111–120 of 139 posts
Earlier quoted context omitted.
I object! The average person has less than one breast and less than one testicle.
I object again! Statistics say there are slightly less men than women in the states. Therefore, the average person has more than one breast and less than one testicle. http://en.wikipedia.org/wiki/Demographics_of_the_United_Stat...
Surprised this isn't already here, pretty popular at my college: I wish I was your derivative, so I could lie tangent to your curves.
And the companion bio-joke: I wish I was mRNA polymerase, so I could unzip your genes.
Front: (a^n) + (b^n) = (c^n) true only when n Back: I have discovered a truly marvelous proof but this shirt is too too narrow to contain it
It was something like ten years Andrew Wiles spent on it, wasn't it?
Three mathematicians and three accountants are traveling by train to a conference. At the station, the three accountants each buy tickets and watch as the three mathematicians buy only a single ticket.
"How are three people going to travel on only one ticket?" asks an accountant. "Watch and you'll see," answers an mathematician. They all board the train. The accountants take their respective seats but all three mathematicians cram into a restroom and close the door behind them.
Shortly after the train has departed, the conductor comes around collecting tickets. He knocks on the restroom door and says, "Ticket, please." The door opens just a crack and a single arm emerges with a ticket in hand. The conductor takes it and moves on.
The accountants saw this and agreed it was quite a clever idea. So after the conference, the accountants decide to copy the mathematicians on the return trip and save some money (being clever with money, and all). When they get to the station they buy a single ticket for the return trip.
To their astonishment, the mathematicians don't buy a ticket at all. "How are you going to travel without a ticket?" asked one perplexed accountant.
"Watch and you'll see," answered an mathematician. When they board the train the three accountants cram into a restroom and the three mathematicians cram into another one nearby. The train departs.
Shortly afterward, one of the mathematicians leaves his restroom and walks over to the restroom where the accountants are hiding. He knocks on the door and says, "Ticket, please."
An engineer, a physicist, and a mathematician go to a conference together, and split a room in the hotel. They check in and go to sleep. The hotel is old (they couldn't afford anything better on professors' salaries), so the room is heated by a fireplace. A spark jumps out of the fire and catches the rug on fire. The engineer smells the smoke and wakes up. He jumps out of bed, sees the fire, and looks around. The fir…
A mathematician, an engineer and a physicist were travelling through Scotland when they saw a black sheep through the window of the train.
"Aha", says the engineer, "I see that Scottish sheep are black."
"Hmm", says the physician, "You mean that some Scottish sheep are black".
"Nonono", says the mathematician, "All we know is that there is at least one sheep in Scotland, and that at least one side of that one sheep is black."
There are two hard problems in computer science. Naming, cache invalidation, and off-by-one errors.
Make that 10 hard problems. Similar to - There are 10 kinds on people. Those who understand binary and those who don't.
There are 1 kinds of people in this world. Those who know how to save bits. The rest are the default case.
Earlier quoted context omitted.
That one is so bad, because it doesn't work. Presumably the derivative is used to calculate a tangent line. However, a tangent line touches the curve only at one point, unless the curve is totally flat. Either way, this is not what you want. You want fuckin' Taylor series approximations. (They won't work on curves that aren't smooth, but that's all right, because those are the curves you want to touch anyway.) Aww ye…
Strictly speaking, the Taylor series won't even do what you're wanting them to do for smooth functions. (Consider the function e^{-1/x^2}, whose Taylor series is zero at the origin. In fact, given any sequence of numbers, you can cook up a smooth function which has that sequence as the coefficients of its Taylor series at zero.) If you want the Taylor series to agree with the function in an entire neighborhood of a p…
And as long as we are going for perfect rigor, the function you describe is undefined at x=0; you have to make it piecewise and say "if x=0, then this function is 0". (And I could say something about "you want natural curves, not this artificial pieced-together stuff.")
Also, I believe the Taylor series for that function is well-defined and accurate at all points other than x=0, so you can just pick one of those points. Are there smooth functions whose Taylor series are wrong everywhere? I doubt it. And one might argue that it's pretty problematic that, e.g., with the function "0 if x≤0, e^{-1/x^2} if x>0", a Taylor series at any positive point will be wrong for all negative numbers, and a Taylor series at any nonnegative point will be wrong for all positive numbers... well, I dunno, touching half of the entire range of a curve is still a lot (an infinite amount, in fact).
One could imagine a bunch of functions pieced together to make a smooth function (e.g. "e^{-1/x^2} * e^{-1/(x-1)^2} for 0<x<1, e^{-1/(x-1)^2} * e^{-1/(x-2)^2} for 1<x<2, ...") whose Taylor series are all accurate only over a small, finite domain. I guess that kinda answers my question. But at this point I would apply the "natural curves" objection. I wonder, are there smooth non-piecewise functions whose Taylor series are all wrong like that? (Absolute value is a piecewise function, so I'd probably reject anything with absolute value.)