1. [V] In this problem we will make a picture of the surface y = x 2 in R 3 as follows. The approach below can be used to plot any surface in space whose equation is independent of one of the variables, (x, y, z).

(a) In R 2 the equation y = x 2 represents a curve in the plane. Sketch it.

(b) However, the set of points (x, y, z) in R 3 for which y = x 2 represents a surface in space. Why do we expect an equation of the form F (x, y, z) = 0 to be a surface (i.e., a curved sheet) in space? [Hint: Imagine you could manipulate the equation F (x, y, z) = 0 to express one of the variables as a function of the other two.] For y = x 2 what is the function F ?

(c) To visualize this surface, translate the curve your sketched in (a) up and down along the z-axis to form a surface.

(d) What would you need to do to make a picture of the surface x 2 + 4z 2 = 1?