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cept was introduced with activities using a base two exponential function, that is, doubling the pop ulation using the folding paper activity and the grains of wheat on a chessboard activity. We used the exponential model to explore bacteria in a bottle and population growth. When consider ing populations, the “ cause ” is represented by the per - capita reproduction rate, and the “ effect ” was represented by the total population at any given time. From this point of view, the emphasis turned to the main thrust of the article, namely, to quanti fy the nature of consumption of finite energy re sources, and how a variety of simple models can reveal the fallacy of maintaining sustainable growth based on dwindling finite resources. The exponential function models showed the urgency we all should have to address renewable resources. Time is running out, and it is running out quickly when we recognize our location on the exponential graph. We encourage mathematics teachers at all levels to include a mathematics lesson that chal lenges the notion that fossil fuels are unlimited. It is a great way to show exponential functions are useful and currently relevant. Appendix: Proof of the result (2b) for continu ous growth

Bartlett, A. A., (1978). The Exponential Function – Part VI, The Physics Teacher 16, 23 – 24. Bartlett, A. A., (1978). The Exponential Function – Part VII, The Physics Teacher 16, 92 – 93. Bartlett, A. A., (1978). The Exponential Function – Part VIII, The Physics Teacher 16, 158 – 159. The Exponential Function – Part IX, The Physics Teacher 17, 23 – 24. Bartlett, A. A., (1990). A World Full of Oil: The Exponential Function – Part X, The Physics Teacher 28, 540 – 541. Bartlett, A. A., (1996). The Exponential Function – Part XI, The New Flat Earth Society, The Physics Teacher 34, 342 – 343. Bartlett, A.A., Fuller, R.G., Plano Clark, V.L. & Rogers, J.A. (2004). The Essential Expo nential! For the Future of Our Planet. Cen ter for Science, Mathematics and Computer Education, University of Nebraska – Lin coln. Gören, G. (2011). Quantify! A Crash Course in Smart Thinking . (p.90, p.205.) The Johns Hopkins University Press. Martin, W., (2011). Mathematics for the Environ ment. CRC Press. Winicur, D.H., More on Coal Reserves, Physics Today 30( 5 ), May 1977. Bartlett, A. A., (1979).

Suppose that the quantity C ( t ) = C 0

doubles every time interval τ i.e.,

Then

References

Bartlett, A. A., (1976). The Exponential Function – Part I, The Physics Teacher 14, 393 – 401. Bartlett, A. A., (1976). The Exponential Function – Part II, The Physics Teacher 14, 485, 518. Bartlett, A. A., (1977). The Exponential Function – Part III, The Physics Teacher 15, 37, 38, 62. Bartlett, A. A., (1977). The Exponential Function – Part IV, The Physics Teacher 15, 98. Bartlett, A. A., (1977). The Exponential Function – Part V, The Physics Teacher 15, 225 – 226.

John Adam Professor Old Dominion University jadam@odu.edu

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