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Visualizing Partial Derivatives without Graphs.

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explain and use free graphing technology (Winplot) to illustrate. how to visualize ... The treatment is suitable for any introductory treatment of the concept. ... – PowerPoint PPT presentation

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Title: 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.

3
FoundationsMapping (Transformation) Figures
  • Visualize functions f R ? R
  • y f(x)
  • Winplot examples
  • linear
  • nonlinear

4
y 2x -1 y x2
5
Dynamic interpretation
  • Dynamic interpretation of the derivative
    visualized using
  • ?y/?x
  • rates
  • Illustrate using Winplot

6
Visualizing Multi-Variable Functions
  • Visualize functions f Rn ? R
  • y f(x1,x2,,xn)
  • Visualize f R2 ? R
  • y f(x1,x2)

7
y 2(x1-1) 3(x2-1) y x12 - x23
8
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.

9
y 2(x1-1) 3(x2-1) y x12 - x23
10
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.

11
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)

12
Time!
  • Questions?
  • Responses?
  • Further Communication by e-mail
  • flashman_at_humboldt.edu
  • These notes will be available at
  • http//www.humboldt.edu/mef2

13
Thanks-
  • The end!
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