# Thinking about the mean value theorem without geometry

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• archaic
In summary: The theorem states that there must be at least one point where the derivative of a function is equal to the average rate of change of that function over a given interval. This can be applied to real-world problems, such as finding the average velocity of an object during a certain time period. In summary, the mean value theorem can be understood and applied without visualizing a graph, by thinking of it as a formula for finding the "center of gravity" of a function.
archaic
Hello guys, is it possible to "see" the mean value theorem when one is only thinking of numerical values without visualizing a graph? Perhaps through a real world problem?

archaic said:
Hello guys, is it possible to "see" the mean value theorem when one is only thinking of numerical values without visualizing a graph? Perhaps through a real world problem?
What do you want to know? A real world problem which can be solved due to the main value theorem? What should numerically mean? Numerically we always only get an approximation, as the likely mean value isn't rational.

fresh_42 said:
What do you want to know? A real world problem which can be solved due to the main value theorem? What should numerically mean? Numerically we always only get an approximation, as the likely mean value isn't rational.
By numerically I meant reasoning without resorting to geometry. In other words, how can one foster an intuition for this without seeing a graph?

Well, it is a direct consequence of Rolle, so the question reduces to: Can it be "seen" that given ##f(a)=f(b)## we have ##f'(c)=0## somewhere in between. E.g. if the temperature today at 3 p.m. is equal to the temperature yesterday at 1 p.m., do you consider it logical or seen without a graph, that there must have been a minimum or maximum in the meantime?

Here's another example (question 5):
https://www.physicsforums.com/threads/math-challenge-december-2018.961292/but this is geometry, even if without a graph.

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mathwonk and archaic
if you think of the derivative as the speed of a moving particle, the mean value theorem concludes that at some point during a trip, the instantaneous speed must equal the average speed for the whole trip. Assuming speed varies continuously, (or just that it satisfies the intermediate value theorem, which is true for derivatives), this is obvious. I.e. the average speed must be somewhere between the maximum and minimum speeds, hence, by virtue of the derivative satisfying the intermediate value theorem, must equal the instantaneous speed at some point.

Stephen Tashi and archaic

## 1. What is the mean value theorem without geometry?

The mean value theorem without geometry is a mathematical theorem that states that for a continuous function on a closed interval, there exists a point within the interval where the slope of the tangent line is equal to the average rate of change of the function over the interval.

## 2. What is the significance of the mean value theorem without geometry?

The mean value theorem without geometry is significant because it provides a way to relate the average rate of change of a function to the instantaneous rate of change at a specific point. This allows us to make more accurate predictions and calculations in various fields such as physics, economics, and engineering.

## 3. How is the mean value theorem without geometry used in calculus?

The mean value theorem without geometry is used in calculus to prove other important theorems, such as the fundamental theorem of calculus. It is also used to find the maximum and minimum values of a function, and to determine the concavity and inflection points of a curve.

## 4. Can the mean value theorem without geometry be applied to all functions?

No, the mean value theorem without geometry can only be applied to continuous functions on a closed interval. If a function is not continuous or the interval is not closed, the theorem does not hold.

## 5. How is the mean value theorem without geometry related to the intermediate value theorem?

The mean value theorem without geometry is a special case of the intermediate value theorem. It can be thought of as a "mean value" version of the intermediate value theorem, where instead of finding a specific value, we are finding a specific slope.

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