Functions Copyright Cengage Learning All rights reserved 2

Functions Copyright © Cengage Learning. All rights reserved.

2. 4 Average Rate of Change of a Function Copyright © Cengage Learning. All rights reserved.

Objectives ► Average Rate of Change ► Linear Functions Have Constant Rate of Change 3

Average Rate Of Change Of A 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. 4

Average Rate of Change 5

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 6

Average Rate of Change We graph the function s as shown in Figure 1. Average speed Figure 1 7

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. 8

Average Rate of Change Thus, your average speed is The average speed that we have just calculated can be expressed by using function notation: 9

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 10

Average Rate of Change Finding average rates of change is important in many contexts. 11

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 12

Example 1 – Solution (a) Average rate of change = Definition Use f (x) = (x – 3)2 13

Example 1 – Solution (b) Average rate of change = cont’d Definition Use f (x) = (x – 3)2 14

Linear Functions Have Constant Rate of Change 15

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. 16

Example 4 – Linear Functions Have Constant Rate of Change Let f (x) = 3 x – 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? 17

Example 4 – Solution (a) Average rate of change 18

Example 4 – Solution cont’d (b) Average rate of change 19

Example 4 – Solution cont’d (c) Average rate of change 20

Example 4 – Solution cont’d 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. 21