AP Calculus BC 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.
How does a 'general solution' differ from a 'particular solution' for a differential equation?
A general solution represents a family of functions that satisfy the differential equation, while a particular solution is a single function from that family determined by a specific initial condition.
Card 1 of 10
All Flashcards (10)
How does a 'general solution' differ from a 'particular solution' for a differential equation?
A general solution represents a family of functions that satisfy the differential equation, while a particular solution is a single function from that family determined by a specific initial condition.
If a population's growth is modeled by $y = y_0 e^{kt}$, what does the term $y_0$ represent?
The term $y_0$ represents the initial condition, which is the initial size of the population at time $t=0$.
What is a 'particular solution' to a differential equation?
A particular solution is a specific solution to a differential equation, determined by using an initial condition to solve for any unknown constants.
Exponential Growth and Decay Model
This model arises from the statement “The rate of change of a quantity is proportional to the size of the quantity,” represented by the differential equation $\frac{dy}{dt} = ky$.
Name two specific applications where finding general and particular solutions to differential equations is useful.
Finding solutions to differential equations is applied in problems involving motion along a line and in models of exponential growth and decay.
A scientist states that the rate of decay of a radioactive isotope is proportional to the amount present. How would you write this as a differential equation?
This relationship is written as the differential equation for exponential decay, $\frac{dy}{dt} = ky$, where 'y' is the amount of the isotope.
What is the differential equation that models a quantity whose rate of change is proportional to its size?
The model for a quantity whose rate of change is proportional to its size is the differential equation for exponential growth and decay, $\frac{dy}{dt} = ky$.
What is the first step in solving a word problem involving a quantity whose rate of change is proportional to its size?
The first step is to interpret the problem and set up the differential equation that models the situation, which is $\frac{dy}{dt} = ky$.
In the exponential model $\frac{dy}{dt} = ky$, what do the variables represent in the context of a problem?
In this model, 'y' represents the size of the quantity, 't' represents time, and 'k' is the constant of proportionality for the rate of change.
What is the general form of the solution to the differential equation $\frac{dy}{dt} = ky$ with an initial condition $y=y_0$ at $t=0$?
The solution to the exponential growth and decay model with a given initial condition is of the form $y = y_0 e^{kt}$.