Summary: Math 2200 Homework 6 (Finish by: Friday October 29)
Remember to show your work!
Problem 1 Two planes are parallel if they fail to intersect. Find the tangent plane to
the sphere x2
= 6 at (1,-2,1). Find another point of tangency so that the tangent
plane there is parallel to the first one you found.
Solution: Use implicit differentiation:
2x + 2zzx = 0 and 2y + 2zzy = 0
2(1) + 2(1)zx = 0 and 2(-2) + 2(1)zy = 0
zx = -1 and zy = 2
so z - 1 = -(x - 1) + 2(y + 2) and the first tangent plane is z = -x + 2y + 6.
The other point of tangency would be on the opposite side of the sphere at (-1,2,-1),
yielding a parallel tangent plane.
Problem 2 For each of the following functions find the critical points and use the second
derivative test to classify them as maxima, minima, or saddle points. Find also the value
of the function at the critical point.
(i) f(x, y) = x2