secant

The word secant ( Latin : secare = "cut") describes a straight line in planar geometry and analysis that goes through two points on a curve .

Circular edge

Three layers of straight lines to form a circle: secant, tangent , passer-by

In elementary geometry , a secant is a straight line that intersects a circle at two points . A straight line that has exactly one point in common with the circle is called a tangent ; a straight line that has no point in common with the circle is called a passer-by . A secant that goes through the center of the circle is called the center .

A straight line is a secant of a given circle if and only if the distance between the center of the circle and the straight line is smaller than the radius of the circle. If the distance is equal to the radius, it is a tangent; if it is larger than the radius, it is a passerby.

The section of the secant that lies within the circle is called the tendon . The longest chords of a circle are those that go through the center of the circle . These and their lengths are called the diameter of the circle.

The secant theorem describes the relationship between the lengths of two segments of a circle that intersect outside the circle, the secant-tangent theorem describes the relationship between the intersecting tangent and the secant.

Edge of the curve

Secant through two points of a function graph

For x ₁ against x ₀ , the secant approaches the tangent at x ₀

In more general terms, a straight line that runs through (at least) two points on a curve , for example a function graph , is called a secant. Its slope is called the secant slope . The slope of the secant through two points and the graph of the function is given by ${\displaystyle (x_{0}|f(x_{0}))}$ ${\displaystyle (x_{1}|f(x_{1}))}$ ${\displaystyle f}$

${\displaystyle m_{\text{S}}={\frac {f(x_{1})-f(x_{0})}{x_{1}-x_{0}}}}$.

This is precisely the difference quotient of the function in the interval . It plays an important role in differential calculus when defining the derivative: if you hold the position and let the position “wander” against , the secant of a differentiable function approaches the points and the tangent approaches the function graph in the point . The secant slope converges to the slope of the tangent, which is the derivative of the function at the point . ${\displaystyle f}$ ${\displaystyle [x_{0},x_{1}]}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{0}}$ ${\displaystyle f}$ ${\displaystyle (x_{0}|f(x_{0}))}$ ${\displaystyle (x_{1}|f(x_{1}))}$ ${\displaystyle (x_{0}|f(x_{0}))}$ ${\displaystyle f'(x_{0})}$ ${\displaystyle f}$ ${\displaystyle x_{0}}$

The secant method is a numerical approximation method for determining a zero with the aid of curve edges.

Web links

Wiktionary: secant  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : Secant . In: MathWorld (English).
• From the secant to the tangent (animation in which a curve edge can be changed to the tangent with sliders)