1. Slope Is a Rate of Change Section 2.4

2. Lehmann, Intermediate Algebra, 4ed Section 2.4 The ratio of a to b is the fraction A unit ratio is a ratio written as with Suppose the sea level increases steadily by 12 inches in the past 4 hours as it approaches high tide. We can compute how much sea level change per hour by finding the unit ratio of the change in sea level to the change in time: Slide 2 Definition Calculate the Rate of Change Definition Example

3. Lehmann, Intermediate Algebra, 4ed Section 2.4 So, sea level increases by 3 inches per hours. This is an example of rate of change We say, “The rate of change of sea level with respect to time is 3 inches per hour.” The rate of change is constant because sea level increases steadily Slide 3 Definition Calculate the Rate of Change Solution

4. Lehmann, Intermediate Algebra, 4ed Section 2.4 Examples of rates of changes: The number of members of a club increases by five people per month. The value of a stock decreases by $2 per week. The cost of a gallon of gasoline increases by 10¢ per month. Slide 4 Examples of Rates of Change Calculate the Rate of Change Examples

5. Lehmann, Intermediate Algebra, 4ed Section 2.4 Suppose that a quality y changes steadily form y 1 to y 2 as a quality x changes steadily from x 1 to x 2. Then the rate of change of y with respect to x is the ratio of the change in y to the change in x: If either quantity does not change steadily, then this formula is the average rate of change of y with respect to x. Slide 5 Formula for Rate of Change and Average Rate of Change Calculate the Rate of Change Definition

6. Lehmann, Intermediate Algebra, 4ed Section 2.4 1.The number of fires in U.S. hotels declined approximately steadily from 7100 fires in 1990 to 4200 in 2002. Find the average rate of change of the number of hotel fires per year between 1990 and 2002. Slide 6 Finding Rates of Change Calculate the Rate of Change Example Solution

7. Lehmann, Intermediate Algebra, 4ed Section 2.4 The average rate of change of the number of fires per year was about –241.67 fires per year. So, on average, the number of fires declined yearly by about 242 fires. Slide 7 Finding Rates of Change Calculate the Rate of Change Solution Continued

8. Lehmann, Intermediate Algebra, 4ed Section 2.4 2.In San Bruno, California, the average value of a two-bedroom home is $543 thousand, and the average value of a five-bedroom home is $793. Find the average rate of change of the average value of a home with respect to the number of bedrooms. Slide 8 Finding Rates of Change Calculate the Rate of Change Example Continued

9. Lehmann, Intermediate Algebra, 4ed Section 2.4 Consistent in finding signs of the changes Assume that number of bedrooms increases form two to five Assume that the average value increases from $543 thousand to $793 thousand Slide 9 Finding Rates of Change Calculate the Rate of Change Solution

10. Lehmann, Intermediate Algebra, 4ed Section 2.4 Average rate of change of the average value with respect to the number of bedrooms is about $83.33 thousand per bedroom Average value increases by about $83.33 thousand per bedroom Slide 10 Finding Rates of Change Calculate the Rate of Change Solution Continued

11. Lehmann, Intermediate Algebra, 4ed Section 2.4 Suppose that a quantity p depends on a quantity t: If p increases steadily as t increases steadily, then the rate of change of p with respect to t is positive If p decreases steadily as t increases steadily, then the rate of change of p with respect to t is negative Slide 11 Increasing and Decreasing Quantities Calculate the Rate of Change Properties

12. Lehmann, Intermediate Algebra, 4ed Section 2.4 Suppose that a student drives at a constant rate. Let d be the distance (in miles) that the student can drive in t hours. Some values of t and d are shown in the table. Slide 12 Comparing Slope with a Rate of Change Slope Is a Rate of Change Example 1.Create a scattergram. Then draw a linear model.

13. Lehmann, Intermediate Algebra, 4ed Section 2.4 Draw a scattergraph that contains the points 2.Find the slope of the linear model. Slide 13 Comparing Slope with a Rate of Change Slope Is a Rate of Change Solution Slope formula is, replacing y and x with d and t, respectively, we have: Example Continued Solution

14. Lehmann, Intermediate Algebra, 4ed Section 2.4 Arbitrarily use the points (2, 120) and (3, 180) to calculate the slope: Slide 14 Comparing Slope with a Rate of Change Slope Is a Rate of Change Solution Continued The slope is 60 Checks with calculations shown in the scattergraph

15. Lehmann, Intermediate Algebra, 4ed Section 2.4 3.Find the rate of change of distance per hour for each given period. Compare each result with the slope of the linear model. a.From b.From Slide 15 Comparing Slope with a Rate of Change Slope Is a Rate of Change Example Continued Calculate rate of change of the distance per hour from Solution

16. Lehmann, Intermediate Algebra, 4ed Section 2.4 The rate of change (60 miles per hour) is equal to the slope (60) For part b., calculate the rate of change of distance per hour from Slide 16 Comparing Slope with a Rate of Change Slope Is a Rate of Change Solution Continued

17. Lehmann, Intermediate Algebra, 4ed Section 2.4 The rate of change (60 miles per hour) is equal to the slope (60) Slide 17 Comparing Slope with a Rate of Change Slope Is a Rate of Change Solution Continued

18. Lehmann, Intermediate Algebra, 4ed Section 2.4 If there is a linear relationship between quantities t and p, and if p depends on t, then the slope of the linear model is equal to the rate of change of p with respect to t. Slide 18 Slope is a Rate of Change Slope Is a Rate of Change Property

19. Lehmann, Intermediate Algebra, 4ed Section 2.4 Suppose that a quantity p depends on a quantity t: If there is a linear relationship between t and p, then the rate of change of p with respect to t is constant. If the rate of change of p with respect to t is constant, then there is a liner relationship between t and p. Slide 19 Constant Rate of Change Slope Is a Rate of Change Property