12.5 due April11

This is my LAST reading assignment for a math class EVER. At BYU. Who knows. Maybe I'll go on to get a masters degree is math someday. For now, though...this is it. What a bittersweet moment.

Continuity is a condition we have been discussing in class. We have had to be pretty careful because you can't say that x is in a certain range that covers the function at a point where it isn't continuous. So now we have defined the three conditions which must be satisfied for a function to be continuous at a certain point. The third condition (to me) is really the only necessary condition. Because if the limit as x-->a = f(x) = f(a), then that implies that the limit exists and that f is defined at a. But I guess we can go step by step if math says so.

Continuity is proving to be harder than expected. I guess because I figured all we had to show was that the limit exists. Which is true, but to match it up with the function is harder. Essentially we need the limit and the point in order to say that they are continuous, and I get it now... Just because there's a point there doesn't mean there were points before that one that lead up to it (the limit) and just because all the points are leading up to it doesn't mean that there's a point there. Fun.

12.4 due Wednesday April 9

It's always a good feeling when the beginning of a section starts with the idea of doing what we just did only we're going to make it easy now.

It makes sense that the limit of two added functions would be the addition of their limits. I say this because if you add two functions together, then the result is the sum of each individual result, which is what you would be approaching through the limit. As for the limit of the multiplication of functions, I would say that this is less intuitive, but we've all seen it before, and it makes sense. I especially enjoyed the case when M=0. I understood that one the first time through.

Probably the most confusing thing for this section was reading about the three deltas. I thought I understood min(1,a(epsilon)) was like bounds...where 1<delta<a(epsilon). But then we all of a sudden have delta=min(delta1,delta2,delta3)...are those three dimensional bounds. So I'm confused. Then in the proof, I'm super confused with how we got to epsilon/2 for the second part. I'm assuming we will derive this in class though...? Hopefully a good assumption.

The good news is that I genuinely believe that this box of tools is going to make math easier. :)

12.3 due Monday April 7

Alright. Office hours with Dr. Doud went well. He writes his proofs a little different than we do in class, and I guess I needed that added dimension. I now know what's going on. I think. And now a play-by-play of section 12.3:

SO 12.3 is just taking what we've learned already about finding limits (from 12.1) and applying it as a limit at n--> a, where a is an integer value, rather than infinity. So we can find a limit for a divergent series. That makes sense, and the proofs appear to be largely the same as before as far as what we're proving. We're proving that if you go out a certain distance (delta...?) from a, then you will always be within epsilon of the value a.  [[I really hope that is the definition of delta.. if it is, then I think I actually am starting to get this stuff!]]

We choose our distance delta the same way that we chose the value N. By solving for f(x)<epsilon. And our new assumption for this section is that 0<f(x)-L<delta. And since we define delta in terms of epsilon, we can basically say delta(epsilon). Things are going well.

For the quadratic (and other) proofs, the only thing was defining the trick that helped us find delta. I could follow it all, but I probably wouldn't have thought of any of that stuff with the lim as x-->3 of x^2 is 9. I hope that in the future the tricks are similar so I can actually find it. ok \qed for now.

12.2 due April 4

So the homework was pretty confusing and I've concluded that maybe I just didn't understand the last section...but here's a shout out to section 12.2...

From my observation, the proof strategy is essentially to take a summation and try to reduce it down to a limit, and then prove it the same as we did in section 12.1. I get how we choose the N and I understand how it works for n > N...I guess I just don't understand what the point of solving for epsilon (sorry, I'm not going to go on a greek letter hunt) is. Maybe I don't understand what E is at all. However, if we can prove the limit of the sequence, we can say that thus we have proved the summation as well.

I'm now going to use my words to try to understand what epsilon is. So. We are trying to prove a limit. If we can show that for some arbitrary integer less than 0, a sequence will equal a certain result less than epsilon, then we have shown that the sequence approaches a certain number. (...?) I guess that makes sense. So then we use the value we chose for N to solve for n and show that the sequence is less than epsilon for that sequence. I really hope that's right.

Now back to section 12.2. I am pretty fascinated that you can turn an infinite series problem into a limit/sequence problem. Are we just supposed to write out the first few elements of the sum and then make an estimation of what the nth term in the set would be, and then take the limit and solve?

I will most likely be taking a trip to office hours with your friend tomorrow.

12.1 due April 2

So that test was super fair, I thought. Then I came out and found out I did way worse than I thought I did. In fact, I found out I did 10% worse than last time. So....hopefully we don't repeat this pattern on the final.

Regarding section 12.1...I've never heard of the ceiling of x. Is it the same as what something converges to, but just in different notation? I like this idea of limits. I never liked it until recently, but while we were doing number theory, I couldn't help but think about limits. Turns out the real numbers are awesome (even though they  make proofs way harder.) I suppose that us having a point that numbers converge (or diverge) to is useful. Perhaps later in the chapter I will fully understand how this relates to calculus.

Why do I never remember the rules for inequalities. Don't you think I should know these things by now? I didn't realize that if you take the inverse of both sides, you have to switch the sign. Learned that in an example. If that's not true, then I'm officially confused.

I think my confusion for this section boils down to the fact that I can follow the proof, but I have no idea what we're proving or how the proof really proves anything. Usually problems of this variety solve themselves in lecture and when it's not midnight.

Again, nice to see you tonight! Hope your scouts enjoyed the show!

Thoughts, but mostly a question.

While studying for the test today, Ryan and I had a question.

How is it possible that (N) X (N) is countable, but 2^N is uncountable...? Because, isn't 2^N just (N) X {0,1}? That is all.

Studying for Exam 3 due March 31

Well, honestly I haven't spent a lot of time studying for the test yet. I will, though. I've been doing a lot of studying as we go though, because this unit has been quite hard for me (as you know). Based on what we've been doing in class and on homework, I would say that the most important things from Chapter 11 would be finding the gcd and the smallest prime factor. This is easy enough to be tested, but hard enough that we would have to think in order to do it. These seem like valid multiple choice questions. For Chapter 10, I would expect much simpler questions than the ones that we did on the homework. I might be wrong here, but some of those homework questions were ridiculously hard and long. So...simplistically, I think we could show what sets are denumerable and which are uncountable. We probably are going to be accountable for the definitions as always.

Hopefully I will do alright on the exam this time around. Monday-Wednesday are by far the worst days for me.  >.<

Also, I had tickets to Calculus the Musical and totally mixed up the times and missed it. My weekend was ruined by this. I hope you got to attend and enjoy it for me.