Gradients and Directional Derivatives - PowerPoint by 62I0JCj

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									Gradients and Directional
      Derivatives




                     Chp 15.6
      Putting it all together…
• Over the past several classes you have
  learned how to use and take partial
  derivatives
• Today we look at the essential difference
  between defining slope along a 2D curve
  and what it means on a 3D surface or
  curve
                                                  x2
                                             10e y
Example…What’s the slope of f ( x, y )                   at (0,1/2)?
                                           1  x2  y 2




  What’s wrong with the way
  the question is posed?




  What’s the slope along                          What’s the slope
  the direction of the x-                         along the direction of
  axis?                                           the y-axis?
Quick estimate from the contour
plot:

  z  (0  4)  4
  y  (0.5  0)  0.5            z
  m  8

 Look at this in Excel:
     The Directional Derivative
• We need to specify the direction in which the
  change occurs…
• Define, via a slightly modified Newton quotient:

  Du f ( x, y)  f x ( x, y)a  f y ( x, y)b
• This specifies the change in the direction of the
  vector u = <a,b>
                 The Gradient
• We can write the Directional derivative as:


      Du f ( x, y, z )  f x , f y , f z   u1 , u2 , u3



   Gradient of f(x,y,z)
                          f ( x , y , z )
            A Key Theorem
• Pg 982 – the Directional derivative is
  maximum when it is in the same direction
  as the gradient vector!
• Example: If your ski begins to slide down
  a ski slope, it will trace out the gradient for
  that surface!

								
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