Mean Value Theorem
Definition
The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on an open interval (a, b), then there exists at least one point c in (a, b) where the instantaneous rate of change (derivative) equals the average rate of change over the interval.
Related terms
Secant Line: A secant line is a straight line connecting two points on a curve. In the context of the Mean Value Theorem, it represents the average rate of change over an interval.
Concavity: Concavity refers to whether or not a graph curves upward or downward. It plays an important role in understanding how derivatives behave and relates to the second derivative test.
Critical Point: A critical point occurs when either the derivative of a function equals zero or does not exist. In relation to the Mean Value Theorem, critical points can help identify where instantaneous rates of change equal average rates of change.
"Mean Value Theorem" appears in:
Study guides (1)
AP Calculus AB/BC - 5.1 Using the Mean Value Theorem
Additional resources (2)
AP Calculus AB/BC - 2024 AP Calculus AB Exam Guide
AP Calculus AB/BC - 2024 AP Calculus BC Exam Guide
Practice Questions (13)
Which of the following conditions must be met for the Mean Value Theorem to apply?
What does the Mean Value Theorem state?
Consider the function f(x) = x^2 on the interval [0, 2]. Which of the following is guaranteed by the Mean Value Theorem for this function?
Consider the function g(x) = 3x^2 on the interval [-1, 1]. Which of the following is guaranteed by the Mean Value Theorem for this function?
Consider the function h(x) = x^3 on the interval [0, 3]. Which of the following is guaranteed by the Mean Value Theorem for this function?
Consider the function f(x) = sin(x) on the interval [0, π/2]. Which of the following is guaranteed by the Mean Value Theorem for this function?
Consider the function g(x) = 2x on the interval [1, 5]. Which of the following is guaranteed by the Mean Value Theorem for this function?
Consider the function h(x) = cos(x) on the interval [-π/2, π/2]. Which of the following is guaranteed by the Mean Value Theorem for this function?
Consider the function f(x) = x^2 on the interval [-2, 2]. Which of the following is guaranteed by the Mean Value Theorem for this function?
Consider the function g(x) = 3x on the closed interval [0, 4]. Which of the following is guaranteed by the Mean Value Theorem for this function?
Consider the function h(x) = sin(x) on the interval [0, π]. Which of the following is guaranteed by the Mean Value Theorem for this function?
Consider the function f(x) = x^4 on the interval [-1, 1]. Which of the following is guaranteed by the Mean Value Theorem for this function?
Consider the function g(x) = e^x on the interval [0, 2]. Which of the following is guaranteed by the Mean Value Theorem for this function?
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About Fiveable
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Privacy Policy
CCPA Privacy Policy
Resources
Cram Mode
AP Score Calculators
Study Guides
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Glossary
Cram Events
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Crisis Text Line
Help Center
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.