tangenttisuora

Tangenttisuora is the straight line that touches a plane curve at a point and, at that point, shares the same direction as the curve. In mathematical terms, if a curve is differentiable at a point, the tangenttisuora exists and is unique.

For a function y = f(x), the tangenttisuora at x0 has the equation y = f(x0) + f'(x0)(x − x0). The slope of the line is the derivative f'(x0), and the line provides a linear approximation to the function near x0: f(x) ≈ f(x0) + f'(x0)(x − x0).

For a curve given implicitly by F(x, y) = 0, the tangenttisuora at a point (x0, y0) with ∇F(x0, y0) ≠ 0 is given by ∂F/∂x (x0, y0) (x − x0) + ∂F/∂y (x0, y0) (y − y0) = 0. Thus the tangent line is perpendicular to the gradient of F at the point of tangency.

In the special case of a circle with equation x^2 + y^2 = r^2, the tangenttisuora at a point (x1, y1) on the circle is x x1 + y y1 = r^2. This line is perpendicular to the radius from the origin to (x1, y1).

Applications of tangenttisuora include linear approximation, curve sketching, and solving optimization problems where a local linear model around a point is useful. Example: the tangent to y = x^2 at x0 = 2 is y = 4x − 4, since f(2) = 4 and f'(2) = 4.