Analysis
Here are the analysis pictures!
Here is our class picture -- the smartest students at Lamar!
First day of class.
Lindsey's Proof that 2.5 is a limit point of (2,3).
Bekah's Proof that no point outside of a closed interval is a limit point of the closed interval.
Kimberley's Proof that every set that has a limit point has at least two points.
Kimberley's Proof that the endpoints of an open interval are limit points of the open interval.
Chris' argument that Z+ has no limit points (he's going to write that up and distribute).
Jennifer's (yes, Jennifer's, not Jessica's) start to the proof that if p is a limit point of the
intersection of two sets, then p is a limit point of each set. We left her with the question,
"why is (a,b) a subset of H?"
Jennifer's proof that if p is a limit point of the
intersection of two sets, then p is a limit point of each set.
Lisa tackles problem 6!
My doodlings on functions and Megan's definition for "first point to the left."
Megan is pending one lemma in her effort to show that every set which has the property that every point in the set
has a first point to the left in the set and a first point to the right in the set has no limit points.
Class! Hang in there -- I know that we struggled to understand two proofs today and did not quite get to fully correct presentations, but we will! You are doing great -- keep working. Gotta' go grade!
Ted gives small convergence exercise to help understand the definition of convergence. Work on it for Monday.
Lisa makes nice attempt at P6. Almost there (really!).
Callie makes a nice attempt at P7.
My apologies to the class for forgetting to post on Monday!
Kimberly knocks out P3.
Lisa presents P6.
Ted's chicken scratches on convergence, Chevy's, Ford's, and limit points.
Callie gets very close to P7 -- she'll put it up before class on Friday, right Callie?
Allison tackles P8.
Ted talks more about limit points, range of sequences, and convergence.
Humberto! Hope all of you all avoided damage!
Callie finishes P7.
Meagan finishes Question 1
Ted presents simple convergence argument and discusses "there exists" vs. "let..." Note how much prettier your
arguments are than mine! Good job!
Jeremy handles P9.
Jennifer handles P10.
I stated something incorrectly in class! Let me correct it here in print! If you are close to solving a problem which is about to be presented at the board and you wish to receive credit for it as a WRITE-UP, then you may step out of the room so that it is still a "new" problem to turn in on that Friday. In class I said you could turn it in for "presentation credit" which was a mistake on my part. I apologize for confusing everyone! I also said that one could not receive both presentation credit and written credit on a problem, which is wrong as you often receive credit for preseting something one week that you wrote up during the last week. If you have any questions after reading this, we can talk on Monday!
Lyndsey presents P11.
Ana presents P13.
Jeremy makes it half way through P12 which he will get on the board early on Monday! For some reason, I don't seem
to have a picture of the second board he was presenting which had two cases p_k = p and p_k not equal p.
Jeremy tackles second case of P12.
Discussion of P14.
Recall that on Friday, you will place homework in my box on my office door by 1:30. Chris will take pictures *and* Chris will break ties if there is difficulty deciding who presents what! Start with P16 by Jillian. Move to P14 with whoever has least presentations or Jeremy if no one has it other than Jeremy. Then move to the lowest numbered problem that someone has -- we know Megan has P21 and P29. Each of you *must* email me a report on what occurred during class. Jennifer, did I get a copy of P10? I tried taking the pictures without a flash to see if it would cure the angle/reflection problem. I won't do it again -- too dark!
Jeremy tackles second case of P12 again -- ask if you have questions about this on Monday.
Chris tackles P17 -- first points to the right are limit points. Nice job of defending the argument!
Jillian tackles P16 -- no set has a FPR and a RMP.
Wow! It looks like you all tackled a lot of problems!
Ted discovers miracle Cheeto which looks just like Komodo Dragon!
Jillian wraps up P16 -- no set has both an FPR and a RMP.
Megan bumps Jeremy and tackles P14.
Lisa tackles P15.
Allison tackles P18.
Class -- I am sorry -- I forgot to take the last round of pictures!
Ted discusses induction, as related to P19.
Allison completes P18, but Ted forgot picture!
Jill starts P19, but Ted forgot picture!
Jill works on P19 again. Ted forgets to take pictures, again. Bad Ted. Please remind me in class! Chris works on P20 and every one agrees he has case one, but has not written a proof, although he gave one via pictures and words. Both of them are first-order-of-business on Friday!
Jill completes case two of P19.
Lisa presents P15 and Ted discusses.
Chris presents P20. Class is doing a nice job of asking questions! :)
Jennifer presents P22.
Lyndsey B. presents P23.
Except for the excessive use of duct tape, class is now rolling and doing very well. Some are starting to see the progress on their write-ups! Good work!
Chris defends P21.
Jeremy defends P24 -- we'll print it out for the next class and let him defend it verbally to save time.
Who has P25?
Jeremy handed out P24, so it's not on the web. Bekah began P25 so it will be completed on Friday!
Bekah concluded P25.
I discussed the three equivalent definitions of continuity.
Lisa handled begain to prove that every closed bounded point set has a right most point and a left most point. She handled the case where the set is finite.
Kimberly handles P26.
Lindsey T. handles 28.
Jessica McGee handles 29.
My quick notes on limits made precise and on upcoming class events like midterms and practice sessions.
The class asked for photos of the "negation" seminar. Mostly we talked about negating the definitions of
limit points, convergence, and continuity. We also reviewed contradiction and Jillian's proof that no set
has both a right most point and a first point to the right.
Allison puts P30 on the board and defends and I forget to photograph second board -- she will provide me with a copy
and I will make copies and pass them out.
Jillian puts P31 on the board -- to be completed tomorrow.
Jillian conclues P31 from yesterday (see pix above). Megan tackles P32.
Allison starts P35.
Megan concludes case 2 of P32.
Allison hits P35 again.
Julie puts up P27.
Jennifer puts up P36.
Here is what went on the board during the second problem session.
(1) Negation of "f is a first point to the right of M."
(2) Proof that every finite set has a right most point.
(3) Rough sketches for proofs involving convergence.
Jeremy puts up 33 and asks, full credit or not? Hmmm. Minor glitch in proof meets with minor glitch in problem statement!
Also, there is one photo about tangent lines mixed in.
Megan puts up 37.
Allison completes P35.
I sketch that if f is cont on [a,b] and p is in [a,b] then the set of all points in the domain mapping to f(p) is closed.
Jennifer handles 39.
Lyndsey B. starts 41.
Ted scribbles about P41, P42, P43.
Ted rambles about differentiability.
Jill handles P42. Lyndsey passes out 41.
Ted talks about integration.
Megan handles 45 and Ted promises to talk more about it on Monday.
Lisa starts 43 -- class requests proof of "f not continuous at x."
Lisa makes a second attempt at 43 -- class accepts proof of "f not continuous at x" but case where x is FRP needs
to be completed and written up for passing out.
Ted shows how T43 may be used to immediately get T44.
Allison handles 46 - a function trapped between -x^2 and x^2 has derivative 0 at the origin.
Bekah writes up 47 -- we'll copy it on Monday and she can defend first thing Monday!
Bekah rewrites 47 (that's dedication) and defends it. Interlaced in her proof is the same proof using the "epsilon-delta"
definition for derivatives.
Kimberly handles 48 -- all four boards worth.
Ted talks about countability.
Julie tackles 49, it is the first order of business for Friday's class.
Bekah tackles 50 and has the case c > 0 on the board -- we'll address questions about this case on Friday and she'll
provide a write-up of c=0 and c < 0.
Julie completes 49.
Jennifer presents 52.
Ted recaps Jennifer's argument of 52 which went by in 5 minutes last period! Then discusses 61, 62, and 72.
Lindsey T. presents 61 -- Lindsey's Lemma still outstanding: LL: very continuous function on an interval has a maximum on that interval.
Final Problem Session...
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