Graphing Antiderivatives

A car moves on a highway with changing velocity. This change is described by a velocity function and a position function. Differentiating the position function gives velocity, which is the slope of the position graph. The antiderivative of velocity gives position. Using ideas from the first and second derivatives, the velocity graph shows how the position antiderivative rises, falls, and bends.

From time zero to t₁, velocity—or the slope—is positive and increasing, meaning the position graph is increasing and concave up.

In the next interval, velocity is constant and positive. The position graph has a steady, positive slope forming a straight line.

Next, velocity is positive but decreasing. The slope decreases, so the position graph is increasing and concave down.

When velocity approaches zero, the slope becomes zero, and the position remains nearly constant.

Next, velocity is negative and decreasing over the following interval, resulting in a position graph that is decreasing and concave down.

In the final interval, velocity is negative but increases toward zero. The position graph is decreasing, concave up, and approaching a constant.