Dave

btw, whenever you have a limit that ends with 0/0 or inf/inf, you need to simplify it (use L'hopital rule or anything else to continue). 0/0 for limit is meaningless.

 

Or.. uhm.. correct me if I'm wrong? that's the way I studied it, and that's what jumped right away when I first saw the post.

 

So I'm not quite sure how I would continue this limit, but I wouldn't settle if it ended like that..

That's what I've done?

Of course. My objection was that the Taylor series is dependant on knowing the derivative. Depending on how you define e^x then I think the proof that e-coli linked to is basically sufficient.

Sure, the proof in ecoli's link is sufficient (if a little non-rigourous) but it's not the answer to the original question, which was to calculate the derivative using limits.

In this case, I believe that if you want to do this problem properly then you should start from the ground up and use the minimal amount of mathematics possible. (If you didn't want to do this, then why would you be considering this specific problem in the first place?) That means defining the function properly; most commonly, we can do that with a limit form, such as

[math]e^x = \lim_{n\to\infty} \left( 1 + \frac{x}{n} \right)^n[/math]

which you can show to be equivalent to the power series with a little work. This enables you can to calculate the derivative knowing very little at all about power series and nothing at all about logarithms.

If you're going to do that, then wouldn't it be simpler just to look at the derivative of each term in the series and that it is the previous term?

No, because you would have to prove that you can do such a thing. It is not trivial that

[math]f(x) = \sum_{n=0}^{\infty} a_n x^n \Rightarrow f'(x) = \sum_{n=1}^\infty n a_n x^{n-1}[/math]

I have been trying to teach myself calculus and I came upon this:

[math]\frac{d}{dx}e^{x}=e^{x}[/math]

I have been struggling to understand why this is. From my basic knowledge I have got this far.

[math]\frac{d}{dx}e^{x}=\frac{e^{x+h}-e^{x}}{(x+h)-x}[/math]

[math]\frac{d}{dx}e^{x}=\stackrel{Lim}h{\rightarrow}0\frac{e^{x+h}-e^{x}}{(x+h)-x}=\frac{e^{x}-e^{x}}{(x)-x}=\frac{0}{0}[/math]

So I keep coming up with this intermediate form and I cannot figure out how to get rid of it. I might be doing this completely wrong so could some one please point me in the right direction.

Your logic is not correct; specifically, the denominator of your limit is incorrect. It is, in fact, perfectly possible to obtain the derivative through the limit definition. Let [imath]f(x)=e^x[/imath], then we have that:

[math]f'(x) = \lim_{h\to 0} \frac{e^{x+h} - e^x}{(x+h) - x} = \lim_{h\to 0} \frac{e^x(e^h - 1)}{h} = e^x \lim_{h\to 0} \frac{e^h - 1}{h}[/math]

Now, if you accept that I can write [imath]e^x[/imath] in the common Taylor series form:

[math]e^{h} = 1 + h + \frac{h^2}{2!} + \frac{h^3}{3!} + \cdots = \sum_{k=0}^{\infty} \frac{h^k}{k!}[/math]

then

[math]e^{h} - 1 = h + \frac{h^2}{2!} + \frac{h^3}{3!} + \cdots = \sum_{k=1}^{\infty} \frac{h^k}{k!}[/math]

and so

[math]\frac{e^{h} - 1}{h} = 1 + \frac{h}{2!} + \frac{h^2}{3!} + \cdots = \sum_{k=0}^{\infty} \frac{h^k}{(k+1)!}[/math]

giving

[math]\lim_{h\to 0} \frac{e^{h} - 1}{h} = 1 + 0 + 0 + \cdots = 1[/math]

Hence,

[math]f'(x) = e^x \cdot 1 = e^x[/math].

I don't have the time to fully explain the problem, but essentially you have two surfaces here which intersect one another at a number of points. The idea here is to define a curve by the intersection of the two surfaces.

Drawing in 3D is pretty tricky. The easiest thing to do is to set particular values for x and y, and then figure out what z does. So, for example, with the first surface, if I set x=0, I get z = y^2 which is a simple parabola. Similarly, set y=0 and you get a parabola. In fact, that surface is known as a paraboloid; the rotation of a parabola around the z-axis. The second surface is clearly a plane.

Capn's quite right: for example, VeriSign charge $400/year for an SSL certificate, which is something that we could rightly do without.

Well, try and prove this.

I just read this and it made me actually laugh out loud. Love that one, ajb.

Fixed your link

Damnit.

Note to self: consume less beverages.

I am having similar problems. I suspect that something like zlib or mcache is misfiring on the server, but it could be any of a number of things. Possibly it is a server caching problem.

I'll have a look today, but I have to say that I'm not seeing any of these problems?

You'll also notice that times on the forum are no longer 17 minutes slow.

You can thank NTP for that

You need to give a quantitative description of what you mean by 'best'; what precisely are you looking for?

I guess I could rm -Rf /, that might screw things up a bit.

As said before, if you're reading this, you're on the new server! Let us know of any problems in the suggestions and comments forum.

No, you can derive [math]\frac{d}{dx}\sin{x}=\cos{x}[/math] just from the definition of the derivative and the angle sum formulas.

 

i.e. using [math]\frac{df(x)}{dx} = \lim{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}[/math]

To do this, you will also need to prove that:

[math]\lim_{x\to 0} \frac{\sin x}{x} = 1[/math]

which is a little tricky.

Well the changeover might be quite a bit before I anticipated. The server has been delivered and is now fully up and running. I'm getting initial software installed, and right now I'm planning on taking the forums down at some point tomorrow. I might do it early UK time since that's quite a quiet period.

The server itself is a Dell PowerEdge R200 with dual core xeons and 2gb of RAM. We're colocating it in a data centre in Maidenhead just outside of London.

Really that kind of capacity is overkill for SFN but we wanted room for expansion and also the server was on special offer at Dell

A little update: server was shipped today, and will arrive tomorrow. Depending on how quickly I can get things moving, we might do the changeover on Saturday.

This must be the first time in a while I've posted any kind of announcement, but at least I have good news! Well, there's a bit of bad news as well, but it pales in comparison to the good.

The bad news is: SFN will be down anywhere between a couple hours and a day. The good news: we're going to be moving to a much faster dedicated server. Hopefully, this will make browsing the site an order of magnitude faster and will be provide us with a bit more breathing room in terms of resources.

Now, we have no certain date that this will happen yet as I'm still to get the server shipped down to the data centre. However, I do anticipate that it will arrive on Thursday. The downtime should probably be either on Sunday or Monday. I will post more details as they come in, so stay tuned!

Sure. You can record the number of CPU clock cycles used, and that should give you the number of flops, but I'm not sure how one does this in Fortran.

Wooh I didn't mean that it will, or that it might... I am just putting forth my opinion on the matter, and - btw - I really don't mean to get ppl on the defensive, so please don't read what I'm saying as if I'm accusing the management here of bad judgment. I care about the forums, that's all. We always have room for improvement....

 

~moo

I, for one, think that this thread highlights some very good points and feeding trolls is certainly something we should all try to avoid doing. I don't think anybody is accusing anybody else of bad judgement.

This problem has been around ever since I joined SFN about a million years ago, and it's very difficult to deal with in a fair way. We've tried a lot of different approaches to tackle the problem, but right now I think there's a fairly good balance between getting rid of the trolls and allowing legitimate users to post what appear to be trolling questions.

That is no longer trolling, it's ridiculous silliness.

Yup. When it becomes clear that someone is clearly not interested in taking constructive criticism, SFNers need to stop responding and the mods need to step in and take care of it.

I'm not talking about the 'behind the scenes' actions, though, I'm talking about the actions that are SEEN by the n00bs. I think that we - the regular users - know by experience and by our time here - that the mods are keeping order, but the new users, who are here for their own preaching-havoc, might not.

This is a tough one. The flip side of the coin is that if we highlight the moderation action more, we come over as being heavy-handed and it puts people off posting in the first place. We've always been a bit secretive about the moderation of the forum, mostly because generally people shouldn't have to put up with the trolling and also because it's pretty dull at the end of the day.

Our general process begins with a user highlighting the problem via the reported post button, or a resident expert/moderator starting a thread about it. Once it's clear they're causing trouble, the staff discuss it and reach some kind of consensus on what the best approach is. I've always felt we should always give benefit of doubt about these things, even if it means a little inconvenience in the interim period. Some users have really split the staff on what to do (one such user managed to illicit well over 100 posts between about 6 staff members). Ultimately for the trolls, it'll end in a ban or suspension.

Clearly the users aren't really going to see a lot of the process. I've always taken the view that we should hide as much of this as possible as it distracts from the real reason that people are here: to talk about science. We've changed a few things (the thread in announcements forum for banned users, for instance) but if we need to be a bit more transparent then we'll have to change.

Seriously, if anybody else has threads such as these or suggestions they want to make then go ahead, because we're perfectly happy discussing them and they can only make things better.

Hmm. I'm not convinced this is a complete list, but I think it's almost there.

BASIC, Visual Basic, PHP, Tcl/Tk, C, C++, Fortran 95 (77 is evil), bash/awk and friends, Perl, Python, Java, JavaScript.

C has to be my favourite language, followed very closely by Python and PHP. I agree that C++ is mostly a waste of time; however the main scientific code that I use is written in it so I dwindle on

I've found Wikipedia is a good reference for many general equations of the Fourier transform (although you have to be a little careful of the normalization that you're using). Anyway, yes, the convolution identity you've got up there will work in reverse and I believe the proof will work in exactly the same way.

Certainly it's true in my experience. I've found that mental arithmetic is of very little consequence in a mathematics degree, since the emphasis is on proof. So as long as you know how to do addition, you're fine; whether or not you can necessarily do it is not really the point.

Indeed. My mental arithmetic has declined immensely since completing my degree