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In mathematics, particularly calculus, a vertical tangent is a tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency.
A function ƒ has a vertical tangent at x = a if the difference quotient used to define the derivative has infinite limit:
The first case corresponds to an upward-sloping vertical tangent, and the second case to a downward-sloping vertical tangent. The graph of ƒ has a vertical tangent at x = a if the derivative of ƒ at a is either positive or negative infinity.
For a continuous function, it is often possible to detect a vertical tangent by taking the limit of the derivative. If
then ƒ must have an upward-sloping vertical tangent at x = a. Similarly, if
then ƒ must have a downward-sloping vertical tangent at x = a. In these situations, the vertical tangent to ƒ appears as a vertical asymptote on the graph of the derivative.
Closely related to vertical tangents are vertical cusps . This occurs when the one-sided derivatives are both infinite, but one is positive and the other is negative. For example, if
then the graph of ƒ will have a vertical cusp that slopes up on the left side and down on the right side.
As with vertical tangents, vertical cusps can sometimes be detected for a continuous function by examining the limit of the derivative. For example, if
then the graph of ƒ will have a vertical cusp at x = a that slopes down on the left side and up on the right side. This corresponds to a vertical asymptote on the graph of the derivative that goes to on the left and on the right.
The function
has a vertical tangent at x = 0, since it is continuous and
Similarly, the function
has a vertical cusp at x = 0, since it is continuous,
and
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In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0. These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered.
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