Durchschnittsänderungsrate

Durchschnittsänderungsrate refers to the average rate at which a quantity changes over a specific interval. It is a fundamental concept in calculus and mathematics, providing a measure of the overall trend of a function between two points.

To calculate the Durchschnittsänderungsrate of a function f(x) over an interval [a, b], one subtracts the function's value at the beginning of the interval, f(a), from its value at the end of the interval, f(b). This difference, f(b) - f(a), represents the total change in the function's output. This total change is then divided by the length of the interval, which is calculated as b - a. The formula is thus: (f(b) - f(a)) / (b - a).

The result of this calculation is a single numerical value. A positive Durchschnittsänderungsrate indicates that the function increased on average over the interval. A negative rate signifies an average decrease, and a rate of zero suggests that the function's value at the beginning and end of the interval were the same, though it may have fluctuated within the interval.

This concept is distinct from the instantaneous rate of change, which is represented by the derivative of the function at a specific point. The Durchschnittsänderungsrate provides a broader, average perspective of how a function behaves over an entire segment of its domain, making it useful in fields such as economics, physics, and engineering for analyzing trends and average performance.