ATAR Notes: Forum

VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: TrueTears on July 21, 2011, 02:53:54 am

Title: Resources/Guides: For advanced students
Post by: TrueTears on July 21, 2011, 02:53:54 am: So here I'll post some guides on some topics that are above VCE level for those keen students, however you can grasp these concepts to a sufficient degree with VCE knowledge. However some topics actually aren't necessarily "harder" than your VCE maths, rather they emphasize rigour and solid proofs.

To get started, here's a nice guide on epsilon-delta proofs. Used quite a lot in real analysis.

Before you get into it, here's a bit of background on what these are.

I'm sure most of you guys have done differentiation by first principles, you know, where you limit the h to 0 and etc. Yes, that stuff. Now remember the last step? Where you guys just thought "hmm $\lim_{h \to 0}$ just means I can sub h = 0 in, after canceling out the h's" Well a really basic way of extending that limit stuff to a more rigorous approach is using epsilon delta proofs. What you will find is that you are actually not "substituting h = 0 everytime you see h", rather you are using a very intuitive approach formalised by using an epsilon-delta proof.

Now go read.

Also I attached them as pics because VN won't let me attach as a rar file o.O

Vic Mod edit: re-added attachments
Title: Re: Resources/Guides: For advanced students
Post by: taiga on July 21, 2011, 03:01:59 am: Finally some of your maths I can understand :P
Title: Re: Resources/Guides: For advanced students
Post by: TrueTears on July 21, 2011, 03:08:52 am: haha :P

Anyways, for those who are quite keen, here's a challenge question, the method to prove this one is a bit different from the examples that you've read above, however the general jist is the same, if you understood the overall concept, have a good at this one, post a solution even if you think it's not right! Have a shot!

Show that $\lim_{x \to 0} \frac{1}{x^2} = \infty$
Title: Re: Resources/Guides: For advanced students
Post by: xZero on July 21, 2011, 02:40:15 pm: if $|x-a|<\delta$ then $|\frac{1}{x^2}-c|<\eps$ , since $c=\infty$ , we have to adjust the epsilon part of the proof, let M=any positive number, f(x)>M (this means that as you approach the point a, f(x) becomes larger than any finite number)

$f(x)>M \Rightarrow \frac{1}{x^2}>M$
$\Rightarrow \frac{1}{M}>x^2$
$\Rightarrow |x|<\frac{1}{\sqrt{M}}$

let $\delta=\frac{1}{\sqrt{m}}$
$|x|<\delta \Rightarrow |x|<\frac{1}{\sqrt{M}}$ , a=0
$\Rightarrow x^2<\frac{1}{M}$
$\Rightarrow \frac{1}{x^2}>M$
$\Rightarrow f(x)>M$
This completes the proof, hence $\lim_{x \to 0} \frac{1}{x^2} = \infty$

Think this is correct :S
Title: Re: Resources/Guides: For advanced students
Post by: TrueTears on July 22, 2011, 02:05:02 am: Nice :]
Title: Re: Resources/Guides: For advanced students
Post by: TrueTears on July 22, 2011, 11:59:00 pm: wdf? why is this thread "doubled"
Title: Re: Resources/Guides: For advanced students
Post by: Hutchoo on July 23, 2011, 12:09:47 am: Quote from: TrueTears on July 22, 2011, 11:59:00 pm
wdf? why is this thread "doubled"
Two times the fun.
Title: Re: Resources/Guides: For advanced students
Post by: TrueTears on July 23, 2011, 12:11:24 am: Quote from: Hutchoo on July 23, 2011, 12:09:47 am
Quote from: TrueTears on July 22, 2011, 11:59:00 pm
wdf? why is this thread "doubled"
Two times the fun.
loving ur dp brah, makes it triple the fun
Title: Re: Resources/Guides: For advanced students
Post by: TrueTears on July 23, 2011, 12:12:24 am: also wat the shit, where did xZero's post go.

WDF, why are so many threads are all screwed up on vn
Title: Re: Resources/Guides: For advanced students
Post by: Mao on July 23, 2011, 12:13:38 am: wait, where did all the attachments go?

years of collection of useful things. GONE!

Thank god for agora. Not all is lost.
http://vce.atarnotes.com/agora/index.php
Title: Re: Resources/Guides: For advanced students
Post by: xZero on July 23, 2011, 12:51:13 am: Quote from: Hutchoo on July 23, 2011, 12:09:47 am
Quote from: TrueTears on July 22, 2011, 11:59:00 pm
wdf? why is this thread "doubled"
Two times the fun.
Is your dp yoona? And wtf how come my post didn't get doubled?
Title: Re: Resources/Guides: For advanced students
Post by: Lasercookie on August 30, 2011, 10:09:11 pm: I found where the notes that were attached to the first post went:
Re: Chat.

Forum must have glitched lol.
Title: Re: Resources/Guides: For advanced students
Post by: TrueTears on August 30, 2011, 11:29:03 pm: Quote from: laseredd on August 30, 2011, 10:09:11 pm
I found where the notes that were attached to the first post went:
Re: Chat.

Forum must have glitched lol.
LOOOOOOOOOOOOOL NICE FIND
Title: Re: Resources/Guides: For advanced students
Post by: Mao on August 30, 2011, 11:58:57 pm: Quote from: laseredd on August 30, 2011, 10:09:11 pm
I found where the notes that were attached to the first post went:
Re: Chat.

Forum must have glitched lol.

WUT.

So what you mean is, there is a whole heap of practice exams littered in random threads all over the forum? Lawl'd.
Title: Re: Resources/Guides: For advanced students
Post by: Lasercookie on August 31, 2011, 08:01:02 am: Quote from: Mao on August 30, 2011, 11:58:57 pm
Quote from: laseredd on August 30, 2011, 10:09:11 pm
I found where the notes that were attached to the first post went:
Re: Chat.

Forum must have glitched lol.

WUT.

So what you mean is, there is a whole heap of practice exams littered in random threads all over the forum? Lawl'd.
Lol, nah I think it's just stuff that was uploaded after the upgrade. I remember I did upload some stuff that did go missing - that's probably attached to some random threads now lol.
Title: Re: Resources/Guides: For advanced students
Post by: pi on June 20, 2012, 04:55:56 pm: Re-uploaded the attachments :)
Title: Re: Resources/Guides: For advanced students
Post by: jia0020 on February 09, 2015, 08:52:01 am: Quote from: Mao on July 23, 2011, 12:13:38 am
wait, where did all the attachments go?

years of collection of useful things. GONE!

Thank god for agora. Not all is lost.
http://vce.atarnotes.com/agora/index.php

No it's gone.....