Math 53 Final, 12/18/03, 8:00 AM – 11:00 AM
No calculators or notes are permitted. Each of the 12 questions is worth
10 points. Please write your solution to each of the 12 questions on a separate
sheet of paper with your name and your TA’s name on it. Please put a box
around the final answer. To maximize credit, please show your work, and if
you have extra time, double check that you got the correct answer and didn’t
misunderstand the question. Good luck!
1. Let r(t) be a parametrized curve in the x, y plane satisfying r(0) = h1,2i
and r0(0) = h3,4i. Let f(x, y) = exy. Calculate
d
dtf(r(t))
t=0.
2. (a) Find a normal vector to the surface x2+y2+z2= 14 at the point
h3,2,1i.
(b) Find an equation for the tangent plane to the surface z=x2−y2
at the point h2,1,3i.
3. Find the absolute minimum and maximum values of f=x2−4x+y2
subject to the constraint x2+ 2y2≤1.
4. Calculate R RSF·dS, where Sis the portion of the paraboloid
z= 9 −x2−y2
with z≥0, oriented using the upward pointing normal, and
F=hx, y, zi.
5. Calculate Z2
−2Z4
y2
ysin(x2)dx dy.
6. Find the area of the part of the cone z2=x2+y2between the planes
z= 1 and z= 2.
7. Calculate the volume of the region consisting of all points that are
inside the sphere x2+y2+z2= 4, below the cone z=px2+y2, and
above the cone z=−px2+y2.
