A curve called the folium of Descartes is defined by the parametric equations
- (a) Show that if (a, b) lies on the 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?
- (b) Find the points on the curve where the tangent lines are horizontal or vertical.
- (c) Show that the line y = −x − 1 is a slant asymptote.
- (d) Sketch the curve.
- (e) Show that a Cartesian equation of this curve is x3 + y3 = 3xy.
- (f) Show that the polar equation can be written in the form
- (g) Find the area enclosed by the loop of this curve.
- (h) Show that the area of the loop is the same as the area that lies between the asymptote and the infinite branches of the curve. (Use a computer algebra system to evaluate the integral.)
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Single Variable Calculus: Early Transcendentals
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