Rate of Change at a Point

Zoom into any smooth curve far enough and it starts looking like a straight line. That's the whole idea behind rate of change at a point. You can't divide rise by run at a single point, since there's no run, so instead you look at a nearby point, draw the secant line connecting the two, and compute its slope. That secant slope is an average rate of change over the interval between them. As you slide the second point closer and closer to the first, the interval shrinks, the secant line pivots, and its slope settles toward one specific number. That number is the rate of change at the point.

To compare steepness at two different spots on a curve, you do this shrinking process at each one separately, using small intervals around each point and watching where the secant slopes head. On a parabola like f(x) = x², the average rate of change from x = a to x = a+h works out to 2a + h after expanding and simplifying. As h shrinks toward 0, that expression settles on 2a, so the instantaneous rate of change at x = a is 2a. Notice the rate depends on where you are on the curve, which is exactly why a parabola bends: its steepness keeps changing as a changes.

This sets up everything that follows in the unit. Once you can find the rate of change at a point, you can track how that rate itself changes across the domain, which is what reveals concavity and the bending behavior of polynomial graphs. It also draws a sharp line between a function's value and its behavior, a distinction that matters constantly once rates of change become a tool for describing curves rather than just points on them.