tangenttiviivalla

Tangenttiviivalla is the Finnish term meaning “on the tangent line” and is used to refer to the tangent line of a curve at a point. In geometry and calculus, a tangent line is a straight line that touches a curve at a single point and shares the same instantaneous direction as the curve there.

For a differentiable function f, the tangent line to the graph of f at x = a has the equation y = f(a) + f'(a)(x − a). This line is the linear approximation to f near a and is the limit of the secant lines through (a, f(a)) as the second point approaches a. The slope of the tangent line equals the derivative f'(a).

Geometrically, the tangent line represents the best linear approximation to the curve at the point, and it locally describes the curve’s behavior to first order. At smooth points the tangent is unique; at non-smooth points, such as cusps, a tangent may not exist.

Special cases include curves given implicitly. If F(x, y) = 0 defines a curve and (x0, y0) is a point on it, the tangent line can be described by F_x(x0, y0)(x − x0) + F_y(x0, y0)(y − y0) = 0, provided the gradient is nonzero there. For a circle x^2 + y^2 = r^2, the tangent at a point is perpendicular to the radius to that point.

Examples: the tangent to y = x^2 at x = 1 is y = 2x − 1. For the circle x^2 + y^2 = 1 at (1, 0), the tangent is x = 1. Tangenttiviivalla thus denotes this fundamental concept in analytic geometry.