1. Functions are often used to model changing quantities.
Average Rate of Change

2. We are all familiar with the concept of speed.
Average Rate of Change
We are all familiar with the concept of speed.
If you drive a distance of 120 miles in 2 hours, then your average speed, or rate of travel, is:

3. Average Rate of Change
Now, suppose you take a car trip and record the distance that you travel every few minutes.
The distance s you have traveled is a function of the time t: s(t) = total distance traveled at time t

4. We graph the function s as shown.
Average Rate of Change
We graph the function s as shown.
The graph shows that you have traveled a total of: 50 miles after 1 hour 75 miles after 2 hours 140 miles after 3 hours and so on.

5. Average Rate of Change
To find your average speed between any two points on the trip, we divide the distance traveled by the time elapsed.
Let’s calculate your average speed between 1:00 P.M. and 4:00 P.M.
The time elapsed is 4 – 1 = 3 hours.

6. Average Rate of Change
To find the distance you traveled, we subtract the distance at 1:00 P.M. from the distance at 4:00 P.M., that is, – 50 = 150 mi

7. Thus, your average speed is:
Average Rate of Change
Thus, your average speed is:

8. Average Rate of Change
The average speed we have calculated can be expressed using function notation:

9. Average Rate of Change
Note that the average speed is different over different time intervals.

10. For example, between 2:00 P.M. and 3:00 P.M., we find that:
Average Rate of Change
For example, between 2:00 P.M. and 3:00 P.M., we find that:

11. Average Rate of Change—Significance
Finding average rates of change is important in many contexts.
For instance, we may be interested in knowing:
How quickly the air temperature is dropping as a storm approaches.
How fast revenues are increasing from the sale of a new product.

12. Average Rate of Change—Significance
So, we need to know how to determine the average rate of change of the functions that model these quantities.
In fact, the concept of average rate of change can be defined for any function.

13. Average Rate of Change—Definition
The average rate of change of the function y = f(x) between x = a and x = b is:

14. Average Rate of Change—Definition
The average rate of change is the slope of the secant line between x = a and x = b on the graph of f.
This is the line that passes through (a, f(a)) and (b, f(b)).

15. E.g. 1—Calculating the Average Rate of Change
For the function f(x) = (x – 3)2, whose graph is shown, find the average rate of change between the following points:
x = 1 and x = 3
x = 4 and x = 7

16. E.g. 1—Average Rate of Change
Example (a)

17. E.g. 1—Average Rate of Change
Example (b)

18. E.g. 2—Average Speed of a Falling Object
If an object is dropped from a tall building, then the distance it has fallen after t seconds is given by the function d(t) = 16t2.
Find its average speed (average rate of change) over the following intervals: (a) Between 1 s and 5 s (b) Between t = a and t = a + h

19. E.g. 2—Avg. Spd. of Falling Object
Example (a)

20. E.g. 2—Avg. Spd. of Falling Object
Example (b)

21. E.g. 2—Avg. Spd. of Falling Object
Example (b)

22. Difference Quotient & Instantaneous Rate of Change
The average rate of change calculated in Example 2(b) is known as a difference quotient.
In calculus, we use difference quotients to calculate instantaneous rates of change.

23. Instantaneous Rate of Change
An example of an instantaneous rate of change is the speed shown on the speedometer of your car.
This changes from one instant to the next as your car’s speed changes.

24. The graphs show that, if a function is:
Average Rate of Change
The graphs show that, if a function is:
Increasing on an interval, then the average rate of change between any two points is positive.
Decreasing on an interval, then the average rate of change between any two points is negative.

25. E.g. 3—Average Rate of Temperature Change
The table gives the outdoor temperatures observed by a science student on a spring day.

26. E.g. 3—Average Rate of Temperature Change
Draw a graph of the data.
Find the average rate of change of temperature between the following times:
(a) 8:00 A.M. – 9:00 A.M.
(b) 1:00 P.M. – 3:00 P.M.
(c) 4:00 P.M. – 7:00 P.M.

27. E.g. 3—Average Rate of Temperature Change
A graph of the temperature data is shown.
Let t represent time, measured in hours since midnight.
Thus, 2:00 P.M., for example, corresponds to t = 14.

28. E.g. 3—Average Rate of Temperature Change
Define the function F by: F(t) = temperature at time t

29. E.g. 3—Avg. Rate of Temp. Change
Example (a)
The average rate of change was 2°F per hour.

30. E.g. 3—Avg. Rate of Temp. Change
Example (b)
The average rate of change was 2.5°F per hour.

31. E.g. 3—Avg. Rate of Temp. Change
Example (c)
The average rate of change was about –4.3°F per hour during this time interval.
The negative sign indicates the temperature was dropping.

32. Linear Functions Have
Constant Rate of Change

33. Linear Functions
For a linear function f(x) = mx + b, the average rate of change between any two points is the same constant m.
This agrees with what we learned in Section 2.4: The slope of a line is the average rate of change of y with respect to x.

34. Linear Functions
On the other hand, if a function f has a constant rate of change, then it must be a linear function.
You are asked to prove this fact in Exercise 34.

35. E.g. 4—Linear Functions Have Constant Rate of Change
Let f(x) = 3x – 5.
Find the average rate of change of f between the following points.
x = 0 and x = 1
x = 3 and x = 7
x = a and x = a + h
What conclusion can you draw from your answers?

36. E.g. 6—Linear Functions
Example (a)

37. E.g. 6—Linear Functions
Example (b)

38. E.g. 6—Linear Functions
Example (c)

39. E.g. 6—Linear Functions Have Constant Rate of Change
It appears that the average rate of change is always 3 for this function.
In fact, part (c) proves that the rate of change between any two arbitrary points x = a and x = a + h is 3.