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The Folium of Descartes is an implicitly-defined curve, significant in the history of mathematics. (See he Learning Resources on Section 4.1 for more.) It is defined as the set of all pairs (r, y) that solve the

The Folium of Descartes is an implicitly-defined curve, significant in the history of mathematics. (See he Learning Resources on Section 4.1 for more.) It is defined as the set of all pairs (r, y) that solve the

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 38E
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1. The Folium of Descartes is an implicitly-defined curve, significant in the history of mathematics. (See the Learning Resources on Section 4.1 for more.) It is defined as the set of all pairs (x, y) that solve the equation r³3-3axy + y³ = 0, for a given choice of parameter a. A figure is provided below. 4 2 0 Z- セー -4 -2 0 2 4 In your answers to the questions that follow, please focus on explaining your reasoning carefully: your e-assessment (see below) is aimed at checking whether you can get to the right answers. a. Compute y' in terms of a, x 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. d. Hence find the first three non-zero terms in the Taylor approximation to y around the point you identified in part (b).
Transcribed Image Text:1. The Folium of Descartes is an implicitly-defined curve, significant in the history of mathematics. (See the Learning Resources on Section 4.1 for more.) It is defined as the set of all pairs (x, y) that solve the equation r³3-3axy + y³ = 0, for a given choice of parameter a. A figure is provided below. 4 2 0 Z- セー -4 -2 0 2 4 In your answers to the questions that follow, please focus on explaining your reasoning carefully: your e-assessment (see below) is aimed at checking whether you can get to the right answers. a. Compute y' in terms of a, x 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. d. Hence find the first three non-zero terms in the Taylor approximation to y around the point you identified in part (b).
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