Secant right-hand side

In geometry, the relative position of two lines, or a line and a Curve, can be qualified by the adjective secant . This one comes from the Latin secare , which means to cross. In mathematical terms, a line is secant on another line, or more generally with a curve, when it has a nonempty intersection with this one.

To carry out the study of a curve in the vicinity of one of its points P , it is useful to consider the secants resulting from P , i.e. the lines passing by P and another point Q of the curve. It is starting from these secants that the concept is defined of tangent with the curve at the point P : it is about secant the line limit, when it exists, resulting from P when the second point Q approaches P along the curve.

So when Q is sufficiently close to P , the secant can be regarded as an approximation of the tangent.

In the particular case of the curve representative of a numerical function y=f (X) , the Pente of the tangent is the limiting of the slope of the secants, which gives a geometrical interpretation of the Dérivabilité of a function.

Bond between the concepts of secant function and secant line

Let us consider a reality θ. Let us draw a secant line with the Cercle unit (centered with the origin) which passes by the origin and the point (cos θ, sin θ), not of the circle whose vector image forms an angle θ with the directing vector of the x-axis. The absolute Value of the secant trigonometrical of θ is equal to the length of the segment secant line energy of the origin until the line of equation X = 1. If the segment passes by the point (cos θ, sin θ), then the trigonometrical secant of θ is positive, if it passes by the antipodean Point, then the secant of θ is negative.

Approximation by a secant

Let us consider the curve of equation there = F ( X ) in a Cartesian Frame of reference, and consider a point P of coordinated ( C , F ( C )), and another point Q of coordinates ( C + Δ X , F ( C + Δ X )). Then the Slope m secant line, not passing P and Q , is given by:

$m = \ frac {\ Delta there} {\ Delta X} = \ frac {F (C + \ Delta X) - F (c)} {(C + \ Delta X) - C} = \ frac {F (C + \ Delta X) - F (c)} {\ Delta X}$

The member of right-hand side of the preceding equation is the report/ratio of Newton in C (or rate of increase). When Δ X approaches zero, this report/ratio approaches the derived number F ( C ), by supposing the existence of the derivative.

See too

Infinitesimal calculus
Derivation

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