Final Exam Review, Due December 10

• Which topics and theorems do you think are the most important out of those we have studied?

I think the most important topic is, frankly, how to prove things! The different methods that can be used (direct, contrapositive, contradiction, induction, etc.) as well as the basic structure of how proofs should look (logic, quantifiers, assuming the premise, etc.)

Also, the whole concept of sets was a relatively new topic for me, and pretty much made everything in this course possible.

• What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

- How to prove that things are unique. (For example, the intermediate value theorem, proving that there exists one root in a certian interval)

- Review well-ordered vs well-defined and when we use these. 

12.4, Due December 8

Difficult: Well, I do understand how to use these theorems. And I suppose I understood what was going on when they demonstrated the proofs, but on the division and multiplication one, the lemmas they chose to construct were quite.. overly convenient. Like I have no idea how they would do it, they would just prove random stuff, and then it all magically turns to epsilon!

I guess it would be helpful to help understand why and how they would bound f(x) and g(x). Maybe we could discuss what it means in general to have things "bounded".

Interesting: Like I said, it is really interesting and cool to use these proofs once you get through the gross epsilon-delta proofs. But I guess that's what we're here to learn, the gross stuff that makes it all work!

12.3, Due December 5

Professor Jenkins, I had this blog post typed last night, but I guess I accidentally saved it as a draft instead of publishing it. I apologize, but don't worry about changing it in the gradebook, since it was my mistake. 

Difficult: Holy cow, this section was a whole lot to take in!! Given I've never dealt with epsilon before, it was quite easy to get lost in the Greek letters. I understand the logic of what we must prove (I was able to write it using quantifiers! Yeah!) However, the biggest difficulty for me was figuring out how to choose each delta. It makes sense for linear expressions, but once we get to quadratic, polynomial, and rational expressions, I got lost where we were going.

What do they mean by upper bound? They never clarified what that was exactly referring to.

Interesting: This is really interesting, because in my Calculus AB class, to find limits we just looked at graphs and said the limit existed. But now we can prove it!! I feel pretty cool when I win at these.

12.2, Due December 3

Difficult: Awesome, so I understood why we use sequences of partial sums, but will we always need to prove two lemmas to prove convergence? The first lemma to prove {s_n} can be expressed in terms of n and the second to prove that the limit is convergent to a limit? I guess once we have those, we just plug those limits in and the proof is pretty easy.

And, holy cow, it looks tricky to prove a sequence diverges! Maybe it was just because the harmonic sequence is specifically difficult to prove, But those algebraic manipulations didn't seem intuitive at all. Is there a general structure for a proof that a series is divergent? Or does it always just depend on what the series is?

Interesting: Well, I've always used the words series and sequences interchangeably, but now I know the difference! A series is just the numbers in a row, where a series adds them all together! I've also hear of the harmonic sequence before, so it's really cool to finally know what it is. I can't believe it diverges, it seems like it wouldn't! Oh well, my intuition has definitely deceived me before.