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Thread: Damped trigonometric functions

  1. #1
    May 2011

    Damped trigonometric functions

    f(x) = x sin(x). I understand that this function is to be regarded as the product of two functions (y = x and y = sin(x)), but what I don't not understand is this bit:

    "Using properties of absolute and the fat that |sin x| </= 1, you have
    0 </= |x||sin x| </= |x|. Consequently, -|x| </= x sin (x) </= |x|."

    What is the meaning of using absolute value to prove that f(x) = x sin (x) resides within the lines y= -x and y = x? What is the justification?

    Thank you, I would really like to understand this.
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  2. #2
    Super Member Quacky's Avatar
    Nov 2009
    Windsor, South-East England

    Re: Damped trigonometric functions

    Not sure I understand your confusion.

    Do you understant that as \sin{x}\leq{1}, then, for positive x,

    You could conceptualize it this way:
    So what happens when you multiply x by a number less than 1? You get an answer that is less than x for positive x. And as |\sin{x}|\leq{1}, then, for positive x and positive \sin{x}, you you can conclude that:

    0\leq{x\sin{x}}\leq{x}. The absolute value signs are included because the opposite is true for negative numbers:

    -10\times{\frac{1}{2}}=-5 multiplying a negative number by a number less than 1 gives you a larger number.

    This means that x\sin{x}>x for negative x.

    Or, " -|x|\leq x\sin{x} for all x" is to say the same thing.

    So, regardless of the value for x,

    Last edited by Quacky; Nov 25th 2011 at 07:37 AM.
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