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
$$\frac { d B } { d t } = \frac { 1 } { 5 } ( 100 - B ) .$$
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 $\frac { d y } { d x } = y ^ { 2 } ( 2 x + 2 )$. 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 $h ( 1 )$.

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 $f ( 1 ) = 4$.

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 $H(0) = 4$.

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.