So the other day I was making a cup of tea, and asked my sister whether she would like some as well. Since I didn’t mind leaving the teabag in my own cup while I drank it, I realized that the easiest way to make two cups of tea of approximately the same strength would be to first put the teabag in her cup, and then to move it to my cup for the remainder of the time. This is because the teabag lets flavor out the fastest at the beginning, and gets increasingly slow at it as time progresses. Of course the tea flavor always increases; there’s no point at which the teabag starts reclaiming flavor for itself.

With all these words like “increases” and “increasingly slow” this sounds like a good opportunity for some calculus.

The graph above represents f(t), the derivative of tea flavor with respect to time. Note that df/dt (which will now be referred to as Tea/Time) is always positive, but decreases in value approaching zero as time approaches infinity. The strength of the tea–denoted F(t)–is the definite integral of this graph over some time interval.

Let’s make some assumptions to make this problem easier. Since I occasionally forget about my tea as I let it cool down before drinking, let’s assume that I don’t start drinking my tea until time t=infinity. Thus, the strength of her tea is equal to the integral from zero to whatever time I remove the teabag (called t=a), and the strength of my tea is equal to the tail integral from a to infinity.

We can of course solve for a algebraically:

Of course, this simply verifies our original observation that I want to remove the teabag at the point at which her tea is half as strong as the teabag would be if I waited until time infinity before removing the teabag from a single cup. We are, however, happy to see our integrals come out to a statement we already knew to be true.

–

I guess what I’m asking is, “Is it a bad thing that I started forming integrals in my head when my sister said she wanted tea?”

Yeah, I don’t really feel like writing a lot about orientation. I had a good time, so call me a bad blogger or something, but I’m not sure what there is to say. Very quick summary: Randomly ran into CJ on Monday. Hung out with Wells Scholars much of the time. Had a good meeting with my adviser on Wednesday morning, then a very nice talk with the math department about what math courses to take, and then a decent talk with the chem department about taking organic chem first semester.

My first semester courses are:

Mathematics S343 – Honors Differential Equations I
Chemistry C341 – Organic Chemistry I
Honors H205 – Origins and History of the Universe
History and Philosophy of Science X102 – Science Revolutions: From Plato to Nato
Telecommunications T160 – Videogames: Historical & Social Impact

Yeah, you read that last one right.

In terms of my schedule:

Monday: 11:15 CHEM, 2:30 CHEM
Tuesday: 9:30 HPSC, 11:15 MATH, 4:00 HON, 5:45 TEL
Wednesday: 11:15 CHEM
Thursday: 9:30 HPSC, 11:15 MATH, 4:00 HON, 5:45 TEL
Friday: 11:15 CHEM

Yes, my days are kinda lopsided… >_<

So John Wiltshire-Gordon and I realized we don’t actually have any idea how matrices work. Rather, why they work.

We figured out how Gaussian elimination works generally. It looks like if you have an augmented matrix [A | B] and you Gaussian-eliminate it into [C | D], that AD = BC, always. That’s why it works to find the multiplicative inverse as well as the solution to a linear system. We also traced the determinant throughout, and saw that each time you multiply the pivot, you change the determinant. (Well, we knew that already.)

The real question is, what really is a determinant?

I’m sure this will make some great Monday Mathematics columns later.

Yeah, I’ve been really busy lately. Give me ideas for more Monday Mathematics columns, and I’ll write more.

http://www.carloangiuli.com/graphics

LiveGraphics3D is an awesome Java applet that allows real-time rotation of 3D objects in a web browser. I have written this Perl/HTML wrapper to make it usable for anyone, not just people who can figure out how to run Java classes. A much better explanation and a whole bunch of graphics can be found at the URL above.

If you have some graph you’re just dying to see, but you don’t have Mathematica, you can leave a comment here and I’ll put it up for you. I’m nice like that.

Okay, I have some things to ask of you guys. In order of priority:

I am thinking of entering the New Trier Science Research Competition, perhaps both as an individual and team. If you would like to work with me on a team project, please get in touch with me, preferably with some ideas. I’m thinking these would likely be in the chemistry, mathematics, or computer science categories, because I’m probably not interested in your research project that requires me to cross-pollinate three thousand pea plants. Feel free, however, to propose any reasonable or unreasonable ideas for which you would like to collaborate with me. Surprise me.

I am thinking of writing AP study guides, particularly for Biology and Chemistry, and perhaps also for Physics C and Calculus AB and BC. They would be available online for a fee, and include a lot of helpful, detailed review information, unlike commercially-available AP review books, which have astonishingly little useful course content. (They do, however, suggest that you eat a good breakfast that morning.) Give me any input you have about this idea, especially how much you or others would be willing to pay. Remember that my first-year Chemistry review packet was ten pages for third quarter alone; these AP guides would be easily over 40 dense pages. I would probably only be able to finish one or two of them this year.

If you are interested in joining Scholastic Bowl and you are not on the mailing list, contact myself or David Reinstein now. Our first official practice is Wednesday in room 359.

Probably for Patrick only: If you help me get dvipng to work on my website, I will set up an online TeX rendering service with indefinite storage. You want me to do this.

If you are in Aegis Questions, write questions now.

If you have the 2.6.17 Linux kernel, I would like you to compile a few kernel modules for me. Alternatively, tell me how I can get them to compile on Knoppix 5.0.1, as it does not contain all the necessary C include files.

That is all. I hope to hear from some of you, especially about the science project and AP guides. If you read my blog but haven’t commented, now is a great time to start.

Monday Mathematics #2. Because .9 repeating really does equal 1.

I think Labor Day is a good time for the first issue of Monday Mathematics. The BC class is going to begin delta-epsilon, math team is going to start. Everything is timed so well.

This week’s issue is my thoughts on how to approach math, as an introduction to the series. Don’t worry, the next one will actually have math. But read this one too: it’s probably the most important. Because I know you sorts of people are responsible for the parabola shenanigans. I’d be interested to hear what anybody thinks, or still, any ideas people have for future issues.

I’m doing epsilon-delta proofs in MV, which is nice because pretty soon I’m going to have to explain epsilon-delta to the BC class I’m TA-ing. Particularly nice was #6 (lim{x->0} sin(x)/x = 1) because it had an elegant geometric proof. Add to that the fact that its veracity isn’t obvious until you know the series expansion for sine (or l’Hopital’s rule), and you have a pretty interesting problem.

I know why everyone has problems with epsilon-delta. They panic because they can’t find a pattern in all the proofs: each one works differently. But as long as you understand what’s going on (i.e., as long as you understand the definition of a limit) then you should be alright. I’ll try insisting that everyone attempt to conceptualize, but it never works–there are two good types of math students, the type that understands very well, and the type that apes very well. Proofs separate the thinkers from the apers. We shall see.

Until next time, may you always/sometimes/never evaluate the limit definition of a derivative with l’Hopital’s rule.

I opened my shiny new AP Chemistry textbook to the table of contents, looking for the section on organic chemistry. Expectantly, I flipped to the section on nomenclature, and looked for examples using infix numbers…and saw “2-propanol” and the same sort of crap from sophomore year.

You’d think that while teaching the nomenclature rules, they’d want to teach the right rules. I even checked the IUPAC Blue Book to make sure, and yes, it’s infix. So why, if it should be called “propan-2-ol,” do high school textbooks tell us it’s “2-propanol”? The textbook is a new edition, and the nomenclature was last updated in 1993.

Is infix notation just too much for highschoolers? Is putting numbers in the middle too radical? Last time, we conveniently glossed over the fact that a molecule might have more than one group that needs numbering. “Well, that’s more complicated.” Not really, if we had learned the right rules in the first place.

My problem isn’t that we’re not learning the right naming method, my problem is that the book is teaching the wrong one. I mean, how many of us are actually going to need to know how to name organic molecules in our lives? But why teach the standards in a non-standard way, the one time that most of us will ever learn them?

Until next time, may you always be able to draw out 4,5-Dichloro-2- [4-chloro-2-(hydroxymethyl)-5-oxohexyl] cyclohexane-1-carboxylic acid.