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Difference between revisions of "Tangent line"

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[[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.pdf|pdf]]) ]]
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[[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]]) ]]
 
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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 $M$.
 
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">
 
<span id="Fig2">
[[File:Tangent-line-2.png| right| frame| Figure 2. A point $M$ on a curve $L$ without a tangent([[Media:Tangent-line.pdf|pdf]]) ]]
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[[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]]) ]]
 
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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
 
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

Revision as of 19:39, 24 April 2012


to a curve $ \newcommand{\vect}[1]{\mathbf{#1}} $


Figure 1. The tangent $MT$ (in red) to the line $L$ at the point $M$ (pdf)

A straight line representing the limiting position of the secants. Let $M$ be a point on a curve $L$ (Fig. 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 $M$.

Figure 2. A curve $L$ and point $M$ on it without a tangent (pdf)

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 $L$ (Fig. 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 to $S$ at $M$.

References

[BeGo] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces", Springer (1988) (Translated from French)
[Co] H.S.M. Coxeter, "Introduction to geometry", Wiley (1961)
[Gu] H.W. Guggenheimer, "Differential geometry", McGraw-Hill (1963)
[HiCo] D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination", Chelsea (1952) (Translated from German)
How to Cite This Entry:
Tangent line. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_line&oldid=25291
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article