110. Mean Value Theorem

Let 𝑓:[𝑎,𝑏]→ℝ be a function that satisfies the following conditions:

  1. 𝑓 is continuous on the closed interval [𝑎,𝑏]

  2. 𝑓 is differentiable on the open interval (𝑎,𝑏)

Then there exists at least one point 𝑐∈(𝑎,𝑏) such that

𝑓′(𝑐)=𝑓(𝑏)−𝑓(𝑎)𝑏−𝑎

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

Example

Consider 𝑓(𝑥)=𝑥2 on [1,3]

  • The average rate of change is:
𝑓(3)−𝑓(1)3−1=9−12=4

  • The derivative for 𝑓(𝑥) is 𝑓′(𝑥)=2𝑥

  • Setting 𝑓′(𝑐)=4, we solve:

2𝑐=4⇒𝑐=2

Thus, at 𝑐=2, the instantaneous rate of change matches the average rate of change