incorrect diffrentiation

Discussion in 'Math' started by recca02, Apr 3, 2007.

  1. recca02

    Thread Starter Senior Member

    Apr 2, 2007
    1,211
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    this is an old trick at least for the biggies here but please care to elaborate on the below example

    1+1+1......n times=n
    1+1+1... x times = x ...x is a variable -----(1)
    x+x+x....n times = x.n
    x+x+x....x times = x^2
    diff w.r.t x
    1+1+1... x = 2.x
    from (1)
    x = 2.x
    or
    1 = 2
    i know diff isnt allowed since limit does not exist in the case of
    x+x+x....x times
    but somehow i want a little detailed explanation
    wud highly appreciate it if someone can help.
     
  2. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
    144
    d/dx(x) = 1 not 2.x

    i.e. the rate of change of x is 1, which from 1+1+1... x is clearly true.

    Dave
     
  3. recca02

    Thread Starter Senior Member

    Apr 2, 2007
    1,211
    0
    but d/dx(x^2) = is 2.x (that is what i intended)
    i mean diff both rhs and lhs
    there is nothin wrong with diff the only thing is it is not allowed in this case.
     
  4. DrNick

    Active Member

    Dec 13, 2006
    110
    2
    I think your problem is that you are using a dummy variable as part of the limit in your sum, making the summation non-linear. So you have
    sum(x, i=0, x)

    if we differentiate both sides you will notice that since the summation is no longer linear we cant interchange the order of summation and differentiation. SO,

    d/dx ( sum(x, i=0, x) ) = d/dx(x^2) != [ sum(d/dx(x), i=0, x) = sum(1, i=0, x) ]

    that x that is the upper limit must also be considered when differentiating (this is why we can't bring the derivative inside of the sum).
     
  5. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
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    Apologies I misinterpreted your initial question. I concur with the answer offered by DrNick.

    Dave
     
  6. recca02

    Thread Starter Senior Member

    Apr 2, 2007
    1,211
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    thanks mr. dave and drnick
    anyway can anyone come up with the graph of such function
    i mean x+x+x+.....x times
    wud it be a step funtion with a linear component
     
  7. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
    144
    Plotting (sum of x) against (i = 1 to x) gives a straight line of gradient x. Try it in Excel with x arbitrarily set to say 10.

    Dave
     
  8. recca02

    Thread Starter Senior Member

    Apr 2, 2007
    1,211
    0
    i'll bother u just once more.
    how much sense does it make that a variable x which is a real number lets say
    1.786 to be added 1.786 times ( X+X+..X times)
    its been years since i learned summation but if i m not wrong it adds only
    whole numbers it does not take into account fractional values.
     
  9. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
    144
    Although it sounds wierd, 1.786 to be added 1.786 times is the same as 1.786^2 - think of it in terms of integers for a proof, and think that the rules apply equally to all real numbers.

    Dave
     
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