MATH 1920 SUMMER 2009 PRELIM 2
Please write your name on all of the exam booklets you use. Show
all your work and put all your work in the exam booklet. Circle your final
answers and be sure that you have explained them in detail. No calculators
are permitted. Good luck!
1. (8 pts) Consider the function f(x, y) = ln xy + ln yz + ln xz.
(a) Find the directions in which f(x, y, z) increases and decreases most
rapidly at P0(1,1,1).
(b) Find the derivatives of f(x, y, z) in these directions.
2. (7 pts) Find parametric equations for the line tangent to the curve of
intersection of the surfaces xyz = 1 and x2+ 2y2+ 3z2= 6 at the point
(1,1,1).
3. (8 pts) Find all critical points of the function f(x, y) = x2y+xy2−3xy
and determine their nature. Justify your answer.
4. (7 pts) Find the points on the surface z2=xy + 4 closest to the origin.
5. (8 pts) Find the volume of the solid that is bounded above by the surface
z=x2exy and below by the triangular region Rin the xy-plane enclosed
by the lines y=x,y= 0, and x= 1.
6. (7 pts) Evaluate the double integral
Z2
0Z√1−(x−1)2
0
y
x2+y2dy dx.
7. (9 pts) Let Dbe the region in the first octant that is bounded below by
the cone z=px2+y2and above by the sphere x2+y2+z2= 9.
(a) Express the volume of Das an iterated triple integral in
(i) spherical,
(ii) cylindrical, and
(iii) rectangular coordinates.
(b) Find the volume of D.
8. (6 pts) Evaluate ZC
(x+y+z)ds, where Cis the straight line segment
from (1,2,3) to (0,−1,1).