Slope Is a Rate of Change Section 2.4. Lehmann, Intermediate Algebra, 3ed Section 2.4 The ratio of a to b is the fraction A unit ratio is a ratio written.

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Slope Is a Rate of Change

Section 2.4

Lehmann, Intermediate Algebra, 3edSection 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:

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DefinitionCalculate the Rate of Change

.ab

Definition

ab

1.b

Example


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

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DefinitionCalculate the Rate of Change

12 inches 3 inches4 hours 1 hours

Solution


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.

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Examples of Rates of ChangeCalculate the Rate of Change

Examples


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.

Slide 5

Formula for Rate of Change and Average Rate of ChangeCalculate the Rate of Change

Definition

2 1

2 1

change in change in

y yyx x x


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.

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Finding Rates of ChangeCalculate the Rate of Change

Example

Solution


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.

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Solution 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 $793. Find the average rate of change of the average value of a home with respect to the number of bedrooms.

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Example Continued


• 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

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Solution


•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

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Solution Continued


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

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Increasing and Decreasing QuantitiesCalculate the Rate of Change

Properties


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 ChangeSlope Is a Rate of Change

Example

1. Create a scattergram. Then draw a linear model.


• Draw a scattergraph that contains the points

2. Find the slope of the linear model.

Slide 13


Solution

Slope formula is , replacing y and x with

d and t, respectively, we have:

Example Continued

Solution2 1

2 1

y ym

x x


Arbitrarily use the points (2, 120) and (3, 180) to calculate the slope:

Slide 14


Solution Continued

• The slope is 60• Checks with calculations shown in the scattergraph

2 1

2 1

d dm

t t

180 120 6060

3 2 1m


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

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Example Continued

• Calculate rate of change of the distance per hour from

0 to t=4t 0 to t=3t

Solution

0 to t=3t


• 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

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Solution Continued

0 to t=4t


• The rate of change (60 miles per hour) is equal to the slope (60)

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Solution Continued


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.

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Slope is a Rate of ChangeSlope 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.

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Constant Rate of ChangeSlope Is a Rate of Change

Property


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 2005.

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

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Finding a ModelFinding an Equation of a Linear Model

Example

Solution


2. 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)


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Example Continued

Solution

Example Continued


• Rate of change of profit per year is $3 million per year• So, the slope of the linear model is 3

4.Find an equation of the linear model.

• Since p-intercept is (0, 10) and slope is 3, the linear model is:

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Solution

Example Continued

Solution

3 10p t


• Verify the ordered pair (0, 10)• Verify that as the input increases by 1, the output

increases by 3

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Graphing Calculator


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.

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Definition: Unit AnalysisUnit Analysis of a Linear Model

Definition


A driver fills her car’s 12-gallon gasoline tank and drives as a constant speed. The car consumes 0.04 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

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Finding a ModelUnit Analysis of a Linear Model

Example

Solution


2. Find the G-intercept of a linear model.

• Tank is full at 12 gallons: ordered pair (0, 12)


• Gasoline remaining in the tank with respect to distance traveled is:

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Example Continued

Solution

Solution

0.04m

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 :

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Example Continued

Solution

Solution

0.04 12G t

Example Continued

0.04 12G t


• 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

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Solution Continued


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.

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Analyzing a ModelTwo Variables That Are Approximately Linearly Related

Example

0.17 2.15s t


• 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?

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Solution

Example Continued


• 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.

Slide 31


Solution0.17 2.15s t y mx b

Example Continue


• Rates of change are shown in the table – all are close to 0.17

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Solution


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.

Slide 33


Solution

Example Continue

Property

Slope Is a Rate of Change Section 2.4. Lehmann, Intermediate Algebra, 3ed Section 2.4 The ratio of a to b is the fraction A unit ratio is a ratio written.

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