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Math Help - Modeling a Guitar Note

  1. #1
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    Modeling a Guitar Note

    Hey everyone!

    So, I have a mini-assignment in my functions class and it is a trig project on modeling a guitar note. I did the project, but I had my math teacher look at it, and he said I missed a few things. so he's letting my re-do it.

    The assignment is a bit long, but i'll type it all out so I can ask my question.

    Modeling a Guitar Note

    The waveform of a guitar note is characterized by an initial sharp peak that falls off rapidly. Guitar effect pedals electronically modify that waveform. We can mathematically modify the oscillations of the sine wave to get the waveform of a guitar note as well as many others. The basic sine wave oscillates between the two horizontal lines y = 1 and y = -2. If we multiply sin(x) by any other function g(x) we will get a curve that oscillates between the graphs of y = g(x) and y = -g (x).

    a) Graph y1= x*sin(x), y2= x, y3=-x. For what exact values of x is the x*sin(x)= x? For what values of x is x*sin(x)= -x?

    b) For what exact values of x˛ is x˛*sin(x)= x˛? For what exact values of x is x˛*sin(x)= -x˛? Support your conclusions with a graph.

    c) Graph y1= 1/x*sin(x) for 0< x < 10 and -2 < y 2. Is it true that for x > 0, -1/x < 1/x*sin(x) < 1/x? Prove your answer.

    d) Graph f(x) = 1/x*sin(x) for -2 < x < 2 and -2 < y < 2. What is f(0)? Is it true for all x in the interval [-0.1, 0.1] for which f(x) is defined, f(x) satisfied 0.99 < f(x) < 1? Explain.


    So yah, like I said, a long question. I could not figure out how to do c and d.
    And with a and b, I got these exact values.

    a) X sin (x) = x at 0, 1.5-1.9, 8, 14.2, 20.5, 26.7, 33, 39.3, 45.6, and it continues to touch the line in intervals of about 6.2. All the negative’s of these values are also true as exact values of x in which x sin (x)=x.

    My math teacher suggested that I turn these numbers into pi radians, and I would see a definite pattern, and that I would be able to solve b with this pattern as well. I'm not sure how to do this, can anyone help me? He means not only in radians, but in a π/2, 11π/6 (thats supposed to be a pi symbol).

    Help would be appreciated! Thanks.


    Also, anyone have any suggestions for d? I think there is a point discontinuity, but i'm not sure what its asking me.
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  2. #2
    GAMMA Mathematics
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    c) Graph y1= 1/x*sin(x) for 0< x < 10 and -2 < y 2. Is it true that for x > 0, -1/x < 1/x*sin(x) < 1/x? Prove your answer.

    d) Graph f(x) = 1/x*sin(x) for -2 < x < 2 and -2 < y < 2. What is f(0)? Is it true for all x in the interval [-0.1, 0.1] for which f(x) is defined, f(x) satisfied 0.99 < f(x) < 1? Explain.
    f(x)=\frac{sin(x)}{x}

    \frac{-1}{x} < \frac{sin(x)}{x} < \frac{1}{x} for x>0

    As long as x \ne 0, you can multiply out by x:

    -1 < sin(x) < 1

    This is false, but the following is true:

    -1 \le sin(x) \le 1


    Did you mean this? f(x)=\frac{1}{xsin(x)}
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  3. #3
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    Uhmm, I believe the question is asking whether it is true or false, so it could very well be false. And no, the question says what I typed.1/xsin(x).

    Is d true? One of my friends stated that f(O)= indeterminate and then went on to say that d was actually true though.
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  4. #4
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    Quote Originally Posted by Slipery View Post
    [snip]
    d) Graph f(x) = 1/x*sin(x) \left [ \text{Mr F EDIT: } f(x) = \frac{\sin x}{x} \right] for -2 < x < 2 and -2 < y < 2. What is f(0)? Is it true for all x in the interval [-0.1, 0.1] for which f(x) is defined, f(x) satisfied 0.99 < f(x) < 1? Explain.
    [snip]
    f(x) = \frac{\sin x}{x} is not defined at x = 0. However, \lim_{x \rightarrow 0} f(x) = 1.
    Note:
    1. f(x) symmetric.
    2. f(x) is a decreasing function over (0, 0.1].
    3. 0.99 < f(0.1) < 1.
    So the answer to (d) is .....?
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