lively applications examples

© 2009 Reva Narasimhan All Rights Reserved

Lively Applications to Introduce

Topics in College Algebra and Precalculus

Reva Narasimhan Kean University

Bid increments on Ebay to introduce piecewise functions

Application

On the online auction site Ebay, the next highest amount that one may bid is based on the current price of the item according to this table. The bid increment is the amount by which a bid will be raised each time the current bid is outdone

Here is how increments are determined:

Current Price Bid

Increment

$ 0.01 - $ 0.99 $ 0.05

$ 1.00 - $ 4.99 $ 0.25

$ 5.00 - $ 24.99 $ 0.50

$ 25.00 - $ 99.99 $ 1.00

$ 100.00 - $ 249.99 $ 2.50

For example, if the current price of an item is $7.50, then the next bid must be at least $0.50 higher. 1. Explain why the bid increment, I, is a function of

the price, p. 2. Find I(2.50) and interpret it. 3. Find I(175) and interpret it. 4. What is the domain and range of the function I? 5. Graph this function. What do you observe? 6. The function I is given in tabular form. Is it

possible to find just one expression for I which will work for all values of the price p? Explain.

Follow up 1. Introduce the idea of piecewise functions. 2. Introduce the function notation associated with

piecewise functions. Use a simple case first, and then extend. Relate back to the tabular form of functions.

3. Practice the symbolic form of piecewise functions.

4. Graph more piecewise functions. Relate to the table and symbolic form for piecewise functions.

Web Applications The entire table from which the table to the left was excerpted from the Ebay site at http://pages.ebay.com/help/buy/bid-increments.html

Ask students how the example can be extended using the extra data. Other models Here are some possibilities for other models similar to this one: Use the current rates for first class mail as a

function. Ask students to write it in tabular, symbolic, and graphical form.

Ask students to formulate a piecewise function for hotel rates for off-peak and peak times.


Phone plan comparison to introduce linear inequalities

Application The Verizon phone company in New Jersey has two plans for local toll calls: Plan A charges $4.00 per month plus 8 cents per

minute for every local toll minute used per month.

Plan B charges a flat rate of $20 per month regardless of the number of minutes used per month.

Your task is to figure out which plan is more economical and under what conditions. To analyze this problem mathematically, we need to break it down into the following steps. 1. Write an expression for the monthly cost for Plan

A, using the number of minutes as the input variable. What kind of function did you obtain? What is the y-intercept of this function and what does it signify? What is the slope of this function and what does it signify?

2. Write an expression for the monthly cost for Plan B, using the number of minutes as the input variable. What kind of function did you obtain? What is the y-intercept of this function and what does it signify? What is the slope of this function and what does it signify?

3. Complete the following table for the monthly cost of the two plans:

# of minutes per month

Monthly cost Plan A

Monthly cost Plan B

0

50

100

150

200

250

300

4. Use your table to help graph the functions in (1)

and (2) on the same plot. What do you observe? 5. From your table and graph, can you determine

when Plan A would be cheaper? When Plan B would be cheaper?

6. Can you set up an algebraic expression that will help you to answer the question in (5)?

Follow up

1. Introduce new algebraic skills to proceed further.

2. Practice algebraic skills 3. Revisit problem and finish up 4. Develop other what-if scenarios which build

on this model. 5. If technology is used, how would it be

incorporated within this unit? Web Applications Interactive spreadsheet models can be found on a web site designed by the author at: http://www.collegemath.info Click on the link for College Algebra/precalculus and follow the link to Mathematical models in Excel Other models Here are some possibilities for other models similar to this one:

Modify Plan A to read “$10 per month with 100 free minutes and $0.06 per minute thereafter” and perform the comparisons, with Plan B unchanged. Note now that Plan A will give a piecewise linear function and will not have the same slope throughout the interval.

Compare costs of wireless phone plans. Ask students to research the rate plans on the Internet. Since these plans have a fairly complex structure, ask students how they may analyze the different plans.

Compare rental car costs – one with unlimited mileage and another with cost for variable mileage.


Rainforest decline to introduce exponential functions

Application The total area of the world’s tropical rainforests have been declining at a rate of approximately 8% every ten years. Put another way, 92% of the total area of rainforests will be retained 10 years from now. For illustration, consider a 10000 square kilometer area of rainforest. (Source: World Resources Institute) 1. Assume that the given trend will continue. Fill in

the following table to see how much of this rainforest will remain in 90 years.

Years in the

future Forest acreage

(sq km)

0 10000

10

20

30

40

50

60

70

80

90

2. Plot the points in the table above, using the

number of years in the horizontal axis and the total acreage in the vertical axis. What do you observe?

3. From your table, approximately how long will it

take for the acreage of the given region to decline to half its original size?

4. Can you give an expression for the total acreage

of rainforest after t years? (Hint: Think of t in multiples of 10.)

5. Use your expression in part(4) to predict the

acreage of the given region in 120 years. 6. Use your expression in part(4) to predict the

acreage of the given region in 175 years.

Follow up 1. Connect the table with symbolic and graphical

representations of the exponential function. 2. Discuss exponential growth and decay, with

particular attention to the effect of the base. 3. Discuss why the decay can never reach zero. 4. Expand problem to introduce techniques for

solutions of exponential equations. 5. If using technology, incorporate it from the

outset to explore graphs of exponential functions and to find solutions of exponential equations.

Web Applications More data about the state of tropical forests can

be found at the web site for the World Resources Institute, http://www.wri.org/

Google Earth at http://earth.google.com

can be used to study landscapes over time. Data for population growth over time can be

found at web site for the U.S. Census Bureau http://www.census.gov/population/www/ censusdata/hiscendata.html Other models Here are some possibilities for other models similar to this one: Model continuously compounded growth of an

investment. Ask students to construct table and then graph and formulate an exponential expression.

Model inflation by using an average annual rate of inflation of 4%. Ask students to construct table and then graph and formulate an exponential expression.

Ask students to research the value of a specific car model, such as the Subaru Outback. Find its original price and its value after 4 years. Data can be found online at www.edmunds.com. Model the depreciated value of the car using an exponential model and a linear model. Compare and contrast the two models.

Note that none of these models use regression capabilities.


Examining real world data to introduce piecewise polynomials Application The following graph and table give the attendance at Yellowstone National for selected years between 1960 and 2000.

0.00.0

0.50.5

1.01.0

1.51.5

2.02.0

2.52.5

3.03.0

2000200019901990198019801970197019601960

Attendance in Yellowstone National Park

(in millions)

Year Attendance

1960 1,443,288 1970 2,297,290 1980 2,000,273 1990 2,823,572 2000 2,838,233

1. Explain why a linear or a quadratic function

would not model this data set well. 2. Describe the trend in the data. What do you

observe? 3. Plot the set of points given in the plane on the x-

y coordinate plane. Connect the points with a smooth curve. What do you observe?

4. Find a best fit cubic for the data between 1960 and 1990.

5. Since the attendance did not change appreciably between 1990 and 2000, this portion can be modeled by a constant function. How would you choose this constant?

6. Write down the expression for the piecewise function modeling this data set from 1960 to 2000.

Follow up

1. Ask why just a polynomial model may not be

suitable in the long run. 2. Ask whether the choice of model is unique. 3. Ask to compare values from the fitted function

to the actual values to judge effectiveness of the model.

4. Introduce solutions of equations within this context. How would the expressions for setting up the equations change?

Technology Applications An Excel spreadsheet which details how

regression can be performed using Excel can be found at:

http://www.kean.edu/~rnarasim/ collegemath/modeling_with_excel.xls

http://www.mathdemos.org has many demos using the Internet and Excel

More information and resources can be found on http://www.mymathspace.net Questions or comments can be emailed to the author at [email protected]


Video Games and Dot Products (Excerpted from Narasimhan, Precalculus, p. 575, Brooks-Cole/ Cengage, 2009)

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