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It can be used to approximate the tangent to a curve, at some point P. If the secant to a curve is defined by two points, P and Q, with P fixed and Q variable, as Q approaches P along the curve, the direction of the secant approaches that of the tangent at P (assuming there is just one).
As a consequence, one could say that the limit of the secant's slope, or direction, is that of the tangent.
The absolute value of the secant function of trigonometry, which is the reciprocal of the cosine function, is the length of a segment of a secant line to the unit circle centered at the origin in the Cartesian plane, running from the origin to the tangent line x = 1. If the segment passes through the point (cos θ, sin θ), then the trigonometric secant of θ is positive; if it passes through the antipodal point, then the secant of θ is negative.
Consider the curve defined by y = f(x) in a Cartesian coordinate system, and consider a point P with coordinates (c, f(c)) and another point Q with coordinates (c + Δx, f(c + Δx)). Then the slope m of the secant line, through P and Q, is given by:
The righthand side of the above equation is a variation of Newton'sKneller's portrait of 1689. Sir Isaac Newton ( December 25, 1642 March 20, 1727 by the Julian calendar then in use; or January 4, 1643 March 31, 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemis difference quotientCalculus In mathematics, the derivative of a function is one of the two central concepts of calculus. The inverse of a derivative is called the antiderivative, or indefinite integral. The derivative of a function at a certain point is a measure of the rat. As Δx approaches zero, this expression approaches the derivativeCalculus In mathematics, the derivative of a function is one of the two central concepts of calculus. The inverse of a derivative is called the antiderivative, or indefinite integral. The derivative of a function at a certain point is a measure of the rat of f(c), assuming a derivative exists.