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Homework Help: Addition formula for f(x+y) using mean-value theorem

  1. Feb 23, 2010 #1
    1. The problem statement, all variables and given/known data
    suppose f and g are two differentiable functions with f(0)=0, g(0)=1 and f'(x)=g(x) and g'(x)=-f(x). For a fixed y in R put

    F(x) = f(x+y) - f(x)f(y) - g(x)f(y)

    Compute F'(x) and F''(x)
    Then let

    E(x) = [F(x)]^2 + [F'(x)]^2


    Apply Mean Value Theorem to E and hence prove the addition formulae for f(x+y) and g(x+y)

    2. Relevant equations



    3. The attempt at a solution
    I don't think what I've computed it correct, can anyone kindly please check?
    http://img192.imageshack.us/img192/5237/rimg0036.jpg [Broken]
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Feb 23, 2010 #2
    First thing I notice is that you take a derivative with respect to y in F'(x). You are holding y fixed... that is treat it like a constant. It is equivalent to taking the partial derivative w.r.t. x.

    Hope this gets you somewhere.
     
  4. Feb 23, 2010 #3
    that's what I thought but I was unsure about it
    does that mean f(y) and g(y) will be treated as constants?
    and is it correct that derivative of [F(x)]^2 is 2F(x).F'(x)?
     
  5. Feb 23, 2010 #4
    Yes to all of the above :).
     
  6. Feb 23, 2010 #5
    I'm still stuck :( got loads of fs f's f''s with x y @_@ squares and all.. T_T
     
  7. Feb 23, 2010 #6
    Post your stuff. I'll take a look.
     
  8. Feb 23, 2010 #7
    http://img641.imageshack.us/img641/3946/rimg0001m.jpg [Broken]
    http://img291.imageshack.us/img291/5116/rimg0003a.jpg [Broken]
    it's messy though...
     
    Last edited by a moderator: May 4, 2017
  9. Feb 23, 2010 #8
    I would say everything looks great after the first page you posted.

    Before getting started on the second page, have a look at the last line on your first page. You see the expression:

    [tex] (F(c) + F''(c)) ?[/tex]

    Using the fact that f'(x) = g(x) and g'(x) = -f(x), and considering your formulas on lines 1 and 3 of the first page, you should be able to simplify this expression considerably.

    Once you move on from here, you should consider taking a = 0 for your MVT and then use your information about f(0) and g(0) to further simplify the problem.

    Unfortunately, I have to run! Good luck :).
     
  10. Feb 23, 2010 #9
    http://img52.imageshack.us/img52/5700/rimg0001.jpg [Broken]
    http://img188.imageshack.us/img188/5554/rimg0002q.jpg [Broken]

    still stuck :(
     
    Last edited by a moderator: May 4, 2017
  11. Feb 24, 2010 #10
    You should be able to find (provided you follow the steps given above) that E(b) = 0. Are you able to get this?
     
  12. Feb 25, 2010 #11
    yeah.. so the addtion formula of f(x+y) = f(x)g(y) + f(y)g(x) right? =)
     
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