Motion and Net Change

AP Calculus · Axiom Academy

Connecting position, velocity, and acceleration through integrals In calculus, motion along a line is described by three related functions. Position tells where an object is. Velocity tells how fast and in what direction it moves. Acceleration tells how the velocity is changing. Derivatives move down the chain (position to velocity to acceleration). Integrals move back up, recovering the original quantity plus an initial condition. Step 2: Displacement vs. Total Distance These two quantities are the most commonly confused concepts on the AP exam. They use the same velocity function but measure fundamentally different things. This can be positive, negative, or zero. It measures where you ended up relative to where you started. This is always non-negative. It measures the total ground covered, regardless of direction. Step 3: The Net Change Theorem The displacement formula is actually a special case of a much more powerful result. The Net Change Theorem says that integrating any rate of change gives the net change in the original quantity. In words: the integral of a rate of change over equals the net change in the quantity from to . If is the rate water flows into a tank (liters/min), then is the net change in water volume over that interval. If is a population growth rate (people/year), then is the net change in population. If is a marginal cost function (dollars/unit), then is the increase in cost when production goes from 100 to 200 units.

This is the written version of the interactive lesson above. See the full AP Calculus course.