View source for Tangent line

''to a curve''

{{MSC|53|58}}
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$\newcommand{\vect}[1]{\mathbf{#1}}$
<span id="Fig1">
[[File:Tangent-line-1.png| right| frame| Figure 1. The tangent $MT$ (in red) to the line $L$ at the point $M$ ([[Media:Tangent-line-1.pdf|pdf]]) ]]
</span>
The tangent line to a curve is
a straight line representing the limiting position of the secants. Let $M$ be a point on a curve $L$ ([[#Fig1|Fig.&nbsp;1]]). A second point $M_1$ is chosen on $L$ and the straight line $M_1$ is drawn. The point $M$ is regarded as fixed, and $M_1$ approaches $M$ along the curve $L$. If, as $M_1$ goes to $M$, the line $MM_1$ tends to a limiting line $MT$, then $MT$ is called the tangent to $L$ at $M$.

<span id="Fig2">
[[File:Tangent-line-2.png| right| frame| Figure 2. A curve $L$ and point $M$ on it without a tangent ([[Media:Tangent-line-2.pdf|pdf]]) ]]
</span>
Not every continuous curve has a tangent, since $MM_1$ need not tend to a limiting position at all, or it may tend to two distinct limiting positions as $M_1$ tends to $M$ from different sides of $L$ ([[#Fig2|Fig.&nbsp;2]]). 

If a curve in the plane with rectangular coordinates is defined by the equation $y=f(x)$ and $f$ is differentiable at the point $x_0$, then the slope of the tangent at $M$ is equal to the value of the derivative $f^\prime(x_0)$ at $x_0$; the equation of the tangent at this point has the form
$$
y - f(x_0) = f^\prime(x_0)(x - x_0).
$$
The equation of the tangent to a curve $\vect{r} = \vect{r}(t)$ in space is
$$
\vect{t}(\lambda) = 
\vect{r} + \lambda \frac{\mathrm{d}\vect{r}}{\mathrm{d}t}, \quad
\lambda \in \R.
$$

By a tangent to a surface $S$ at a point $M$ one means a straight line passing through $M$ and lying in the [[Tangent plane|tangent plane]] to $S$ at $M$.

====References==== 

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|valign="top"|{{Ref|BeGo}}||valign="top"| M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces", Springer (1988) (Translated from French)  {{MR|0917479}}  {{ZBL|0629.53001}} 
|-
|valign="top"|{{Ref|Co}}||valign="top"| H.S.M. Coxeter, "Introduction to geometry", Wiley (1961)  {{MR|1531486}} {{MR|0123930}}  {{ZBL|0095.34502}} 
|-
|valign="top"|{{Ref|Gu}}||valign="top"| H.W.  Guggenheimer, "Differential geometry", McGraw-Hill (1963)  {{MR|0156266}}  {{ZBL|0116.13402}} 
|-
|valign="top"|{{Ref|HiCo}}||valign="top"| D.  Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination", Chelsea (1952) (Translated from German)  {{MR|0046650}}  {{ZBL|0047.38806}} 
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