MATH 138 Winter 2012
Assignment 8
Topics: Sequences (11.1) and Series (11.2)
Due: 11 am Friday, March 16th.
Instructions:
Print your name and I.D. number at the top of the rst page of your solutions, and underline your last name.
Submit your solutions i
Math 136
Term Test 1 Solutions
1. Short Answer Problems
[1] a) State the denition of a subspace of Rn .
Solution: A non-empty subset S of Rn is a subspace of Rn if it satises all ten properties
of addition and scalar multiplication of vectors in Rn .
[1]
Math 136
Term Test 2 Solutions
1. Short Answer Problems
[1] a) What is the denition of a subspace S of a vector space V?
Solution: S is a subspace of V if it is a subset of V and it is a vector space under the same
operations as V.
12
1 3
01
,
,
01
00
03
Math 136 - Sample Final Exam 1
NOTE: The questions on this exam does not exactly reect which questions will be on this terms
exam. That is, some questions asked on this exam may not be asked on our exam and there may
be some questions on our exam not aske
Math 136 - Sample Final Exam 2
NOTE: The questions on this exam does not exactly reect which questions will be on this terms
exam. That is, some questions asked on this exam may not be asked on our exam and there may
be some questions on our exam not aske
MATH 138 Winter 2012
Assignment 2
Topics: Trigonometric Integrals, Trigonometric Substitutions
Due: 11 am Friday, January 20.
Instructions:
Print your name and I.D. number at the top of the rst page of your solutions, and underline your last name.
Submi
MATH 138 Winter 2012 Assignment 5 Topics: Differential Equations (Chapter 9) Due: 11 am Friday, February 17th.
Instructions: Print your name and I.D. number at the top of the first page of your solutions, and underline your last name. Submit your solution
MATH 138 Winter 2012
Assignment 6
Topics: Parametric Equations and Curves (Chapter 10)
Due: 11 am Friday, March 2nd.
Instructions:
Print your name and I.D. number at the top of the rst page of your solutions, and underline your last name.
Submit your so
MATH 138 Winter 2012
Assignment 7
Topics: Polar Coordinates (10.3,10.4) and Sequences (11.1)
Due: 11 am Friday, March 9th.
Instructions:
Print your name and I.D. number at the top of the rst page of your solutions, and underline your last name.
Submit y
SAMPLE FINAL 1 ANSWERS
1. Short Answer Problems
a) Find the inverse of A =
Solution: A1 =
1
5
12
.
1 3
3 2
.
11
b) Let L1 , L2 : R3 R3 where L1 is a reection in the x2 x3 -plane and L2 is a reection
in the x1 x2 -plane. Find the matrices for L1 , L2 and L
Math 136
Sample Final 1
1. Short Answer Problems
a) Find the inverse of A =
12
.
1 3
b) Let L1 , L2 : R3 R3 where L1 is a reection in the x2 x3 -plane and L2 is a reection
in the x1 x2 -plane. Find the matrices for L1 , L2 and L2 L1 .
c) Show that if C is
MATH 136
Winter 2005
Final Exam
Monday 11 April 2005, 7:00 p.m. to 10:00 p.m.
1 6
1
1 12
13
0
0
2 3 7 10
.
[8] 1. (a) Let A =
1
6
0
0 3
2
2 12 1
1
7 3
1 6
0
0
The reduced row echelon form of A is
0
0
0
0
Determine a basis for each of Nul A, Col A and
MATH 138 Winter 2012
Assignment 9
Topics: Integral Test (11.3), Comparison Test (11.4), Alternating Series Test (11.5)
Due: 11 am Friday, March 23rd.
Instructions:
Print your name and I.D. number at the top of the rst page of your solutions, and underlin
MATH 138 Winter 2012
Assignment 10
Topics: Ratio/Root test (11.6), Power series (11.8, 11.9) and Taylor series (11.10)
Due: Not to be handed in.
1. (11.6, Q32) A series
n=1
an is dened by the equations,
a1 = 1,
an+1 =
2 + cos n
an .
n
Determine whether th
Assignment 10: Solutions MATH 138 2012
2b. (11.8, Q31) If k is a positive integer, show that the radius of convergence of the series
n=0
(n!)k n
x
(kn)!
is k k .
Solution: Applying the ratio test, we want to nd the domain of x that ensures,
an+1
[(n + 1)!
Assignment 3: Solutions MATH 138 2012 Section 7.4: 14) Evaluate the integral, 1 dx. (x + a)(x + b) Solution: If a = b then we have the following decomposition, 1 1 = (x + a)(x + b) b-a 1 1 - x+a x+b
With this it is easy to integrate the problem above, x+a
Assignment 7: Solutions MATH 138 2012
Section 10.3:
1: Identify the curve by nding a Cartesian equation for the curve.
a) (Q19)
r2 cos 2 = 1.
b) (Q20)
r = tan sec .
Solution:
a) We use the formula that cos 2 = cos2 sin2 and then use the denitions for x an
Assignment 8: Solutions MATH 138 2012
Section 11.1:
1: Determine whether the sequence converges or diverges. If it converges, nd the limit.
a) (Q34)
an =
(1)n+1 n
,
n+ n
b) (Q42)
an = ln(n + 1) ln(n),
c) (Q56)
an =
(3)n
.
n!
Solutions:
a) Before we consid
Assignment 9: Solutions MATH 138 2012
Section 11.3:
1: (Q26) Determine whether the series is convergent or divergent,
n=1
n4
n
,
+1
using a) the integral test and b) using the limit comparison test.
Solution:
a) Dene the function f (x) = x/(x4 + 1), which