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HSC 2013-14 MX1 Marathon (archive) (2 Viewers)

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seanieg89

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Re: HSC 2013 3U Marathon Thread

And these sort of expressions are usually integrated by cases.
 

RealiseNothing

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Re: HSC 2013 3U Marathon Thread

The insides being trigonometric has nothing to do with it, it is just that

.
Is there anything you can do to get around this or compensate for the extra positive area you get?
 

seanieg89

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Re: HSC 2013 3U Marathon Thread

Yeah, you can use floor functions to find an expression for the DEFINITE integral between 0 and a for an arbitrary a. Let this expression be F(a), extend it to the whole real line by asserting that it must be odd, and add an arbitrary constant. This will be a primitive valid over the real line.
 

anomalousdecay

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Re: HSC 2013 3U Marathon Thread

I was in 2-unit one day and discovered a new way to use Newton's method of approximation, to get an exact answer of something.
How ironic, using an approximation to get the exact answer.
This is only a 2-unit question, but by being limited to using Newton's method of Approximation, it becomes extension 1 work.


Newton's method of Approximation.png
Given that the curve is y=x^2, find the equation of the tangent, given it touches y = x^2 at point A.

Hence, find the x-value of the intersection of the tangent with the curve, given that the tangent touches the x-axis at x=2, Using only Newton's approximation method (You must use it because I am a harsh marker) :chainsaw::awesome:

I discovered this result on my own, hopefully you guys can too. Image made using Geogebra.
 
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RealiseNothing

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Re: HSC 2013 3U Marathon Thread

I was in 2-unit one day and discovered a new way to use Newton's method of approximation, to get an exact answer of something.
How ironic, using an approximation to get the exact answer.
This is only a 2-unit question, but by being limited to using Newton's method of Approximation, it becomes extension 1 work.


View attachment 28150
Given that the curve is y=x^2, find the equation of the tangent, given it touches y = x^2 at point A.

Hence, find the x-value of the intersection of the tangent with the curve, given that the tangent touches the x-axis at x=2, Using only Newton's approximation method (You must use it because I am a harsh marker) :chainsaw::awesome:

I discovered this result on my own, hopefully you guys can too. Image made using Geogebra.
Isn't this essentially the proof of Newton's method but working backwards?
 

Makematics

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Re: HSC 2013 3U Marathon Thread

That isn't correct.
after checking the answer using 3 different methods, i can confirm it is indeed correct. well technically there are two answers if you take different cases, but sqrt (1+sin2x) +c is one of them at least :p if you change 1+sin2x into sin ^2 x +cos ^2 x + 2sinxcosx, it becomes sinx + cosx +c, which is the simplified answer.
 

Sy123

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Re: HSC 2013 3U Marathon Thread

after checking the answer using 3 different methods, i can confirm it is indeed correct. well technically there are two answers if you take different cases, but sqrt (1+sin2x) +c is one of them at least :p if you change 1+sin2x into sin ^2 x +cos ^2 x + 2sinxcosx, it becomes sinx + cosx +c, which is the simplified answer.
My mistake
 

anomalousdecay

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Re: HSC 2013 3U Marathon Thread

Isn't this essentially the proof of Newton's method but working backwards?


i.e. we want to find










That's correct. Well done.

It is similar to the proof, but it doesn't actually prove it, unless you already have in teh question.

But I just found it interesting and a much more elegant way of answering the question. I did that in my 2-unit exam, because I got bored with the extra time.
 

anomalousdecay

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Re: HSC 2013 3U Marathon Thread

Isn't this essentially the proof of Newton's method but working backwards?


i.e. we want to find










That's correct. Well done.

It is similar to the proof, but it doesn't actually prove it, unless you already have in the question.

But I just found it interesting and a much more elegant way of answering the question. I did that in my 2-unit exam, because I got bored with the extra time.
 

Sy123

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Re: HSC 2013 3U Marathon Thread

 

Sy123

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Re: HSC 2013 3U Marathon Thread



i dont need to put absolute because its not area yeh?
Nope the absolute value needs to be there, and you did the right thing by splitting into 4 integrals (only 2 is needed though).

 

HeroicPandas

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Re: HSC 2013 3U Marathon Thread

Nope the absolute value needs to be there, and you did the right thing by splitting into 4 integrals (only 2 is needed though).

Sorry i wasnt specific

Pretend, I = Z

Split 4 times,

I = |A|+ |B| + |C| + |D|

I = A + B + |- C - D| (i was talking about these absolute values)

i can eliminate each absolute value from each (sinx) because of ASTC

For, 0 ≤x≤ pi/2, sinx >0 (A)
For, pi/2 ≤ x ≤ pi, sinx >0 (S)
For, pi ≤ x ≤ 3pi/2, sinx <0 (T)
For, 3pi/2 ≤ x ≤ 2pi, sinx <0 (C)

and i realised i shouldnt have splitted it into 4 lol


what abotu this?
 

Sy123

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Re: HSC 2013 3U Marathon Thread

Sorry i wasnt specific

Pretend, I = Z

Split 4 times,

I = |A|+ |B| + |C| + |D|

I = A + B + |- C - D| (i was talking about these absolute values)

i can eliminate each absolute value from each (sinx) because of ASTC

For, 0 ≤x≤ pi/2, sinx >0 (A)
For, pi/2 ≤ x ≤ pi, sinx >0 (S)
For, pi ≤ x ≤ 3pi/2, sinx <0 (T)
For, 3pi/2 ≤ x ≤ 2pi, sinx <0 (C)

and i realised i shouldnt have splitted it into 4 lol


what abotu this?
Yep and

So the answer should be 2 :p
 
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Re: HSC 2013 3U Marathon Thread

Or just draw a picture lol
 

Sy123

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Re: HSC 2013 3U Marathon Thread



 
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