problem 1: Give a complete description (in words) of the surface (x - 1)^{2} + y^{2 }+ 4(z- 6)^{2}= 16.
problem 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).
problem 3: Define f(x, y) = (x^{3}y)/(x^{6}+ y^{2})
Investigate limf(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.