1
a
We nd critical points by setting
f =0
f = 4u3 4v, 4v 3 4u = 0
and so v = u3 and u = v 3 . Substituting the rst into the second, we get
u9 u = 0
which we can factor repeatedly.
u(u8 1) = u(u4 1)(u4
Exam 3 - Solutions
Spring 2012
1.
2. (a) Plot the region of integration in the xy-plane
2.5
2.0
1.5
+
y
x
y
=
2
1.0
0.5
0.0
0.0
0.5
1.0
1.5
2.0
2.5
x
(b)
u=x
y
v =x+y
u+v
2
v u
2y ! y =
2
u + v = 2x !
Exam 2 - Solutions
Spring 2012
1. f (u, v) =
u3
3
+
v3
3
v2
2
u+2
(a) Determine the location of all critical points of f (u, v).
f = u2 1, v 2 v = 0, 0
u2 = 1 u = 1
v 2 = v v = 0, 1
Critical points ar
Solution: APPM 2350
Exam 2
Summer 2014
1. (30 points) Suppose we want to calculate
y xy
e dA
x
R
Where R is the region in the rst quadrant bounded by the curves y =
1
4
x
, y = 2x, y = , and y = .
2
x
Solution: APPM 2350
Final
Summer 2014
1. (50 points) Suppose you nd yourself in a parallel universe where the local force due to gravity takes the
x 2
y 2
z 2
improbable form F = y, x + ez , yez . Con
APPM 2350
FINAL EXAM
SPRING 2014
INSTRUCTIONS: Electronic devices, books, and crib sheets are not permitted. Write your name and your instructors
name on the front of your bluebook. Work all problems.
APPM 2350
Spring 2012
Exam 1 Solutions
February 15, 2012
Problem 1
a
In order to determine the standard equation of the plane of intersection of two surfaces, we must nd a
relationship between x, y, a