Geometric meaning of Mean Value Theorem

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Discussion Overview

The discussion centers around the geometric interpretation of the Mean Value Theorem (MVT) in the context of a specific function representing velocity over a time interval. Participants explore the implications of MVT for understanding instantaneous and average acceleration, as well as the geometric relationships involved.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks clarification on the geometric meaning of MVT, specifically regarding the relationship between instantaneous and average acceleration over the interval [0,4].
  • Another participant explains that the instantaneous acceleration must equal the average acceleration at some point in the interval, emphasizing that this occurs at least once according to MVT.
  • A different perspective suggests that if MVT did not hold, the continuous nature of the derivative would imply it could not consistently be above or below the average acceleration, leading to a contradiction.
  • Another participant proposes a geometric interpretation involving integration, arguing that if the derivative were consistently above or below the average, the integrals would differ, reinforcing the necessity of MVT.
  • A later post questions whether the method discussed could be used to find the average of a function.

Areas of Agreement / Disagreement

Participants generally agree on the basic implications of the Mean Value Theorem, but there are varying interpretations of its geometric meaning and implications, particularly regarding the relationship between instantaneous and average values.

Contextual Notes

Some assumptions about the continuity and differentiability of the function are taken for granted, and the discussion does not resolve the specifics of how to apply the concepts to find averages.

Who May Find This Useful

This discussion may be useful for students and educators interested in the geometric interpretation of calculus concepts, particularly the Mean Value Theorem and its applications in physics.

WiFO215
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I'd like the geometric meaning of the Mean Value Theorem. Say for instance I had a function of velocity that varied as [tex]t^{3}[/tex] + [tex]3t^{2}[/tex] + 3t +1. I consider the interval [0,4].
So by MVT, I have a number c in [0,4] such that f'(c)(4) = f(4) - f(0). What does that mean? That there is an accelaration in that interval equal to the net change in accelaration? Meaning that the slope at 4 minus the slope at 0 is equal to the net change in slope?

Could someone PLEASE explain?
 
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The physical meaning you are looking for is that, between 0 and 4 seconds, the instantaneous acceleration must be equal to the average acceleration at some time. This can happen multiple times, but the mean value theorem guarantees it happens at least once. Geometrically, the average acceleration is the slope of the secant line between (0, 1) and (4, 125) in your example. The instantaneous acceleration at a point c is the slope of the tangent line at c, or f'(c).

Intuitively, this makes sense, because the average acceleration should be less than the maximum acceleration and more than the minimum acceleration in that time period (unless it is constant). It seems reasonable that at some time in that time period, the instantaneous acceleration would hit every value in between the minimum and maximum, thus it will hit the average at some time c.
 
Another way of thinking about it is this. Let's say that there were some function f (that fulfills all necessary requirements for MVT to hold) such that MVT didn't hold. Then since f' is continuous (or assume that if that isn't in the hypotheses, since it's always true in "normal" phenomena), it must either always be less than the average acceleration or always greater than the average acceleration (since it can't jump from one side to the other). But in that case it would have to shoot too short or too far from where the average would take you...but that contradicts where the object ends (by choice of f).
 
Yet another even more geometric way of looking at it would be just straight integrate your unknown function and the "average" function. If you assume (like my last post) that f' is always less than the average or always greater than the average, then the integrals of the f and the average function will (strictly) differ over the time period (because we're talking continuous functions) and thus would once again give us a contraction.
 
Can I find the average of a function using this method?
 
Last edited:

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