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

March 2006 Mathematics Notebook

An Example of a "Learning Process" Journal

7:35 am Ballina NSW Australia

This is a continuation of my notes for "Dr. Riemann's Zeros".

Chapter 4 Indian summer

• "In 1900, the Riemann Hypothesis was given new prominence in a famous lecture by the great German mathematician David Hilbert. ... Hilbert decided to use the occasion of the new century to look at the topic of unsolved problems in mathematics, ... He believed that 'A branch of science is full of life as long as it offers an abundance of problems; a lack of problems is a sign of death.' " [p. 52 - 53]
• "... and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor wish it ... " (written by Hilbert in 1900) [p. 54]
• "A transcendental number is one that is neither an integer, nor a fraction, nor the root of an algebraic equation ... It's possible to prove that there are far more transcendental numbers than any other numbers on the line of real numbers." (e.g. pi, e ) [p. 56]

• "Ramanujan was interested in what are called 'partitions'. This is one of the easiest concepts in number theory to describe ... A partition of a whole number N is any selection of positive numbers which, added together, make N." [p. 64]

Here are the main points from a review of these notes:

• Ideas in mathematics (and in calculus) can be represented visually, analytically (algebraically), numerically (tables) and linguistically.
• Rene Descartes [1596 - 1650] created analytic geometry.
• When constructing a graph one needs to pay particular attention to the "viewing window" of the graph. This involves a consideration of both the scale of the axes as well as the minimum and maximum values for each axis. A graph may "appear" quite different depending on the values chosen for the axes.
• The idea of a linear model (an analytic/algebraic idea) is equivalent to the geometric idea of a straight line.

• The important property of a straight line is its slope. The slope of a line is determined by any two points on the line. Once one knows the slope, there are two main forms for the equation of the line, one where one also knows a point that the line passes through, and the other when one knows the value of the y-intercept (which is a special point that the line passes through).
  
  
• Use of Mathematica:
  • Steps to obtain a graph of a function of x:
    • Set y equal to the expression involving x
      • y = Sqrt[x^3 - 4x]
    • Create a plot using the Plot command
      • p1 = Plot[y, {x, -5, 5}, PlotStyle -> Red];
    • Combine two or more graphs using the Show command
      • Show[p1, p2];
  • Steps to obtain a fitted curve to a set of data:
    • Create a list of data points
      • acres = {213, 297, 374, 426, 461, 478}
    • Create a plot of the points using ListPlot
      • plotacres = ListPlot[acres, Prolog -> AbsolutePointSize[5]];
    • Fit a curve to the points using the Fit command, specifying the structure of the curve
      • f2 = Fit[acres, {1, x, x^2}, x]
    • Plot the curve using the Plot command
      • pf2 = Plot[f2, {x, 0, 6}, PlotStyle -> Blue];
        
• The three forms of the equation of a straight line are:
  • point-slope form
  • slope-intercept form
  • general form
    
    
• The two most important ideas for having a deep understanding of calculus are function and limit.
  
  
• Leonhard Euler [1707 - 1783] was one of first to apply calculus to real-life problems in physics.
  
  
• Functions can be specified in a variety of ways: language (giving a rule for the correspondence), set theory notation, an equation (analytic approach) or a set of equations (piecewise definition), a table (numeric approach) or a diagram (visual approach - e.g. a Venn diagram). The set theoretic approach is used in analysis, but in basic and applied settings one usually uses equations.
  
  
• If the equation is of the form y = f(x), then this is called an explicit form. This form is relatively easy to evaluate and graph. If the equation is of the form f(x,y), it is called an implicit form and is usually much more difficult to evaluate or graph.
  
  
• Since a function is defined as a unique mapping of x into y, a vertical line can intersect the graph of a function only once. This is called the vertical line test for a function. Many equations give rise to graphs that fail this test for a function. The equation and the graph are still meaningful, but it is not proper to refer to these as functions.
  
• There are many types of function:

  Definition: A polynomial function is of the form .

  Definition: A radical function is of the form .

  Definition: A rational function is of the form .

  Definition: A composite function is of the form and is usually written or

  Definition: A function is even if f(-x) = f(x).

  Definition: A function is odd if f(-x) = -f(x).

• The general idea of function is straight forward. Most introductory calculus material focuses on the elementary functions. The idea of a composite function is simply the idea of recursion!
  
  
• Calculus is the study of change (algebra) and continuity (geometry) when extended to infinitely small values.

Here is the table from January 6 that attempts to monitor my progress:

This has been a good beginning to getting back to calculus. 12:15 PM

2:40 PM I just realized that it has been almost exactly two months since I last looked at the Larson book.

Here are the section sub-headings that I was using for making notes for each section:

• Introduction
• Keywords
• History
• Description
• pdf Files
• Mathematica
• Summary
• Reflection
• Difficulties

1.3 Evaluating Limits Analytically [p. 56 - 66]

Introduction: Whereas the previous section provided an intuitive approach to finding limits numerically (using a table of values) or graphically (using graphing software), this section provides precise algebraic approaches.

Keywords: Properties of Limits, A Strategy for Finding Limits, Cancellation and Rationalization Techniques, The Squeeze Theorem

History:

Description: The limits for all of the following functions can be obtained by direct substitution:

• polynomial functions
• rational functions (with denominator not equal to 0)
• radical functions
• composite functions
• trigonometric functions

If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, then try to find a function g that agrees with f for all x other than x = c. (i.e. see if you can cancel the denominator with a factor of the numerator).

A second approach for some rational functions involves "rationalizing the numerator". This means mutiplying both numerator and denominator by the same expression which will then allow one to remove the radical symbol and turn the numerator into a rational expression.

Two special trigonometric limits

pdf Files: Gearhart, W. B. & Shultz, H. S. (1990, March) The Function (sin x)/x. The College Mathematics Journal.

Mathematica: This appears to be very simple. There is a command called Limit which takes three arguments: the function, the value that the independent variable is approaching, and the direction that it is approaching from. (Web, Mathematica)

Summary: An introduction to simple algebraic and trigonometric limits.

Reflection: So far the discussion has been about simple limits involving direct substitution or a couple of simple ways to handle indeterminate forms. However a couple of "facts" look like they will be useful in the future: namely the two trigonometric limits. The idea behind "rationalizing the numerator" is now clear to me, as is the significance of the phrase. In general I feel I understand the idea of limit but am rusty when it comes to applying it to tricky special functions. Difficulties: None so far, but the real test will be the next session when I attempt the problems posed on pages 64 - 65.