AP Calculus AB Flashcards: Exponential Models with Differential Equations
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What is the differential equation that models the statement, “The rate of change of a quantity is proportional to the size of the quantity”?
The model for this statement is the differential equation dy/dt = ky, where y is the quantity, t is time, and k is the constant of proportionality.
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What is the differential equation that models the statement, “The rate of change of a quantity is proportional to the size of the quantity”?
The model for this statement is the differential equation dy/dt = ky, where y is the quantity, t is time, and k is the constant of proportionality.
The amount of a radioactive substance, A, decays at a rate proportional to the amount present. Write a differential equation to model this scenario.
This scenario is modeled by the differential equation dA/dt = kA, where k would be a negative constant representing decay.
What is the specific solution to the differential equation dy/dt = ky, given the initial condition that y = y₀ when t = 0?
The particular solution for this initial value problem is y = y₀e^(kt).
A population of cells, P, doubles every hour. If the initial population is 500, what is the particular solution modeling its growth?
The particular solution is P = 500e^(kt). The information about doubling time would be used to solve for the constant k, but the form of the solution is determined by the initial condition.
How do you interpret the meaning of the differential equation dy/dt = ky in a real-world context?
It means that the speed at which the quantity y is changing is directly dependent on its current size; larger quantities will grow or decay faster than smaller ones.
In the exponential growth/decay solution y = y₀e^(kt), what do the variables y₀ and k represent?
The variable y₀ represents the initial size of the quantity at time t=0, while k is the proportionality constant that determines the rate of growth (if k > 0) or decay (if k < 0).
What is the difference between a general and a particular solution to a differential equation?
A general solution represents a family of functions that solve the equation and includes an arbitrary constant. A particular solution is a single function derived by using an initial condition to find the specific value of that constant.
In the model dy/dt = ky, what determines whether the equation represents exponential growth or exponential decay?
The sign of the proportionality constant k determines the behavior: a positive k results in exponential growth, while a negative k results in exponential decay.
What is an initial condition in the context of a differential equation?
An initial condition is a known value of the function at a specific point, such as y=y₀ when t=0, which is used to determine a particular solution from a general solution.
Besides exponential growth and decay, what is another specific application mentioned for finding solutions to differential equations?
Another specific application is modeling motion along a line, where differential equations can relate a particle's position, velocity, and acceleration.