1. A curve called the folium of Descartes is defined by the parametric equations 3t ¤(t) = 3t2 y(t): 1+ t3 ' 1+t3 (a) Show that if the point (a,b) lies on this curve, then so does (b, a); that is, the curve is symmetric with respect to the line y = x. Where does the curve intersect this line? Hint. Assume that (a, b) is the point at "time" tı and consider another time moment t2 = 1/t1. (b) Find the points on the curve where the tangent lines are horizontal or vertical. (c) Analyze the behavior of the curve when t → -1 from the left and from the right. Also consider cases t → ±x∞. (d) Show that the line x + y +1= 0 is an asymptote. Hint: Use the previous point.
1. A curve called the folium of Descartes is defined by the parametric equations 3t ¤(t) = 3t2 y(t): 1+ t3 ' 1+t3 (a) Show that if the point (a,b) lies on this curve, then so does (b, a); that is, the curve is symmetric with respect to the line y = x. Where does the curve intersect this line? Hint. Assume that (a, b) is the point at "time" tı and consider another time moment t2 = 1/t1. (b) Find the points on the curve where the tangent lines are horizontal or vertical. (c) Analyze the behavior of the curve when t → -1 from the left and from the right. Also consider cases t → ±x∞. (d) Show that the line x + y +1= 0 is an asymptote. Hint: Use the previous point.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. A curve called the folium of Descartes is defined by the parametric equations
3t
3t2
¤(t) :
1+ t3 '
y(t)
1+ t3
(a) Show that if the point (a, b) lies on this curve, then so does (b, a); that is, the curve is
symmetric with respect to the line y = x. Where does the curve intersect this line? Hint:
Assume that (a, b) is the point at "time" tı and consider another time moment t2 = 1/t1.
%3D
(b) Find the points on the curve where the tangent lines are horizontal or vertical.
(c) Analyze the behavior of the curve when t → -1 from the left and from the right. Also
consider cases t → ±o.
(d) Show that the line x +y +1=0 is an asymptote. Hint: Use the previous point.
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