Math 300 Writing for Mathematics
This essay continues the scenario of Essay 1. You
will now describe some further counter-intuitive properties of infinite sets
for an audience that has had exposure to calculus II.
Recall that your essay should have a title,
introduction and summary, as well as a main body.
The
Blocks Unlimited store has expanded its offerings to include an Ultra Deluxe
Set of blocks.
Show that this set contains the Deluxe Set as a subset (which in turn, as you should have noted in Essay 1, contains the Starter Set as a subset). Thus argue that the Ultra Deluxe Set, when stacked, would reach infinitely high.
Recall the special paint from Essay 1. Show that this set cannot be painted with a finite amount of the paint. Show that, in contrast, a finite amount of ordinary paint (in fact less than 3 cubic feet) would suffice to fill the blocks if they were hollow.
Compare the Ultra Deluxe Set with the infinite horn generated by revolving the
graph of from to about the x-axis.
Show that this horn has finite volume but infinite surface area and so is somewhat like the Ultra Deluxe Set. Give the value of the volume.
The
examples given in the previous paragraph are to be discussed in detail in Essay
2. You should include in your essay
some additional examples of your own creation or a discussion of the following:
Consider
the horn generated by revolving the graph of from to about the x-axis. How
does this relate to the Ultra Deluxe Set?
An
outline for this essay is due in class the seventh week of the term.
1. This assignment is a sequel to earlier assignments of Baldwin, Berman, Thulin and Radford and is based eventually
on an exercise in “Writing in the teaching and learning of mathematics”, J. Meier and T. Rishel, MAA 1998.