Wikipedia:Reference desk archive/Mathematics/2006 August 30

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[edit] coplanar

my math teacher today was explaining collinear and coplanar today. I get the collinear part fine, but the coplanar just goes over my head. Do you think that you could explain coplanar to me, because I'm trying to do this assignment and I can't do it, because I don't get it. Your help would be greatly appreciated!! thank you!! kirsten

Have you taken a look at coplanarity for a definition? How about collinearity? What you need is in those articles. --HappyCamper 01:34, 30 August 2006 (UTC)
The coplanarity article may be too terse for a beginner. Question: do you understand the geometric concept of a plane? A plane in space is like a thin flat sheet of glass. Any three points in space necessarily share a common plane. (This is why a 3-legged stool always sits steady.) You already know that any two points share a common line, and a third point may be on that same line (collinear) or not. Similarly, a fourth point may share a common plane with the first three (coplanar) or not. (This is why a 4-legged stool may not always sit steady.) It is highly unlikely that four or more points will be coplanar. Many geometric arguments or algorithms assume that this unlikely coincidence is prohibited. Sometimes the assumption is stated explicitly, sometimes not.
Why is this important? Consider the claim that "two points determine a line"; it implicitly assumes that the two points are distinct. The claim that "three points determine a plane" likewise assumes that the three points are not collinear. The assumption that there are no such coincidences is vital.
Going the other direction, we have theorems that assert that sometimes collinearity or coplanarity must occur. Examples include Desargues' theorem, Monge's theorem, and this page of generalizations. --KSmrqT 16:03, 30 August 2006 (UTC)
Another simple example is Pasch's axiom. In the geometric axiom system Coxeter describes in his Introduction to Geometry, plane is a defined term, and the existence of planes as we know them is guaranteed by Pasch's axiom. (This system is similar to Tarski's axioms. Contrast this to Hilbert's axioms described in A geometria és határterületei by Reiman István, where planes and incidence of a point and a plane are primitives.) – b_jonas 17:18, 31 August 2006 (UTC)


Thank you so much!!!! i understand it now, and that really helped!! Thanks once agian!!

[edit] Calculus help

We are working on limits in class. We learned the Squeeze Theorum today in class and the teacher gave us these problems. I do not understand them at all. Any help would be greatly appreciated. THank you!

  1. If 1 \le x^2 + 2x + 2 for all x, find \lim_{x\rightarrow 1} f(x)\,
  2. Prove that \lim_{x\rightarrow 0} x^4 \cos(2/x) = 0
  3. \lim_{x\rightarrow 2} \frac{x-2}{x-2}
Well, let's see here. Have you taken a look at the article yet? I think only the second question makes sense to use the squeeze theorem. In fact, take a look at example 1 in that article. The rest of them, you can find the answer by inspection. Also, take a look at the first question - the way it is written, it is not clear what the question is asking. For example, what is f(x)?. --HappyCamper 01:43, 30 August 2006 (UTC)
I assume that problem 1 is supposed to have the condition 1 \le f(x) \le x^2 + 2x + 2 for all x, otherwise it does not make sense. All three can be solved by applying the squeeze theorem, although for 1 and 3 there are obviously easier ways of finding the solution. It is not exactly clear what you do not understand: Is it the squeeze theorem itself, or is it what you are supposed to solve in these problems, or is it how to apply the theorem to the problems? --LambiamTalk 03:14, 30 August 2006 (UTC)
Good thinking. But then, it should be 1 \le f(x) \le x^2 - 2x + 2 and it would make even more sense. – b_jonas 12:53, 30 August 2006 (UTC)
Or the limit is for x\rightarrow -1.  --LambiamTalk 15:04, 30 August 2006 (UTC)

For the third one, that's an indeterminate form of type \frac{0}{0} . But it's simple; just cancel out the factors, ending up with \frac{1}{1} = 1,. So the function would be:

f(x)=\begin{cases} 1, & x < 2 \\ \textrm{undefined}, & x = 2 \\ 1, & x > 2 \end{cases}

So that means \lim_{x\rightarrow 2} \frac{x-2}{x-2} = 1 --Ķĩřβȳ♥ŤįɱéØ 03:34, 30 August 2006 (UTC)

Right, but you did not apply the squeegee theorem. Also, we're not supposed to solve other people's homework for them here. --LambiamTalk 05:59, 30 August 2006 (UTC)
Now as for the second task, the squeeze theorem is indeed a good way to solve it. To choose the bounding functions, notice that the value of the cosine function is bounded between − 1 and 1. – b_jonas 12:48, 30 August 2006 (UTC)
I suppose (3) could be considered a degenerate case of the squeeze theorem by saying 1 \le \frac{x-2}{x-2} \le 1. Confusing Manifestation 13:55, 30 August 2006 (UTC)

[edit] Calculus help, again

Is there a method to be able to know how to integrate some functions, where integration by parts or substitution doesn't help ? I mean, for example, integrating e^x \ln \, x or \frac{e^x}{x} (which is basically the same problem). Furthermore, what would be the recommended method in cases where integration by parts and substitution fail ? I know many function are not really integrable, and leads to the invention of new functions (such as the error function). Xedi

Ok, had a look at the list of integrals \int\frac{e^{cx}\; dx}{x} = \ln|x| +\sum_{i=1}^\infty\frac{(cx)^i}{i\cdot i!} Xedi 14:28, 30 August 2006 (UTC)
You may find that ecx = sinh(cx) + cosh(cx). Also \int_{x=0}^t \frac{\sinh(x)}{x}dx is hyperbolic sine integral, shi(t) (which sounds weird when you say it). (Igny 16:22, 30 August 2006 (UTC))
Thanks Xedi 18:52, 30 August 2006 (UTC)
What do you mean when you say it? I find it worrying that they've created a notation which, if it ever appeared in a high school mathematics textbook, would cause many students to roll on the floor laughing. Confusing Manifestation 13:24, 31 August 2006 (UTC)
In such case there are still some advanced techniques you may use, like trigonometric substitution(for integral like \int\sqrt{x^2+a^2} dx), substitution by tangent half-angle formula, partial fraction, taylor series, or in case of definite integral, there's tricks like exploiting symmetry, conversion of coordinate system(into polar coordinate), or use numerical method. BTW, you can "cheat" by using computer algebra system(Risch algorithm) ;) --Lemontea 14:37, 31 August 2006 (UTC)

See also exponential integral. Conscious 18:40, 1 September 2006 (UTC)

[edit] Question on the derivative market

Question on the derivative market and particularly on the process with several volatilities:

I would like to know if any process with more than one volatility for an underlying asset exist.

Nicolas

Your question is not clearly posed. If you have a stochastic differential equation with two volatilities, say,
dX = αXdt + σ1XdW1 + σ2XdW2
where α, σ1 and σ2 are constants and W1 and W2 are independent Brownian motions, then this process has variance/volatility
\sigma_1^2+\sigma_2^2,
and has the same distributions as
dX=\alpha X dt+\sqrt{\sigma_1^2+\sigma_2^2} X dW.
Is that what you wanted to know about? Or do you mean that the volatility is changing in time? –Joke 01:54, 31 August 2006 (UTC)

Are you trying to model it? Joke is just saying that adding one piece of random noise to a function is the same as adding any number. And do you mean volatility or a periodic variation? Volatility isn't a 'real' quantity anyway, just a measure inferred from historical data. Rentwa 06:28, 31 August 2006 (UTC)