The shape of a curve

A derivative describes what a function is doing at one input, but when we follow those rates across an interval, a larger picture appears. Derivative information can reveal the shape and behavior of a curve, guide decisions about it, and provide useful approximations near a chosen input. We’ll begin by differentiating rates themselves, then use the first two derivatives to move between local information and broader conclusions. Together, these ideas show how the behavior of a function near each input can guide reasoning about the function as a whole.