College Math Teaching

March 29, 2020

A change of variable to determine if growth is still exponential

This video is pretty good, and I thought that I’d add some equations to the explanation:

So, in terms of the mathematics, what is going on?

The graph they came up with is “new confirmed cases” on the y-axis (log scale) and total number of cases on the x-axis. Let’s see what this looks like for exponential growth.

Here, letting the total number of cases at time t be denoted by P(t) , the number of new cases is P'(t) , the first derivative.

In the case of exponential growth, P(t) = Ae^{kt} where k is positive.

P'(t) = Ake^{kt} which is what is being plotted on the y-axis. So with the change of variable we are letting u = Ae^{kt} and our new function is F(u) = ku , which, of course, is a straight line through the origin. That is, of course, IF the growth is exponential.

To get a feel for what this looks like, suppose we had polynomial growth; say P(t) = At^k . Then P'(t) =Akt^{k-1} = ak\frac{t^{k}}{t} =ak\frac{u}{u^{\frac{1}{k}}} =aku^{\frac{k-1}{k}} In the case of linear growth we’d have F(u) =ak (constant) and for, say, k = 3, F(u) =3au^{\frac{2}{3}} or a “concave down” function.

Now for the logistic situation in which the number of cases grows exponentially at first and then starts to level out to some steady state value, call it L, the relationship between the number of cases and the new number of cases looks like P'(t) = akP(L-P)) so our F(u) =aku(L-u) which is a quadratic which opens down.

Yes, this gets studied in differential equations class when we study autonomous differential equations.

Now for some graphs:

This is exponential growth vs. logistic growth; we get something similar to the latter when cases start to peak.

Here, I tweaked the logistic model to have the same derivative as the exponential model near t = 0 .

Here: we have linear growth P(t) = 5t vs the F(u) = 5

Here: cubic growth P(t) = 5t^3 vs. F(u) = 5u^{\frac{2}{3}}

March 26, 2020

My review lessons online

Filed under: applications of calculus, COVID19, differential equations, linear albegra — collegemathteaching @ 11:04 am

We had an extra week to prepare to teach online, so I put notes from the previous few weeks up in blog form:

https://bradleylinearalgebra2020.wordpress.com/blog-2/

https://bradleyappliedcalculus.wordpress.com/blog-2/

https://bradleymth224differentialequations.wordpress.com/blog-2/

That was quite a bit of work, but I did find some cool videos out there and embedded them in my lessons.

March 24, 2020

My teaching during the COVID-19 Pandemic

My university has moved to “online only” for the rest of the semester. I realize that most of us are in the same boat.
Fortunately, for now, I’ve got some academic freedom to make changes and I am taking a different approach than some.

Some appear to be wanting to keep things “as normal as possible.”

For me: the pandemic changes everything.

Yes, there are those on the beach in Florida. That isn’t most of my students; it could be some of them.

So, here is what will be different for me:
1) I am making exams “open book, open note” and take home: they get it and are given several days to do it and turn it back in, like a project.
Why? Fluid situations, living with a family, etc. might make it difficult to say “you HAVE to take it now…during period X.” This is NOT an online class that they signed up for.
Yes, it is possible that some cheat; that can’t be helped.

Also, studying will be difficult to do. So, getting a relatively long “designed as a programmed text” is, well, getting them to study WHILE DOING THE EXAM. No, it is not the same as “study to put it in your brain and then show you know it” at exam time. But I feel that this gets them to learn while under this stressful situation; they take time aside to look up and think about the material. The exam, in a way, is going through a test bank.

2) Previously, I thought of testing as serving two purposes: a) it encourages students to review and learn and b) distinguishing those with more knowledge from those with lesser knowledge. Now: tests are to get the students to learn..of course diligence will be rewarded. But who does well and who does not..those groups might change a little.

3) Quiz credit: I was able to sign up for webassign, and their quizzes will be “extra credit” to build on their existing grade. This is a “carrot only” approach.

4) Most of the lesson delivery will be a polished set of typeset notes with videos. My classes will be a combination of “live chat” with video where I will discuss said notes and give tips on how to do problems. I’ll have office hours ..some combination of zoom meetings which people can join and I’ll use e-mail to set up “off hours” meetings, either via chat or zoom, or even an exchange of e-mails.

We shall see how it works; I have a plan and think I can execute it, but I make no guarantee of the results.
Yes, there are polished online classes, but those are designed to be done deliberately. What we have here is something made up at the last minute for students who did NOT sign up for it and are living in an emergency situation.

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