Throughout this module, if something does not exist, write DNE in the answer box.

Recap Video

Take a look at the following video which recaps the ideas from the section.

Example Video

Below is a video showing a worked example.

Problems

If , evaluate using the limit definition above.
If , find the equation of the tangent line to at .
If , then the limit definition of the derivative tells us that .
In the previous problem, the domain of was:
We found a derivative of . The domain of is:
In particular, does not exist. If we go to the limit definition of the derivative with , we would get (in the second to last step): Since the graph of is only defined for , we need , so in reality we have What does this tell you about the slopes of the secant lines as the point gets closer to ?
This means the lines themselves are:
Consider the following graph of a function .
PIC
  • We have and .
  • The limits and .
  • The limit .
  • Does have any points of discontinuity? .
  • How many horizontal tangent lines does the graph of have? .
  • List the points where in increasing order:
  • The number is zero.
  • Which of the following is the graph of ?

Suppose .
  • The limit definition of the derivative tells us that .
  • The graph of has a horizontal tangent line at .
  • The point on the graph where the tangent line is parallel to is at .