The Theorem
The Mean Value Theorem (MVT) states that if a function is continuous and differentiable on an interval \([a, b]\), then there exists at least one point \(c\) in that interval where the instantaneous rate of change equals the average rate of change:
\[f'(c) = \frac{f(b) - f(a)}{b - a}\]
In physical terms: if you drive from point A to point B and your average speed was 60 mph, at some point during the trip, your speedometer must have read exactly 60 mph.
Worked Examples
Example 1: Finding 'c'
For \(f(x) = x^2\) on the interval \([0, 2]\), find the point \(c\) guaranteed by the MVT.
- Average rate: \(\frac{f(2) - f(0)}{2 - 0} = \frac{4 - 0}{2} = 2\).
- Instantaneous rate: \(f'(x) = 2x\).
- Set them equal: \(2c = 2 \implies c = 1\).
- Result: \(c = 1\) is in the interval \((0, 2)\).
The Bridge to Quantum Mechanics
The Mean Value Theorem is the foundation of many proofs in physics. In Quantum Mechanics, we talk about "Expectation Values"—the average result of many measurements. While the MVT is a classical theorem, its spirit lives on in Ehrenfest's Theorem, which shows that the averages of quantum operators follow classical equations of motion. The "mean" behavior of a quantum wavepacket is exactly what we observe as classical physics.