Wikipedia:Reference desk/Archives/Mathematics/2011 May 23

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May 23

Partial differentiation under the integral sign

Find all second partials of the function f(x,y)= integrate from(x/y) to (xy) e^(-t^2) dt

i can do this question if it is f(x) = integrate from 0 to x e^(-t^2) dt because that would simply be e^(-x^2) - 1, but the two variables + partial derivatives here in this question are throwing me off. how would i differentiate this function? any help is appreciated. —Preceding unsigned comment added by 169.232.101.13 (talk) 05:00, 23 May 2011 (UTC)

In general, ${\frac {d}{dx}}\int _{f(x)}^{g(x)}h(t)\ dt=h(g(x))g'(x)-h(f(x))f'(x)$. Use this, and remember that partial differentiation just means differentiating wrt one variable while the other is considered constant. -- Meni Rosenfeld (talk) 08:28, 23 May 2011 (UTC)

I think the relevant articles are Differentiation under the integral sign and Leibniz integral rule. 130.88.73.71 (talk) 12:03, 23 May 2011 (UTC)

so does that mean d/dx(integrate from(x/y) to (xy) e^(-t^2) dt)= e^(-xy^2) d/dx (xy) - e^(-x/y^2) d/dx(x/y) where d/dx denote partial differentiation with respect to x? —Preceding unsigned comment added by 169.232.101.13 (talk) 14:55, 23 May 2011 (UTC)

This sounds about right, though you have some missing parentheses. Please see Help:Displaying a formula and use LaTeX so we can understand you better. -- Meni Rosenfeld (talk) 08:05, 24 May 2011 (UTC)

Simple differentiation

I'm still revising my maths, and the book I'm looking at says that if y=x^n then dy/dx=nx^(n-1). It gives as an example y=4, with dy/dx=0.

According to the formula, shouldnt the answer be 1 rather than 0, as 4 is 4^1, and x^0 is 1? Thanks 92.29.113.29 (talk) 11:03, 23 May 2011 (UTC)

The answer 0 is correct because 4 is a constant. The derivative of x to the 1 power is indeed 1, but that's because x isn't a constant. The derivative of a constant is always zero. Consider a modification to your "4 is 4^1": The number 4 is also 2^2. Then your reasoning would say the derivative is 2*2^1=4. Or 4 is also (√2)^4, which gives you yet another answer, etc... Staecker (talk) 11:19, 23 May 2011 (UTC)

Another way of thinking about this: y=4, as it is currently written, is not in the form y=x^n (there are no variables on the right hand side), so you can not apply that rule. However, you can rewrite the equation as y=4x^0 (since x^0=1 for all nonzero x), which fits the pattern with n=0. Applying the rule gives dy/dx=0. Our article on differentiation might be helpful. 130.88.73.71 (talk) 12:55, 23 May 2011 (UTC)

Yet another way of looking at it is that after finding 4 · 4^1 − 1, you still have to multiply by the derivative of 4 because of the chain rule. And that is 0. Michael Hardy (talk) 03:06, 26 May 2011 (UTC)

dervative

what is the sixth derivative for f(x)=1/(x2-4x+3) at x=2?

I know I can slope the function six times but since the quiz didn't have much time I know there must be an easier way, but all I've found out till now is that f'(x)=(x-2)f2(x)--Irrational number (talk) 14:19, 23 May 2011 (UTC)

Using partial fraction decomposition, you can write the function as ${\frac {a}{x-1}}+{\frac {b}{x-3}}$ for some values of $a$ and $b$, then it is easy to differentiate six times. —Кузьма^討論 14:53, 23 May 2011 (UTC)

Idid it and it worked! thanks a lot.--Irrational number (talk) 16:49, 24 May 2011 (UTC)

hyperbola

What is the integral for a hyporbole(a vertical hyperbola) If we only look at one side of it. I mean what is the integral for (1+4x2)1/2?--Irrational number (talk) 14:37, 23 May 2011 (UTC)

I assume that you want to integrate √(1 + 4x²) with respect to x. The evaluation of this integral is a standard exercise. Could you please show us what you have tried to do so far? You want to calculate

$\int \!\!{\sqrt {1+4x^{2}}}\,\operatorname {d} x\,.$

First of all, make the substitution u = 2x. Then follow your course notes. Get as far as you can and we'll help you finish off the question. Take a look at hyperbolic sine and inverse hyperbolic function too. — Fly by Night (talk) 20:21, 23 May 2011 (UTC)

actually all I know is the integral for polinomial functions,sine,cosine and +-u'/(1-u2)1/2 and +-u'/1+u2 and nothing else (this question is not a homework. this is just something I'm very eager to know because I can calculate the length of a parabolic curve with it--Irrational number (talk) 12:21, 24 May 2011 (UTC)

Can you do Integration by substitution? Follow the link to hyperbolic functions and let $x={\tfrac {1}{2}}\sinh t$. -- Meni Rosenfeld (talk) 13:52, 24 May 2011 (UTC)

Is what I'm looking for in List of integrals of irrational functions?--Irrational number (talk) 17:21, 24 May 2011 (UTC)

Yes, but you'll need to do the substitution Fly by Night suggested first. --Tango (talk) 17:51, 24 May 2011 (UTC)

MATHEMATICS FOR PHYSICS

solve;

  (X+iy)÷(x-iy)  —Preceding unsigned comment added by Mnihu (talk • contribs) 14:43, 23 May 2011 (UTC) 

This is not an equation, so it cannot be "solved." If you are asking how to perform division with complex numbers, see Complex number#Multiplication and division. —Bkell (talk) 14:55, 23 May 2011 (UTC)

It's a bit like rationalizing a surd. ${\frac {X+iy}{x-iy}}={\frac {(X+iy)(x+iy)}{(x-iy)(x+iy)}}={\frac {Xx-y^{2}+(X+x)yi}{x^{2}+y^{2}}}$ Widener (talk) 23:27, 25 May 2011 (UTC)

Student's t-distribution - for dummies

Hello Refdeskers. I need to run a Student's t-distribution for my (friend's) masters, and I can't really make heads or tails of it. I know Wikipedia has articles, but we seem to not have the basics required to grasp what it's about. Here's the thing. We ran a couple of tests on 15 deaf schoolgirls, 15 boys, and the same amounts of girls and boys without hearing disorders, and we calculated the standard deviations for the population, and afterwards we were told to run a t-test on the data. We know nothing more specific. We are using MS Excel. It demands from us two sets of data and some kind of traces, and a type. For us this really makes limited sense, since neither of us had ever anything to do with statistics. Can anyone, in plain, simple, short words, explain what we are looking for? Thanks to everyone and anyone for their input, we will be grateful! --Ouro (blah blah) 15:01, 23 May 2011 (UTC)

There are two types of t test, paired and unpaired. A paired test is used when both data sets have the same number of samples, and each data point from one set can be matched up with a specific data point from the other set. If there is no natural pairing of data points, an unpaired test is usually used. For your data you should probably use a paired design, pairing each deaf boy against a hearing boy and each deaf girl against a hearing girl. In any case, the setup in Excel is quite different for paired versus unpaired t tests. For a paired test, the data should be arranged in two columns, with each row containing the two data points that are paired with each other. I don't have Excel installed on this computer so I can't verify what I am saying, but I don't think you should need to give any further information. Looie496 (talk) 04:37, 24 May 2011 (UTC)

Looie, that is utter nonsense. Pairing is used only when there is pairing. How would you decide which boy to pair with which? If there's not a crystal-clear answer to that, then you're talking nonsense.

Ouro, you're being cryptic. You say you have deaf girls, deaf boys, hearing girls and hearing boys, and you "were told to run a t-test on the data". I disbelieve that that's all you were told. A t-test is a test of a hypothesis. Are you looking at whether deaf people differ from hearing people, or whether girls differ from boys, or what???? You haven't told us! You need to say something more specific. Michael Hardy (talk) 04:52, 24 May 2011 (UTC)

Well, the premise of the entire paper is to compare individual physical characteristics (that is i. e. number of repetitions for one test or the time taken to run a certain distance in another) of a mixed-gender group of hearing vs a mixed-gender group of deaf people. We ran a few tests, got nice results, put them in nice tables. And the professor who's taking care of my friend just said - really - calculate the SD and do the t-test on the results. That's all the information that was given. We're looking for how the deaf ones match up to the hearing ones. Okay, now I've actually provided you with more information. Sorry about that. I was in a bit of a hurry when writing my question yesterday. However, is this enough for you so you could more or less tell me what I'm supposed to do? Thanks! --Ouro (blah blah) 05:00, 24 May 2011 (UTC)

Oh, and one more thing: Looie's assumption actually is false - we are not pairing individual people, just comparing the groups. --Ouro (blah blah) 06:18, 24 May 2011 (UTC)

The t-test tests the hypothesis that two samples come from the same parent distribution. More specifically, it tests the hypothesis that the mean values of two samples are the same. Say, you measure body height for deaf kids and for hearing kids and you obtain mean values $h_{1}$ and $h_{2}$ and standard deviations $\sigma _{1}$ and $\sigma _{2}$. Then the statistic ${\frac {h_{1}-h_{2}}{\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}}}}$ follows a t distribution (under certain assumptions), i.e. you can easily obtain the probability that this the difference between the two mean values differs from 0 by more than a certain amount under the null hypothesis. If that difference is too large (i.e. the corresponding probability is too low), then you can reject the null hypothesis and claim you've found a significant difference between the two groups. This is the principle of the test, you'll have to check the details and how to do it in excel (probably just a button click...). --Wrongfilter (talk) 07:35, 24 May 2011 (UTC)

"Wrongfilter", could I talk you into using $\sigma$ for population SD and S for sample SD? That is standard. Michael Hardy (talk) 18:08, 27 May 2011 (UTC)

Michael, please always use one level of indentation higher than the comment to which you are replying. Please try to use more gentle terms than "utter nonsense". Correcting people's mistakes is good, being rude isn't. -- Meni Rosenfeld (talk) 08:09, 24 May 2011 (UTC)

Well, I kind of understand Wrongfilter's explanation (I know more or less how to do it from the mathematical point of view now). The thing is, Excel (or Calc) give too little explanation, and Wikipedia's article gives too much - I've been hoping to kind of get an Ouro-level explanation on the RD ;) but thanks for everyone's input. --Ouro (blah blah) 15:23, 24 May 2011 (UTC)

I think I understand it now, based on help from Calc, Wikipedia and thanks to you guys. This is what I came up with. Calc and Excel require four components: dataset one, dataset two, selection whether the test is one-sided or two-sided, and a selection whether we are dealing with paired result sets, sets of same variance or sets of different variances. So I take my two samples, say the results of a certain trial for group A and for the same trial for group B. This will be my data for the calculation. Now, I will use a two-sided test because I want Calc to cut off as much on both ends, so my third component is 2. For the fourth, I will select the third option, so - sets of results characterised by different variances, because the results are different in both sets. So the entry in Calc will be something like (D4:D18;D4:D18;1;3). Can anyone just quickly tell me if I got this at least a little bit right? ;) --Ouro (blah blah) 15:38, 24 May 2011 (UTC)

I think you are a little unclear on what one-sided and two-side tests are. If your hypothesis is something like "Deaf children are faster than hearing children" then that is one-sided. If your hypothesis is something like "The speed of deaf children is different to than of hearing children" then it is two-sided. Also, it may well be reasonable to assume they have the same variance (the samples will have different variances, but the populations may have the same variance), but you would need to do an F-test to determine the reasonableness of that assumption, so let's not complicate things. --Tango (talk) 18:05, 24 May 2011 (UTC)

Oh. Then it is two-sided still, even though my reason for choosing it was incorrect. Thanks :) --Ouro (blah blah) 18:16, 24 May 2011 (UTC)

OK, if I'm understanding this correctly, the null hypothesis is that whatever it is you're measuring is on average the same for deaf children as for hearing children. The division into males and females then amounts to blocking. To be continued.... Michael Hardy (talk) 02:46, 26 May 2011 (UTC)

The idea was to analyse the values separately for both genders, not in general. That's the way it's usually broken down in literature on the subject. And that's what my friend's tutor had expected her to do. --Ouro (blah blah) 05:00, 26 May 2011 (UTC)

At the risk of making it more confusing: You have a variable describing the outcome of some test given to your subjects, such as how quickly they can calculate something. The subjects vary by gender (male/female) and hearing (deaf/not deaf). You want to see if any difference in the test variable is due to the gender or hearing differences and not just chance. A t-test would be the standard way to do this for one of the subject variables (either gender or hearing). It will tell you whether the mean difference in the values of the tested variable is likely due to gender (or hearing) or due to chance. From your description, you want a two-tail test, since you don't assume up front that the difference can only be one way (i.e. that boys are faster than girls, or vice versa). You want the unpaired test, and as you have said, assuming unequal variances. To test the impact of both variables (gender and hearing) you can run two seperate t-tests or the more general one-way ANOVA,which will tell you whether the mean difference between all four subject variables (boy-hearing, boy-deaf, girl-hearing, and girl-deaf) are likely due to chance. The ANOVA can also indicate whether there is an interaction between gender and hearing, such as whether boys do worse than girls when they can hear but not when they are deaf. Hope this helps. — Preceding unsigned comment added by 12.186.80.1 (talk) 18:59, 26 May 2011 (UTC)

At the risk of letting you down - this didn't make it complicated, it actually rounds up everything nicely. All you said makes sense to me, except that I didn't need such an in-depth analysis. We didn't need to cross-analyse all four variables. So the t-test is actually enough for us. Thanks loads! I'm almost ready to mark this thread resolved, but am waiting whether Michael Hardy comes back with something more on the subject :) --Ouro (blah blah) 05:54, 27 May 2011 (UTC)

proof?

Has there ever been a proof created to prove that any even number can be the sum of two prime numbers for all numbers greater then 4 (although if you count 1 as prime, then all even numbers might apply and thinking about it 4 would work as 2 + 2 since 2 is prime)? Googlemeister (talk) 20:33, 23 May 2011 (UTC)

That would be Goldbach's conjecture, I think, in which case the answer is a resounding "no". Grandiose (me, talk, contribs) 20:36, 23 May 2011 (UTC)

It ain't one of the Millennium Prize Problems but you'd be extremely famous if you could prove it! Dmcq (talk) 20:51, 23 May 2011 (UTC)

No one will create such a proof, since the conjecture is false. 188.156.63.247 (talk) 10:28, 25 May 2011 (UTC)

Really ? Do you have a counter example ? Or maybe a nonconstructive disproof ? Gandalf61 (talk) 10:32, 25 May 2011 (UTC)

I think it has been manually proven by computers for even numbers of up to 15 digits. While not a proof, it is at the very least circumstantial evidence that the conjecture is true. Googlemeister (talk) 14:23, 25 May 2011 (UTC)

You can do better than that. The primes act in many ways as though each number N is prime with probability 1/log(N), or better, each odd N is prime with probability 2/log(N). Of course this is not easy to give any very precise meaning, as really any number is either prime or it's not, but typically it makes a lot of correct predictions.

Then you ask, given an even number M, what is the probability that it's to be the sum of two primes? Well, for each odd N less than M, you evaluate the probability that N and M−N are both prime, something on the order of 1/log(M)^2. That goes to zero as M goes to infinity, but not very fast. Not nearly as fast as the number of possible N to try goes up.

So you do a few estimates and you find that the probability that M is a counterexample to Goldbach goes to zero very fast, fast enough that the infinite product of the probability of M not being a counterxample converges to something very very close to 1, given that you have checked that there are no counterexamples less than some large value.

This is of course not a proof. However to me it's fairly convincing. The Goldbach conjecture is true; a proof, however, is still lacking. --Trovatore (talk) 05:46, 26 May 2011 (UTC)

Hi, I'm another person who saw this question and was interested by it. Can someone with more knowledge of math tell me why this is a horrible idea and would never work?

Assertion: All primes greater than 2 are odd. Let m = 2j+1 where j is any natural number and m is prime Let n = 2k + 1 where k is any natural number and n is prime Then for all even natural numbers x > 4, there exist primes m and n such that x = m+n. Proof by induction: Base case: x = 6, then m = 3 and n = 3. Inductive step: Given x' = x + 2, where x = m+n where m & n are prime, prove there exist some primes m' and n' such that x' = m'+n' (the hard part nobody knows how to do)20.137.18.50 (talk) 14:48, 25 May 2011 (UTC)

The idea in proof by induction is generally to use the induction hypothesis to show the inductive step. However, knowing primes m and n such that x = m+n does not appear helpful in finding or showing existence of primes m' and n' such that x+2 = m'+n'. If m or n happens to be the lower member of a twin prime pair then we can use x+2 = (m+2)+n or x+2 = m+(n+2), but otherwise the inductive step looks exactly like the original problem. PrimeHunter (talk) 03:33, 26 May 2011 (UTC)