AP Calculus BC Flashcards: Working with the Intermediate Value Theorem (IVT)
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.
In the context of the IVT, what is the relationship between the number 'd' and the function values at the endpoints of the interval?
The number 'd' must be a value that is between the function's values at the endpoints, f(a) and f(b).
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In the context of the IVT, what is the relationship between the number 'd' and the function values at the endpoints of the interval?
The number 'd' must be a value that is between the function's values at the endpoints, f(a) and f(b).
What are the two primary conditions that a function must satisfy for the Intermediate Value Theorem to apply?
The function must be continuous, and it must be on a closed interval [a, b].
What is a primary use of the Intermediate Value Theorem?
The Intermediate Value Theorem is used to explain the behavior of a continuous function on an interval, specifically that it takes on all intermediate values.
A function f is continuous on [1, 10], with f(1) = 5 and f(10) = 20. Does the IVT guarantee a value c such that f(c) = 12?
Yes, because f is continuous on a closed interval and 12 is a number between f(1)=5 and f(10)=20, the IVT guarantees such a value c exists between 1 and 10.
Does the Intermediate Value Theorem provide a method for finding the value of 'c'?
No, the IVT is an existence theorem; it guarantees that 'c' exists but does not provide a way to calculate its exact value.
If a function g is continuous on [-5, 5] with g(-5) = -10 and g(5) = 10, what does the IVT guarantee about a root of the function on this interval?
Since 0 is between -10 and 10, the IVT guarantees there is at least one number c between -5 and 5 such that g(c) = 0, meaning there is at least one root.
State the Intermediate Value Theorem (IVT).
If a function f is continuous on the closed interval [a, b] and d is a number between f(a) and f(b), then the IVT guarantees there is at least one number c between a and b such that f(c)=d.
For the IVT to apply to a function f on an interval from a to b, what type of interval must it be?
The function f must be on a closed interval, denoted as [a, b].
Does the IVT guarantee that there is only one value 'c' such that f(c) = d?
No, the IVT only guarantees that there is *at least one* number 'c'; there could be more than one.
What does the Intermediate Value Theorem guarantee?
The IVT guarantees the existence of at least one number 'c' in the interval (a, b) where the function's value f(c) is equal to an intermediate value 'd'.