# Mean Value Theorem

The Mean Value Theorem (MVT) is a foundational theorem in calculus that relates the average rate of change of a continuous and differentiable function over an interval to the instantaneous rate of change at a specific point within that interval.

Mathematically, the MVT states that if a function `f(x)`

is continuous on the interval `[a, b]`

and differentiable on the open interval `(a, b)`

, then there exists at least one point `c`

in the open interval `(a, b)`

such that the derivative of the function at that point, `f'(c)`

, is equal to the average rate of change of the function over the interval, or `(f(b) - f(a)) / (b - a)`

.

The Mean Value Theorem has many applications in calculus, including providing a basis for other important theorems like Rolle’s Theorem and the Intermediate Value Theorem. Additionally, the MVT is utilized in the analysis of monotonic functions and for establishing bounds on the error in numerical integration techniques such as the trapezoidal rule and Simpson’s rule.