## Friday, January 18, 2008

### Back on My Grizzie

Well I didn't make a post about the first week of classes and there is a simple reason for that. Nothing happened!

In French that week was dedicated to reviewing material from the previous semester. Luckily we had this since over break I had forgotten some of the basic things we learned near the beginning of the year. Until Thursday I possibly had the most interesting professor. Not interesting in a positive way, she spoke French but with a Spanish accent. It wasn't bad enough that my comprehension of spoken French had drastically dropped thanks to the break but her accent certainly was not helping things (for those of you that are not aware, I took three years of Spanish in high school). Doesn't take a genius to figure that I switched into a different section as soon as one became available. The current teacher is the least organized organization freak. She prepares power point presentations for every topic we cover, and she occasionally forgets to bring them with her to class. She says the funniest things while teaching. Occasionally the projector would not work so she would need to write on the black board and would always mention how her spelling is horrendous, regardless of which language; English, French, or German, she chose to use. Getting another A in French shouldn't be too hard since she forces us to learn the vocabulary with her daily quizzes so that just leave me to learn the grammar, which is usually something trivial or close to it.

Managerial Economics is going to be a lot of work, not because of the difficulty of the material but rather because of the amount of "busy work" he is making us do. In addition to watching lectures and doing problem sets we must also read his stupid articles and post a comment. I think the worst part is that the class only meets two hours every Tuesday and Thursday. If it were my call I would choose to meet for one hour every day but Friday. That way it is a little harder to fall behind on the lectures since you don't have to watch a two hour video which requires you to set aside a good portion of your "free time". So far it looks like the only reason people have trouble with the class is because of the math. During the introductory lecture he said that he would try to use as little math as possible so it disturbs me how the math is what people find hard. They way he has chosen to teach the class requires only that one knows high school algebra and basic calculus, the class even has calculus 1 as a prerequisite. In any case getting an A in this class should not be a difficult task, just one filled with bull shit assignments.

Abstract Algebra at the moment is quite boring since we are going over basic number theory concepts. In addition to that the textbook really doesn't fit the kind of upper division books I which I prefer, the theorem-proof format. Advanced Calculus jumped right into material from the first second of class. This was expected since it is the second portion of a two semester course. We began with the proof of the generalized man value theorem (1). Currently we have just started looking at the Riemann-Stieltjes integral. These two classes, especially Abstract Algebra, should be my easiest two A's of the semester. Even though getting an A in Advanced Calculus should be easy I will continue to put a great deal of work into covering more than the professor does in our class because I wish to take the graduate level Analysis class next fall.

Of all the problems I have solved so far this semester, the following was the most pleasing, even though it was not the most difficult or interesting.

Suppose $f$ is a real, three times differentiable function on $[-1,1]$, such that
$f(-1)=0$, $f(0)=0$, $f(1)=1$, and $f'(0)=0$.
Prove that $f'''(s)\ge{3}$ for some $s\in (-1,1)$.

On nonacademic topics I have given up on someone. Since I don't feel like retyping the major argument I shall just copy it from another of my blogs.

"So I've gotten sick of being a second option. That's the main reason I've decided to say no. (I know that one person out there will know what I mean)
It all wasn't a big deal at first since I understand that sometimes emergencies happen. But the rate at which these "emergencies" occurred you start to wonder how many are real emergencies and how many are people bitching and moaning and not knowing how to handle their own "problems". Either way eventually you feel like you're always at the bottom of their list, even when you had a previously planned engagement.
Well I guess the last two days are what really did it for me. For starters the conversation we had wasn't the most comfortable thing for me, and I'm not referring to the portion about periods. I am honestly starting to believe that no good deed goes unpunished. Today you call of something we planned because you didn't realize that trying to do all your homework in the few hours u would be home on Monday really wasn't going to work. Guess you should have realized that before we made plans. However, I probably would not have been as ticked about this if I was informed before wasting 3-4 hours of my day waiting for you to cancel at the last minuet.
Either way I'm saying fuck it. I am no longer going to go out of my way just to be with you if it's obviously not as important to you. Maybe you will notice, maybe you won't, either way I'm better off since I'll be with people who want my company just as much as I want theirs."
As previously stated this is one of the two main reasons I have decided to no longer "pursue" her. Currently not many people know about this so worst case scenario is that we remain friends. Some of you might say that I am being selfish but here is my opinion on the matter. Always make number one first in your life. Until someone gets to the point where they have earned the number one status thats how it will be. Don't get me wrong I give people many opportunities to earn that status, the girl in the above section had probational number one status, but as you can clearly see that is no longer the case. Currently I can think of one person who is around 1.2 status, so attaining that precious number one spot is possible even for mortal women.
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(1) Let $f$ and $g$ be continuous real-valued functions on $[a,b]$ and differentiable on $(a,b)$. Then there exist a point $c\in (a,b)$ such that $\frac{f(a)-f(b)}{g(a)-g(b)}=\frac{f'(c)}{g'(c)}$. (2)

(2) Now some of you might think that Rolle's theorem is just a special case of the mean value theorem, but it turns out that Rolle's theorem is an essential tool for proving this theorem.

## Tuesday, January 01, 2008

### Convexity and Continuity

Not to disappoint I shall at least make one new post during my winter break. However, do not look for this to be a recap of how things went. If I could I would rather just forget that these three weeks or so ever happened. On the bright side I did however, manage to get some productive things done. I finished reading the chapter on continuity and have almost finished reading the chapter on differentiation. While working through the continuity exercised I ran into one that was particularly interesting problem. Before going any further I would like to mention that I had to ask for help in finding this solution.

Theorem:
Every real-valued convex function is continuous.

A real-valued function
$f$ defined in $(a,b)$ is said to be convex if

$\displaystyle f(\lambda x + (1-\lambda)y)\le \lambda f(x) + (1-\lambda)f(y)$

whenever $x,y \in (a,b)$ and $0<\lambda<1$.

Lemma: If $f$ is continuous in (a,b) and if $a< s< t< u< b$ then

$\displaystyle\frac{f(t)-t(s)}{t-s}\le\frac{f(u)-f(s)}{u-s}\le\frac{f(u)-f(t)}{u-t}.$

Proof of lemma: This is not to difficult to prove so I shall just give the major idea and leave the rest to the reader. Let $t=\lambda s + (1-\lambda)u\Longrightarrow \lambda = \frac{t-u}{s-u}$. Obviously $0<\lambda<1$, so use this new way to express $f(t)$ and the rest follows with ease.

Proof of theorem:

First we choose a point
$x\in (a,b)$ and a neighborhood of $x$ such that $N_{\delta}(x)\subset (a,b)$. Now using the lemma we have that

$\frac{f(x)-f(x-\delta)}{\delta}\le\frac{f(y)-f(x)}{y-x}\le\frac{f(x+\delta)-f(x)}{\delta}$, for $y\in (x,x+\delta)$.

Now from this we see that there is a positive number $C$ such that $|f(x)-f(y)|\le C|x-y|$. Let $K=\inf\{C:|f(x)-f(y)| . Thus if $|x-y|<\frac{\epsilon}{K}$ we have that $|f(x)-f(y)|<\epsilon$.