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Difference between revisions of "Tangent line"

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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 $M_M_1$ 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 $L$ (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 $M$.
  
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 $f'(x_0)$ 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 $L$ (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'(x_0)(x - x_0).
+
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
 
The equation of the tangent to a curve $\vect{r} = \vect{r}(t)$ in space is

Revision as of 19:29, 24 April 2012

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

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 $M$.

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 planeto $S$ at $M$.

References

[a1] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961)
[a3] H.W. Guggenheimer, "Differential geometry" , McGraw-Hill (1963)
[a4] D. Hilbert, S.E. Cohn-Vossen, "Geometry and the imagination" , Chelsea (1952) (Translated from German)
[a5] B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)
How to Cite This Entry:
Tangent line. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_line&oldid=25288
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article