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Learning:
The Journey of a Lifetime
A Cloud Chamber of the Mind

January 2006 Mathematics Notebook

Introduction      
Goals
   
     

An Example of a "Learning Process" Journal

Tuesday January 3, 2006
Learning Log Number 1

Chapter 1 Limits and Their Properties [p. 39 - 88]

Overview

The two key ideas for understanding calculus are those of a function and the idea of a limit. Functions were reviewed in the previous chapter. This chapter focuses on the idea of a limit and the idea of a continuous function.

Topic: 1a A Preview of Calculus [p. 41 - 46]

Introduction: Calculus is the study of change (algebra) and continuity (geometry) when extended to infinitely small values.

Keywords: What is calculus? The Tangent Line Problem, The Area Problem

History: none

Description: The tangent line problem may be stated as follows: "Given a function f and a point P on its graph, find the equation of the tangent line to the graph a the point P". The area problem may be stated as follows: "find the area of a plane region bounded by the graph of a function".

pdf Files: none

Mathematica: review University of Illinois website!! (see link at bottom of this page)

Summary: The idea of a tangent line as a limiting case of a secant line is straight forward. Similarly, the idea of the area under a curve as a limiting case of a group of rectangles is straight forward.

Reflection: Both of the above situations are straight forward so long as the function (curve) "behaves itself" (i.e. doesn't have unusual properties such as an infinite oscillation or a vertical line or a discontinuous nature.

I would like to be able to use Mathematica to graph these limiting situations, but am afraid that the effort will outweigh the advantage of seeing what I can already visualize in my mind. Nonetheless, I will see what I can find on the web about this

Links: Google [mathematica "area under a curve" approximation ]

Comment: The University of Illinois web site is worth a long visit. But for the moment I am going to stay with the text book. Nonetheless, the U of I site makes excellent use of Mathematica to explore ideas in calculus. My next activity will be to play around with this site and see what happens.


10:30 am Ballina NSW Australia

Blue sky and hot. Wonderful. The first goal for the new year is to get back on top of my calculus Learning. The Learning Log entries are simply a way of keeping track of the big picture of what I am trying to Learn plus a few reflective comments. The following table is a way of noting the various books and resources that I am using.

I have added the books on number theory that I have purchased since arriving in Australia. I have three books on the go at the moment: one on calculus, one on prime numbers, and one on chaos and fractals. That seems about right. Today, the goal is to have a good look at calculus and the Larson book and my notes on it.

Mathematics
Description
Start
End
Done
Calculus
Calculus. 6th ed. (1998). Roland Larson, Robert Hostetler & Bruce Edwards Nov 09/05    
Chap. P Preparation for the calculus [p. 1 - 38]
Nov 09/05 Nov 13/05 Yes
Chap. 1.1 A Preview of Calculus [p. 39 - 46]
Jan 01/06 Jan 01/06 Yes
Chap. 1.2 Finding Limits Graphically and Numerically
     
 
     
University of Illinois Urbana-Champagne web site      
0.1 PlotFunctions.nb
Jan 01/06 Jan 01/06 Yes
0.2 ParametricPlot2D.nb
Jan 01/06 Jan 01/06 Yes
       
Mathematica
Mathematica Navigator. 2nd ed. (2004). Heikki Ruskeepaa      
Fractals & Chaos
The Computational Beauty of Nature. (1998). Gary William Flake      
       
Coincidences, Chaos, and All That Math Jazz. (2005). Edward Burger & Michael Starbird Dec 03/05    
Read chaps 1 - 6
Dec 03/05 Dec 15/05 Yes
Make MindGenius map for chaps 1 - 2
Dec 15/05 Dec 15/05 Yes
Make MindGenius map for chap 3
Dec 16/05 Dec 16/05 Yes
       
Number Theory
Meta Math! - The Quest for Omega. (2005). Gregory Chaitin      
       
Everything and More: A Compact History of Infinity. (2003). David Foster Wallace      
       
Dr. Riemann's Zeros. (2002). Karl Sabbagh Dec 29/05    
Read chap 1 "Prime Time"
Dec 29/05 Dec 29/05 Yes
Read chap 2 " 'Gorgeous Stuff' "
Dec 30/05 Dec 30/05 Yes
Read chap 3 "New Numbers for Old"
     
       
Fundamentals of Number Theory. (1977). William J. LeVeque      
       
The Divine Proportion. (1970). H. E. Huntley      
       
Gamma: Exploring Euler's Constant. (2003). Julian Havil      

 

     

This has been an engrossing session! Finding the UIUC site for Calculus and Mathematica was stunning. It has been a long time since I had so much fun. 1:45 PM

Total elapsed time for the day: 3 hr 15 min.