Lagrange's Mean Value Theorem (MVT): Average and Instantaneous Rates of Change

Understand Lagrange's Mean Value Theorem (MVT) in calculus, which relates a function's average rate of change over an interval to its instantaneous rate of change at some point within that interval. This guide provides a clear explanation and proof of the MVT.

Lagrange's Mean Value Theorem

Statement of the Theorem

Lagrange's Mean Value Theorem (LMVT), also known as the Mean Value Theorem (MVT), states: If a function f(x) is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that:

f(b) - f(a) = f'(c)(b - a)

This theorem essentially says that there's a point within the interval where the instantaneous rate of change (the derivative) is equal to the average rate of change over the entire interval.

Proof of Lagrange's Mean Value Theorem

The proof uses Rolle's theorem. We construct an auxiliary function F(x) = f(x) + λx, where λ is chosen such that F(a) = F(b). Since F(x) satisfies the conditions of Rolle's theorem, there exists a point c in (a, b) where F'(c) = 0. Solving for λ and substituting gives us the LMVT.

(The detailed steps of the proof, including the definition of the auxiliary function F(x) and the application of Rolle's theorem, would be included here.)

Example: Verifying LMVT

Let's verify the LMVT for f(x) = x² on the interval [4, 6].

  1. Check for continuity and differentiability: f(x) = x² is continuous and differentiable everywhere.
  2. Calculate f(4) and f(6): f(4) = 16, f(6) = 36
  3. Calculate f'(x): f'(x) = 2x
  4. Apply LMVT: 36 - 16 = f'(c)(6 - 4) => 20 = 2c * 2 => c = 5

Since c = 5 lies in the interval (4, 6), the LMVT holds.

Geometric and Physical Interpretations

Geometric Interpretation: The slope of the secant line connecting the points (a, f(a)) and (b, f(b)) is equal to the slope of the tangent line at some point (c, f(c)) within the interval (a, b). The tangent line is parallel to the secant line.

Physical Interpretation: If f(t) represents the position of an object at time t, then f'(t) is its instantaneous velocity, and [f(b) - f(a)] / (b - a) is its average velocity. The LMVT states that there exists a time c where the instantaneous velocity equals the average velocity.

Consequences of LMVT

  • Rolle's Theorem as a Special Case: If f(a) = f(b), then the LMVT implies there's a point c where f'(c) = 0 (Rolle's theorem).
  • Constant Function: If f'(x) = 0 for all x in [a, b], then f(x) is a constant function on this interval.

Applications of LMVT

The LMVT is used extensively in various areas of mathematics and science, from calculus to physics.

Examples: Applying LMVT

(Examples applying LMVT to determine average rate of change, verify the theorem for a specific function, and finding an instantaneous velocity equal to the average velocity would be included here, with detailed solutions provided.)

Conclusion

Lagrange's Mean Value Theorem is a powerful result in calculus that connects the average rate of change of a function to its instantaneous rate of change at some point within the interval. It has significant theoretical and practical implications.