What does the difference quotient represent?

The difference quotient represents the secant slope to a curve between two points on that curve. It is defined mathematically as the expression ((f(x + h) - f(x)) / h), where (f) is a function, (x) is a point on the curve, and (h) is a small change in (x).

When evaluating the difference quotient as (h) approaches zero, it leads to the derivative, which provides the instantaneous rate of change of the function at a specific point. However, the difference quotient itself, before taking the limit, specifically refers to the average slope between two distinct points on the function, forming a secant line.

This concept is crucial in calculus, as it lays the foundation for understanding derivatives and the behavior of functions between points. It captures the essence of how functions change and allows for deeper analysis of their properties. In sum, the difference quotient clearly corresponds to the secant slope connecting two points on the curve of a function, illustrating the average rate of change over that interval.