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Math Help - directional derivatives and the gradient vector

  1. #1
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    directional derivatives and the gradient vector

    Suppose you are climbing a hill whose shape is given by the equation
    z = 1000 - 0.005x^2 -0.01y^2, where x,y,z are measured in meters, and you are standing at a point with coorindates (60,40,966). The positive x-axis points east and the positve y axis points north.

    a. if you are walk due south, will you start to ascend or descend? At what rate?

    b. if you are walk northwest, will you start to ascend or descend? At what rate?

    c. in which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?

    Any help is greatly appreaciated!!!!!!
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  2. #2
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    Quote Originally Posted by panic!@thediscoanytime View Post
    Suppose you are climbing a hill whose shape is given by the equation
    z = 1000 - 0.005x^2 -0.01y^2, where x,y,z are measured in meters, and you are standing at a point with coorindates (60,40,966). The positive x-axis points east and the positve y axis points north.
    You titled this "directional derivatives and the gradient vector" so I would guess that somewhere in the chapter in which this problem appears you are told that if \vec{u} is a unit vector, then the derivative of f in the direction of \vec{u} is equal to the dot product \vec{u}\cdot \nabla f.
    What is \nabla z here?

    a. if you are walk due south, will you start to ascend or descend? At what rate?
    What is a unit vector pointing "due south", that is, in the negative y direction?

    b. if you are walk northwest, will you start to ascend or descend? At what rate?
    what is a vector pointing "northwest", that is, at 45 degrees between the negative x-axis and the positive y-direction?

    c. in which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?
    The gradient of z, \nabla z always points in the direction of "fastest ascent". What is the length of the gradient vector? What angle does it make with the x-axis?

    [quote]Any help is greatly appreaciated!!!!!!
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