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 interval where the slope of the tangent line is equal to the slope of the secant line connecting the endpoints.
To solve for f(8) - f(2), we first need to find the derivative of the function at a point between 2 and 8 using the Mean Value Theorem. This derivative, also known as the average rate of change, can then be used to find the slope of the secant line connecting f(2) and f(8). Finally, we can use this slope to calculate the difference between f(8) and f(2).
Solving for f(8) - f(2) using the Mean Value Theorem allows us to find the average rate of change of a function on a specific interval. This can be helpful in many real-world applications, such as calculating the average velocity of a moving object or the average growth rate of a population.
No, the Mean Value Theorem is only applicable to continuous functions on a closed interval that are differentiable on the open interval. This means that there cannot be any breaks or discontinuities in the function on the given interval, and it must have a well-defined derivative at every point within the interval.
No, the Mean Value Theorem can only give an estimate of the average rate of change between two points. It does not provide the exact value of f(8) - f(2), but rather a value that is guaranteed to be between the maximum and minimum values of the derivative on the given interval.