Vertical Tangents and Cusps

In this topic:

    1. Define Vertical Tangents and Cusps.

    2. Illustrate with the functions and

There are four possibilities for unbounded behavior of a derivative around a given real number They are:

vertical tangents and cusps _gr_5.gif]

which are called vertical tangents; and when these limits differ in sign they are:

which are called cusps. Here are the formal definitions of vertical tangents and cusps followed by an example of each.

Definition (Vertical Tangent) Suppose the function is continuous at the point Then the graph of has a vertical tangent at if one of the following holds:

Example (Vertical Tangent) Sketch the graph of and explain why there is a vertical tangent at .

    Solution. Using the product rule we find the derivative as


To determine any vertical tangents we consider where is undefined. Notice that at the derivative is undefined but Thus, the point is a candidate for a being a vertical tangent to the graph of We check the following limits to determine if is a vertical tangent.

    Since and as


    Since and as

By the definition of a vertical tangent is a vertical tangent which can be seen from the sketch of the graph of

vertical tangents and cusps _gr_35.gif]

Definition (Cusp) Suppose the function is continuous at the point Then the graph of has a cusp at if one of the following holds:

Example (Cusp) Sketch the graph of and explain why there is a cusp at .

    Solution. Using the product rule we find the derivative as


vertical tangents and cusps _gr_49.gif]

To determine any cusps we consider where is undefined. Notice that at the derivative is undefined but Thus, the point is a candidate for a being a cusp for the graph of the function . We check the following limits to determine if is a cusp.

    Since and as

vertical tangents and cusps _gr_59.gif]

    Since and as

By the definition of a cusp is a cusp which can be seen from the sketch of the graph of