. Differential and integral calculus. Fig. 43- 5. y 4- x2y = x. Fig. 44- 6. xz — 2 x?y — 2 x2 — 8y = o.. Fig. 45. Loci 223 7. y {x — a) = x (x — 2 a). 8. af = x4 4- x5. 9. y (a — x) = x2 (a + x). 10. y = x2 (1 — jt)3. 11. y*(*2 a2) = x 12. r3 = rt;3 — ji:3. 13. y (a2 — x2) = a3. POLAR EQUATIONS.156. Suggestions. 1. Determine as far as possible the form and properties ofthe curve from its equation. 2. Deduce the first derivative of r with respect to 0 fromthe equation of the curve. (a) Investigate for asymptotes, Cf. § 79. (/>) Investigate for maximum and minimum points. Cf. § 112.(V) Inve

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. Differential and integral calculus. Fig. 43- 5. y 4- x2y = x. Fig. 44- 6. xz — 2 x?y — 2 x2 — 8y = o.. Fig. 45. Loci 223 7. y {x — a) = x (x — 2 a). 8. af = x4 4- x5. 9. y (a — x) = x2 (a + x). 10. y = x2 (1 — jt)3. 11. y*(*2_ a2) = x 12. r3 = rt;3 — ji:3. 13. y (a2 — x2) = a3. POLAR EQUATIONS.156. Suggestions. 1. Determine as far as possible the form and properties ofthe curve from its equation. 2. Deduce the first derivative of r with respect to 0 fromthe equation of the curve. (a) Investigate for asymptotes, Cf. § 79. (/>) Investigate for maximum and minimum points. Cf. § 112.(V) Investigate for points of inflexion. Cf. § 131.(d) Investigate for direction of curvature. Cf. § 130. EXAMPLES.1. Trace the curve r = a sin 3 6. r = o, when 0 = o°, 0 = 6o°, 0 = 1200, $ = 1800, etc.; hence, the curve repeatedly passes through the origin. r = a (a maximum value, since sin 3 0 cannot exceed unity)when 0 = 300, 0 = 1500. 6 = 2700. r = — a, a minimum value, when # = —30°, 0 = —1500, 0=-27O°. As 6 increases from o° to 300, r increases from o to a as 6increases from 3