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Last week, the New York Times Online published a few pieces providing information on the Debt Crisis in America. My favorite was Charting the American Debt Crisis. The third panel (reproduced below) breaks down our debt by who holds the debt (the left side) and how we acquired the debt (the right side).
In my summer school PreCalculus class today we continued our discussion of Polar Coordinates. I began the class by asking the students to draw the graph of:
r = 2sin(θ)+2 by hand. Their gaph should have looked like this:
We discussed the process of translating between rectangular coordinates and polar coordinates when a student asked “can we just transform the entire function?” This led to a geeky, yet exciting, hour discussion and investigation.
First, we simplified the problem by dropping the “+2” from r = 2sin(θ)+2 (don’t worry we’ll go back to it…). So we algebraically converted the equation r = 2sin(θ) into rectangular coordinates (x and y). After substituting the conversions in (y=rsin(θ) and r2 =x2+y2), the algebra cleans up nicely by completing the square and a circle with center (0,1) and radius of 1 emerges. After half the class had already plugged the polar coordinate equation into their calculator, we all agreed that this process worked for r=2sinθ, but what about r = 2sin(θ)+2?
Here’s the input into Wolfram Alpha: graph y=2sin^2(aCos(x/(sqrt(x^2+y^2))))+2sin(aCos((x/(sqrt(x^2+y^2))))
Anyway, here was the graph Wolfram Alpha spit out:
Where’s the bottom!!!!
So, in our equation, to find the angle of the polar coordinate in terms of x and y, we had to use inverse cosine, whose range is only zero to pi. In order to get the bottom, we needed angles from pi to 2pi. To find that, we had to shift our cosine by pi; that is, we had to add pi to each of the angles:
Wolfram Alpha input: graph y=2sin^2(aCos(x/(sqrt(x^2+y^2)))+pi)+2sin(aCos(x/(sqrt(x^2+y^2)))+pi)
Wolfram Alpha output:
But how do we get both? A simple google search (remember this is the middle of class and my computer’s monitor is projected to the entire class, so a little bit of a gamble… good thing Wolfram Alpha isn’t a subsidary of GoDaddy!) landed me on this blog: Gower’s. Somewhere in the comments, I found that a simple comma would allow my to graph both equations together, so i input the following into Wolfram Alpha:
graph y=2sin^2(aCos(x/(sqrt(x^2+y^2))))+2sin(aCos(x/(sqrt(x^2+y^2)))), y=2sin^2(aCos(x/(sqrt(x^2+y^2)))+pi)+2sin(aCos(x/(sqrt(x^2+y^2)))+pi)
Wolfram Alpha thought for awhile, and returned nothing.
PLEASE HELP US if you know how to get wolfram alpha to graph two intense equations. (note: I can graph two simple graphs: input: graph y=2x,y=x^2)
I updated the article I’m reading and the site I’m visiting. In order to archive:
Article I’m Reading: Madoff’s Curveball- Jeffery Toobin Jeffery Toobin’s piece about Mets owner Fred Wilpon and Bernie Madoff. Warning: long
Site I’m visiting: MLB Trade Rumors The best site for up to the minute trade rumors for every team
With the MLB trade deadline approaching and the NFL off-season condensed into one week, there are some interesting trade rumors flying around in Philly.
For the Eagles, Dominique Rodgers-Cromartie (DRC) would turn a weakness into a strength for sure and he would help the Eagles win now. The Eagles need to think about winning this year, because Mike Vick will not be the same next year or the year after. Full NFL seasons are going to start to take a toll on Vick and the Eagles will soon be on the look-out for a new QB. There’s no doubt in my mind that the Eagles backup will start at least 3 games this year. I’d like to see the Eagles trade Kolb for DRC, but also go after a guy like Brian Hoyer the #2 for the Patriots. With their pickup of Ryan Mallet, they have a solid backup for the future, so Hoyer, a good backup himself would be expendable. In any case, the Eagles need to have a backup plan at QB once Kolb is out of the picture.
Over at CItizens Bank Park, the Phillies rumors surround a bat. Renting Carlos Beltran has a nice ring to it, but I don’t think he is the answer. On the other hand, Hunter Pence would fill the gaping offensive hole left by Jayson Werth’s departure. Pence is in the last year of his contract, but if the Phillies were confident that they could re-sign him (they have Ibanez, Lidge and Oswalt’s contracts coming off the books… 33 million), so they can afford an expensive addition. Plus with Pence signed, the Phillies would have more leverage in negotiations with Jimmy Rollins this winter; his offense could be considered expendable.
The debate about acquiring Pence surrounds trading Dominic Brown. Over the past few years, prospects have become extremely over valued. I heard Brown compared to Darryl Strawberry recently. I think a better peak projection would be exactly what Hunter Pence is right now: .300 average, 25 home runs, and 15-20 stolen bases. Pence has a proven track record of success, while Brown only has question marks. Some trades work well for both teams (see Red Sox/Marlins deal featuring Josh Beckett, Hanley Ramirez, et al.). I think a deal of Pence for Brown gives both teams what the desperately need, as long as Pence is not a 3 month player (yes, I am counting October…).
This weekend, the New York Times included a section called “Education Life,” which featured some discussion of the purpose, role, and value of grad school. One of their “Most Emailed” articles is titled “Master’s as the New Bachelor’s” by Laura Pappano. I wanted to share a few comments given some research I had done on the subject and I also thought this was appropriate given the article I shared at the right in the “Article I’m Reading” section of my blog: Louis Menand’s New Yorker Piece Debating the Value of College.
These articles consist almost entirely of anecdotal evidence. I am unaware of an empirical analysis, but last year, I presented a paper by Dominic Brewer, et al. titled “Does it Pay to Attend an Elite Private College.” The paper considered the different economic returns of different types of colleges (private, public, small, and big). The punch line is:Yes, an Elite Private College has significantly more economic return than other colleges. BUT, the paper considered data from the seventies and eighties. This study needs to be reproduced given the vastly different college environment almost thirty years after the data was collected (the paper is from 1998).
Specifically, with the explosion of tuition, is an elite private undergraduate education worth the money. Also, with such a high proportion of high school graduates going on to college, is the elite undergraduate degree worth the same, or more depending on graduate school experiences? Also, thirty years ago students chose their college, while over the past few decades, colleges have much more control over who is admitted. Finally, since the recession and (hopefully?) post-recession, are labor market returns any different given workers’ educational background?
So, is an undergraduate degree “worth” the skyrocketing cost? I would agree with the paper I cited earlier by Brewer, et al. that only elite private colleges garner a return worth the significant tuition differences to large public universities. That being said, according to Ms. Pappano’s NYT article and Menand’s “First Theory” , maybe undergraduate degrees will not provide the signal that we prescribe to them. Is acquiring a Master’s or other graduate degree just the new sorting mechanism for potential employers?
Each week I post an article I read recently or that I think deserves a read. This past week, I shared a New Yorker article that considered the value of college by Louis Menand. I will post some comments in a separate post, but in order to archive this article, here’s the link:
This morning I was thinking about the value of an MVP award in sports. Specifically, I wondered whether two star players with similar stats over their careers would garner the same salary if one had an MVP trophy and the other did not. Similarly, I wondered if a team with an MVP player would garner more interest, sell more tickets, and ultimately make more money than a team with a similar player without the trophy. I am not quite sure how I would go about testing either question empirically, but it is something I will look into in the future…
As I was checking up on my geeky fantasy baseball team (NL-only, auction, keeper, contracts, minor leaguers), I found that the Padres sent Anthony Rizzo to the Minors one day before he lost rookie eligibility. He has 44 days of Major League service and the cut-off is 45 days (excluding September 40-man roster expansion service). While I know, understand, and appreciate why teams wait to bring up a prospect to extend arbitration (http://phuturephillies.com/2011/05/13/domonic-browns-service-time-a-consideration/) I wonder if the cut-off for rookie status impacted the Padres decision at all.
Taking a slight variation from the original questions I posed in this post: Does a team with a Rookie of the Year candidate garner more interest, sell more tickets, and ultimately make more money? Was Anthony Rizzo’s 44 days of service a coincidence or were the Padres trying to preserve his ability to make a run at next year’s rookie of the year award? As I said earlier, I’m not quite sure how to test this yet, but as the link above referenced, Dominic Brown of the Phillies had significant service time last season, but preserved his status as a rookie for 2011. I’m just not sure that Dominic Brown is the driving force behind the Phillies selling out every night…