Special Interest Group in AOLA (SIGAOLA) of an e-Learning Platform in Mathematics Education
Hosting of Mathematics Contents by Professor Ivan Cnop
Please read the calculus-setup motivation first. [Calculus setup]
The first notebook immediately sets the scope of continuity in a technology environment. It gives a first constructive proof. In fact we are zooming in on the graph around the x-axis.
A tangent line should for now be introduced by zooming in at a point on a smooth graph. This is done in many other texts, and I am not claiming originality.
This text offers some animations that are not new either. They help to get a intuitive feeling of Rolle theorem, the proof of which is difficult in the general situation: the formal proof uses completeness of the real number system. It is possible to use the intermediate value theorem if one assumes the derivative is continuous.
The text also shows a first application in economy.
The Rolle text should be remembered for later use: its "zero growth" comes before growth, it helps find solutions to maximum (or minimum) problems, and it is a theoretical basis for polynomial approximations.
While rewriting calculus material I want to convey the message that in many cases the way a limit is approached (a metric property) is as important as the existence and its value (the topological properties).
To show the importance of the behavior near a limit, this notebook
- shows how the limitations of a CAS (even a powerful one) gives misleading information;
- shows how simple mathematical reasoning can get around these shortcomings.
It is by no mean the first in a calculus sequence (rather one of the final ones).
Limits At Infinity
The notebook about growth of functions being large I find it better to collect the material about fitting infinite graphics on screen separately.
The main effort is in setting tick marks on a non-linearly compressed half line. This is what makes the compression visually apparent, and it can be used independently. Without ticks it is too hard to believe that we have obtained what is called a "one-point compactification". It remains possible to hide the programming for the students.
I have been using such unevenly spaced ticks for Listplotting sequences for a long time now, and my students have been convinced more easily than by using the standard ListPlot of the first 10 or 25 first terms of a sequence, as is done in most textbooks.
The notebook will prepare students for the introduction of big-O growth orders ("eventually dominated by a multiple of") to be introduced next. It is one of the more difficult concepts for them. Most of the time you have to look for very large x (outside the usual plot ranges) before you actually see the dominating property.
For website applications the final part is the most useful since it shows how to compactify the full coordinate plane into a square.
Using the tan function for compacting reals we have obtained striking symmetry between decay near zero and the growth at infinity for powers. Both are shown by the order of contact with corresponding horizontal (entire powers) or vertical lines (roots). It sheds new light on "order of growth". Try more functions as in the other notebook (ln and square root) and animate the graphs to convince oneself of differences in their asymptotic growth.
Uniform continuity on intervals, is a much more important concept than pointwise continuity and should be learned in calculus instead of the pointwise epsilon - delta stuff.
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