Mean value theorem

The mean value theorem is simple, yet powerful: Between any two points on a continuous function, the derivative will somewhere equal the slope of the line connecting the points.

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    Why is it called Rolle's theorem?

    The Frenchman Michel Rolle (1652-1719) was the son of a shopkeeper. Having received only some elementary schooling, he learned mathematics on his own.

    He made some important contributions to the field of mathematics, out of which the most famous one bears his name: Rolle's theorem, a special case of the mean value theorem.

    This theorem is fundamental for many basic proofs in calculus - ironically so, as Rolle was back then one of the loudest opponents of calculus itself!

    An example of the mean value theorem

    Imagine standing on one side of a valley, in the West village. You want to go to the East town on the other side. Either you take the cable way over the valley, or you walk.

    At some point along the hike, the slope would be equal to the slope of the cable way.

    In the mean value theorem, we liken the function graph to the mountain slope. At some point between our starting point and our end point, the derivative will equal the mean slope,

    where is the change in altitude and is the change in distance.

    And if West village and East town were on the same altitude, there'd even be some point where the mountain was all flat!

    What are the requirements for the mean value theorem?

    In the picture below, West village translates to the point , and East village translates to .

    If the mountain has a precipice, we cannot warrant that the slope of the cable way equals the average slope at some point. So in the mean value theorem, we require that the function be continuous between and .

    Also, we must be able to find the slope of the mountain at every spot. This means that the derivative must exist anywhere between and .

    If so, there exists a point in the open interval such that:

    Rolle's theorem is a special case of the mean value theorem, corresponding to the case where both villages are on the same altitude i.e. if , then .

    What is the mean value theorem?

    The mean value theorem states that for a curve stretching from one point to another, there will be at least one other point on the curve where its tangent line is parallel to the straight line between the endpoints.

    As a visual interpretation, image a hiker making her way up a mountain.

    In at least one point along the way, the slope of the ground will be parallel to the tightened cable of a zip-line running from the bottom of the mountain to the top.

    The straight line connecting two points on a graph is called a secant, and the slope of the secant is given by the rise over run of the function between the points. As we know by now, the slope of a function's tangent is given by the function's derivative.

    Therefore, we can define the mean value theorem in a more formal manner:

    Let be a real-valued function. If is continuous on and differentiable on , then there exists a point in such that:

    Without the conditions of continuity and differentiability, we cannot know for sure that the point will exist. To see why, let's return to the hiker.

    The mountain surface is not as smooth as we imagined before, but instead contain only sharp edges.

    This would violate the criterion of being a differentiable function. Although there could potentially still be a point with the properties of , we cannot conclude that it must exist. The same inconclusiveness happens if the function is not continuous on the interval.

    What is Rolle's theorem?

    Rolle's theorem is a special case of the mean value theorem for which the function's value is the same for the two end points.

    Let be a real-valued function. If is continuous on and differentiable on with , then there exists a point in such that

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