown that $H(t) > 1$. Find the value of $t$, for $0 < t < 5$, at which $H$ has a critical point. Determine whether the critical point corresponds to a relative minimum, a relative maximum, or neither a relative minimum nor a relative maximum of the depth of seawater at the location. Justify your answer.
(c) Use separation of variables to find $y = H(t)$, the particular solution to the differential equation $\frac{dH}{dt} = \frac{1}{2}(H - 1)\cos\left(\frac{t}{2}\right)$ with initial condition $H(0) = 4$.

Let $f$ be the function satisfying $f'(x) = -3x f(x)$, for all real numbers $x$, with $f(1) = 4$ and $\lim_{x \rightarrow \infty} f(x) = 0$.
(a) Evaluate $\displaystyle\int_{1}^{\infty} -3x f(x)\, dx$. Show the work that leads to your answer.
(b) Use Euler's method, starting at $x = 1$ with a step size of $0.5$, to approximate $f(2)$.
(c) Write an expression for $y = f(x)$ by solving the differential equation $\dfrac{dy}{dx} = -3xy$ with the initial condition $f(1) = 4$.

The number of gallons, $P(t)$, of a pollutant in a lake changes at the rate $P'(t) = 1 - 3e^{-0.2\sqrt{t}}$ gallons per day, where $t$ is measured in days. There are 50 gallons of the pollutant in the lake at time $t = 0$. The lake is considered to be safe when it contains 40 gallons or less of pollutant.
(a) Is the amount of pollutant increasing at time $t = 9$? Why or why not?
(b) For what value of $t$ will the number of gallons of pollutant be at its minimum? Justify your answer.
(c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify your answer.
(d) An investigator uses the tangent line approximation to $P(t)$ at $t = 0$ as a model for the amount of pollutant in the lake. At what time $t$ does this model predict that the lake becomes safe?

Consider the differential equation $\frac { d y } { d x } = 2 y - 4 x$.
(a) The slope field for the given differential equation is provided. Sketch the solution curve that passes through the point $( 0, 1 )$ and sketch the solution curve that passes through the point $( 0 , - 1 )$.
(b) Let $f$ be the function that satisfies the given differential equation with the initial condition $f ( 0 ) = 1$. Use Euler's method, starting at $x = 0$ with a step size of 0.1, to approximate $f ( 0.2 )$. Show the work that leads to your answer.
(c) Find the value of $b$ for which $y = 2 x + b$ is a solution to the given differential equation. Justify your answer.
(d) Let $g$ be the function that satisfies the given differential equation with the initial condition $g ( 0 ) = 0$. Does the graph of $g$ have a local extremum at the point $( 0, 0 )$? If so, is the point a local maximum or a local minimum? Justify your answer.

A population is modeled by a function $P$ that satisfies the logistic differential equation
$$\frac { d P } { d t } = \frac { P } { 5 } \left( 1 - \frac { P } { 12 } \right) .$$
(a) If $P ( 0 ) = 3$, what is $\lim _ { t \rightarrow \infty } P ( t )$ ?
If $P ( 0 ) = 20$, what is $\lim _ { t \rightarrow \infty } P ( t )$ ?
(b) If $P ( 0 ) = 3$, for what value of $P$ is the population growing the fastest?
(c) A different population is modeled by a function $Y$ that satisfies the separable differential equation
$$\frac { d Y } { d t } = \frac { Y } { 5 } \left( 1 - \frac { t } { 12 } \right)$$
Find $Y ( t )$ if $Y ( 0 ) = 3$.
(d) For the function $Y$ found in part (c), what is $\lim _ { t \rightarrow \infty } Y ( t )$ ?

5. Consider the differential equation $\frac { d y } { d x } = \frac { 1 + y } { x }$, where $x \neq 0$.
(a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated. (Note: Use the axes provided in the pink exam booklet.) [Figure]
(b) Find the particular solution $y = f ( x )$ to the differential equation with the initial condition $f ( - 1 ) = 1$ and state its domain.

At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function $W$ models the total amount of solid waste stored at the landfill. Planners estimate that $W$ will satisfy the differential equation $\frac { d W } { d t } = \frac { 1 } { 25 } ( W - 300 )$ for the next 20 years. $W$ is measured in tons, and $t$ is measured in years from the start of 2010. (a) Use the line tangent to the graph of $W$ at $t = 0$ to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of 2010 (time $t = \frac { 1 } { 4 }$ ). (b) Find $\frac { d ^ { 2 } W } { d t ^ { 2 } }$ in terms of $W$. Use $\frac { d ^ { 2 } W } { d t ^ { 2 } }$ to determine whether your answer in part (a) is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time $t = \frac { 1 } { 4 }$. (c) Find the particular solution $W = W ( t )$ to the differential equation $\frac { d W } { d t } = \frac { 1 } { 25 } ( W - 300 )$ with initial condition $W ( 0 ) = 1400$.

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
$$\frac { d B } { d t } = \frac { 1 } { 5 } ( 100 - B ) .$$
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 $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ in terms of $B$. Use $\frac { d ^ { 2 } B } { d t ^ { 2 } }$ 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 $B ( 0 ) = 20$.

Consider the differential equation $\frac{dy}{dx} = 6x^{2} - x^{2}y$. Let $y = f(x)$ be a particular solution to this differential equation with the initial condition $f(-1) = 2$.
(a) Use Euler's method with two steps of equal size, starting at $x = -1$, to approximate $f(0)$. Show the work that leads to your answer.
(b) At the point $(-1, 2)$, the value of $\frac{d^{2}y}{dx^{2}}$ is $-12$. Find the second-degree Taylor polynomial for $f$ about $x = -1$.
(c) Find the particular solution $y = f(x)$ to the given differential equation with the initial condition $f(-1) = 2$.

Consider the differential equation $\frac{dy}{dx} = 1 - y$. Let $y = f(x)$ be the particular solution to this differential equation with the initial condition $f(1) = 0$. For this particular solution, $f(x) < 1$ for all values of $x$.
(a) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f(0)$. Show the work that leads to your answer.
(b) Find $\lim_{x \to 1} \frac{f(x)}{x^3 - 1}$. Show the work that leads to your answer.
(c) Find the particular solution $y = f(x)$ to the differential equation $\frac{dy}{dx} = 1 - y$ with the initial condition $f(1) = 0$.

At the beginning of 2010, a landfill contained 1400 tons of solid waste. The increasing function $W$ models the total amount of solid waste stored at the landfill. Planners estimate that $W$ will satisfy the differential equation $\frac{dW}{dt} = \frac{1}{25}(W - 300)$ for the next 20 years. $W$ is measured in tons, and $t$ is measured in years from the start of 2010.
(a) Use the line tangent to the graph of $W$ at $t = 0$ to approximate the amount of solid waste that the landfill contains at the end of the first 3 months of 2010 (time $t = \frac{1}{4}$).
(b) Find $\frac{d^2W}{dt^2}$ in terms of $W$. Use $\frac{d^2W}{dt^2}$ to determine whether your answer in part (a) is an underestimate or an overestimate of the amount of solid waste that the landfill contains at time $t = \frac{1}{4}$.
(c) Find the particular solution $W = W(t)$ to the differential equation $\frac{dW}{dt} = \frac{1}{25}(W - 300)$ with initial condition $W(0) = 1400$.

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 $B(0) = 20$.

Consider the differential equation $\frac { d y } { d x } = y ^ { 2 } ( 2 x + 2 )$. Let $y = f ( x )$ be the particular solution to the differential equation with initial condition $f ( 0 ) = - 1$.
(a) Find $\lim _ { x \rightarrow 0 } \frac { f ( x ) + 1 } { \sin x }$. Show the work that leads to your answer.
(b) Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f \left( \frac { 1 } { 2 } \right)$.
(c) Find $y = f ( x )$, the particular solution to the differential equation with initial condition $f ( 0 ) = - 1$.

Consider the differential equation $\frac { d y } { d x } = x ^ { 2 } - \frac { 1 } { 2 } y$.
(a) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$.
(b) Let $y = f ( x )$ be the particular solution to the given differential equation whose graph passes through the point $( - 2,8 )$. Does the graph of $f$ have a relative minimum, a relative maximum, or neither at the point $( - 2,8 )$ ? Justify your answer.
(c) Let $y = g ( x )$ be the particular solution to the given differential equation with $g ( - 1 ) = 2$. Find $\lim _ { x \rightarrow - 1 } \left( \frac { g ( x ) - 2 } { 3 ( x + 1 ) ^ { 2 } } \right)$. Show the work that leads to your answer.
(d) Let $y = h ( x )$ be the particular solution to the given differential equation with $h ( 0 ) = 2$. Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $h ( 1 )$.

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 $t = 3$?

The depth of seawater at a location can be modeled by the function $H$ that satisfies the differential equation $\frac{dH}{dt} = \frac{1}{2}(H - 1)\cos\left(\frac{t}{2}\right)$, where $H(t)$ is measured in feet and $t$ is measured in hours after noon $(t = 0)$. It is known that $H(0) = 4$.
(a) A portion of the slope field for the differential equation is provided. Sketch the solution curve, $y = H(t)$, through the point $(0, 4)$.
(b) For $0 < t < 5$, it can be shown that $H(t) > 1$. Find the value of $t$, for $0 < t < 5$, at which $H$ has a critical point. Determine whether the critical point corresponds to a relative minimum, a relative maximum, or neither a relative minimum nor a relative maximum of the depth of seawater at the location. Justify your answer.
(c) Use separation of variables to find $y = H(t)$, the particular solution to the differential equation $\frac{dH}{dt} = \frac{1}{2}(H - 1)\cos\left(\frac{t}{2}\right)$ with initial condition $H(0) = 4$.

Let $y = f ( x )$ be the particular solution to the differential equation $\frac { d y } { d x } = ( 3 - x ) y ^ { 2 }$ with initial condition $f ( 1 ) = - 1$.
A. Find $f ^ { \prime \prime } ( 1 )$, the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 1 , - 1 )$. Show the work that leads to your answer.
B. Write the second-degree Taylor polynomial for $f$ about $x = 1$.
C. The second-degree Taylor polynomial for $f$ about $x = 1$ is used to approximate $f ( 1.1 )$. Given that $\left| f ^ { \prime \prime \prime } ( x ) \right| \leq 60$ for all $x$ in the interval $1 \leq x \leq 1.1$, use the Lagrange error bound to show that this approximation differs from $f ( 1.1 )$ by at most 0.01.
D. Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 1.4 )$. Show the work that leads to your answer.

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.

Main topics covered: Differential equations; exponential function.
We consider the differential equation
$$\text { (E) } y ^ { \prime } = y + 2 x \mathrm { e } ^ { x }$$
We seek the set of functions defined and differentiable on the set $\mathbb { R }$ of real numbers that are solutions to this equation.

1. Let $u$ be the function defined on $\mathbb { R }$ by $u ( x ) = x ^ { 2 } \mathrm { e } ^ { x }$. We admit that $u$ is differentiable and we denote $u ^ { \prime }$ its derivative function. Prove that $u$ is a particular solution of $( E )$.
2. Let $f$ be a function defined and differentiable on $\mathbb { R }$. We denote $g$ the function defined on $\mathbb { R }$ by: $$g ( x ) = f ( x ) - u ( x )$$ a. Prove that if the function $f$ is a solution of the differential equation $( E )$ then the function $g$ is a solution of the differential equation: $y ^ { \prime } = y$. We admit that the converse of this property is also true. b. Using the solution of the differential equation $y ^ { \prime } = y$, solve the differential equation (E).
3. Study of the function $u$ a. Study the sign of $u ^ { \prime } ( x )$ for $x$ varying in $\mathbb { R }$. b. Draw the table of variations of the function $u$ on $\mathbb { R }$ (limits are not required). c. Determine the largest interval on which the function $u$ is concave.