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Difference between revisions of "Tangent line"

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''to a curve''
 
''to a curve''
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<span id="Fig1">
 
<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]]) ]]
 
[[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 $M$.
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The tangent line to a curve is
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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 $M$.
  
 
<span id="Fig2">
 
<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]]) ]]
 
[[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>
 
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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
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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]]).  
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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).
 
y - f(x_0) = f^\prime(x_0)(x - x_0).
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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)
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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}}
 
|-
 
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|valign="top"|{{Ref|Co}}||valign="top"| H.S.M. Coxeter, "Introduction to geometry", Wiley (1961)
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|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)
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|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)
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|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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|}
 
|}

Latest revision as of 00:50, 25 April 2012

to a curve

2020 Mathematics Subject Classification: Primary: 53-XX Secondary: 58-XX [MSN][ZBL]

$\newcommand{\vect}[1]{\mathbf{#1}}$

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

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 $L$ (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) MR0917479 Zbl 0629.53001
[Co] H.S.M. Coxeter, "Introduction to geometry", Wiley (1961) MR1531486 MR0123930 Zbl 0095.34502
[Gu] H.W. Guggenheimer, "Differential geometry", McGraw-Hill (1963) MR0156266 Zbl 0116.13402
[HiCo] D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination", Chelsea (1952) (Translated from German) MR0046650 Zbl 0047.38806
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