Section 14.5 Gradients and Directional Derivatives in Space - PowerPoint PPT Presentation

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Section 14.5 Gradients and Directional Derivatives in Space

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Section 14.5 Gradients and Directional Derivatives in Space Directional Derivative of Function of 3 Variables Let w = f(x,y,z) and let the partial derivatives of f ... – PowerPoint PPT presentation

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Title: Section 14.5 Gradients and Directional Derivatives in Space


1
Section 14.5Gradients and Directional
Derivatives in Space
2
  • Directional Derivative of Function of 3 Variables
  • Let w f(x,y,z) and let the partial derivatives
    of f exist at (a,b,c) in its domain. Let
    be a unit vector.
    Then
  • Now, using a similar argument to last section we
    can show that
  • where ? is the angle between the gradient and
    the unit vector

3
  • Similar to before we have
  • If ? 0, then our direction is the same as the
    gradient and
  • If ? p, then our direction is the opposite of
    the gradient and
  • If ? p/2, then our direction is the
    perpendicular to the gradient and

4
  • When we had a function of 2 variables, if the
    directional derivative was zero we were moving in
    a direction tangent to the level curve and the
    gradient was perpendicular to the level curve
  • With a function of 3 variables, if the
    directional derivative is zero we are moving in a
    direction tangent to the level surface and the
    gradient is perpendicular to that level surface

5
Example
  • Find
  • Find the derivative in the direction of
  • Find the maximum rate of change of f at (3,4,5)
  • Find the vector in the direction of the maximum
    rate of change at (3,4,5)

6
Finding a Tangent Plane
  • Determine the equation of the plane tangent to
    at (1,1)
  • Now consider f the function of a level surface of
    where w 0
  • Now we know that the gradient of w at a point
    (a,b,c) is perpendicular to w at that point
  • Therefore we have
  • Now find the tangent plane using the gradient
  • Lets make sure they are the same
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