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Types of Non-Differentiable Functions: Corners, Cusps, Vertical Tangents, Discontinuities, Summaries of Differential Equations

Four types of functions that do not have derivatives at certain points: corners, cusps, vertical tangents, and discontinuities. what each type is and provides examples. It also covers the relationship between continuity and differentiability.

What you will learn

  • What is a cusp in the context of functions?
  • What are the four types of functions that do not have derivatives at certain points?
  • What is a corner in the context of functions?

Typology: Summaries

2021/2022

Uploaded on 09/27/2022

anarghya
anarghya 🇺🇸

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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.
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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:

  1. Corners
  2. Cusps
  3. Vertical tangents
  4. 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