Calculus 231
Michael Pogwizd
Calculus Project 1
The chaos theory suggests that even the smallest action can lead to major change. The most common hypothesis is that a butterfly flapping its wings could cause a hurricane on the other side of the world.
In this project, we will be looking mathematically how even the smallest changes can lead to a dramatic change.

We will be using a mathematical technique called iterating. This is done by taking a function f(x) and finding the answer when x = 0. Then, the answer from that same function is plugged into the function to get a new answer; this step is repeated again and again, showing the new answers.
Here how it is done:
then,
and so on
Example:
etc.
The function we will be using will include a variable. This variable will be inputted with real numbers to see how the function changes when the variables are different.
First, we will start with the function:
c represents the variable that we will be changing.
By using MathCAD, we can analyze the graph and numerical information by inputting the information.

We will start by expressing the domain (i):
We can increase the domain to find larger numerical values; however, we will restrict it to {1,5} for now.




Before we enter the function, we must first define the variable c:
We will start at zero and work our way up.
Now we define the iterating function, it appears different from the original function because it also iterates the function:
Finally, we show that f(0) = 0
The entire information looks like this:
The variable
The domain
Defining f(0)
The iterating function
Now we show the process numerically and graphically:
Numeric:
Graph:
Notice that when c = 0 the graph only shows points on the x-axis starting at 1 and going to infinity. This is because the iterating only expresses integers on the domain, and the function only produces 0 for every answer, like this:
Input 0 in the next function:
and so on.
Now we will take a look at the graph as c goes from 0 to 1. To save time and space, I have prepared an animation that will go over the graph. We will then break it down and discuss the relationships.
File name: NEWANIMATION (0,1)
Double Click to Play Animation
File name: Animation (0,1)
Now let's break this graph down. Play it again and notice how the graph remains level until it becomes large than 0.25. This is because when c is between 0 and 0.25, the graph has a limit.
Tale a look at the animation when c is between 0 and 0.25:
Double Click to Play Animation
File name: NEWANIMATION (0, .25)
Notice how the answer becomes closer and closer to a single constant on the table. Therefore, we can conclude that when c is between 0 and 0.25, the graph will approach a limit.
Now well observe the animation showing the graph when c is greater than 0.25