Tangenten

Tangenten are lines that touch a curve at a point without crossing it, and that match the curve’s direction to first order at the point of contact. In the plane, a tangent line to a differentiable curve at a point P is the unique line that has the same slope as the curve’s graph at P; equivalently, it is the limit of secant lines as another point on the curve approaches P.

Algebraically, if the curve is given as the graph y = f(x) and x0 is a point in its domain, the tangent line at x0 is y = f(x0) + f′(x0)(x − x0). If the curve is defined implicitly by F(x, y) = 0, with gradient ∇F(x0, y0) ≠ (0, 0), the tangent line at (x0, y0) is ∂F/∂x(x0, y0)(x − x0) + ∂F/∂y(x0, y0)(y − y0) = 0.

For a circle with center (h, k) and radius r, the tangent at a point (x1, y1) on the circle is perpendicular to the radius to that point; in Cartesian form it can be written as (x1 − h)(x − x1) + (y1 − k)(y − y1) = 0.

Tangents are central to calculus as the linearization of a function at a point. They provide the first-order approximation y ≈ y0 + f′(x0)(x − x0), and they underpin many geometric constructions, such as finding the point of tangency to a given curve or determining the slope of a curve at that point.

In German, Tangens denotes the tangent function, a different concept from tangent lines, though related by historical origins of the word.