How do I do this math problem?

Discussion in 'Odds & Ends' started by Manus Domini, Oct 22, 2011.

  1. Manus Domini

    Manus Domini Hall of Fame

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    Before I give the problem, I want posters to know I am NOT asking for the answer, so don't give it to me. Nor should you give away the entire process, because all I have to then do is plug in numbers, so you are giving me the answer anyway. I just want where to start and to be let known if I am doing the problem incorrectly.

    So, the problem I am assigned is:

    "analytically simplify the following limit, which represents the definition of the derivative f'(a) for the function f(x)=sinx and x=a."

    My teacher doesn't give the limit, and so I don't know where to start. The substitute (since the teacher wasn't there yesterday) said something about about the "angle addition" rule or something, so what I have so far is:

    (sin(x+dx)-sinx)/(dx)-->(sinxcosdx+sindxcosx-sinx)/(dx)-->(sinx-sinx+sindxcosx)/(dx)-->(sindxcosx)/(dx)-->(sin0cosx)/0=limit does not exist

    but the limit should definitely exist at all times except when sinx-->0, which it isn't necessarily doing.
     
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  2. Hot Sauce

    Hot Sauce Hall of Fame

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    Dunno.....
     
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  3. Claudius

    Claudius Professional

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    The limit as dx approaches 0 exists, and is equal to the derivative of sin(x), which is cos(x).

    To simplify, you will need to use these two facts:

    1.) lim x--> 0 sin(x)/x = 1
    2.) sin(x) - sin(y) = 2cos(1/2(x + y))sin(1/2(x-y))

    So,

    lim dx --> 0 (sin(x+dx) - sin(x))/dx = lim dx--> 0 ((2cos(x + dx/2)sin(dx/2))/dx = lim dx --> 0 cos(x+dx/2)sin(dx/2)/(dx/2) = cos(x + 0/2) * 1 = cos(x).

    Hence, d/dx (sin(x)) = cos(x).
     
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  4. Tammo

    Tammo Banned

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    is this Calculus?
     
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  5. Slayer_of_Kings

    Slayer_of_Kings New User

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    It's good 'ol Trig.
     
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  6. Frankenstine

    Frankenstine New User

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    No. It's definitely calculus.
     
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  7. The Bawss

    The Bawss Banned

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    Yes. 10truths
     
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  8. Polaris

    Polaris Hall of Fame

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    I would have said, "You're on the right track, but your evaluation of the limit of (sin x)/x needs more attention."

    Instead, Manus now has the answer in its full glory, which he did not want :) .
     
    Last edited: Oct 23, 2011
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  9. sureshs

    sureshs Bionic Poster

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    This question is extensively discussed on the Web, including the fact that the proof of the limit of sin(x)/x as x->0 is 1 should come from some method (like geometric analysis) rather than L'Hospital's rule, otherwise it will become a circular proof.
     
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