StellarA rate of change describes how one quantity changes compared to another. In calculus, we usually study how a function’s output changes as the input changes. The average rate of change of a function on an interval measures the overall change in f(x) divided by the overall change in x, and it is represented by the slope of the secant line through two points on the graph. If a function changes position, height, population, temperature, or any other quantity, the average rate of change tells us how fast it changed over that interval as a whole, not at one exact instant. The formula is (f(b)-f(a))/(b-a). This is algebraically familiar from slope, but in calculus it becomes the stepping stone to a deeper idea.
The instantaneous rate of change measures how fast a function is changing at one specific input value. Since a single point does not create an interval, we cannot directly use the average rate formula at just one point. Instead, we take average rates of change over smaller and smaller intervals around that point and see what value they approach. That limiting value, when it exists, is the instantaneous rate of change, and geometrically it is the slope of the tangent line to the graph at that point. So Unit 2 begins by connecting a familiar algebraic idea, slope over an interval, to the new calculus idea of a slope at an exact moment.