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| [[Image:Vertical tangent.svg|thumb|Vertical tangent on the function ''ƒ''(''x'') at ''x''=''c''.]]
| | The name of the author is Nestor. My job is a messenger. Delaware is our birth location. Camping is something that I've done for many years.<br><br>Check out my web page ... [http://Louisianastrawberries.net/ActivityFeed/MyProfile/tabid/61/UserId/95947/Default.aspx Louisianastrawberries.net] |
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| 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.
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| == Limit definition ==
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| A function ƒ has a vertical tangent at ''x'' = ''a'' if the [[difference quotient]] used to define the derivative has infinite limit:
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| :<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>
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| 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. | |
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| For a [[continuous function]], it is often possible to detect a vertical tangent by taking the limit of the derivative. If
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| :<math>\lim_{x\to a} f'(x) = {+\infty}\text{,}</math>
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| then ƒ must have an upward-sloping vertical tangent at ''x'' = ''a''. Similarly, if
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| :<math>\lim_{x\to a} f'(x) = {-\infty}\text{,}</math>
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| 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.
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| == Vertical cusps ==
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| 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
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| :<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>
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| then the graph of ƒ will have a vertical cusp that slopes up on the left side and down on the right side.
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| As with vertical tangents, vertical cusps can sometimes be detected for a continuous function by examining the limit of the derivative. For example, if
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| :<math>\lim_{x \to a^-} f'(x) = {-\infty} \quad \text{and} \quad \lim_{x \to a^+} f'(x) = {+\infty}\text{,}</math>
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| 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.
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| == Example ==
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| The function
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| :<math>f(x) = \sqrt[3]{x}</math>
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| has a vertical tangent at ''x'' = 0, since it is continuous and
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| :<math>\lim_{x\to 0} f'(x) \;=\; \lim_{x\to 0} \frac{1}{\sqrt[3]{x^2}} \;=\; \infty.</math>
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| Similarly, the function
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| :<math>g(x) = \sqrt[3]{x^2}</math> | |
| has a vertical cusp at ''x'' = 0, since it is continuous,
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| :<math>\lim_{x\to 0^-} g'(x) \;=\; \lim_{x\to 0^-} \frac{1}{\sqrt[3]{x}} \;=\; {-\infty}\text{,}</math>
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| and
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| :<math>\lim_{x\to 0^+} g'(x) \;=\; \lim_{x\to 0^+} \frac{1}{\sqrt[3]{x}} \;=\; {+\infty}\text{.}</math>
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| == References ==
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| [http://www.sosmath.com/calculus/diff/der09/der09.html Vertical Tangents and Cusps]. Retrieved May 12, 2006.
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| [[Category:Mathematical analysis]]
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The name of the author is Nestor. My job is a messenger. Delaware is our birth location. Camping is something that I've done for many years.
Check out my web page ... Louisianastrawberries.net