Math 241
Name: _
Homework 1
_
1. One method of defining a sequence +8 8" is to specify the first term, and then give a recursive
formula for the subsequent terms. For example, the equations
+" "
and
+8 #+8" for 8 "
define the sequence e" # % ) "' f. The f
Math 241
Name: _
Homework 4
1. The figure to the right shows a regular hexagon in
three-dimensional space. Find the coordinates of the
point T .
7, 1, 2
P
2, 5, 1
3, 7, 4
0, 0, 1
2. The figure to the right shows a a pyramid in
three-dimensional space.
(a)
Math 241
Name:
Quiz 7
1. [10 points] Find the length of the curve
1
x(t) = 1 + 4t 3 t 3 , 2t 2
from the point (1, 0) to the point (4, 18).
2. [10 points] Find the ow line x(t) of the vector eld
F(x, y) = (2x, 3)
satisfying x(0) = (4, 2).
Practice Problems: Final Exam
1. Let f (x, y) = x3 3x2 2y2 . Find the critical points of f , and classify each critical point as a
local max, a local min, or neither.
2. Let S be the portion of the cone z =
x2 + y2 for which z 10, with upward-pointing nor
Math 241
Name: _
Homework 6
z
1. The figure to the right shows a polyhedron with five vertices,
eight edges, and five faces.
0,1,1
1,0,1
(a) Find the equation for the plane containing the points
a" ! "b, a! " "b, and a" " !b.
0,1,0
1,0,0
y
x
1,1,0
(b) Ske
Math 241
Name: _
Homework 3
1. The figure to the right shows a tiling of the
plane by congruent regular hexagons. Find
the coordinates of the point T .
P
0, 0
1, 0
2. In the figure to the right, a rectangle with a
length of "! units and a width of & units
Math 241
Name: _
Homework 5
1. A copper rod with a length of )!.! cm is placed along the B-axis, with the left end at B !. The
middle portion of the rod is heated, and then the rod is thermally insulated from its surroundings.
(a) After being heated, the
Math 241
Name:
Midterm Exam
1. [12 points (4 pts each)] Find the sum of each of the following series.
(a) 1
1
1
1
1
1
+
+
+
3
9
27
81
243
(b)
x 2n+5
n=1
(c)
1
1
1
1
1
1
+
+
+
0!
1!
2!
3!
4!
5!
2. [12 points] State whether each of the following series
Math 241
Name:
Quiz 8
1. [10 points] Find the critical point of the function
f (x, y) = x 3 y + 3x 2 y,
and determine whether it is a local maximum, a local minimum, or neither. You must show your work
to receive full credit.
2. [5 points] Is the quadrati
Practice Problems: Final Exam
1. Let f (x, y) = x3 3x2 2y2 . Find the critical points of f , and classify each critical point as a
local max, a local min, or neither.
2. Let S be the portion of the cone z =
x2 + y2 for which z 10, with upward-pointing nor
Math 241
Name:
Quiz 10
1. [10 points] Evaluate
x sin z dV , where T is the triangular
T
0, 2,
2, 0,
prism shown in the gure to the right.
2, 2,
0, 2, 0
2, 0, 0
2, 2, 0
xz 2 dV , where R is the region dened by
2. [10 points] Use cylindrical coordinates
Math 241
Name:
Quiz 9
1. [8 points] Evaluate
(1, 2).
T
y2 dA, where T is the triangular region in R2 with vertices at (0, 0), (1, 0), and
x2
1
2. [5 points] Compute
0
3y2 dy dx.
0
3. [7 points] Use four rectangles (two in each direction) and the midpoint
Study Guide for Quiz 10
Note: The following formulas will be written on the board during the quiz:
x = r cos
x = sin cos
x2 + y 2 + z 2 = 2
y = r sin
y = sin sin
r2 + z 2 = 2
x2 + y 2 = r 2
z = cos
dA = r dr d
dV = r dr d dz
dV = 2 sin d d d
r = sin
Math 241
Name: _
Homework 2
1. The Koch snowflake is the fractal shape shown in the figures to the right. The
snowflake is constructed from a large equilateral triangle, using the following
process:
i) To each side of the large triangle, attach a smaller