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Math Help - Upper Lower Limit and intermediate value

  1. #1
    Senior Member Shanks's Avatar
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    Upper Lower Limit and intermediate value

    Suppose f(x)=\cos{x^2}-\cos{(x+1)^2}.
    (1)Find the upper and lower limit of f(x) \text{ as }x\to +\infty.
    (2)Let A be the upper limit, B be the lower limit. prove or disprove that
    for any real number c in the closed interval [B,A], there is a number sequence \{x_n\} \text{ such that }x_n\to +\infty,f(x_n)\to c.
    Last edited by Shanks; December 19th 2009 at 01:31 AM.
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    Senior Member Shanks's Avatar
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    OK, the folowing is what I thought:
    1)I guess the upper limit is 2, and the lower limit is -2, I believe my conjecture is true, But I can't convince myself by analysis proof. I think it has something to do with the spectul of transcendental number in Number Theory.
    2)If (1) is proved, (2) follows immediately by the intermediate value theorem.
    Last edited by Shanks; December 19th 2009 at 07:19 AM.
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  3. #3
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    Quote Originally Posted by Shanks View Post
    Suppose f(x)=\cos{x^2}-\cos{(x+1)^2}.
    (1)Find the upper and lower limit of f(x) \text{ as }x\to +\infty.
    (2)Let A be the upper limit, B be the lower limit. prove or disprove that
    for any real number c in the closed interval [B,A], there is a number sequence \{x_n\} \text{ such that }x_n\to +\infty,f(x_n)\to c.
    Using the formula \cos\theta-\cos\phi = 2\sin\tfrac{\theta+\phi}2\sin\tfrac{\phi-\theta}2, you see that f(x) = 2\sin\bigl(x^2+x+\tfrac12\bigr)\sin\bigl(x+\tfrac1  2\bigr). Whenever x + \tfrac12 = \bigl(k+\tfrac12\bigr)\pi, the second of those two sines is equal to \pm1. The x^2 term in the expression for the first sine increases very much faster than x, as x gets larger. So the first sine will oscillate between \pm1 very rapidly. See the attachment, which shows the "envelope" of the functions \pm2\sin\bigl(x+\tfrac12\bigr), with \sin\bigl(x^2+x+\tfrac12\bigr) oscillating rapidly between them. Thus f(x) (which obviously has to lie between –2 and +2) will come close to both those values in each interval of length \pi. Since f is continuous, it will also take all intermediate values.
    Attached Thumbnails Attached Thumbnails Upper Lower Limit and intermediate value-cosine.jpg  
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    Senior Member Shanks's Avatar
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    I have three question:
    1)How do you get the attached graph? any math software work for you?
    2)Can you attach a file about the graph of the given function?
    3)Let (t) be the decimal part of the positive number t. Prove or disprove that if x is irrational number,then the set {(nx):n is natural number} is dense in the open interval (0,1).
    What about the set {(nx):n is square number}? What about the set {(nx): n belongs to a given infinite subset of the natural number}
    add one more question:
    What is the "envelope" of function ?
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    Quote Originally Posted by Shanks View Post
    I have three question:
    1)How do you get the attached graph? any math software work for you?
    2)Can you attach a file about the graph of the given function?
    The new iMac that I got as an early Christmas present came loaded with an application called Grapher, hidden in the Utilities folder and not advertised at all, so it came as a nice surprise. It produces excellent graphs both for functions in 2D and for surfaces in 3D. I'm attaching an enlarged version of the graph that it produces for f(x) = \cos x^2 - \cos(x+1)^2.

    Quote Originally Posted by Shanks View Post
    What is the "envelope" of function ?
    You can see very clearly from the graph how the function oscillates between the limits \pm2\sin\bigl(x+\tfrac12\bigr). That is what I meant by these functions forming an "envelope" for f(x).
    Attached Thumbnails Attached Thumbnails Upper Lower Limit and intermediate value-cosine.jpg  
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  6. #6
    Senior Member Shanks's Avatar
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    WAWAh, iMac,I can't afford it! Is the Grapher programmable?
    Please enlighten me on the third question, aprreciation more than I can say!
    3)Let (t) be the decimal part of the positive number t,for example, (\pi)=0.1415926....
    Prove or disprove that if x is irrational number,then the set {(nx):n is natural number} is dense in the open interval (0,1).
    What about the set {(nx):n is square number}? What about the set {(nx): n belongs to a given infinite subset of the natural number}

    The third question arise in the process of my way to solve it.
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  7. #7
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    Quote Originally Posted by Shanks View Post
    3)Let (t) be the decimal part of the positive number t,for example, (\pi)=0.1415926....
    Prove or disprove that if x is irrational number,then the set {(nx):n is natural number} is dense in the open interval (0,1).
    The fractional parts of the numbers nx must all be different (otherwise x would be rational). Since they all lie in the bounded interval [0,1), some of them must get arbitrarily close together. If mx is very close to nx, then (nm)x will be very close to an integer. Then the fractional parts of k(nm)x (for k=1,2,3,...) will go through the unit interval in very small steps. If you make that argument a bit more precise, it will show that the set {(nx)} is dense in the interval.

    Quote Originally Posted by Shanks View Post
    What about the set {(nx):n is square number}?
    I guess that these numbers will also be dense in the unit interval, but I don't know of a proof.

    Quote Originally Posted by Shanks View Post
    What about the set {(nx): n belongs to a given infinite subset of the natural number}.
    This will clearly not hold for a general infinite subset of the natural numbers, because you can find a subsequence of {(nx)} that converges to any given element of the unit interval. The set of elements of that subsequence obviously won't be dense in the interval.
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