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Graphs of Functions & Their Derivatives

Sarah Miller

Sarah Miller

7 min read

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Study Guide Overview

This study guide covers the Mean Value Theorem, including its introduction, conditions (continuity and differentiability), and the theorem statement. It provides a worked example and practice questions. The guide also explains Rolle's Theorem as a special case, detailing its conditions and providing an example. Finally, it includes a summary and glossary of terms.

Mean Value Theorem

Table of Contents

  1. Introduction to the Mean Value Theorem
  2. Conditions for the Mean Value Theorem
  3. Understanding the Theorem
  4. Worked Example
  5. Practice Questions
  6. Rolle's Theorem
  7. Summary and Key Takeaways
  8. Glossary

Introduction to the Mean Value Theorem

The mean value theorem is a fundamental result in calculus that links the average rate of change of a function over an interval to the instantaneous rate of change at some point within that interval.

Conditions for the Mean Value Theorem

To apply the mean value theorem, the following conditions must be satisfied:

  1. The function ff must be continuous over the closed interval [a,b][a, b].
  2. The function ff must be differentiable over the open interval (a,b)(a, b).
Exam Tip

Always ensure to justify that the function satisfies both conditions of continuity and differentiability before applying the theorem in an exam.

Understanding the Theorem

If the above conditions are met, then there exists at least one point cc in the open interval (a,b)(a, b) such that:

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

This means that there exists a point within the interval where the instantaneous rate of change (the derivative) is equal to the average rate of change over the interval.

Key Concept

The mean value theorem guarantees the existence of such a point but does not specify where it is.

Worked Example

A social sciences researcher is using a function m(t)m(t) to model the total mass of all the garden gnomes appearing on l...
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Question 1 of 11

Which of the following is a requirement for a function to be eligible for the Mean Value Theorem?

The function must be discontinuous over the closed interval [a,b][a, b]

The function must be differentiable over the closed interval [a,b][a, b]

The function must be continuous over the closed interval [a,b][a, b] and differentiable over the open interval (a,b)(a, b)

The function must have equal values at the interval endpoints, i.e., f(a)=f(b)f(a) = f(b)