what is lagrange's mean value theorem
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Mean value theorem of Lagrange states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there is at least one pointx=ξ on this interval, such that
f(b)−f(a)=f′(ξ)(b−a).
This theorem (also known as First Mean Value Theorem) allows to express the increment of a function on an interval through the value of the derivative at an intermediate point of the segment.
ITS evidence IS
Consider the auxiliary function
F(x)=f(x)+λx.
We choose a number λ such that the condition F(a)=F(b) is satisfied. Then
f(a)+λa=f(b)+λb,⇒f(b)−f(a)=λ(a−b),⇒λ=−f(b)−f(a)b−a.
As a result, we have
F(x)=f(x)−f(b)−f(a)b−ax.
The function F(x) is continuous on the closed interval [a,b],differentiable on the open interval (a,b) and takes equal values at the endpoints of the interval. Then there is a point ξ in the interval (a,b) such that
F′(ξ)=0.
It follows that
f(b)−f(a)=f′(ξ)(b−a).
Lagrange’s mean value theorem has a simple geometrical meaning. The chord passing through the points of the graph corresponding to the ends of the segment a and b has the slope equal to
k=tanα=f(b)−f(a)b−a.
Then there is a point x=ξ inside the interval [a,b], where the tangent to the graph is parallel to the chord (Figure 2).
The mean value theorem has also a clear physical interpretation. If we assume that f(t) represents the position of a body moving along a line, depending on the time t, then the ratio of
f(b)−f(a)b−a
is the average velocity of the body in the period of time b−a. Since f′(t)is the instantaneous velocity, this theorem means that there exists a moment of time ξ, in which the instantaneous speed is equal to the average speed.
Lagrange’s mean value theorem has many applications in mathematical analysis, computational mathematics and other fields. Let us further note two remarkable corollaries.
Mean value theorem of ladrange is analytically beneficial to us ..
Mean value theorem of Lagrange states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), then there is at least one pointx=ξ on this interval, such that
f(b)−f(a)=f′(ξ)(b−a).
This theorem (also known as First Mean Value Theorem) allows to express the increment of a function on an interval through the value of the derivative at an intermediate point of the segment.
ITS evidence IS
Consider the auxiliary function
F(x)=f(x)+λx.
We choose a number λ such that the condition F(a)=F(b) is satisfied. Then
f(a)+λa=f(b)+λb,⇒f(b)−f(a)=λ(a−b),⇒λ=−f(b)−f(a)b−a.
As a result, we have
F(x)=f(x)−f(b)−f(a)b−ax.
The function F(x) is continuous on the closed interval [a,b],differentiable on the open interval (a,b) and takes equal values at the endpoints of the interval. Then there is a point ξ in the interval (a,b) such that
F′(ξ)=0.
It follows that
f(b)−f(a)=f′(ξ)(b−a).
Lagrange’s mean value theorem has a simple geometrical meaning. The chord passing through the points of the graph corresponding to the ends of the segment a and b has the slope equal to
k=tanα=f(b)−f(a)b−a.
Then there is a point x=ξ inside the interval [a,b], where the tangent to the graph is parallel to the chord (Figure 2).
The mean value theorem has also a clear physical interpretation. If we assume that f(t) represents the position of a body moving along a line, depending on the time t, then the ratio of
f(b)−f(a)b−a
is the average velocity of the body in the period of time b−a. Since f′(t)is the instantaneous velocity, this theorem means that there exists a moment of time ξ, in which the instantaneous speed is equal to the average speed.
Lagrange’s mean value theorem has many applications in mathematical analysis, computational mathematics and other fields. Let us further note two remarkable corollaries.
Mean value theorem of ladrange is analytically beneficial to us ..
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