1. TARUN GEHLOT (B.E, CIVIL, HONOURS)
Vertical Tangents and Cusps
In the definition of the slope, vertical lines were excluded. It is customary not to assign a
slope to these lines. This is true as long as we assume that a slope is a number. But
from a purely geometric point of view, a curve may have a vertical tangent. Think of a
circle (with two vertical tangent lines). We still have an equation, namely x=c, but it is not
of the form y = ax+b. In fact, such tangent lines have an infinite slope. To be precise we
will say:
The graph of a function f(x) has a vertical tangent at the point (x0,f(x0)) if and only
if
Example. Consider the function
We have
Clearly, f'(2) does not exist. In fact we have
So the graph of f(x) has a vertical tangent at (2,0). The equation of this line is x=2.
2. TARUN GEHLOT (B.E, CIVIL, HONOURS)
In this example, the limit of f'(x) when is the same whether we get closer to 2
from the left or from the right. In many examples, that is not the case.
Example. Consider the function
We have
So we have
3. TARUN GEHLOT (B.E, CIVIL, HONOURS)
It is clear that the graph of this function becomes vertical and then virtually doubles back
on itself. Such pattern signals the presence of what is known as a vertical cusp. In
general we say that the graph of f(x) has a vertical cusp at x0,f(x0)) iff
or
In both cases, f'(x0) becomes infinite. A graph may also exhibit a behavior similar to a
cusp without having infinite slopes:
Example. Consider the function
f(x) = |x3
- 8|.
Clearly we have
Hence
4. TARUN GEHLOT (B.E, CIVIL, HONOURS)
Direct calculations show that f'(2) does not exist. In fact, we have left and right
derivatives with
So there is no vertical tangent and no vertical cusp at x=2. In fact, the phenomenon this
function shows at x=2 is usually called a corner.
