Cobweb Applet Exploration Worksheet
Name:
Learning Objectives: Explore cobweb plots for a variety of simple functions and orbits for a number of seeds,
identify fixed points, observe attracting and repelling fixed points, recognize periodic cycles, see mo
The Mandelbrot Set and Iterating a Complex Function Worksheet
Complex Function: Just as we have real-valued functions where a real input x produces a real output f(x) we
can have complex-valued functions where a complex input z produces a complex output f
Complex Numbers Worksheet
Complex Numbers: a + bi where a and b are real numbers and i 2 = 1 .
Complex Plane: Complex numbers can be visualized with the horizontal axis representing the real component
and the vertical axis the imaginary i component. Here
Block Fractals Worksheet
The following pictures show the first couple of iterations of a fractal made from cubes. By examining the
changes between Stage 1 and Stage 2 you should be able to identify how to proceed to subsequent stages and
you should be abl
The Quadratic Family Cobweb Exploration Worksheet
Objectives: In this activity you will explore iteration and orbits with quadratic functions, fixed and periodic
points, attracting and repelling fixed points, and cobweb plots.
The Cobweb Applet is found a
Sierpinski Triangle Area Worksheet
Directions: Complete the table and answer the following questions. Assume the area of the original triangle is 1
square unit.
Iteration n
0
1
2
3
4
5
Number of triangles removed at stage n
Area of one removed triangle at
Koch Curve Length Worksheet
Directions: Complete the table and answer the following questions. Assume the length of the original segment
is 1 unit.
Iteration n
Number of segments added at stage n
Length of one segment added at stage n
Total length added a
Iterating Functions Worksheet: When a butterfly flaps its wings
Consider the function f ( x ) = x 2 + 0.1875 . Using Excel or your calculator (especially the ANS button), find the first
few iterates of each of the following seeds:
1. Consider the seed
x0
Iterating Functions Worksheet: Exploring Orbits
Objectives: In this worksheet we explore several functions, look at orbits using Excel, and describe the
behavior we observe. In this activity we become more comfortable with ideas of iteration, orbits, peri
Iterating Functions Handout
Function: A machine for doing things
with numbers. Given a function f and an
input x the output is call f(x).
Iterating a Function: Evaluating a function over and over using the previous output as
the new input.
Note: Iterating
Fractal Workshop
Dimension Worksheet
Name_
Recall that the an object scaled by by a factor of S generates N copies of the
original object then the dimension D satisfies the relationship N = S D . Find a suitable S
and N for the fractals described below an
Dimension and Fractals
We say a line is a one-dimensional object, a plane is a two-dimensional object, and a
cube is a three-dimensional object. Consider these modes of transportation:
A train moves in one
dimension on its linear
track.
A boat moves in tw
Definition of a Fractal
Sources:
Hans Lauwerier. Fractals: Endlessly Repeated Geometrical Figures. Princeton, NJ: Princeton
University Press. 1991.
Robert L. Devaney. A First Course in Chaotic Dynamical Systems. Cambridge, MA: Perseus
Books. 1992.
http:/e
Block Fractals Worksheet
The following pictures show the first couple of iterations of a fractal made from cubes. By examining the
changes between Stage 1 and Stage 2 you should be able to identify how to proceed to subsequent stages and
you should be abl