Suppose s(t) = 2t 3 represents the position of a race car along a straight track, measured in feet from the starting line at time t seconds. What is the average rate of change of s(t) from t = 2 to t = 3 ?
The average rate of change is equal to the total change in position divided by the total change in time:
|= 38 ft per second|
In physics, velocity is the rate of change of position. Thus, 38 feet per second is the average velocity of the car between times t = 2 and t = 3 .
What is the instantaneous rate of change of the same race car at time t = 2 ?
The instantaneous rate of change measures the rate of change, or slope, of a curve at a certain instant. Thus, the instantaneous rate of change is given by the derivative. In this case, the instantaneous rate is s'(2) .
|s'(2)||=||6(2)2 = 24 feet per second|
Thus, the derivative shows that the racecar had an instantaneous velocity of 24 feet per second at time t = 2 .