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In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method , so it is considered a quasi-Newton method .
As Q approaches P along the curve, if the slope of the secant approaches a limit value, then that limit defines the slope of the tangent line at P. [1] The secant lines PQ are the approximations to the tangent line. In calculus, this idea is the geometric definition of the derivative. The tangent line at point P is a secant line of the curve
The slope of this line is (+) (). This formula is known as the symmetric difference quotient. In this case the first-order errors cancel, so the slope of these secant lines differ from the slope of the tangent line by an amount that is approximately proportional to .
Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
Geometrically, this difference quotient measures the slope of the secant line passing through the points with coordinates (a, f(a)) and (b, f(b)). [10] Difference quotients are used as approximations in numerical differentiation, [8] but they have also been subject of criticism in this application. [11]
If the two points that the secant line goes through are close together, then the secant line closely resembles the tangent line, and, as a result, its slope is also very similar: The dotted line goes through the points ( 2 , 4 ) {\displaystyle (2,4)} and ( 3 , 9 ) {\displaystyle (3,9)} , which both lie on the curve y = x 2 {\displaystyle y=x^{2}} .
Domain of tangent and secant : The domains of and are the same. They are the set of all angles θ {\displaystyle \theta } at which cos θ ≠ 0 , {\displaystyle \cos \theta \neq 0,}