The Mean Value Theorem is a fundamental theorem in calculus that states that for a function that is continuous on a closed interval and differentiable on the open interval, there exists a point within the interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints.
To verify the Mean Value Theorem for a given function, you need to first ensure that the function is continuous on the closed interval and differentiable on the open interval. Then, you can use the formula f'(c) = (f(b) - f(a))/(b - a), where c is the point within the interval that satisfies the theorem.
The function used to verify the Mean Value Theorem is f'(x), the derivative of the given function.
No, the Mean Value Theorem can only be applied to functions that are continuous on a closed interval and differentiable on the open interval.
The Mean Value Theorem is used in real-world applications, such as in physics and engineering, to determine average rates of change and to approximate values of functions at specific points. It is also used in economics to calculate marginal rates of production and in statistics to analyze data.