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Rates of Change
If we take care of the moments, the years will take care of themselves – Maria Edgeworth
Two and a half acres, or nearly the equivalent size of two football fields, of rain forest are consumed every second causing 137 plant and animal species to be lost every single day – source: RAN (Rainforest Action Network. In the U.S., the average motorist spends 46 hours per year in traffic jams and some 5.7 billion gallons of gas per year are wasted in those same traffic jams. Some 400 illegal aliens will walk across the 375 mile U.S. – Mexico border per day. (September 20th, 2004 issue of Time magazine). Nolan Ryan, who threw seven no-hitters, had a fast ball which traveled over 100 mph. 2003’s number one ranked tennis player, Andy Roddick’s serve was clocked over 150 mph. Wade Boggs batted over .400 for 162 games, but this spanned two seasons so the record books do not reflect it. In a college football game against BYU in 1968, UTEP quarterback Brooks Dawson threw for 304 yards in the final 10:21 of the fourth quarter. At this rate, he could have thrown for nearly 1800 yards in a single game. Teenagers are known for rapid growth spurts, but that ‘is nothing’ compared to a teen age Tyrannosaurus Rex, who gained 4.6 pounds per day during the tender ages of 14 to 18, their most rapid period of growth. (Time. Aug. 23rd, 2004). The number of calories consumed per average woman has increased 22% since 1971 in this country. (Time Feb, 16th, 2004). From slowest to largest, a comparison of rates of change for some geologic processes shows striking contrast, the average erosion of a continent is 0.03 mm per year while the cutting of the Grand Canyon is 0.7 mm per year; the postglacial rise of sea level is 5 mm per year while the advancing of the Tigris-Euphrates delta is nearly 25,000 mm per year. According to Willamette Industries, Inc., to produce Sunday newspapers in this country, the rate at which trees are consumed is currently at half-a-million trees per week. The Siberian pipeline leaks oil into the surrounding Soviet water table at a rate of nearly three million tons each year. A committee is assigned to predict Nike’s annual gain in profit, cost and revenue; a meteorologist follows the advance of Hurricane Ivan; an engineer is assigned the duty of investigating the rate an oil spill is advancing toward shore. What do all these scenarios have in common? A rate of change. A measure of the change of one parameter when compared to another. We are driving 50 mph, so our position on a map is changing 50 miles every hour. And if we slow a car from 57 to 50 miles per hour, we will get much better gas mileage, and the average American driver would save about $200 per year.
Every 8 seconds a birth, every 13 seconds a death, every 25 seconds 1 new international migrant. On September 19th, 2004, if you were 20 years old, you were witness to a percent increase in US population of 24 % in your lifetime. In other words, in your lifetime, the population of the US increased by 24 %. If you were 40, the percent change was 57 % in your lifetime, and if you were 90, it was a gargantuan 205 %. According to the US Population Clock and the US Census Bureau, Population Division, the population of the United States on Sunday, September 19th, 2004, at 12:59:47 PM EDT was 294,311,858. The agencies estimated that in the United States, every 8 seconds there is a birth, every 13 seconds there is a death and every 25 seconds one more international immigrant. They tell us this amounts to a net gain in population for the United States of one person every 10 seconds. Updating the population count is done by adding births, subtracting deaths, and adding net migration. So, in 10 years, what will be the popualtion of the United States, if the present growth rate remains the same?
Why should we care about this question, that is, why should we care about the rate of change of the population of the United States? The subtle question is crucial for politicians and policy makers, who race around like thoroughbreds chomping at their bits trying to move our society forward. This change reaches out, touches and infiltrates the very fabric of our society; we are all affected by our countries growth rate. On a national level, such predictions impacts policies on sustainability issues, like hunger and famine, ecology and the environment, agriculture and food supply, petroleum and other energy sources. In the United States, the rate of growth is the best single measure of the burden humans place on the environment. Let’s follow a singular thread of cause and effect. Population growth causes greater individual consumption. Economic growth is then led by this wave of individual consumption because the economy is consumption-driven. As the economy changes, the tentacles of such change cause policy makers’ and community leaders to focus on such impacted issues such as health care, welfare reform, retirement planning, educational funding, import and export laws, marketing strategies, agricultural demands, and community planning. But, the litany of changes does not end here. With their hands seemingly already full, politicians, policy leaders, community leaders must not be blind to how these societal changes would then in turn come full circle and impact the health of the nation’s forest and oceans, rivers and mountains. The fragile oceanic ecosystem where demand is outrunning sustainable stocks, or the fading forests where the shrinkage of forest cover means that capacity give us air or supplies of wood products is rapidly being shrunk in return. From over-fishing to urbanizing our farmland, from increases in carbon emissions to increases in grain productions, from overcrowding to failure to recycle, such disregard will affect flood control, soil protection, water purification and quality of air. And all this will in turn affect the future rate of growth of our country’s population. Full circle. Taking this one step further, if we begin to also concern ourselves about the growth of the world’s 6.4 billion people, the significance of the impact of each of these issues is magnified greatly.
How do we begin to attack the question, how to predict the future population of the United States? For if we can accomplish this, we may attack any question involving a rate of change. Rates of change are used to predict future earnings, predicted exports or imports, rates of cholesterol, heart rates, populations, cancer death rates, unemployment rates, rates of bird migration and on and on.
Luckily, the US Bureau of the Census and the US Population Clock can provide us with a tremendous amount of data so we may model this problem. Below, we summarized for you the population of the United States chronologically backwards. We borrowed, from the websites of the US Bureau of the Census and US Population Time Clock, the population for the US at different times. We started the recording of the U.S. population September 19th, 2004, 12:59:48 Eastern Time. The first set of population data was recorded mere seconds apart. Working backwards, the second branch of data was recorded monthly. The last set of data recorded yearly, stopping on September 1st, 2000.
09/19/2004 All Times EDT (Eastern Standard Time)
12:59:48 PM EDT 294,311,858
12:59:38 PM EDT 294,311,857
12:59:24 PM EDT 294,311,856
12:59:16 PM EDT 294,311,855
12:59:06 PM EDT 294,311,854
12:58:56 PM EDT 294,311,853
12:58:42 PM EDT 294,311,852
12:58:28 PM EDT 294,311,850
09/18/2004 All Times EDT (Eastern Standard Time)
05:59:48 PM EDT 294,305,143
05:59:38 PM EDT 294,305,142
05:59:24 PM EDT 294,305,140
05:59:06 PM EDT 294,305,138
05:58:56 PM EDT 294,305,137
Monthly U.S. Population (thousands): September 1, 2000 - September 1, 2003
| |Population is in thousands | |
|Month |Year | | | |
| |2003 |2002 |2001 |2000 |
|1-Dec | |289,717 |286,686 |283,628 |
|1-Nov | |289,470 |286,434 |283,374 |
|1-Oct | |289,208 |286,168 |283,107 |
|1-Sep |291,946 |288,934 |285,890 |282,819 |
|1-Aug |291,659 |288,646 |285,598 | |
|1-Jul |291,384 |288,369 |285,318 | |
|1-Jun |291,116 |288,100 |285,044 | |
|1-May |290,855 |287,837 |284,777 | |
|1-Apr |290,619 |287,600 |284,535 | |
|1-Mar |290,366 |287,344 |284,282 | |
|1-Feb |290,148 |287,124 |284,065 | |
|1-Jan |289,950 |286,923 |283,867 | |
Yearly US Population. Numbers in thousands.
2000 September 1 275,857
1999 September 1 273,439
1998 September 1 271,007
1997 September 1 268,563
1996 September 1 265,998
1995 September 1 263,559
1994 September 1 261,104
1993 September 1 258,619
Using the same methods we developed in Chapter One, we will estimate the rate of change for the U.S. population with two types of growth in mind, numerical difference (linear) and percent change (exponential). We will then re-discuss these two types of growth so that we may explore two different models used to predict future populations. This will enable us to get at the heart of rates of change. Let’s pore over the calculations carefully.
Numerical
Based on 7 year data estimates:
Numerical growth between 2000 and 1993, 275,857,000 – 258,619,000 = 17,238,000.
Average Numerical Growth per year 17,238,000/7 = 2,462,571 people per year
Predicted population for September 1st, 2014:
2000 population + 2,462,571 people per year x 14 years = 310,332,993 people living in the United States of America.
Based on 1 year data estimates:
Numerical growth between 2000 and 1999, 275,857,000 – 273,439,000 = 2,418,000.
Numerical Growth per year 2,418,000 people per year.
Predicted population for September 1st, 2014:
2000 population + 2,418,000 people per year x 14 years = 309,709,000
Based on 1 month data estimates:
Numerical growth from September 1st, 2003 - August 1st, 2003,
291,946,000 – 291,659,000 = 287,000.
Estimated Numerical Growth per year 287,000 people per month x 12 months per year = 3,444,000 per year
Predicted population for September 1st, 2014:
2003 population + 3,444,000 people per year x 11 years = 329,830,000
Based on 19 hour data estimates:
Numerical growth from 9/19/04, 12:59:48 PM EDT – 9/18/04, 05:59:48 PM EDT,
294,311,858 - 294,305,143 = 6715.
Estimated Numerical Growth per year: 6715 people / 19 hours x 24 hours per day x 365 days per year = 3,095,968 per year
Predicted population for September 1st, 2014:
9 years, 11 ½ months is 9 years + 11.5/12 or 9.96 years
September 19, 2004 population + 3,095,143 people per year x 9.96 years = 294,311,858 + 3,095,143 people per year x 9.96 years = 325,139,482
Based on 10 second data estimates:
Numerical growth from 09/19/2004, 12:59:48 PM EDT - 12:59:38 PM EDT, 294,311,858 - 294,311,857 = 1
Estimated Numerical Growth per year: (1 person per 10 seconds) x (60 seconds per 1 minute) x (60 minutes / 1 hour) x (24 hours per day) x (365 days per year) = 3,153,600 per year
Predicted population for September 1st, 2014:
9 years, 11 ½ months is 9 years + 11.5/12 or 9.96 years
September 19, 2004 population + 3,153,600 people per year x 9.96 years = 294,311,858 + 3,153,600 people per year x 9.96 years = 325,721,714
Average Percent Growth
We will now employ the same data and use an alternative technique in analyzing the data. We will use the ratio [pic].
Based on 7 year data estimates:
Percent Change from 1993 to 2000,
(275,857,000 - 258,619,000)/258,619,000 = 0.0667 or 6.67% change over 7 years.
Average Percent Change per year was 0.0667/7 = 0.009522 or 0.9522 percent increase per year
Predicted population for September 1st, 2014 using a growth factor of 1.009522:
2000 population(1.009522)^14 = 314,995,999
Based on 1 year data estimates:
Percent Change from 1999 to 2000, (275,857,000 - 273,439,000)/273,439,000 = 0.0088
Estimated Percent Change per year 0.0088/1 = 0.0088 percent per year
Predicted population for September 1st, 2014 using a growth factor of 1.0088:
September 1, 2000 population x (1.0088)^14 = 311,856,671
Based on 1 month data estimates:
Percent Change from August 1st, 2003 – September 1st, 2003.
(291,946,000 - 291,659,000) / 291,659,000 = 0.0010
Estimated Percent Change per year 0.0010 per month x 12 months per year = 0.0118 per year
Predicted population for September 1st, 2014 using an annual growth factor 1.0118:
September 1, 2003 population x (1.0118)^11 = 291946000(1.0118)^11 = 332,157,417
Based on 19 hour data estimates:
Percent Change from 9/18/04, 05:59:48 PM EDT – 9/19/04, 12:59:48 PM EDT.
(294,311,858 - 294,305,143)/ 294,305,143 = 1.000023
Percent Change per year: (0.000023 / 19 hours) x (24 hours per day) x (365 days per year) = 0.0105 per year.
Predicted population for September 1st, 2014:
9 years, 11 ½ months is 9 years + 11.5/12 or 9.96 years
September 19, 2004 population x (1.0105)^9.96 = 294,311,858 (1.0105)^9.96 = 326,579,926
Based on 10 second data estimates:
Percent Change from 09/19/2004, 12:59:48 PM EDT - 12:59:38 PM EDT, (294,311,858 - 294,311,857)/ 294,311,857 = 0.000000003
Percent Change per year: (0.000000003 person per 10 seconds) x (60 seconds per 1 minute) x (60 minutes per 1 hour) x (24 hours per day) x (365 days per year) = 0.0107 per year
Predicted population for September 1st, 2014:
9 years, 11 ½ months is 9 years + 11.5/12 or 9.96 years
September 19, 2004 population(1.0107)^9.96 = 294,311,858 (1.0107)^9.96 = 327,224,284
Let’s summarize our findings in a table, so we may discuss which predictions we like, and which we do not like.
Table 1. Predicted September 1st, 2014 US Population, using Numerical Differences and Percent Change taken from data extracted during different time intervals.
| |7 year |1 year |1 month |19 hour |10 second |
|Constant Average Change |310,332,993 |309,709,000 |329,830,000 |325,139,482 |325,721,714 |
|Proportional Change |314,995,999 |311,856,671 |332,157,417 |326,579,926 |327,224,284 |
Which number in the above table is the best prediction for the U.S. population on September 1st, 2014? To answer this crucial question, let us examine the observed pattern displayed in the numbers with eyes attending to detail and ask our selves some key questions. Why is the Percent Change always higher than the Numerical Change? Which is the better model for population, numerical or percent change? Is it better to look at estimates over longer time intervals or shorter time intervals? And finally, which population prediction do you feel is the most accurate?
We will answer the first two questions simultaneously. But first, some clarification.
• A population increasing due to numerical differences is assumed to increase the same amount each year. We call this a constant rate of change because the rate of change is specifically the predicted increase in population each year and we assume this number to be the same from year to year. Constant changes from year to year is logically coined a constant rate of change. We refer to this type of growth rate as linear growth.
• A population that increases according to percent change increases at a rate that is in proportion to its current population. This means, the predicted increase is not the same each year, rather the population increase is based on the population from the prior year. We refer to this type of growth as proportional growth.
To answer the question, which growth rate is more likely, let’s visualize a population’s growth from the perspective of a single family. When we compare linear growth with exponential growth, we will assume we have two children. As we outline both growth rates, ask yourself which is more likely to occur, the linear scenario or the exponential scenario.
For linear growth, we first assume you have two children. For a constant rate of change, this means each successive generation will increase by two children. What would this look like? Your two children would have a total of two children from both their families combined. Maybe they each have one child. Maybe one has two children, the other has none. Either way, now you have two grand children. Those two grandchildren have between them a total of two children. You now have two great grand children. And the pattern continues; 2 + 2 + 2 + ….
For exponential growth, again, you begin with two children. But, for exponential growth, the family’s size will depend upon it’s size from the prior generation. So, this means each of your two children have two children, giving you four grand children. Each grand child will have two children, giving you 8 great grand children. And so on; 2 + 4 + 8 + ….
For a single family, certainly either growth rate is possible. But, which growth rate is more likely? The latter, exponential growth. (Right?) Now, again for a single family, if we list the subsequent children, grand children and so on for each successive generation, for linear growth we have {2, 4, 6, 8, … } as compared to {2, 4, 8, 16 … } for exponential growth. Which growth rate exhibits the most rapid growth? Exponential. For both answers, we arrive at exponential growth as the logical growth pattern. This means, in ordinary English, that we expect the percent change to have larger predictions than the numerical change and we think these larger predictions are more likely going to be more accurate.
Now, let’s continue our scrutiny of Table 1. Basically, the gist of our discussion so far is that we need to concentrate on the bottom row of Table 1. These predictions are better than those from the top row. Next, we ask, is it better to use longer or shorter time intervals to make our predictions. In short, which cell on the second row of Table 1 is the best predicted population for September 1, 2014? What does your intuition tell you? The detail involved when using populations that are mere seconds apart should have less error that populations taken years apart. This intuition is the basic premise when science employs rate of change to make predictions. You can get rough approximations from populations that are 7 years apart, better approximations from populations that are a year apart, but the smaller you make the interval, the better the approximation should be because there is less margin for error. When you estimate population changes (rates of change) with smaller and smaller intervals, we are actually coming close to approximating the rate of change at a given instant. So, we expect the true population on September 1st, 2014 to be close to but certainly not exactly 327,224,284 because the best estimate will come from the percent change taken from populations according to the shortest time intervals, mere seconds apart.
To summarize, the most efficient method to predict a future population is to use population differences from the smallest time interval feasible and assuming the percent change, that is exponential growth, is the best model. In another context, linear growth might be preferable and for other settings, exponential growth may be best. There are many types of growth models other than these two, but for now we will confine our comparison to use only these two growth rate models.
Exercise Set
1. Below are world population estimates. If the first three estimates taken in July, August and September of 2004 are accurate, use percent change model to determine if the rest of the populations quoted are accurately based the percent change model or merely a fabrication?
Monthly World population figures:
07/01/04 6,377,641,642
08/01/04 6,383,805,814
09/01/04 6,389,969,987
10/01/04 6,395,935,316
11/01/04 6,402,099,489
12/01/04 6,408,064,817
01/01/05 6,414,228,990
02/01/05 6,420,393,163
03/01/05 6,425,960,803
04/01/05 6,432,124,976
05/01/05 6,438,090,304
06/01/05 6,444,254,477
07/01/05 6,450,219,806
2. The table below depicts predictions about the U.S. population. (Adapted from the US Census Bureau’s home Page.)
|Population in thousands |2000 |2020 |2040 |2050 |
|and race or Hispanic | | | | |
|origin | | | | |
|POPULATION | | | | |
|TOTAL |282,125 |335,805 |391,946 |419,854 |
|White alone |228,548 |260,629 |289,690 |302,626 |
|.Black alone |35,818 |45,365 |55,876 |61,361 |
|.Asian Alone |10,684 |17,988 |27,992 |33,430 |
|.All other races |7,075 |11,822 |18,388 |22,437 |
|.Hispanic (of any race) |35,622 |59,756 |87,585 |102,560 |
|.White alone |195,729 |205,936 |210,331 |210,283 |
|not Hispanic | | | | |
If these rates of change reflected in the table above come to fruition, what changes in society do you foresee for the year 2010? 2050? What should politicians, policy makers and community leaders do now to prepare for then?
For questions 3 and 4, use the table below that displays the 20 largest countries in 2004: Source: U.S. Census Bureau, International Database. Taken from
World’s population in 2004.
|Rank |Country |Population |
|1 |China |1,298,847,624 |
|2 |India |1,065,070,607 |
|3 |United States |293,027,571 |
|4 |Indonesia |238,452,952 |
|5 |Brazil |184,101,109 |
|6 |Pakistan |159,196,336 |
|7 |Russia |143,782,338 |
|8 | Bangladesh |141,340,476 |
|9 |Nigeria |137,253,133 |
|10 |Japan |127,333,002 |
|11 |Mexico |104,959,594 |
|12 |Philippines |86,241,697 |
|13 |Vietnam |82,689,518 |
|14 |Germany |82,424,609 |
|15 |Egypt |76,117,421 |
|16 |Turkey |68,851,281 |
|17 |Ethiopia |67,851,281 |
|18 |Iran |67,503,205 |
|19 |Thailand |64,865,523 |
|20 |France |60,424,213 |
It is predicted that for 2004 to 2050, the fastest growing region will be Middle Africa, which is expected to increase at a population rate of 190 percent. The next fastest growing region is expected to be Western Africa, and Western Asia, which will increase at a rate of 140 percent. Next is Central America, expected growth by 60 percent, the United States by 50 percent, South America by 42 percent, the Caribbean by 36 percent, Northern Europe by 6 percent, Eastern Asia by 5 percent, and the rest of Europe falls by 3 percent. Southern Africa, due to HIV/AIDS epidemic is expected to fall by 20 percent.
3. Re-rank the 20 largest countries, using the given growth rates for 2004 to 2050, to estimate the populations for the year 2050.
4. Politicians make policies. If you were a senator, what policies and changes would you suggest to prepare for tomorrow’s global community, given the changing global populations you found in problem 3?
For questions 5 to 10: World Population – online activity. Go to the US Census Bureau
5. Find the current world population
6. How much did the population grow last year? in the last 10 years? 50 years?
7. What do you think influences population growth?
8. What do you think are some of the important consequence of population growth?
9. How many years did it take for the population to increase from 2 to 3 billion?
10. How many years will it take for the population to double its current level if you use the percent growth method with the current population and last year’s population?
For problems 11 to 13, the following data for Median Income was extracted from the US Census Bureau,
Table 2. Yearly Median Income in dollars for a household by race, United States
|Race |White |African |Hispanic |Asian |
|Median | |American | | |
|Income | | | | |
|2003 |45,572 |29,689 |32,997 |55,699 |
|2002 |45,994 |29,845 |33,103 |53,832 |
|2001 |45,225 |29,939 |34,099 |54,488 |
|2000 |45,860 |30,980 |34,636 |58,225 |
|1999 |45,673 |30,118 |33,178 |54,991 |
|1998 |45,077 |27,932 |31,214 |51,385 |
|1997 |43,544 |27,989 |29,752 |50,558 |
|1996 |42,407 |26,797 |28,422 |49,386 |
11. Using percent change and only the data from 2002 and 2003, predict the median income for each of the four stated races for the presidential election year of 2008.
12. Using average percent change and the data from 1996 and 2003, predict the median income for each of the four stated races for the presidential election year of 2008.
13. State whether the data from 2002 & 2003 or 1996 & 2003 is a better predictor of median income. Why?
Linear versus Exponential Growth.
Let’s consider the old problem of the little boy wanted to save money. He awakes one morning, pulls out an old dusty checkerboard from under his bed and carefully sets it up on the small antique nightstand next to his bed. He places two cents on the first square in the corner of the board and pledges to himself, a Boy Scout pledge, that he will place money on each square every week, double the amount of money he placed the week before. How long can he do this, how much money will he have placed on the last square alone? Well, first of all, the little boy knows he has 64 squares because he knows the board is 8 by 8. The realization then dawns that this task will require his due diligence for a little over a year.
Okay, lets figure this out. The rate here based on doubling the amount of money each week. So, for each successive week, he will place, in dollars, 0.02, 0.04, 0.08, 0.16, 0.32, 0.64, 1.28, and then $2.56 on the eighth week, finalizing the first row of the board. This type of growth is called exponential, and so far the little boy is thinking this isn’t so bad. But by the time the second row is being developed, this successive doubling is proving to be too much. Week nine, 5 dollars and 12 cents, Week ten, $10.24, then $20.48, $40.94, $81.92, $163.84, $327.68 and by the end of the 16th week, he places $655.36. He stops and thinks. He will need to save $655.36 in one week.
By the time this savings plan reaches the third row, over a thousand dollars will be placed on the first square and the dollar amount will continue to successively double. So, for this growth, we need to duplicate the pattern. The pattern created by this growth is 0.02, 0.02 x 2, 0.02 x 2 x 2, 0.02 x 2 x 2 x 2, … , 0.02 x 263. Notice each successive term on changing by the growth factor of 2. This tells us two vital things, first, successive doubling involves a base of two. Thus, successive tripling would involve a base of three, successive quadrupling a base of four and so forth. Secondly, the exponent represents the week we placed money on a square. To figure out how much we placed on the square in the fourth row, third column, it is week 24 + 3 or 27. So, we would place 0.02 x 2^26 or $1,342,177.28. We would have placed half that amount on week 26 and twice that on week 28. So, how much money would we need to place on the square representing week 64, the week of the last square on the board? Well, 0.02 x 2 ^ 63. Do you have a feel as to how large that number is? 18446744073709551616 or more clearly represented as $ 184,467,440,737,095,516.16 or loosely speaking, nearly two hundred million billion dollars.
Realizing the magnitude of the task he had set for himself, the little boy gives up this strategy to save money and invents a new one. He begins as he did before, placing two cents on the first square in the corner of the board, makes the same pledge to himself, and then places 2 more cents on each square weekly. On week two, he places 4 cents, on week three, 6 cents, and so on. Now, after 64 weeks, he places [pic]cents, a far cry from two hundred million billion dollars.
(See chapter 5 for a formula to efficiently find the total sum of the money saved.)
• Exponential Growth is considerably faster than Linear Growth
Exercise Set
For questions 1 to 10, determine whether the growth is linear or exponential.
1. By the end of 1998, the number of people living with HIV, the virus that causes AIDS, had grown to an estimated 33.4 million, which was 10 percent more than one year before.
2. The Food and Agriculture Organization (FAO) estimates that 53,000 square miles of tropical forests (rain forest and other) were destroyed each year during the 1980s. Of this, they estimate that 21,000 square miles were deforested annually in South America, most of this in the Amazon Basin.
3. Some pond lilys double everyday in size. It may take them 40 days to completely cover a pond.
4. According to FAO (FAO 1997), Mexican deforestation in the period 1990-1995 averaged 510,000 hectares annually.
5. In 2004, according to the US Census Bureau, the global population was 6,377,641,642, and it had a growth rate of 1.13 %.
6. On September 16th, 2004, the southern US state of Alabama has been hit full-on by
Hurricane Ivan, with winds of up to 135mph battering the coastline, according to the BBC.
7. The handheld multimedia player market is poised to grow rapidly over the next five years, according to a report from In-Stat/MDR. The report projects 700 percent growth in 2004, and a CAGR (compound annual growth rate) of 179 percent through 2008.
8. It is said that a person’s heart rate increases 140% in the anticipation of the start of a race.
9. Humidity in dry air causes an elevated heart rate because each 1 pound of body weight loss corresponds to 15 ounces (450 mL) of dehydration.
10. Feelings of loves usually result in braycardia, which is a dropping of the heart rate, to 90 percent of its original rate.
11. Add 10 two’s. Then multiply 10 two’s together. How much larger is the exponential sum than the linear sum?
Linear Models
The teen age tyrannosaurus was estimated to gain 4.6 pounds of weight per day during those tumultuous teen years from 14 to 18 years of age.
All linear models have a starting value and a constant rate of change.
This growth spurt can be viewed mathematically as a rate of change because it is a measure of the change of weight per one day. It is a linear growth because the change is itself is constant from day to day; the colossal dinosaur gained 4.6 pounds each day. So, the natural question arises, how much did the T. Rex weigh after these four 4 years passed, given that the pre-teen weighed about a ton before the gigantic growth spurt. Intuitively, you would add 4.6 pounds to itself for the number of days that would pass in those four years and add that amount to one ton. This repeated addition is the same as multiplication. This intuition is dead on accurate and is an example of a linear growth model. We will multiply the number of days of the growth spurt to 4.6 pounds per day.
We have 4.6 pounds per day x 4 years x 365 days per year = 6716 pounds. A ton is 2000 pounds and 6716 / 2000 is 3.358 or just greater than 3 tons of weight gain. Thus, 1 ton plus 3 tons is 4 tons of T. Rex. Who says size doesn’t matter?
A linear model has the form y = b + mx
where
• the starting value is b
• the constant rate of change is m
We summarize our numerical calculation by writing y = 2000 + 4.6 x (4 x 365) lbs and we have the weight of T Rex on her 18th birthday. Mathematically speaking, the constant rate of change is referred to as the slope, and is traditionally denoted by the variable m. The variable b represents the initial condition and in this case, the starting weight of T Rex, ie, her weight on her 14th birthday.
Finally, note that linear models, because they have a constant rate of change, either always increase or decrease. This behavior in a model is coined monotonic, meaning the change is either increasing or decreasing. Linear models exhibit monotonic behavior. They will either always increase or always decrease. We often use a continuous linear model to represent the behavior of discrete data points.
Example One
Initially, you have lost $20,000 in an investment and you continue to lose $350 per month. In how many months will it be before your losses total $26,300, thus your balance is – 26,300?
Solution
First, let’s define our variables. Let x be what we are looking for, which is the number of months since the initial loss of $20,000. Let y be the output; the output is always dependent on the value of x. In this example, the monthly balance is dependent on the number of months that pass. We let y represent the total amount of money lost in the investment. Thus, your balance is a loss, a negative number, right?
• What is the starting point? In other words, what is the value of y when x = 0, that is, the time in months is 0? Well, initially, there is $20,000 lost in the stock market, so your balance, thus the starting point b, is - 20,000.
• What is the constant rate of change? Remember, we are losing a constant amount of money each month, $350. So, in this scenario, the constant rate of change is the constant loss of $ 350 per month. And since we are losing money each month, this is a negative number. We say the slope is negative and we have m = - 350
In ordinary English, we know our starting value is - $20,000 and the rate of change is –350. Our linear model looks like this, y = - $20,000 – 350x. Now, to answer our question, we need to solve for x when y is – 26,300? Substituting we have:
- 26,300 = - 20,000 – 350x
-6,300 = – 350x
[pic]. Our loss will reach $26,300 after 18 months.
Example Two
It is noon, on September 22, 2004 and you decide to take a hike up Mount Everest. Create a linear equation from the table below, where x represents numbers of hours since noon, September 22, 2004 and y represents the elevation you are at on the mountain after you hiked x hours.
|x |0 |1 |2 |5 |
|y |20,000 |20,225 |20,450 |21,125 |
Solution
We need to create a mathematical model, will a linear model fit the data in the table?
• To find b, the starting elevation clearly occurs is 20,000 feet (at x = 0), thus b = 20,000
• To identify m, the constant rate of change, we look for a common difference among the data. This means we glance at y values associated with consecutive values of x.
20,000 20,225 20,450
225 225
So, m = 225 and our linear model has the form y =b+ mx or y = 20,000 + 225x.
When presented with a table of data:
• The starting point occurs whenever the input variable (often referred to as the x variable) is zero.
• The constant rate of change is the difference in values of the out put variable (often referred to as the y variable) that are constant for equal differences in the input.
Example Three
It is noon, on September 23, 2004 and you are flying at 20,000 feet above the ground. You begin your decent down. Create a graph from table below, where x represents numbers of minutes since noon, September 23, 2004 and y represents the elevation you are at in your plane. When would the distance between the plane and the ground be zero?
|x |0 |1 |2 |5 |
|y |20,000 |19,875 |19,750 |19,375 |
Solution
On a graph:
• The starting point graphically is the y – intercept which in this example represents the altitude when time is 0. Thus, b = 20,000.
• What is the constant rate of change? Recall, graphically, the slope tells us two things.
the sign of m tells us whether the line rises or falls as we move from left to right parallel to the horizontal axis. Since the altitude (y- values) decrease for every increase in time (x-value), the sign is negative, [pic].
the absolute value of m gives us a sense of just how steep that line is and since, the common difference is 125, we are descending 125 ft for every one unit of time, that is, for every minute; m = - 125.
So, we know, the y-intercept is 20,000 and for every one unit we move to the right, the line drops 125 units down. So, how long before you land the plane? Well, your plane has landed when the elevation above the ground is at zero. Graphically, we are asking, what is x when y = 0? What is the x – intercept?
So, we graph the line and find the x – intercept.
Thus, in 160 minutes from noon, September 23rd, 2004, we land the plane. We have [pic]or at 2:40 pm, September 23rd, 2004. It’s a slow decent down to the ground; [pic].
Example Four
Which of the following graphs most closely represents a woman paying off a $20,000 debt by making equal payments of $ 250 per month, where x is the number of monthly payments since the debt was first assumed and y is the amount of money left on the debt?
a) [pic] b) [pic]
c) [pic] d) [pic]
Solution
• What is the starting point? It occurs when x = 0, right? Graphically, the starting point is the y- intercept. Thus, the y-intercept is 20,000.
• What is the constant rate of change? Graphically, the slope tells us two things.
o the sign of m tells us whether the line rises or falls as we move from left to right.
o the absolute value of m gives us a sense of just how steep that line is.
Let’s visually explore this concept. Picture this:
For every payment, there is less on left on the loan. We are starting with $20,000 and for every payment, we remove $250 from the $20,000. Are you visualizing 20,000 marked on the y-axis, and the line representing the linear model going down $250 for every one month shift to the right. This reduces our choices above to either c) or d), because those lines have negative slopes.
Next: we need to determine how steep the line should be. Well, for every 1unit right (1
payment), we drop $250 (amount removed from the debt.)
[pic]
Only choice d) has this slope.
Example Five
|Year |square mi |
|1997 |- 5034 |
|1998 |- 6501 |
|1999 |- 6663 |
|2000 |- 7658 |
|2001 |- 7027 |
|2002 |- 9845 |
|2003 |- 9343 |
The Disappearing Amazon Rainforest. Below is a table representing the year and the square miles of forest lost to deforestation for the Amazon Basin for that given year. Assuming a linear growth model, predict how much of the forest would be lost in total since 1997 for the years 2004, 2005 and 2006.
Solution
Initially for this problem represents the year 1997. Initially, in the year 1997, the Amazon Basin lost 5034 square miles ( - 5034).
If we plot the data on the horizontal axis with respect to the given Year and on the vertical axis the Square Miles of forest that was cut down that year, we note the points do not fall along a perfect line. Real world data rarely does. But, the points do seem to ‘roughly’ model linear growth, If the data fit a linear model perfectly we would have each data point showing up along a horizontal line. Why? The annual data does seem to be growing or it could be simply maintaining between the bounds of -5000 and -10,000 square miles. We are not so much interested in the annual loss but rather the accumulated loss over time. Assuming a steady or ‘average’ annual loss will allow us to use a linear model to make a prediction about the future loss of forest.
First, let’s define our variables. Let the accumulated loss of forest be what we are looking for; we will assign the independent variable to the year and the dependent variable or output will be the total amount of forest lost. When using a linear model the output is always dependent on the independent–value. In this problem, this means the year controls the amount of square miles removed from the forest in the Amazon Basin since 1997. We must be careful here. We are not directly given this information, we must find this information.
|Year |Total square miles |
| |lost since 1997 |
|1997 |- 5034 |
|1998 |- 11535 |
|1999 |- 18198 |
|2000 |- 25856 |
|2001 |- 32883 |
|2002 |- 42728 |
|2003 |- 52071 |
We construct a new table, where the independent variable x is the year and the dependent variable is what we are searching for, the total square miles lost to deforestation since 1997 in the Amazon Basin.
We let y be the amount of the forest lost to deforestation in square miles. Thus, the amount of land is lost, thus a negative number, right?
• What is the starting point? In other words, what is y when x = 0? Well, initially, there is - 5034 square miles lost in the year 1997. Your starting point b, - 5034.
• What is the constant rate of change? Remember, for linear growth, we must assume we are losing a constant amount of forest each year. So, in this problem, the constant rate of change is the constant loss of forest. If we subtract –5034 from - 52071 we get the difference in square miles lost for those 6 years between 2003 and 1997. Completing this calculation, we obtain –47037 square miles lost. Next, we divide by 6, and find the average number of square miles lost per year is - 7839.5 square miles.
Plotting the data from this new table, again allowing the horizontal axis to be the independent variable, the year, and the vertical axis to be the dependent variable, the total square miles of forest lost since 1997, we have a new graph which also closely resembles a linear collection of data.
Because we have characterized this situation as the Amazon Basin is losing forest land each year, the slope is negative to represent the notion of ‘losing’. So, we have m = - 7839.5. In ordinary English, we know our starting value is - 5034 and the rate of change is –7839.5. Thus, the model we use starts with a negative 5034 square miles and this number reduces by 7839.5 square miles each year.
y = - 5034 – 7839.5.6(x-1997)
Graphing, we have:
[pic]
Now to answer our original question. We need to find y when x is 2004, 2005 and 2006 respectively. Substituting we have
y = - 5034 – 7839.5 (2004-1997) = - 59910.5 square miles;
y = - 5034 – 7839.5 (2005-1997) = - 67,750 square miles;
y = - 5034 – 7839.5 (2006-1997) = - 75,589.5 square miles.
Why is this question important, that is why we should care about deforestation of the rainforests? What difference does it make if a few plants and animals die? For most people, the forests are not all that pleasant to visit in the first place. Hot. Humid. Remote. Too many bugs. Too many snakes.
On a global scale, rainforests once covered 14 % of the earth’s surface. Now they cover 6 %. It is estimated that nearly half of the world’s species of plants, animals and microorganisms will be severely threatened or destroyed over the next 25 years due to deforestation. Tropical rain forest are home to over half of all plant and animal species. Their elimination due to tropical deforestation would greatly threaten the fine balance that is the carbon cycle. At least 75 percent of deforestation in these areas is due to burning, which releases about over 2,000,000,000 tons of Carbon Dioxide into the atmosphere each year. Rainforests are home to 70 % of the nearly 3000 plants that the U.S. National Cancer Institute identified as active against cancer cells. Vincristine is one of the world’s most powerful anticancer drugs and it is extracted from the Periwinkle, a rainforest plant. Finally, deforestation is the single largest reason for extinctions over the last 65 million years.
The Amazon Rainforest is the most diverse and biologically active wonder on this planet. If this timeless beauty were a country, it would be the ninth largest country in the world, representing 54 percent of the rainforests left on the earth. One pond in Brazil has a larger variety of fish than is found in all of the rivers in Europe. A twenty five acre of rain forest in Borneo may contain the same number of different tree types as is found in all of North America. Over 10,000,000 Indians lived in the Amazonian Rainforest 500 years ago. Currently, this number is less than 200,000. More than half of the world’s 10,000,000 species of plants, animals and microorganisms as well as two-thirds of all the world’s fresh water supply is found in the Amazon Rainforest.
One sobering note. The Amazon Rainforest are the lungs of our plant. We risk our very survival by eliminating the natural mechanism nature has provided for us to breath. Our plant life creates the very air we inhale and exhale every moment of every day through photosynthesis by continuously recycling carbon dioxide into oxygen. Over 20 percent of the earth’s oxygen is produced in the Amazon Basin. If this largest of the rain forests were gone tomorrow, the world’s atmosphere would not have the necessary air for us to breath. We don’t breath, we don’t live, it is that simple. For a final dark remark, independent research institutions have forecast if the government continues with its deforestation in the Amazon region, up to 40% of the total rainforest in the basin will be destroyed within 20 years. So, pause. Relax. Take a breath. And just realize, we may not be able to have this conversation in 2024.
Example Six
Life Expectancy Life expectancy represents the average number of years people are projected to live. According to the “Universal Almanac 1990”, Edited by John W. Wright, “life expectancy has improved steadily over the years, largely due to a decline in deaths during childhood.” Often the word “steadily” is an indicator that the growth rate is linear. This is because if we examine the word ‘steadily’ in context, “improved steadily” can be often be reworded as “a steady rate of growth” and it is likely that the writer of the sentence may in fact mean a “constant rate of growth.”
Below is a table for the Life Expectancy at Birth for selected years since 1940 for the United States, all races, both genders. Source: Adapted from the following publications: “Universal Almanac 1990”, Edited by John W. Wright; US life expectancy at all-time high, but infant deaths up – CDC, Agence France-Presse - February 11, 2004 ;; “Life Expectancy Hits 76.9 in U.S.” By ROSIE MESTEL LA Times, September 13, 2002
Table: Life Expectancy at Birth
|Year |1940 |1960 |1970 |1980 |1986 |1991 |
|y |2 |5 |8 |17 |23 |26 |
4. Find an equation that best represents the relationship between x and y?
|x |1 |2 |3 |5 |8 |10 |
|y |200 |185 |170 |140 |95 |65 |
5. For the two graphs depicted below, where the horizontal axis represents miles and the vertical axis represents dollars,
[pic]
which real world situation is represented?
a. Two competing rent-a-car companies have the following options: Co A costs 25 cents per mile, Co B costs 40 dollars a day. How many miles should one need to drive if they were to decide to rent from Co B?
b. Two competing rent-a-car companies have the following options: Co A costs 25 cents per mile, Co B costs 40 dollars a day and 20 cents per mile. How many miles should one need to drive if they were to decide to rent from Co B?
c. Two competing rent-a-car companies
have the following options: Co A costs 25 cents per mile, Co B costs 40 dollars a day and 25 cents per mile. How many miles should one need to drive if they were to decide to rent from Co B?
d. Two competing rent-a-car companies have the following options: Co A costs 25 cents per mile, Co B costs 40 dollars a day and 10 cents per mile. How many miles should one need to drive if they were to decide to rent from Co B?
6. Match the four graphs with the 4 real world explanations that most closely depicts the graph.
a. [pic]
b. [pic]
c. [pic]
d. [pic]
i) Over a 6 year period, a company that initially showed a loss of 2 million dollars, lost $ 667,000 per year.
ii) Over a 6 year period, a company that initially showed a profit of 2 million dollars, lost $ 667,000 per year.
iii) Over a 6 year period, a company that initially showed a profit of 3 million dollars, lost $ 2 million per year.
iv) Over a 6 year period, a company that initially showed a profit of $ 2 million dollars, lost $ 1,000,000 per year.
7. For the depicted graph below, where the horizontal axis is time t and the vertical axis is distance, d.
[pic]
which of the following scenarios would be most accurately described with the above graph?
a. A man leaves home to drive to work, speeding up with a constant acceleration for 4 minutes, hits a constant speed for the two minutes, speeds up with a constant acceleration for the next four minutes, then hits a constant speed for the last 12 minutes.
b. A man leaves work and drives home, fast, then slow, fast then slow.
c. A man leave his home to go to work. He drive for ½ hour, get a flat tire, pulls over, spends 15 minutes fixing the flat tire. He then speeds off to work, driving for another half an hour twice as fast as he did earlier. He then stays at work for 90 minutes.
d. A man leave his home to go to work. He drive for ½ hour, get a flat tire, coasts for 15 minutes, and then fixes the flat tire immediately. He then speeds off to work, driving for another half an hour twice as fast as he did earlier. He then stays at work for 90 minutes.
8. Draw a Cartesian coordinate system (x and y axes that intersect.) How would you have to scale your axes if you were sketching a line and the rate of linear change you were modeling was:
a) 10? b) 100? c) 1000?
d) 1,000,00? e) One Billion?
f) One Trillion? g) Negative one trillion?
9. Draw 5 Cartesian coordinate systems, each time scaling the units from –100 to 100 on the x axis and –100 to 100 on the y axis. Count by 20’s. Draw five linear models, one where the rate of change is 0.01, one where the rate of change is 1, one where the rate of change is 5 one where the rate of change is -1 and one where the rate of change is - 2.
10. On one (the same set) of axes, assume Y1, Y2, Y3, and Y4 are linear models. Fill in the table, write the equation for the four linear models and then sketch the models on the domain specified.
|x |Y1 |Y2 |Y3 |Y4 |
|0 |5 |5 |5 |10 |
|1 |10 |2 ½ | | |
|2 | | |10 | |
|3 | | | |2 ½ |
For problems 11 to 24, use the following information regarding Heart Disease, Table A and Table B: Though estimates vary slightly, nearly 1 million Americans die each year from heart and heart related diseases, accounting for approximately 40 percent of the deaths in the United States. It is estimated that close to 62 million Americans have some form of cardiovascular disease. In 1999, for instance, Cardiovascular disease accounted for almost 960,000 deaths; cancer nearly 550,000 deaths, accidents nearly100,000; Alzheimer's disease 44,500 and HIV/AIDS close to 14,800. Thus it may be surprising to note that death rates from heart disease has dropped steadily since 1960. Below are two tables.
The first table, table A, shows the Death Rates from Heart Disease for Males ages 45-54 in the United States for selected years, all races. Use this table for problems 11-17.
The second table, Table B, shows the Death Rates from Heart Disease for both genders ages 45-54 in the United States for selected years, all races. Use this table for problems 18-24.
Adapted from the following publications: “Universal Almanac 1990”, Edited by John W. Wright ; Dr. Joseph Mercola, Author of the Total Health Program ;
Table A: Death Rates for Males age 45-54 from Heart Disease per 100,000 population in the United States
|Year |1960 |1970 |1980 |1985 |
|Death Rates per |420.4 |376.6 |282.6 |236.9 |
|100,000 | | | | |
11. Given the year, we want to predict the death rates per 100,000 for a male ages 45-54 in the United States. What would be a good choice for input and out put for this model?
12. For each time interval provided, calculate the change in death rates per year.
13. State in a complete sentence the information you calculated in part b).
14. What is the average of these constant ‘rates of change’ you found in part b). State a complete sentence for this information.
15. Create a best fit linear model to fit this data.
16. Use this model to predict what the death rate per 100,000 for males ages 45-54 in the year 2010 and 2050 would be in the United States.
17. Are there limitations to the usefulness of linear models?
Table B: Death Rates for both genders age 45-54 from Heart Disease per 100,000 population in the United States
|Year |1979 |1995 |1996 |
|Death Rates per |184.6 |111 |108.2 |
|100,000 | | | |
18. Given the year, we want to predict death rate per 100,000 for ages 45-54 in the United States. What would be a good choice for input and out put for this model?
19. For each time interval provided, calculate the change in death rate per 100,000 per year.
20. State in a complete sentence the information you calculated in part b).
21. What is the average of these constant ‘rates of decay’ you found in part b). State a complete sentence for this information.
22. Create a best fit linear model to fit this data.
23. Use this model to predict what the death rate per 100,000 for ages 45-54 would be in the year 2010 and 2050 would be in the United States.
24. Are there limitations to the usefulness of linear models?
25. On-line Project: Research the land area in square miles of the Amazon Basin’s Rainforest today. Use the linear growth model developed in Problem 5 from this section of the text to answer “assuming deforestation continues along this linear model derived in Problem 5, in what year would the Rain Forest in the Amazon Basin be gone?”
26. On-Line Project: Research polices and practices regarding Aging, Medicare, and Medicaid. Start with the United States Department of Health and Human Services and go from there. Then use the linear growth model found in Problem Six of this section of the text to predict the Life Expectancy for each year for the next ten years. Write a paragraph elaborating on changes you think could be instituted regarding how we care for our elderly. Discuss any relevant topics, such as aging policies and practices, Medicare, health insurance, pharmaceutical needs, nursing home providers, in home providers or catastrophic insurance policies.
For problems 27 to 32, match the starting value and rate of change with the graph. Assume the below graphs a through f are linear models.
27. b = 0, m = 100
28. b = 0, m = - 100
29. b = 100, m = 100
30. b = 100, m = - 100
31. b = -100, m = 100
32. b = -100, m = - 100
a. [pic]b. [pic]
c. [pic]d. [pic]
e. [pic]f. [pic]
Exponential Models
Rules of Exponents: Assume all bases are positive.
• To multiply identical bases, add the exponents. EX: x5x4 = x5+4 = x9
• To divide identical bases, subtract the exponents. EX: [pic]
• When an exponent is raised with an exponent, multiply the exponents.
EX: (x5)4 = x20
• Any base raised to the zero power is 1. EX: [pic]
• Any base raised to a negative exponent can be rewritten as the reciprocal of the base raised to the absolute value of the exponent. EX: [pic]
Radical notation: x is a real number, and n and m are integers: [pic]
Perhaps one of the most striking well known examples to visualize the power of exponential growth is to simply fold a sheet of paper. Take an 8 ½ by 11 inch piece of paper, and fold it in half. The result will be thicker than the original sheet of paper. How much thicker? Twice as thick. Now, the original sheet of paper was about 1/100 of a cm or 0.004 inches thick, and the folded paper is 0.004” + 0.004” = 2*0.004” = 0.008 inches thick. Fold the paper in half again; its been folded in half twice. The paper is 2 * 0.008 inches thick. Or more succinctly, using exponential notation, 0.004”*2*2 or 0.004*22 inches thick, or four times as thick as it was originally. You are thinking, big deal, it was awfully thin originally. Keep the attitude in check until we finish this example. So, fold it in half a third time, and after three folds, the folded paper is as thick as your finger nail, roughly 0.004*2*2*2 = 0.004*23 inches thick or 0.032 inches thick. If we fold again and again and again and again, you have now folded it in half seven times, and the folded paper is 27 or 128 times as thick as it once initially. Now, it is roughly 0.004*128 (0.004*27) = 0.512 inches thick, it is the same thickness as your cell phone. This may be your limit here in terms of actually folding the paper. But, if you could continue folding it in half, after 10 folds it would be 0.004*210 = 4.096 inches thick, a little over 4 inches thick. Now, the folded paper is as thick as your calculator is wide. Two more folds, it is 0.004*212 inches thick, or 16 inches thick, about 1 1/3 feet. The height of a stool. After 17 folds, it would be 0.004*217 inches thick, or 524 inches or about 524/12 = 43 ½ feet. Taller than a house. Five more folds and that sheet of paper is now 0.004*222 inches thick, about 16,777 inches thick or nearly 1400 feet. It is about as thick as the Empire State Building is high, thicker actually. Ten more folds, 0.004*232 = 17179869 inches or 1431656 feet or 1431656/5280 271 miles thick. The paper has left our atmosphere. Twenty more folds, 0.004*252 inches, 1.8 x 1013 inches, or 284,318,158 miles thick. The paper is burning up now because it has just touched the sun, punctured it, pierced through its middle and burst out the other side. At sixty folds, our paper, though awfully narrow by now, is the diameter of the solar system and at 100 folds it has the radius of the universe.
And that is just 100 folds! Not a 1000. Not a million. One hundred. So, growth defined by successively doubling is kinda quick, huh. If successive doubling is such a fast (powerful) growth rate, imagine what successive tripling would be like? This type of growth rate needs its proper reverence.
We will analyze exponential growth using the exact same lingo and jargon we used with linear growth rates.
Recall:
All linear models have a starting value and a constant rate of change.
Exponential models possess the same traits as linear growth models. Each growth rate, linear or exponential, acts on an initial value with a growth or decay rate that is expected per unit increment. They have the same feelings and emotions as any other growth model. Their job is to model growth. But, an exponential growth model will model the growth in proportion to the original amount or quantity present.
All models for growth rates have a starting value and
a rate of change that needs to be expressed.
An exponential model has the form
y =a r x
where
• the starting value is a
• the rate of change r is based on
a proportion of its current value
In our example, y = 0.004 * 2x, where y is the thickness in inches of the folded piece of paper, a = 0.004 is the paper’s initial thickness in inches and the growth rate, the successive doubling is representing by 2x, where x is the number of times you fold the paper. So, r = 2 means the base of two is multiplied to each subsequent answer, giving you a new value that is in proportion (twice as large) to the current value.
All exponential models have a starting value and
a rate of change based on a proportion of it’s current value.
Finally, note that exponential models, like linear models, either always increase or always decrease. Recall, this behavior in a model is called monotonic, meaning the change is either increasing or decreasing. Exponential models exhibit monotonic behavior. They will either always increase or always decrease.
Lastly, exponential models that exhibit increasing behavior are said to exhibit exponential growth, while exponential models that exhibit decreasing behavior are said to exhibit exponential decay.
Example One
According to the US Census Bureau, the population of the state of Arizona in 1990 was 3,665,228 and in 2000 was 5,130,632. Assuming an exponential growth rate, predict the population of the state of Arizona in the year 2008.
Solution
For this exponential model, a plausible average annual growth factor is found by dividing the percentage of growth by the number of years for this growth. In this example we divide the ratio of the population in 2000 by the population in 1990 by 10. So, we have 5,130,632/3,665,228 = 1.399813, which means it grew by 0.399813 percent over a time period of 10 years. Roughly, this translates to 0.399813 /10 pr 0.0399813 percent change per year. In 2000, Arizona’s population was 5,130,632 and we want the population eight years later, assuming an average percent growth rate of 0.0399813 for each year. We have
• x = 8, the year is 2008 and the population of the state of Arizona to be predicted to be 5,130,632(1.0399813)8 = 7,020,614 people.
Another more precise method for predicting the population would be to find the growth factor so that when it is repeatedly multiplied with itself within the growth model we obtain the same ten-year growth rate. To see this we need to consider the ratios [pic]. When we solve for the variable [pic] we will have the appropriate growth factor:
[pic].
• x = 8, the year is 2008 and the population of the state of Arizona to be predicted to be 5,130,632(1.03421)8 = 6,714,702 people.
Example Two
Since 1999, the percentage of 8th-grade students who reported having five or more drinks in a row in the past 2 weeks has been decreasing. In 1999, 15.2 % reported to have drunk five or more drinks in a row for 2 weeks prior to the poll. The percentage dropped to 14.1 % in 2000, 13.2 % in 2001, 12.4 % in 2002 and 11.9 % in 2003. SOURCE: Johnston, L.D., O'Malley, P.M., and Bachman, J.G. (2003). Monitoring the Future national survey results on drug use, 1975-2002 Volume I: Secondary School Students (NIH Publication No. 03-5375). Bethesda, MD: National Institute on Drug Abuse Tables 2-2 and 5-3. Data for 2003 are from a press release of December 19, 2003, and demographic disaggregations are from unpublished tabulations from Monitoring the Future, University of Michigan
a) Assuming the growth rate is exponential, predict the percent of eighth graders who would drink those five drinks in 2004 and 2005. Use the time periods 2002-2003 to make the prediction.
b) Assuming the growth rate is exponential, predict the percent of eighth graders who would drink those five drinks in 2003 using time period from 2001-2002. How accurate was the prediction?
c) Why do you think we assumed an exponential growth rate to make the predictions?
Solution
a) For 2002-2003, the percent decrease is measured by the 2003 percent divided by the 2002 percent, 11.9/12.4 = 0.9597. So, we have y =arx = 11.9(0.9597)x. For the 2004 prediction, we have have y =11.9(0.9597)1= 11.42. For the 2005 prediction, we have
have y =11.9(0.9597)2 = 10.96. Great, the drop in percent of those eighth graders with a potential drinking problem continues.
b) For 2001-2002, the percent decrease is measured by the 2002 percent divided by the 2001 percent, 12.4/13.2 = 0.9394. So, we have y =arx = 12.4(0.9394)x. For the 2003 prediction, we have y =12.4(0.9394)1= 11.64. The true percent was 11.9 the degree of accuracy was 0.03 out of 11.9 or within 0.0025. Not bad.
c) We could have assumed various models for our prediction. If the data fluctuated up and down, meaning that the percent of 8th-graders who drank rose and fell over the years documented, then other models may have served to be better predictors. Since the percent of 8th-graders who ‘drank’ appeared to consistently fall from 15.2 % to 14.1 % to 13.2 % to 12.4 % and then to 11.9 %, this trend suggested that the rate of change was monotonic. So, in choosing between our two models, linear or exponential, one could argue exponential for several reasons. First, the change did not appear to be constant. Take any two successive years, and the differences between the two percents did not remain the same. For example, the difference between 15.2 % and 14.1 % was 1.1 %; while the difference between 14.1 % and 13.2 % was 0.9 %. Now, since we are dealing with real world data, the constant rate of change usually won’t be exact. So, there must be other reasons. So, secondly then, the problem is dealing with populations, who traditionally tend to behave exponentially. Population growth is usually exponential because you have two children who have two children and so on. Now, we are not dealing with births, we are dealing with a population’s behavior, so the natural question arises, would the same assumption hold, population’s behavior models exponential growth? Often, yes. If 15.2 % of the population were ‘drinkers’, the next year, a portion of the next generation, the 8th-graders, would behave as their peers did, but a segment of the population may choose not to follow the same ‘drinking’ behavior. Changes in a population’s behavior follow the pattern that one person changes, this person influences some one else, who influences some one else. And so on. By definition, exponential growth or decay is based on the proportion of the existing population changing. Exponential modeling is a reasonable choice for this prediction.
Example Three
Below is a table of the world’s population taken from the US Census Bureau by year for the 21st century. To understand whether a linear or an exponential model is preferred, we compared the change of population from the previous year to the ratio of each year’s population to the previous year.
a) Which model, linear or exponential is preferred?
b) Predict the world’s population for the years 2005, 2010, and 2050.
Table 1. The population of the world.
|Year |Population of the world |Ratio of each year’s |Change in population from |
| | |population |the previous year |
| | |by the previous year | |
|2000 |6,085,478,778 | | |
|2001 |6,159,699,306 |1.01 |74,220,528 |
|2002 |6,232,702,169 |1.01 |73,002,863 |
|2003 |6,305,144,680 |1.01 |72,442,511 |
|2004 |6,377,641,642 |1.01 |72,496,962 |
Solution
For a linear model to be preferred, the rate of change must be constant. But, the difference in population between two successive years is not the same. For example, 6,377,641,642 – 6,305,144,680 = 72,496,962 while 6,305,144,680 – 6,232,702,169 = 72,442,511. For an exponential model to be preferred, the growth must be in proportion to its current population [pic] and [pic]
Both calculations give us a common base, called a common ratio of 1.01. This common ratio has remained the same throughout the early part of the 21st century. This constant growth factor for an exponential model allows us to find the population for the world in the future.
b) If we assume the same growth factor, or common ratio, will hold until 2050, we can predict the world’s population for any year between 2004 and 2050. Let x represent the year for the world’s population since 2000.
• When x = 5, the year is 2005 and the population of the world will be predicted to be 6,085,478,778(1.01)5 = 6,395,899,355.
• When x = 10, the year is 2010 and the population of the world will be predicted to be 6,085,478,778(1.01)10 = 6,722,154,502.
• When x = 50, the year is 2050 and the population of the world will be predicted to be 6,085,478,778(1.01)50 = 10,008,837,205 or a little over 10 billion people. How many people do you think this planet can sustain?
Exercise Set
For problems 1 to 5, use the following data. In the United States, the data below gives the number of live births to unmarried women per 1000 woman that are of ages 20-24 years old for the years 1980 to 2000. In 1980, 40.9 out of every 1000 live births were to unmarried women. The number rose to 46.5 in 1985, rose again to 65.1 in 1990, rose to 68.7 in 1995, rose again to 70.7 in 199 9 rose to 72.1 in 2000. Note, the difference in successive five year intervals are not constant. Therefore, we will not be prone to use a linear model. Also notice the number of live births per 1000 woman is increasing through out the 20 year period. So, for this problem, can we assume the growth rate is exponential?
SOURCE: Johnston, L.D., O'Malley, P.M., and Bachman, J.G. (2003). Monitoring the Future national survey results on drug use, 1975-2002 Volume I: Secondary School Students (NIH Publication No. 03-5375). Bethesda, MD: National Institute on Drug Abuse Tables 2-2 and 5-3. Data for 2003 are from a press release of December 19, 2003, and demographic disaggregations are from unpublished tabulations from Monitoring the Future, University of Michigan.
1. Use 1980 and 2000 data to predict the number of live births per 1000 unmarried women of ages 20-24 for 2005 and comment on the accuracy of the prediction.
2. Use 1990 and 2000 data to predict the number of live births per 1000 unmarried women of ages 20-24 for 2005 and comment on the accuracy of the prediction.
3. Use 1995 and 2000 data to predict the number of live births per 1000 unmarried women of ages 20-24 for 2005 and comment on the accuracy of the prediction.
4. Use 1999 and 2000 data to predict the number of live births per 1000 unmarried women of ages 20-24 for 2005 and comment on the accuracy of the prediction.
5. Which of the five predictions in problems 1 to 4 were the most accurate? Were any of the predictions accurate? Why or Why Not?
For problem 6 and 7, use table 1.
6. Predict the world’s population for the year 2008.
7. Predict the world’s population for the year 2060.
Use the following data for problems 8 to 10. The population of Mexico in the mid-1980’s was roughly 74,660,000 in 1984, 76,600,000 in 1985 and 78,590,000 in 1986.
8. Find the difference in population from one year to another and then find the ratio of the year’s population to the previous year. Which growth model do you feel would more accurately predict the future population of Mexico, linear or exponential?
9. Assuming the same pattern of growth for the next thirty years, predict the population of Mexico for the year’s 1990, 2000, 2004 and 2010.
10. Go online, and find the true population of Mexico for the year’s 1990, 2000 and 2004. What does this evidence suggest as for the population you found for 2010 in the previous problem?
For problems 11 to 13, use the following data. The number of people estimated to be living with HIV/AIDS, Globally, from 1998-2002 was respectively, 33.4 million, 33.6 million, 36.1 million, 40 million and 42 million.
11. Is the behavior of the data monotonic? Why?
12. Is an exponential model a good choice for this data? Why?
13. If you did use an exponential model and you used the data from 2001-2002, predict the number of people would be living with HIV/AIDS in 2006 and 2010.
14. In the United States, the number of people living with HIV/AIDS was estimated to be 850,000 in 1999 and 900.000 in 2001. Assuming an exponential model, predict the number of people in the United States that would be living with HIV/AIDS in the year 2006.
15. In the United States, the number of deaths due to HIV/AIDS was estimated to be 20,000 in 1999 and 15,000 in 2001. Assuming an exponential model, predict the number of deaths from HIV/AIDS in the United States in the year 2006.
16. Look at the data from problem 14 and 15 then categorize each model you found as either exponential growth or exponential decay. Discuss your findings. The annual rate of death in the United States for AIDS patients in 1987 was 59 per 100, and a little over a decade later in 1998 this number was estimated to be 4 per 100 patients, would this support or refute your findings.
17. On one set (the same set) of axes, sketch both linear models as well as both exponential models. The first column represents the value of the input (x-variable) and all other columns represent the value of the out put (y-variable). Be careful when scaling the units on the axes.
| |Linear |Linear |Exponential |Exponential |
| |Model |Model |Model |Model |
|0 |10 |10 |10 |10 |
|1 |12 |8 |20 |5 |
|2 | | | | |
|3 | | | | |
18. You are filling up a bottle with water. As you fill it up, you double the amount of water every minute. It takes you ten minutes to fill it up.
a) Is this a linear model or exponential model?
b) When is the bottle half way full?
Local Rates of Change
Why do we need to study ‘rates of change’? “The price of unleaded is $1.99 today, this seems high, so what can I expect to pay tomorrow or next week?” For the American consumer, the most frequently viewed energy statistic is the retail price of gasoline. And while this average consumer probably has a fair understanding that gasoline prices are related to many factors, he or she more often than not has little idea of how those factors are connected to create the ripple effect that finally results in the price per gallon at the pump. Moreover, sizeable gasoline price changes do occur. Natural questions abound. Does the price depend upon petroleum refiners, petroleum marketers, the relationship between wholesale and retail distributors? How is the price per gallon affected by Middle East politics, like the war in Iraq or the removal of a dictator from power? When the price per gallon races skyward, are allegations of impropriety, like price gouging, fair? The answers to these questions require us to understand intricacies in the petroleum business, intricacies that are both complex and fluid over time. Most of us are not in the petroleum business and certainly don’t have the time to investigate these questions. But, when the average consumer drives up to the pump, they would like to have an understanding of why they are paying these prices, and equally as important, what could the price of gasoline be expected to be next week? And though this book will in no way explore the intricacies of the petroleum business, it will give you the tools to attack the latter desire. We can show you how to reasonably predict the price of gasoline, despite its wave of rises and falls that seems to appear almost weekly.
According to the California Energy Commission. The table below reflects the State of California’s Average Weekly Retail Gasoline Prices, per gallon, in 2004, and they are not adjusted for inflation.
|Date | Regular | | Premium |
| | |Mid-grade| |
|1990 |44,599 |1997 |42,013 |
|1991 |41,508 |1998 |41,501 |
|1992 |39,250 |1999 |41,717 |
|1993 |40,150 |2000 |41,945 |
|1994 |40,716 |2001 |42,196 |
|1995 |41,817 |2002 |42,815 |
|1996 |42,065 | | |
Source: According to the Source book of Criminal Justice Statistics Online:
The number of people who die each year in motor vehicle accidents in this country is given in the table to the left.
a) Sketch a scatter plot for the data presented in the data.
b) Find the actual change in fatalities for the years 1990 to 2002 and interpret.
c) Use the data to find the average rate of change for the years 1990 to 2002 and interpret.
d) Find the actual change in fatalities for the years 2000 to 2002 and interpret.
e) Use the data to find the average rate of change for the years 2000 to 2002 and interpret
f) Find the actual change in fatalities for the years 2001 to 2002 and interpret.
g) Use this information you found to find the predicted number of traffic fatalities for the year 2005 and interpret.
h) Interpret the data. When were the number of traffic fatalities increasing most rapidly? Least rapidly? When were the number of traffic fatalities decreasing most rapidly? Least rapidly?
Solution
a)
b) The actual change from 1990 to 2002 was 42,815 – 44,599 = - 1784. This means that there were 1,784 less people who died in traffic accidents in 2002 compared to 1990.
c) The average rate of change from 1990 to 2002 was (42,815 – 44,599)/12 = - 1784/12 = 148 2/3. This means that if there was a yearly trend to the loss of lives in the US from traffic accidents between 1990 to 2002, an average of 148 (or 149) less people were killed in traffic accidents from one year to the next during these years.
d) The actual change from 2000 to 2002 was 42,815 – 41,945 = 870. This means that there were 870 more people who died in traffic accidents in 2002 compared to 2000.
e) The average rate of change from 2000 to 2002 was (42,815 – 41,945)/2 = 870/2 = 435. This means that if there was a yearly trend to the loss of lives in the US from traffic accidents between 2000 to 2002, an average of 435 more people were killed in traffic accidents from one year to the next during those two years.
f) The actual change from 2001 to 2002 was 42,815 – 42,196 = 619. This means that there were 619 more people who died in traffic accidents in 2002 compared to 2001.
g) If we use the most recent data and the shortest time interval, we will assume there are 619 more people who will die each year in traffic accidents from 2002 to 2005. So, in 2002, there were 42,8145 people who died in traffic accidents in the U.S. So, 42,815 + 3(619) = 44,672 people. This means we means that if the current trend continues, if nothing is done to make the roads safer, there will be a predicted 44,672 people who will die in traffic accidents in 2005.
h) First we notice the overall trend from the scatter plot. There appears to be a high in the number of fatalities recorded for year 1990, then the number of fatalities appeared to drop in the early 90’s and then slowly rise there after. More specifically, we create a table, showing a third column, the number of more (or less) traffic fatalities in a given year compared to the previous year.
|Year |Fatalities |Difference |
|1990 |44,599 | |
|1991 |41,508 |-3091 |
|1992 |39,250 |-2258 |
|1993 |40,150 |900 |
|1994 |40,716 |566 |
|1995 |41,817 |1101 |
|1996 |42,065 |248 |
|1997 |42,013 |-52 |
|1998 |41,501 |-512 |
|1999 |41,717 |216 |
|2000 |41,945 |228 |
|2001 |42,196 |251 |
|2002 |42,815 |619 |
We can quickly see that the largest decrease in traffic fatalities in a given year compared to the previous year was in 1991. The smallest decrease was in traffic fatalities in a given year compared to the previous year was in 1997. Similarly, the largest increase occurred in 1995, the smallest increase occurred in 2000.
The concept touched upon in part (h) previews a concept termed by mathematicians as a change of concavity. Here, we simply give you a brief preview of the notion of increasing at various rates, a concept we will later define in terms of concavity. In the next section, we will define what it means to increase the fastest and to increase the slowest.
Exercise Set
For problems 1 – 5, use the table for the State of California’s Average Weekly Retail Gasoline Prices, per gallon, in 2004. Calculate the average rate of change in the price of a gallon of regular for the indicated dates. Write your answer in a complete sentence. For example, you would write “The price for a gallon of regular gas in the state of California increased at an average rate of 0.005 dollars per gallon per day from the 4th to the 19th day of June.”
1. from June 21st to July 12th
2. from June 28th to July 12th
3. from July 5th to July 12th
4. What is the best approximation for the Local Rate of Change for a gallon of regular gas on July 12th
5. Predict the price of a gallon of regular gas for July 19th based on the Local Rate of Change from problem 4 and compare your answer to the true price of a gallon of gas on July 19th from the table.
6. Use the table for the State of California’s Average Weekly Retail Gasoline Prices, per gallon, in 2004. Calculate the average rate of change in the price of a gallon of regular from January 5th, 2004 to September 20th , 2004. Would this average rate of change be a good or bad predictor of the price of gas for September 27th, 2004?
7. Which of the following expressions correctly calculates the average rate of change in the company’s revenue (sales), in million’s per item, from the sale of 1,040 items to 2,005 items?
a) [pic] b) [pic]
c) [pic] d) [pic]
e) [pic]
8. According to the CIA World Factbook, the Unemployment Rate (%) in United States was 4.2 in 1999, 4.0 in 2000, 5.8 in 2002 and 6.2 in 2003. Which of the following correctly interpretation of this data.
a) The Unemployment Rate dropped at an average rate of 0.5 (%) per year from 1999 to 2003.
b) The Unemployment Rate increased at an average rate of 0.5 (%) per year from 1999 to 2003.
c) The Unemployment Rate increased at an average rate of 2 (%) per year from 1999 to 2003.
d) The Unemployment Rate increased at 2 (%) per year from 1999 to 2003.
e) The Unemployment Rate dropped at 2 (%) per year from 1999 to 2003.
9. According to the CIA World Factbook, in the United States, there were 20,000 HIV/AIDS deaths in 1999 and 15,000 in 2001. Interpret this statement in terms of
a) average rate of change
b) actual change.
10. According to the CIA World Factbook, in the United States, there were 850,000 people living with HIV/AIDS in 1999. The number grew to 900,000 in 2001. Interpret this statement in terms of
a) average rate of change
b) actual change.
For problems 11 to 14, use the data: According to the CIA World Factbook, the Life Expectancy in the United States, at birth, in 2000 was 77.12 years, in 2001, was 77.26 years, in 2002 was 77.4 years, in 2003 was 77.14 years and in 2004 was 77.43 years.
11. Find the average rate of change of life expectancy from 2000 to 2004.
12. Find the actual change of life expectancy from 2000 to 2004.
13. Find the average rate of change of life expectancy from 2002 to 2003.
14. Find the actual change of life expectancy from 2002 to 2003.
For problems 15 to 18, use the following information: They say people are smoking less these days. We hope so. Cigarette smoking is a major cause of cancers of the lungs, and other organs, such as the esophagus, larynx and oral cavity. It contributes to development of cancers of the pancreas, kidney and the bladder. Lung cancer risk increases steadily with the number of cigarettes smoked per day, case in point, those who smoke 2 or more packs a day have nearly 20 times the risk of developing cancer as nonsmokers. Below is estimated data for the percent of 8th graders in the United States who regularly smoke cigarettes. SOURCE: Johnston, L.D., O'Malley, P.M., and Bachman, J.G. (2003). Monitoring the Future national survey results on drug use, 1975-2002 Volume I: Secondary School Students (NIH Publication No. 03-5375). Bethesda, MD: National Institute on Drug Abuse Tables 2-2 and 5-3. Data for 2003 are from a press release of December 19, 2003, and demographic disaggregations are from unpublished tabulations from Monitoring the Future, University of Michigan. The respondents were considered regular smokers if they reported smoking cigarettes daily in the previous 30 days of when the poll was taken, by selected years 1992-2003
1992 – 7.0 1993 – 8.3
1994 – 8.8 1995 – 9.3
1996 - 10.4 1997 – 9.0
1998 - 8.8 1999 – 8.1
2000 – 7.4 2001 - 5.5
2002 - 5.1 2003 - 4.5
15. Find the actual change for the percent of 8th graders who smoke regularly from 1992 to 2003 and interpret.
16. Find the average rate of change for the percent of 8th graders who smoke regularly from 1992 to 2003. Use this rate of change to predict the percent of 8th graders who would smoke in 2005 and interpret.
17. Find the average rate of change for the percent of 8th graders who smoke regularly from 2002 to 2003. Use this “short term” rate of change predict the percent of 8th graders who would smoke in 2005 and interpret.
18. Interpret the data. When were the percent of 8th graders in the United States who regularly smoke cigarettes increasing most rapidly? Least rapidly? When were the number of percent of 8th graders in the United States who regularly smoke cigarettes decreasing most rapidly? Least rapidly?
For problems 19 to 22, use the following information: Now, let’s re-ask the same question as we asked in the problem set above for the older teenagers, in other words, let’s see if the 12th graders smoking less too. Below is estimated data for the percent of 12th graders in the United States who regularly smoke cigarettes. SOURCE: Johnston, L.D., O'Malley, P.M., and Bachman, J.G. (2003). Monitoring the Future national survey results on drug use, 1975-2002 Volume I: Secondary School Students (NIH Publication No. 03-5375). Bethesda, MD: National Institute on Drug Abuse Tables 2-2 and 5-3. Data for 2003 are from a press release of December 19, 2003, and demographic disaggregations are from unpublished tabulations from Monitoring the Future, University of Michigan. The respondents were considered regular smokers if they reported smoking cigarettes daily in the previous 30 days of when the poll was taken, by selected years 1992-2003
1992 – 17.2 1993 – 19.0
1994 – 19.4 1995 – 21.6
1996 – 22.2 1997 - 24.6
1998 – 22.4 1999 – 23.1
2000 – 20.6 2001 – 19.0
2002 – 16.9 2003 – 15.8
19. Find the actual change for the percent of 12th graders who smoke regularly from 1992 to 2003 and interpret.
20. Find the average rate of change for the percent of 12th graders who smoke regularly from 1992 to 2003. Use this rate of change to predict the percent of 12th graders who would smoke in 2005 and interpret.
21. Find the average rate of change for the percent of 12th graders who smoke regularly from 2002 to 2003. Use this “short term” rate of change predict the percent of 12th graders who would smoke in 2005 and interpret.
22. Interpret the data. When were the percent of 12th graders in the United States who regularly smoke cigarettes increasing most rapidly? Least rapidly? When were the number of percent of 12th graders in the United States who regularly smoke cigarettes decreasing most rapidly? Least rapidly?
For problems 23 to 30, use the following data. Source: According to the Source book of Criminal Justice Statistics Online: the total fatalities in alcohol related motor vehicle crashes per year in the United States.
|Year |alcohol related |
|1990 |22,587 |
|1991 |20,159 |
|1992 |18,290 |
|1993 |17,908 |
|1994 |17,308 |
|1995 |17,732 |
|1996 |17,749 |
|1997 |16,711 |
|1998 |16,673 |
|1999 |16,572 |
|2000 |17,380 |
|2001 |17,400 |
|2002 |17,419 |
23. Sketch a scatter plot for the data presented in the data.
24. Find the actual change in alcohol related traffic fatalities for the years 1990 to 2002 and interpret.
25. Use the data to find the average rate of change in alcohol related traffic fatalities for the years 1990 to 2002 and interpret.
26. Find the actual change alcohol in related traffic fatalities for the years 2000 to 2002 and interpret and interpret.
27. Use the data to find the average rate of change in alcohol related traffic fatalities for the years 2000 to 2002 and interpret.
28. Find the actual change in alcohol related traffic fatalities for the years 2001 to 2002 and interpret.
29. Use this information you found to find the predicted number of alcohol related traffic fatalities for the year 2005 and interpret.
30. Interpret the data. When were the number of alcohol related traffic fatalities increasing most rapidly? Least rapidly? When were the number of alcohol related traffic fatalities decreasing most rapidly? Least rapidly?
World Population and Sustainability
|Year |Mid-Year Population |
|1950 |2,555,360,972 |
|1960 |3,039,669,330 |
|1970 |3,708,067,105 |
|1980 |4,454,389,519 |
|1990 |5,284,679,123 |
|1990 |5,284,679,123 |
|2000 |6,085,478,778 |
|2010 |6,812,248,283 |
|2020 |7,510,699,958 |
|2030 |8,111,883,766 |
|2040 |8,623,136,543 |
|2050 |9,050,494,208 |
[pic]
And now we have come full circle in our conversation of growth rates and rates of change, back to population. The world population is growing exponentially. Above is a table showing the world’s population for a century, from 1950 to 2000 and projected until 2050. As we did earlier in the chapter, we can project the world’s population from 2010 onward using an growth rate with the common ratio or base of 1.01 (or more precisely, 1.013). If we graph the world’s population over this same 100 year span, we have a nice exponential model.
Obviously the earth cannot continue to endure population growth at the current rate without it letting up. How many people can our fragile planet support? Ecologists have often made use of the concept of carrying capacity in tackling the stress and strain that populations put on the environment. Carrying capacity is simply the largest number of any given species that a habitat can support indefinitely.
There are many factors that influence carrying capacity of the earth. Ecologists look at net primary productivity (NPP), which measure the amount of solar energy converted into other forms of energy by plant photosynthesis minus the energy needed by those plants. They say it serves as a representation of the total food resource on earth. By deforestation and destruction of vegetation, humans have destroyed about 12% of the terrestrial NPP, and now use an additional 27%.
Though estimates vary, all our efforts have already resulted in the erosion of 40-50 % of the Earth's land surface through agriculture and urbanization, we use more than half of the accessible fresh water and 8% of the ocean’s productivity, we have increased atmospheric CO2 concentration by 30% and we are responsible for the extinction of about 20% of bird species in the past 200 years as well as overexploited about 22% of marine fisheries.
Many ecologist and global organizations estimate the world’s population will rise to somewhere between 9 and 12 billion people by 2050 and then stabilize. But, this notion itself carries many problems. The prediction would make sense if the remainder of life on earth remained stable. But, in order to sustain a human population of 6 billion, it is estimated we are currently losing roughly 70,000 species a year. One could argue that we are in a era of mass extinction with the human population chiefly responsible. Common sense should prevail; a human population of just 6 billion is not even sustainable because the world’s wildlife cannot indefinitely sustain a loss of 70,000 species a year. And as our population increases, won’t the number of extinctions? Doesn’t our survival depend on the earth’s survival? Our population might become stable somewhere between 9 and 12 billion, but that does not mean the world’s ecosystem’s would be stable too.
|Year |Mid-Year Pop |
|-8000 BC |5,000,000 |
|-2000 BC |27,000,000 |
|1 AD |170,000,000 |
|1000 |254,000,000 |
|1500 |425,000,000 |
|1900 |1,550,000,000 |
|2000 |6,085,478,778 |
|2010 |6,812,248,283 |
|2020 |7,510,699,958 |
|2030 |8,111,883,766 |
|2040 |8,623,136,543 |
| 2050 |9,050,494,208 |
[pic]
Now, let’s re-examine the world’s population from a historical prospective. Below is a rough estimate of the world’ population from – 8000 BC to 2000 and then projected out until 2050.
Roughly speaking, the population of the world took 1,650 years (from Year 0 AD to year 1650 AD) to double from ¼ billion people to ½ billion people. It took only 200 years (from 1650 to 1850) to double again from ½ billion people to 1 billion. It took only 80 years (from 1850 to 1930) to double again from 1 to 2 billion and just 45 years (from 1930 to 1975) to double again from 2 to 4 billion.
The graph to follow shows this phenomenon of growth with striking clarity. If we graph the historical estimates of the world population over a time line that spans – 8000 BC to 2050 AD, we find a glaring demonstration of the sheer power of exponential growth. In short, the graph appears rather steep in recent years. But, for populations, exponential growth is usually only good for ‘short term’ models. Populations cannot grow at an increasing rate forever.
Clearly, the implication of the graph is that population growth is seemingly unlimited and this phenomenon is associated with exponential functions. But, we know the environment is not unlimited in the resources it can provide. As we know, populations in actuality do not increase forever because eventually, a combination of factors will hold down population growth. Usually, this ‘hold’ comes in one of two ways, either the population stabilizes and approaches a natural limit or the population overgrows the capacity of the environment and then collapses, spiraling toward extinction.
Logistic Curves
Typically, at the beginning, the growth of a population is usually exponential, increasing at an increasing rate, close to an exponential model based on a common ratio. At some point though, we would expect the rate of increase to begin to slow, and the population should plateau. Then the population will probably still increase, but this increase will be at a slower rate. In mathematical terms, the population will grow exponentially at an increasing rate, then continue to grow but at a decreasing rate. The point in time where the rate of change changes from an increasing rate to a decreasing rate is called the “point of inflection.” At that point, the concavity of the graph changes from concave up to concave down; the population begins to stabilize. The graph itself takes on the appearance of an elongated S. Will this S-shaped model, called a logistic model, occur with our world’s population?
[pic]
The S-shaped or logistic model differs from the exponential curve in two ways:
1. It has an upper limit (asymptote) and this is where population stabilizes.
2. The population increases at an increasing rate at first, but then increases at a decreasing rate until the stabilizing population is reached.
Population Collapse
Some populations, though, behave more catastrophically. A second possibility is that a population will continue to grow exponentially at an increasing rate, where upon it reaches a critical value. At this time the population cannot survive since it can not sustain itself, so its collapses. Could our world population be destined for such a collapse, and why would the world population possibly do this? Maybe we as humans have wreaked so much havoc on the environment, the global ecosystem can not support our growth. Famine. Lack of water. Lack of quality air. Food sources gone due to extinction. Water supplies saturated or polluted. Deforestation depletes the action of photosynthesis depleting breathable air. Energy supplies over exploited. You pick the reason. But, populations that collapse this way do so because their exponential growth occurs at an shocking rate and as a result, the population collapses. Quickly.
[pic]
DEER POPULATION
Source: Missouri Department of Conservation: ONLINE:
The Missouri Department of Conservation approximated the number of deer in thousands for a twenty-two year span for a particular habitat. The table below gives the approximate cumulative deer population in thousands for two-year intervals. We in turn created a third column, with the calculated yearly rate of change for each two-year interval.
|Year |Pop in |Average Yearly Rate |
| |thousands |of Change |
|0 |10 | |
|2 |14 |2 |
|4 |20 |3 |
|6 |35 |7.5 |
|8 |60 |12.5 |
|10 |90 |15 |
|12 |110 |20 |
|14 |127 |8.5 |
|16 |140 |6.5 |
|18 |152 |6 |
|20 |158 |3 |
|22 |162 |2 |
[pic], [pic]
Note that the deer population increases throughout the twenty-two year span. This is indicated as one glances at the numbers in the second column, they increase steadily. The third column represents how fast the deer population is increasing. These numbers increase until year 12, at which point the numbers in the third column then begin to decrease. This means, that while the deer population is increasing for all twenty-two years, the rate (how fast) the population increases has changed. The deer population increased at an increasing rate until year 12. At year 12, the rate of increase began to drop, or decrease. The habitat was seeing more deer but at a slower rate of increase. This is how most populations behave naturally, without outside influences greatly enhancing or diminishing their growth. No population will increase at an increasing rate indefinitely.
At some point, a population’s growth levels off, stabilizes. This is called the carrying capacity of the population. If the carrying capacity is exceeded, often populations drop drastically. Extinction sometimes follows.
Below is a scatter plot for the table above. The curve you see is called a logistic growth curve or S-shaped curve of growth. Actual populations do follow the basic logistic growth curve below, and so this curve can provide us with valuable guidance in managing populations.
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Where the curve is increasing at an increasing rate, we call the curve concave up and where the curve is increasing at a decreasing rate, we call the curve concave down. This change in rate for a curve that is increasing (or decreasing) is called a change in concavity. The point on the graph where this change occurs is called the inflection point.
Let’s re-explore these terms with our deer population. From both the table and the graph above, we see the deer population is always increasing, from Year 0 to Year 22. From the table, we see the average rate of change increase from year 0 to Year 12 and on the graph, we observe the increasing population at an increasing rate (a steeper graph as one reads the graph from left to right). Over these first 12 years, we see the graph is concave up. At Year 12, the behavior of the graph changes. While the population continues to grow after year 12, it grows at a slower rate, on the table we see the average rate of change drop. It is increasing at a decreasing rate from Year 12 to Year 22 and on the graph we see a flattening of the curve for these years. This part of the curve is concave down. The 12th year, where this change occurred is where the point of inflection occurs.
You may ask, what happens after the inflection point. Many options could occur to explain this behavior on the graph. For our deer population, we could speculate that if the deer population in this Missouri habitat were to grow past the inflection point, the habitat would experience heavy grazing. That in turn could destroy the habitat. As a result, the physical condition of the deer would deteriorate and then a lower reproductive rate could follow. Ultimately, growth rates and deer numbers stabilize at some lower concentration. An associated impact is that when deer numbers exceed the inflection point, the habitat destruction affects other species. And a habitat’s recovery is notoriously slow. Sometimes hunters will make this argument to justify curtailing the deer population.
Many factors other than the scenario above certainly could cause the decreasing rate of change and thus the occurrence of the inflection point on the graph of a deer population. Some species will undergo a natural slowing down of their increasing population, call it Mother’s natures way of handling the growth rate. Still other species experience a slow down of their rate of increasing growth because of external forces. Contributors from pollution to pesticides to over hunting, as well as other forms of human impact can cause such a change. Introduction of new or perhaps just more predators into the habitat or introduction of a new or more species into the habitat that compete for the deer’s food supplies could change the deer population’s rate of growth. Environmental calamities like diseases, draught or famine do not always cause a population to decrease in size, often they simply contribute to the population’s slower growth rate. But, if the deer are lucky, the population will rise into the asymptotic or stabilizing behavior as seen earlier, as opposed to a catastrophic collapse.
Exercise Set
Use this information for problems 1 – 3. Experts in population theory disagree as to what the largest sustainable population could be globally. The experts predictions range as low as 6 billion and as high as 12 billion. Generally, 9 to 10 billion tend to be the most agreed upon maximum sustainable population.
1. If we assume the largest world’s sustainable population to be 9 ½ billion, at the present grow rate, assume a logistic model and build a table with predictions for the global population every five years from 2005 to 2050, roughly the year the population will reach 9 ½ billion. Account for the fact that soon the population’s growth rate will have to increase at a decreasing rate.
2. What factors may influence the world’s population if the collapse model takes into effect.
3. Predict the world’s population for the year 3000. How do you process this in the light of the sustainability predictions stated above.
For problems 4 to 8, use the table below that was adopted from the NLREG Homepage: Source: NLREG Home Page: The table below reveals the number of new cases of AIDS in the United States by year, starting 1981.
|Year |Total Number |
| |Aids Cases |
|1981 |500 |
|1982 |750 |
|1983 |1500 |
|1984 |3500 |
|1985 |7000 |
|1986 |12,500 |
|1987 |20,500 |
|1988 |31,500 |
|1989 |34,000 |
|1990 |42,000 |
|1991 |41,500 |
|1992 |46,000 |
|1993 |46,500 |
|1994 |47,000 |
|1995 |47,500 |
4. Create a third column and find the average rates of change for the number of cases of AIDS per year.
5. Sketch a scatter plot for the data. Label the horizontal axis the Year, starting with 1981 and the vertical axis the number of new AIDS cases.
6. Find the years where the graph is concave up and write a sentence that pertains to the real world situation describing what it means for the graph to be concave up.
7. Find where years where the graph is concave down and write a sentence that pertains to the real world situation describing what it means for the graph to be concave down.
8. Approximate which year contains the inflection point. Write a sentence that pertains to the real world situation describing what is happening during this year. Specifically, why do you think the inflection point occurred during that year? What may have contributed in the United States that year that lead the data to change it rate of growth.
One last note: By the end of 2003, an estimated 35.7 million adults and 2.1 million children younger than 15 years were living with this HIV/AIDS virus. Approximately two-thirds of these people live in the Sub-Saharan Africa. More than 20 million have died people with HIV/AIDS have died since 1981.Source:
For problems 9 - 13, used the data from the table below. Source The Otter Project
{1032ABCB-19F9-4CB6-8364-2F74F73B3013}/PopulationAverage04.gif
California Sea Otter Population The first column gives the number of years since 1999, the second column gives the California Sea Otter’s population. Note: The actual population of the sea otter is given until 2003. The predicted population of the sea otter was made by the authors, based on a logistic model for population growth.
|0 |2150 |
|1 |2160 |
|2 |2190 |
|3 |2240 |
|4 |2300 |
|5 |2500 |
|6 |2600 |
|7 |2650 |
|8 |2680 |
|9 |2700 |
|10 |2705 |
9. Create a third column and find the average rates of change for the sea otter population.
10. Sketch a scatter plot for the data. Label the horizontal axis the Number of years since 1999 and the vertical axis the number of California Sea Otters.
11. Find the years where graph is concave up and write a sentence that pertains to the real world situation describing what is means to be concave up.
12. Find where years where the graph is concave down and write a sentence that pertains to the real world situation describing what is means to be concave down.
13. Approximate which year contains the inflection point. Write a sentence that pertains to the real world situation describing what is happening during this year.
For questions 14-18 use the data presented below.
|Year |Population |
|1980 |2500 |
|1983 |2700 |
|1986 |3100 |
|1989 |3600 |
|1992 |4500 |
|1995 |5500 |
|1998 |6700 |
|2001 |8000 |
|2004 |9500 |
The table represents population every three years for a gander of seagulls that inhabit an island.
14. Decide if the model is logistic or not. If is not, only answer the questions below that would apply.
15. Sketch the scatter plot. If it has an inflection point, estimate the year pertaining to the point.
16. Find the years where graph is concave up and write a sentence that pertains to the real world situation describing what is means to be concave up.
17. Find where years where the graph is concave down and write a sentence that pertains to the real world situation describing what is means to be concave down.
18. Approximate which year is at the inflection point. Write a sentence that pertains to the real world situation describing what is happening during this year.
For questions 19-23 use the data presented below
|Year |Population |
|1980 |2500 |
|1983 |2480 |
|1986 |2450 |
|1989 |2410 |
|1992 |2300 |
|1995 |2200 |
|1998 |2150 |
|2001 |2120 |
|2004 |2100 |
The table above represents population every three years for a gander of seagulls that inhabit an island.
19. Decide if the model is logistic or not. If is not, only answer the questions below that would apply.
20. Sketch the scatter plot. If it has an inflection point, estimate the year pertaining to the point.
21. Find the years where graph is concave up and write a sentence that pertains to the real world situation describing what is means to be concave up.
22. Find where years where the graph is concave down and write a sentence that pertains to the real world situation describing what is means to be concave down.
23. Approximate which year is at the inflection point. Write a sentence that pertains to the real world situation describing what is happening during this year.
The table below represents the population every three years for a gander of seagulls that inhabit an island.
|Year |Population |
|1980 |2500 |
|1983 |2600 |
|1986 |2800 |
|1989 |3500 |
|1992 |4500 |
|1995 |4800 |
|1998 |4900 |
|2001 |4950 |
|2004 |4955 |
24. Decide if the data represents a logistic growth pattern. If not, only answer the questions below that would apply.
25. Sketch the scatter plot. If it has an inflection point, estimate the year pertaining to the point.
26. Find the years where graph is concave up and write a sentence that pertains to the real world situation describing what is means to be concave up.
27. Find the years where the graph is concave down and write a sentence that pertains to the real world situation describing what is means to be concave down.
28. Approximate which year is at the inflection point. Write a sentence that pertains to the real world situation describing what is happening during this year.
For problems 29 to 34, write TRUE or FALSE, as it pertains. If it is false, correct the statement so that it is true.
29. An exponential growth model’s graph never has an inflection point.
30. A logistic growth model’s graph always has an inflection point.
31. The average yearly rate of change for a population that is growing exponentially is always increasing from year to year.
32. If a population grows according to a logistic model, it’s rate of growth will increase until it reaches the inflection point, at then it will decrease.
33. If a population declines according to a logistic model, it’s rate of growth will decrease until it reaches the inflection point, at then it will increase.
34. You would prefer your salary to be linear adjustments to grow linearly, not exponentially.
35. Population A increases forever linearly at a constant rate of 1,000,000 new creatures per year and population B increases exponentially at a growth rate of 0.00001 % per year. If the two populations are the same size this year, Population A will always be larger than Population B from next year forward.
36. An economist writes “there is every indication that the world population is currently following a logistic model.” This economist is looking at the data we provided for you in this text. Argue the economist position, you pick pro or con.
37. Can any population grow exponentially forever? Why or why not?
Concavity
Imagine an unusual shaped bottle as it is being filled with water. The water is being poured into the bottle at a constant rate. Below is a scatter plot representing the height of the water inside the bottle in cm versus the time in seconds.
Notice as time goes on, the water level rises, but it does so at a varying rate. From 0 seconds to 4 seconds, it seems that the water level height is increasing at a faster rate. This means the height of the water level from 0 to 4 seconds increases more with each passing second. Each second, the bottle’s increase in height of the water was greater than for the second before. In other words, from 1 to 2 seconds, the water level height rises 10 cm, from 2 to3 seconds, the water level height rises 15 cm. At 4 seconds, though, this trend changes. Though the water level continues to rise, it does so at a slower rate. For each subsequent second, the water level still rises, but its increase is smaller than it was for the second before. So, from 4 to 5 seconds, it rose 10 cm, from 5 to 6 seconds, it rose just 4 cm.
We do not want you to infer all growth rates are either linear or exponential. In fact, not all growth rates are linear or exponential. We next view the same event from two mathematical perspectives. For this example, we will be filling a container with a steady stream of water. The two parameters we are observing are the increase in height over time and the increase in volume over time. In all cases, we will fill the containers at a constant rate. We will discuss a linear growth rate and see how the effect of this constant rate of change on one parameter will reflect on a non-linear growth rate for another parameter.
If we were to sketch the graph representing the increase in volume with respect to time, this graph represents linear growth. However the parameter of the height of the water level does not necessarily increase at constant rate. What would cause the rate of the water to increase at an increasing rate? Decreasing rate? Steady rate?
Let’s start with a cylinder. Imagine water being poured into the cylinder, like a soda can, at a constant rate. What does filling at a constant rate now mean? This tells us the same amount of water is going into the cylinder every second. The volume increases at a constant rate (Plot 1). Because all cross sections of the cylinder are identical, the rise in water level is also occurring at a constant rate. Therefore, if we were to sketch a graph representing the increase in height over time, this graph would also be linear in nature (Plot 2).
Plot 1. Plot 2.
[pic][pic]
Now let’s change the container to a child’s sand bucket. If you were to chart or measure the increase in volume versus time, this graph is again best described as being linear (Plot 3). When we focus our attention to the height with respect to time graph, this picture will not be linear. As we examine cross sections as we move up the bucket, each cross section is a circle whose radius is increasing. Now each cross section, given a little thickness will require more water to fill. If we were to chart this, this gives us a graph with the characteristic of increasing at a decreasing rate, concave down.
Plot 3. Plot 4.
[pic] [pic]
Next, consider a fish bowl. Round in shape, somewhat resembling a sphere. If we fill the fish bowl at a constant rate, the volume will increase at a constant rate. But imagine the rate at which the height of the water is increasing. Starting at the base of the fish bowl and moving up to the half way point, each cross section gets larger and larger, therefore, the height at which the water level is increasing is occurring at a decreasing rate. Once the water level reaches the mid point this characteristic changes. Each consecutive cross section of the fish bowl from the mid level to the top is a smaller circle. The graph representing the changing height over the second half of the filling process would be represented by a curve that is concave up. The rate of change for the height is described as increasing at an increasing rate.
Test yourself. Envision filling an hour glass completely with water. If you are filling this container with water at a constant rate, volume with respect to time will be a linear growth. The graph representing an increase in volume is simply a line. But, what about the graph of the height of the water level over time, can you do this?
Now, let us come full circle. If we are observing a linear growth rate, what is the concavity? When asking about concavity, we are basically asking is the rate of increase increasing or decreasing. Or is the rate of decrease increasing or decreasing? Well, for a linear growth rate, which is it? A linear growth rate has a constant rate of change, thus it is either increasing or decreasing at the same rate. What would its concavity be? Zero.
Exercise Set
For problems 1 to 7, imagine a unusual shaped bottle filled or drained of water. Below is a scatter plot that plots the height of the bottle in cm versus the time in seconds. For each scatter plot, discuss where the function (in terms of seconds) has a rate of change that is increasing at an increasing rate, a rate of change that is increasing at a decreasing rate of change, a rate of change that is decreasing at an increasing rate and a rate of change that is decreasing at a decreasing rate.
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]
7. [pic]
For problems 8 to 10, use the following scenario: a rocky road ice cream cone is filled up. So, imagine a cone and a semi-sphere for shapes. A child is eating it at a constant rate.
8. Draw a graph plotting volume versus time. Label it “linear” or “non-linear”
9. Draw a graph plotting height versus time. Label it “linear” or “non-linear”
10. Discuss the change of concavity in a physical sense. Be precise. Where in the scenario is the graph changing concavity.
For problems 11 to 13, use the following scenario: a soda can is filled to the top. So, imagine a cylinder that is filled up. A child sucks soda through a straw at a constant rate and does not stop until the can is empty.
11. Draw a graph plotting volume versus time. Label it “linear” or “non-linear”
12. Draw a graph plotting height versus time. Label it “linear” or “non-linear”
13. Discuss the change of concavity in a physical sense. Be precise. Where in the scenario is the graph changing concavity.
For problems 14 to 16, use the following scenario: an upside down pyramid is being filled with sand at a constant rate.
14. Draw a graph plotting volume versus time. Label it “linear” or “non-linear”
15. Draw a graph plotting height versus
time. Label it “linear” or “non-linear”
16. Discuss the change of concavity in a physical sense. Be precise. Where in the scenario is the graph changing concavity.
For problems 17 to 19, use the following scenario: a right side up pyramid being filled with sand at a constant rate.
17. Draw a graph plotting volume versus time. Label it “linear” or “non-linear”
18. Draw a graph plotting height versus time. Label it “linear” or “non-linear”
19. Discuss the change of concavity in a physical sense. Be precise. Where in the scenario is the graph changing concavity.
For problems 20 to 22, use the following scenario: an hourglass is being filled with sand that is poured at a constant rate.
20. Draw a graph plotting volume versus time. Label it “linear” or “non-linear”
21. Draw a graph plotting height versus time. Label it “linear” or “non-linear”
22. Discuss the change of concavity in a physical sense. Be precise. Where in the scenario is the graph changing concavity.
For problems 23 to 25, use the following scenario: a wine glass is being filled with wine at a constant rate.
23. Draw a graph plotting volume versus time. Label it “linear” or “non-linear”
24. Draw a graph plotting height versus time. Label it “linear” or “non-linear”
25. Discuss the change of concavity in a physical sense. Be precise. Where in the scenario is the graph changing concavity.
For problems 26 to 28, use the following scenario: a filled wine glass is drank with the aid of a straw at a constant rate.
26. Draw a graph plotting volume versus time. Label it “linear” or “non-linear”
27. Draw a graph plotting height versus time. Label it “linear” or “non-linear”
28. Discuss the change of concavity in a physical sense. Be precise. Where in the scenario is the graph changing concavity.
29. Earlier, we looked at the following data. In the United States, the data below gives the number of live births to unmarried women per 1000 woman that are of ages 20-24 years old for the years 1980 to 2000. In 1980, 40.9 out of every 1000 live births were to unmarried women. The number rose to 46.5 in 1985, rose again to 65.1 in 1990, rose to 68.7 in 1995, rose again to 70.7 in 199 9 rose to 72.1 in 2000. Note, the difference in successive five year intervals are not constant. Therefore, we will not be prone to use a linear model. Also notice the number of live births per 1000 woman is increasing through out the 20 year period Sketch a scatter plot of the data and then state reasons why the logistic curve you see exists. These reasons should explain the perceived ‘glass ceiling’ you see in the scatter plot. These reasons we assume should be societal. SOURCE: Johnston, L.D., O'Malley, P.M., and Bachman, J.G. (2003). Monitoring the Future national survey results on drug use, 1975-2002 Volume I: Secondary School Students (NIH Publication No. 03-5375). Bethesda, MD: National Institute on Drug Abuse Tables 2-2 and 5-3. Data for 2003 are from a press release of December 19, 2003, and demographic disaggregations are from unpublished tabulations from Monitoring the Future, University of Michigan.
For problems 30 to 35, create a table with two columns, one for the dependent variable, one for the independent variable that when plotted, shows the indicated behavior.
30. concave up
31. concave down
32. increasing at an increasing rate
33. increasing at a decreasing rate
34. decreasing at an increasing rate
35. decreasing at a decreasing rate
Media
One last note on the topic of “changing rates of change”. When the newspaper headlines report such tales such as the Congress has cut the defense budget, often this is a smoke screen in the form of well selected words. The headlines may well just mean the rate at which the defense budget was increasing was cut. In other words, the budget was still increasing, just at a slower rate. Statements such as “the rate of the spread of AIDS is on the decline” or “the rate of growing casualties is now decreasing” need to be understood for what they are; they are statements referring to an increasing phenomena that is now increasing, but at a slower rate. Most situations have fluctuating ratios of change; it is normal for many different populations or parameters to increase at decreasing rates or decrease at an increasing rate.
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