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CALCULUS 265 Honors
Fall Semester 2000
8:00 to 8:50 MTTF
Carver 0002

Instructor: Hentzel
Office: 432 Carver
Office Hours: 9:00-10:00 MTTF

E-mail  hentzel@iastate.edu
phone  294-8141
Website

http://www.math.iastate.edu/hentzel/honors.265

Tuesday, November 7:    Section  8.2


Main idea: Once you believe that 1+1/2+1/4+1/8 ... stops at 2, you

have a whole lot of other suspicious things that you have to

accept as well. 

Key Words: snowflake, countable, diagonalization process,  

Goal: Expand your concepts of mathematical reality.


For the snowflake curve.  What is the length of the perimeter.

(a)  The number of edges is

    3, 3(4), 3(4)^2, 3(4)^3, ... ,3(4)^n

(b)  The lengths of the edges is

    1, 1/3, (1/3)^2, (1/3)^3, ... ,(1/3)^n.

(c) The perimeter is

    3, 3(4/3), 3(4/3)^2, 3(4/3)^3, ..., 3(4/3)^n

    So the perimeter goes off to infinity since (4/3)^n goes

    off to infinity.


(d) The area of an equilaterial triangle is s^2 Sqrt[3]/4.  Thus

the area grows by  (the number of sides)  (side/3)^2 Sqrt[3]/4.

                  |      n  | 2
               n  | (1/3)   |
           3(4)   | ------- | Sqrt[3]/4   = 
                  |   3     |
                  |         |


         Sqrt[3]    3 4^n      Sqrt[3]  3 4^n           Sqrt[3]   4^n
        ---------  -------   = ------- ------------ = ---------  ---------
            4      3^(2n+2)       4     9 3^(2n  )         12     9^n


        Sqrt[3]
        -------( 1 + 4/9 + (4/9)^2 + (4/9)^3 + ... + (4/9)^n + ... )
           12


          Sqrt[3]     1           Sqrt[3]      9      3 Sqrt[3]
        ---------- -------   =   -----------  ---  = -----------
           12       1-4/9           12         5         20


This is how the area grows.  The total area including the original

triangle of area  Sqrt[3]/4  is

         2 Sqrt[3]
       -------------      = 0.69282
             5


     (e) We show that the augmentations never touch.  We claim that

you never can get out of the circumcircle of the original triangle.

At each step, the added bump is an equilaterial triangle. The 

circumcircle of this equilaterial triangle is tangent to the previous

circumcircle and on the inside.  Continuing in this way, all subsequent

constructions take place inside a sequence of circles, each tangent

to the privious and on the inside.  Since all of these  circles are

disjoint, the triangular extensions are disjoint. 




Sue needs a medicine in her body.  During each day her metabolism reduces

the amount of medicine in her body by 60 %.  That is, only 40% of what is

in her body today will be there tomorrow.  Each pill has 10 mg of medicine.  

There must be 16 mg in her body for the medicine to be effective.  How many 

doses till this level is reached?  The medicine has moderate side effects 

when the dose in the body reaches 16.6 mg.  How many doses till this is 

reached?  The medicine has horrible side effects when the medicine in the

body reaches 16.7 mg.  How many doses till this is reached?