Math 53 Midterm #1, 10/8/07, 3:10 PM – 4:00 PM
(please do not leave the exam between 3:50 and 4:00)
No calculators or notes are permitted. Each of the 5 questions is worth
10 points. Please write your solution to each of the 5 questions on a separate
sheet of paper with your name, SID number, and GSI’s name on it. For each
question, to get full credit, you must put a box around your final answer and
show correct work or justification. Good luck!
1. Find the area of the region enclosed by the polar curves r= 5 sec θand
θ=−π/4 and θ=π/4.
2. Find the tangent plane to the surface
z=9
x+y
at the point (1,2,3). Write your answer as an equation of the form
ax +by +cz =d.
3. Does the following limit exist? If so, what is it? Justify your answer.
lim
(x,y)→(0,0)
px2+y2+xy2
px2+y2
4. The surfaces x2+y2= 2 and y=zintersect in a curve C. Find a unit
tangent vector to the curve Cat the point (1,1,1).
5. Let r(t) be a vector-valued function of t. Suppose that r(0) = h2,2,1i
and r0(0) = h1,1,2i. Compute the derivative
d
dtkr(t)k
t=0.