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Thread: Calculus & Analysis Marathon & Questions

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    not actually a porcupine porcupinetree's Avatar
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    Calculus & Analysis Marathon & Questions

    Calculus & Analysis Marathon & Questions
    This is a marathon thread for single-variable calculus & analysis (mainly real and maybe complex analysis). Please aim to pitch your questions for first-year/second-year university level maths. Excelling & gifted/talented secondary school students are also invited to contribute.

    (mod edit 7/6/17 by dan964)

    ===============================

    Thought it'd be a good idea to start a marathon for users to post and answer first year uni calculus problems.

    First question:

    Last edited by dan964; 7 Jun 2017 at 4:40 PM.
    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

    I can't word properly yet.









    Last edited by leehuan; 24 Apr 2016 at 5:05 PM.
    porcupinetree likes this.

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    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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    Administrator Trebla's Avatar
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    Re: First Year Uni Calculus Marathon

    lel these look like tute or lecture qns

    (I actually remember the above question haha)

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    Re: First Year Uni Calculus Marathon












    =======================




    Last edited by leehuan; 24 Apr 2016 at 5:40 PM. Reason: I may not have put in the add on question myself after Trebla...even though he meant USyd

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    Rambling Spirit
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    Re: First Year Uni Calculus Marathon

    Is this thread's main intention as a Q&A, or as a challenge thread?

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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
    Is this thread's main intention as a Q&A, or as a challenge thread?
    I think exercise/challenge

    My cries for help are going to remain in seperate threads

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

    (A bit tricky)

    For which positive integers k is it possible to find a continuous function f:R->R that such that

    f(x)=y

    has exactly k solutions x for every real number y?

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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
    (A bit tricky)

    For which positive integers k is it possible to find a continuous function f:R->R that such that

    f(x)=y

    has exactly k solutions x for every real number y?
    Is there any answer besides k=1?

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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
    Is there any answer besides k=1?
    Indeed that is the essence of the question, I don't want to spoil it too quickly. It is very good exercise to reach a conclusion yourself and try to rigorously justify it. (The main tools at your disposal being properties of continuous functions like the intermediate and extreme value theorem).

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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 Trebla View Post
    lel these look like tute or lecture qns

    (I actually remember the above question haha)
    There was a tute question similar to that question, with x^2 instead of x^3. (But it's the same general process.)

    Quote Originally Posted by InteGrand View Post
    Is this thread's main intention as a Q&A, or as a challenge thread?
    Challenge/exercise, just like the other maths marathons
    Bachelor of Science (Advanced Mathematics) @ USYD

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    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    Is there any answer besides k=1?
    But what about k = ∞ ?
    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


















    I hope...

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

    Quote Originally Posted by Paradoxica View Post
    But what about k = ∞ ?
    Infinity is a concept not a number right

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    Rambling Spirit
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post

















    I hope...
    For one thing, what if f need not be differentiable?

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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post











    =======================




    Part 1 (not a very rigorous proof, though):



    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 InteGrand View Post
    For one thing, what if f need not be differentiable?
    Hmm...





    At least, I don't think a cusp/corner can't be a local extrema here?

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    Supreme Member Flop21's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Paradoxica View Post
    But what about k = ∞ ?
    ∞ is not a number, so k cannot equal ∞ right?
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    Ancient Orator leehuan's Avatar
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    Re: First Year Uni Calculus Marathon



    For completeness sake

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    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post
    Infinity is a concept not a number right
    Quote Originally Posted by Flop21 View Post
    ∞ is not a number, so k cannot equal ∞ right?
    An equation can still have infinite solutions.
    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

    Quote Originally Posted by Paradoxica View Post
    An equation can still have infinite solutions.
    For this question, because infinity is not a measurable quantity (and I'm not sure if you can assume that it is an integer) we can't say for sure what the behaviour of f is if f(x)=y yields an infinite number of solutions.

    At least, my simple brain can't visualise f anymore

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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by leehuan View Post











    =======================




    Also, regarding your proof, I think you need to the read the question again. (Particularly the definition of f(x))
    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

    Not even sure what I was thinking at the time now

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    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon

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



    Here's my attempt:

    Since x is positive, we can invoke the following inequalities:





    It is easy to show that the left hand side is bounded from above by 3/2, and the right hand side is bounded from below by the same value.

    It is then sufficient to show the left bound is monotonically increasing, and the right bound is monotonically decreasing, for sufficiently large x.
    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

    Quote Originally Posted by Paradoxica View Post
    Here's my attempt:

    Since x is positive, we can invoke the following inequalities:





    It is easy to show that the left hand side is bounded from above by 3/2, and the right hand side is bounded from below by the same value.

    It is then sufficient to show the left bound is monotonically increasing, and the right bound is monotonically decreasing, for sufficiently large x.
    Believe it or not my workbook was happy if you just used L'Hopitals
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