Lesson 3: Average & Instantaneous Rates of Change

Average Rate of Change 

The table shows the results of a student recording temperature every 3 s.   

Time(s)	Temperature (F)
0	66.756
3	89.330
6	91.544
9	92.678
12	93.542
15	94.010
18	94.496
21	94.550

How can you estimate the rate of change of temperature at exactly 15 s?   

A closer look at the temperatures in the table shows that for every 3 s the temperature is recorded from 0 to 21 s to the nearest thousandths.  The temperature amounts are increasing but are not a constant rate between time intervals. 

In other words, the rate of change of temperature is different for different intervals. 

The table below shows the first difference and the average rate of change for each 3 s interval.   

The average rate of change appears to decrease as time increases.  The average rate of change is  from 12 to 15 s and the average rate of change is  from 15 to 18s.  So it is difficult to estimate the rate of change at exactly 15 s, however, a sequence of average rates of change can be used to estimate the rate of change at an exact time. 

Solution:

b. The time interval is 1< t > 2. 

The average rate of change of height with respect to time is 3.9 m/s during the 2^nd second of the model’s flight.