Difference between revisions of "Tangent (geometry)"

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Locally, a tangent line intersects a curve in a single point.  However, if a curve is neither [[convex]] nor [[concave]], it is possible for a tangent line to intersect a curve in additional points.  For instance, the tangent line of the curve <math>y = \sin x</math> at <math>(0, 0)</math> intersects it in 1 point, while the tangent line at <math>(\frac{\pi}4, \frac{1}{\sqrt 2})</math> intersects it in 2 points and the tangent line at <math>(\frac{\pi}2, 1)</math> intersects it in [[infinite]]ly many points (and is in fact the tangent line at each point of intersection).
 
Locally, a tangent line intersects a curve in a single point.  However, if a curve is neither [[convex]] nor [[concave]], it is possible for a tangent line to intersect a curve in additional points.  For instance, the tangent line of the curve <math>y = \sin x</math> at <math>(0, 0)</math> intersects it in 1 point, while the tangent line at <math>(\frac{\pi}4, \frac{1}{\sqrt 2})</math> intersects it in 2 points and the tangent line at <math>(\frac{\pi}2, 1)</math> intersects it in [[infinite]]ly many points (and is in fact the tangent line at each point of intersection).
  
At a given point, a curve may have either 0 or 1 tangent lines.  The condition "having a tangent line at a point" is equivalent to "being [[differentiable]] at a point."  It is a fairly strong condition on a [[function]] -- only [[continuous]] functions may have tangent lines, and there are many continuous functions which fail to have tangent lines either at some points (for instance, the [[absolute value]] function <math>y = |x|</math> at <math>x = 0</math>) or even at all points!   
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At a given point, a curve may have either 0 or 1 tangent lines.  For the graph of a [[function]], the condition "having a tangent line at a point" is equivalent to "being a [[differentiable]] function at that point."  It is a fairly strong condition on a function -- only [[continuous]] functions may have tangent lines, and there are many continuous functions which fail to have tangent lines either at some points (for instance, the [[absolute value]] function <math>y = |x|</math> at <math>x = 0</math>) or even at all points!   
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== See also ==
 
== See also ==
 
* [[Calculus]]
 
* [[Calculus]]

Revision as of 15:15, 12 August 2006

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A tangent line' is a linear approximate to a curve. That is, if you zoom in very closely, the tangent line and the curve will become indistinguisable from each other.

Locally, a tangent line intersects a curve in a single point. However, if a curve is neither convex nor concave, it is possible for a tangent line to intersect a curve in additional points. For instance, the tangent line of the curve $y = \sin x$ at $(0, 0)$ intersects it in 1 point, while the tangent line at $(\frac{\pi}4, \frac{1}{\sqrt 2})$ intersects it in 2 points and the tangent line at $(\frac{\pi}2, 1)$ intersects it in infinitely many points (and is in fact the tangent line at each point of intersection).

At a given point, a curve may have either 0 or 1 tangent lines. For the graph of a function, the condition "having a tangent line at a point" is equivalent to "being a differentiable function at that point." It is a fairly strong condition on a function -- only continuous functions may have tangent lines, and there are many continuous functions which fail to have tangent lines either at some points (for instance, the absolute value function $y = |x|$ at $x = 0$) or even at all points!


See also