Discovering Differentiability
1.
For to exist, which of the following must be true?
2.
Can a function be discontinuous at a point if the limit is defined at that point?
3.
For a function to be differentiable at a point x=a, the value of the derivative must exist there. How do we define the derivative of a function at a point?
4.
At what x-value (approximately) is the derivative of the function in Example 1 undefined? Why?
5.
If the limit exists at a hole on a function, how come the derivative doesn't?
6.
At what x-value is the derivative of the function in Example 2 undefined? Why?
7.
At the "vertex" of the graph in Example 3, we essentially have two lines meeting. Estimate the slope of the line to the right of the vertex. Estimate the slope of the line to the left of the vertex.
8.
At what x-value (approximately) is the derivative of the function in Example 3 undefined? Why?
9.
Can a function be continuous everywhere but not differentiable everywhere?
10.
What happens to the slope of the graph in Example 4 as you approach the minimum point from either direction?
11.
At what x-value (approximately) is the derivative of the function in Example 4 undefined? Why?
12.
Why is the slope of a vertical line undefined?
13.
At what x-value (approximately) is the derivative of the function in Example 5 undefined? Why?
14.
What are the different characteristics of a graph that will show you that it is not differentiable at that point?