Third Class (Wednesday 13 September 1995)

Goals to accomplish during class

  1. Learn from experience
  2. Build a basic World-Wide Web home page
  3. Increase your fluency with LaTeX

Homework for after class

Of course, you may start on the homework during class if you have time!

  1. A Maple puzzle.

    Take a look at Exercise 12 on page 46 of Chapter 3 of the CalcLabs with Maple V manual: 

     
  A cylindrical can with a top and bottom is
  to contain 1000 cubic centimeters. Find
  the dimensions of the can if its surface
  area is 600 square centimeters.
  Note: there are two answers.

    Consider the following alleged Maple solution to the problem. 

      Let r and h denote the radius and the height of
  the cylinder. We set up a pair of simultaneous
  equations and solve them for r and h.

  > eq1:=Pi*r^2*h=1000; eq2:=2*Pi*r^2+2*Pi*r*h=600; fsolve({eq1,eq2},{r,h});

  eq1 := Pi*r^2*h = 1000

  eq2 := 2*Pi*r^2+2*Pi*r*h = 600

  {r = -11.13867027, h = 2.565567417}

    Obviously something is wrong here, because it does not make sense for the radius to be negative!

    Write a Maple session, addressed to an audience of first-year calculus students, that explains the error and exhibits a correct solution. Turn in hard copy at the next class.

    A requirement of this assignment is that your paper include a Maple plot. (That's also a hint: look at a suitable graph to see what is going on.) To get a plot into a Maple worksheet, select "Edit" and "Copy" from the plot window, move the cursor back into the main Maple window, and select "Edit" and "Paste".

  2. Texas A&M University keeps track of your name, telephone number, e-mail address, and so forth in an on-line phone book database named PH. There is an informational page about Locating People within the Texas A&M University System that enables you to update your PH entry. Add your World-Wide Web home-page address to your PH entry. Your URL (Uniform Resource Locator) probably has one of the following forms:
    http://calclab.math.tamu.edu/~your-logon-id/
    (if your home page is on the calclab machines)
    /~first-name.last-name/
    (if your home page is on the main mathematics server)
    http://http.tamu.edu:8000/~your-login-id/
    (if your home page is on tam2000)
  3. Intelligent mail programs can automatically include a "signature" file into each mail message you send. (For instance, pine and emacs have this capability; mailtool, as far as I know, does not.) Use a text editor to make yourself a signature file. I change my signature file from time to time; currently it looks like this: 
           :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
       Harold P. Boas                       E-mail: boas@tamu.edu
         World-Wide Web URL:     /~Harold.Boas/
       :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

    Generally, a signature file should contain your name and some contact information. Some people include little ASCII pictures or witty sayings. It is recommended that your signature file not exceed four lines. (If you send mail from within Netscape, it will complain if your signature file exceeds four lines, even if some of the lines are blank.)

    Save your signature file as .signature (that file name starts with a period.) Then open a terminal window and use the command chmod a+r .signature to make the file world readable. Now try sending yourself mail to see if the signature does get included in your mail message.

Activities to do during class

Learn from experience

I would like to share with all of you some of the ideas, problems, and errors that I encountered in reading your homework papers.

Why study math?

I enjoyed reading your papers on the subject "Why study mathematics?" All of them met or exceeded expectations.

Content

I read your papers first of all for content, to see what interesting ideas you had. Here are some of the common themes mentioned.

  1. Practical considerations
    1. Mathematics is needed in everyday life, from such mundane situations as making change in the grocery store and understanding mortgage payments to more sophisticated concerns such as modeling world population growth and evaluating the data on global warming.
    2. Mathematics is fundamental to modern technological developments that make possible everything from cordless telephones to CAT scans.
    3. Studying mathematics can lead to many careers, not just in scientific disciplines like particle physics and marine biology, but also in business and industry. (Of my three college roommates---all mathematics majors---one became chairman of the statistics department at a major state university, one works at Bell Laboratories, and one went to law school and got a job at a Manhattan law firm. I also know a PhD mathematician who went on to medical school.)
  2. Spiritual considerations
    1. Just as you might go to the recreation center to exercise your body, studying mathematics exercises your mind. A good workout makes you feel good.
    2. If we accept that abstract reasoning is a feature of our intelligence that distinguishes us from lower animals, then we can infer that studying mathematics is a means to developing our full potential as human beings. Without stretching a point too much, we could develop this argument to make the case that studying mathematics is a religious activity. In this vein, one person quoted the proverb, "With clarity comes peace of mind."
  3. Moral considerations. Mathematics, like music, is a universal language, and learning it is a way to develop commonalities with people from other cultures. You can elaborate this point to make an argument for the study of mathematics as a contribution to world peace.
Style

Secondarily I looked for good style and organization. I would like to reinforce two points about communicating ideas on paper. (Three of my four grandparents were English teachers, so good writing is appreciated in my family.)

  1. A well-organized essay should have an introduction that states the thesis, a middle section that develops the argument, and a concluding section that recapitulates the theme. The way I learned it in high school, "first you say what you're going to say, then you say it, and then you say what you said."

    Technical research papers often do not follow this model, but they probably should. I always start off a research paper with an introductory section that summarizes the results. I like to close a research paper with a section on open problems and directions for future investigations.

  2. It is often said, with justification, that you should write from experience. Give concrete examples to support your points. Instead of writing "mathematics is a foundation for many careers," write "studying mathematics can lead to careers in accounting, economics, chemistry, ...." Better yet, write "Arthur Pancoe, one of the nation's ten best stockbrokers, attributes his success to the analytical skills he learned as a mathematics student."

Here are two books I recommend on style.

Language

When I read your papers, I marked errors in grammar, spelling, punctuation, and so forth. Many of you are not native speakers of the English language, and I expected you to make some mistakes. Your grade will not suffer if your English is imperfect, but I assume that you are interested in improving your English, so I will continue to correct language errors on your papers.

I always feel embarrassed that I cannot communicate with my German in-laws in their mother tongue. I hope that those of you who are foreign-born will not feel embarrassed about your English. After all, in being able to speak more than one language you are more capable than many of your fellow students. In this connection, here are two points to keep in mind.

  1. I like to tell students that the most successful students are those who make every possible mistake---once.
  2. There is a joke that Europeans tell.

Those of you who are native speakers of English---watch out! I have a talent for proofreading.

LaTeX

Finally, I marked errors stemming from improper LaTeX mark-up. Since this was your first LaTeX paper, I expected you to make some mistakes in mark-up. If your code was perfect, you exceeded my expectations. Here is a list of some LaTeX idiosyncrasies that you should know about.

LaTeX subtleties

Maple tricks

  1. In revising your Maple worksheets, some of you wanted to insert a new region before the first line of the worksheet. You found that "Insert Prompt" on the "Edit" menu did not work: it made a new line after the first line. Here is the solution. Put the cursor at the very beginning of the first line, and then select "Split Region at Cursor" from the "Edit" menu. That will move the first line down and make a new line above. You can change the new region to a text region by putting the cursor there, clicking the "Input" button, and clicking "Text."
  2. You get valid, but weird-looking results if you differentiate a combination of arrow-defined functions by using the D operator. Try converting to Maple expressions and using diff instead. For example, in exercise 5 on page 43 of the CalcLabs with Maple V manual, 
           > f:=x->x^3+3*x+7; g:=x->tan(x^2+1);
       
                                        3
                             f := x -> x  + 3 x + 7
       
                                            2
                             g := x -> tan(x  + 1)

    you get a strange result if you ask Maple for D(f*g); (although D(f*g)(x); is fine). What looks better is diff(f(x)*g(x),x); or unapply(diff(f(x)*g(x),x),x);

  3. Here is a little quirk of Maple. 
           > a:=5; b:=7; f:=x->a*x+b;

                                     a := 5

                                     b := 7

                               f := x -> a x + b

    Why didn't Maple respond f := x -> 5x + 7 instead? This is another situation in which Maple behaves better with expressions than with functions. To get the expected result (as an arrow-defined function), you have to say f:=unapply(a*x+b,x);

A basic World-Wide Web home page

Your first project for the semester is to build a World-Wide Web home page and to help someone else build one. The page is supposed to contain at least one link and one image.

Today we are going to get started by building a simple home page. During the semester, you will add links and images and further information to the page, but today the goal is just to get a page that works.

World-Wide Web pages are written in Hyper Text Mark-up Language (HTML). This language is similar to LaTeX in spirit. You prescribe the logical design of the page, but not the visual design. Thus, you are allowed to say, "This is a level-three heading," but you are not allowed to say, "This is centered text in a fourteen point bold sans serif font."

In HTML, most of the mark-up instructions come in grouped pairs, like LaTeX's \begin{math} ... \end{math} structure. A typical example of an HTML "tag" is <EM>this is emphasized text</EM> (emphasized text is typically displayed in italics, but the font choice is left up to the individual Web browser). Notice that the start tag is surrounded by angle brackets, and the end tag looks just the same except that it also has a slash. It does not matter whether the tags are upper case or lower case.

Here is a template for a very basic Web page. Use the mouse to cut this template and paste it into a text editor. Then modify the information to apply to yourself. Notice that blank lines do not start new paragraphs, so you can use blank lines to make your input file more readable. (You must use an explicit <P> tag to start a new paragraph.)

<html>

<head>

<title>A sample home page</title>

</head>

<body>

<h1>My home page</h1>

<P> Here is some information about me. Notice that the paragraph tag P does not need a matching end tag.

<h2>This is a level two header</h2>

<OL> <LI>This is the first item of an ordered list.
<LI>The list item tag LI is another one that has an optional end tag.</LI> </OL>

<h3>Information about me</h3>

<ul>
<li>My e-mail address
<li>My telephone number
<li>And so on; this is an unordered list.
</ul>

</body>

</html>

Here is what will display on the screen:

My home page

Here is some information about me. Notice that the paragraph tag P does not need a matching end tag.

This is a level two header

  1. This is the first item of an ordered list.
  2. The list item tag LI is another one that has an optional end tag.

Information about me

After you edit the HTML template, you need to save it as a file and be sure that Web browsers can find it. Each computer system has a special location where it expects to find publicly accessible HTML files. On the calclab machines and the main mathematics server, the location for HTML files is a subdirectory named public_html in your home directory; this subdirectory has probably already been created for you with the correct access permissions by the system administrator. (On tam2000, the subdirectory is named .public_html with an initial period; you will have to create it yourself and make it world-readable and world-executable.)

Save your basic home page in this special subdirectory with the name index.html (this is a standard name for home pages). Use the "Save As" feature on your text editor: you can probably navigate to the subdirectory public_html with a mouse click, and then type in the name index.html.

There is one more step: you need to change the access permissions on index.html to make it a world-readable file. Open a terminal window, execute the command cd public_html (on tam2000, cd .public_html) and then the command chmod a+r index.html.

Now test it out. Open a Web browser (Mosaic or Netscape), click on the "Open" button, and type in http://calclab.math.tamu.edu/~your_user_id/ (if your home page is on the calclab machines) or /~your_user_id/ (if your home page is on the main mathematics server) or http://http.tamu.edu:8000/~your_user_id/ (if your home page is on tam2000). Your home page should display in the browser. Now have your neighbor try to display your home page (this is an extra check to confirm that the file permissions are correct).

When you have time to go further, you can look at some information for building home pages from the Web Beginner's Corner at Texas A&M University, and you can investigate some of the html resources on our class list of resource materials. Other tips:

A LaTeX exercise

This exercise was originally planned as homework last week, but since we ran out of time, it is now part of today's classwork.

Recall that in the Maple assignment from last class, you determined a certain third degree polynomial p. A little experimentation with Maple will convince you that p(n) is always an integer when n is an integer. Find a proof of this fact, write in LaTeX an explanation suitable for high school students, and turn in hard copy by the end of class.

Remark. You can write the polynomial as a fraction whose numerator is a polynomial q(n) with integral coefficients, and whose denominator is 6. Hence what you need to show is that the numerator is divisible by both 2 and 3 (and hence 6) when n is an integer. If you were writing a solution for a graduate class, you might (for example) invoke Fermat's little theorem, which says that a^p is congruent to a modulo p when a is an integer and p is a prime. For a high-school audience, however, you need to give a more elementary, bare-hands presentation.

Finished?

If you have done all the above activities, and there is still time left, here are some other activities you can do.

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Harold P. Boas
Last modified: Sat Jul 30 10:07:18 EDT 2022