Cusps and noncusps a. Graph the curve r1t2 = 8t3, t39.

Chapter 11, Problem 89

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Cusps and noncusps a. Graph the curve r1t2 = 8t3, t39. Show that r_102 = 0 and the curve does not have a cusp at t = 0. Explain. b. Graph the curve r1t2 = 8t3, t 29. Show that r_102 = 0 and the curve has a cusp at t = 0. Explain. c. The functions r1t2 = 8t, t 29 and p1t2 = 8t 2, t49 both satisfy y = x 2. Explain how the curves they parameterize are different. d. Consider the curve r1t2 = 8tm, tn9, where m 7 1 and n 7 1 are integers with no common factors. Is it true that the curve has a cusp at t = 0 if one (not both) of m and n is even? Explain. 9

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