First-Order Differential Equations (College Board AP® Calculus AB): Exam Questions

A disease spreads among a population of $N$ people at a rate which is proportional to the product of the number of people infected by the disease and the number of people not infected by the disease. If $u$ denotes the number of people infected by the disease, which of the following differential equations could be used to model this situation with respect to time, $t$, where $k$ is a positive constant.

If $y$ satisfies $\frac{dy}{dt}=ky$, where $k$ is a non-zero constant, then $y$ could be

The slope field shown above corresponds to which of the following differential equations?

Which of the following is the solution to the differential equation $\frac{dy}{dt}=4y$ with the initial condition $y\left(0\right)=3$?

If $y$ satisfies $\frac{dy}{dx}=\frac{1}{2y}$, then $y$ could be

The growth of population $y$ is described by the differential equation $\frac{dy}{dt}=ky$, where $k$ is a constant and $t$ is measured in hours. If the population doubles every 8 hours, then the value of $k$ is

Which of the following is the solution to the differential equation $\frac{dy}{dx}=\frac{x^3}{y}$ with the initial condition $y\left(2\right)=-3$?

Shown above is a slope field for which of the following differential equations?

In the differential equation $\frac{dy}{dx}=x^k{y}^2$, $k$ is a positive integer. Which of the following is the solution to the differential equation with the initial condition $y\left(0\right)=1$?

The number of radioactive atoms, $N$, in a sample is described by the differential equation $\frac{dN}{dt}=-kN$, where $k$ is a positive constant and $t$ is measured in years. If the number of radioactive atoms is only one tenth of the original number after 1.255 years, then the value of $k$ is