Applying the Mean Value Theorem

In summary, the conversation is about understanding and applying the Mean Value Theorem (MVT) to a given graph of the derivative of a function. The graph shows information about the function for a specific range of x values, and the goal is to use the MVT to make estimations and explain the reasoning behind them. The conversation also touches on the Intermediate Value Theorem (IVT) and the behavior of the function based on its derivative. The questions ask to explain why certain statements are true and to use the information obtained to make further conclusions about the function.
  • #1
soe236
24
0
Hey,
I know the basic definition of the MVT, but I'm having a lot of trouble applying it to this problem. I would greatly appreciate any kind of help or guidance.

A graph of the derivative of f(x) is displayed below. Information about the function f(x) is known only for -2.5 < x < 3.5. Also f(-2) = 1. Consider the graph carefully, and consider the information in both the numbers and the shapes of the graph (both "quantitative" and "qualitative" information).

the graph:
http://img148.imageshack.us/img148/5224/graphfx6.jpg

a) Explain why -2 < f(0) < -1. Look carefully at the graph and make estimates using the MVT. Explain the steps of your reasoning in detail.
b) Explain why f(3) > 4 +f(1). Again, use the MVT and explain your reasoning in detail.
c) How big and how small can f(1) - f(0) be?
d) Use the information in a), b), and c) to explain why f(3) must be positive.
e) Explain why f(x) = 0 must have a solution between 0 and 3. Use the IVT (intermediate value th.) and the
information obtained in previous parts of this problem.
 
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  • #2
Starting from x = -2, how does f(x) behave, can you tell by looking at f'(x)?
 
  • #3
EnumaElish said:
Starting from x = -2, how does f(x) behave, can you tell by looking at f'(x)?

well from x=-2 to about x=.5, f'(x) is negative which means f(x) is decreasing and f(x) is increasing on approx [0.5,3.5] ..do i do something with the first derivative test? I just don't know what the question is asking for or atleast what I have to find to answer it
 
  • #4
and the critical point is at 0.5 if that's of any use?
 

FAQ: Applying the Mean Value Theorem

What is the Mean Value Theorem?

The Mean Value Theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the open interval where the slope of the tangent line is equal to the average slope of the function on the closed interval.

What is the significance of the Mean Value Theorem?

The Mean Value Theorem is significant because it provides a mathematical guarantee for the existence of a point where the instantaneous rate of change of a function is equal to its average rate of change. This point is known as a critical point and is used to find important information about the behavior of a function, such as maximum and minimum values.

How is the Mean Value Theorem applied in real-life situations?

The Mean Value Theorem has many practical applications in fields such as physics, economics, and engineering. For example, it can be used to determine the average velocity of an object over a given time interval, or the average rate of change of a company's stock prices over a period of time.

What are the conditions for applying the Mean Value Theorem?

In order to apply the Mean Value Theorem, the function must be continuous on a closed interval and differentiable on the open interval. Additionally, the endpoints of the closed interval must have the same function values, meaning that the function must be connected and have no breaks or gaps in the interval.

How is the Mean Value Theorem related to other calculus concepts?

The Mean Value Theorem is closely related to other important calculus concepts such as the derivative and the concept of critical points. It is also a key component in the proof of the Fundamental Theorem of Calculus. Additionally, it is often used in conjunction with other theorems and techniques, such as the Intermediate Value Theorem and Rolle's Theorem, to solve more complex problems in calculus.

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