Abstract
A standard part of the calculus curriculum is learning exponential growth models. This paper, designed to serve as a teaching aid, extends the standard modeling by showing that simple exponential models, relying on two points to fit parameters do not do a good job in modeling population data of the distant past. Moreover, they provide a constant doubling time. Therefore, the student is introduced to hyperbolic modeling, and it is demonstrated that with only two population data points, an amazing amount of information can be obtained, such as reasonably accurate doubling times that are a function of t, as well as accurate estimates of such entertaining topics as the total number of people that have ever lived on earth.
Highlights
This year, the world’s population passed the 7 billion mark
The student is introduced to hyperbolic modeling, and it is demonstrated that with only two population data points, an amazing amount of information can be obtained, such as reasonably accurate doubling times that are a function of t, as well as accurate estimates of such entertaining topics as the total number of people that have ever lived on earth
It is possible to construct an exponential growth model of population, which begins with the assumption that the rate of population growth is proportional to the current population: dP dt where k is the rate of population growth, and P
Summary
This year, the world’s population passed the 7 billion mark. Being able to forecast population in the future, and even being able to answer some interesting questions about population in the past, depends on developing accurate mathematical models of population growth. The hyperbolic growth model differential equation is developed and solved, once again estimating parameters in a very hands-on simple fashion This model is compared to real data. Given only the simple tools of an introductory calculus course, we would rely on an exponential model to forecast future population growth, but on a hyperbolic growth model to answer questions about the past. To achieve these ends, the paper begins by asking students to collect world population data from 1 AD to the present. The student is pointed to the notion that the simplistic models presented here are insufficient for truly accurate projections, and that such projections would need to take numerous additional factors into account
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