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[[Image:Vertical tangent.svg|thumb|Vertical tangent on the function ''ƒ''(''x'') at ''x''=''c''.]] | |||
In [[mathematics]] and [[Calculus]], a '''vertical tangent''' is [[tangent]] line that is [[Vertical direction|vertical]]. Because a vertical line has [[Infinity|infinite]] [[slope]], a [[Function (mathematics)|function]] whose [[graph of a function|graph]] has a vertical tangent is not [[differentiable]] at the point of tangency. | |||
== Limit definition == | |||
A function ƒ has a vertical tangent at ''x'' = ''a'' if the [[difference quotient]] used to define the derivative has infinite limit: | |||
:<math>\lim_{h\to 0}\frac{f(a+h) - f(a)}{h} = {+\infty}\quad\text{or}\quad\lim_{h\to 0}\frac{f(a+h) - f(a)}{h} = {-\infty}.</math> | |||
The first case corresponds to an upward-sloping vertical tangent, and the second case to a downward-sloping vertical tangent. Informally speaking, 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 | |||
:<math>\lim_{x\to a} f'(x) = {+\infty}\text{,}</math> | |||
then ƒ must have an upward-sloping vertical tangent at ''x'' = ''a''. Similarly, if | |||
:<math>\lim_{x\to a} f'(x) = {-\infty}\text{,}</math> | |||
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. | |||
== Vertical cusps == | |||
Closely related to vertical tangents are '''vertical [[cusp (singularity)|cusps]]'''. This occurs when the [[one-sided derivative]]s are both infinite, but one is positive and the other is negative. For example, if | |||
:<math>\lim_{h \to 0^-}\frac{f(a+h) - f(a)}{h} = {+\infty}\quad\text{and}\quad \lim_{h\to 0^+}\frac{f(a+h) - f(a)}{h} = {-\infty}\text{,}</math> | |||
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 | |||
:<math>\lim_{x \to a^-} f'(x) = {-\infty} \quad \text{and} \quad \lim_{x \to a^+} f'(x) = {+\infty}\text{,}</math> | |||
then the graph of ƒ will have a vertical cusp 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 <math>\infty</math> on the left and <math>-\infty</math> on the right. | |||
== Example == | |||
The function | |||
:<math>f(x) = \sqrt[3]{x}</math> | |||
has a vertical tangent at ''x'' = 0, since it is continuous and | |||
:<math>\lim_{x\to 0} f'(x) \;=\; \lim_{x\to 0} \frac{1}{\sqrt[3]{x^2}} \;=\; \infty.</math> | |||
Similarly, the function | |||
:<math>g(x) = \sqrt[3]{x^2}</math> | |||
has a vertical cusp at ''x'' = 0, since it is continuous, | |||
:<math>\lim_{x\to 0^-} g'(x) \;=\; \lim_{x\to 0^-} \frac{1}{\sqrt[3]{x}} \;=\; {-\infty}\text{,}</math> | |||
and | |||
:<math>\lim_{x\to 0^+} g'(x) \;=\; \lim_{x\to 0^+} \frac{1}{\sqrt[3]{x}} \;=\; {+\infty}\text{.}</math> | |||
== References == | |||
[http://www.sosmath.com/calculus/diff/der09/der09.html Vertical Tangents and Cusps]. Retrieved May 12, 2006. | |||
[[Category:Mathematical analysis]] |
Revision as of 17:05, 21 June 2013
In mathematics and Calculus, a vertical tangent is 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.
Limit definition
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. Informally speaking, 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.
Vertical cusps
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 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.
Example
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
References
Vertical Tangents and Cusps. Retrieved May 12, 2006.