A cusp on a graph is a distinctive point where a curve sharply changes direction, forming a pointed tip. It's a specific type of singular point on a curve.
Understanding a Cusp
More precisely, a cusp, also known as a spinode, is a point on a curve where two distinct branches of the curve converge. A defining characteristic of a cusp is that the tangents to each branch of the curve at this meeting point are equal. This means that as you approach the cusp from either side along the curve, the direction of the curve (represented by its tangent line) becomes the same, even though the curve does not pass smoothly through the point.
While a function describing a curve with a cusp is continuous at that point, it is not differentiable. This means that a unique tangent line with a finite slope cannot be precisely defined at the cusp itself, although the tangents approaching it from both sides will share the same, often vertical, orientation.
Cusps vs. Corners
It's important to distinguish a cusp from a more general "corner" on a graph. While both represent points of non-differentiability where a function changes direction sharply, their precise characteristics differ:
| Feature | Cusp | Corner |
|---|---|---|
| Tangent Lines | The tangents from both branches of the curve meet and are equal (same slope). | The tangents from different directions exist but are not equal (different slopes). |
| Shape | Resembles a pointed tip where the curve momentarily reverses direction. | Forms a sharp angle; the curve changes direction abruptly. |
| Differentiability | Continuous but not differentiable. | Continuous but not differentiable. |
| Derivative | The derivative approaches positive or negative infinity from both sides, or both approach the same finite value from a vertical tangent. | The derivative approaches different finite values from different sides (left-hand vs. right-hand derivatives do not match). |
| Example Function | (y = x^{2/3}) at (x=0), (y^2 = x^3) (cusp at origin) | (y = |
Examples of Cusps in Graphs
Cusps appear in various mathematical curves and equations:
- Standard Cusp: The most common example is the graph of (y = x^{2/3}). At (x=0), the graph forms a sharp, upward-pointing cusp. The derivative (dy/dx = \frac{2}{3}x^{-1/3}) becomes undefined at (x=0), indicating a vertical tangent.
- Cubic Cusp: The equation (y^2 = x^3) (or (x^{2/3})) also demonstrates a cusp at the origin, often called a "cusp of the first kind."
- Other Forms: Cusps can also be found in more complex parametric equations or implicitly defined curves, where a curve might "loop back" on itself, forming a sharp point.
Understanding cusps is crucial in calculus and curve analysis, as they represent points where standard differentiation rules don't apply directly and indicate a significant change in the curve's behavior.
