Calc

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The directional derivative of z = f(x,y) at (2,1) in the direction toward the point (1,3) is -2/sqrt(5), and the directional derivative in the direction toward the point (5,5) is 1. Compute dz/dx and dz/dy at (2,1).

I have no idea how to solve this problem. Help, please? Thank you.

  • Calc -

    From point (2,1) to (1,3):
    u = <1-2,3-1> = <-1,2>
    Note that vector u above is not a unit vector, so we need to make this a unit vector by dividing it with its magnitude.
    u = <-1,2>/sqrt(5)

    <dz/dx,dz/dy>•<-1/sqrt(5),2/sqrt(5)> = -2/sqrt(5)
    -dz/dx + 2dz/dy = -2. ..(1)

    From point (2,1) to (5,5):
    v = <5-2,5-1> = <3,4>
    Again, this is not a unit vector.
    v = <3,4>/5 = <3/5,4/5>

    <dz/dx,dz/dy>•<3/5,4/5> = 1
    3dz/dx + 4dz/dy = 5. ..(2)

    Using elimination and substition to both equations (1) and (2), you'll get the value for dz/dx and dz/dy

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