The average velocity is the ratio between the displacement of an object in space \(\mathbb R^n\) (e.g. for \(n=3\)), given by the difference of the departure and destination point \(\Delta x:=x_2-x_1\), and the duration of its movement, given by the difference of the departure and destination time \(\Delta t:=t_2-t_1\): \[\bar {v}=\frac{\text{destination point}-\text{departure point}}{\text{destination time}-\text{departure time}}=\frac {\Delta x}{\Delta t}.\]
For a time interval \(I\subset \mathbb R\), it is convenient to model the position of the object in space \(x\) as a function of time \(t\): \[t:\cases{I\to\mathbb R^n\\t\to x(t),}\] called a curve (or the trajectory function) of the object in space. At the departure time \(t\in I\), the object is located at the "departure" position \(x(t)\). After the time period \(h > 0\) with \(t+h\in I\), the object will move to the position \(x(t+h)\). In this case, the average velocity of the object is given by
\[\bar {v}=\frac {x(t+h)-x(t)}{h}.\]
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