# Understanding Exponential Growth with Technology

### Jaime Carvalho e Silva

Departamento de Matematica
3000
Coimbra,
Portugal
Phone: 351-39-4191199

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## ABSTRACT

One of the concepts related to the exponential function that is less well understood is the concept of 'exponential growth'. A lot of people in everyday life speak about it since Thomas Malthus (1766-1834) used it to describe population growth, but very few understand it; they want to say that the growth is very fast, but they confuse growth with the real value and ignore that there are other kinds of growth that are also very fast. It is my idea, and I experienced it with students and secondary school teachers, that technology (calculators or computers) can contribute to a better understanding of 'exponential growth' and how it can be compare d with others.

I think a student learns better a difficult concept when he studies, with some detail, significant examples or problems. I will discuss several interesting examples that cannot be fully understood without an extensive use of technology.

1. Using a graphic calculator or a computer, compare the graphs of 2^x and x^2. The results are puzzling because in the interval [0,1] the graph of the exponential is 'over' the other, but then for a long time the graph of the power function is 'over' the graph of the exponential. Exponential growth is not a big growth everywhere, and 'bigger growth' and 'bigger value' are two very different things. The same question can be asked about the graphs of 2^x and x^5, with similar conclusions, but it will be much more difficult to discover when the graph of the exponential will become finally over the graph of the power function.
2. Using a graphic calculator or a computer, compare the graphs of 2^x and x^100. With x^100 or a bigger power function a graph will be useless, because an overflow will happen. But then we can use a Computer Algebra System; to compare 2^x and x^100 directly for big x, will be cumbersome; so we can see if the fraction 2^x / x^100 is bigger or smaller than one. The alternative of a log-graph will be discussed.
3. Using a graphic calculator or a computer analyze the chessboard legend. Using a table of values or a spreadsheet we can see that the usual procedure of multiplying by two gives impractical results. Other problems can be used to obtain the same shocking result.
4. Using a graphic calculator or a computer analyze the popular way of getting a lot of money by mail (pyramid procedure). Again, using a table of values or a spreadsheet we can see that the usual propaganda that says that if you send money or letters to 5 or ten people, and expect thousands back cannot work more than one year even if all the population of the earth participates