SOLUTIONS FOR A SAMPLE EXAM.
Problem 1. We use the divergence theorem. Note that div F = 2x. Then
F dS =
div FdV =
1x2 y 2
Here D = cfw_(x, y )|x2 + y 2 1.
(2x2x3 2xy 2 )dA =
(r2 cos sin3 cossin2 )|1 d
Problem 1. Find the limit limn+ an , where an = (1 +
n2 ) .
Problem 2. Determine either or not the innite series converges
n=1 (ln(n)+1)2 .
Problem 3. Find the radius of convergence for the power series
ANSWERS FOR EXAM 1.
Problem 1. We should minimize the function f (x, y, z ) = x2 + y 2 + z 2
under the constrain equation h(x, y, z ) = x + y + z 1 = 0. Note that
f (x, y, z ) = (2x, 2y, 2z ),
h(x, y, z ) = (1, 1, 1).
By the method of Lagrange multipliers
1. Determine convergence or divergence of the innite series
cos( n ).
2. Determine convergence or divergence of the innite series
3. Find all values of the variable x such that the innite series
4. Find the radius of convergence of
Problem 1. Find the directional derivative for the mapping f (x, y, z ) =
sin(xyz ) + x2 + y 2 + z 3 in the direction of the unit vector u = (3 , 3 , 3 ).
Problem 2. Find the tangent plane to the unit sphere centered at (0, 0, 0)
EXAM 1 (PRACTICE)
Problem 1. Find all points on the unit sphere x2 + y 2 + z 2 = 1 such that
a normal lines at these points are parallel to the line
x = 1 + t, y = 1 + t, z = ln 1 + t.
Problem 2. Find a total dierential of the function w = sin(xyz ) + exy
Evaluate the line integral C (x2 + y 2 )dx + 2xydy where the curve C is the
circle x2 + y 2 = 5 oriented counterclockwise.
The surface S is represented by the function r(u, v ) = (u + v )i + (u
v )j + sinv k where 0 u 2, 0 v 2. Find the unit norm
1. The surface is given by equation z 2 = x2 + y 2 . Find all points on this
surface such that the tangent plane at these points is parallel to the plane
x + y + z = 3/sin(1).
2. Let F (x, y, z ) = x2 + y 2 + z 2 25 = 0. Dierentiate the functio
Problem 1.Rewrite the integral
x3 y 2 dzdydx
using the order of integration dzdxdy. Sketch the region of the integration.
Problem 2. Evaluate the integral
cfw_(x, y, z )|x2 + y 2 + z 2 9.
(x2 + y 2 + z 2 ) 3 dV where D =
SOLUTIONS FOR A SAMPLE EXAM.
Problem 1. Note that the vector eld F = (x2 + y 2 )i + 2xy j is conservative. Really
F = f , where f =
+ xy 2 .
Therefore since the circle is the closed curve, by the fundamental theorem of
the line integrals
F dr = 0.
SOLUTIONS FOR THE SAMPLE EXAM 4.
Problem 1. Instead of looking for the limit of the sequence cfw_an we try
to nd the limit of the sequence ln(an ).
ln(an ) = 2n ln(1 +
ln(1 + 21 +
+ 2) =
lim ln(an ) = lim
Problem 1. Evaluate the double integral
region bounded by lines y = 0, x = 1, y = x.
x2 dA, where D is the
Problem 2. Find the area of the plane x + y + z = 1000 above the disk
x2 + y 2 1.
Problem 3. Find the volume of the solid which is
Problem 1. Evaluate the ux
F dS , F = 2z i + 2xy j + y 2 k where
the surface S is the boundary of the solid E with the negative orientation(
the vector eld of unit normals directed inward). The solid E is bounded by
the paraboloid z = 1 x
ANSWERS FOR EXAM 1 (SAMPLE)
Problem 1. Du = cos(
) + 3 + 1.
Problem 2. The tangent plane is given by formula x + y + z = 3.
Problem 3. There are two points P0 = (0, 0, 2) and P1 = (0, 0, 2).
Problem 4. dz = 0.
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