Sustainability and Changing Rates of Change

Christopher Coughenour, The Evergreen State College
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In this module, students explore variable rates of change and applications to sustainability. The first component of this module consists of two primers on rates of change for students and instructors to work through. The first primer discusses basic concepts related to quantifying rates of change. Another primer is provided that outlines the approach to mathematically modeling rates of change, particularly for non-renewable resources. Exponential growth and the logistic model of growth are presented, with two examples provided for students to work through. These primers could be given as individual learning assignments or could be integrated into a classroom discussion of rates of change (and even derivatives) with the questions from the primers given as homework assignments. These are appropriate for undergraduates of any level who have proficiency in algebra and modest experience with spreadsheet software.

The second component of this module provides two exercises related to crude oil production rates in order to explore relevant sustainability issues. The only prerequisite needed to complete these exercises is an understanding of basic rates of change concepts, the logistic growth model, and spreadsheet proficiency at the level needed to complete the primers. Separate roblem sets are provided for students to analyze petroleum production rates first in the United States and then globally. These exercises guide students through the process of fitting a mathematical model to historical oil data and then critically analyzing the validity of such a model and its possible utility in predicting future oil production rates.

All exercises are designed so that students who have not yet taken calculus may work through them and, thus, the 'discrete' case of the derivative (differences method) in analyzing rates of change and inflection points is utilized.

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Learning Goals

Rates of change
Introduction to derivatives
Introduction to mathematical modeling in science (exponential and logistic growth models)
Critical analysis of hydrocarbon consumption and the societal, political, and environmental implications of past and future reliance on a non-renewable resource

Context for Use

Possible courses in which module could be utilized:
Physical Geology
Introductory Economic or Petroleum Geology
Environmental Science
Environmental Policy
Introductory Data Analysis for Scientists
Economics (Supply/Demand of Commodities)

Teaching situations:
Discussions of the geology of hydrocarbons
Sustainability and environmental impacts of hydrocarbons
Non-renewable resources

Primers can be discussed and worked through in class or computer lab. Primers are written so that students may utilize them for independent study.
Exercises were designed as take-home exercises, but could be done in a computer lab setting (it may take 3 or more hours to work through the exercises or longer if students are not proficient in algebra or in using a spreadsheet program).

Educational level:All exercises are designed so that calculus is not a prerequisite and, thus, students from all levels (Freshman-Senior) can work through the problems if they are proficient in algebra. For more advanced students, a formal discussion of derivatives could easily be inserted and the solutions to the exponential growth and logistic equation models could be pursued from first principles.

Description and Teaching Materials

Teaching Notes and Tips

The more formal definition of a derivative could easily be inserted when giving the primer to students. The author has done this on previous occasion with positive results for an audience whose members had mostly not taken calculus previously.

The analysis of "peak oil" and future oil production rates can be supplemented with a worthwhile discussion of the merits and pitfalls of the model and the assumptions used to generate such predictions.


Grading the petroleum production rates exercises would provide valuable insights into students' strengths and weaknesses on concepts of rate of change, handling and manipulating data, and synthesizing their analysis of the data with larger conceptual issues.

References and Resources

Acknowledgment: I would like to thank Dr. Robert Cole of The Evergreen State College for his generosity in sharing time and previous work on oil production rates and how such data may be applied in the classroom. Our discussions set me down the path to create the crude oil production rates component of the learning module.


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Campbell, C.J. and Laherrere, J.H., 1998. The End of Cheap Oil. Scientific American, v. 278, no. 3, 78-83.

Caruso, G., 2005. "When Will World Oil Production Peak?" 10th Annual Asia Oil and Gas Conference, Kuala Lumpur, Malaysia, June 13, 2005. Accessed April 11, 2011.

Deffeyes, K.S., 2005. Beyond Oil: The View from Hubbert's Peak. Hill and Wang, 224 pp.

Hubbert, M.K., 1956. "Nuclear Energy and the Fossil Fuels." Spring Meeting of the Southern District Division of Production, American Petroleum Institute, San Antonio, Texas, March 7-9, 1956: In Pub. 95, Shell Development Co., 40 pp.

Hubbert, M.K., 1971. "The Energy Resources of the Earth," in: Energy and Power, A Scientific American Book, pp. 31-40.

Longwell, H.J., 2002. "The Future of the Oil and Gas Industry: Past Approaches, New Challenges." World Energy, v. 5, n. 30, 100-104.

Simmons, M., 2005. Twilight in the Desert. Wiley, 448 pp.

United States Energy Information Administration (EIA). International Petroleum Production Data. Accessed March 31, 2011.

USGS, 2000. U.S. Geological Survey World Petroleum Assessment. [] Accessed March 31, 2011.

USGS, 2005. U.S. Geological Survey Assessment Concepts for Conventional Petroleum Accumulations. In: Petroleum Systems and Geologic Assessment of Oil and Gas in the Southwestern Wyoming Province, Wyoming, Colorado, and Utah. Accessed March 31, 2011.