. An elementary treatise on the differential calculus founded on the method of rates or fluxions. preceding article for determining theexistence of an asymptote. In this case we have [rsin^-0)] ad2 6 sin(i - ey 6 = — ha. The angle 0 = 1 corresponds to 570 18, nearly, and sincethe expression for the perpendicular on the asymptote is neg-ative its direction is 6X + 900 = 1470 18; consequently, theasymptote is laid off as in Fig. 27. Numerically equal positive and negative values of 6 give thesame values for r; hence the curve is symmetrical with refer-ence to the initial line. While 6 varies fro
. An elementary treatise on the differential calculus founded on the method of rates or fluxions. preceding article for determining theexistence of an asymptote. In this case we have [rsin^-0)] ad2 6 sin(i - ey 6 = — ha. The angle 0 = 1 corresponds to 570 18, nearly, and sincethe expression for the perpendicular on the asymptote is neg-ative its direction is 6X + 900 = 1470 18; consequently, theasymptote is laid off as in Fig. 27. Numerically equal positive and negative values of 6 give thesame values for r; hence the curve is symmetrical with refer-ence to the initial line. While 6 varies from o to 1, r is negative and varies from Oto co, giving the infinite branch in the thirdquadrant. As 0 passes the value unity, and increasesindefinitely, r becomes positive and decreases,approaching indefinitely to the limiting valueay which we obtain from (1) by making 6 in-finite. Hence the curve describes an infinitenumber of whorls approaching indefinitely tothe circle r = a, which is therefore called an /asymptotic circle. The points of inflexion in this curve aredetermined in Art. Fig. 27. 272 CURVE TRACING. [Ex. XXIX. Examples XXIX. i. Trace the curve r = a cos3 \ 0. Show that, to describe the curve, 0 must vary from o to 37? ; alsothat the curve is symmetrical to the initial line. Find the values of 6which correspond to the maxima and minima ordinates and abscissas,the initial line being taken as the axis of x. 2. Trace the curve r — a (2 sin 0 — 3 sin30). Show that the entire curve is described while 0 varies from o to 7t,and that the curve is symmetrical with reference to a perpendicularto the initial line. 3. Trace the curve r = 2 + sin 30. A maximum value of r (equal to 3) occurs at 0 = 300 ; a mini-mum (equal to 1) at 0 = 900. The curve is symmetrical with refer-ence to lines inclined at the angles 300, 900, and 1500 to the initialline. 4. Trace the curve r = 1 + sin curve consists of five equal loops. 5. Trace the curve r3 = a5 sin 30. The curve
Size: 1235px × 2024px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1870, bookpublishernewyo, bookyear1879
