Introduction and Preview

Calculus is all about change. Calculus provides the mathematical tools to examine important questions about dynamic behavior; e.g. how fast is the world population increasing? If we continuously release a pollutant into a lake at a known rate, what's the total amount of pollutant that will be dumped into the water in the next five years? How long will the nonrenewable supplies of coal and oil last if we maintain the current per capita use but population continues to grow? How long will supplies last if industrialization and a "rising standard of living" push per capita usage ever higher? If we disturb the natural population dynamics of a salmon species by fishing, how should we regulate our removal of fish to provide a maximal sustainable yield in the future? Although we will study many different applications of calculus in this introductory course, we will place a particular emphasis on the problem of sustainability. Sustainability means meeting the needs of the present without compromising the ability of future generations to meet their own needs. Our definition of sustainability comes from the 1987 publication Our Common Future by the United Nations World Commission on Environment and Development (also known as the Brundtland commission after its chair, Norwegian diplomat Gro Harlem Brundtland) We will highlight two different approaches to using calculus to investigate sustainability questions. Here we will briefly describe each approach and list some types of questions Calculus will help us answer. Approach I: Start with Data

Calculus works best if we have a specific formula in functional form for a particular relationship. Once we have a formula, we can compute rates of change or measures of accumulation using, respectively, derivatives and integrals, the two basic tools of calculus. What we often have in the real world, however, is data, not formulas. In the first approach, we explore how we might get formulas that do a good job of representing observed data. We will import data in the program Excel and ask the program to show us the best linear, quadratic, cubic or higher power polynomial formulas that represent that data. We can also examine the best logarithmic and exponential approximating the data. As an example, the table below shows the number of metric tons of Carbon Dioxide (C02) released into the atmosphere by the United States and China each year for the 25-year period beginning in 1980. The data comes from the Energy Information Administration of the U. S. Department of Energy (http://tonto.eia.doe.gov/country/index) Year| Years after 1980| United States| China|

1980| 0| 17407.88| 5333.73|

1981| 1| 16961.33| 5209.81|

1982| 2| 16070.15| 5459.92|

1983| 3| 15935.8| 5782.81|

1984| 4| 16834.22| 6262.33|

1985| 5| 16784.45| 6741.07|

1986| 6| 16822.04| 7148.84|

1987| 7| 17417.09| 7627.39|

1988| 8| 18220.6| 8126.43|

1989| 9| 18555.14| 8220.55|

1990| 10| 18339.66| 8217.71|

1991| 11| 18161.26| 8600.78|

1992| 12| 18544.67| 8879.84|

1993| 13| 18906.28| 9484.89|

1994| 14| 19190.74| 10257.86|

1995| 15| 19393.95| 10430.05|

1996| 16| 20079.87| 10553.59|

1997| 17| 20329.25| 11252.8|

1998| 18| 20465.27| 10884.43|

1999| 19| 20728.56| 10749.34|

2000| 20| 21352.76| 10679.49|

2001| 21| 20984.79| 11186.57|

2002| 22| 21131.52| 12379.22|

2003| 23| 21313.65| 14604.76|

2004| 24| 21763.38| 17428.87|

2005| 25| 21842.26| 19516.53|

Our first job is to graph the data to see if we can see any trends from a visual inspection of the results. The graphs of CO2 emissions look like this:

What similarities do the emissions of the U.S. and China share? Are there qualitative differences in the trends of the two nations?...