MATH 082, EXAM 3b (solutions), 11 APRIL, 2008, NAME:
(Each of the following four problems is worth 15 points.)
(1) Consider the function f (x, y) = x2 + y 2 , dened on the elliptical disk R = cfw_(x, y) :
4x2 + y 2 4.
(a) Find all critical points for f th
MATH 082, EXAM 2a (WITH SOLUTIONS), 15 OCT., 2004, NAME:
(There are six problems, each worth 10 points.)
(1) A solution to the wave equation a2 zxx = zyy (where a is a positive constant) is a function
f (x, y) such that a2 fxx (b, c) = fyy (b, c) holds fo
MATH 082, EXAM 1a, 24 SEPT., 2004, NAME:
(There are six problems, each worth 10 points.)
(1) Given the equation x2 + 2y 2 z 2 = 1:
(a) Sketch the intersection of the surface with the plane y = 0.
This is the curve x2 z 2 = 1, in the xz-plane. Its a hyperb
MATH 082, SAMPLE PROBLEMS FOR EXAM 1, 08 FEB, 2008
(Here is a sampling of problems that may be included in Fridays exam. In order to get the
most out of them, hide the solutions until you have worked the problems yourself.)
(1) Let v = 2i j + 2k. Find the
MATH 082, EXAM 2b SOLUTIONS, 05 MARCH, 2008
(Each of the following four problems is worth 15 points.)
(1) Consider the vector-valued function r(t) = (3 cos t)i + (3 sin t)j.
(a) Compute the unit tangent vector T ().
This is counterclockwise motion on a ci
MATH 082, EXAM 2a SOLUTIONS, 05 MARCH, 2008
(Each of the following four problems is worth 15 points.)
(1) Consider the vector-valued function r(t) = (2 cos t)i + (2 sin t)j.
(a) Compute the unit tangent vector T (/2).
This is counterclockwise motion on a
MATH 082, EXAM 3a (WITH SOLUTIONS), 15 NOV., 2004, NAME:
(There are six problems, each worth 10 points.)
(1) Let f (x, y) = y 3 + x2 x 3y.
(a) Find all critical points of f .
Were looking for points that make the gradient zero. Thus we need to solve
the s
MATH 082, EXAM 1a, 21 (SOLUTIONS) SEP., 2005
(1) Given the surface represented by the equation xyz 2 = 1:
(a) Sketch the intersection of the surface with the plane z = 1. (Be sure to label axes.)
This is the graph of the equation xy = 1 in the xy-plane. I
MATH 082, EXAM 2a, 17 OCT., 2005, NAME:
(There are six problems, each worth 10 points.)
(1) Given the two vectors u = 2i j + 3k, v = i + 2j k in space:
(a) Find the area of the triangle with vertices (0, 0, 0), (2, 1, 3), (1, 2, 1).
A=
1
2
u v , where u v
MATH 082, EXAM 2a (WITH SOLUTIONS), 15 OCT., 2004, NAME:
(There are six problems, each worth 10 points.)
(1) A solution to the wave equation a2 zxx = zyy (where a is a positive constant) is a function
f (x, y) such that a2 fxx (b, c) = fyy (b, c) holds fo
MATH 082, EXAM 2a (WITH SOLUTIONS), 07 MAR., 2005, NAME:
(There are ve problems, each worth 12 points.)
(1) Given the three points P (0, 2, 1), Q(1, 1, 0), R(2, 1, 1) in space:
(a) Find the area of the triangle with vertices P, Q, R.
Area =
1
2
PQPR =
1
2
MATH 082, FINAL EXAM a (WITH SOLUTIONS), 17 DEC., 2004, NAME:
(There are nine problems, each worth 10 points.)
(1) Find an equation for the plane containing the three points P (1, 2, 1), Q(1, 2, 1), and
R(0, 1, 3) in R3 .
First you need a normal vector, w
MATH 082, EXAM 3a, 11 NOV., 2005, NAME:
(1) Let f (x, y) = 2x2 + y 2 , with R the elliptical disk cfw_(x, y) : 4x2 + y 2 4.
(a) (5 points). Find all critical points of f that lie in the interior of R (i.e., where
4x2 + y 2 < 4).
f (x, y) = 4xi + 2y j = 0
MATH 082, EXAM 1b SOLUTIONS, 08 FEBRUARY, 2008
(Each of the following four problems is worth 15 points.)
(1) (a) Find the vector u that has the same direction as that of v = 3i 4j 12k, and
three times the magnitude.
u = 3v = 9i 12j 36k.
(b) Find the vecto
MATH 082, EXAM 3a (solutions), 11 APRIL, 2008, NAME:
(Each of the following four problems is worth 15 points.)
(1) Consider the function f (x, y) = x2 + y 2 , dened on the elliptical disk R = cfw_(x, y) :
x2 + 4y 2 4.
(a) Find all critical points for f th
MATH 082, SAMPLE PROBLEMS FOR EXAM 2, 05 MAR, 2008
(Here is a sampling of problems to prepare you for Exam 2. In order to get the most out of
them, hide the solutions until you have worked the problems yourself.)
(1) Given the position vector function r(t
MATH 082, SAMPLE PROBLEMS FOR EXAM 3, 11 APR, 2008
(Here is a sampling of problems to prepare you for Exam 3. In order to get the most out of
them, hide the solutions until you have worked the problems yourself.)
(1) Let f (x, y) = y 3 + x2 x 3y.
(a) Find
MATH 082, SAMPLE PROBLEMS FOR THE FINAL EXAM, 09 MAY, 2008
(Here is a sampling of problems to prepare you for the nal exam. These are taken from the
material covered since Exam 3; namely 11.7, 12.112.7. Expect six multi-part problems.
The rst three will b
MATH 082, EXAM 1a SOLUTIONS, 08 FEBRUARY, 2008
(Each of the following four problems is worth 15 points.)
(1) (a) Find the unit vector u that has direction opposite to that of v = 3i 4j 12k.
v =
3
32 + 42 + 122 = 13. So u = 13 i +
4
j
13
+
12
k.
13
(b) Fin