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AP Calculus AB Practice Quiz: Modeling Situations with Differential Equations

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 7 questions to check your progress.

Question 1 of 7

The rate of change of the population, P, of a city is directly proportional to the population itself. Which of the following differential equations best models this relationship, where k is a positive constant?

All Questions (7)

The rate of change of the population, P, of a city is directly proportional to the population itself. Which of the following differential equations best models this relationship, where k is a positive constant?

A) dP/dt = kP

B) dP/dt = k/P

C) P = kt

D) dP/dt = k

Correct Answer: A

The phrase 'the rate of change of the population, P' is represented by the derivative dP/dt. The phrase 'is directly proportional to the population itself' means that this rate is equal to a constant of proportionality, k, multiplied by the population, P. Therefore, the correct differential equation is dP/dt = kP. [cite: 2759]

According to Newton's Law of Cooling, the rate at which an object cools is proportional to the difference between its temperature, T, and the constant ambient temperature, A. If the object is cooling, which differential equation models this situation, where k is a positive constant?

A) dT/dt = k(T + A)

B) dT/dt = k(A - T)

C) dT/dt = kT

D) T = k(T - A)

Correct Answer: B

The 'rate at which an object cools' is its rate of temperature change, dT/dt. This rate is 'proportional to the difference between its temperature, T, and the ambient temperature, A,' which can be written as k(T - A). Since the object is cooling, its temperature T is greater than the ambient temperature A, and the rate of change dT/dt must be negative. To ensure dT/dt is negative when k is a positive constant and T > A, the relationship must be dT/dt = -k(T - A), which is equivalent to dT/dt = k(A - T). [cite: 2759]

In a fixed population of 1,200 people, the rate at which a disease spreads is proportional to the product of the number of people who are infected, I, and the number of people who are not yet infected. Which differential equation describes this relationship, where k is a constant?

A) dI/dt = kI

B) dI/dt = k(1200 - I)

C) dI/dt = kI(1200 - I)

D) dI/dt = kI / (1200 - I)

Correct Answer: C

The 'rate at which a disease spreads' is dI/dt. The number of infected people is I. The number of people not yet infected is the total population minus the infected, which is (1200 - I). The statement says the rate is proportional to the 'product' of these two quantities. Therefore, the differential equation is dI/dt = k * I * (1200 - I). [cite: 2759]

Which of the following statements provides the fundamental definition of a differential equation?

A) An equation that can only be solved by finding an antiderivative.

B) An equation that relates an independent variable to a dependent variable using only algebraic operations.

C) An equation that contains a function of an independent variable and one or more of that function's derivatives.

D) An equation that models the slope of a line at a single point.

Correct Answer: C

By definition, differential equations are equations that relate a function of an independent variable to the function's own derivatives. This distinguishes them from algebraic equations. [cite: 2760]

A tank contains 200 gallons of water. A brine solution with a concentration of 3 pounds of salt per gallon is pumped into the tank at a rate of 4 gallons per minute. The well-mixed solution is pumped out at the same rate. If S(t) represents the amount of salt in the tank at time t, which differential equation models this scenario?

A) dS/dt = 12 - 0.02S

B) dS/dt = 3 - 4S

C) dS/dt = 12

D) dS/dt = 0.02S

Correct Answer: A

The rate of change of salt, dS/dt, is the rate salt enters minus the rate salt leaves. Rate In = (concentration in) × (flow rate in) = (3 lb/gal) × (4 gal/min) = 12 lb/min. Rate Out = (concentration out) × (flow rate out). The concentration in the tank is S(t) pounds / 200 gallons. So, Rate Out = (S/200 lb/gal) × (4 gal/min) = 4S/200 lb/min = 0.02S lb/min. Therefore, dS/dt = Rate In - Rate Out = 12 - 0.02S. [cite: 2759]

The rate of change of a quantity y with respect to x is inversely proportional to the cube of y. Which differential equation models this relationship, where k is a constant?

A) dy/dx = ky^3

B) dy/dx = k/y^3

C) dy/dx = k/x^3

D) y = k/x^3

Correct Answer: B

The 'rate of change of a quantity y with respect to x' is written as the derivative dy/dx. 'Inversely proportional to the cube of y' means that this rate is equal to a constant of proportionality, k, divided by y^3. Combining these gives the differential equation dy/dx = k/y^3. [cite: 2759]

The differential equation dV/dt = -k√V, where k is a positive constant, can be used to model the volume of water V in a draining tank. Which of the following verbal statements best describes this relationship?

A) The volume of water in the tank is decreasing at a constant rate.

B) The rate at which the volume of water is changing is directly proportional to the square root of the volume.

C) The volume of water in the tank is directly proportional to the square root of the time.

D) The rate at which the volume of water is changing is inversely proportional to the square root of the volume.

Correct Answer: B

The term dV/dt represents 'the rate at which the volume of water is changing.' The term √V represents 'the square root of the volume.' The equation dV/dt = -k√V shows that the rate of change is equal to a constant (-k) multiplied by √V. This is the definition of a direct proportion. The negative sign indicates the volume is decreasing. [cite: 2759, 2760]