Exponential Models

Exponential models use mathematical functions that look like f(x) = b^x where b is some number, the base, and x is the argument of the function. You have probably seen functions like f(x) = x², in this presentation we'll explore functions that look like f(x) = 2^x. Undoubtedly you've seen and experienced applications of exponential models, the Golden Gate Bridge and the St. Louis Arch are two examples. The suspension cables on the Golden Gate Bridge are curved, they take on the shape of a special exponential function's curve. A similar function gives us the shape of the St. Louis Arch!

To understand the concepts involved with exponential functions, let's consider a couple of examples.

Exponential Growth
We begin with an experiment. Take a piece of paper and fold it in half. Now fold it in half again and keep folding it in half. How many times could you fold it in half? To get a sense of how thick that folded paper is getting consider these facts:

At 7 folds it is as thick as a notebook.
If you would have been able to fold it 10 times, it would be as thick as the width of your hand.
At 17 folds it would be taller than your average house.
20 folds and that sheet of paper is a quarter way up the Sears tower.
30 folds and it has crossed the outer limits of the atmosphere.
50 folds and it has reached the sun from the earth.
60 folds it has the diameter of the solar system.
100 folds it has the radius of the universe.

How can it be that something as thin as a sheet of paper can be compared to the radius of the universe? This concept is at the heart of exponential growth. Notice that there is a substantial increase in length with each fold and that length increases rapidly.

Let's take a look at a picture.

folds
Given what you now know, let's consider a question.
Keith is looking for someone to watch his dog, Cimarron, for two weeks. He has offered two choices:
1. $150 per week
or
2. One penny for the first day, two pennies for the second day, four pennies for the third day, eight pennies for the fourth day, and so on for 14 days.
Which option should you choose?

Now you are ready for the MAIN CONCEPT.

Exponential Decay
Now let's apply this concept to a new problem, one that involves a decreasing amount as opposed to an increasing amount.

As soon as a cup of hot chocolate is poured, it begins to cool. After a long period of time, the temperature of the hot chocolate eventually stops decreasing and stabilizes at the same temperature of the room. How does it cool? Is the rate of cooling constant? What about the myth?

The model (mathematical equation) we will use is:

y = Room temp. + y₀e^{-0.6 t}

Here's a plotting tool for our model.

http://amath.colorado.edu/java/mvt2.0/

Let's create a plot and answer those questions.

For more applications see the problem page.

Careers in Applied Mathematics

Have you ever wondered what kind of a job you could get with a degree in applied mathematics?

Studying turbulence around airplanes.
Managing complicated networks: routing telephone calls
Some people who do mathematics for a living
Resources from MAA (Mathematical Association of America)

Women in Mathematics

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