Three Down One Left

Three midterms completed, one is left next week. I’ve got two major assignments due next week, but they are all but completed. Another one is due on the week of the 27th of March. I still have to start that one.

I got B in my first one midterm. I lost most of my points in the first exercise which I had trouble with. It was worth a quarter of the exam. I did the second midterm on Thursday. It went very well. My third was on Friday. It went OK. There was one exercise that I couldn’t complete. In retrospect, it was an easy one. I just needed to use some analytical tools to analyze a mathematical series. What messed me up is that this was a Numerical Methods class, not analysis. I didn’t think of using le Théorême des deux gendarmes or the Squeeze Theorem.

Squeeze Theorem

An example of a squeezed function

The squeeze theorem is formally stated as follows.

Let I be an interval containing the point a. Let f, g, and h be functions defined on I, except possibly at a itself. Suppose that for every x in I not equal to a, we have:

g(x) \leq f(x) \leq h(x)

and also suppose that:

\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L.

Then \lim_{x \to a} f(x) = L.

  • The functions g(x) and h(x) are said to be lower and upper bounds (respectively) of f(x).
  • Here a is not required to lie in the interior of I. Indeed, if a is an endpoint of I, then the above limits are left- or right-hand limits.
  • A similar statement holds for infinite intervals: for example, if I = (0, ∞), then the conclusion holds, taking the limits as x → ∞.

It’s a very powerful analytical tool to deal with infinite mathematical series, since it’s easy to major and minor them.

I’m getting very good at Analysis, I should have known to use those theorems. Then again, you can’t always get everything right. I still have half of the semester left to really do my best.

My brain was mush yesterday. I managed to get home and cook some goulash in the slow cooker. It took 7 hours to cook and was delicious. I’ll be posting photos and my recipe later on. Slow cookers are fantastic. I’ll be buying one when I get home to Taiwan. You just combine the ingredients, after having browned them in a pan, and let it simmer for hours.

Wednesday’s classes got canceled. That was a problem actually. I would have liked to ask some questions during my Numerical Methods class. It would have cleared up a few things. Live and learn.

The classes got canceled because of the snow storm. On Thursday, after my Analysis I midterm, I went back to my faculty building to find it evacuated. After waiting for over an hour, we learned that part of the roof had crashed in. There was some water damage. Afternoon classes were canceled. I went grocery shopping and went back home for a few minutes. I went to work at around 3:30PM and arrived back home at around 6PM.

I spent the night studying. I went to bed late, got up at around 5AM and continued studying. I missed my morning classes and went to my exam.

Afterwards, I didn’t take a nap, but lounged about in my bedroom for hours. I couldn’t really do anything. There was supposed to be a big storm. I was disappointed to see that it hadn’t yet happened.

I’ve been taking it easy today. I’ve done my washing, spent some time on the internet and thought about my stories. I’ve started a new story, but I’m going to start writing Symria again.

I’m pretty sure that I won’t publish it on my blog, until it’s completed and edited. However, I am thinking about publishing an excerpt and a synopsis in the coming weeks.

Author: range

I'm mathematician/IT strategist/blogger from Canada living in Taipei.

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