Download Types of Non-Differentiable Functions: Corners, Cusps, Vertical Tangents, Discontinuities and more Summaries Differential Equations in PDF only on Docsity!
3.2 Notes
There are four different types of functions that do not have derivatives at certain points.
Graph | |
If the derivative is If the derivative is
What is the derivative at Since the left derivative does not equal the right derivative, the derivative does not exist.
This is called a corner.
Graph √
What is the derivative when Negative What is the derivative when Positive
What is the derivative when Since the left derivative does not equal the right derivative, the derivative does not exist.
This is called a cusp.
Graph (^) √
What is the derivative when Positive What is the derivative when Positive When , the derivative looks like a vertical line which is undefined.
This is called a vertical tangent.
Graph any type of discontinuity.
It should make sense that if there is value for an x, there is no derivative for the x.
These are called discontinuities.
The four types of functions that are not differentiable are:
- Corners
- Cusps
- Vertical tangents
- Any discontinuities
Let’s look at the curve What is the derivative?
I have created two tangent lines. One line goes through (-1, 1) and the other goes through (2, 4).
I have created two tangent lines. One line goes through (-1, 1) and the other goes through (2, 4).
What is the slope at? What is the slope at? ( ) ( )
Between ( ) & ( ), we can find every number. There is some between that will create every derivative between
This is called the Intermediate Value for Derivatives.
What do you know if ( )^ Draw a graph to disprove it if you can.
Is ( ) continuous at? No. There could be a hole.
Is ( ) differentiable at? If it is differentiable, it must be continuous, but it might not be.
Is ( )? No, if there is a piecewise function, the value might not equal the limit.
Does ( )? If the limit exists, the left side limit must equal the right side limit.
Does ( )? Same as above.
What do you know if?
This is the definition of the derivative in the form of ( ) (^) ( ) ( )
Is ( )^? Yes, by this definition.
Is ( ) differentiable at? Yes, differentiability implies continuity.
Is ( )? We know nothing about the value at
Does ( )? We know nothing about the limit at
Does ( ) We know nothing about the limit at
