Difference between revisions of "Tangent (geometry)"
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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 indistinguishable from each other at the point in which they intersect. | |
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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 | ||
| + | ==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>\left(\frac{\pi}4, \frac{1}{\sqrt 2}\right)</math> intersects it in 2 points and the tangent line at <math>\left(\frac{\pi}2, 1\right)</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>\left(\frac{\pi}4, \frac{1}{\sqrt 2}\right)</math> intersects it in 2 points and the tangent line at <math>\left(\frac{\pi}2, 1\right)</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. 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! | 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! | ||
| + | {{stub}} | ||
== See also == | == See also == | ||
* [[Calculus]] | * [[Calculus]] | ||
Revision as of 22:25, 24 November 2007
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 indistinguishable from each other at the point in which they intersect.
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 
at 
intersects it in 1 point, while the tangent line at 
intersects it in 2 points and the tangent line at 
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 
at 
) or even at all points!
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