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Functions

Copyright © Cengage Learning. All rights reserved.

2.4 Average Rate of Change of a Function

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Objectives

►  Average Rate of Change

►  Linear Functions Have Constant Rate of Change

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Average Rate Of Change Of A Function

Functions are often used to model changing quantities.

In this section we learn how to find the rate at which the values of a function change as the input variable changes.

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Average Rate of Change

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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 = 60 mi/h.

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

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Average Rate of Change We graph the function s as shown in Figure 1.

Figure 1

Average speed

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Average Rate of Change 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.

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. 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, 200 – 50 = 150 mi.

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Average Rate of Change Thus, your average speed is

The average speed that we have just calculated can be expressed by using function notation:

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Average Rate of Change Note that the average speed is different over different time intervals. For example, between 2:00 P.M. and 3:00 P.M. we find that

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Average Rate of Change Finding average rates of change is important in many contexts.

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Example 1 – Calculating the Average Rate of Change

For the function f (x) = (x – 3)2, whose graph is shown in Figure 2, find the average rate of change between the following points:

(a) x = 1 and x = 3

(b) x = 4 and x = 7

f (x) = (x – 3)2

Figure 2

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Example 1 – Solution

(a) Average rate of change = Definition

Use f (x) = (x – 3)2

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Example 1 – Solution

(b) Average rate of change =

cont’d

Definition

Use f (x) = (x – 3)2

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Linear Functions Have Constant Rate of Change

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Linear Functions Have Constant Rate of Change

For a linear function f (x) = mx + b the average rate of change between any two points is the same constant m.

The slope of a line y = mx + b is the average rate of change of y with respect to x. On the other hand, if a function f has constant average rate of change, then it must be a linear function.

In the next example we find the average rate of change for a particular linear function.

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Example 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.

(a) x = 0 and x = 1 (b) x = 3 and x = 7 (c) x = a and x = a + h

What conclusion can you draw from your answers?

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Example 4 – Solution

(a) Average rate of change

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Example 4 – Solution

(b) Average rate of change

cont’d

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Example 4 – Solution

(c) Average rate of change

cont’d

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Example 4 – Solution 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.

cont’d