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Author Archives: Persiflage
Some problems to think about
I had hoped to write some more problems and will return to this as soon as I can, but for now here are some (harder) problems to help you think about the course as well as some more recent material. … Continue reading
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Local versus Global
This is just to reinforce today’s lecture. The idea is that many concepts: The limit of \(f(x)\) of a limit at \(x = a\), The continuity of \(f(x)\) at \(x= a\), The differentiability of \(f(x)\) at \(x=a\), The point \(a\) … Continue reading
Another midterm.
Here are the solutions by ChatGPT to this week’s midterm. What grade would you give it? Give your grade in the comments! (Hint: there are definitely some things which are wrong here.) Problem 1 We need to show that \[ … Continue reading
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ChatGPT
I gave ChatGPT the midterm. It scored around 60/100. (Update: I just graded the midterm and was more generous, so 65/100.) It failed miserably on the graphing question, which is maybe expected since it doesn’t have the ability scan in … Continue reading
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The game of infinite intelligence
I’ve talked about how analyzing games naturally leads to inductive arguments, and gave a few examples. Here is a tricky game problem which is a little mind bending. It’s much tricker than anything else I have suggested, so consider it … Continue reading
The Sticks Game
Although I did this in class, I thought it might be useful to write out a proof here. There are many variations, but it should look something quite similar to this. Claim: For \(n = 0,1,2,\ldots\) a non-negative integer, the … Continue reading
Constructing Irrational Numbers using Cantor’s Argument
Cantor’s argument also allows us to “explicitly” construct irrational numbers. Let’s take a list which includes all the rational numbers between \(0\) and \(1\). First we add \(1\), and then we take \(a/b\) first over the possible denominators \(b > … Continue reading
Rational Numbers
In class we gave the “classic” proof that \(\sqrt{2}\) is irrational, but we also gave a different proof using analysis. For each pair of positive integers \([a,b]\) with \(a/b \in [0,1]\) with \(b > 1\), we considered the interval \[ … Continue reading
Hello World!
This will be the class blog for 16100 in Autumn 2024, and will be an occasional supplement to the regular course lectures. (I started blogging about courses during the pandemic, but students find the blog useful so have continued to … Continue reading
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