Geometric meaning of Mean Value Theorem

In summary, the Mean Value Theorem states that between two points on a curve, there must be a point where the instantaneous rate of change is equal to the average rate of change between those two points. This can happen multiple times, but the theorem guarantees it happens at least once. This has a geometric interpretation in terms of the slope of the secant and tangent lines on a graph, and it makes intuitive sense that the instantaneous rate of change would hit all values between the minimum and maximum rates of change. The theorem also has implications for finding the average of a function.
  • #1
WiFO215
420
1
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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  • #2
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.
 
  • #3
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).
 
  • #4
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.
 
  • #5
Can I find the average of a function using this method?
 
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1. What is the Mean Value Theorem?

The Mean Value Theorem is a fundamental theorem in calculus that states that for any function that is continuous on a closed interval, there exists at least one point within that interval where the derivative of the function is equal to the slope of the secant line connecting the endpoints of the interval.

2. What is the geometric interpretation of the Mean Value Theorem?

The geometric interpretation of the Mean Value Theorem is that it guarantees the existence of a point on the graph of a function where the tangent line is parallel to the secant line connecting the endpoints of the interval.

3. How is the Mean Value Theorem used in calculus?

The Mean Value Theorem is used to prove other important theorems in calculus, such as the First and Second Derivative Tests for extrema, and the Fundamental Theorem of Calculus. It is also used to solve problems involving rates of change and optimization.

4. What are the conditions for the Mean Value Theorem to be applicable?

The conditions for the Mean Value Theorem to be applicable are that the function must be continuous on a closed interval and differentiable on the open interval. Additionally, the endpoints of the interval must have the same function value.

5. Can the Mean Value Theorem be applied to all functions?

No, the Mean Value Theorem can only be applied to functions that meet the necessary conditions, such as being continuous and differentiable on a given interval. Additionally, the Mean Value Theorem is not applicable for functions that have vertical tangent lines or are not continuous on a closed interval.

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