Tangentlína

Tangentlína is the tangent line to a curve at a specified point. It is a straight line that touches the curve at that point and, at the contact point, shares the same slope as the curve. In calculus, tangentlína provides the first-order, or linear, approximation of the curve near the contact point.

For a differentiable function y = f(x), the tangentlína at x = a has equation y = f(a) + f'(a)(x - a). For a parametric curve r(t) = (x(t), y(t)), the tangentlína at t0 uses the tangent vector r'(t0) and can be written as r(t0) + s r'(t0). For an implicit curve F(x,y) = 0 with ∇F(x0,y0) ≠ 0, the tangentlína at (x0, y0) is given by F_x(x0,y0)(x - x0) + F_y(x0,y0)(y - y0) = 0.

Examples: the tangentlína to y = x^2 at a = 2 is y = 4x - 4. The tangentlína to the circle x^2 + y^2 = 1 at (1,0) is the vertical line x = 1.

Applications include local linearization, approximation of functions, and the analysis of curves in geometry, physics, and engineering. The concept is a fundamental tool in calculus and analytic geometry, enabling a linear view of nonlinear relations over small intervals. Tangentlína, as the geometric analogue of the derivative, captures the instantaneous rate of change and direction of motion of a curve at a given point.