. Differential and integral calculus, an introductory course for colleges and engineering schools. Y V \ vi p^ \ / A £ T| K / 0 X a b o Fig. 1. Fig. 2. gent to the curve. From the figures it can be seen that, if the arcconvex be , as this tangent rolls along the curve, the angle (concave) i • i • i • , ^r • ii \ increases ) , which it makes with OX continually , and that (decreases) .-. x . (an increasing ).,.,„ ^ therefore f (x) is function along the arc. From f a decreasing ) the figures it can be seen that the converse of this is also true, viz., increasing ) . ( convex ) , . \ throughout a
. Differential and integral calculus, an introductory course for colleges and engineering schools. Y V \ vi p^ \ / A £ T| K / 0 X a b o Fig. 1. Fig. 2. gent to the curve. From the figures it can be seen that, if the arcconvex be , as this tangent rolls along the curve, the angle (concave) i • i • i • , ^r • ii \ increases ) , which it makes with OX continually , and that (decreases) .-. x . (an increasing ).,.,„ ^ therefore f (x) is function along the arc. From f a decreasing ) the figures it can be seen that the converse of this is also true, viz., increasing ) . ( convex ) , . \ throughout an arc, that arc ^s decreasing ) ( concave ) when viewed from below. Applying corollary 1 of theorem 3, Art. 54, we have Theorem 4- Iff(%) is ] [at every point of an arc, that arc If f(x) is IS convexconcave j when viewed from below. (convex ) arcs of a curve, y = fix). (concave) u Jy n Hence, to determine the concave we have only to determine the intervals within which fix) or y is 72 DIFFERENTIAL CALCULUS §56. 56. The Flex.* A flex is a point at whicn a curve changes fromconvex to concave or from concave to curve in the figure has flexes at A, B, and C. Since the curve bends in opposite direc-tions on opposite sides of theflex, it is geometrically evidentthat the flex-tangent crosses thecurve at the flex. There are cases in which thecurve is convex on one side of apoint and concave on the otherside, and yet the point is notconsidered to be a flex. For example, the tangent curve, y = tan x, changes from convex to concave at the points x = ± -, ± -^-, . . (see figure in Art. 8), but since the ordinates of these points areinfinite, that is, since tan x becomes discontinuous at these points,they are not classed as flexes. Again, the curve in the accompanying figure changes from con-vex to concave at P, and, moreover, has no discontinuity P is not regarded as a flex. The peculiarity is that/(a;) hasat P a discontinuity like the discon-tinuity at D
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