Mathematics Chronology

6:50 am It has been almost two months since I last played with some math. Today I want to briefly review notes for 2007 and then have a start at looking at the book "The Humongous Book of Calculus Problems" (2006) by W. Michael Kelley which I bought a few days ago.

The first task is to update my chart to include this book.

Done. The Kelley book is now listed as one of the textbooks for calculus.

Here is an image of the front cover:

and the back cover:

Now to sit back and have a first read of chapter 1 Linear Equations and Inequalities.

W. Michael Kelley (2006) The Humungous Book of Calculus Problems Chapter 1 Linear Equations and Inequalities Here are the main ideas mentioned in this chapter: graph of a line slope of a line x- and y-intercepts of a line mid-point of a line slope-intercept form of a line standard form of a line parallel lines perpendicular lines interval notation inequalities graphs of inequalities absolute value equations absolute value inequalities systems of equations: solve using substitution method systems of equations: solve using elimination method This chapter only discusses equations of the form ax + b = c. The fact that the variable x is not raised to any power other than one, means that the equation is "linear".

At this point I want to review my other calculus resources and synthesize important points from their opening chapters.

Most of these books begin with a review of important ideas, concepts and procedures that are pre-requisite to an understanding of calculus.

The most basic of these ideas are:

• the idea of a variable
• the idea of an equation
• the idea of a function
  • notation
  • graphical
  • numerical
  • technical
• analytic approaches (i.e. algebraic, symbolic)
• numerical approaches (i.e. tables of values)
• graphical approaches
• technical approaches (i.e. use of calculators and computers)

All of the four approaches provide a slightly different perspective on the same situation. Wherever possible one should try playing with ALL of these approaches to fully understand the situation under consideration. Mathematics is a subtle combination of all approaches. One needs to both understand the ideas involved in a particular situation as well as having the necessary skills (both mental and algorithmic as well as technical) in order to fully explore a problem. Knowing a few algorithms for solving two equation in two unknowns is useless if one does not understand the underlying ideas of an equation and a solution.

A study of functions usually begins with the simplest case, that of linear functions. This is what the Kelley book does in chapter 1. Larson takes a similar approach but in more detail. My other books consider this to be already understood and begin with a more general approach to all functions. One of the values of beginning with linear functions is that it permits one to develop algebraic skills for manipulating expressions which will be essential later. In addition to developing such "manual" skills, it is now important to develop the necessary technical skills for using particular software packages to explore mathematical situations. In my case, I am interested in using Mathematica to explore the problems that I will encounter.

A final issue is that of using technology for making notes. In general this is straight forward - just type. But if one wants to type a mathematical expression, then the situation becomes more difficult and one must learn an additional technique for doing this (e.g. learn to use a software package called MathType). Fortunately this is not too difficult, but it does involve an additional step.

One final comment. While exploring functions and the ideas of calculus (limit, derivative, integral) one is usually using some form of number system. Thus a clear understanding of number is also critical. The Spivak book begins with such a consideration, but the other books take our general sense of number as a given and proceed without a deeper review of this idea. Such approaches are usually referred to as "analysis" and emphasize the analytic nature of the topic rather than a more applied perspective.

The idea of a variable is very straight forward. A variable is simply anything that may take different values. In mathematics it is usually represented by means of a letter. Thus the letter x may represent a variable and it may, in a particular situation, take specific values (such as 2, 3 and 4).

An equation is also very straight forward. An equation is an algebraic expression where the values on each side of the equal symbol can be shown to be the same.

Here is a trivial example:

Find all values of x that satisfy the equation 3x + 2 = 7.

Performing the same operation on each side of an equation leaves the equality intact. Thus subtract 2 from each side gives 3x = 5.

Similarly dividing each side by 3 gives x = 5/3. This is the only value of x that satisfies the equation.

One may follow the material in either Kelley or Larson to see a fuller description of linear functions and some of the important ideas (slope-intercept form, graphs of linear functions, x- and y-intercepts, inequalities, absolute values, solution by substitution and solution by elimination).

I see three important activities before leaving the topic of linear equations:

• completing a number of exercises by hand
• completing a number of exercises using MathType to make the notes
• completing a number of exercises using Mathematica

Recipe Notes

Original recipe in http://www.spinachrecipes.org/baked-creamed-spinach.html

Ingredients (serves 4) (I halved the original recipe which called for two pounds of spinach)

Main Ingredients	Actual
2 lb fresh spinach	1 lb fresh spinach
1.5 cups milk	0.75 cups 1% milk
1 cup chedder cheese, grated	1 cup cheddar cheese, grated (I like cheese, so decided not to halve this item)
2 tbsp butter	1 tbsp butter
2 tbsp flour	1 tbsp flour
1 teaspoon salt	omit
pinch of pepper	pinch of pepper
pinch of nutmeg	pinch of nutmeg

Steps

1	Trim and wash the spinach. This took awhile to cut off all the stems from the spinach leaves.
2	Place in a large pot with water still on the leaves.
3	Cover and cook until wilted.
4	Allow to cool, then squeeze out the excess water.
5	In a saucepan, melt the butter; then add the flour.
6	Whisk and gently cook for 3 - 4 minutes.
7	Add milk and bring to a boil.
8	Add the pepper and nutmeg.
9	Cook for about 5 - 6 minutes, stirring often.
10	Add the spinach and 1/2 cup cheese to the sauce. Mix thoroughly.
11	Spoon into a baking dish, then top with 1/2 cup of cheese.
12	Bake covered at 375 F (190 C) for 20 minutes.
13	Serve as desired.

Here are a few photos: