. Differential and integral calculus. dxdu 4 x* — 3 x2! o27 j0i7o Ty_ 0,0 dx 12 X2— 6 x~dx 0 dy 2~dx0,0 lt* IdyV dy , . ••• Uj = °- •••^=±°>attheongm. Therefore, at the origin (o, o), the curve has two tangentscoinciding with the .ar-axis. From the equation of the curve, we have, y = ± y/x3(i — x). Hence, since x cannot be negative, the curve is situated inthe first and fourth quadrants, and is symmetrical with respectto the #-axis. Hence the origin is a cusp of the first species. 4. Show that the cissoid (2 a — x)y2 = Xs has a cusp of thefirst species at the origin. 5. Show that (y — x2)2
. Differential and integral calculus. dxdu 4 x* — 3 x2! o27 j0i7o Ty_ 0,0 dx 12 X2— 6 x~dx 0 dy 2~dx0,0 lt* IdyV dy , . ••• Uj = °- •••^=±°>attheongm. Therefore, at the origin (o, o), the curve has two tangentscoinciding with the .ar-axis. From the equation of the curve, we have, y = ± y/x3(i — x). Hence, since x cannot be negative, the curve is situated inthe first and fourth quadrants, and is symmetrical with respectto the #-axis. Hence the origin is a cusp of the first species. 4. Show that the cissoid (2 a — x)y2 = Xs has a cusp of thefirst species at the origin. 5. Show that (y — x2)2 — xb has a cusp of second species atthe origin. 6. Show that the semi-cubic parabola ay1 = xs has a cusp ofthe first species at the origin. Singular Points 215 7. Show that the cycloid x = a vers-1 - — V2 ay — f has an J a infinite number of cusps of the first species. 8. Show that the curve a (x2 + y*) = xs has a conjugatepoint at the origin. u = ax2 -f- af- — xs dudx du = 2 ax Sx2 = o. dy = 2 .. (o, o) is a critical point. dy dx 3* 2 ay ;~| _ 6 jp — 2 rt- -^ 2a£ i, or -f = ± Fig. 38 I Hence, the origin is a conjugate point. See (e) § , thus: Solving the equation, we have, y = ± x i This equation is satisfied for the point (o, o). But y isimaginary for any negative value of x and for any positivevalue of x less than a; hence (o, o) is isolated from the curve. 9. The curve y1 = x (x + df has a conjugate point at(— a, o). 10. The origin is a conjugate point of the curve y2 (x2 — a2)= x*. 11. Show that the point (a, o) is a conjugate point of thecurve ay — x3 -f- 4 ax2 — 5 a2x -j- 2 a3 = o. 216 Differential Calculus 12. Show that the curve azf — 2 aboPy — x5 = o has a doublepoint of osculation at the origin and that one branch of thecurve has a point of inflexion at that point. 13. Show that the curve y = x cot-1 x has a point saillantat the origin. Since y is positive, and has only onevalue for any value of x,
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
