Wikipedia:Reference desk/Archives/Mathematics/2010 October 18

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October 18

Displacement –> Velocity

If you have a time dependent displacement vector, how do get the velocity vector?
Consider for example the time dependent displacement vector $s(t)=(2t+2)i+t^{2}j+(2-2t)k$
115.178.29.142 (talk) 03:07, 18 October 2010 (UTC)

If you're asking this, you probably know that the derivative is the slope of a curve at some point, i.e., the rate of change of a function at that point. If your function is displacement, what's another term for rate of change of position? Feezo (Talk) 03:26, 18 October 2010 (UTC)

Take a look at the velocity vector article. — Fly by Night (talk) 08:53, 18 October 2010 (UTC)

trouble with differentiating something using only first principles

Hi all,

I have this function f(x) = cos (sin x) and I am told to differentiate it using first principles only. That means that I cannot bring in the chain rule and I will have to go through the different trig formulas.

I tried doing this but I had trouble arriving at the answer that I want (-sin (sin x) cos x). Can somebody please teach me how to do this?

Thanks a bunch! —Preceding unsigned comment added by 164.67.84.95 (talk) 05:40, 18 October 2010 (UTC)

We have

$f(x)=\cos(\sin(x)),\qquad f(x+h)=\cos(\sin(x+h))=\cos(\sin(x)\cos(h)+\sin(h)\cos(x)).$

For the derivative, we take the difference quotient

$\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=\lim _{h\to 0}{\frac {\cos(\sin(x)\cos(h)+\sin(h)\cos(x))-\cos(\sin(x))}{h}}.$

As A goes to zero, cos(A) goes to 1 and sin(A) goes to A; we apply this for h, to get:

$\lim _{h\to 0}{\frac {\cos(\sin(x)\cos(h)+\sin(h)\cos(x))-\cos(\sin(x))}{h}}=\lim _{h\to 0}{\frac {\cos(\sin(x)+h\cos(x))-\cos(\sin(x))}{h}}.$

Expanding again, we have

$\lim _{h\to 0}{\frac {\cos(\sin(x)+h\cos(x))-\cos(\sin(x))}{h}}$

$=\lim _{h\to 0}{\frac {\cos(\sin(x))\cos(h\cos(x))-\sin(\sin(x))\sin(h\cos(x))-\cos(\sin(x))}{h}}.$

As h goes to 0, hcos(x) goes to 0, so that cos(hcos(x)) goes to 1 and sin(hcos(x)) goes to hcos(x):

$\lim _{h\to 0}{\frac {\cos(\sin(x))\cos(h\cos(x))-\sin(\sin(x))\sin(h\cos(x))-\cos(\sin(x))}{h}}$

$=\lim _{h\to 0}{\frac {\cos(\sin(x))-h\cos(x)\sin(\sin(x))-\cos(\sin(x))}{h}}$

$=\lim _{h\to 0}{\frac {-h\cos(x)\sin(\sin(x))}{h}}$

$=-\cos(x)\sin(\sin(x)),\,\!$

which is the result we were looking for. —Anonymous Dissident^Talk 06:36, 18 October 2010 (UTC)

Another way (depending on precisely which first principles are allowed) is to convert the trig formulas into exponential ones using Euler's formula. This is basic if you take the usual Taylor series for sin, cos, and the exponential function as their definitions, since Euler's formula is an immediate corollary of these definitions. Wolfram Alpha can do these manipulations (see the "alternate form" entry). From there you have a couple of terms of the form $e^{f(x)}$. You'll require the sum of differentiable functions to be the differential of the sum, and scaling functions scales the derivative, both of which are arguably very basic, and, if not, their proofs are very short. You can compute the derivative of these terms $e^{f(x)}$ directly. First, there's the approximation

$e^{f(x+h)}\approx e^{f(x)+hf'(x)}$

$=e^{f(x)}e^{hf'(x)}$

$=e^{f(x)}(1+(hf'(x))+(h^{2}f'(x)^{2})/2!+...)$

where the ...'s give terms with the degree of n larger than 2. Thus $e^{f(x+h)}-e^{f(x)}\approx e^{f(x)}(hf'(x)+...)$ so

${\frac {d}{dx}}e^{f(x)}=\lim _{h\to 0}{\frac {e^{f(x+h)}-e^{f(x)}}{h}}$

$=\lim _{h\to 0}{\frac {e^{f(x)}(hf'(x)+...)}{h}}$

$=e^{f(x)}\lim _{h\to 0}{\frac {hf'(x)}{h}}$

$=e^{f(x)}f'(x)$.

Now you have the desired derivative in terms of exponentials. If you want, you can convert back to trig, most likely using very mundane transformations, which will give the usual result. IMO the advantage of this method over the above is that you get as a lemma the derivative of $e^{f(x)}$ and you avoid some of the more hand wavey "function goes to V as h goes to 0" arguments (though not all, particularly the initial approximation above--which does, however, fit perfectly with the intuition behind derivatives). The disadvantage is that it's less basic, and less direct. —Preceding unsigned comment added by 67.158.43.41 (talk) 09:28, 18 October 2010 (UTC)

Thank you all! Really appreciate your help! —Preceding unsigned comment added by 164.67.84.28 (talk) 15:19, 18 October 2010 (UTC)

Note the connection with #The limit of sin x below. Specifically, Anonymous Dissident's "As A goes to zero, cos(A) goes to 1 and sin(A) goes to A" is a subtler statement than it appears. He knows that h is linear in the denominator, so he is expanding sin and cos to the linear power of their argument. -- 124.157.218.53 (talk) 17:58, 21 October 2010 (UTC)

Statistics

Four random variables were cast 5 times, and these were the results:

X	1	2	3	4	5	6
A	96	98	100	97	96	?
B	26.1	24.9	27.1	24.2	21.8	?
C	21.7	21.9	24.5	19.8	20.6	?
D	20.7	21.0	21.9	19.1	20.2	?

NOTE THAT THEY ARE CORRELATED. What is the probability, on the sixth trial, that A > B + C + D + 44.6 ?
--220.253.253.75 (talk) 09:58, 18 October 2010 (UTC)

With so few samples, and absent any information about the process that created the values, it seems to be futile to try to model their interrelations. I'd just compute $A-B-C-D-44.6$ for each column, assume (for want of a better option) that this was normal distributed, and estimate the mean and variance. –Henning Makholm (talk) 15:53, 18 October 2010 (UTC)

If we could actually assume a normal distribution, maybe we could say something. Michael Hardy (talk) 19:11, 18 October 2010 (UTC)

The condition (A > B + C + D + 44.6) was true 0 times and false 5 times. The probability, on the sixth trial, that A > B + C + D + 44.6, is 1/7. Bo Jacoby (talk) 21:39, 18 October 2010 (UTC).

That reasoning would seem to imply that we should expect that A > B + C + D + googol on the next try with probability 1/7. That does not look like a useful estimate to me. Do you have any particular statistical model in mind for your calculation? –Henning Makholm (talk) 22:41, 18 October 2010 (UTC)

You are right. You sketched a sophisticated model but provided no estimate. The question was if A > B + C + D + 44.6, not if A > B + C + D + googol. The statistical model for my calculation is this. Let the number of items in the population be N, the number of items in the sample be n, the number of true answers in the population be K, and the number of true answers in the sample be k. The conditional probability for K knowing N and n and k is

$(K|N,n,k)={\frac {(k|N,n,K)(K|N,n)}{(k|N,n)}}$

where

$(k|N,n,K)={\frac {{\binom {K}{k}}{\binom {N-K}{n-k}}}{\binom {N}{n}}}$

is the hypergeometric distribution, and the prior probabilities are uniformly

$(K|N,n)={\frac {1}{N+1}}$

$(k|N,n)={\frac {1}{n+1}}$

for k=0..n and K=0..N. So the general formula is

$(K|k,N,n)={\frac {{\binom {K}{k}}{\binom {N-K}{n-k}}}{\binom {N+1}{n+1}}}$

Substituting k=0, N=6, n=5 you get the results

$(K=0|k=0,N=6,n=5)={\frac {{\binom {0}{0}}{\binom {6-0}{5-0}}}{\binom {6+1}{5+1}}}={\frac {6}{7}}$

$(K=1|k=0,N=6,n=5)={\frac {{\binom {1}{0}}{\binom {6-1}{5-0}}}{\binom {6+1}{5+1}}}={\frac {1}{7}}$

Bo Jacoby (talk) 06:21, 19 October 2010 (UTC).

My last sentence was unproductively snarky and should have been left out, sorry. What I mean is that by reducing the observations to a yes-no question you're throwing away information that intuively ought to be useful, which is cruder than I think the OP was expecting (and had some right to expect). –Henning Makholm (talk) 06:50, 19 October 2010 (UTC)

Yes, we should use more information to estimate the probability. The sad fact is that none of us did. I hoped that someone smarter than me would provide the ultimate answer, but that has so far not happened. The expression X=A−B−C−D assumes the 5 values X=(27.5 30.2 26.5 33.9 33.4), which are obviously all less than 44.6. All we need to know is the number of values (S₀=5), their sum (S₁=151.5) and the sum of squares (S₂=4635.31). Let's assume that the numbers are samples from a normal distribution. In order to compute the distribution for the unknown parameters (μ,σ | S₀, S₁, S₂) we need the known distribution (S₁, S₂ | S₀, μ, σ) but we also need prior distributions

$(\mu ,\sigma |S_{0},S_{1},S_{2})={\frac {(S_{1},S_{2}|S_{0},\mu ,\sigma )(\mu ,\sigma |S_{0})}{(S_{1},S_{2}|S_{0})}}$

But uniform prior distributions are not defined as continuous normalized probability distributions. So here I am stuck. I think that the standard way of approaching the problem is to use the Student's t-distribution. The section Prediction interval#Unknown mean, unknown variance treats the traditional approach to this problem. Not being an expert on frequentist statistics, I am not the right person to pursue this task. Bo Jacoby (talk) 18:39, 19 October 2010 (UTC).

how to refute someone's life work nicely

I already have sponsorship based on my first draft and am almost done typesetting my paper, but how can I pull the trigger on someone, now 70, whose name is intimately associated with what I just disproved? There have been doubts, to be sure, his work while having great mind share has been coming apart seriously at the seam, but his work is still a bedrock, and now I'm about to publish a disproof.

How can I do that humanely? It can only be a terrible feeling for him if he reads my paper. What have I ever done that is equivalent to providing a leading model in a discipline for decades, and being synonymous with that model? Nothing. What right do I have to do that to a man's life's work? Should I even do it?93.186.23.237 (talk) 17:56, 18 October 2010 (UTC)

Not to worry. Stick to the facts and keep it non-personal. If you're right, the blame belongs to truth itself and you're only a messenger; if you're wrong, the face with egg on it will be yours anyway.

By the way, if whatever you're "disproving" is something people widely consider to be proved, rather than just a conjecture, it would be prudent to adopt a humbler attitude to the possibility that your disproof, and not the original proof, is the one with a subtle flaw not visible to its creator. –Henning Makholm (talk) 18:42, 18 October 2010 (UTC)

Important proofs that last a long time but then are disproved are quite interesting - sometimes more so than the original proof was! But yes I wouldn't count my chickens and I'd keep it non-personal. I don't think you need worry much and they might like a copy to check out themselves. Dmcq (talk) 19:35, 18 October 2010 (UTC)

Write the person in question requesting his comments on your work. Then the two of you are working together in finding the truth rather than fighting one another in order to be right. Bo Jacoby (talk) 21:18, 18 October 2010 (UTC).

Agreed - contact the person, try to enlist their help. Even if they aren't interested in helping you, at least you have had the courtesy to let them know what you are planning to publish, so they hear it from you and not from a third party. And if you believe you have a disproof or counterexample to an open conjecture, then the originator of the conjecture should find this just as interesting as a proof. Gandalf61 (talk) 12:52, 19 October 2010 (UTC)

Notice, that you use of the word "disproof" automatically got a number of the commenters to assume that you have found an error in a claimed proof. If you mean that you have given a negative answer to an open question put by this person, don't worry; this is reasonably common, and does not at all crush a person in the view of colleagues. Important conjectures stimulate mathematical progress; and both negative and positive answers may stimulate further progress. (For instance, why did it take such a while to find a counterexample? Is the statement true under some circumstances, but not under other? What are the most natural way to frame the conditions, under which the statement is true?)

Please remember that a person is not proven to be in error, because an open question put by that person has been answered in the negative! If (s)he has made a clear conjecture, i.e., declared him/herself to guess it to be true, then that guess is proved to be wrong; that's all. JoergenB (talk) 18:56, 19 October 2010 (UTC)

The limit of sin x

In calculus textbooks it is often asserted that the limit of sin h as h tends to zero is h, so that for small h, sin h ≈ h. But the limit of h as h tends to zero is zero. Why then is dividing sin h by h meaningful as h tends to 0, whilst 0/0 is not meaningful? jftsang 19:48, 18 October 2010 (UTC)

The notation $\lim _{h\to 0}\sin h=h$ is formally meaningless. It is just a sloppy shorthand for $\lim _{h\to 0}{\frac {\sin h}{h}}=1$. You're allowed to divide by $h$ because the expression inside the $\lim _{h\to 0}$ operator is only ever required to make sense for small but nonzero $h$.

You may have a rule that says $\lim {\frac {x}{y}}={\frac {\lim x}{\lim y}}$. This is not a definition of what the left-hand side means; just a useful short-cut for calculating it that can be used iff the right-hand side makes sense (that is, the limits exist and $\lim y\neq 0$. If these premises are not true, the rule is invalid and unhelpful, but it is still possible that you can evaluate the LHS from first principles (i.e. the ε–δ definition). –Henning Makholm (talk) 20:14, 18 October 2010 (UTC)

(edit conflict) This is a very good question, but be careful with your wording. The limit of a function is a number, a constant, not another function. When you say "the limit of sin h as h tends to zero is h," what you mean is that $\lim _{h\to 0}{\frac {\sin h}{h}}=1$. The ratio between the two functions approaches 1, so sin h is a good approximation to h if h is small. (The statement $\lim _{h\to 0}\sin h=h$ doesn't make sense.) The reason a limit like this is meaningful is that we are discussing the behavior of the ratio (sin h)/h as h approaches 0, but is still nonzero; we are not concerned with what happens when h actually equals 0. See also L'Hôpital's rule. —Bkell (talk) 20:19, 18 October 2010 (UTC)

Another way to explain this behavior is to note that the function x is a first order approximation of sin x for x, and "near" zero all the correction terms required to reconcile x and sin x are "small". For instance, sin 0.01 = 0.0099998... which is ridiculously close to the input, 0.01. The infinite series definition for sin x is

$\sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots$

If we plug in a value near zero, the cube of that value is *very* near zero, so those infinitely many correction terms which use higher and higher exponents nearly cancel to zero. Plugging this series in to find the limit of $\sin x/x$ as $x\to 0$ gives 1 immediately. 67.158.43.41 (talk) 21:38, 18 October 2010 (UTC)

So, Henning Makhom and Bkell are quite right... but I'm really interested in what precisely your text book states. If it does claim that the limit of sin h is h when h tends to zero, without any further qualification, then I suspect that it is erroneous. (Alas, this would not be the first time there is an error in school textbooks for mathematics!) However, you may have mis-read it.

Could you please give an exact quote (with all qualifications)? JoergenB (talk) 21:56, 18 October 2010 (UTC)

All good textbooks on introductory calculus begin the series of theorems on limits by presenting two trivial theorems. The first states that the limit of a constant is that constant. The second states that the limit of a polynomial (or any function with a denominator that doesn't contain a variable) can be found by direct substitution. This second theorem tells me that the limit of sin h as h approaches zero is zero, not h.

It is true that as h gets smaller, sin h can be approximated by h. But that statement is not the same as saying the limit of sin h as h approaches zero is h (which would not be compatible with the second of the two trivial theorems.) Dolphin (t) 22:10, 18 October 2010 (UTC)

The second "theorem" you mention is actually the definition of continuity in the general case, and requires someone to show sin x continuous before it shows $\lim _{x\to 0}\sin x=\sin 0=0$. But that's a small quibble. 67.158.43.41 (talk) 22:41, 18 October 2010 (UTC)

jftsang, was your question prompted by Anonymous Dissident's answer to the #trouble with differentiating something using only first principles question above in which he wrote "As A goes to zero, ... sin(A) goes to A"? -- 119.31.121.84 (talk) 00:22, 19 October 2010 (UTC)

Partially, yes. Bkell's response above clarifies the issue for me, but given that, is it still justifiable to say "As A goes to zero, ... sin(A) goes to A"? 80.46.78.71 (talk) 19:09, 22 October 2010 (UTC)

I hope you get a better answer before this archives, but as with JoergenB, I'd be very interested in seeing an explicit quote from a text book that says "... sin(x) goes to x". When speaking about such things, I say that "As x goes to zero, sin(x) goes as x", meaning that $sin(x)=0+x+O(x^{2}){\mbox{ as }}x\to 0$. Likewise, I say "As x goes to zero, cos(x) goes as 1 - x²". I don't know how formal or proper such language is. -- 58.147.59.32 (talk) 03:33, 23 October 2010 (UTC)

If you have to ask then no. In other words: The statement is technically meaningless. If either you or your correspondent doesn't have a very solid grasp of the matter, then using it will only cause confusion and should be avoided. However, people who understand something very well can allow themselves to be sloppy, because they know what they can be sloppy about and how to correct the sloppiness. So in a conversation between mathematicians, "as A goes to 0, sin(A) goes to A" will be understood to mean $\lim _{A\to 0}{\frac {\sin A}{A}}=1$ and can be used if it's easier to say. -- Meni Rosenfeld (talk) 13:56, 24 October 2010 (UTC)

Combinatorics

Yesterday I did a math contest and one of the question I didn't get. It said something like: one marble is placed in a bag with the label "1", 2 with lable "2", ..., and 50 with label 50, such that there are 1275 marbles in the bag. How many must be taken out of the bag to guarantee at least 10 with the same number. How should I have solved this? Thanks. 24.92.78.167 (talk) 21:58, 18 October 2010 (UTC)

Well, what's the largest number of marbles you can take from the bag and have at most 9 of each number? One more than that will guarantee 10. Algebraist 22:02, 18 October 2010 (UTC)

Note also that the question is not about probability at all. It is purely a combinatorics problem: What is the least number N such that every N-element subset of your set of marbles contains 10 equinumbered ones? –Henning Makholm (talk) 22:17, 18 October 2010 (UTC)

The worst case scenario is that you pick all the marbles numbered 1 through 8, and 9 marbles of each of the numbers 9 through 50, When you take one more marble you are guaranteed to have 10 with the same number. Bo Jacoby (talk) 00:48, 21 October 2010 (UTC).

Test guessing Probability

Today in class we were talking about a certain multiple choice test, on which there is no guessing penalty. The teachers advised us to guess the same letter on all of the problems we could not solve or could not get to to maximise correct guesses, which intuitively makes sense since a, b, c, and d will each be right about 1/4 of the time and guessing the same letter for each more or less guarantees that ~1/4 of the guesses will be right whereas guessing different letters creates a chance that none of them are right. However I'm having trouble proving this mathematically, since if I guess randomly each guess has a 1/4 chance of guessing right. How does guessing the same each time increase my chances? Thanks. 24.92.78.167 (talk) 22:57, 18 October 2010 (UTC)

It wouldn't. Maybe the idea is to avoid wasting time thinking about your guess.--RDBury (talk) 23:06, 18 October 2010 (UTC)

It may be that the test was deliberately constructed such that the correct answers are equidistributed among the four positions for even small ranges of the questions. So if you can only do, for example, the first 60 of 100 questions, guessing the remaining 40 with the same letter would guarantee that at least 9 of them hit correctly; whereas guessing randomly would risk getting none right. This would make sense if you're concerned about optimising for the worst mark you can possibly get, which is probably a reasonable assumption in many cases.

One can imagine other, desperate, situations where a different strategy would be rational. For example, for some reason you might need 75% correct answers, to the extent that you don't care about the difference between 0% and 74% because either will get you killed / thrown out of school / disowned by your family. Then guessing randomly would give a small but nonzero chance of reaching the threshold, whereas marking all the same would be certain loss. –Henning Makholm (talk) 23:21, 18 October 2010 (UTC)

If you guess completely randomly (which is hard for a person to do, by the way), the number of correct answers you get would follow a Binomial distribution, with p=0.25 and n = however many questions you guess on. If the teacher distributed the correct answer completely randomly among a,b,c, and d, then guessing the same letter each time would also give you a binomial distribution with the same parameters. However, unless the teacher made a point of distributing the correct answer randomly, I can guarantee that he did not. Like I said, humans aren't good at doing things randomly. If the teacher noticed that a lot of the correct answers were a, he would intentionally make a couple questions where a isn't the correct answer. It's unlikely that the teacher would make each answer correct exactly 1/4 of the time, but you can be pretty sure that the distribution is not random. I don't think that guessing the same answer each time will increase your Expected value for the number of correct answers, but it will almost certainly (unless the teacher really, really did things truly randomly) decrease the Variance. In other words, you'd be leaving less up to chance. From the perspective of an economist, people are Loss averse; the value of the points you might potentially lose are probably worth more than the value of the points you might potentially gain (there are exceptions, as Henning Malkolm points out). In that respect, decreasing your uncertainty is a smart move, and it is indeed in your interest to choose the same answer each time when you guess. Buddy431 (talk) 02:54, 19 October 2010 (UTC)

I disagree. It is very probable that the teacher will have "favorite letters" which will be the correct answer in a higher proportion. If you don't know what these letters are, sticking with a single letter will usually increase your variance, not decrease it. -- Meni Rosenfeld (talk) 09:10, 19 October 2010 (UTC)

I would expect, given that the teacher explicitly gives this advice, that he/she has deliberately made sure that the cumulative counts of each letter in questions 1 through n is close to n/4 for all n. (That may be too big an assumption). –Henning Makholm (talk) 10:10, 19 October 2010 (UTC)

If the teacher deliberately makes the answers 'random' as in about the same number for each choice of a,b,c,d rather than actually random you can take advantage of this by counting the various answers you give and choose the one with the minimum frequency for all the rest. Of course that's exactly the opposite answer to if the teacher has favourite answers and doesn't check for an equal number each ;-) Dmcq (talk) 10:32, 19 October 2010 (UTC)

The obvious next step is to analyze the correct answer distribution in the teacher's past exams. Many teachers post their previous exams (and answer keys) online, making such analysis easy. If he doesn't, you may want to become friends with a fraternity brother, who often keep copies of past exams Buddy431 (talk) 14:31, 19 October 2010 (UTC)

One factor that may influence this is situations where the real answer falls into a neat pattern with the distractor answers (to take a simple example, if the answers to a run of questions were compass directions, it would be natural [and intuitive for the student as well as the teacher] to use A North, B East, C South, D West each time, but there are plenty of similar situations). If one of the answers in that pattern is never actually correct for whatever reason (but still sensible to have there to catch people out), it would skew the distribution of correct answers.

In fact, I'm almost certain that the vast majority of people setting these sorts of tests are much more concerned about making the wrong answers plausible enough to catch out the people who've misunderstood the material/been lazy about checking their working thoroughly/etc. than not leaving themselves open to the tiny minority of test takers who are applying probabilistic analysis to try and improve their grade marginally. --81.153.109.200 (talk) 20:10, 19 October 2010 (UTC)

User:Buddy431 is dead right that the most direct answer to your question is the binomial distribution, but you might want to look into the idea of expectation value as well.--81.153.109.200 (talk) —Preceding undated comment added 20:16, 19 October 2010 (UTC).

Oops I probably should have mentioned that this is a standardized test, so there's no advantage to be had from guessing from past tests or favorite answers or probabalistic analysis. 24.92.78.167 (talk) 02:46, 21 October 2010 (UTC)

On a standardized test, the test publisher will be very careful to avoid any "letter bias", so the remark about the binomial distribution seems more likely to hold.

Of course, your best chance of getting the right answer is to know the right answer. Study. 68.36.117.147 (talk) 01:48, 22 October 2010 (UTC)