1. Visualizing Partial Derivatives without Graphs. Martin Flashman. Humboldt State University and Occidental College. flashman@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. Foundations Mapping (Transformation) Figures ► Visualize functions f : R  R y = f(x) ► Winplot examples:  linear  nonlinear

4. y = 2x -1 y = x 2

5. Dynamic interpretation ► Dynamic interpretation of the derivative visualized using   y/  x  rates ► Illustrate using Winplot

6. Visualizing Multi-Variable Functions ► Visualize functions f : R n  R y = f(x 1,x 2,…,x n ) ► Visualize f : R 2  R y = f(x 1,x 2 )

7. y = 2(x 1 -1) – 3(x 2 -1) y = x 1 2 - x 2 3

8. Visualizing The Partial Derivative for y = f(x 1,x 2 ) ► Dynamic interpretation of the partial derivatives for f : R 2  R y = f(x 1,x 2 ) visualized using  y/  x 1 and  y/  x 2. ► Static picture next slide. ► Winplot dynamic visualization.

9. y = 2(x 1 -1) – 3(x 2 -1) y = x 1 2 - x 2 3

10. Visualizing The Partial Derivative for y = f(x 1,x 2,…,x n ) ► Dynamic interpretation of the partial derivative for f : R n  R y = f(x 1,x 2,…,x n ) visualized using  y/  x 1, …,  y/  x n. ► Static picture here. ► Winplot dynamic visualization.

11. Conclusion ► Can use this visualization for other aspects of functions ► f : R n  R k f (x 1,x 2,…,x n )=(y 1,y 2,…,y k ) f (x 1,x 2,…,x n )=(y 1,y 2,…,y k ) where y k = f k (x 1,x 2,…,x n )

12. Time!Time! ► Questions? ► Responses? ► Further Communication by e-mail: flashman@humboldt.edu ► These notes will be available at http://www.humboldt.edu/~mef2

13. Thanks- The end!