MATH 162.02
QUIZ 1
SHOW ALL YOUR WORK! GOOD LUCK!
Sept 29, 2011
NAME: Key
1.
a) (4 pts.) Determine whether the sequence an = cos(1/2n) converges or diverges.
1
= cos
2x
lim cos
x
lim
x
1
2x
= cos 0 = 1
Thus, the sequence cfw_an converges to 1.
b) (3 pts.
MATH 162.02
Quiz 6 Solutions
Name:
1. [10 Points] Find r(t) if r (t) = 2ti + 3t2 j +
tk and r(1) = i + j.
Solution: We have that
r(t) =
=
r (t) dt
2t dt i +
3t2 dt j +
= (t2 + C1 )i + (t3 + C2 )j +
t dt k
2 3/2
t + C 3 k.
3
But
r(1) = i + j = (1 + C1 )i +
MATH 162.02
Quiz 2 Solutions
Name:
Determine whether the following series absolutely converge, conditionally
converge or diverge.
1. [8 points] Consider the following series:
n=1
(1)n n
n2 + 2
a. Does the series converge absolutely?
n=1
Lets compare using
MATH 162.02
QUIZ 1
SHOW ALL YOUR WORK! GOOD LUCK!
Oct 13, 2011
NAME:
cn (4)n is convergent, what can be said about the convergence or divergence of the
1. ( 6pts.) If
n=0
following series?
(2)n cn
a) (3 pts.)
n=0
Converges; the above statement says that a
MATH 162.02
Solutions for HW #6
Exercise 13.1.4: Find the limit.
et 1 1 + t 1 3
lim
,
,
t0
t
t
1+t
Solution:
et 1
3
1+t1
et 1 1 + t 1 3
,
,
=
lim
, lim
, lim
lim
t0
t0
t0 1 + t
t0
t
t
1+t
t
t
et
lim , lim
t0 1 t0
=
=
1
2 1+t
1
,3
1, 1/2, 3
Exercise 13.1.1
MATH 162.02
QUIZ 3
SHOW ALL YOUR WORK! GOOD LUCK!
Nov 10, 2011
NAME:
1. ( 6pts.) Find the values of x such that the vectors 3, 2, x and 2x, 4, x are orthogonal.
The vectors are orthogonal if their dot product is 0. Thus, we need x such that
3, 2, x 2x, 4,
MATH 162.02
Solutions for HW #1
Exercise 10: Find a formula for the general term an of the following sequence,
assuming that the pattern of the rst few terms continues.
12
34
, , , ,.
4 9 16 25
Solution: I like to write all of the symbols that appear in
MATH 162.02
Solutions for HW #2
Exercise 11.5.26: Show that the series is convergent. How many terms of the
series do we need to add in order to nd the sum to the indicated accuracy?
(1)n1 nen
(|error| < 0.01)
n=1
Solution: That the series converges follo
MATH 162.02
Solutions for HW #4
Exercise 10.2.32: Find the area enclosed by the curve x = t2 2t, y =
and the y -axis.
t
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
Figure 1. The area in question
Solution: We integrate from one root of x to the