tangent line

If the curve   y=f(x)  of xy-plane is sufficiently smooth in its point  (x0,y0)  and in a neighborhoodMathworldPlanetmath of this, the curve may have a tangent line (or simply ) in  (x0,y0).  Then the tangent line of the curve  y=f(x)  in the point  (x0,y0)  is the limit position of the secant lineMathworldPlanetmath through the two points  (x0,y0)  and  (x,f(x))  of the curve, when x limitlessly tends to the value x0 (i.e.  xx0).  Due to the smoothness,


and the slope m of the secant (http://planetmath.org/SecantLine) tends to


which will be the slope of the tangent line.

Note.  Because the tangency is a local property on the curve, the tangent with the tangency point(x0,y0)  may intersect the curve in another point, and then the tangent is a secant (http://planetmath.org/SecantLine), too.  For example, the curve  y=x3-3x2  has the line  y=0  as its tangent in the point  (0, 0)  but this line the curve also in the point  (3, 0).

Title tangent line
Canonical name TangentLine
Date of creation 2013-03-22 14:50:31
Last modified on 2013-03-22 14:50:31
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 12
Author Mathprof (13753)
Entry type Definition
Classification msc 26B05
Classification msc 26A24
Synonym tangent
Synonym tangent of the curve
Synonym tangent to the curve
Related topic Curve
Related topic TangentOfConicSection
Related topic Hyperbola2
Defines tangency point