MATH 3251, Midterm 3, Fall 1996
November 21, 1996.
50 points are divided between 4 problems (one page).
No books, no notes. Calculators are permitted.
1 (12 points). Compute 
for the function
2 (12 points). Let a function F(x,y,z) be defined by
. Suppose that the equation F(x,y,z)=0
implicitly defines each of three variables x,y, and z as
functions of the other two
.
Compute
3(a) (10 points). Find the tangent planes to each of two surfaces
at the same point P(1/2,1/4,1/4).
(b) (4 points). Find symmetric equations of the line of intersection of these two planes (which is nothing else but the tangent line to the curve defined as intersection of the above surfaces).
4 (12 points). Find all points on the plane curve 
different from (0,0) for which
Reminder:
The tangent plane to the level surface f(x,y,z)=k at a point 
is given by
or put otherwise, if is the position vector of
, then the tangent plane is given by
Answers for this midterm will be posted at
http://www.math.umn.edu/ 
krylov/Math
at about noon Nov 21