Lively Applications Examples
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1. The document describes an application using data on attendance at Yellowstone National Park between 1960 and 2000 to introduce piecewise polynomials. 2. The attendance data is best modeled by a cubic polynomial from 1960 to 1990, and a constant function from 1990 to 2000 as attendance did not change much in that time period. 3. The piecewise function uses a cubic from 1960 to 1990 and a constant from 1990 to 2000 to model the Yellowstone National Park attendance data over that time period.
Lively Applications Examples
- 1. Lively Applications to Introduce Topics in College Algebra and Precalculus Reva Narasimhan Kean University Bid increments on Ebay to introduce piecewise functions Application Follow up On the online auction site Ebay, the next highest 1. Introduce the idea of piecewise functions. amount that one may bid is based on the current price 2. Introduce the function notation associated with of the item according to this table. The bid increment piecewise functions. Use a simple case first, and is the amount by which a bid will be raised each time then extend. Relate back to the tabular form of the current bid is outdone functions. 3. Practice the symbolic form of piecewise Here is how increments are determined: functions. 4. Graph more piecewise functions. Relate to the Bid table and symbolic form for piecewise functions. Current Price Increment Web Applications $ 0.01 - $ 0.99 $ 0.05 The entire table from which the table to the left was $ 1.00 - $ 4.99 $ 0.25 excerpted from the Ebay site at $ 5.00 - $ 24.99 $ 0.50 http://pages.ebay.com/help/buy/bid-increments.html $ 25.00 - $ 99.99 $ 1.00 Ask students how the example can be extended using the extra data. $ 100.00 - $ 249.99 $ 2.50 Other models For example, if the current price of an item is $7.50, then the next bid must be at least $0.50 higher. Here are some possibilities for other models similar 1. Explain why the bid increment, I, is a function of to this one: the price, p. 2. Find I(2.50) and interpret it. Use the current rates for first class mail as a 3. Find I(175) and interpret it. function. Ask students to write it in tabular, 4. What is the domain and range of the function I? symbolic, and graphical form. 5. Graph this function. What do you observe? Ask students to formulate a piecewise function 6. The function I is given in tabular form. Is it for hotel rates for off-peak and peak times. possible to find just one expression for I which will work for all values of the price p? Explain. © 2009 Reva Narasimhan All Rights Reserved
- 2. Phone plan comparison to introduce linear inequalities Application The Verizon phone company in New Jersey has two Follow up plans for local toll calls: Plan A charges $4.00 per month plus 8 cents per 1. Introduce new algebraic skills to proceed minute for every local toll minute used per further. month. 2. Practice algebraic skills Plan B charges a flat rate of $20 per month 3. Revisit problem and finish up regardless of the number of minutes used per 4. Develop other what-if scenarios which build month. on this model. Your task is to figure out which plan is more 5. If technology is used, how would it be economical and under what conditions. To analyze incorporated within this unit? this problem mathematically, we need to break it down into the following steps. Web Applications 1. Write an expression for the monthly cost for Plan A, using the number of minutes as the input Interactive spreadsheet models can be found on a variable. What kind of function did you obtain? web site designed by the author at: What is the y-intercept of this function and what does it signify? What is the slope of this function http://www.collegemath.info and what does it signify? 2. Write an expression for the monthly cost for Plan Click on the link for College Algebra/precalculus B, using the number of minutes as the input and follow the link to Mathematical models in Excel 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 Other models and what does it signify? 3. Complete the following table for the monthly cost Here are some possibilities for other models similar of the two plans: to this one: # of minutes Monthly cost Monthly cost Modify Plan A to read “$10 per month with per month Plan A Plan B 100 free minutes and $0.06 per minute 0 thereafter” and perform the comparisons, with Plan B unchanged. Note now that Plan 50 A will give a piecewise linear function and 100 will not have the same slope throughout the interval. 150 Compare costs of wireless phone plans. Ask students to research the rate plans on the 200 Internet. Since these plans have a fairly 250 complex structure, ask students how they may analyze the different plans. 300 Compare rental car costs – one with unlimited mileage and another with cost for 4. Use your table to help graph the functions in (1) variable mileage. 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)? © 2009 Reva Narasimhan All Rights Reserved
- 3. Rainforest decline to introduce exponential functions Application Follow up The total area of the world’s tropical rainforests have 1. Connect the table with symbolic and graphical been declining at a rate of approximately 8% every representations of the exponential function. ten years. Put another way, 92% of the total area of 2. Discuss exponential growth and decay, with rainforests will be retained 10 years from now. For particular attention to the effect of the base. illustration, consider a 10000 square kilometer area 3. Discuss why the decay can never reach zero. of rainforest. (Source: World Resources Institute) 4. Expand problem to introduce techniques for solutions of exponential equations. 1. Assume that the given trend will continue. Fill in 5. If using technology, incorporate it from the the following table to see how much of this outset to explore graphs of exponential functions rainforest will remain in 90 years. and to find solutions of exponential equations. Years in the Forest acreage Web Applications future (sq km) More data about the state of tropical forests can 0 10000 be found at the web site for the World Resources Institute, http://www.wri.org/ 10 Google Earth at http://earth.google.com 20 can be used to study landscapes over time. 30 Data for population growth over time can be 40 found at web site for the U.S. Census Bureau 50 http://www.census.gov/population/www/ 60 censusdata/hiscendata.html 70 Other models 80 Here are some possibilities for other models similar 90 to this one: 2. Plot the points in the table above, using the Model continuously compounded growth of an number of years in the horizontal axis and the investment. Ask students to construct table and total acreage in the vertical axis. What do you then graph and formulate an exponential observe? expression. Model inflation by using an average annual rate 3. From your table, approximately how long will it of inflation of 4%. Ask students to construct take for the acreage of the given region to decline table and then graph and formulate an to half its original size? exponential expression. Ask students to research the value of a specific 4. Can you give an expression for the total acreage car model, such as the Subaru Outback. Find its of rainforest after t years? (Hint: Think of t in original price and its value after 4 years. Data multiples of 10.) can be found online at www.edmunds.com. Model the depreciated value of the car using an 5. Use your expression in part(4) to predict the exponential model and a linear model. Compare acreage of the given region in 120 years. and contrast the two models. 6. Use your expression in part(4) to predict the Note that none of these models use regression acreage of the given region in 175 years. capabilities. © 2009 Reva Narasimhan All Rights Reserved
- 4. Examining real world data to introduce piecewise polynomials Application Follow up The following graph and table give the attendance at Yellowstone National for selected years between 1. Ask why just a polynomial model may not be 1960 and 2000. suitable in the long run. 2. Ask whether the choice of model is unique. Attendance in Yellowstone National Park 3. Ask to compare values from the fitted function (in millions) to the actual values to judge effectiveness of the 3.0 model. 4. Introduce solutions of equations within this 2.5 context. How would the expressions for setting up the equations change? 2.0 Technology Applications 1.5 An Excel spreadsheet which details how 1.0 regression can be performed using Excel can be found at: 0.5 http://www.kean.edu/~rnarasim/ collegemath/modeling_with_excel.xls 0.0 1960 1970 1980 1990 2000 http://www.mathdemos.org has many Year Attendance demos using the Internet and Excel 1960 1,443,288 1970 2,297,290 1980 2,000,273 More information and resources can be 1990 2,823,572 found on 2000 2,838,233 http://www.mymathspace.net 1. Explain why a linear or a quadratic function Questions or comments can be emailed to the would not model this data set well. author at 2. Describe the trend in the data. What do you rnarasim@kean.edu 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. © 2009 Reva Narasimhan All Rights Reserved
- 5. Video Games and Dot Products (Excerpted from Narasimhan, Precalculus, p. 575, Brooks-Cole/ Cengage, 2009) © 2009 Reva Narasimhan All Rights Reserved