110. Mean Value Theorem

Let be a function that satisfies the following conditions:

  1. is continuous on the closed interval

  2. is differentiable on the open interval

Then there exists at least one point such that

This means that the instantaneous rate of change (derivative) at some point is equal to the average rate of change over the entire interval

Example

Consider on

  • The average rate of change is:

  • The derivative for is

  • Setting , we solve:

Thus, at , the instantaneous rate of change matches the average rate of change