Visualizing Partial Derivatives without Graphs.

1
Visualizing Partial Derivatives without Graphs.

• Martin Flashman.
• Humboldt State University and Occidental
  College.
  
• flashman_at_humboldt.edu

2
Abstract

• In this presentation the author will
• explain and use free graphing technology
  (Winplot) to illustrate
  
• how to visualize the partial derivative without
  graphs.
  
• The treatment is suitable for any introductory
  treatment of the concept.
  
• Based on mapping (transformation) figures this
  approach
  
• allows students to understand the concepts in an
  n-dimensional context
  
• without any change in presentation from that
  given for the ordinary derivative.

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FoundationsMapping (Transformation) Figures

• Visualize functions f R ? R
• y f(x)
• Winplot examples
• linear
• nonlinear

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y 2x -1 y x2
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Dynamic interpretation

• Dynamic interpretation of the derivative
  visualized using
  
• ?y/?x
• rates
• Illustrate using Winplot

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Visualizing Multi-Variable Functions

• Visualize functions f Rn ? R
• y f(x1,x2,,xn)
• Visualize f R2 ? R
• y f(x1,x2)

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y 2(x1-1) 3(x2-1) y x12 - x23
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Visualizing The Partial Derivative for y
f(x1,x2)

• Dynamic interpretation of the partial
  derivatives for f R2 ? R
  
• y f(x1,x2)
• visualized using ?y/?x1 and ?y/?x2.
• Static picture next slide.
• Winplot dynamic visualization.

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y 2(x1-1) 3(x2-1) y x12 - x23
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Visualizing The Partial Derivative for y
f(x1,x2,,xn)

• Dynamic interpretation of the partial derivative
  for f Rn ? R
  
• y f(x1,x2,,xn)
• visualized using ?y/?x1, ,?y/?xn.
• Static picture here.
• Winplot dynamic visualization.

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Conclusion

• Can use this visualization for other aspects of
  functions
  
• f Rn ? Rk
• f (x1,x2,,xn)(y1,y2,,yk)
• where yk fk(x1,x2,,xn)

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Time!

• Questions?
• Responses?
• Further Communication by e-mail
• flashman_at_humboldt.edu
• These notes will be available at
• http//www.humboldt.edu/mef2

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Thanks-

• The end!