High School Thinking about the mean value theorem without geometry

[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?

 

[fresh_42]

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.

 

[archaic]

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?

 

[fresh_42]

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.

 

[mathwonk]

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.

 

[Svein]

Think of the mean value theorem as a formula to find the "center of gravity" (or more correctly "center of mass").