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Thread: Calculus & Analysis Marathon & Questions

  1. #51
    not actually a porcupine porcupinetree's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by seanieg89 View Post
    Do you have any hints as to what the 'suitably chosen function' is? I'm struggling to find it
    Bachelor of Science (Advanced Mathematics) @ USYD

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    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by porcupinetree View Post
    Do you have any hints as to what the 'suitably chosen function' is? I'm struggling to find it
    It is slightly tricky to spot, but the core idea is the same as in the proof of the mean value theorem. So perhaps I should make this a leadup exercise.

    The mean value theorem asserts we can find a c in (a,b) with f'(c)=(f(b)-f(a))/(b-a) if f is a differentiable function on [a,b].

    B1. (Rolle's Theorem) Prove that if g is a differentiable function on [a,b] with g(a)=g(b), then g'(c)=0 for some c in (a,b). (Use the extreme value theorem to do this)

    B2. (MVT) By choosing g appropriately show that f'(c)=(f(b)-f(a))/(b-a) for some c in (a,b). (The appropriate choice is easier to see here than in my original question, I think you are capable of finding it. We can also interpret this statement geometrically, which might help).

    Then attempt the questions in my previous post. Aim to use Rolle's in a similar way to how it is used in Q2 of this post. With this target in mind, the choice of function should be easier for you to find.
    Last edited by seanieg89; 30 Apr 2016 at 12:35 PM.

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    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Note: To be clear on nomenclature, when I say a function is differentiable on [a,b], I am being lazy. I actually mean that f is a continuous function on [a,b] that is differentiable on (a,b). No assumptions are made about the existence of one-sided derivatives or anything at the boundary of the interval. I doubt this will affect the way anyone approaches the problem though.

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    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    I kinda want to make this thread more accessible to some people (where possible) so here's a somewhat easy question. If it's ignored too bad back to reasonable difficulty.



    Notes:
    Spoiler (rollover to view):
    Some other theorem should be assumed.
    Last edited by leehuan; 30 Apr 2016 at 11:49 AM.

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    Rambling Spirit
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    I kinda want to make this thread more accessible to some people (where possible) so here's a somewhat easy question. If it's ignored too bad back to reasonable difficulty.



    Notes:
    Spoiler (rollover to view):
    Some other theorem should be assumed.
    This is basically a hint to one of seanieg89's hint exercises.

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    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by InteGrand View Post
    This is basically a hint to one of seanieg89's hint exercises.
    I, did not see that coming.

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    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    It's just providing the function for my question B2 (2nd question in second post) which is not too much of a spoiler, but now someone should definitely be able to prove that.

    B1 is a bit trickier, but for any student who wants to assume Rolle's and doesn't care where it comes from, you can move straight on to the questions in my first post which was the main point (to prove L'Hopital's).

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    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    And some much easier ones for students who don't want to do the above questions:

    E1. Prove that if a function f: (a,b) -> R is differentiable and f'(x) is non-negative in this interval, then f is non-decreasing in this interval.

    E2. Prove that if a function f: (a,b) -> R is differentiable and f'(x) = 0 in this interval, then f is constant.

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    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    I thought E2 was kinda intuitive but I'm finding it hard to word my final bit.

    Rolle's theorem dictates that if f is continuous on [a,b] and differentiable on (a,b), and we have f(a)=f(b), then there exists at least one value of c such that f'(c)=0 for c in (a,b)

    But if f'(c)=0 for all (a,b) (or alternatively f has a horizontal tangent for all points on the interval), then as we have an infinite number of values satisfying f'(c)=0, f must be constant for every c in [a,b]

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    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    But if f'(c)=0 for all (a,b) (or alternatively f has a horizontal tangent for all points on the interval), then as we have an infinite number of values satisfying f'(c)=0, f must be constant for every c in [a,b]
    What do you mean by saying f must be constant for every c? What does it mean to be constant at a point?

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    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    I wasn't too sure how to argue it cause both statements seemed trivial corollaries of specific theorems or definitions. I just wasn't sure how to argue about how if the tangent is always horizontal, the curve never increases or decreases

    (And yeah that language was what I meant by I don't know how to word it)


    Sent from my iPhone using Tapatalk

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    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    I wasn't too sure how to argue it cause both statements seemed trivial corollaries of specific theorems or definitions. I just wasn't sure how to argue about how if the tangent is always horizontal, the curve never increases or decreases

    (And yeah that language was what I meant by I don't know how to word it)


    Sent from my iPhone using Tapatalk
    I wouldn't say they are trivial. (But they are easy consequences of specific theorems. A correct proof should be very short.)

    You can't rely on intuition too much here as funny things can happen in analysis. I can tell you that just changing your wording in that final paragraph won't really lead to a proof. Also, you mention Rolle's theorem in your second para but then make no use of it later...did you mean to?

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    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by seanieg89 View Post
    I wouldn't say they are trivial. (But they are easy consequences of specific theorems. A correct proof should be very short.)

    You can't rely on intuition too much here as funny things can happen in analysis. I can tell you that just changing your wording in that final paragraph won't really lead to a proof. Also, you mention Rolle's theorem in your second para but then make no use of it later...did you mean to?
    Oh. Whoops at not properly using the theorem I quoted.

    But nah I clicked submit with the mindset that the proof was invalid so I was open to comment. I was too stuck on wording

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    not actually a porcupine porcupinetree's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by seanieg89 View Post
    A1:








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    Bachelor of Science (Advanced Mathematics) @ USYD

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    Re: First Year Uni Calculus Marathon

    That's more like it .

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    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    This question got asked in my calculus tutorial. Took me 30 seconds of staring at it before I realised what was going on but Paradoxica could probably do it in 1 second.
    (Especially since I have a feeling this got asked on BoS before...)

    Paradoxica likes this.

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    Rambling Spirit
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    This question got asked in my calculus tutorial. Took me 30 seconds of staring at it before I realised what was going on but Paradoxica could probably do it in 1 second.
    (Especially since I have a feeling this got asked on BoS before...)

    Where on BOS do you think it got asked before? As in in some marathon from the past?

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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by InteGrand View Post
    Where on BOS do you think it got asked before? As in in some marathon from the past?
    Feels like some marathon or extracurricular thread. Might not have been the exact same one but it looked too familiar.

    @Para yes the answer was 0

  19. #69
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    This question got asked in my calculus tutorial. Took me 30 seconds of staring at it before I realised what was going on but Paradoxica could probably do it in 1 second.
    (Especially since I have a feeling this got asked on BoS before...)

    I don't see the answer jumping out at me.

    Imaginary Error Function lol

    I'm thinking of using a bounding argument on the integral and using that to find the limit.
    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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    Rambling Spirit
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Paradoxica View Post
    I don't see the answer jumping out at me.

    Imaginary Error Function lol

    I'm thinking of using a bounding argument on the integral and using that to find the limit.
    Spoiler (rollover to view):
    LH
    leehuan likes this.

  21. #71
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    Feels like some marathon or extracurricular thread. Might not have been the exact same one but it looked too familiar.

    @Para yes the answer was 0
    Well that was based on me accidentally seeing that as -t2

    So no.
    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.

  22. #72
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by InteGrand View Post
    Spoiler (rollover to view):
    LH
    *blindly bashes everything with L'hôpital's Rule*
    leehuan likes this.
    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.

  23. #73
    not actually a porcupine porcupinetree's Avatar
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    Re: First Year Uni Calculus Marathon

    Bachelor of Science (Advanced Mathematics) @ USYD

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    Loquacious One Drsoccerball's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by porcupinetree View Post
    Has USYD already started Taylor series ?

  25. #75
    not actually a porcupine porcupinetree's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Drsoccerball View Post
    Has USYD already started Taylor series ?
    Yup. I'm guessing you guys haven't?
    Bachelor of Science (Advanced Mathematics) @ USYD

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