PlanetMath: tangent line

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tangent line

(Definition)

If the curve of -plane is sufficiently smooth in its point $(x_0,\,y_0)$ and in a neighborhood of this, the curve may have a tangent line (or simply tangent¹) in $(x_0,\,y_0)$ . Then the tangent line of the curve in the point $(x_0,\,y_0)$ is the limit position of the secant line through the two points $(x_0,\,y_0)$ and $(x,\,f(x))$ of the curve, when limitlessly tends to the value (i.e. $x\to x_0)$ . Due to the smoothness,

$\displaystyle f(x)\to f(x_0) = y_0,$

$\displaystyle (x,\,f(x))\to (x_0,\,y_0),$

and the slope

of the secant tends to

$\displaystyle \lim_{x\to x_0}\frac{f(x)\!-\!f(x_0)}{x\!-\!x_0} = f'(x_0)$

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 $(x_0,\,y_0)$ may intersect the curve in another point, and then the tangent is a secant, too. For example, the curve $y = x^3\!-\!3x^2$ has the line as its tangent in the point $(0,\,0)$ but this line cuts the curve also in the point $(3,\,0)$ .

Footnotes

...tangent ¹: The word is initially a participial form tangens (its genitive: tangentis) of the Latin verb tangere `to touch'.

"tangent line" is owned by Mathprof. [ owner history (1) ]

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See Also: curve, tangent of conic section, hyperbola

Other names:

tangent, tangent of the curve, tangent to the curve

Also defines:

tangency point

This object's parent.

Attachments:: tangent plane (elementary) (Topic) by rspuzio
limited tangent (Definition) by mathcam
normal line (Definition) by pahio
envelope (Definition) by pahio
tangent of circle (Definition) by pahio

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Cross-references: line, intersect, property, slope, secant line, limit, neighborhood, point, smooth, curve
There are 56 references to this entry.

This is version 9 of tangent line, born on 2004-11-22, modified 2007-05-10.
Object id is 6511, canonical name is TangentLine.
Accessed 11808 times total.

Classification:

AMS MSC:	26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems)
	26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions)

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