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Change in 2 variable function

  1. May 30, 2013 #1
    1. The problem statement, all variables and given/known data

    f(x,y) = (x-y) / (x+y)

    Calculate Δf and df for the change of point (3,2) to point (3.2, 2.1)


    2. Relevant equations


    3. The attempt at a solution

    I guess that I have to use limits, but don't know how to begin with.
     
  2. jcsd
  3. May 30, 2013 #2

    HallsofIvy

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    Are you saying you do not know what [itex]\Delta f[/itex] and df mean or are you simply unable to differentiate?

    [tex]\Delta f= f(3.2, 2.1)- f(3, 2)[/tex]
    [tex]df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}dy[/tex]
     
  4. May 30, 2013 #3
    Yes, I know what they mean, but I didn't know where to start.

    So Δf should be:
    Δf = (3.2 - 2.1)/(3.2 + 2.1) - (3 - 2)/(3 + 2)
    Δf = 1.1/5.3 - 1/5
    Δf = 0.007547...

    And df should be:
    df = ((2y)/(x+y)^2)*dx + ((2x)/(x+y)^2)*dy

    Are my attempts correct ?
     
  5. May 30, 2013 #4

    epenguin

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    I think the first part is right. With hindsight you shouldn't see it anyhow difficult - there is the function, and the change in the function is the change in the function - the difference between its values for that x,y and this x,y.

    The second part is not part of the question but good to do. I think your second term has a mistake of sign. To see this don't just do the differentiation again, but notice the symmetry, that f(x, y) = - f(y, x) . Such symmetries are often useful checks for errors and shortening calculations.

    Then if you put in place of dx, dy Δx, Δy you ought to see whether you get a fair approximation to the previous result, though it will not be exact (except by accident).
     
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