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,
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 add
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
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 ex
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 ex
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
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 soluti
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 s
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
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
i
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 th
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
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 equation
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
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
W
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 tha
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),
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.
S