1. Give a complete description (in words) of the surface (x - 1)^{2} + y^{2} + 4(z - 6)^{2} = 16.

2. (a) Build an equation for a hyperboloid of two sheets with the following properties:

- The central axis of the hyperboloid is the x-axis

- The two sheets are 12 units apart (at closest), and are mirror images of each other across the yz-plane.

- The hyperboloid intersects the plane x = 10 in a circle of radius 4.

(b) Find a function y = f(x) whose graph in the xy-plane, when rotated around the x- axis, would produce the hyperboloid built in (a). (Hint: what equation is true about all points in the xy-plane?)

3. Define f(x, y) = (x^{3}y)/(x^{6} + y^{2})

Investigate lim f(x, y).

_{ (x,y)→ (0,0)}

(a) Show that the limit is zero when (0, 0) is approached along any line. (Consider x = 0 separately from y = mx.)

(b) Show that the limit is zero when (0, 0) is approached along any parabola of the form y = kx^{2}.

(c) Find an approach to (0, 0) where the limit is non-zero, thereby demonstrating that The overall limit doesn't exist

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