Shown above is a slope field for which of the following differential equations?
(A) $\frac { d y } { d x } = x y + x$
(B) $\frac { d y } { d x } = x y + y$
(C) $\frac { d y } { d x } = y + 1$
(D) $\frac { d y } { d x } = ( x + 1 ) ^ { 2 }$

Shown above is a slope field for which of the following differential equations?
(A) $\frac { d y } { d x } = \frac { y - 2 } { 2 }$
(B) $\frac { d y } { d x } = \frac { y ^ { 2 } - 4 } { 4 }$
(C) $\frac { d y } { d x } = \frac { x - 2 } { 2 }$
(D) $\frac { d y } { d x } = \frac { x ^ { 2 } - 4 } { 4 }$

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 $f(1.2)$.

21. If $\frac { d y } { d t } = k y$ and $k$ is a nonzero constant, then $y$ could be
(A) $2 e ^ { k r y }$
(B) $2 e ^ { k t }$
(C) $e ^ { k t } + 3$
(D) $k t y + 5$
(E) $\frac { 1 } { 2 } k y ^ { 2 } + \frac { 1 } { 2 }$

Consider the differential equation $\frac { d y } { d x } = \frac { 3 x ^ { 2 } } { e ^ { 2 y } }$.
(a) Find a solution $y = f ( x )$ to the differential equation satisfying $f ( 0 ) = \frac { 1 } { 2 }$.
(b) Find the domain and range of the function $f$ found in part (a).

Consider the differential equation $\frac { d y } { d x } = \frac { - x y ^ { 2 } } { 2 }$. 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.
(Note: Use the axes provided in the test booklet.)
(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 $f ( - 1 ) = 2$.

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.

A particle moves along the $x$-axis with position at time $t$ given by $x(t) = e^{-t}\sin t$ for $0 \leq t \leq 2\pi$.
(a) Find the time $t$ at which the particle is farthest to the left. Justify your answer.
(b) Find the value of the constant $A$ for which $x(t)$ satisfies the equation $Ax^{\prime\prime}(t) + x^{\prime}(t) + x(t) = 0$ for $0 < t < 2\pi$.

Consider the differential equation $\frac { d y } { d x } = \frac { 1 } { 2 } x + y - 1$. (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 $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$. Describe the region in the $x y$-plane in which all solution curves to the differential equation are concave up. (c) Let $y = f ( x )$ be a particular solution to the differential equation with the initial condition $f ( 0 ) = 1$. Does $f$ have a relative minimum, a relative maximum, or neither at $x = 0$ ? 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 $\frac { d y } { d x } = - \frac { 2 x } { y }$.
(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated.
(Note: Use the axes provided in the pink test booklet.)
(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 $f ( 1 ) = - 1$.

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 $f(1) = 2$.

As a pot of tea cools, the temperature of the tea is modeled by a differentiable function $H$ for $0 \leq t \leq 10$, where time $t$ is measured in minutes and temperature $H(t)$ is measured in degrees Celsius. Values of $H(t)$ at selected values of time $t$ are shown in the table below.

\begin{tabular}{ c } $t$

(minutes)

& 0 & 2 & 5 & 9 & 10 \hline

$H(t)$

(degrees Celsius)

& 66 & 60 & 52 & 44 & 43 \hline \end{tabular}
(a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time $t = 3.5$. Show the computations that lead to your answer.
(b) Using correct units, explain the meaning of $\frac{1}{10}\int_{0}^{10} H(t)\,dt$ in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate $\frac{1}{10}\int_{0}^{10} H(t)\,dt$.
(c) Evaluate $\int_{0}^{10} H'(t)\,dt$. Using correct units, explain the meaning of the expression in the context of this problem.
(d) At time $t = 0$, biscuits with temperature $100^{\circ}\mathrm{C}$ were removed from an oven. The temperature of the biscuits at time $t$ is modeled by a differentiable function $B$ for which it is known that $B'(t) = -13.84e^{-0.173t}$. Using the given models, at time $t = 10$, how much cooler are the biscuits than the tea?

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 $B ( 0 ) = 20$.

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

Which of the following is the solution to the differential equation $\frac { d y } { d x } = 2 \sin x$ with the initial condition $y ( \pi ) = 1$ ?
(A) $y = 2 \cos x + 3$
(B) $y = 2 \cos x - 1$
(C) $y = - 2 \cos x + 3$
(D) $y = - 2 \cos x + 1$
(E) $y = - 2 \cos x - 1$

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 $f ( 2 ) = 3$.

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 $t = 3$?