Part 1/6:

Understanding the Mean Value Theorem: A Detailed Exploration

The Mean Value Theorem (MVT) is a fundamental concept in calculus that bridges the gap between instantaneous and average rates of change of a function. This article delves into the theorem, providing clear examples to illustrate its application.

Recap of the Mean Value Theorem

To begin, the Mean Value Theorem states that for a function ( f(x) ) that is continuous on a closed interval ([a, b]) and differentiable on the open interval ((a, b)), there exists at least one point ( C ) in the interval such that the derivative of the function at that point is equal to the average rate of change of the function over the interval. Mathematically, this can be expressed as:

[

f'(C) = \frac{f(b) - f(a)}{b - a}

]