Question on Derivatives Real Analysis
Applying MVT and IVT (?) to show something about f'''? (edit: actually, Taylor's Theorem using Lagrange remainder)
How do you do Impersonation in .NET? Is there a simple out of the box way to impersonate a user in .NET?  So far I've been using  for all my impersonation requirements.  Is there a better way to do it by using .NET Framework?  I have a user credential set, (username, password, domain name) which represents the identity I need to impersonate. Why is it string.join(list) instead of list.join(string)? Quotes insert problem while working with webservice quotes are not inserting may i know what i have to do. my code is     for(integer l=0;l&lt;reqquotes;l++){          qu = new Quote();          qu.Name = qName;          //opp.CloseDate =system.today();           qList.add(qu);             insert oppList;          insert qList;     } Why is $1$ not a prime number? As soon as I start SQL Server with no application querying, in the task manager it starts to consume memory very fast till it consumes around 15GB. How can I resolve this issue? What's the canonical way to check for type in Python? Hot questions no longer work on iPad (ok on iPhone) (Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: . Evaluate $6^{433} \pmod {21}$ and a proving question Question 1: Denote $a \mod b$ as $a \% b$, where $a$ and $b$ are some integers  Evaluate 12^32475 % 21    The following is what I tried:    12^32475 % 21 = (12^3 % 21)^10825 % 21 = 6^10825 % 21 = ... = 6^433 % 21   But how to continue ?  Question 2:  I don't understand this: n-1= -1 mod n. Where is this from? And does this line become the next line?    Context: my class is discrete math for computer science. So far for related topics, it covers modular inverse, (extended) GCD algorithm, Fermat's Little Theorem and Chinese Remainder Theorem (CRT). Big-O Notation Proof Given the definition of Big-O, prove that $f(n) = n^2 - n$ is $O(n^2)$.  When I use the given definition, I get $n^2 - n \leq n^2 - n^2$ which means that $n^2 -n \leq 0$, which is not true. Is there some step I'm missing? How to sort lines by float number I've such a file:  name: xxx --- time: 5.4 seconds name: yyy --- time: 3.2 seconds name: zzz --- time: 6.4 seconds ...   Now I want to sort this file by these float numbers to generate a new file as below:  name: yyy --- time: 3.2 seconds name: xxx --- time: 5.4 seconds name: zzz --- time: 6.4 seconds ...   I've tried the command awk '{print $5}' myfile | sort -g but this will show me ONLY the float numbers. Let users editing a question know if editing it would add it to the reopen queue This is sort of similar to , which was positively received, but was declined for technical reasons. I propose that users who are editing a post that was closed within the last five days be shown a notice on top of the edit form indicating that the post will be added to the reopen queue if edited, and that if one does not plan on editing the post into a state where it can be reopened, one should wait until five days after closure to make the edit. Closed questions are only added to the review queue once per closure by editing, and if a user comes by and makes a minor (yet benign) edit, the diff that reopen reviewers see is just that of the minor edit, and as such they dismiss the task as &quot;Leave Closed&quot;, which prevents it from being added to the queue again by another edit. This deprives the post author (and others interested in seeing the question reopened) of their chance to get the post reopened by editing. While some of these edits may be performed by users using (semi-)automated tools who would thus not see or overlook the warning, this will be a big help because 99%+ of users make edits solely manually and would thus see and possibly respond to such a notice. To help make this request more technically feasible, I propose that it instead only be loaded with the edit form, and that it should only be shown to users who can make binding edits. (Suggested edits occur at the time of approval, and the request to let the reviewers know has already been declined.) Here's an example of what I'm expecting:  This question was recently [closed/marked as duplicate]. This question will be added to a queue for possible reopening if you make this edit. If you are not planning to edit this post into a state where it can be reopened, please wait until [date and time after 70 days] to make your edit. If you are only making edits to the question's tags, please ignore this notice. What can I do on a Visualforce page that is a renderAs="pdf" that will result in the name of the file that shows up when the user clicks Save As... to not be the name of the Visualforce page.  The page is generated via a GET request coming from clicking a custom button.  Below is some code that is representative of the PDF.   &lt;apex:page standardController="Account" extensions="AccountPdf"           showHeader="false"           sidebar="false"           renderAs="PDF"          title="AccountReport.pdf"          contentType="application/pdf#AccountReport.pdf"          &lt;head&gt;              &lt;title&gt;AccountReport.pdf&lt;/title&gt;          &lt;/head&gt;          &lt;apex:outputText value="Hello World"/&gt;  &lt;/apex:page&gt;   The browser bar does not have the title value in it and when the user clicks Save As... the name of the file is the name of the Visualforce page (e.g., myPageName).  If I remove the renderAs="pdf" the file downloads automatically with the correct file name due to the contentType, but the actual content type doesn't appear to be a PDF. If $I$ and $J$ are ideals in a ring $R$ with $1$ such that $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n \in \mathbb{N}$ If $I$ and $J$ are ideals in a ring $R$ with 1 which are  co-maximal, i.e $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n$ in $\mathbb{N}$  Work done:     Should I proceed using    By taking $$K=I^m+J^m \neq R$$  Let $$S=\{I,J \in R \mid I^m+J^n \neq R \}$$ be the required partially order set ordered by inclusion then  $$I+J \subseteq I^n+J \subseteq I^n+J^m \subseteq \ldots $$  Then, I think this chain doesn't have an upper bound...as $I^m+J^n \neq R$ for all $ m,n \in \mathbb N$. Am I right??  Thanks for any help!! Should laptops remain plugged in when their battery is 100% charged? When your laptop's battery is 100% charged, should you leave it plugged in so any battery power doesn't get used, or will that cause overcharging, overheating, etc.? Should the laptop be unplugged when the battery level is 100%?  I'm asking because my laptop's screen tends to get dim when unplugged, so I don't like to run it on battery. I want to update my sources.list file with the fastest server from the command line in a fresh Ubuntu Server install. I know this is trivially easy with the GUI, but there doesn't seem to be a simple way to do it from from the command line? How To Show Node Comment Count in node.tpl? I have had the bright idea of changing my user permissions so as guests cannot see the comments on a node. I have no problems with Views around my site, as they can clearly see there are comments on nodes, and they will be more inclined to register. The problem seems to be when they arrive on my site directly to a node, they don't see any comment, and think "that's a bit crap; let's go."  Is there a way I could show the amount of comments below a node? Find all the numbers $n$ such that $\frac{4n-5}{60-12n}$ can't be reduced. Find all the numbers $n$ such that $\frac{4n-5}{60-12n}$ can't be reduced.   Attempt:  $$\gcd(4n-5,60-12n)=(4n-5,-8n+55)=(4n-5,-4n+50)=(4n-5,45)$$  $$n=1: (4-5,45)=1\quad \checkmark\\ n=2: (3,45)=3\quad \times\\ n=3: (7,45)=1\quad \checkmark\\ n=4: (11,45)=1\quad \checkmark\\ n=5: (15,45)=15\quad \times\\ n=6: (19,45)=1\quad \checkmark\\ n=7: (23,45)=1\quad \checkmark\\ n=8: (27,45)=9\quad \times\\ \vdots$$ So the answer is that it can't be reduced for $n=1,3,4,6,7,..$ i.e  $$\bigg\{n\bigg|n\notin \begin{cases}a_1=2\\a_n=a_{n-1}+3\end{cases}\bigg\}$$  I want to verify that my solution is correct Case of the 'mice problem' for $n=3$ Abstract algebra book recommendations for beginners. I am taking abstract algebra in this semester. I find that my lecturer didn't provide enough examples for understanding. So I would like to ask what are the recommended books which has lots of solved problems and exercises? The lecturer said in this semester, he will cover group theory, ring theory and a bit of field theory. Which book contains extensive information in these three fields?