Answer
See the explanation
Work Step by Step
The Mean Value Theorem (MVT) states that if a function \(f(x)\) is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one value \(c\) in the open interval \((a, b)\) such that:
\[f'(c) = \frac{f(b) - f(a)}{b - a}\]
Here are the hypotheses and conclusion of the Mean Value Theorem:
Hypotheses:
1. \(f(x)\) is continuous on the closed interval \([a, b]\).
2. \(f(x)\) is differentiable on the open interval \((a, b)\).
Conclusion:
There exists at least one value \(c\) in the open interval \((a, b)\) such that the instantaneous rate of change of the function at \(c\) (given by \(f'(c)\)) is equal to the average rate of change of the function over the interval \([a, b]\).
Physical Interpretations:
1. Instantaneous Rate of Change: The Mean Value Theorem provides a link between the average rate of change over an interval and the instantaneous rate of change at some point within that interval. In physical terms, this can be related to motion, where average velocity over an interval is equated to instantaneous velocity at some point during that interval.
2. Speed and Velocity: If \(f(x)\) represents the position of an object at time \(x\), then the Mean Value Theorem can be used to show that there exists a moment in time where the instantaneous velocity (derivative of position with respect to time) is equal to the average velocity over a given time interval.
3. Economics and Rates of Change: In economic contexts, where functions represent quantities such as profit or production, the Mean Value Theorem can be applied to demonstrate the existence of a time when the rate of change of the quantity is equal to the average rate of change over a specified time period.
In summary, the Mean Value Theorem provides a mathematical tool for connecting average and instantaneous rates of change in various real-world scenarios, making it a valuable concept in calculus with practical applications in physics, economics, and other fields.
