Speed, Velocity, and Acceleration
Speed, velocity, and acceleration seem to be everyone's favorite example for teaching calculus.
Definitions
For two given positions, there is a vector r⃗ between them. Taking the distance of r⃗, i.e. ||r⃗||, gives the... distance.
In R1, velocity is the rate of change in position; speed would then be an equivalent concept. However, in higher dimensions velocity is a vector (e.g., v⃗ = dr⃗/dt = [dx/dt dy/dt dz/dt] in R3) and speed is given by ||v⃗||.
Acceleration is the rate of change in velocity. In higher dimension where velocity is a vector, so too is acceleration. For example, a⃗ = dv⃗/dt = [d2x/dt2 d2y/dt2 d2z/dt2].