What sets does the Mean Value Theorem apply to?

In summary, the Mean Value Theorem makes a universal claim for all real numbers a and b, as long as the function f(x) is continuous on the interval [a,b] and f'(x) exists on (a,b). This means that the theorem applies to all possible values of a and b, as long as the given conditions are met. This conclusion was reached by examining the hypotheses of the theorem and considering the definitions of continuity and differentiability.
  • #1
zeion
466
1

Homework Statement



What set(s) are a and b assumed to be elements of? Does the mean Value Theorem make a universal claim (for all a ... for all b) or an existential claim (there exists a ... there exists b) about a and b? Explain how you came to this conclusion.


Homework Equations



We were told to look at the wikipedia page:

http://en.wikipedia.org/wiki/Mean_value_theorem

The Attempt at a Solution



Initially I thought that a and b are assumed to be elements of the domain of the function in question... so assumed that f(x) fulfills the antecedent of the theorem, then for every a, b in the domain of f(x) the consequent is true.

But then can I also assume that a and b are all elements of the real numbers, then for every function that happens to be continuous on this interval, the consequent is true?

Does that matter? Am I confusing myself? lol.
 
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  • #2
The hypotheses are that a and b are real numbers and f is a function such that:

f(x) is continuous on [a,b]
f'(x) exists on (a,b)

Does that help you think through your question?
 
  • #3
LCKurtz said:
The hypotheses are that a and b are real numbers and f is a function such that:

f(x) is continuous on [a,b]
f'(x) exists on (a,b)

Does that help you think through your question?

Yeah I think you just answered my question.
 

1. 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 within the interval where the slope of the tangent line is equal to the average rate of change of the function over the interval.

2. What is the significance of the Mean Value Theorem?

The Mean Value Theorem is significant because it provides a way to connect the concepts of continuity and differentiability, which are fundamental in calculus. It also allows us to prove important results, such as the existence of critical points and the relationship between the average and instantaneous rates of change.

3. How is the Mean Value Theorem used in real-life applications?

The Mean Value Theorem has many practical applications in the real world, such as in physics, economics, and engineering. It is used to analyze motion and rates of change, optimize functions, and make predictions based on data. For example, the Mean Value Theorem can be used to determine the average velocity of an object over a given time interval, or to find the optimal production level for a company.

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

No, the Mean Value Theorem can only be applied to functions that meet the conditions of being continuous on a closed interval and differentiable on the open interval. If a function is not continuous or differentiable, the Mean Value Theorem cannot be used to find a point where the slope of the tangent line equals the average rate of change.

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

The Mean Value Theorem is closely related to other fundamental concepts in calculus, such as the Intermediate Value Theorem, Rolle's Theorem, and the First and Second Derivative Tests. These theorems all build upon the ideas of continuity and differentiability, and can be used to prove and understand important results in calculus.

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