1. The Folium of Descartes is an implicitly-defined curve, significant in the history of mathematics. (Seethe Learning Resources on Section 4.1 for more.) It is defined as the set of all pairs (r, y) that solve theequationr³-3ary+y³ = 0,for a given choice of parameter a. A figure is provided below.420-2T-2024a. Compute y' in terms of a, z and y.b. There are two points on the curve where y' = 0. One is at (0,0). Find the other, in terms of theparameter a.c. Compute the second and third derivatives of y in terms of a, z, y and lower-order derivatives of y.d. Hence find the first three non-zero terms in the Taylor approximation to y around the point youidentified in part (b).

As per the company rule, we are supposed to solve the first three sub-parts of a problem. Therefore…

a. Compute y' in terms of a, z and y. b. There are two points on the curve where y' = 0. One is at (0,0). Find the other, in terms of the parameter a. c. Compute the second and third derivatives of y in terms of a, z, y and lower-order derivatives of y.