Sheet #0281: Differentiability of Piecewise Functions			

Names___________________________________________Period_____________

Roles for your group members:   Recorder, Function Grapher, Derivative Grapher
Your mission is to graph the given piecewise function on the grid on the left and to graph the derivative of the piecewise function on the grid on the right.  You must also write a piecewise function for the derivative.

Warm-up
Graph  and zoom in several times on the point (0, 0)
Graph  and zoom in several times on the point (0, 0)
What do you notice?

Now graph on the axes below: 			Also graph:  
	
Write a piece-wise function for  and discuss the continuity of 

Now write a piece-wise function for  and discuss the continuity of 

1.

	

Discuss the continuity of 

Discuss the continuity of 

2.  					

	

Discuss the continuity of 

Discuss the continuity of 

3.						

	

Discuss the continuity of 

Discuss the continuity of 

4.						

	

Discuss the continuity of 

Discuss the continuity of 

Making piecewise functions differentiable:
I.		Find what you need to make the function continuous first
II.	Find what you need to make the function differentiable
III.	Use what you found to set up a system of equations which you will solve to find 		any missing values that will make your function, not only continuous, but 			differentiable
IV.	You should check your solution(s)

5.	Find the values for  and  that make the piecewise function differentiable for 		all real numbers

Written by:  Denise Fuji McCleary for http://www.zendog.org/homework
2009diffpiece.doc