Displacement, Velocity, and Acceleration

This is where calculus enters the course. Velocity is the rate of change of position: $v(t) = dx/dt$, the slope of $x(t)$ at one instant. Acceleration is the rate of change of velocity: $a(t) = dv/dt$, the slope of $v(t)$ at one instant. Run the operations in reverse and you get integrals: $\Delta x = \int v(t)\,dt$ is the area under $v(t)$, and $\Delta v = \int a(t)\,dt$ is the area under $a(t)$. Every motion problem in mechanics passes through these four operations.

Most of the trouble comes from reading $dx/dt$ as an ordinary quotient of small numbers, or from computing $\Delta v / \Delta t$ when what was asked is $dv/dt$ at a single instant. The misconceptions in this topic drill those exact moves: confusing an average with an instantaneous rate, dropping the constant of integration, mixing up which derivative gives which extremum, and treating velocity and acceleration as one quantity instead of two independent vectors.