1. Slope Is a Rate of Change
Section 2.4
Slope Is a Rate of Change

2. Calculate the Rate of Change
Definition
Calculate the Rate of Change
Definition
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:
Example
Section 2.4
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3. Calculate the Rate of Change
Definition
Calculate the Rate of Change
Solution
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
Section 2.4
Slide 3

4. Examples of Rates of Change
Calculate the Rate of Change
Examples
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.
Section 2.4
Slide 4

5. Definition Formula for Rate of Change and Average Rate of Change
Calculate the Rate of Change
Definition
Suppose that a quality y changes steadily form y1 to y2 as a quality x changes steadily from x1 to x2. 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.
Section 2.4
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6. Finding Rates of Change
Calculate the Rate of Change
Example
1. The number of fires in U.S. hotels declined approximately steadily from 7100 fires in 1990 to in Find the average rate of change of the number of hotel fires per year between and 2002.
Solution
Section 2.4
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7. Finding Rates of Change
Calculate the Rate of Change
Solution Continued
The average rate of change of the number of fires per year was about – fires per year. So, on average, the number of fires declined yearly by about fires.
Section 2.4
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8. Finding Rates of Change
Calculate the Rate of Change
Example Continued
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 $ Find the average rate of change of the average value of a home with respect to the number of bedrooms.
Section 2.4
Slide 8

9. Finding Rates of Change
Calculate the Rate of Change
Solution
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
Section 2.4
Slide 9

10. Finding Rates of Change
Calculate the Rate of Change
Solution Continued
Average rate of change of the average value with respect to the number of bedrooms is about $ thousand per bedroom
Average value increases by about $83.33 thousand per bedroom
Section 2.4
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11. Increasing and Decreasing Quantities
Calculate the Rate of Change
Properties
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
Section 2.4
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12. Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Example
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.
1. Create a scattergram. Then draw a linear model.
Section 2.4
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13. Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Solution
Draw a scattergraph that contains the points
2. Find the slope of the linear model.
Example Continued
Solution
Slope formula is , replacing y and x with
d and t, respectively, we have:
Section 2.4
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14. Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Solution Continued
Arbitrarily use the points (2, 120) and (3, 180) to calculate the slope:
The slope is 60
Checks with calculations shown in the scattergraph
Section 2.4
Slide 14

15. Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Example Continued
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
Solution
Calculate rate of change of the distance per hour from
Section 2.4
Slide 15

16. Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Solution Continued
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
Section 2.4
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17. The rate of change (60 miles per hour) is equal to the slope (60)
Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Solution Continued
The rate of change (60 miles per hour) is equal to the slope (60)
Section 2.4
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18. Slope is a Rate of Change
Property
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.
Section 2.4
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19. Constant Rate of Change
Slope Is a Rate of Change
Property
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.
Section 2.4
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20. Finding an Equation of a Linear Model
Finding a Model
Finding an Equation of a Linear Model
Example
A company’s profit was $10 million in 2005 and has increased by $3 million per year. Let p be the profit (in millions of dollars) in the year that is t years since
1. Is there a linear relationship between t and p?
Since the profit is a constant $3 million per year, the variables t and p are linearly related
Solution
Section 2.4
Slide 20

21. Finding an Equation of a Linear Model
Finding a Model
Finding an Equation of a Linear Model
Example Continued
Find the p-intercept of a linear model
Profit was $10 million in 2005
2005 is 0 years since 2005
This gives the ordered pair (0, 10)
So, the p-intercept is (0, 10)
3. Find the slope of the linear model.
Solution
Example Continued
Section 2.4
Slide 21

22. Finding an Equation of a Linear Model
Finding a Model
Finding an Equation of a Linear Model
Solution
Rate of change of profit per year is $3 million per year
So, the slope of the linear model is 3
Find an equation of the linear model.
Since p-intercept is (0, 10) and slope is 3, the linear model is:
Example Continued
Solution
Section 2.4
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23. Finding an Equation of a Linear Model
Finding a Model
Finding an Equation of a Linear Model
Graphing Calculator
Verify the ordered pair (0, 10)
Verify that as the input increases by 1, the output increases by 3
Section 2.4
Slide 23

24. Definition: Unit Analysis
Unit Analysis of a Linear Model
Definition
We perform a unit analysis of a model’s equations by determining the units of the expression on both sides of the equation. The simplified units of the expressions on both sides of the equation should be the same.
Section 2.4
Slide 24

25. Unit Analysis of a Linear Model
Finding a Model
Unit Analysis of a Linear Model
Example
A driver fills her car’s 12-gallon gasoline tank and drives as a constant speed. The car consumes gallon per mile. Let G be the number of gallons of gasoline remaining in the tank after she has driven d miles since filling up.
1. Is there a linear relationship between d and G?
Rate of change is a constant –0.04 gallons per minute, so d and G are linearly related
Solution
Section 2.4
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26. Unit Analysis of a Linear Model
Finding a Model
Unit Analysis of a Linear Model
Example Continued
2. Find the G-intercept of a linear model.
Tank is full at 12 gallons: ordered pair (0, 12)
3. Find the slope of the linear model.
Gasoline remaining in the tank with respect to distance traveled is:
Solution
Example Continued
Solution
Section 2.4
Slide 26

27. Unit Analysis of a Linear Model
Finding a Model
Unit Analysis of a Linear Model
Example Continued
4. Find the equation of the linear model.
Since p-intercept is (0, 12) and slope is –0.04, the linear model is:
5. Perform a unit analysis of the equation
Here a unit analysis on the equation :
Solution
Example Continued
Solution
Section 2.4
Slide 27

28. Unit Analysis of a Linear Model
Finding a Model
Unit Analysis of a Linear Model
Solution Continued
We use the fact that = 1 to simplify the units of
the expression on the right-hand side of the equation:
Units on both sides are gallons: Suggesting correct
Section 2.4
Slide 28

29. Two Variables That Are Approximately Linearly Related
Analyzing a Model
Two Variables That Are Approximately Linearly Related
Example
Yogurt sales (in billions of dollars) in the United States are shown in the table for various years.
Let s be yogurt sales (in billions of dollars) in the year that is t years since 2000.
A model of the situation is:
1. Use a graphing calculator to draw a scattergram and the model in the same viewing window. Check whether the line comes close to the data.
Section 2.4
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30. Two Variables That Are Approximately Linearly Related
Analyzing a Model
Two Variables That Are Approximately Linearly Related
Solution
Draw in the same screen using a graphing calculator
See Sections B.8 and B.10
2. What is the slope of the model? What does it mean in this situation?
Example Continued
Section 2.4
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31. Two Variables That Are Approximately Linearly Related
Analyzing a Model
Two Variables That Are Approximately Linearly Related
Solution
which is of the form
Since m is the slope, the slope is 0.17
Sales increase by 0.17 billion dollars per year
3. Find the rates of change in sales from one year to the next. Compare the rates of change with the results in Problem 2.
Example Continue
Section 2.4
Slide 31

32. Rates of change are shown in the table – all are close to 0.17
Analyzing a Model
Two Variables That Are Approximately Linearly Related
Solution
Rates of change are shown in the table – all are close to 0.17
Section 2.4
Slide 32

33. Two Variables That Are Approximately Linearly Related
Analyzing a Model
Two Variables That Are Approximately Linearly Related
Example Continue
4. Predict the sales in 2010.
Substitute the input of 10 for t:
If two quantities t and p are approximately linearly related, and if p depends on t, then the slope of a reasonable linear model is approximately equal to the average rate of change of p with respect to t.
Solution
Property
Section 2.4
Slide 33