# Tangent line

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

to a curve

A straight line representing the limiting position of the secants. Let be a point on a curve (Fig. a). A second point is chosen on and the straight line is drawn. The point is regarded as fixed, and approaches along the curve . If, as goes to , the line tends to a limiting line , then is called the tangent to at .

Figure: t092170a

Figure: t092170b

Not every continuous curve has a tangent, since need not tend to a limiting position at all, or it may tend to two distinct limiting positions as tends to from different sides of (Fig. b). If a curve in the plane with rectangular coordinates is defined by the equation and is differentiable at the point , then the slope of the tangent at is equal to the value of the derivative at ; the equation of the tangent at this point has the form

The equation of the tangent to a curve in space is

By a tangent to a surface at a point one means a straight line passing through and lying in the tangent plane to at .