AP Calculus BC Flashcards: Using the Mean Value Theorem
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.
If a function satisfies the Mean Value Theorem's conditions, what equality is guaranteed to be true for some point c in (a, b)?
The theorem guarantees that for some point c in the interval (a, b), the equation f'(c) = (f(b) - f(a)) / (b - a) holds true.
Card 1 of 10
All Flashcards (10)
If a function satisfies the Mean Value Theorem's conditions, what equality is guaranteed to be true for some point c in (a, b)?
The theorem guarantees that for some point c in the interval (a, b), the equation f'(c) = (f(b) - f(a)) / (b - a) holds true.
What is the main conclusion of the Mean Value Theorem?
It guarantees there is at least one point within the open interval where the instantaneous rate of change equals the average rate of change over the interval.
In the context of the Mean Value Theorem, what is the 'average rate of change' over an interval [a, b]?
It is the slope of the secant line connecting the endpoints of the interval, calculated as (f(b) - f(a)) / (b - a).
Where does the Mean Value Theorem guarantee the point 'c' will be located?
The theorem guarantees the point 'c' is located somewhere within the open interval (a, b), not including the endpoints.
In the context of the Mean Value Theorem, what is the 'instantaneous rate of change' at a point c?
It is the value of the derivative of the function at that point, denoted as f'(c).
How is the Mean Value Theorem used to justify conclusions about a function?
By confirming a function is continuous and differentiable on an interval, the theorem justifies the conclusion that a specific instantaneous rate of change must occur at some point.
State the Mean Value Theorem.
If a function f is continuous on [a, b] and differentiable on (a, b), then there is a point c in (a, b) where the instantaneous rate of change equals the average rate of change.
What is the key difference between the intervals required for continuity and differentiability in the MVT?
The MVT requires continuity on the closed interval [a, b], which includes the endpoints, but only requires differentiability on the open interval (a, b), which excludes them.
A particle's position is given by a continuous and differentiable function f(t) on [0, 5]. If f(0)=10 and f(5)=60, what conclusion can be justified using the MVT?
The MVT justifies that at some time t in (0, 5), the particle's instantaneous velocity (rate of change) was exactly equal to its average velocity of 10 units/time.
What are the two conditions a function must satisfy on an interval [a, b] for the Mean Value Theorem to apply?
The function must be continuous over the closed interval [a, b] and differentiable over the open interval (a, b).