1 give a complete description in words of the

Question

1. Give a complete description (in words) of the surface (x - 1)² + y² + 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) =    (x³y)/(x⁶ + y²)

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 = kx².

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