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Advanced Higher Maths 2011-2012 : Discussion and Help

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    I'm in AH maths just now too. c:

    We're just about to start the volume of revolution stuff, but I've already taught myself all of the calculus stuff in the course (wait until you start on integration by parts and implicit differentiation - both lots of fun!), so it's not too bad.

    On that note...

    A question from last year's prelim:

    Integrate 5ln(x^2+9) dx

    Was very fun. Integration by parts with an inverse trig substitution (when you simplified it down to 1/(x^2+9), the integral of that is 1/3 arctan (x/3)).

    ...I'll leave now.

    What is everyone learning from? My teacher's using "Teejay" notes with us, but I find the Maths In Action textbooks are pretty useful too.
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    Another lovely question that I'm stuck on.

    The Diagram shows the graph x^2y=4a^2(2a-y).
    This classical graph is known as 'The Witch of Agnesi'

    a) Show that y=\frac{8a^2}{x^2 +4a^2}
    (done this - no problem)

    b) Hence show that the gradient of the tangent at (2a,a) is -\frac{1}2{}

    It's probably real straight forward but I just don't see it at the moment.

    Thanks in advance.
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    (Original post by laughylolly)
    Another lovely question that I'm stuck on.

    The Diagram shows the graph x^2y=4a^2(2a-y).
    This classical graph is known as 'The Witch of Agnesi'

    a) Show that y=\frac{8a^2}{x^2 +4a^2}
    (done this - no problem)

    b) Hence show that the gradient of the tangent at (2a,a) is -\frac{1}2{}

    It's probably real straight forward but I just don't see it at the moment.

    Thanks in advance.
    Differentiate, and solve it equal to 0 when x=2a ?

    Not sure if that's how you do it though
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    (Original post by laughylolly)
    Another lovely question that I'm stuck on.

    The Diagram shows the graph x^2y=4a^2(2a-y).
    This classical graph is known as 'The Witch of Agnesi'

    a) Show that y=\frac{8a^2}{x^2 +4a^2}
    (done this - no problem)

    b) Hence show that the gradient of the tangent at (2a,a) is -\frac{1}2{}

    It's probably real straight forward but I just don't see it at the moment.

    Thanks in advance.
    Maybe you could try finding what value(s) of a allow the fact that 2x = y (since point is 2a,a) and find out gradient that way?
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    (Original post by Evan7788)
    Differentiate, and solve it equal to 0 when x=2a ?

    Not sure if that's how you do it though
    Putting when it's equal to zero is only when it's a stationary point but we know the gradient is - 1/2 ... hmm wait that gives me an idea.
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    You mean  8a^3/(x^2+4a^2)

    Differentiate this.

    Edit: yep, I officially fail at using Latex.

    Okay okay, I've done this, so I'll try and write it out on paper (LaTex is a bad call).
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    (Original post by JordanR)
    You mean  8a^3/(x^2+4a^2)

    Differentiate this.

    Edit: yep, I officially fail at using Latex.
    How though? How can you differentiate with all the a's in there too?

    Here's a website that will help you with latex : http://www.codecogs.com/latex/eqneditor.php

    (Original post by soup)
    Maybe you could try finding what value(s) of a allow the fact that 2x = y (since point is 2a,a) and find out gradient that way?
    It's in the differential calculus chapter so I'm pretty sure you have to differentiate.
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    (Original post by laughylolly)
    How though? How can you differentiate with all the a's in there too?



    It's in the differential calculus chapter so I'm pretty sure you have to differentiate.
    a is a constant, yes? Treat it as such and differentiate it normally.

    Would you like me to post my full solution now, or do you want to work from here? c:



    Now we know that m=-\frac{16a^3x}{4a^2+x^2}

    Substitute in your value for x and you'll get...

    Sorry, I feel like I've given too much of it away. ><

    Edit: does anyone have some specific music that they like to listen to when they do maths? I'm currently enjoying Big Booty Bitches and most of Mozart's work, as well as some fast paced 80s shred guitar metal. I think it has to be something lively!
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    (Original post by JordanR)
    a is a constant, yes? Treat it as such and differentiate it normally.

    Would you like me to post my full solution now, or do you want to work from here? c:



    Now we know that m = -16 a^3x/(4a^2 + x^2)^2

    Substitute in your value for x and you'll get...

    Sorry, I feel like I've given too much of it away. ><

    Edit: does anyone have some specific music that they like to listen to when they do maths? I'm currently enjoying Big Booty Bitches and most of Mozart's work, as well as some fast paced 80s shred guitar metal. I think it has to be something lively!
    Sorry, I don't understand that at all. And how come the fraction suddenly became negative "/ ?
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    (Original post by laughylolly)
    Sorry, I don't understand that at all. And how come the fraction suddenly became negative "/ ?
    Chain rule.

     \frac{d}{dx} (x^2+4) = \frac{d}{du} * \frac{1}{u} * \frac{du}{dx}

    u = x^2 + 4

    \frac{1}{u} = u^-1

    \frac{d}{du}*u^-1 = -\frac{1}{u^2}

    Sorry, hope that makes sense. c:
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    (Original post by JordanR)
    Chain rule.

     d/dx 1/(x^2+4) = d/du * 1/u  * du/dx



u = x^2 + 4



1/u = u^-1



d/du u^(-1) = -1 * u^(-2)

    Sorry, hope that makes sense. c:

    Edit: oh dear... latex and I just do not get along...

    d/du u^(-1) = -1 * u^(-2) which of course simplifies to just -1/u^2
    Oh right okay.

    But I still don't get how you make -16a^3x etc... to show -1/2

    oh wait... never mind. I got it.

    I'm pretty sure I had -16a^3x something written down before but for some reason it just didn't work. Thank you!
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    Do you think Universities want to see A grades or do you think they would accept B´s / C´s?
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    (Original post by laughylolly)
    Oh right okay.

    But I still don't get how you make -16a^3x etc... to show -1/2

    oh wait... never mind. I got it.

    I'm pretty sure I had -16a^3x something written down before but for some reason it just didn't work. Thank you!
    Sorry.

    Apparently I do a lot of things well (exaggeration), but my explanatory skills certainly leave a lot to be desired.

    Where is this question from? AH maths syllabus or is it from a STEP paper or something?

    You're welcome! Glad to help (albeit murkily ).

    To be honest, I'm quite interested in studying maths at uni, but since I was an idiot at standard grade and got terrible higher grades, I doubt that's going to be possible (3/4 at SG, B at higher maths). Teacher says I'm well on track for an A at AH though, so I have absolutely no idea what to do.


    (Original post by nerd434)
    Do you think Universities want to see A grades or do you think they would accept B´s / C´s?
    Well of course unis want As, but some course accept Bs and Cs too, but obviously these courses are less prestigious/demanding, as are the universities themselves.
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    How many Advanced Highers are you all doing?
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    (Original post by JordanR)
    We're just about to start the volume of revolution stuff, but I've already taught myself all of the calculus stuff in the course (wait until you start on integration by parts and implicit differentiation - both lots of fun!), so it's not too bad.
    Have a go at this one:

    \displaystyle \int \dfrac{dx}{1 + \sin x}

    The steps in Wolfram Alpha is not taught in AH but there is a method which is entirely within AH. See if you can find it
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    (Original post by nerd434)
    How many Advanced Highers are you all doing?
    3.

    English, maths, physics.
    (Original post by ukdragon37)
    Have a go at this one:

    \displaystyle \int \dfrac{dx}{1 + \sin x}

    The steps in Wolfram Alpha is not taught in AH but there is a method which is entirely within AH. See if you can find it

    I'll give it a go. Seems like a lot of trig substitutions (I hope!), which I'm admittedly not too bad at. Although I'm not too sure at how well I can handle trig substitutions on a few hours of sleep at half ten on a Sunday.


    Guessing you're from Scotland and did AH maths, yeah?
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    (Original post by JordanR)
    Guessing you're from Scotland and did AH maths, yeah?
    Well, "live in Scotland" is more accurate. Yep I did AH Maths.
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    (Original post by JordanR)
    Sorry.

    Apparently I do a lot of things well (exaggeration), but my explanatory skills certainly leave a lot to be desired.

    Where is this question from? AH maths syllabus or is it from a STEP paper or something?

    You're welcome! Glad to help (albeit murkily ).

    To be honest, I'm quite interested in studying maths at uni, but since I was an idiot at standard grade and got terrible higher grades, I doubt that's going to be possible (3/4 at SG, B at higher maths). Teacher says I'm well on track for an A at AH though, so I have absolutely no idea what to do.

    Well of course unis want As, but some course accept Bs and Cs too, but obviously these courses are less prestigious/demanding, as are the universities themselves.
    I just didn't get the method you used at first but now I get it haha. I had originally tried differentiating with the quotient rule but I wasn't 100% sure what to do with the a's. I probably did have \frac{-16a^32a}{(4a^2 +4a^2)^2} written out but just didn't realize that I could simplify it. I can do like all the really complicated things in maths but when it comes to 1 + 1 I suddenly think it equals 3 or something.

    It was in my AH maths textbook that question. Was like 2nd last question of an exercise in the differential calculus chapter.

    I'm going to be applying for Maths at uni, I got A* at IGCSE and A1 at higher. Just hope I do good this year! Some questions are so easy but just really lengthly, some aren't so nice but I've only been doing AH maths for a week so I guess I'll get better at it as I go along.

    Maybe you could do a subject with joint honours in maths and then maybe you'll be able to go to a better university?

    (Original post by nerd434)
    How many Advanced Highers are you all doing?
    I'm doing three. AH Maths, Stats and Physics.
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    (Original post by ukdragon37)
    Well, "live in Scotland" is more accurate. Yep I did AH Maths.
    Ah, I see. c:

    How is AH maths? Probably a stupid question since it's going to vary for absolutely everyone, but I was expecting to go into the classroom and leave there unable to sit down comfortably for a few periods, but so far it's been pretty cushy. Just a natural progression from Higher, really.

    I'm sure proofs will manage to break me though...


    (Original post by laughylolly)
    I'm going to be applying for Maths at uni, I got A* at IGCSE and A1 at higher. Just hope I do good this year! Some questions are so easy but just really lengthly, some aren't so nice but I've only been doing AH maths for a week so I guess I'll get better at it as I go along.

    Maybe you could do a subject with joint honours in maths and then maybe you'll be able to go to a better university?

    I'm doing three. AH Maths, Stats and Physics.
    Best of luck! Although you won't need it with those grades.

    Yeah, maybe. I have no idea. I'm mostly annoyed at myself. My attendance for my Standard Grades was low. Like, <20% low. My work ethic for 5th year was terrible, and I did less than an hour's revision for each subject before the exams, so it's not surprising I got what I did (despite being predicted AAAAA...).

    I know I could've gotten AAAAA and straight 1s at SG, but ah well. It's done now. Believe it or not, I'm not entirely stupid, although you wouldn't be the first to tender the stupidity claim!
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    (Original post by JordanR)
    I'll give it a go. Seems like a lot of trig substitutions (I hope!), which I'm admittedly not too bad at. Although I'm not too sure at how well I can handle trig substitutions on a few hours of sleep at half ten on a Sunday. ?
    Oh I nearly forgot. Try doing it without explicitly using trig substitutions
Updated: August 12, 2012
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