Posts
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Approximating π, part 2
Last time we met, we found a Taylor series for the arctangent function, and we used the fact that π = 4 arctan(1) to find approximations to π. Unfortunately, this representation had really poor convergence behavior, because 1 is way the hell out at the edge of the interval of convergence for this series. I’m loath to abandon a fun trick, though; is there a way we can iterate on this idea?
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Approximating π
A fun thing to do at the end of a Calculus II class is to use Taylor series to compute values of various functions. For instance, I have a whole problem set about the “68-95-99.7” rule, where we find a Taylor series for the normal distribution and then integrate it. Here’s another fun application:
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What shape is an hourglass?
If I ask you what shape an hourglass is, there is a very specific shape that leaps to mind. (Maybe there is also theme music.)
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GPAs are weird
The other day in class we were talking about something having to do with weighted averages1. As a familiar example of weighted averages, I had people calculate a weighted GPA for warmup (with weights given by credit hours). While we were discussing this calculation, I mentioned that I think GPAs are wack and we shouldn’t use them, but that I wasn’t going to get on my soapbox about this. So then a student asked about this on my daily exit quiz:
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The center of mass of a system of point masses in one dimension. If you don’t want to say a bunch of physics words about net torque, you can instead think of this as a weighted average of the locations of the masses with weights given by each mass. ↩
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Function transformations: new and old
This is one of the most annoying things to teach in any functions-based class: Supposing $a, b > 0$, why does the transformation $y=f(x-a)+b$ move the graph of $f$ to the right by $a$ and up by $b$, even though the signs are different? Why does the thing that’s happening horizontally work “backwards” from the thing that’s happening vertically? Prompted by a nice activity from Matt Enlow, I finally have an explanation that feels persuasive. I road-tested this with my “Functions Modeling Change” class this week, and I now am excited to share it more broadly.
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Pride is a protest
And it always has been
I had a great time marching in Westminster’s entry in the pride parade the other weekend. The focus of our entry was our upcoming name change from Westminster College to Westminster University, so we got Westminster University shirts and drawstring bags and signs to wave around, and here’s what all the pictures look like. Look at all these happy people happily waving happy signs with happy rainbow flags!
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MathFest in Tampa: Upholding Our Values
I was recently invited to write a piece for MAA Focus responding to the decision to keep MathFest in Tampa in the midst of numerous legislative attacks against trans existence in the state. It’s out now in the June/July issue, and I invite you to read it there in context with several other pieces.1 However, I’m republishing it here, because there is one place I’d like to add a very specific link in response to an editorial decision. See if you can, y’know, spot it, and see if you can tell what specific word the editor asked me to avoid saying.
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Look for a future blog here responding in particular to the well-intentioned remarks from Hortensia Soto, MAA President. ↩
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Pappus's theorem
In my Calculus II class, we were talking about solids of revolution, and how calculus allows us to pretty easily solve a suite of problems that the Greek philosopher-mathematicians spent a lot of time thinking about (mostly because it was economically useful to have accurate computations of the volumes of pithoi and amphorae, both of which are solids of revolution). I mentioned something about how Archimedes had solved a special case (the volume of a paraboloid), and that there was some other guy who did something cool in this regard, but I couldn’t pull the name immediately.
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Integrating something two ways
Today in Calculus II we were playing with the Wolfram Problem Generator, which gave us the following very tricky integral:
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The graph Laplacian and cohomology
I’m leading a directed study with a student about spectral graph theory this semester. (Long story short, he was in my linear algebra class, another student saw a cool thing about clustering using eigenvalues of the graph Laplacian in Tim Chartier’s book, and the three of us did some summer research work about it.) This means that I am learning a lot of stuff about graph theory and linear algebra, and this often bleeds out into other weird branches of mathematics. Here’s a good example, the moral of which is that if something is hard or annoying, you may be looking at it wrong.
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My new jekyll blog!
Here is some kind of intro post for this jekyll blog. Mostly it is the boilerplate that comes with a fresh jekyll install, which I am keeping around for my own reference.
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