### Electronic Proceedings of the Seventh Annual Conference on
Technology in Collegiate Mathematics

*CONTRIBUTED PAPER: 7-FC2*
### Understanding exponential growth with technology

Jaime Carvalho e Silva

Departamento de Matemtica

Universidade de Coimbra

Apartado 3008, 3000 Coimbra, Portugal

Phone: 351-39-4191199

E-mail: jaimecs@mat.uc.pt

#### 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.

- 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.
- 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.
- 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.
- 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