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4. Derivatives
4.1. Velocities
Suppose an object moves along a straight line according to an equation
of motion s = f(t),wheres is the displacement (directed distance)
of the object from the origin at time t. The function s is called the
position function of the object.
In the time interval from t to t +∆t the change in position is
∆s = f(t +∆t) − f(t). The average velocity over this time interval
is
average velocity =
displacement
time
=
f(t +∆t) − f(t)
∆t
=
∆s
∆t
.
(If we draw the curve s = f(t) in the t, s–plane, then this number is
the slope of the secant PQ,with
P =(t, f(t)),Q=(t +∆t, f(t +∆t)).
Suppose now that we compute the average velocities over shorter
and shorter time intervals [t, t +∆t]. In other words, we let ∆t
approach 0. We define the velocity (or instantaneous velocity) v(t)
at time t to be the limit of these average velocities:
v(t) = lim
∆t→0
f(t +∆t) − f(t)
∆t
.
This means that the velocity at time t is equal to the slope of the
tangent line at P , see Section 4.3.
Example 9
Suppose an object moves vertically (the Galileo’s Law reads:
s =1/2gt
2
here a constant g equals 9.8m/sec
2
). Find its velocity at
t;att =2.
15
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