Problem 41E

41. Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems.

(a) Graph the curve with equation




At how many points does this curve have horizontal tangents? Estimate the $x$ -coordinates of these points.

(b) Find equations of the tangent lines at the points $(0,1)$ and $(0,2)$.

(c) Find the exact $x$ -coordinates of the points in part (a).

(d) Create even more fanciful curves by modifying the equation in part (a).


Step-by-Step Solution