Answer the three background questions first, then explore the applet. [br]The checkboxes in the applet below will show you five different functions. Each function has at least one point at which it is not differentiable. Examine the functions one by one and determine the point(s) at which they are not differentiable. There is a hint if you need it. Answer the questions that follow the applet as you go.
1.
For [math]lim_{x\longrightarrow a}f\left(x\right)[/math] 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?
Yes, if there is a hole at that point, the limit will be defined there but the function doesn't have to be.
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?
The derivative of a function at a point is the slope of the line tangent to the curve at that point.
4.
At what x-value (approximately) is the derivative of the function in Example 1 undefined? Why?
At about x=0.5, because there's a hole there.
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?
At x=0, because the left- and right-hand limits don't match, you can't have a tangent line there that approximates the curve.
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.
right: ~1 left:~-1
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?
The slopes get steeper, more and more negative on the left, more and more positive on the right.
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?
Because we calculate slope by dividing the change in y by the change in x. In a vertical line, there is no change in x-values, so we're dividing by 0, which is 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?