Editing Tangent (section)

{{Use dmy dates|date=September 2021}}
{{Short description|In mathematics, straight line touching a plane curve without crossing it}}
{{About||the tangent function|Tangent (trigonometry)|other uses|Tangent (disambiguation)}}
[[Image:Tangent to a curve.svg|220px|right|thumb|Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.]]
[[Image:Image Tangent-plane.svg|220px|right|thumb|Tangent plane to a sphere]]

In [[geometry]], the '''tangent line''' (or simply '''tangent''') to a [[plane curve]] at a given [[Point (geometry)|point]] is, intuitively, the [[straight line]] that "just touches" the curve at that point. [[Leibniz]] defined it as the line through a pair of [[infinitesimal|infinitely close]] points on the curve.<ref>In "[[Nova Methodus pro Maximis et Minimis]]" (''[[Acta Eruditorum]]'', Oct. 1684), Leibniz appears to have a notion of tangent lines readily from the start, but later states: "modo teneatur in genere, tangentem invenire esse rectam ducere, quae duo curvae puncta distantiam infinite parvam habentia jungat, seu latus productum polygoni infinitanguli, quod nobis curvae aequivalet", ie. defines the method for drawing tangents through points infinitely close to each other.</ref><ref>{{cite book | page = 23 | title = Science and the Enlightenment | author = Thomas L. Hankins | isbn = 9780521286190 | year = 1985 | publisher = Cambridge University Press}}</ref> More precisely, a straight line is tangent to the curve {{nowrap|''y'' {{=}} ''f''(''x'')}} at a point {{nowrap|''x'' {{=}} ''c''}} if the line passes through the point {{nowrap|(''c'', ''f''(''c''))}} on the curve and has [[slope]] {{nowrap|''f''{{'}}(''c'')}}, where ''f''{{'}} is the [[derivative]] of ''f''. A similar definition applies to [[space curve]]s and curves in ''n''-dimensional [[Euclidean space]].

The point where the tangent line and the curve meet or [[intersection (geometry)|intersect]] is called the '''''point of tangency'''''. The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
The tangent line to a point on a differentiable curve can also be thought of as a ''[[tangent line approximation]]'', the graph of the [[affine function]] that best approximates the original function at the given point.<ref>Dan Sloughter (2000) . "[https://math.dartmouth.edu/opencalc2/dcsbook/c3pdf/sec31.pdf Best Affine Approximations]"</ref> 

Similarly, the '''tangent plane''' to a [[Surface (topology)|surface]] at a given point is the [[Plane (mathematics)|plane]] that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in [[differential geometry]] and has been extensively generalized; {{Crossreference|see [[Tangent space]]}}.

The word "tangent" comes from the [[Latin]] {{lang|la|[[wikt:en:tangere#Latin|tangere]]}}, "to touch".