. A new treatise on the elements of the differential and integral calculus . towards the line, it is said to be concave, or to have its con-cavity turned towards the line; but, if the sense in which itbends from the tangent is from the line, it is said to be convex,or to have its convexity turned towards the line. To find the conditions of the concavity or convexity of acurve towards a given line, take that line for the axis of x,and let P, of which the co-ordinates are x and y, be the pointat which the curve is to be examined with reference to theseproperties. Draw thetangent at F: then,from
. A new treatise on the elements of the differential and integral calculus . towards the line, it is said to be concave, or to have its con-cavity turned towards the line; but, if the sense in which itbends from the tangent is from the line, it is said to be convex,or to have its convexity turned towards the line. To find the conditions of the concavity or convexity of acurve towards a given line, take that line for the axis of x,and let P, of which the co-ordinates are x and y, be the pointat which the curve is to be examined with reference to theseproperties. Draw thetangent at F: then,from our definition, ifat P the curve be con-vex to the axis of x,the ordinates of thecurve for the abscissaex -\- h, X — 7i, must begreater than the corresponding ordinates of the tangent at P;h having any value between some small but finite limit andzero. But, if the curve be concave towards the axis of x, thereverse must be the case. If the equation of the curve is ?/ = F(x), the ordinate cor-responding to the abscissa x -\- h is 7 2 y + Ay = F{x) + JiF\x)+f:, F-[x) + .... + F^\x-\-Oh). The equation of the tangent to the curve at the point {X, i/)is 2/1 - y = F^(x) {x,-x), or y, = A-r) -f .r^ F\x)-xr{x).Observing that .r, //, are tlie co-ordinates of the point of tan- 264 DIFFERENTIAL CALCULUS. gencj, the ordinate of tlie tangent corresponding to tlie ab-scissa x-\-hh Vi + ^y\ — F{x) + xF{x) + liF{x) — xF{x)— F{x)^liF{x): hence, if 5 denote the difference y A^ Ly — (y^-f~ ^^i); ^^have « = ^ F{x) + ... + ~^^ F^\x + eh). The sign of this difference, when 7i is very small, is the same as that of —-F^i^x), which has the sign of F{x) whether h be positive or negative : therefore, if F{x) be positive, thecurve is convex to the axis oi x; and it is concave if F[x) benegative. We have supposed the point of the curve at which its con-vexity or concavity was examined to be above the axis of ic,or to have a positive ordinate. Had the point been below theaxis, F {x) posi
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