AP Calculus BC Unit 7 Practice Test & Exam | 22 Questions

1. A rumor is spreading through a small town with a fixed population of $P$ people. Let $N(t)$ represent the number of people who have heard the rumor at time $t$. The rate at which the rumor spreads is directly proportional to the product of the number of people who have heard the rumor and the number of people who have not yet heard the rumor. Which of the following differential equations best models this situation, where $k$ is a positive constant?

(A)$\frac{dN}{dt} = k(P - N)$(B)$\frac{dN}{dt} = k N (P - N)$(C)$\frac{dN}{dt} = k N (N - P)$(D)$\frac{dN}{dt} = k P (N - P)$

2. An object moves along a straight line with velocity $v(t)$ for time $t > 0$. The rate of change of the velocity with respect to time is inversely proportional to the square of the velocity. Given that the velocity is always positive and increasing, which of the following differential equations describes this relationship, where $k$ is a constant?

(A)$\frac{dv}{dt} = \frac{k}{v^2}$(B)$\frac{dv}{dt} = k v^2$(C)$\frac{dv}{dt} = k - v^2$(D)$\frac{dv}{dt} = \frac{k}{\sqrt{v}}$

3. For what values of the constant $r$ is the function $y = e^{rx}$ a solution to the differential equation $y'' - y' - 6y = 0$?

(A)$r = -2$ and $r = 3$(B)$r = -3$ and $r = 2$(C)$r = 2$ and $r = 3$(D)$r = -6$ and $r = 1$

4. Which of the following functions is a solution to the differential equation $y'' + 16y = 0$?

(A)$y = e^{4x}$(B)$y = \sin(16x)$(C)$y = \cos(4x)$(D)$y = 4x^2$

5. Which of the following functions $y$ satisfies the differential equation $x \frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$ for all $x > 0$?

(A)$y = x^2$(B)$y = e^x$(C)$y = \ln(x)$(D)$y = \frac{1}{x}$

Refer to the figure below.

6. A slope field for a differential equation is shown in the figure above. Which of the following could be the differential equation represented by the slope field?

(A)$\frac{dy}{dx} = x + y$(B)$\frac{dy}{dx} = x - y$(C)$\frac{dy}{dx} = xy$(D)$\frac{dy}{dx} = \frac{x}{y}$

Refer to the figure below.

7. The slope field for the differential equation $\frac{dy}{dx} = \frac{1}{4}y(4-y)$ is shown in the figure above. If $y = f(x)$ is the particular solution to the differential equation with the initial condition $f(0) = 1$, which of the following statements describes the behavior of $f(x)$ as $x \to \infty$?

(A)$f(x) \to \infty$(B)$f(x) \to 4$(C)$f(x) \to 0$(D)$f(x) \to -\infty$

8. Which of the following differential equations corresponds to the slope field described below? [Image Cue]: A slope field in the xy-plane. The slope segments are horizontal (slope of 0) along the line y = x. In the region where x > y (below the line y = x), the slope segments have a positive slope. In the region where x < y (above the line y = x), the slope segments have a negative slope.

(A)$\frac{dy}{dx} = x + y$(B)$\frac{dy}{dx} = x - y$(C)$\frac{dy}{dx} = y - x$(D)$\frac{dy}{dx} = xy$

9. Let $y = f(x)$ be the particular solution to the differential equation $\frac{dy}{dx} = 2x - y$ with the initial condition $f(1) = 3$. What is the approximation for $f(1.4)$ obtained by using Euler's method with two steps of equal length?

(A)2.60(B)2.72(C)2.80(D)3.00

10. Let $y = f(x)$ be the particular solution to the differential equation $\frac{dy}{dx} = g(x,y)$ with the initial condition $f(1) = 2$. The table below gives selected values of $g(x,y)$. | $x$ | $y$ | $g(x,y)$ | | :--- | :--- | :--- | | 1 | 2 | 0.5 | | 1.5 | 2.25 | 0.8 | | 1.5 | 2.5 | 0.6 | | 2 | 2.65 | 0.4 | What is the approximation for $f(2)$ obtained by using Euler's method with two steps of equal length?

(A)2.50(B)2.55(C)2.65(D)3.30

11. Which of the following is the general solution to the differential equation $\frac{dy}{dx} = 2xy^2$?

(A)$y = -\frac{1}{x^2 + C}$(B)$y = \frac{1}{x^2 + C}$(C)$y = \sqrt{\frac{2x^3}{3} + C}$(D)$y = Ce^{x^2}$

12. Which of the following is the general solution to the differential equation $\frac{dy}{dx} = e^{x-y}$?

(A)$y = x + C$(B)$y = \ln(e^x + C)$(C)$y = e^x + C$(D)$y = \ln(e^x) + C$

13. Which of the following is the particular solution $y = f(x)$ to the differential equation $\frac{dy}{dx} = x e^{-y}$ with the initial condition $f(0) = 0$?

(A)f(x) = $\ln\left$($\frac{x^2}{2} $+ 1$\right$)(B)f(x) = $\ln\left$($\frac{x^2}{2}\right$)(C)f(x) = $\frac{x^2}{2} $+ 1(D)f(x) = e^{$\frac{x^2}{2}$}

14. Consider the differential equation $\frac{dy}{dx} = y^2$. Let $y = f(x)$ be the particular solution to the differential equation with the initial condition $f(0) = 1$. Which of the following is the expression for $f(x)$ and its correct domain of validity?

(A)f(x) = $\frac{1}{1-x} \text{ for } $x $\neq $1(B)f(x) = $\frac{1}{1-x} \text{ for } $x < 1(C)f(x) = $\frac{1}{1-x} \text{ for } $x > 1(D)f(x) = $\frac{-1}{1+x} \text{ for } $x > -1

15. The mass $M(t)$ of a radioactive substance remaining at time $t$ years decreases at a rate proportional to the amount present. At time $t=0$, the mass is 50 grams. At time $t=10$, the mass is 40 grams. Which of the following differential equations, with the correct constant of proportionality $k$, describes this relationship?

(A)$\frac{dM}{dt} = \left( \frac{1}{10} \ln(0.8) \right) M$(B)$\frac{dM}{dt} = \left( 10 \ln(0.8) \right) M$(C)$\frac{dM}{dt} = \left( \frac{1}{10} \ln(1.25) \right) M$(D)$\frac{dM}{dt} = -0.1 M$

16. A population of bacteria $P(t)$ grows at a rate proportional to the size of the population, where $t$ is measured in hours. At time $t=0$ hours, the population is 2,000. At time $t=3$ hours, the population is 6,000. What is the population at time $t=6$ hours?

(A)10,000(B)12,000(C)18,000(D)36,000

17. [Skill: 3.F | Topic: 7.9] The population of a certain species of fish in a lake is modeled by the logistic differential equation $$\frac{dP}{dt} = 0.04P(500 - P)$$ where $t$ is measured in months and $P(t) > 0$. If the initial population is $P(0) = 50$, what is the limit of the population as $t$ approaches infinity?

(A)250(B)500(C)2500(D)5000

18. [Skill: 3.D | Topic: 7.9] The rate of change of a quantity $y$ with respect to time $t$ is modeled by the logistic differential equation $$\frac{dy}{dt} = \frac{1}{10}y(60 - y)$$ At what value of $y$ is the rate of change of the quantity maximized?

(A)30(B)60(C)90(D)300

19. A tank initially contains 200 liters of water. A brine solution containing 2 grams of salt per liter is pumped into the tank at a rate of 5 liters per minute. The well-mixed solution is pumped out of the tank at a rate of 7 liters per minute. Let $S(t)$ represent the number of grams of salt in the tank at time $t$. Which of the following differential equations describes the rate of change of the amount of salt in the tank with respect to time?

(A)$\frac{dS}{dt} = 10 - \frac{7S}{200 - 2t}$(B)$\frac{dS}{dt} = 10 - \frac{7S}{200}$(C)$\frac{dS}{dt} = 10 - \frac{5S}{200 - 2t}$(D)$\frac{dS}{dt} = 10 - \frac{7S}{200 + 2t}$

AP Calculus BC Unit 7: Differential Equations Practice Exam

Exam Details

A. Multiple-Choice Questions (19 Questions)

B. Short Answer Questions (3 Questions)

AP Calculus BC Unit 7: Differential Equations Practice Exam

Exam Details

A. Multiple-Choice Questions (19 Questions)

B. Short Answer Questions (3 Questions)