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HSC 2012 MX1 Marathon #2 (archive) (2 Viewers)

barbernator

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Re: HSC 2012 Marathon :)

find the inverse function of. and state its domain and range
<a href="http://www.codecogs.com/eqnedit.php?latex=y=e^{4x}@plus;5e^{2x}@plus;1" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y=e^{4x}+5e^{2x}+1" title="y=e^{4x}+5e^{2x}+1" /></a>
 
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bleakarcher

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Re: HSC 2012 Marathon :)

find the inverse function of. and state its domain and range
<a href="http://www.codecogs.com/eqnedit.php?latex=y=e^{4x}@plus;5e^{2x}@plus;6" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y=e^{4x}+5e^{2x}+6" title="y=e^{4x}+5e^{2x}+6" /></a>
Cbf getting a piece of paper and answering this right now but just to help out you have to restrict the function to a particular domain for there to exist an inverse function.
 

Sy123

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Re: HSC 2012 Marathon :)

find the inverse function of. and state its domain and range
<a href="http://www.codecogs.com/eqnedit.php?latex=y=e^{4x}@plus;5e^{2x}@plus;6" target="_blank"><img src="http://latex.codecogs.com/gif.latex?y=e^{4x}+5e^{2x}+6" title="y=e^{4x}+5e^{2x}+6" /></a>


I sketched the original function in Geogebra, and it turns out it is monotonic. So no need to restrict domain, and I have clarified it with ln(g(x))>0 since we cant have -5 - sqrt{4y+21} since that will always be negative
 
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barbernator

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Re: HSC 2012 Marathon :)

different constants, as the restricted domains of the original functions vary by pi/2

simplify the following trig expression.

sin(x)+cos(2x)+sin(3x)+cos(4x) damnit that doesn't even come out good at all.

Here is a conceptual one :)

Find the volume created by rotating y=x^2 - 3 about the x axis. Find this volume using simpsons rule and 5 values. state why they are the same/
 
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bleakarcher

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Re: HSC 2012 Marathon :)



I sketched the original function in Geogebra, and it turns out it is monotonic. So no need to restrict domain, and I have clarified it with ln(g(x))>0 since we cant have -5 - sqrt{4y+21} since that will always be negative
With this question, how come the domain of the inverse function isnt identical to the range of the function?

The range of the function is y>1 and the domain of the inverse function is x>=-21/4
 
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Sy123

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Re: HSC 2012 Marathon :)

Hmm, the volume is not defined?

Because the parabola is fully above the x-axis, and there is no enclosed part to rotate about. Or is that the trick?


EDIT: I have no idea why the inverse function turned out like that, Ive rechecked my working out, and my algebra seems fine.

When you sketch both functions, they are near perfect reflections of each other except for the fact, that the inverse function is cut off at -21/4

But, it has something to do with the square root since,

when you test

You get x>1 (domain) while the range of the original function is indeed y>1
 
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barbernator

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Re: HSC 2012 Marathon :)

Hmm, the volume is not defined?

Because the parabola is fully above the x-axis, and there is no enclosed part to rotate about. Or is that the trick?
whoops, sorry it was supposed to be x^2 - 3 haha
 

bleakarcher

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Re: HSC 2012 Marathon :)

Hmm, the volume is not defined?

Because the parabola is fully above the x-axis, and there is no enclosed part to rotate about. Or is that the trick?


EDIT: I have no idea why the inverse function turned out like that, Ive rechecked my working out, and my algebra seems fine.

When you sketch both functions, they are near perfect reflections of each other except for the fact, that the inverse function is cut off at -21/4

But, it has something to do with the square root since,

when you test

You get x>1 (domain) while the range of the original function is indeed y>1
oh yes of course. lol, I got confused when barbenator said the range was y>-21/4 and didnt bother working it out on my own..
 

Timske

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Re: HSC 2012 Marathon :)

different constants, as the restricted domains of the original functions vary by pi/2

simplify the following trig expression.

sin(x)+cos(2x)+sin(3x)+cos(4x) damnit that doesn't even come out good at all.

Here is a conceptual one :)

Find the volume created by rotating y=x^2 - 3 about the x axis. Find this volume using simpsons rule and 5 values. state why they are the same
/
V = pi x^2 find intercepts y^2=(x^2-3)^2
Simpsons rule uses parabolic
 

Sy123

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Re: HSC 2012 Marathon :)

Also I dont understand your question, do you want me to find the volume, rotated by x-axis via normal method, AND simpsons rule, and explaining why they are the same volume?

I did that and I got

They are the same, because Simpsons Rule is approximating via sketching parabolas around the curve and the such, and since this is a perfect parabola, Simpson's Rule will give an exact answer?

Also I have a 3U test on Applications of Calculus to the Physical world and Inverse Functions, can someone post a question from those topics, if you could?
Thanks!
 

barbernator

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Re: HSC 2012 Marathon :)

Also I dont understand your question, do you want me to find the volume, rotated by x-axis via normal method, AND simpsons rule, and explaining why they are the same volume?

I did that and I got

They are the same, because Simpsons Rule is approximating via sketching parabolas around the curve and the such, and since this is a perfect parabola, Simpson's Rule will give an exact answer?
yeh, that is correct. The question was aimed at just testing the understanding that simpsons rule uses parabolas to estimate that is all :)
 

barbernator

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Re: HSC 2012 Marathon :)

Ok here is another question. Prove any product-sum or sum-product trig ratio from the startpoint of sin(x+y) and sin(x-y)
 

Sy123

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Re: HSC 2012 Marathon :)

Ok here is another question. Prove any product-sum or sum-product trig ratio from the startpoint of sin(x+y) and sin(x-y)
Heh, what is the question asking?

I have not heard of product-sum or sum-product ratios, and when I google it, I get sinx cos y= 1/2(sin(x+y)+sin(x-y))
Is that what I have to prove?
 

SpiralFlex

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Re: HSC 2012 Marathon :)

I'll join after English tomorrow, so regret playing TF2. I may find some awesome inverse questions.
 

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