1. Greg Kelly, Hanford High School, Richland, Washington

2. The functions that we studied last year were all functions of one independent variable: eg: In real life, functions often have more than one independent variable: eg:

3. Functions with two independent variables can be represented graphically. This is not easy to do by hand, and our calculators do not do a great job either.

4. y x z 10 100 sketch of graph

5. y x z x y 10 100 sketch of graphlevel curves Level curves are drawn by holding the z value constant (similar to contour lines on a topographic map.)

6. y x z 10 100 sketch of graph Let’s look at the same graph plotted on the TI-89: First change the mode to 3D. Then go to the Y= screen and enter the equation.

7. To graph “wire frame” on TI-89 1.Go to mode and set the mode to 3-D mode. 2.The default setting for your TI-89 should be wire frame. 3.To select this setting. Go to the y= page and select tools scroll down to 9 “format”

8. By pressing the arrow keys, you can rotate this graph!

9. Definition of a Function of Two Variables

10. Domain and Range

11. Example 1 a Find the domain of the function: (Solution on next slide)

13. Example 1 b Find the domain of the function. (solution on next slide)

15. Example 2 Find the range of Describe the graph

18. The problem of sketching the graph of a surface in space can be simplified by the use of a computer or graphing calculator

19. Rotating Graphs I Graph on your TI 89 calculator and practice getting a good window.

20. Rotating Graphs II

21. Rotating Graphs III

22. Here is a proof that any two numbers are equal. Can you find the error(s)? Start with any two random numbers a and b a + b = t (a + b)(a - b) = t(a - b) a 2 - b 2 = ta - tb a 2 - ta = b 2 - tb a 2 - ta + (t 2 )/4 = b 2 - tb + (t 2 )/4 (a - t/2) 2 = (b - t/2) 2 a - t/2 = b - t/2 a = b