What are the three formulas of the mean value theorem?

The mean value theorem provides three equivalent formulas, describing the relationship between the average velocity between two points on the function graph and the instantaneous velocity of the function at a certain point: f(b) - f(a) = f'(c) * (b - a)f(c) = (f(a) f(b)) / 2f'(c) = (f(b) - f(a)) / (b - a)

Three formulas of the mean value theorem

The mean value theorem is an important theorem in mathematical analysis. It describes the relationship that under certain conditions, the average velocity between two points on the function graph is equal to the instantaneous velocity of the function at a certain point. The mean value theorem has three equivalent formulas:

Formula 1:

Let the function f(x) be continuous on the closed interval [a, b], and on the open interval Differentiable on the interval (a, b). Then there exists a c ∈ (a, b) such that:

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

Formula 2:

Suppose the function f(x) is on the closed interval [a, b] Can be guided. Then there exists a c ∈ (a, b) such that:

f(c) = (f(a) + f(b)) / 2

Formula 3:

Suppose the function f(x) is on the closed interval [a, b] Can be guided. Then there exists a c ∈ (a, b) such that:

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

These three formulas are equivalent, and they may be more convenient in different situations. Among them, Equation 1 is usually used to calculate the average rate between two points, while Equations 2 and 3 are used to find stationary points or extreme points on the function graph.

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