The Mean Value Theorem
If 𝑓 is continuous on [𝑎, 𝑏] and differentiable on (𝑎, 𝑏), then there exists some 𝑐 with 𝑎 < 𝑐 < 𝑏 such that
𝑓′(𝑐) = [𝑓(𝑏) − 𝑓(𝑎)] / (𝑏 − 𝑎)
This means the instantaneous rate of change at 𝑐 equals the average rate of change of 𝑓 over [𝑎, 𝑏].