Let $f$ be a function with $f(1) = 4$ such that for all points $(x, y)$ on the graph of $f$ the slope is given by $\dfrac{3x^2 + 1}{2y}$.
(a) Find the slope of the graph of $f$ at the point where $x = 1$.
(b) Write an equation for the line tangent to the graph of $f$ at $x = 1$ and use it to approximate $f(1.2)$.
(c) Find $f(x)$ by solving the separable differential equation $\dfrac{dy}{dx} = \dfrac{3x^2 + 1}{2y}$ with the initial condition $f(1) = 4$.
(d) Use your solution from part (c) to find $f(1.2)$.

The function $f$ is differentiable for all real numbers. The point $\left(3, \dfrac{1}{4}\right)$ is on the graph of $y = f(x)$, and the slope at each point $(x, y)$ on the graph is given by $\dfrac{dy}{dx} = y^{2}(6 - 2x)$.
(a) Find $\dfrac{d^{2}y}{dx^{2}}$ and evaluate it at the point $\left(3, \dfrac{1}{4}\right)$.
(b) Find $y = f(x)$ by solving the differential equation $\dfrac{dy}{dx} = y^{2}(6 - 2x)$ with the initial condition $f(3) = \dfrac{1}{4}$.

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 $\dfrac{dy}{dx} = x^{4}(y-2)$.
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.
(b) While the slope field in part (a) is drawn at only twelve points, it is defined at every point in the $xy$-plane. Describe all points in the $xy$-plane for which the slopes are negative.
(c) Find the particular solution $y = f(x)$ to the given differential equation with the initial condition $f(0) = 0$.

Consider the differential equation $\frac { d y } { d x } = - \frac { 2 x } { y }$.
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.
(b) Let $y = f ( x )$ be the particular solution to the differential equation with the initial condition $f ( 1 ) = - 1$. Write an equation for the line tangent to the graph of $f$ at $( 1 , - 1 )$ and use it to approximate $f ( 1.1 )$.
(c) Find the particular solution $y = f ( x )$ to the given differential equation with the initial condition $f ( 1 ) = - 1$.

Consider the differential equation $\dfrac{dy}{dx} = \dfrac{-xy^2}{2}$. Let $y = f(x)$ be the particular solution to this differential equation with the initial condition $f(-1) = 2$.
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.
(b) Write an equation for the line tangent to the graph of $f$ at $x = -1$.
(c) Find the solution $y = f(x)$ to the given differential equation with the initial condition $f(-1) = 2$.

Consider the differential equation $\dfrac{dy}{dx} = \dfrac{1+y}{x}$, where $x \neq 0$.
(a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated.
(b) Find the particular solution $y = f(x)$ to the differential equation with the initial condition $f(-1) = 1$ and state its domain.

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 $\lim _ { x \rightarrow \infty } f ( x )$.

Solutions to the differential equation $\frac{dy}{dx} = xy^3$ also satisfy $\frac{d^2y}{dx^2} = y^3\left(1 + 3x^2y^2\right)$. Let $y = f(x)$ be a particular solution to the differential equation $\frac{dy}{dx} = xy^3$ with $f(1) = 2$.
(a) Write an equation for the line tangent to the graph of $y = f(x)$ at $x = 1$.
(b) Use the tangent line equation from part (a) to approximate $f(1.1)$. Given that $f(x) > 0$ for $1 < x < 1.1$, is the approximation for $f(1.1)$ greater than or less than $f(1.1)$? Explain your reasoning.
(c) Find the particular solution $y = f(x)$ with initial condition $f(1) = 2$.

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 $\frac { d y } { d x } = e ^ { y } \left( 3 x ^ { 2 } - 6 x \right)$. Let $y = f ( x )$ be the particular solution to the differential equation that passes through $( 1, 0 )$.
(a) Write an equation for the line tangent to the graph of $f$ at the point $( 1, 0 )$. Use the tangent line to approximate $f ( 1.2 )$.
(b) Find $y = f ( x )$, the particular solution to the differential equation that passes through $( 1, 0 )$.

Consider the differential equation $\dfrac { d y } { d x } = ( 3 - y ) \cos x$. Let $y = f ( x )$ be the particular solution to the differential equation with the initial condition $f ( 0 ) = 1$. The function $f$ is defined for all real numbers.
(a) A portion of the slope field of the differential equation is given below. Sketch the solution curve through the point $( 0, 1 )$.
(b) Write an equation for the line tangent to the solution curve in part (a) at the point $( 0, 1 )$. Use the equation to approximate $f ( 0.2 )$.
(c) Find $y = f ( x )$, the particular solution to the differential equation with the initial condition $f ( 0 ) = 1$.

Consider the differential equation $\dfrac{dy}{dx} = 2x - y$.
(a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated.
(b) Find $\dfrac{d^2y}{dx^2}$ 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 $y = mx + b$ is a solution to the differential equation.

Consider the differential equation $\frac { d y } { d x } = \frac { y ^ { 2 } } { x - 1 }$.
(a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated.
(b) Let $y = f ( x )$ be the particular solution to the given differential equation with the initial condition $f ( 2 ) = 3$. Write an equation for the line tangent to the graph of $y = f ( x )$ at $x = 2$. Use your equation to approximate $f ( 2.1 )$.
(c) Find the particular solution $y = f ( x )$ to the given differential equation with the initial condition $f ( 2 ) = 3$.

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 $f ( 1 ) = 0$.

A medication is administered to a patient. The amount, in milligrams, of the medication in the patient at time $t$ hours is modeled by a function $y = A(t)$ that satisfies the differential equation $\frac{dy}{dt} = \frac{12 - y}{3}$. At time $t = 0$ hours, there are 0 milligrams of the medication in the patient.
(a) A portion of the slope field for the differential equation $\frac{dy}{dt} = \frac{12 - y}{3}$ is given. Sketch the solution curve through the point $(0, 0)$.
(b) Using correct units, interpret the statement $\lim_{t \rightarrow \infty} A(t) = 12$ in the context of this problem.
(c) Use separation of variables to find $y = A(t)$, the particular solution to the differential equation $\frac{dy}{dt} = \frac{12 - y}{3}$ with initial condition $A(0) = 0$.
(d) A different procedure is used to administer the medication to a second patient. The amount, in milligrams, of the medication in the second patient at time $t$ hours is modeled by a function $y = B(t)$ that satisfies the differential equation $\frac{dy}{dt} = 3 - \frac{y}{t + 2}$. At time $t = 1$ hour, there are 2.5 milligrams of the medication in the second patient. Is the rate of change of the amount of medication in the second patient increasing or decreasing at time $t = 1$? Give a reason for your answer.

Consider the differential equation $\dfrac{dy}{dx} = \dfrac{1}{2}\sin\!\left(\dfrac{\pi}{2}x\right)\sqrt{y+7}$. Let $y = f(x)$ be the particular solution to the differential equation with the initial condition $f(1) = 2$. The function $f$ is defined for all real numbers.
(a) A portion of the slope field for the differential equation is given. Sketch the solution curve through the point $(1, 2)$.
(b) Write an equation for the line tangent to the solution curve in part (a) at the point $(1, 2)$. Use the equation to approximate $f(0.8)$.
(c) It is known that $f''(x) > 0$ for $-1 \leq x \leq 1$. Is the approximation found in part (b) an overestimate or an underestimate for $f(0.8)$? Give a reason for your answer.
(d) Use separation of variables to find $y = f(x)$, the particular solution to the differential equation $\dfrac{dy}{dx} = \dfrac{1}{2}\sin\!\left(\dfrac{\pi}{2}x\right)\sqrt{y+7}$ with the initial condition $f(1) = 2$.

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 $f(1) = 4$.

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.