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Thread: Calculus & Analysis Marathon & Questions

  1. #26
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    Believe it or not my workbook was happy if you just used L'Hopitals
    Not rigorous enough :P

    jks. But I cringe at directly using that on the expression, I would rather use it on the bounds, it's much more nice-looking.
    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.

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

    @IG

    Still suspecting I haven't fully finished seanieg's question though. Anything else to pick up?

  3. #28
    Rambling Spirit
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    Re: First Year Uni Calculus Marathon

    ^ I think you should try invoking the IVT to prove some things I think you're trying to prove.

  4. #29
    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    (Sorry about delayed response, am wrestling with a problem of my own )

    Yep @leehuan, IVT, and EVT are key to rigorous argument here. You are on a nice train of thought and I think you have a good intuitive grasp of why k=2 is impossible, but making this rigorous takes some care.

    The "infinitesimal" part of your proof in particular needs to be made more precise, and things like Rolle's theorem are inapplicable here because the function need not be differentiable.

    From what you are saying though, it seems to me you have the correct "picture" in your head for your k=2 disproof.

    (I am also unconvinced that this "picture" serves to disprove the k>2 possibility so think about explaining that a little clearer).

  5. #30
    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Paradoxica View Post
    But what about k = ∞ ?
    Yes it is possible. Eg: f(x)=x*sin(x).

    Quote Originally Posted by porcupinetree View Post
    Part 1 (not a very rigorous proof, though):



    This is easy to make rigorous. For large x, x^3+1 < 2x^3 and x^2+1 < 2x^2. So

    log(x^3+1)/log(x^2+1) < log(2x^3)/log(x^2) = (log(2)+3log(x))/2log(x).

    and

    log(x^3+1)/log(x^2+1) > log(x^3)/log(2x^2)=3log(x)/(log(2)+2log(x)).

    Squeeze to finish.

    Quote Originally Posted by leehuan View Post
    Hmm...





    At least, I don't think a cusp/corner can't be a local extrema here?
    Corners do not have to be local extrema. Eg f(x)=max(x,2x) at 0.

    Also, lack of differentiability can occur in ways worse than corners. (f(x)-f(a))/(x-a) needs not tend to a limit at all as x tends to a, or it might only tend to a limit from one side.

    It is more trouble than it is worth to appeal to theorems about differentiable functions and then treat the exceptional points separately. (In fact a continuous function might be differentiable nowhere!)
    Last edited by seanieg89; 24 Apr 2016 at 7:45 PM.

  6. #31
    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    Hmmm

    With the IVT I can pretty much find how to apply it. Just get rid of that infinitesimal stuff and properly redefine it (which I might do later). Which gives me some ideas on the EVT but I haven't placed too much thought into it to know what's right and what's flawed.

    May continue this when I have more free time. Gah I hate assignment based subjects

  7. #32
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by seanieg89 View Post
    Yes it is possible. Eg: f(x)=x*sin(x).



    This is easy to make rigorous. For large x, x^3+1 < 2x^3 and x^2+1 < 2x^2. So

    log(x^3+1)/log(x^2+1) < log(2x^3)/log(x^2) = (log(2)+3log(x))/2log(x).

    and

    log(x^3+1)/log(x^2+1) > log(x^3)/log(2x^2)=3log(x)/(log(2)+2log(x)).

    Squeeze to finish.



    Corners do not have to be local extrema. Eg f(x)=max(x,2x) at 0.

    Also, lack of differentiability can occur in ways worse than corners. (f(x)-f(a))/(x-a) needs not tend to a limit at all as x tends to a, or it might only tend to a limit from one side.

    It is more trouble than it is worth to appeal to theorems about differentiable functions and then treat the exceptional points separately. (In fact a continuous function might be differentiable nowhere!)
    *cough*weierstrass*cough*
    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.

  8. #33
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Paradoxica View Post
    *cough*weierstrass*cough*
    Imagine Carrotsticks BOS trial last Q. of 4U paper 2016: 'Sketch the graphs of the Weierstrass and Dirichlet functions' .
    Paradoxica and kawaiipotato like this.

  9. #34
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by InteGrand View Post
    Imagine Carrotsticks BOS trial last Q. of 4U paper 2016: 'Sketch the graphs of the Weierstrass and Dirichlet functions' .
    last part of the question

    "also note significant points on the graph"
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    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. #35
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by InteGrand View Post
    Imagine Carrotsticks BOS trial last Q. of 4U paper 2016: 'Sketch the graphs of the Weierstrass and Dirichlet functions' .
    the final part will be:

    hence, sketch the indefinite integral of the weierstrass and dirichlet functions, with any choice of C you like.
    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.

  11. #36
    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by porcupinetree View Post



    Regarding the continuity part:

    Informally, when x≥0, 0≤f(x)≤x3 and when x≤0, x3≤f(x)≤0

    But in both cases if we take limits to nought, by the squeeze theorem we have lim x->0 f(x) = 0 which incidentally enough equals f(0)
    ______________________

    The answer given to the derivative was wrong because I didn't regard the significance of what h was.


    lim x->0 [ f(0+h)-f(0) ] / h
    = lim x->0 f(h)/h

    but f(h) = h3 when h is rational
    and f(h) = 0 when h is irrational

    Note then that f(h)/h = h2 for h in Q
    or = 0 for h not in Q

    Haven't done questions like this one before yet but just staring at it I'm inclined to say the squeeze theorem can be used again

  12. #37
    Rambling Spirit
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    Re: First Year Uni Calculus Marathon

    Yeah it's basically squeeze law. We can use that to show that f'(0) = 0.

  13. #38
    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    Haven't done questions like this one before yet but just staring at it I'm inclined to say the squeeze theorem can be used again


    (The "=>" comes from the squeeze law. Generally if we suspect something might tend to zero, it is often convenient to take absolute values as then we just need to bound it above by something that tends to zero.)

  14. #39
    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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    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by porcupinetree View Post




    Tightly squeezed LaTeX 101

  16. #41
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post




    Tightly squeezed LaTeX 101
    You moved the limit inside without justification though...
    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.

  17. #42
    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon

    Lol sorry, forgot to say the exponential function was continuous even though I had it written down.

  18. #43
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    Lol sorry, forgot to say the exponential function was continuous even though I had it written down.
    I don't recall that continuity is sufficient to exchange limits.
    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.

  19. #44
    Senior Member KingOfActing's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Paradoxica View Post
    I don't recall that continuity is sufficient to exchange limits.
    I'm pretty sure it's exactly one of the (equivalent) definitions of continuity?
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  20. #45
    Rambling Spirit
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    Re: First Year Uni Calculus Marathon

    Yeah you can move the limit inside for a continuous function. If lim as x -> a of g(x) is b and f is continuous at b, then lim as x -> a of f(g(x)) = f(b).

  21. #46
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    This is not a calculus question, but it's associated so I'll put it here.

    Last edited by Paradoxica; 27 Apr 2016 at 8:04 PM.
    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. #47
    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Paradoxica View Post
    This is not a calculus question, but it's associated so I'll put it here.

    This is only true for sufficiently small epsilon.
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  23. #48
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by seanieg89 View Post
    This is only true for sufficiently small epsilon.
    Tis what I was thinking, but what is sufficient?
    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.

  24. #49
    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Paradoxica View Post
    Tis what I was thinking, but what is sufficient?
    Anything smaller than the first positive solution to



    It's about 0.117041.

  25. #50
    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Last edited by seanieg89; 30 Apr 2016 at 12:36 PM.

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