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  3. ZuAI AP College Board Calculus AB/BC
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Question BankQuestion Bank

Using the Mean Value Theorem

Samuel Baker

Samuel Baker

6 min read

Next Topic - Extreme Value Theorem, Global vs Local Extrema, and Critical Points
Study Guide Overview

This study guide covers the Mean Value Theorem (MVT). It explains the theorem, its conditions (continuity and differentiability), and its formula. It includes a walkthrough example and practice problems applying the MVT to find a point c where the instantaneous rate of change equals the average rate of change over a given interval. The guide also reviews the concepts of continuity and differentiability.

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Previous Topic - Analytical Applications of DifferentiationNext Topic - Extreme Value Theorem, Global vs Local Extrema, and Critical Points

Question 1 of 10

The Mean Value Theorem states that if a function fff is continuous on [a,b][a, b][a,b] and differentiable on (a,b)(a, b)(a,b), then there exists a ccc in (a,b)(a,b)(a,b) such that which of the following is true?

f′(c)=f(b)−f(a)f'(c) = f(b) - f(a)f′(c)=f(b)−f(a)

f′(c)=f(a)−f(b)b−af'(c) = \frac{f(a) - f(b)}{b - a}f′(c)=b−af(a)−f(b)​

f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}f′(c)=b−af(b)−f(a)​

f′(c)=f(a)+f(b)b−af'(c) = \frac{f(a) + f(b)}{b - a}f′(c)=b−af(a)+f(b)​