Position, Velocity and Acceleration as Calculus

One derivative down, one integral up

In Physics C every kinematic statement is a calculus statement. Position $x(t)$, velocity $v(t)$ and acceleration $a(t)$ sit on a single ladder:

Going the other way you integrate, and every integration produces a constant that only the initial condition can fix:

On a graph the two operations become slope and signed area. The area under $v(t)$ is displacement, not distance travelled; area below the axis subtracts. To get distance you must integrate $|v(t)|$, which means splitting the integral at every instant where $v$ changes sign.

Two quantities are routinely confused. Average velocity is $\bar v=\dfrac{\Delta x}{\Delta t}$; average speed is $\dfrac{\int|v|\,dt}{\Delta t}$. They agree only when the motion never reverses.

A particle is speeding up when $v$ and $a$ have the same sign — equivalently when $\dfrac{d}{dt}\left(v^2\right)=2va>0$ — and slowing down when they have opposite signs. The sign of $a$ alone says nothing about speed.