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Rates of Change and Applications to Motion

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Rates of Change and Applications to Motion

Rates of Change and Applications to Motion

Rates of Change and Applications to Motion

Rates of Change and Applications to Motion

Rates of Change and Applications to Motion

Average Rates of Change

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:


Avg Rate   =  
    =  
    =  
    = 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 .

Instantaneous Rates of Change

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'(t) = 6t 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 .