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Using Jacobians to get a Best Linear Approximation

  1. Dec 3, 2011 #1
    1. The problem statement, all variables and given/known data

    Let g(s,t) =<t*sqrt(s),s*cos(t)>

    a) Find the Jacobian matrix of g(s,t).
    b) Using the idea that a derivative is the best linear approximation to a function, use a linear approximation to estimate the value of g(1.02,0.01)

    3. The attempt at a solution
    d(x,y)/d(s,t) = [t/(2sqrt(s)) sqrt(s); cos(t) -s*sin(t)]
    Jg(1,0) = [0 1; 1 0]
    [0 1; 1 0] * [0; 1] = [1; 0]

    To this I add

    [0 1; 1 0] * [0.01; 0.02] = [0.02; 0.01]

    I have,

    [1.02; 0.01]. So I estimate g(1.02,0.01) to be around (1.02,0.01)?
     
  2. jcsd
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