The Mean Value Theorem for Integrals: “If f is continous on [a,b], then there exists a # c in [a,b] such that…”
The Mean Value Theorem for Integrals is a fundamental result in calculus that establishes a relationship between the average value of a function over an interval and the instantaneous value of the function at a specific point within that interval
The Mean Value Theorem for Integrals is a fundamental result in calculus that establishes a relationship between the average value of a function over an interval and the instantaneous value of the function at a specific point within that interval.
Formally, the statement of the Mean Value Theorem for Integrals is as follows:
If a function f is continuous on a closed interval [a, b], then there exists at least one number c in the interval (a, b) such that the average value of the function f(x) over the interval [a, b] is equal to the value of the function f(c) at the point c.
In a more concise notation, it can be written as:
∫[a,b] f(x) dx = f(c) * (b – a)
where ∫[a,b] represents the definite integral of f(x) on the interval [a, b] and (b – a) represents the length of the interval.
The Mean Value Theorem for Integrals essentially states that if a function is continuous on an interval, then at some point within that interval, the instantaneous value of the function must be equal to its average value over that interval.
This theorem is analogous to the Mean Value Theorem for Derivatives, which relates the derivative of a function to the average rate of change of the function over an interval. However, in the case of integrals, the Mean Value Theorem connects the average value and the instantaneous value of a function.
The Mean Value Theorem for Integrals has several important applications, such as finding the average value of a function, estimating definite integrals, and proving important properties of functions. It is a powerful tool in calculus for understanding the behavior of functions over specific intervals.
More Answers:
Evaluating the Right Riemann Sum Approximation for the Function f(x) = x^2 + 1/x√ + x + 5 over the Interval [1, 7]Unlocking the Power of the Second Fundamental Theorem of Calculus | Connecting Differentiation and Integration
Understanding the Power Rule | Evaluating the Integral of a Constant with Respect to x