Suppose that $f$ is a function such that $f'(x) = \frac{1}{x}$ for all    $x &gt; 0$. Prove that if $f(1) = 0$, then $f(xy) = f(x) + f(y)$ for all    $x, y &gt; 0$   I've tried applying the Mean Value Theorem, such that $f'(x) = \frac {f(b) - f(a)}{b - a} = \frac {1}{x}$, where $y = f(x)$ if continuous on $[a,b]$ and differentiable on $(a,b)$ such that there exists at least one point $c ∈ (a,b)$.   I'm probably wrong, but i don't know how do i really approach this question and what they are really asking for (e.g. $f(xy) = f(x) + f(y)$)
Suppose f is a function such that $f'(x) = \frac{1}{x}$ for all $x &gt; 0$. Prove that if $f(1) = 0, then f(xy) = f(x) + f(y)$ for all $x,y &gt; 0$  As stated above, how do i go about doing this question?   Hi all, i would like to apologize if the question is missing context or other details.   This question was part of my calculus tutorial. I was on the topic of application of differentiation when this question was given to me.  Thus far i have learnt about intermediate value theorem, mean value theorem, rolle's theroem, Fermat's thereom.  As of this point in my learning, i have not touched upon integration. The only reason why i accepted the answer with integration in it was because i understood it.  Thus i believe i am suppose to use one the few theorems that i've stated to solve this question. But i have no idea where to start.  At first i differentiated the equation $f(xy)=f(x)+f(y)$ with respect to x  i believed the answer to be $f'(xy)\frac{dy}{dx}=f'(x)+f'(y)\frac{dy}{dx}$  which would be $\frac{1}{xy}\frac{1}{x}=\frac{1}{x}+\frac{1}{y}\frac{1}{x}$  At this point a few questions came to my mind  1) am i suppose to differentiate the eqn in the first place 2) what has $f(1)=0$ got to do with anything 3) what theorem am i suppose to use to solve this question  This was my thought process throughout the question and i hope someone would point out my mistakes. Thank you
How to cleanly launch a GUI app via the Terminal? I am trying to use following apex class with invocable method but its not showing in process builder.  global with sharing class SlackOpportunityPublisher {      private static final String slackURL = 'https://testsalesforceapps.slack.com/';      global class Oppty {         @InvocableVariable(label='Opportunity Name')         global String opptyName;         @InvocableVariable(label='Stage')         global String stage;     }      @InvocableMethod(label='Post to Slack')     global static void postToSlack(List&lt;Oppty&gt; oppties) {         Oppty o = oppties[0]; // If bulk, only post first to avoid overloading Slack channel         Map&lt;String,Object&gt; msg = new Map&lt;String,Object&gt;();         msg.put('text', 'The following opportunity has changed:\n' + o.opptyName + '\nNew Stage: *' + o.stage + '*');         msg.put('mrkdwn', true);         String body = JSON.serialize(msg);             System.enqueueJob(new QueueableSlackCall(slackURL, 'POST', body));     }      global class QueueableSlackCall implements System.Queueable, Database.AllowsCallouts {          private final String url;         private final String method;         private final String body;          global QueueableSlackCall(String url, String method, String body) {             this.url = url;             this.method = method;             this.body = body;         }          global void execute(System.QueueableContext ctx) {             HttpRequest req = new HttpRequest();             req.setEndpoint(url);             req.setMethod(method);             req.setBody(body);             Http http = new Http();             HttpResponse res = http.send(req);         }      }  } Deriving the Coulomb force equation from the idea of photon exchange? Since Newton's law of gravitation can be gotten out of Einstein's field equatons as an approximation, I was wondering whether the same applies for the electromagnetic force being the exchange of photons. Is there an equation governing the force from the exchange of photons? Are there any links which would show how the Coulomb force comes out of the equations for photon exchange? I know that my question is somewhat similar to the one posted here , but it doesn't really have an answer to my question specifically. How do I cd into a directory in the home folder? How can I send mail from an iPhone application I was checking on something in the Harry Potter wiki and saw Harry listed as "Half-Blood."  Both his parents were magical, but, as we know, his grandparents on his Mother's side were muggles.  This reminds me of the racially loaded terms like octoroon.  (In the south, you still hear those terms every once in a while.)  When there was still segregation, having only 1 great-grandparent of an ethnic group was enough to classify a person of that group.  So how far down the line did it go in the Harry Potter universe?  Just how many generations removed from a muggle-born would a person have to be before they're no longer classified as half-blood or something similar? Validity of naively computing the de Broglie wavelength of a macroscopic object Many introductory quantum mechanics textbooks include simple exercises on computing the de Broglie wavelength of macroscopic objects, often contrasting the results with that of a proton, etc.  For instance, this example, taken from a textbook:     Calculate the de Broglie wavelength for      (a) a proton of kinetic energy 70 MeV      (b) a 100 g bullet moving at 900 m/s   The pedagogical motivation behind these questions is obvious: when the de Broglie wavelength is small compared to the size of an object, the wave behaviour of the object is undetectable.  But what is the validity of the actual approach of applying the formula naively? Of course, a 100g bullet is not a fundamental object of 100g but a lattice of some $10^{23}$ atoms bound by the electromagnetic force. But is the naive answer close to the actual one (i.e. within an order of magnitude or two)? How does one even calculate the de Broglie wavelength for a many body system accurately? Travel to US two day before B1 visa expires Live notification shown for own edits When editing an answer, the &quot;new revision&quot; message is shown for a short moment. This makes no sense because the revision you just created is automatically displayed anyway. Undefined reference to `pow' and `floor' When a user is suspended, their account is reduced to 1 rep and all of his privileges are stripped. However, they can still join chat rooms.   This leads to various forms of possible abuse (all of which I've seen by suspended users):   Join/leave spams Expletives in profile messages Hold one sided conversations with users of the room via    Suspended users cannot be kicked, and even moderators can do very little (aside from completely removing the account, which is often overkill).  My proposal is to extend  for suspended users to one of two levels:   Prevent them from being shown in the user list at all. Strip their profile profile picture and description from the profile view visible from chat rooms (not from profile pages). Why can't the Others go around the Wall at sea? As explained in a  earlier , the Wall does not extend all the way to the sea on the East, and the castle there is essentially undefended anyway. Thus, wildlings -- and Stannis' army -- were able to go around it undetected.  We know that the Others and the wights they raise can't cross through the Wall, and I think we can assume that it's built to stand up to any attacks they make trying to bring it down. But there does not appear to be any reason they can't just go around like anyone else.  Possibly related: in the TV Show, we actually see the attempted evacuation of Hardhome, and the Night King allows Jon Snow and his human army and civilians to flee back to their boats to safety, even after raising a massive army of wights. You could interpret this as merely a psychological ploy, but it may be a hint that the dead can't cross the water either.  Has there been any indication (books or show) that the Others/White Walkers and their army of wights are blocked by open water, or are somehow tied to the land of the Frozen North, or that there's any other obstacle keeping them from just sailing/swimming around the Wall? Access a glyph on fontspec using a GID (Glyph ID) I'm trying to access a glyph, but it's only mentioned in the "Basic Latin and Latin 1" set for a font. The glyph has no name or unicode ID, but it does have a glyph ID or GID. fontspec doesn't mention possibilities for using this glyph. Neither does Adobe in the adobe feature file syntax:   I'd still like to be able to access a glyph through its ID, since it's the only way I can access it (I think).  My specific case requires to access the glyph ID 554 from Stevens Titling Pro Sable Brush. I will only post this font if it's strictly necessary, since I don't like to distribute these fonts to the wide world.  I posted an MWE containing another (free) font. This is not the font I intend to use, but perhaps a starting point to try and access glyphs manually.   \documentclass{article}  \usepackage{fontspec} \setmainfont{EB Garamond}  \parindent=0pt \begin{document}  Some text int EB Garamond. How can I access glyph 123 now?  \end{document} Possible Duplicate:          If I am using the word model in the context of financial models and the UK, then which of these words should I use? Also, are there any key difference in the meanings of the two? On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $ To prove Cayley-Hamilton theorem, why can't we substitute $A$ for $\lambda$ in $p(\lambda) = \det(\lambda I - A)$? Cayley Hamilton Theorem states that if $A$ is an $n \times n$ matrix over the field $F$ then $p(A) = 0$.  We note that $p(\lambda) = \det(\lambda I - A)$. Hence, why can't we just substitute $\lambda$ with $A$ and directly prove Cayley-Hamilton Theorem by saying that $p(A) = p(\lambda) = \det(\lambda I - A) = \det(AI - A) = 0$? .NET offers a generic list container whose performance is almost identical (see Performance of Arrays vs. Lists question). However they are quite different in initialization.   Arrays are very easy to initialize with a default value, and by definition they already have certain size:  string[] Ar = new string[10];   Which allows one to safely assign random items, say:  Ar[5]="hello";   with list things are more tricky. I can see two ways of doing the same initialization, neither of which is what you would call elegant:  List&lt;string&gt; L = new List&lt;string&gt;(10); for (int i=0;i&lt;10;i++) L.Add(null);   or  string[] Ar = new string[10]; List&lt;string&gt; L = new List&lt;string&gt;(Ar);   What would be a cleaner way?  EDIT: The answers so far refer to capacity, which is something else than pre-populating a list. For example, on a list just created with a capacity of 10, one cannot do L[2]="somevalue"  EDIT 2: People wonder why I want to use lists this way, as it is not the way they are intended to be used. I can see two reasons:   One could quite convincingly argue that lists are the "next generation" arrays, adding flexibility with almost no penalty. Therefore one should use them by default. I'm pointing out they might not be as easy to initialize. What I'm currently writing is a base class offering default functionality as part of a bigger framework. In the default functionality I offer, the size of the List is known in advanced and therefore I could have used an array. However, I want to offer any base class the chance to dynamically extend it and therefore I opt for a list. Static Classes In Java Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$ Here is an exercise, on analysis which i am stuck.    How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on $\mathbb{R}$? I'm new to feature selection and I was wondering how you would use PCA to perform feature selection. Does PCA compute a relative score for each input variable that you can use to filter out noninformative input variables? Basically, I want to be able to order the original features in the data by variance or amount of information contained.