Let $f(x)$ be a polynomial such that $$\lim_{x \to \infty}\frac{x^4f(x)}{x^8+1} = 3, f(-6)=37, f(-1)=2,f(2)=5, f(3)=10$$ Then:  Find the value of $f'(3)$ Find the value of $$\lim_{x \to -6} \frac{f(x)-x^2-1}{3(x+6)}$$   Both of the sub-questions require finding $f(x)$, so I'll proceed to find that. According to the first limit, $f(x)$ is a fourth-degree polymial with the leading coefficient as $3$. We can rewrite $f(x)$ as: $$f(x) = 3x^4 + ax^3 + bx^2 + cx + d$$ Given four values of $f(x_i) = y_i$, substituting them gives us a system of 4 equations in $a,b,c,d$: $$\begin{align} f(-6) = 37 &amp;= 1296\times3-216a+36b-6c+d \\ f(-1) = 2 &amp;= 3-a+b-c+d \\ f(2) = 5 &amp;= 48 + 8a+4b+2c+d \\ f(3) = 10 &amp;= 243 + 27a + 9b + 3c + d\\ \end{align}$$ This is extremely tedious without using a calculator/computer, so are there any easier ways to compute $f(x)$ or solve the two sub-questions? Preferably a method that does not involve solving the given system of equations. I am aware of Lagrange interpolation, but using that for the given values gives me a third-degree polynomial, which does not satisfy the first condition.
Let f(x) be a  polynomial satisfying $\lim_{x \to\infty} \frac{x^4f(x)}{x^8+1}=3,$  $f(2)=5 $ and $f(3)=10$, $f(-1)=2$ and $f(-6)=37$ . And then there are some question related to this. In the solution it is given that according to the given information, $f(x)$ is a polynomial of degree 4 with leading coefficient 3. $f(x)=3(x-2)(x-3)(x+1)(x+6)+(x^2+1)$ Now what i think is that he got the coefficient 3 using the limits and the rest of the terms$[(x-2)(x-3)(x+1)(x+6)]$ from the respective $f(x)$ values from what was given and also the $(x^2+1)$ was added for the remainders, as they satisfy all the values that have been given to us in the question. I just wanted to know if my approach is right? or is there another elegant way to do it?.
What's the added value of SD line over regression line when examining association between 2 variables? I'm trying to incorporate different practices to use when exploring a new data set. Especially, how to examine the association between two variables.  Steps for example (not necessarily by order):   plot a y-by-x scatter plot of the raw data to see the relationship visually. compute summary statistics for each variable (mean and sd) compute correlation coefficient r draw the OLS regression line, compute its slope and intercept etc....   I've come across the "SD line" in Freedman's Statistics book, which is defined as:     "the line that goes through the point of averages and climbs at the rate of one vertical SD for each horizontal SD" Freedman, D., Pisani, R., &amp; Purves, R. (2007). Statistics (4th edn).   Since this book ("Statistics") is a canonical textbook, I consider its choice to discuss the SD line as an indication for the line's importance. However, a simple google search for the term "SD line" doesn't yield as many independent results. Most of them come directly from Freedman's book. This tells me it's not a central concept in bivariate analyses in general.  When comparing the SD with the OLS regression line, it seems like the regression line is more informative (than the SD line) for predicting y from x. Therefore, I'm wondering if bothering to plot the SD line has any benefit or added value that I would not already have when plotting the regression line.  Example using mtcars dataset, focusing on association between weight and mpg  data(mtcars)  ## calculate means mean_wt &lt;- mean(mtcars$wt) mean_mpg &lt;- mean(mtcars$mpg)  ## calculate standard deviations sd_wt &lt;- sd(mtcars$wt) sd_mpg &lt;- sd(mtcars$mpg)  ## scatter plot plot(x = mtcars$wt, y = mtcars$mpg)  ## add the "point of averages" points(mean_wt, mean_mpg, col = "red", cex = 1.5, pch = 16)  ## calculate the slope of the sd line slope &lt;- -1*sd_mpg/sd_wt  ## plot the sd line curve(expr = x*slope + (mean_mpg - slope*mean_wt), add = TRUE, col = 'blue', lwd = 2, type = "l", lty = 2)  ## plot the regression line model &lt;- lm(mpg ~ wt, data = mtcars) abline(model, col = "orange", lwd = 2)  ## legend legend("topright",        legend = c("Regression line", "SD line"),        col = c("orange", "blue"),        lty = c(1, 2),        lwd = c(2, 2))     Thus, my question: how can the SD line increase one's understanding about the relationship between two variables, in a way that's either adding or complementing on what the regression line already tells? Separate long math text under sum symbol into different lines? I have the following equation, where there is a lot of stuff under the sum symbol:  \begin{equation} d(\vec{x},\vec{y})=   \sum_{Z_{xy}\in\vec{Z}_{xy},\forall x\in\vec{x},\forall y\in\vec{y}}      f(Z_{xy}) \end{equation}   In the resulting document, I find it kind of hard to read. Is there a way to write the equation to make the result more readable, e.g. putting the stuff under the sum symbol on different lines? An "urgent" tag for questions that need answers within minutes Missing entry in flagging history I happened to glance upon my flagging history and I realized that there was an entry missing from the spam section in it. I had marked 6 posts as spam, out of which 4 were deemed helpful and 1 was declined. I'm not sure what happened to the remaining 1 entry and why it is missing from the history summary. Here is a snapshot of the spam section of my flagging history.    Is this behavior a  or a ?  Note: I've checked my flagging history completely and there is no spam flag which is still in the active state. I think we can all agree, this sucks:  If you've been around a little while, you've probably encountered hundreds of answers like this in various forums, some of them even marked as &quot;The Answer&quot; by well-meaning1 forum admins looking to close a thread. We could try to enumerate the commonly-observed problems with it...  Link is probably broken If link isn't broken, it's spam If link isn't spam, it's to some broadly-related resource that the answerer found by pressing &quot;I'm feeling lucky&quot; when googling the question's title.  ...but the real issue here is that we don't know! Even if the link was originally pointing directly to the most awesomeest answer ever for exactly the question being asked, there's nothing in the answer itself to even hint at what direction we're being pointed in. Strip the markup, and this is what you're left with:  i think you should take The tutorial HERE! This will help you a lot  Is there even one tiny speck of information there? No. Even the title of the page being linked to would have provided something to go on, but that was entirely too much work. So let me be clear: this sort of response is not an answer. If you see this, flag it. Moderators, if you see it flagged, delete it. There's really only one valid exception to this rule, and that's when the question is:  Suggest me some tutorials where i can learn quick.  Don't get me wrong, it's still a bad answer - but when the question is kinda asking for bad answers this is to be expected. Close or flag the question and move on - this is much more efficient than hanging around to babysit it by deleting every crap answer that inevitably shows up. If you're still reading, you're probably getting really bored by now. &quot;Gee, Shog - answers that contain ZERO information are terrible, horrible, very-bad things? What shocking revelations will you share next?&quot; Yeah. It's really simple... Except, . See, the problem is that once you step away from black-and-white &quot;answers / doesn't answer&quot; decision-making, things get... Interesting. And all those other answers with links in them See, this is an answer:  And so is this:  Yes, they're both very short, and yes, they contain links. But strip the markup, and you still get at least a little bit of useful information. Does that mean these answers should forever hang around the site? No, not necessarily - if it turns out they're just not that useful, they should probably still be removed - or at very least, down-voted so that they rank below other answers. Just remember: if the text of the post contains an honest attempt at answering the question, then it is an answer - so don't flag it otherwise, and if you do, don't complain if your flag gets declined. Answers &amp; Apples If you're still confused, forget about answers. Think about apples instead:  Answers are just like that, but less tasty in pies.  1 By which I mean to say, &quot;criminally lazy&quot; The Boolean modifier is not working I'm trying to model an , but it's not going so well when it comes to the magazine of the weapon.   Next to the magazine is a mesh. I want to cut holes in the mag with that mesh. But the magazine just goes invisible or connects with the mesh when trying to execute the Boolean Modifier. I have tested to invert the normals and some other stuff, but it isn't working. What does the symbol $\subset\subset$ mean? In some texts (mainly complex analysis or harmonic analysis) I sometimes see the following double subset symbol $\subset\subset$ for inclusion relation of two regions, e.g., $\Omega$ and $\Omega'$ are two regions in $\mathbb{C}$ such that $\Omega \subset\subset \Omega'$. I never figured out what it means exactly; I always interpreted it as the closure $\overline{\Omega}$ is contained in $\Omega'$ (so that some nasty boundary effects can be avoided). Is that right? Or does $\subset\subset$ mean some other kind of inclusion? Thanks. Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$? We know that $f(x)=x^4+1$ is a .  My question is:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ has a linear factor over $\mathbb{Q}$?   Edit: Thanks for @Jyrki Lahtonen's answer, I want to do some modifications:     Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?   Thanks in advance! Find the center of the symmetry group $S_n$.   Attempt:  By definition, the center is $Z(S_n) = \{ a \in S_n : ag = ga \forall\  g \in S_n\}$. Then we know the identity $e$ is in $S_n$ since there is always the trivial permutation. Suppose $a$ is in $S_n$, but not equal to identity. Now we can imagine the permutation as bijective function that maps from $\{1,2,\dotsc,n\}$ to $\{1,2,\dotsc,n\}$.  So suppose $p$ is a permutation map. Then $p$ maps from a location $i$ to a location $j$. Take $p(i) = j$ where $i\neq j$. Let $k$ be in $\{1,2,\dotsc,n\}$, where $k$, $i$ and $j$ are all different elements. The cycle $r = (jk)$, then we will see if this commutes.  $rp(i) = rj$    Can someone please help me, I am stuck?  Thank you. Does $\lim_{n\to \infty} \int^{\infty}_{-\infty} f_n(x) dx\, = \int^{\infty}_{-\infty} f(x)\,dx\,$? Is it considered plagiarism for a professor to use uncited sources in teaching materials? In a theoretical physics homework problem, I have to show the following: $$\partial_\nu T^{\mu\nu} = - j_\nu F^{\mu\nu}$$  Where $T$ is the Energy-Momentum-Tensor, $j$ the generalized current and $F$ the Field-Tensor. We use the $g$ for the metric tensor, I think in English the $\eta$ is more common.  I know the following relationships:   Current and magnetic potential with Lorenz gauge condition: $$\mathop\Box A^\mu = \mu_0 j^\mu$$ Energy-Momentum-Tensor: $$T^{\mu\nu} = \frac1{\mu_0} g^{\mu\alpha} F_{\alpha\beta} F^{\beta\nu} + \frac1{4\mu_0} g^{\mu\nu} F_{\kappa\lambda} F^{\kappa\lambda}$$ Field-Tensor: $$F^{\mu\nu} = 2 \partial^{[\mu} A^{\nu]} = \partial^{\mu} A^{\nu} - \partial^{\nu} A^{\mu}$$ d'Alembert operator: $$\mathop\Box = \partial_\mu \partial^\mu$$ Bianchi identity: $$\partial^{[\mu} F^{\nu\alpha]} = 0$$   So far I have set all the definitions into the formula I have to show, but I only end up a lot of terms from antisymmetrisation and product rule. I also drew all what I have in , but I still cannot see how to tackle this problem.  Could somebody please give me a hint into the right direction? Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$ If the order of $G$ is $p^2$ then how do I show that $G$ is isomorphic to $\mathbb Z_{p^2}$ or $\mathbb Z_p\times\mathbb Z_p$. From what I understand, entropy is a concept defined by the experimentalist due to his ignorance of the exact microstate of a system. To say the number of accessible microstates $W$ of the universe is constantly increasing is nothing more than saying 'ignorance begets ignorance'.  I have often encountered the argument of ever increasing entropy for the presence of an inherent time asymmetry, most prominently in the works of Penrose. It just doesn't seem to make sense.  Let us imagine alien beings who are experiencing time in reverse. For them, increasing ignorance is in the direction of our decreasing ignorance. So how can perpetual increase in entropy indicate an 'arrow of time'?  One possible explanation (I thought of) for this argument was the fact that there may never be a mechanism to reduce ignorance. I will pose this as a question:  An observer determines the number of possible microstates of system+observer to be $W_0$ at time $t_0$. After improving his measurements, can he (at a later time) measure $W'$ (where $W_0&gt;W'$) as the number of possible microstates? Assume that all the microstates are equally probable in this case.  Unless the above in false in general, how else can anyone claim that entropy reveals an arrow of time?  EDIT: I am performing this discussion for isolated systems (the universe or it's parts if relevant). The means I propose to reduce entropy in the present, is by redefinition of macroscopic variables and of microscopic models used to count the number of distinguishable micro-states (that obviously yield the same macro-state). This is essentially an argument against the robustness of entropy via redefinition. Make back/forward work on paginated lists I just bought a controller because I am tired of not to being able to do any tricks with keyboard controls. The problem is I am can't configure controls as FIFA 13 PC version  doesn't consider a controller in their settings.  Does anyone know how to do this? I'd like to create a Yahoo! account. The signup page won't let me proceed without entering a cell phone number, and it won't recognize my Google Voice number as a valid cell phone number. Is there a workaround to be able to create a Yahoo! account without having a cell phone that can receive SMS messages? using trial and error in math problems Suppose one bacterium is in a jar at 12AM, and then suppose there are two bacteria at 12:30AM,  and then there are four bacteria at 1AM, etc.  (a)  How many bacteria will be in the jar at 12PM that day?  (b)  Use trial and error to estimate at what time there will be 1 billion (1,000,000,000) bacteria in the jar.  for a) is it going to be 44 bacterias ? or am i wrong? Please help with b), i have no idea how to figure it out in short term Thanks Should covariates that are not statistically significant be 'kept in' when creating a model? I have several covariates in my calculation for a model, and not all of them are statistically significant.  Should I remove those that are not?  This question discusses the phenomenon, but does not answer my question:   There is nothing in the answer to that question that suggests that non-significant covariates be taken out, though, so right now I am inclined to believe that they should stay in.  Before even reading that answer, I was thinking the same since a covariate can still explain some of the variance (and thus help the model) without necessarily explaining an amount beyond some threshold (the significance threshold, which I see as not applicable to covariates).  There is another question somewhere on CV for which the answer seems to imply that covariates should be kept in regardless of significance, but it is not clear on that.  (I want to link to that question, but I was not able to track it down again just now.)  So...   Should covariates that do not show as statistically significant be kept in the calculation for the model?  (I have edited this question to clarify that covariates are never in the model output by the calculation anyway.)  To add complication, what if the covariates are statistically significant for some subsets of the data (subsets which have to be processed separately).  I would default to keeping such a covariate, otherwise either different models would have to be used or you would have a statistically significant covariate missing in one of the cases.  If you also have an answer for this split case, though, please mention it. Transfer / security in Istanbul Ataturk airport after 29/06/2016