R. Robinson
9/10/97

Start with a differentiable function like f(x)=1/(x^2+1). If you zoom in a few times at any point, the graph looks like line. Try this for the function above. To enter this function push the "graph" menu and select "y(x)=". Type "y1=1/(x^2+1)". Select "wind" (or "rang" on TI-85) and set the window so x=-2...2 and y=0...1.5. Then push "graph" on the "graph" menu. You may need to push "2nd" "graph" if the graph command is on top of another command (probably "select").
   (1.) Draw a picture the graph you get on your paper.
Zoom in (use the "zin" key in the "zoom" menu) a few times at the point (2, 0.2) until the graph looks like a line.
   (2.) How many times did you need to zoom in and what was the range of the window by the time you were finished zooming?
Now return to the "graph" menu and select "y(x)=". Type "y2=der1(y1,x)". You can get the "der1" form the "calc" (i.e., calculus) menu or you can type it out (use "2nd" "alpha" for lower case letters). Reset the window to x=-2...2 and y=0...1.5 and push "graph".
   (3.) Draw a picture the graph you get on your paper.

Now let's try f(x)=x sin( -10 ln |x|) with f(0)=0 (this is similar to the function xsin(1/x) we discussed in class. Why?) . This function continuous. Can you see why? Make sure your calculator is in "radian" mode (press the "mode" key). Enter "y3=x sin (-10*ln abs x)". You can find "abs" in the "math" menu under "ops". Use the "select" key to unselect "y1" and "y2". Draw the graph for x=-2...2 and y=-2...2. If you zoom in on any point on the graph for nonzero x you will eventually get something that looks like a line. This function is differentiable for nonzero x.
   (4.) Try this for x=1. Draw a picture the "line" you get on your paper.
Now reset the window to its original size. Zoom in on the point (0,0) over and over.
   (5.) Describe the pictures you get. What does this say about differentaiblity at x=0?
Note:You can try and plot the derivative of this too. What happens?

Now comes the most interesting part. The Weierstrass function is defined by an infinite series

This function is defined as an infinite series (series are studied in Math 32), and it is known that for appropriate choices of the parameters L and S (eg L=1.5 and S=1.1) that it is continuous for all x but differentiable for no x. This is because the graph of this function is not at all smooth. In fact it is fractal which means a small part of the graph, when blown up, looks a lot like the graph as a whole. As you graph the function, think about how the fractal property leads to the non differentiability.

We will approximate the Weierstrass function by a finite series. To prepare to graph this function, "exit" until you get a blank screen. Then type "L=1.5" "enter", "S=1.1" "enter", "M=L^(S-2)" "enter" and "N=10" "enter". To type upper case letters and "=" press the "alpha" key first. After each "enter" the calculator should say "done". Get into the graph menu and type

"y4=sum seq(M^J sin(L^J x),J,0,N,1)"

The "sum" and "seq" come from the "list "ops" menu, or you can type them in directly. You could also type

"y4=sin(x)+M sin (L x)+M^2 sin (L^2 x)+ ... +M^10 sin (L^10 x)"

including all the in-between terms, but the "sum" and "seq" commands do all this adding for you!
   (6.) Plot this for x=0...6 and y=-2.5...2.5. (Be sure you are in radian mode!).

Note: The calculator needs to do a lot of thinking to draw this; it will be slow! Pick a point on the graph and zoom in twice (yes: slower still). Draw a sketch of what you see on your paper. How does this relate to differentiability?
   (7.) Change the parameters by making R=3. Reset the window and plot. How does the new curve look compared to the old? Draw a sketch.
   (8.) Make L=2 and S=1.1. Set the window for x=0...2*pi and y=-2.5...2.5. Draw what you see. Why is this picture symmetric?
   (9.)What happens if you reduce N?
   (10.) What conclusions can you draw from all this?.

Challenge: On the same screen, draw the pictures for N=6 and for N=7. Also draw M^7 sin (L^7 x). Explain what you see.

Here are a few pictures of the Weierstrass function:

Zoomed: