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Given z=x²+3xy-2y² Estimate the change in z

  1. Feb 14, 2015 #1
    1. Given z=x²+3xy-2y² Estimate the change in z when x changes from 2 to 2,5 and y changes from 3 to 2,5


    2. I am stuck with the math and can't figure it out as what I need to do further.


    3.
    a=2
    b=3
    Δx= +0,5
    Δy= -0,5

    dz/dx = 2x+3y
    =2(2)+3(3)
    =13
    dz/dy = 3x-4y
    = 3(2)-4(3)
    =-6
    Δz=f(a+Δx ; b+Δy) - f(a,b)
    =f
    [2+0,5 ; 3+(-0,5)] - f(2,-0,5)
    =f(2,3)(0,5) + f(2,3)(-0,5)
    =** This is where I am stuck.

    =13(0.5) + (-6)(-0,5)
    =9,5???
     
  2. jcsd
  3. Feb 14, 2015 #2

    stevendaryl

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    You're almost there: To first order, you can approximate: [itex]\delta z = \frac{\partial z}{\partial x} \delta x + \frac{\partial z}{\partial y} \delta y[/itex]

    You know that:
    [itex]\delta x = + 0.5[/itex] (in America, we use "." for decimals, rather than ",")
    [itex]\delta y = - 0.5[/itex]
    [itex]\frac{\partial z}{\partial x} = 13[/itex]
    [itex]\frac{\partial z}{\partial y} = -6[/itex]

    So you just plug in what you know to compute [itex]\delta z[/itex]
     
  4. Feb 16, 2015 #3

    HallsofIvy

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    That's a plausible way to estimate the change in z but the exact value is z(2.5, 2.5)- z(2, 3).
     
  5. Feb 16, 2015 #4

    stevendaryl

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    But the homework asked for an estimate.
     
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