Unit 7 Overview (College Board AP® Calculus AB): Exam Questions

Consider the differential equation $\frac{\text{d}y}{\text{d}x}=\frac{1}{2}\text{sin}\left(\frac{\pi }{2}x\right)\sqrt{y+7}$. Let $y=f\left(x\right)$ be the particular solution to the differential equation with the initial condition $f\left(1\right)=2$. The function $f$ is defined for all real numbers.

A portion of the slope field for the differential equation is given below. Sketch the solution curve through the point $(1,2)$.

Write an equation for the line tangent to the solution curve in part (a) at the point $(1,2)$. Use the equation to approximate $f(0.8)$.

It is known that $f\text{'}\text{'}\left(x\right)>0$ for $-1\le x\le 1$. Is the approximation found in part (b) an overestimate or an underestimate for $f(0.8)$? Give a reason for your answer.

Use separation of variables to find $y=f\left(x\right)$, the particular solution to the differential equation $\frac{\text{d}y}{\text{d}x}=\frac{1}{2}\text{sin}\left(\frac{\pi }{2}x\right)\sqrt{y+7}$ with the initial condition $f\left(1\right)=2$.

The depth of seawater at a location can be modeled by the function $H$ that satisfies the differential equation

$\frac{\text{d}H}{\text{d}t}=\frac{1}{2}(H-1)\cos \frac{t}{2}$

where $H\left(t\right)$ is measured in feet and $t$ is measured in hours after noon ($t=0$). It is known that $H\left(0\right)=4$.

A portion of the slope field for the differential equation is provided below.

Sketch the solution curve $y=H\left(t\right)$ through the point $(0,4)$ on the slope field above.

For $0<t<5$, it can be shown that $H\left(t\right)>1$. 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. Justify your answer.

Use separation of variables to find $y=H\left(t\right)$, the particular solution to the differential equation

$\frac{\text{d}H}{\text{d}t}=\frac{1}{2}(H-1)\text{cos}\frac{t}{2}$

with initial condition $H\left(0\right)=4$.

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 $y=A\left(t\right)$ that satisfies the differential equation $\frac{\text{d}y}{\text{d}t}=\frac{12-y}{3}$. At time $t=0$ hours, there are 0 milligrams of the medication in the patient.

A portion of the slope field for the differential equation $\frac{\text{d}y}{\text{d}t}=\frac{12-y}{3}$ is given below. Sketch the solution curve through the point $(0,0)$.

Using correct units, interpret the statement $\underset{t\to \infty }{\lim }A\left(t\right)=12$ in the context of this problem.

Use separation of variables to find $y=A\left(t\right)$, the particular solution to the differential equation $\frac{\text{d}y}{\text{d}t}=\frac{12-y}{3}$ with initial condition $A\left(0\right)=0$.

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 $y=B\left(t\right)$ that satisfies the differential equation $\frac{\text{d}y}{\text{d}t}=3-\frac{y}{t+2}$. 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.

A cylindrical barrel with a diameter of 2 feet contains collected rainwater, as shown in the figure below. The water drains out through a valve (not shown) 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 modelled by

$\frac{dh}{dt}=-\frac{1}{10}\sqrt{h}$

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 $V=\pi r^{2}h$.

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.

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.

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 $t$.

Consider the differential equation

$\frac{dy}{dx}=\frac{y^2}{x-1}$

On the axes provided, sketch a slope field for the given differential equation at the six points indicated.

Let $y=f\left(x\right)$ be the particular solution to the given differential equation with the initial condition $f\left(2\right)=3$. Write an equation for the line tangent to the graph of $y=f\left(x\right)$ at $x=2$. Use your equation to approximate $f(2.1)$.

Find the particular solution $y=f\left(x\right)$ to the given differential equation with the initial condition $f\left(2\right)=3$.

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 $91 °\text{C}$ at time $t=0$, and the internal temperature of the potato is greater than $27 °\text{C}$ for all times $t>0$. The internal temperature of the potato at time $t$ minutes is modeled by the function $H$ satisfying the differential equation $\frac{dH}{dt}=-\frac{1}{4}(H-27)$, where $H\left(t\right)$ is measured in degrees Celsius and $H\left(0\right)=91$.

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 $t=3$.

Use $\frac{d^{2}H}{d{t}^2}$ to determine whether your answer in part (a) is an underestimate or an overestimate of the internal temperature of the potato at time $t=3$.

For $t<10$, an alternate model for the internal temperature of the potato at time $t$ minutes is the function $G$ satisfying the differential equation $\frac{dG}{dt}=-(G-27{)}^{2/3}$, where $G\left(t\right)$ is measured in degrees Celsius and $G\left(0\right)=91$. Find an expression for $G\left(t\right)$. Based on this model, what is the internal temperature of the potato at time $t=3$?

Consider the differential equation $\frac{dy}{dx}=\frac{1}{3}x(y-2{)}^2$.

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 $(1,0)$.

Let $y=f\left(x\right)$ be the particular solution to the given differential equation with initial condition $f\left(1\right)=0$. Write an equation for the line tangent to the graph of $y=f\left(x\right)$ at $x=1$. Use your equation to approximate $f(0.7)$.

Find the particular solution $y=f\left(x\right)$ to the given differential equation with initial condition $f\left(1\right)=0$.