Vertical Tangent

Vertical Tangent:

In geometry, the tangent line (or simply the tangent) to a curve at a given point is the straight line that "just touches" the curve at that point (in the sense explained more precisely below). As it passes through the point where the tangent line and the curve meet, or the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point.


In mathematics, a vertical tangent is tangent line to be vertical. Since a vertical line contain infinite slope, a function whose graph contain a vertical tangent be not differentiable on the point of tangency. During the definition of the slope, vertical lines are excluded. Although from a simply geometric point of view, a curve might contain a vertical tangent. Imagine of a circle (with two vertical tangent lines). We still contain an equation, namely x=c, except it be not of the form y = ax+b.In fact, such tangent lines contain an infinite slope.
Limit Definition of Vertical Tangent

A function ƒ have a vertical tangent on x = a. Condition the difference quotient use to identify the derivative have infinite limit:

[lim_(h->0)(f(a+h)-f(a))/(h)=+oo] (or [lim_(h->0)(f(a+h)-f(a))/(h)=-oo]

The first case corresponds toward an upward-sloping vertical tangent, with the second case toward a downward-sloping vertical tangent. Easily speaking, the graph of ƒ have a vertical tangent at x = a condition the derivative of ƒ on a be either positive or negative infinity.



Used for a continuous function, it is often possible toward detect a vertical tangent through taking the limit of the derivative.

if [lim_(x->a)f'(x)=+oo]

After that ƒ have to contain an upward-sloping vertical tangent at x = a. In the same way, if

if [lim_(x->a)f'(x)=-oo]

then ƒ have to contain an downward-sloping vertical tangent at x = a. Within these situations, the vertical tangent toward ƒ appears since a vertical asymptote on the graph of the derivative.Let us now look at few examples of Vertical Tangents.

Examples of Vertical Tangent

Example 1 for vertical tangent

The function

[f(x) = root(4)(x)]

have a vertical tangent at x = 0, while it is continuous also

[lim_(x->0)f'(x)=lim_(x->0)(1)/root(4)(x^2)=oo]

[g(x)=root(4)(x^2)]

Have a vertical cusp on x = 0, given that it be continuous,

[lim_(x->0)g'(x)=lim_(x->0)(1)/root(4)(x)=+oo]

[lim_(x->0^-)g'(x)=lim_(x->0^-)(1)/root(4)(x)=-oo]

Example 2 for vertical tangent

What value of x does the function (you mean the graph of the equation) 5x^2+2xy+y^2=36 have a vertical tangent line?

Solution

5x^2 + 2xy + y^2 = 36
and put y = -x
5x^2 - 2x^2 + x^2 = 36
4x^2 = 36
x^2 = 9
x = +3,-3


Hope you like the above example of Vertical Tangent.Please leave your comments, if you have any doubts.