Table of Contents
Rolle’s Theorem and Lagrange’s Mean Value Theorem in Detail
Rolle’s theorem states that a function has a maximum or a minimum at an interior point if and only if the function is differentiable at that point.
Lagrange’s mean value theorem states that, for a function that is differentiable on an interval, there is a point in the interval where the derivative is equal to the average of the function’s derivatives at the two endpoints of the interval.
Lagrange’s Mean Value Theorem Proof
The Lagrange mean value theorem states that if is a function that is continuous on the closed interval and differentiable on the open interval then there exists a point within the open interval where the derivative is equal to the average value of the function on the closed interval.
To prove the Lagrange mean value theorem, we will use the mean value theorem for derivatives. The mean value theorem for derivatives states that if is a function that is continuous on the closed interval and differentiable on the open interval then there exists a point within the open interval where the derivative is equal to the average value of the function on the closed interval.
We will use the mean value theorem for derivatives to show that there exists a point within the open interval where the derivative is equal to the average value of the function on the closed interval.
We will start by assuming that there exists a point within the open interval where the derivative is equal to the average value of the function on the closed interval. We will then use the mean value theorem for derivatives to show that the derivative is also equal to the average value of the function on the open interval.
We will start by assuming that there exists a point within the open interval where the derivative is equal to the average value of the function on the closed interval. We will then use the mean value theorem for derivatives to show that the derivative is also equal to the average value of the function on the open interval.
We will start by assuming that
Geometrical Interpretation of Lagrange’s Mean Value Theorem
The Lagrange’s mean value theorem states that for a given function and a given interval, there exists a point in the interval at which the function takes on its average value.
The theorem can be illustrated geometrically by considering a line segment that connects two points on the graph of the function. The theorem states that there is a point on the line segment where the function takes on its average value.
Rolle’s theorem
A function is continuous on a closed interval if and only if it is continuous at each point in the interval.
Geometrical interpretation of Rolle’s mean value theorem
The Rolle’s mean value theorem states that if a function is continuous on a certain interval and differentiable on the interior of that interval, then there is a point in the interval where the function takes on its average value.
This theorem can be interpreted geometrically as follows: imagine a function represented by a curve drawn on a coordinate plane. The curve will be continuous on a certain interval, and differentiable on the interior of that interval. There will be a point on the curve where the function takes on its average value. This point will be the midpoint of the interval.
Solved Example
Question:
A bus travelling at a constant speed of 72 km/h takes 1 hour to travel 108 km. Find the average speed of the bus.
The average speed of the bus is 36 km/h.
