AP Calculus BC Flashcards: Modeling Situations 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 primary skill involved in modeling a situation with a differential equation?
The primary skill is interpreting a verbal statement of a problem and translating it into a mathematical equation that involves a derivative expression. [cite: 2759]
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What is the primary skill involved in modeling a situation with a differential equation?
The primary skill is interpreting a verbal statement of a problem and translating it into a mathematical equation that involves a derivative expression. [cite: 2759]
A radioactive substance decays at a rate proportional to the amount present. If A is the amount of the substance at time t, write a differential equation to model this situation.
The model is dA/dt = -kA, where k is a positive decay constant and the negative sign indicates the amount is decreasing. [cite: 2759]
What is a differential equation?
A differential equation is an equation that relates a function of an independent variable to one or more of the function's derivatives. [cite: 2760]
Why are differential equations useful for describing real-world phenomena?
They are useful because many physical, biological, and economic phenomena are most easily described by their rates of change. [cite: 2759, 2760]
Translate this statement into a differential equation: "The rate of change of a population P is directly proportional to the population size."
This can be written as dP/dt = kP, where k is the constant of proportionality. [cite: 2759]
The rate at which a rumor spreads is proportional to the product of the number of people who have heard it (H) and the number who have not (N). Write a differential equation for the number of people who have heard the rumor with respect to time t.
The differential equation is dH/dt = k * H * N, where k is a constant. [cite: 2759]
In a differential equation, what does the derivative expression represent?
The derivative expression represents the instantaneous rate of change of the function with respect to its independent variable. [cite: 2759]
What is the relationship between a function and its derivatives within a differential equation?
A differential equation explicitly defines the relationship between the value of a function and the value of its rate(s) of change at any given point. [cite: 2760]
The velocity of a car, v, is decreasing at a rate of 5 m/s². Write a differential equation for the velocity with respect to time t.
Since velocity is the function and its rate of change is acceleration, the differential equation is dv/dt = -5. [cite: 2759]
What are the essential components of a differential equation used for modeling?
The essential components are a function, its independent variable, and at least one of the function's derivatives. [cite: 2760]