Fall 2013 Math 251-502
Exam 1A: Solutions
c 2013 Art Belmonte
Fri, 13/Sep
1. Find parametric equations for the tangent line to the
space curve r (t) = 3 cost 4 sint 1 t at the
2
point corresponding to t = /2.
3. Show that the line of intersection of the p
Fall 2013 Math 251-502
Exam 3B: Solutions
c 2013 Art Belmonte
Fri, 08/Nov
2 + x2 + y2 dV where E is the region
4. Compute
E
inside the sphere x2 + y2 + z2 = 25 and above the
plane z = 4. Use cylindrical coordinates.
When sphere and plane intersect, r2 =
Fall 2012 Math 251-503
Exam 4B: Solutions
c 2012 Art Belmonte
Tue, 11/Dec
3. Compute the surface area of the spiral ramp
s = r cos r sin 2 , 0 r 4,
0 8.
The surface area is
1. Find work done by the force eld w = x y + 2
in moving an object along the curv
Fall 2013 Math 251-502
Exam 4A: Solutions
c 2013 Art Belmonte
Sat, 07/Dec
3. Determine the surface area of the portion of a
hyperboloid parameterized by
1 + z2 cos ,
q=
5 z 5,
NOTE: See your exam for graphics.
1 + z2 sin ,
z ,
0 2.
The surface area is
1.
Fall 2012 Math 251-502
Exam 4A: Solutions
c 2012 Art Belmonte
Tue, 11/Dec
3. Compute the surface area of the spiral ramp
s = r cos r sin , 0 r 4, 0 8.
The surface area is
Work is
C w dg
b
a w (g (t) g
=
1 sr s dr d = .
1 dS =
1. Find work done by the fo
Fall 2012 Math 251-503
Exam 3B: Solutions
c 2012 Art Belmonte
Fri, 16/Nov
1. Find the surface area of the part of the hyperbolic
paraboloid z = f (x, y) = y2 x2 that lies between
the circular cylinders x2 + y2 = 4 and x2 + y2 = 9.
We project the surface
Fall 2013 Math 251-502
Exam 2B: Solutions
c 2013 Art Belmonte
Fri, 04/Oct
Analyze critical points.
(x, y)
(1/2, 1/2)
(0, 0)
f (x, y)
1/4 = 0.25
0
LPMDs
12 36
0 36
Type
local max
saddle pt
4. Find tangent planes to surfaces at specied points P.
1. A recta
Fall 2013 Math 251-502
Exam 3A: Solutions
c 2013 Art Belmonte
Fri, 08/Nov
2 + x2 + y2 dV where E is the region
4. Compute
E
inside the sphere x2 + y2 + z2 = 25 and above the
plane z = 3. Use cylindrical coordinates.
When sphere and plane intersect, r2 =
Fall 2012 Math 251-502
Exam 3A: Solutions
c 2012 Art Belmonte
Fri, 16/Nov
The volume is V =
2
0
1. Find the surface area of the part of the hyperbolic
paraboloid z = f (x, y) = y2 x2 that lies between
the circular cylinders x2 + y2 = 1 and x2 + y2 = 4.
Fall 2012 Math 251-503
Exam 2B: Solutions
c 2012 Art Belmonte
Fri, 12/Oct
4. Use Lagrange multipliers to nd the absolute
maximum and absolute minimum values of
f = x2 + y subject to the constraint x2 + y2 = 1.
Let g = x2 + y2 1. Solve f = g and
g = 0 to
Fall 2013 Math 251-502
Exam 1B: Solutions
c 2013 Art Belmonte
Fri, 13/Sep
3. Find the point of intersection of the line through
points A (1, 3, 1) and B (3, 4, 2), and the plane
x y + z = 2. (Obtain parametric equations for the
line, then substitute into
Fall 2013 Math 251-502
Exam 2A: Solutions
c 2013 Art Belmonte
Fri, 04/Oct
Construct Hessian matrix and LPMDs.
fxx
fyx
H=
fxy
fyy
2 6
6 6y
=
2 12y 36
LPMDs =
1. A rectangular stick of butter is placed in the
microwave oven to melt. Its volume is V = xyz,
Fall 2012 Math 251-502
Exam 2A: Solutions
c 2012 Art Belmonte
Fri, 12/Oct
4. Use Lagrange multipliers to nd the absolute
maximum and absolute minimum values of
f = 2x + y2 subject to the constraint x2 + y2 = 1.
Let g = x2 + y2 1. Solve f = g and
g = 0 to
Fall 2012 Math 251-503
Exam 1B: Solutions
c 2012 Art Belmonte
Fri, 21/Sep
1. Find an equation of the plane that passes through
the point A (9, 7, 1) and contains the line
L (t) = 3t 1 + t 2 t .
Construct an equation of the tangent line.
L (t) = A + tv
=
Fall 2012 Math 251-502
Exam 1A: Solutions
c 2012 Art Belmonte
Fri, 21/Sep
1. Find an equation of the plane that passes through
the point A (1, 2, 3) and contains the line
L (t) = 3t 1 + t 2 t .
Construct an equation of the tangent line.
L (t) = A + tv
=
Fall 2013 Math 251-502
Exam 4B: Solutions
c 2013 Art Belmonte
Sat, 07/Dec
3. Compute S f dS, where f = x2 ez y2 z
and S is the part of the cylinder parameterized as
q = [2 cos , 2 sin , z], 0 z 3, 0 2.
The surface integral of the scalar eld f is
NOTE: Se