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Thread: HSC 2017 MX1 Marathon

  1. #51
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    Re: HSC 2017 MX1 Marathon

    Quote Originally Posted by Drsoccerball View Post
    What's f(x) ?
    Any continuous, once differentiable function satisfying the given conditions.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: HSC 2017 MX1 Marathon

    Quote Originally Posted by InteGrand View Post
    That part was just a general statement.
    my bad...

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    Re: HSC 2017 MX1 Marathon

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  4. #54
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    Re: HSC 2017 MX1 Marathon

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: HSC 2017 MX1 Marathon

    Quote Originally Posted by Paradoxica View Post
    Alphabetic progression.

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    Re: HSC 2017 MX1 Marathon

    Quote Originally Posted by Paradoxica View Post
    Geometric progression.
    Mahan Ghobadi

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    Re: HSC 2017 MX1 Marathon

    Quote Originally Posted by Paradoxica View Post
    The answer is nk.


    by letting and using assumption we get nk.
    Mahan Ghobadi

    Maths Tutor- ESL (80)| 2 Unit maths (96)(2013) | 3 Unit maths (99)| 4 Unit maths(95)| Physics (88)| music1(93)

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    Re: HSC 2017 MX1 Marathon

    Quote Originally Posted by Mahan1 View Post
    Geometric progression.
    You're fun at parties...

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    Re: HSC 2017 MX1 Marathon

    Quote Originally Posted by mini8658 View Post
    A, B are roots of x^2=5x-8.

    Find an expression for a^1/3 + b^1/3


    I presently cannot think of a way to determine the value using only Extension 1 Methods (there is the subtle distinction of principal valued cube roots vs generalised cube roots) as there is a necessary step in deducing the magnitude of the complex cube roots from the arguments...
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  10. #60
    Ancient Orator leehuan's Avatar
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    Re: HSC 2017 MX1 Marathon


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    Re: HSC 2017 MX1 Marathon




    Last one does feel like a cheat though.
    Last edited by Lugia101101; 3 Dec 2016 at 11:07 PM. Reason: Formatting

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    Re: HSC 2017 MX1 Marathon

    Quote Originally Posted by Lugia101101 View Post



    Last one does feel like a cheat though.
    Last one is meant to follow from the previous two. These aren't unconnected problems
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

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    Re: HSC 2017 MX1 Marathon

    This is probably what the question wants, then:

    leehuan likes this.

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    Re: HSC 2017 MX1 Marathon




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    Re: HSC 2017 MX1 Marathon

    For a function to be invertable, it must be defined such that it is one-to-one. If a function is monotone, then every value will result in a single value of , however, the function must be strictly monotonic, else there could exist two values and such that . So if the function is one-to-one, then there exists an inverse, and hence will only have an inverse if it is strictly monotone.

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    Re: HSC 2017 MX1 Marathon

    Quote Originally Posted by Lugia101101 View Post
    For a function to be invertable, it must be defined such that it is one-to-one. If a function is monotone, then every value will result in a single value of , however, the function must be strictly monotonic, else there could exist two values and such that . So if the function is one-to-one, then there exists an inverse, and hence will only have an inverse if it is strictly monotone.
    A bit confused. What do you mean by 'strictly monotonic'? Or were you intending to say strictly increasing/decreasing (i.e. f'(x)≠0)

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    Re: HSC 2017 MX1 Marathon

    A function with no two points and such that . An example would be:

    Which fits the definition for monotonic increasing, but isn't strictly monotonic increasing. (From Wikipedia)
    A function can still have at some point and have an inverse, such as , which is strictly monotonic increasing, has at , and has the inverse function .
    Last edited by Lugia101101; 7 Dec 2016 at 10:05 PM. Reason: Mistake Correction

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    Re: HSC 2017 MX1 Marathon


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    Re: HSC 2017 MX1 Marathon

    Also, we don't need a function to be monotonic for it to be invertible, since we can easily form discontinuous functions that are not monotonic but pass the horizontal line test. However, you can prove as an exercise that a continuous one-to-one function must be monotonic.

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    Re: HSC 2017 MX1 Marathon

    Such as a function defined as ?

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    Re: HSC 2017 MX1 Marathon

    Was basically the point of the question. In addition, the function has to be continuous.



    However, I haven't heard of the "strict monotonicity" grammar though

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    Re: HSC 2017 MX1 Marathon

    The tan function defined above is continuous but not monotonic.

    It's easy to come up with a discontinuous function that's not monotonic but is invertible. It suffices to take a graph that has two branches, one starting at the point (0, 2) and decreasing strictly, smoothly and asymptotically to y = 1, and the other branch a reflection of this about the y-axis and shifted down enough so that the overall graph passes the horizontal line test (and only include one of the points at the discontinuity at x = 0).
    Last edited by InteGrand; 8 Dec 2016 at 1:23 AM.

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    Re: HSC 2017 MX1 Marathon



    Q4 is a trick question

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    Re: HSC 2017 MX1 Marathon

    Quote Originally Posted by leehuan View Post


    Q4 is a trick question
    For question 3:


    For question 4, as we showed there is only one point of intersection, this would occur when x=k.
    Last edited by frog1944; 19 Dec 2016 at 9:34 AM.
    Mathematics Extension 2 - Physics - Chemistry - Economics - English Advanced

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    Re: HSC 2017 MX1 Marathon


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