Mean value theorem

Mean value theorem

The Mean Value Theorem (MVT) is a fundamental theorem in calculus that describes a critical property of functions that are continuous and differentiable over a certain interval. It essentially states that if a function f 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 the interval (a, b) such that the derivative of f at c is equal to the average rate of change of f over [a, b]. Mathematically, this can be expressed as: \[f'(c) = \frac{f(b) - f(a)}{b - a}\] where f'(c) is the derivative of f at c, and a < c < b.

Background

The Mean Value Theorem is an extension of the Intermediate Value Theorem, which asserts that a continuous function over a closed interval takes on every value between its endpoints. While the Intermediate Value Theorem guarantees the existence of a value within the interval, the Mean Value Theorem provides a specific condition related to the function's rate of change.

Applications

The Mean Value Theorem has several important applications in both pure and applied mathematics. It is used to prove the Fundamental Theorem of Calculus, establish the uniqueness of solutions to differential equations, and in the analysis of motion in physics. Additionally, it serves as a foundational concept in proving other significant results in calculus, such as Taylor's theorem.

Example

Consider the function f(x) = x^2 on the interval [1, 4]. The function is continuous and differentiable on this interval. According to the Mean Value Theorem, there exists a c in (1, 4) such that: \[f'(c) = \frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{3} = 5\] Since f'(x) = 2x, we find that 2c = 5, or c = 2.5. This demonstrates that there is indeed a point c = 2.5 in the interval (1, 4) where the instantaneous rate of change of the function equals the average rate of change over the interval.

See Also

• Calculus
• Derivative
• Intermediate Value Theorem
• Fundamental Theorem of Calculus