- #1

archaic

- 688

- 214

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

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.

- #1

archaic

- 688

- 214

Mathematics news on Phys.org

- #2

- 19,516

- 25,504

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.archaic said:

- #3

archaic

- 688

- 214

By numerically I meant reasoning without resorting to geometry. In other words, how can one foster an intuition for this without seeing a graph?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.

- #4

- 19,516

- 25,504

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?

I'm afraid your question cannot be answered. It is too vague.

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.

I'm afraid your question cannot be answered. It is too vague.

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.

Last edited:

- #5

mathwonk

Science Advisor

Homework Helper

- 11,774

- 2,040

- #6

Svein

Science Advisor

- 2,305

- 805

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.

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.

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.

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.

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.

- Replies
- 12

- Views
- 2K

- Replies
- 4

- Views
- 2K

- Replies
- 8

- Views
- 1K

- Replies
- 12

- Views
- 1K

- Replies
- 1

- Views
- 3K

- Replies
- 21

- Views
- 1K

- Replies
- 1

- Views
- 1K

- Replies
- 8

- Views
- 1K

- Replies
- 1

- Views
- 1K

- Replies
- 1

- Views
- 4K

Share: