MTH 101 Study Guide - Final Guide: Quotient Rule, Cartesian Coordinate System
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Spring 2021: (a) answer: in order to determine the signs of x(t) and y(t) as t varies, it helps to make a table: t. 1 + t3 x(t) y(t) ( , 1) ( 1, 0) (0, ) In particular, for both x(t) and y(t) to be positive, t must lie on the interval (0, ). Also notice that x(t) = y(t) = 0 occurs precisely when t = 0. From this analysis, we can deduce that the loop of the folium is realized as t varies over the interval t 0. (b) answer: observe. 3(1 2t3) (1 + t3)2 , dy dt. 3t(2 t3) (1 + t3)2 . dy dx dy/dt dx/dt. = 1. (c) answer: let (x, y) be any point on the folium. If t = 0, then x = y = 0 and clearly the point (0, 0) is symmetric about the line y = x. Next, assume t 6= 0 (and t 6= 1).
