AP Calculus AB Unit 7 Practice Test & Exam | 15 Questions

1. The rate of change of the volume $V$ of a melting snowball with respect to time $t$ is directly proportional to the square of its surface area $S$. Which of the following differential equations describes this relationship, where $k$ is a non-zero constant?

(A)$\frac{dV}{dt} = k S^2$(B)$\frac{dV}{dt} = \frac{k}{S^2}$(C)$\frac{dV}{dS} = k t^2$(D)$\frac{dS}{dt} = k V^2$

2. A cup of coffee with temperature $H$ (in degrees Fahrenheit) is placed in a room that is maintained at a constant temperature of $70^\circ F$. The coffee cools at a rate proportional to the amount by which its temperature exceeds the room temperature. If $t$ is measured in minutes and $k$ is a positive constant, which of the following differential equations describes this situation?

(A)$\frac{dH}{dt} = -k(H - 70)$(B)$\frac{dH}{dt} = k(H - 70)$(C)$\frac{dH}{dt} = k(70 - H)$(D)$\frac{dH}{dt} = -kH(H - 70)$

3. A rumor is spreading through a town with a fixed population of $P$ people. Let $N$ represent the number of people who have heard the rumor at time $t$. The rate at which the rumor spreads is jointly proportional to 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 models this situation?

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

4. Consider the differential equation $y'' - y' - 6y = 0$. For what values of the constant $k$ is the function $y = e^{kx}$ a solution to the differential equation?

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

5. Which of the following functions is a solution to the differential equation $\frac{dy}{dx} = y \cos(x)$ ?

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

Refer to the figure below.

6. The slope field for a certain differential equation is shown in the figure. Which of the following could be the differential equation?

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

7. Which of the following best describes the slope field for the differential equation $\frac{dy}{dx} = (y-1)(y-3)$?

(A)Segments with zero slope lie along the lines $y=1$ and $y=3$, and segments with negative slope lie in the region $1 < y < 3$.(B)Segments with zero slope lie along the lines $y=1$ and $y=3$, and segments with positive slope lie in the region $1 < y < 3$.(C)Segments with zero slope lie along the lines $x=1$ and $x=3$, and segments with negative slope lie in the region $1 < x < 3$.(D)Segments with zero slope lie along the lines $x=1$ and $x=3$, and segments with positive slope lie in the region $1 < x < 3$.

8. [Skill: 1.E | Topic: 7.5] Which of the following is the general solution to the differential equation $\frac{dy}{dx} = \frac{x}{y^2}$ ?

(A)$y = \sqrt[3]{\frac{3}{2}x^2 + C}$(B)$y = \sqrt[3]{3x^2 + C}$(C)$y = \sqrt{\frac{2}{3}x^3 + C}$(D)$y = \frac{3}{2}x^2 + C$

9. [Skill: 1.E | Topic: 7.5] Which of the following is the general solution to the differential equation $\frac{dy}{dx} = (y-2)\cos x$ ?

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

10. Which of the following is the particular solution y = f(x) to the differential equation $\frac{dy}{dx} $= xy^2 with the initial condition f(0) = -1 ?

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

11. Let y = f(x) be the particular solution to the differential equation $\frac{dy}{dx} $= (y+1)e^x with the initial condition f(0) = 0 . Which of the following is the expression for f(x) ?

(A) y = e^{e^x} - 1 (B) y = e^{e^x - 1} - 1 (C) y = e^x - 1 (D) y = e^{e^x} - e

12. The population $P(t)$ of a bacteria culture grows at a rate directly proportional to the size of the population at any time $t \geq 0$, where $t$ is measured in hours. If the population doubles every 5 hours, which of the following differential equations describes this relationship?

(A)$\frac{dP}{dt} = \frac{\ln 2}{5} P$(B)$\frac{dP}{dt} = \frac{5}{\ln 2} P$(C)$\frac{dP}{dt} = 5 (\ln 2) P$(D)$\frac{dP}{dt} = \frac{1}{5} P$

13. A dose of medication is administered to a patient. The amount of medication $M(t)$ in the bloodstream, measured in milligrams, decays at a rate proportional to the amount present at time $t$ hours. If $M(0) = 500$ and $M(2) = 200$, what is the value of $M(4)$?

(A)50(B)80(C)100(D)125

AP Calculus AB Unit 7: Differential Equations Practice Exam

Exam Details

A. Multiple-Choice Questions (13 Questions)

B. Short Answer Questions (2 Questions)

AP Calculus AB Unit 7: Differential Equations Practice Exam

Exam Details

A. Multiple-Choice Questions (13 Questions)

B. Short Answer Questions (2 Questions)