The Mean Value Theorem

The Mean Value Theorem (MVT) is an important theorem in calculus that connects the concepts of derivatives and averages. It states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the interval where the instantaneous rate of change (derivative) is equal to the average rate of change over the interval.

Step 1: Visualizing the Function

To understand the Mean Value Theorem, let's consider a function f(x) that is continuous on the interval [a, b] and differentiable on the interval (a, b).

Step 2: Secant Line

Consider the secant line that connects the points (a, f(a)) and (b, f(b)).

Step 3: Tangent Line at a Point

Within the interval (a, b), there exists at least one point c where the tangent line is parallel to the secant line. This means that the slope of the tangent line is equal to the slope of the secant line.

Step 4: Equal Rates of Change

The tangent line represents the instantaneous rate of change (slope) of the function at the point c. By the Mean Value Theorem, this instantaneous rate of change is equal to the average rate of change over the interval [a, b] given by the secant line.

By the Mean Value Theorem, we can conclude that there exists at least one point c in the interval (a, b) where the instantaneous rate of change (represented by the tangent line) is equal to the average rate of change (represented by the secant line) over the interval [a, b].