Pollock's Fractals

So you go on various news sites while you're at work and read through all the main headline news articles, procrastinating away the morning. You start off alright, with something like this...

New York Times: Obama Reiterates Desire for Comprehensive Budget Package






...buuuuuuut you end up reading increasingly unimportant articles until eventually you end up here...

http://perezhilton.com/category/kim-kardashian

It's okay. We all do it. I love a bit of Perez as much as the next girl...You go into History afterwards and delete the incriminating evidence and then scramble to assuage your guilt over reading such rubbish by finding something intelligent and 'worthy' to read.

It's always good to have quality sites in your History just in case your computer crashes and you have to call the IT guys. Just as your mother may have advised you to always wear nice underwear in case you end up in hospital (yeah... I never quite got that one either. I think doctors in the ER have more pressing things to worry about... But it's a good enough analogy in this case I guess), you should always have a nice web history in case your computer crashes and the IT guys stumble upon it.

Jackson Pollock

So read something you can be proud to leave on your web history. Pollock's Fractals is a few years old now, but as with many of Discover Magazine's articles, its fascinating subject matter is timeless.

As a side note, if you are reading fun but useless articles about Kim Kardashian and leaving inane comments on the absurdity that "this is NEWS?!" please stop. Of course it's not news, but you did click on it, so why the outrage?

Anyway, read Jennifer Ouellette's piece about Pollock's Fractals. Do it. Now. And leave me a comment to tell me what you think!

And maybe afterwards, you can catch up on what Kourtney Kardashian thinks of Kris Jenner's facelift...

Until next time, xxj

Probably the best pizza in Pakistan?

So I wonder if this holds -

P(Pizza Party is the best pizza in Pakistan) > P(any other pizza in Pakistan is better than Pizza Party)

I'm not convinced.

Integration by Parts - KNOW THIS FOR YOUR HSC!!!!!

'Integration by Parts' is a method for finding the integral of two functions multiplied together. If you're doing Extension 2 in your Higher School Certificate (HSC), and in fact most advanced high school math exams for senior year students, this is a crucial rule to know. It will undoubtedly show up in some part of HSC exam Question 1, which usually deals with integration.

The integration by parts rule -


I kinda like the first way of writing it. It's just easier to say in your head and have it stick so you can just have it up in your memory for your exam. The second way with f(x) and g(x) is good when you are first learning this method, perhaps, because it helps visualise two separate functions and what's happening with these functions. It's really up to you which one you like, but I will be using the 'u' and 'v' notation rather than f(x) and g(x).

So here's an example of how to use integration by parts, and then after this example from the 2009 HSC Extension 2 exam, I will then show you where this rule came from. It's nice to see how everything connects, but honestly that is a luxury you may not necessarily have time for if you're just trying to cram in as much stuff as possible before your exams, so I have put it at the end of this post just so you can see how everything comes together and relates.



Now this is where the integration by parts rule comes from...

The product rule for differentiation!

Well, the product rule for differentiation is another rule you will have to know for your exam. If you don't know it, learn it asap. If you do know it, whew! :)

So the rule for integration by parts comes simply from integrating the product rule. Yep, that's it!

So, really, if you are in your exam and forget what the rule for integration by parts is, you can find it by integrating the product rule. I wouldn't recommend this, though. Just memorise them both. Time in an exam can be much better spent.

So here are the two ways of writing the product rule depending on which notation you like.

So if you integrate the first part of the product rule, it's like the integral is undoing the d/dx, because it's like what's the integral of the derivative of u*v? The integration and differentiation undo each other.
Any questions? Issues? Disputes? Confused? Just a bit of a Nigel and need a friend? Comment below!

xj

Google's unusual Nortel bids rooted in mathematics

Check out this article. Yay for math geeks :)

Google's unusual Nortel bids rooted in mathematics

It's so wrong to like a big corporation as much as I like Google, but I can't help it.

Does Origami Help with Medical Research?

The answer is YES it does! Cu-razy, huh?

I watched a fantastic lecture on YouTube last night (another rockin' night for me!) as part of the Museum of Mathematics channel. MIT professor Erik Demaine's lecture "The Geometry of Origami" is part of MoMath's Math Encounters series which aims to diversify and expand the general public's interest and understanding in mathematics.

You can find a direct link to Part 1 of 4 of Erik Demaine's lecture here.

The first three parts deal primarily with the relationship between origami folding and mathematics. There are mentions here and there of applications such as working out mathematically the most efficient way to fold an airbag flat in a car, but it is not until the 4th part of the lecture that he talks about really interesting practical applications relating to biomathematics and medical research.

In particular, Demaine's explains that linkage folding can be used to see how proteins in the body fold in 3 dimensions. Demaine outlines applications in research of diseases and the folding of proteins to capture bad proteins while leaving the others in tact.

So yeah, this is really great stuff, but... well it's all a bit science-y and serious.

You know when else these mathematics-based folding techniques are applied?

Designing Transformers!

Interesting article from the NY Times this week...

This is a really interesting article. I certainly can imagine this becoming an increasingly important idea in education. I certainly wish I had had these sorts of programs, particularly in high school, to build up my instincts when it comes to maths. Amazing. I wonder just how far-reaching the application of this style of learning could be... What do you think?

Brain Calisthenics for Abstract Ideas

http://www.nytimes.com/2011/06/07/health/07learn.html?_r=1

Mmmmm, pi....

TriBeCa Film Festival

So I saw 'Revenge of the Electric Car' at the TriBeCa Film Festival last week. Loved it. It's the sequel to 'Who Killed the Electric Car?' from a few years back.

Follow the link to read my blog post about it below. Just another one of the many interesting career paths involving maths and science. If I hear "Ugh, when will I ever use this?" from a student one more time... !!!

http://addingbliss.typepad.com/adding-bliss/2011/04/how-jess-spent-her-week-week-7-attending-the-tribeca-film-festival.html

xxj

Question 1, HSC 2009 Extension 1 exam

2009 HSC, Extension 1 Mathematics Question 1 (12 marks)
(a) Factorise 8x^3 +27 (2 marks)

Whew! This question is lovely, because in terms of factorising a cubic, this is as easy as it gets. It's of the form (a^3+b^3), where a=2x and b=3. This is really easy if you can remember the formula:

(a^3+b^3)=(a+b)(a^2-ab+b^2)


If you don't remember the formula, don't fret! You can look at this problem and kind of just figure it out by inspection and a little trial and error.

We're dealing with cubics, and we've got 8, x^3, and 27.



So 2 is the cube root of 8, x is the cube root of x^3, and 3 is the cube root of 27. Even if you're just guessing after this, you've got a pretty good idea of where to start.

(2x+3)

The reason I would 'trial' with the (+) is simply because the original polynomial has a (+). If it doesn't work out, try again with a minus sign. As I mentioned before, this is just if you don't remember the formula and you're stuck but still want to spend the time to work it out. I wouldn't do trial and error in an exam unless I was pretty sure I knew I was heading in the right direction. It's not worth spending lots of time on something if you're going in totally blind.

Anyway, so you have (2x+3) as one potential factor. Now let's see what's left over when we pull that out.

(2x+3)(4x^2-6x+9)

8x^3 +12x-12x+18x-18x +27

YAAAAAY!!!




(b) Let f(x)=ln(x-3). What is the domain of f(x)? (1 mark)

Because this is one mark, it means you should be able to do this by inspection based on your knowledge of the domain of the natural log function, or it should take just one or two simple steps. This should not take you longer than a minute.

What do we know about the function 'ln'??

Well, ln0 is undefined. Basically the ln(x) function is saying 'e' to the power of what number equals x? So 'e' to the power of what number equals zero? In this case, that number is undefined. Equally, e to the power of what number equals a negative number? Also undefined.



So now we can very easily answer this question. What are the values for x that satisfy the condition that (x-3) must be greater than zero?

(x-3)>0

Add 3 to both sides to have x as the subject and you get...

Domain: x>3


(c) Find lim ((sin(2x))/x) (1 point)
x->0

That looks confusing trying to type it out, but you can see below. Once again, this is a 1 point question so should take you a minute or so. Just a couple of steps.

Okay, there are a couple of things to note here.

First of all, it's a good idea to just know this:

If you know this, then you should know how to do this question. It is based on l'Hopital's rule.

This working should be sufficient for a 1 point question in the HSC. You may want to just say something about "from l'Hopital's rule..." or whatever but really for 1 point I can't imagine they'd need a proof as long as you know it.

Here is an explanation of how to use l'Hopital's rule:

(d) Solve the inequality (x+3)/(2x) > 1 (3 points)

Okay, so since this is three points we probably need a few steps, but solving inequalities should just be a mechanical process and if you have to stress about this or you're spending too much time thinking about it rather than just going through the motions of solving it, I suggest you just do a whole heap of these in preparation for your exam because it should be easy marks. Anything mechanical like this you want to just be able to churn out with no stress or effort. There will be longer harder questions later that deserve that stress and effort, but not this one.

Okay, so we want to find the critical point. We know that x can't equal zero because the term '2x' is in the denominator and this would be undefined.

So change the inequality into an equality for the time being to find the critical point.


(e) Differentiate x(cos^2(x)) (2 marks)

Once again, a mechanical question. Worth 2 marks, so I guess 1 mark for knowing what the product rule is and another mark for actually getting the correct answer.

The product rule is what you use to differentiate a function that is two functions multiplied together.

In this case, one function is x and the other is cos^2(x).

The product rule is...

f(x) = uv, where u and v are functions

then,

f'(x)= uv'+vu'

Pretty straight forward. You just need to memorise it.

(f) I have no idea how to do integration symbols and whatnot keyboard, so here's part (f) written out. It's worth 3 marks.


So there you go, question 1 of the 2009 extension 1 mathematics exam.

There's nothing in there that should cause any problems if you practice. Don't waste time writing out pretty notes about stuff or doing flashcards or whatever. Just do actual past papers! Do question 1 of 2009, 2008, 2007, 2006, 2005... You will start to see the same types of questions over and over again and this is how you really truly internalise it. Don't sit around memorising what l'Hopital's rule looks like if you have no idea how or when to apply it.

Anyway, with Question 1 it's often all mechanical, going through the motions type maths.

If you have any questions, please let me know I'm happy to help!

xx

How to... find the size of an interior angle of a right triangle using sin/cos/tan

So yesterday's post was working with right triangles and trig functions when you know the interior angles and you have the length of one side of the triangle, but what about if you want to find the size of an interior angle? Well, we use the same trig functions - sin/cos/tan - as we do if we're trying to find the length of a side except we use the inverse.


By using the inverse functions we can find the size of the interior angle. It's important when you're doing this to make sure you still get your SOH CAH TOA correct and pay special attention to which side is the opposite (O) or adjacent (A) side in relation to whichever angle you're wanting to find. It's a common mistake to go "oh, I use sine" and then switch up the O, A or H. This is particularly common if there is a drawing and the right triangle is positioned with the hypotenuse facing the bottom of the page, like this...



Anyway, here's an example of using inverse trig functions -

Now, if you put this into your calculator it will probably return the answer 36.8698 etc.

Generally we don't talk about degrees with decimals, but rather as degrees and minutes. Just like with time, there are sixty minutes in one degree as there are sixty minutes in one hour. To change it on your calculator to read as degrees and minutes rather than with a decimal, find the button on your calculator that looks like it is a couple of apostrophes and degrees signs (the little bubble apostrophe thing :) ). It may be different on different calculators, so play around with it and you'll know when you've found it because it will change your answer to read as degrees and minutes. Don't wait for an exam situation to do this :)

Xx

How to... find information about a right triangle using sin/cos/tan

Before you use these rules, it is important to ensure the triangle you are working with is, in fact, a right triangle. Never assume a triangle is a right triangle! This is a rookie mistake. Unless the angle is marked as 90 degrees, assume it is not until you can prove it.

Okay, so if you really do have a right triangle, you can find relationships between the interior angles and the lengths of the sides of the triangle using the trigonometric functions sine, cosine and tangent (i.e. sin, cos and tan).

Now, PLEASE don't judge my handwriting. I just bought a Bamboo pad and pen. I love it, but it feels pretty wonky for now... I'm hoping I'll get better at it once I get more practice in so you won't have to read math tutorials that appear to be written by a 5 year old...
Ergh, I know it looks awful! But that's all there is to it. If you say "SOH CAH TOA" out loud a couple of times it will just be etched into your brain forever. I'm not sure why, but it's one of those things everyone seems to remember... That doesn't necessarily mean people remember what it means or how to use it, but the sounds are easy enough to remember.

Here's an example... Again, excuse my awful scrawl. Practice makes perfect :)




You might have noticed an interesting relationship between sine and cosine. Oh god, so nerdy. Interesting? Yeah, I know what you're thinking. But the more of this stuff you do the more you will realise how everything is related and you just start to see relationships all over the place. Before you know it you'll be going "Ngaaaa" and yelling out "PI IS EXACTLY THREE!" to shock people into being quiet and getting their attention. Yes, the Simpsons episode where Lisa discovers the bully antidote. Anyway, back to the interesting relationship! In the example above, sin40' turns out to be the same as cos50'.

In fact, sin(x)=cos(90-x) and cos(x)=sin(90-x).

Anyway, this post could continue for pages and pages as I go on and on about different trig functions. I'll post some more stuff in the coming days, but for now I hope this has been of some use to you. Please send through any questions you may have. I'm happy to answer or clarify anything.