Prove Involving Mean Value Theorem for $x \rightarrow a_+$
Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.
"Didn't use to get" or "Didn't used to get"? I just made a minor , and  the grammar correction was unnecessary.  The original text was     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't used to get.   It was corrected to:     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't use to get.    Was this grammatical correction an improvement? Was the original text grammatically correct? Is the edited text grammatically correct? What material fills the synaptic cleft? Is it water? The synaptic cleft is the gap between the pre-synaptic and post-synaptic neurons, and neurotransmitters are transferred between the neurons within this region. What substance exits in this space, is it water? If so, what force makes the neurotransmitters move toward the receptors despite colliding with water molecules? I’ve always understood that the phrase two weeks usually turns into two weeks’ when used as a modifier as in   I’m giving my two weeks’ notice.  I get two weeks’ vacation. (“two weeks’ holiday” for Brits)   with an apostrophe on the word weeks’, indicating that the vacation “belongs” to the weeks.   One way to explain this is the phrase “two weeks of vacation” being contracted to “two weeks’ vacation” – the vacation is “of” the weeks; that is, it is possessed by them.  But I’ve seen a lot of people omit the apostrophe in casual writing, and thinking about it, it seems plausible that the noun vacation would be plainly modified by the adjectival phrase two weeks.  But on the other hand, shouldn’t adjectival phrases be hypenated, namely two-weeks vacation?  Yet that seems wrong, too: the hyphenate should be singular, two-week vacation (like two-tone shoes) because .   touches on this (and references a good ) but doesn’t explain why the plural must be possessive in this case.   suggests that two weeks might be adverbial rather than adjectival, and thus resists being used attributively. But that still doesn’t quite explain this particular muddle of pluralization, possession, and attribution.  Given all this, why should “two weeks vacation” be written “two weeks’ vacation” with a possessive apostrophe? Use of .apply() with 'new' operator. Is this possible? Render just part of a final image from Cycles? When should I use the "strictfp" keyword in java? I've looked up what this does, but does anyone actually have an example of when you would use the strictfp keyword in Java? Has anyone actually found a use for this?  Would there be any side-effects of just putting it on all my floating point operations? Miraak is stealing all my dragon souls Allowing a group Read-Write Access to a directory I am looking for a exact derivation of a relation between redshift $z$ and distance $d$.  What I know is the definition $$z=\frac{\lambda_{\text{observed}}}{\lambda_{\text{unshifted}}}-1=\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}-1$$ and that the Hubble constant $H$ as a function of $z$ is: $$H^2=H_0^2\left(\Omega_m\left(1+z\right)^3+\Omega_{\Lambda}\right)$$  How can I use this to derive the distance? Our Epson dot matrix printer is not printing on the following DOS command.:  Type text.txt&gt;prn   But it does print perfectly on:  open TEXT.txt&gt;File&gt;Print    What is the issue here? MySQL Hashing Function Implementation I know that php has md5(), sha1(), and the hash() functions, but I want to create a hash using the MySQL PASSWORD() function.  So far, the only way I can think of is to just query the server, but I want a function (preferably in php or Perl) that will do the same thing without querying MySQL at all.  For example:  MySQL hash -> 464bb2cb3cf18b66  MySQL5 hash -> *01D01F5CA7CA8BA771E03F4AC55EC73C11EFA229  Thanks! The identity cannot be a commutator in a Banach algebra? The Wikipedia article on  claims, without a proof or reference, that there does not exist a (unital) Banach algebra $B$ and elements $x, y \in B$ such that $xy - yx = 1$. This is surprising to me, but maybe the proof is straightforward; anyone have a proof and/or a reference?   More generally, I would have naively thought that I could embed any ring into a Banach algebra. I guess there are actually serious restrictions to doing this; are these issues discussed anywhere? A certain element which makes functionals positive Suppose $X$ is a possibly non-separable Banach space and let $X^*$ be its dual. Also, let $(f_n)_{n=1}^\infty$ be a sequence of [EDIT: linearly independent] norm-one functionals in $X^*$. Does there exist an element $x\in X$ such that $f_n(x) \in (0,\infty)$ for all $n$? I have run out of space on my WinXP virtual machine, which I only gave 10 GB space for when I created it.  Is there an easy way to increase it to, say, 20 GB?  I can't see any obvious option in VirtualBox settings.    The suggestion below gives this error  wim@wim-ubuntu:/media/data/winxp_vm$ VBoxManage modifyhd wim.vdi --resize 20000 VBoxManage: error: Cannot register the hard disk '/media/data/winxp_vm/wim.vdi' {46284957-2c09-4e70-8a49-bfbe0f7f681d} because a hard disk '/home/wim/VirtualBox VMs/winxp_vm/wim.vdi' with UUID {46284957-2c09-4e70-8a49-bfbe0f7f681d} already exists VBoxManage: error: Details: code NS_ERROR_INVALID_ARG (0x80070057), component VirtualBox, interface IVirtualBox, callee nsISupports Context: "OpenMedium(Bstr(pszFilenameOrUuid).raw(), enmDevType, AccessMode_ReadWrite, fForceNewUuidOnOpen, pMedium.asOutParam())" at line 210 of file VBoxManageDisk.cpp     Removing the .vdi from VirtualBox before calling VBoxManage command, then adding it back in, was successful.  But now I can't boot the virtual machine, I get this worrying screen:    By the way, it says FATAL: Could not read from the boot medium!  System halted.    The vdi must be reattached to the VM after VBoxManage command.  Further, the partition will need to be resized from WITHIN windows, because you will have this empty space:    I was able to resize the partition easily using a bit of freeware called . I am using a linux server which has 128GB of memory and 24 cores. I use top to see how much it is used. Its output is pasted at the end of the post. Here are two questions:  (1) I see that each of the running processes occupies a very small percentage of memory (%MEM no more than 0.2%, and most just 0.0%), but how the total memory is almost used as in the fourth line of output ("Mem:  130766620k total, 130161072k used,   605548k free,   919300k buffers")? The sum of used percentage of memory over all processes seems unlikely to achieve almost 100%, doesn't it?  (2) how to understand the load average on the first line ("load average: 14.04, 14.02, 14.00")?   Thanks and regards!  Edit:  Thanks!  I also really like to hear some rough numbers based on used percentage of memory to determine if a server is heavily loaded, since I once became the one who cramed the server without understanding the current load.  Is swap regarded as almost the same as memory? For example, when memory and swap are almost of same size, if the memory is almost running out but the swap is still largely free, may I just view it as if the used percentage of memory + swap is still not high and run other new processes?  How would you consider together CPU or memory (or memory + swap) usage? Do you become worried if either of them reaches too high or both?  Output of top：  $ top    top - 12:45:33 up 19 days, 23:11, 18 users,  load average: 14.04, 14.02, 14.00 Tasks: 484 total,  12 running, 472 sleeping,   0 stopped,   0 zombie Cpu(s): 36.7%us, 19.7%sy,  0.0%ni, 43.6%id,  0.0%wa,  0.0%hi,  0.0%si,  0.0%st Mem:  130766620k total, 130161072k used,   605548k free,   919300k buffers Swap: 63111312k total,   500556k used, 62610756k free, 124437752k cached    PID USER      PR  NI  VIRT  RES  SHR S %CPU %MEM    TIME+  COMMAND  6529 sanchez   18  -2 1075m 219m  13m S  100  0.2  13760:23 MATLAB 13210 timothy   18  -2 48336  37m 1216 R  100  0.0   3:56.75 absurdity 13888 timothy   18  -2 48336  37m 1204 R  100  0.0   2:04.89 absurdity 14542 timothy   18  -2 48336  37m 1196 R  100  0.0   1:08.34 absurdity 14544 timothy   18  -2  2888 2076  400 R  100  0.0   1:06.14 gatherData  6183 sanchez   18  -2 1133m 195m  13m S  100  0.2  13676:04 MATLAB  6795 sanchez   18  -2 1079m 210m  13m S  100  0.2  13734:26 MATLAB 10178 timothy   18  -2 48336  37m 1204 R  100  0.0  11:33.93 absurdity  12438 timothy   18  -2 48336  37m 1216 R  100  0.0   5:38.17 absurdity 13661 timothy   18  -2 48336  37m 1216 R  100  0.0   2:44.13 absurdity 14098 timothy   18  -2 48336  37m 1204 R  100  0.0   1:58.31 absurdity 14335 timothy   18  -2 48336  37m 1196 R  100  0.0   1:08.93 absurdity 14765 timothy   18  -2 48336  37m 1196 R   99  0.0   0:32.57 absurdity 13445 timothy   18  -2 48336  37m 1216 R   99  0.0   3:01.37 absurdity 28990 root      20   0     0    0    0 S    2  0.0  65:50.21 pdflush 12141 tim       18  -2 19380 1660 1024 R    1  0.0   0:04.04 top  1240 root      15  -5     0    0    0 S    0  0.0  16:07.11 kjournald  9019 root      20   0  296m 4460 2616 S    0  0.0  82:19.51 kdm_greet     1 root      20   0  4028  728  592 S    0  0.0   0:03.11 init     2 root      15  -5     0    0    0 S    0  0.0   0:00.00 kthreadd     3 root      RT  -5     0    0    0 S    0  0.0   0:01.01 migration/0     4 root      15  -5     0    0    0 S    0  0.0   0:08.13 ksoftirqd/0     5 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/0     6 root      RT  -5     0    0    0 S    0  0.0  17:27.31 migration/1     7 root      15  -5     0    0    0 S    0  0.0   0:01.21 ksoftirqd/1     8 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/1     9 root      RT  -5     0    0    0 S    0  0.0  10:02.56 migration/2    10 root      15  -5     0    0    0 S    0  0.0   0:00.34 ksoftirqd/2    11 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/2    12 root      RT  -5     0    0    0 S    0  0.0   4:29.53 migration/3    13 root      15  -5     0    0    0 S    0  0.0   0:00.34 ksoftirqd/3 What is the difference between "I've been to China" and "I've gone to China"? Both are in the present perfect, but one uses the verb 'to go' and the other 'to be'. Is there a rule for this?  Is there any chance that the differences between "I have been" vs. "I have gone", are differences in English dialect (I don't know if this is the right word).  Edit Could I say that North American English speakers use "I have gone", and maybe U.K speakers say "I've been"? Or vice versa. I read it as a paperback book somewhere in the mid-90s, I think. The protagonist lived in an apartment building. He's out on the balcony when a birdlike (but humanoid) alien swooped down and attacked him. The presence of the aliens was commonly known and was a known danger. He kills it with his watering can and is rewarded by either the aliens or the humans for it with money. Other than that, all I remember was that the chapters were started with an internet-like news report which I think chronicled things getting worse and more violent. One of the updates involved the bombing of a bridge with the casualties raised by having two slow-moving trucks moving side by side obstructing traffic and ensuring a full bridge.  The internet system was pretty ubiquitous as I recall. I think that he looked up his watering can shortly after the incident with the aliens, but decided to keep this one, even if it leaks a bit. I want to say that it was by Piers Anthony, but none of the book titles look familiar. I think the cover I had showed the man being attacked by the bird-man, which looked kind of like the illustrations on the cover of The Stand. In how many ways can we choose k items from n distinct items put in a circle.   Then further extension of the question, if I want to find out the number of ways to choose k items from n distinct items in a circle such that no 2 are together.  Thanks "Stack Overflow requires external JavaScript from another domain" message is error 404  Well, it seems that Wget picks it up so the last statement is false and the exact message is:  Error 6 (net::ERR_FILE_NOT_FOUND): The file or directory could not be found.   and it loads in Chrome once after I clear the cache but not after. Where is a good place to learn classical optics for high school competitions?