HSC 2012 MX1 Marathon #2 (archive) (1 Viewer)

[asianese]

Re: HSC 2012 Marathon

Hehe..

ALSO!! OMGAH found some 'proof' or some thingos to do with 'e' and the limiting sum! Totally correlates with what we were discussing before. It's from a 4U 1998 Syd Boys paper but yeah.

http://4unitmaths.com/sydneyboys1998.pdf

8b): People in the know - is what they're doing fine? Or is it completely okay since they didn't want us to 'prove' that it is e???

 

[Carrotsticks]

Re: HSC 2012 Marathon

Hehe..

ALSO!! OMGAH found some 'proof' or some thingos to do with 'e' and the limiting sum! Totally correlates with what we were discussing before. It's from a 4U 1998 Syd Boys paper but yeah.

http://4unitmaths.com/sydneyboys1998.pdf

8b): People in the know - is what they're doing fine? Or is it completely okay since they didn't want us to 'prove' that it is e???

Not 100% happy with it, since it assumed we are taking a discrete limit, but I suppose for the most part it's okay. After all, the idea of the HSC Mathematics is to examine your ability to apply your knowledge, rather than strict formalities and justifications (although ofc those are important too sometimes).

Also, those who attended the BOS MX2 seminar have, included in the booklet I gave them, a proof for the series expansion of e.

Also, the proof for the limit definition of e (not the series one) has been in the HSC twice, once in the 2009 HSC and again in another 3U HSC (forgot which year).

 

[Sy123]

Re: HSC 2012 Marathon

Hehe..

ALSO!! OMGAH found some 'proof' or some thingos to do with 'e' and the limiting sum! Totally correlates with what we were discussing before. It's from a 4U 1998 Syd Boys paper but yeah.

http://4unitmaths.com/sydneyboys1998.pdf


8b): People in the know - is what they're doing fine? Or is it completely okay since they didn't want us to 'prove' that it is e???

I cleared it up with seanieg89 earlier after my post on the e series proof. I questioned him about this:
http://en.wikipedia.org/wiki/Binomial_theorem#Applications

I you look at the section Applications, you will see a part called 'Series for e'. Which they then prove the e infinite series using the Binomial Expansion, however seanieg89 said that they justified what they did by saying that it follows the monotone convergence theorem, hence it is justified to take the limit of each term seperately. (Dont ask me what the theorem actually is, I dont anything about it! heh)

So yeah.

 

[RealiseNothing]

Re: HSC 2012 Marathon

I cleared it up with seanieg89 earlier after my post on the e series proof. I questioned him about this:
http://en.wikipedia.org/wiki/Binomial_theorem#Applications

I you look at the section Applications, you will see a part called 'Series for e'. Which they then prove the e infinite series using the Binomial Expansion, however seanieg89 said that they justified what they did by saying that it follows the monotone convergence theorem, hence it is justified to take the limit of each term seperately. (Dont ask me what the theorem actually is, I dont anything about it! heh)

So yeah.

A monotone series is where the series is either increasing or decreasing (not necessarily strictly though). Convergence occurs when this series is bounded.

 

[Carrotsticks]

Re: HSC 2012 Marathon

I cleared it up with seanieg89 earlier after my post on the e series proof. I questioned him about this:
http://en.wikipedia.org/wiki/Binomial_theorem#Applications

I you look at the section Applications, you will see a part called 'Series for e'. Which they then prove the e infinite series using the Binomial Expansion, however seanieg89 said that they justified what they did by saying that it follows the monotone convergence theorem, hence it is justified to take the limit of each term seperately. (Dont ask me what the theorem actually is, I dont anything about it! heh)

So yeah.

The theorem basically states that if a function is monotone and bounded, then it converges.

We need that condition for their proof for 'e' because since the sequence is monotone, it is in a sense 'well behaved', so we can in fact work with the terms individually rather than as an entire sequence in itself, without having to fear that it does something funky halfway through.

So for example, a sequence that we WOULDN'T treat term-by-term is sin(n)/n because although it converges to 0, it oscillates up and down and most certainly not monotone.

 

[seanieg89]

Re: HSC 2012 Marathon

Hehe..

ALSO!! OMGAH found some 'proof' or some thingos to do with 'e' and the limiting sum! Totally correlates with what we were discussing before. It's from a 4U 1998 Syd Boys paper but yeah.

http://4unitmaths.com/sydneyboys1998.pdf

8b): People in the know - is what they're doing fine? Or is it completely okay since they didn't want us to 'prove' that it is e???

As others have said...the manipulation is legal BUT it is well beyond a four unit student to justify why. (So I think it is a silly question to ask. All it does is lull more students into the trap of thinking things that LOOK true are true whether or not you can prove them!)

 

[seanieg89]

Re: HSC 2012 Marathon




(This is an example of a smooth function which is zero and has all derivatives zero at a certain point, but is NOT the zero function.)

 

[Sy123]

Re: HSC 2012 Marathon

Evaluate:



Giving Reasons.

EDIT: Ive been trying to do seanieg's question for a while now but I cant seem to get it, nor do I understand it completely. Isnt f(0)=0 for all x because is its part of the zero function itself?

 

[RealiseNothing]

Re: HSC 2012 Marathon

Evaluate:



Giving Reasons.

EDIT: Ive been trying to do seanieg's question for a while now but I cant seem to get it, nor do I understand it completely. Isnt f(0)=0 for all x because is its part of the zero function itself?

Change all the sine's of the second half of the expression to cosine's.



Since by squaring both sides we get

So all the sine's and cosine's cancel out and we are left with the answer of 22.

 

[Sy123]

Re: HSC 2012 Marathon

Correct, Ive been working on a Projectile Motion question but that will be for tomorrow since I need to finalise the proofs and stuff. Someone post a question.

 

[barbernator]

Re: HSC 2012 Marathon

<a href="http://www.codecogs.com/eqnedit.php?latex=sin^{2}(2^{\circ})@plus;sin^2(4^{\circ})@plus;...@plus;sin^2(88^{\circ})\\ =sin^{2}(2^{\circ})@plus;sin^2(4^{\circ})@plus;...@plus;sin^2(44^{\circ})@plus;cos^2(90-46^{\circ})@plus;...@plus;cos^2(90-88^{\circ})(sin(\theta)=cos(90-\theta ))\\ =(sin^{2}(2^{\circ})@plus;cos^2(2^{\circ}))@plus;(sin^{2}(4^{\circ})@plus;cos^{2}(4^{\circ}))@plus;...@plus;(sin^{2}(44^{\circ})@plus;cos^{2}(44^{\circ}))\\ =22" target="_blank"><img src="http://latex.codecogs.com/gif.latex?sin^{2}(2^{\circ})+sin^2(4^{\circ})+...+sin^2(88^{\circ})\\ =sin^{2}(2^{\circ})+sin^2(4^{\circ})+...+sin^2(44^{\circ})+cos^2(90-46^{\circ})+...+cos^2(90-88^{\circ})(sin(\theta)=cos(90-\theta ))\\ =(sin^{2}(2^{\circ})+cos^2(2^{\circ}))+(sin^{2}(4^{\circ})+cos^{2}(4^{\circ}))+...+(sin^{2}(44^{\circ})+cos^{2}(44^{\circ}))\\ =22" title="sin^{2}(2^{\circ})+sin^2(4^{\circ})+...+sin^2(88^{\circ})\\ =sin^{2}(2^{\circ})+sin^2(4^{\circ})+...+sin^2(44^{\circ})+cos^2(90-46^{\circ})+...+cos^2(90-88^{\circ})(sin(\theta)=cos(90-\theta ))\\ =(sin^{2}(2^{\circ})+cos^2(2^{\circ}))+(sin^{2}(4^{\circ})+cos^{2}(4^{\circ}))+...+(sin^{2}(44^{\circ})+cos^{2}(44^{\circ}))\\ =22" /></a>

 

[seanieg89]

Re: HSC 2012 Marathon

Evaluate:



Giving Reasons.

EDIT: Ive been trying to do seanieg's question for a while now but I cant seem to get it, nor do I understand it completely. Isnt f(0)=0 for all x because is its part of the zero function itself?

The question you are asking does not make sense, there is no x involved in the equation f(0)=0. My question is just asking to show that the piecewise defined function f is infinitely differentiable at x=0, with all derivatives zero. This is just an exercise in first principles differentiation.

 

[RealiseNothing]

Re: HSC 2012 Marathon

I haven't tried this question so not sure if it'll be easy/hard, but I just thought of it now.

In how many ways can you arrange 10 people if you have 2 tables that both have 10 chairs.

 

[Sy123]

Re: HSC 2012 Marathon

Assuming that the 10 people can sit anywhere of the 20 seats, is it:



Probably wrong, perms and combinations are not my finest topics.

 

[bleakarcher]

Re: HSC 2012 Marathon

Evaluate:



Giving Reasons.

EDIT: Ive been trying to do seanieg's question for a while now but I cant seem to get it, nor do I understand it completely. Isnt f(0)=0 for all x because is its part of the zero function itself?

Define the proposition P(k): f is k times differentiable at 0 with f^(k)(0)=0 for all natural (where 0 is included)
If k=0: It is clear that f is 0 times differentiable at x=0 since a cusp occurs. f^(0)(0)=f(0)=0
Hence, P(0) holds.
I think if you continue the inductive argument it works.

 

[bleakarcher]

Re: HSC 2012 Marathon

I haven't tried this question so not sure if it'll be easy/hard, but I just thought of it now.

In how many ways can you arrange 10 people if you have 2 tables that both have 10 chairs.

Are the tables considered identical?

 

[seanieg89]

Re: HSC 2012 Marathon

Define the proposition P(k): f is k times differentiable at 0 with f^(k)(0)=0 for all natural (where 0 is included)
If k=0: It is clear that f is 0 times differentiable at x=0 since a cusp occurs. f^(0)(0)=f(0)=0
Hence, P(0) holds.
I think if you continue the inductive argument it works.

Induction is a good way to do it, but x=0 is not a 'cusp'.

 

[RealiseNothing]

Re: HSC 2012 Marathon

Are the tables considered identical?

Yes.

 

[bleakarcher]

Re: HSC 2012 Marathon

Induction is a good way to do it, but x=0 is not a 'cusp'.

It's still non-differentiable at x=0 right?

 

[seanieg89]

Re: HSC 2012 Marathon

No. The whole point of the question is to prove that it IS differentiable at zero. And twice differentiable at zero. Etc.