A bottle of milk is taken out of a refrigerator and placed in a pan of hot water to be warmed. The increasing function $M$ models the temperature of the milk at time $t$, where $M(t)$ is measured in degrees Celsius (${}^{\circ}\mathrm{C}$) and $t$ is the number of minutes since the bottle was placed in the pan. $M$ satisfies the differential equation $\frac{dM}{dt} = \frac{1}{4}(40 - M)$. At time $t = 0$, the temperature of the milk is $5^{\circ}\mathrm{C}$. It can be shown that $M(t) < 40$ for all values of $t$.
(a) A slope field for the differential equation $\frac{dM}{dt} = \frac{1}{4}(40 - M)$ is shown. Sketch the solution curve through the point $(0, 5)$.
(b) Use the line tangent to the graph of $M$ at $t = 0$ to approximate $M(2)$, the temperature of the milk at time $t = 2$ minutes.
(c) Write an expression for $\frac{d^{2}M}{dt^{2}}$ in terms of $M$. Use $\frac{d^{2}M}{dt^{2}}$ to determine whether the approximation from part (b) is an underestimate or an overestimate for the actual value of $M(2)$. Give a reason for your answer.
(d) Use separation of variables to find an expression for $M(t)$, the particular solution to the differential equation $\frac{dM}{dt} = \frac{1}{4}(40 - M)$ with initial condition $M(0) = 5$.

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 $H(0) = 4$.

Consider the differential equation given by $\dfrac{dy}{dx} = \dfrac{xy}{2}$.
(a) On the axes provided, sketch a slope field for the given differential equation at the nine points indicated.
(b) Let $y = f(x)$ be the particular solution to the given differential equation with the initial condition $f(0) = 3$. 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 particular solution $y = f(x)$ to the given differential equation with the initial condition $f(0) = 3$. Use your solution to find $f(0.2)$.

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.

Consider the differential equation $\dfrac{dy}{dx} = \dfrac{3 - x}{y}$.
(a) Let $y = f(x)$ be the particular solution to the given differential equation for $1 < x < 5$ such that the line $y = -2$ is tangent to the graph of $f$. Find the $x$-coordinate of the point of tangency, and determine whether $f$ has a local maximum, local minimum, or neither at this point. Justify your answer.
(b) Let $y = g(x)$ be the particular solution to the given differential equation for $-2 < x < 8$, with the initial condition $g(6) = -4$. Find $y = g(x)$.

Consider the differential equation $\frac { d y } { d x } = 2 x - y$.
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point $( 0, 1 )$.
(b) The solution curve that passes through the point $( 0, 1 )$ has a local minimum at $x = \ln \left( \frac { 3 } { 2 } \right)$. What is the $y$-coordinate of this local minimum?
(c) Let $y = f ( x )$ be the particular solution to the given differential equation with the initial condition $f ( 0 ) = 1$. Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f ( - 0.4 )$. Show the work that leads to your answer.
(d) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. Determine whether the approximation found in part (c) is less than or greater than $f ( - 0.4 )$. Explain your reasoning.

Consider the differential equation $\frac{dy}{dx} = 5x^{2} - \frac{6}{y-2}$ for $y \neq 2$. Let $y = f(x)$ be the particular solution to this differential equation with the initial condition $f(-1) = -4$.
(a) Evaluate $\frac{dy}{dx}$ and $\frac{d^{2}y}{dx^{2}}$ at $(-1, -4)$.
(b) Is it possible for the $x$-axis to be tangent to the graph of $f$ at some point? Explain why or why not.
(c) Find the second-degree Taylor polynomial for $f$ about $x = -1$.
(d) 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.

Consider the logistic differential equation $\frac { d y } { d t } = \frac { y } { 8 } ( 6 - y )$. Let $y = f ( t )$ be the particular solution to the differential equation with $f ( 0 ) = 8$.
(a) A slope field for this differential equation is given below. Sketch possible solution curves through the points $( 3,2 )$ and $( 0,8 )$.
(b) Use Euler's method, starting at $t = 0$ with two steps of equal size, to approximate $f ( 1 )$.
(c) Write the second-degree Taylor polynomial for $f$ about $t = 0$, and use it to approximate $f ( 1 )$.
(d) What is the range of $f$ for $t \geq 0$ ?

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 $B(0) = 20$.

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 $B ( 0 ) = 20$.

The points $( - 1 , - 1 )$ and $( 1 , - 5 )$ are on the graph of a function $y = f ( x )$ that satisfies the differential equation $\frac { d y } { d x } = x ^ { 2 } + y$. Which of the following must be true?
(A) $( 1 , - 5 )$ is a local maximum of $f$.
(B) $( 1 , - 5 )$ is a point of inflection of the graph of $f$.
(C) $( - 1 , - 1 )$ is a local maximum of $f$.
(D) $( - 1 , - 1 )$ is a local minimum of $f$.
(E) $( - 1 , - 1 )$ is a point of inflection of the graph of $f$.

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) $\frac { d y } { d t } = k y ( 1 - y )$

Let $y = f ( x )$ be the solution to the differential equation $\frac { d y } { d x } = x - y$ with initial condition $f ( 1 ) = 3$. What is the approximation for $f ( 2 )$ obtained by using Euler's method with two steps of equal length starting at $x = 1$ ?
(A) $- \frac { 5 } { 4 }$
(B) 1
(C) $\frac { 7 } { 4 }$
(D) 2
(E) $\frac { 21 } { 4 }$

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) $\frac { d P } { d t } = 100 P ^ { 2 }$

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 $\frac { d y } { d x } = 2 x - y$.
(a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated.
(b) Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. Determine the concavity of all solution curves for the given differential equation in Quadrant II. Give a reason for your answer.
(c) Let $y = f ( x )$ be the particular solution to the differential equation with the initial condition $f ( 2 ) = 3$. Does $f$ have a relative minimum, a relative maximum, or neither at $x = 2$? Justify your answer.
(d) Find the values of the constants $m$ and $b$ for which $y = m x + b$ is a solution to the differential equation.

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 $t = 3$?

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 $t$.

Let $y = f ( x )$ be the particular solution to the differential equation $\frac { d y } { d x } = y \cdot ( x \ln x )$ with initial condition $f ( 1 ) = 4$. It can be shown that $f ^ { \prime \prime } ( 1 ) = 4$.
(a) Write the second-degree Taylor polynomial for $f$ about $x = 1$. Use the Taylor polynomial to approximate $f ( 2 )$.
(b) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 2 )$. Show the work that leads to your answer.
(c) Find the particular solution $y = f ( x )$ to the differential equation $\frac { d y } { d x } = y \cdot ( x \ln x )$ with initial condition $f ( 1 ) = 4$.