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MAT C01
Groups
From Gallian: Chapters 0,1,2
Note: The enumeration in this document is not the same as in the
book
definition 0.1.
Z = cfw_. . . 3, 2, 1, 0, 1, 2, 3, . . . denotes the integers.
N = cfw_1, 2, 3, 4, . . . denotes the natural numbers.
axiom 1
MAT C01
Permutation Groups
From Gallian: Chapter 5
definition 0.1. A permutation of a set X is a bijective mapping of
X to itself. A bijective mapping is a mapping that is one-to-one
and onto.
We are mainly interested in permutations of finite sets, in pa
MAT C01
Group Homomorphisms and Isomorphisms
From Gallian: Chapters 6 and 10(part of)
definition 0.1. Let G and H be groups. A mapping f : G H is
called a (group) homomorphism if for any , G, f () =
f ()f (). (The operation between and on the left hand si
MAT C01
Finite Groups - Subgroups
From Gallian: Chapter 3
Note: The enumeration in this document is not the same as in the
book
definition 0.1.
The order, |G|, of a group G is its cardinality,
i.e. thenumber of elements of G (finite or infinite). If the o
MAT C01
Cyclic Groups
From Gallian: Chapter 4
definition 0.1. Let G be a group.
1. The set cfw_g1, g2, . . . , gn G is called the set of generators of
G if it is the smaller subset of G such that applications of the
group operation on the generators prod
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
FINAL EXAMINATION
MATB41H Techniques of the Calculus of Several Variables I
Examiner: E. Moore
Date: December 5, 2014
Start Time: 2:00PM
Duration: 3 hours
1. [12 points]
(a)
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
FINAL EXAMINATION
MATB41H Techniques of the Calculus of Several Variables I
Examiner: E. Moore
Date: December 19, 2011
Duration: 3 hours
1. [6 points]
(a) Carefully complete
Dec 28, 2015
MATH 1014 3.0 W2016
APPLIED CALCULUS II - SECTION P
Course material will be online at http:/www.yorku.ca/pat/MATH1014P/
Calendar Description: Calculus in Polar Coordinates. Techniques of Integration. Indeterminate Forms.
Improper Integrals. S
University of Toronto Scarborough
Computer & Mathematical Sciences
MAT 0158 Winter 2014
Problem Set #2
Due: Thursday, February 6, 2014 at the start of lecture
All problems are from text (VANDEN EYNDEN).
1. P 5051: #4, 22, 34
2. P 55: #19, 22
3. P 58—59: #
University of Toronto Scarborough
Computer & Mathematical Sciences
MAT 0158 Winter 2014
Problem Set #5
Due: Thursday, March 27, 2014 at the start of lecture
All problems are from text (VANDEN EYNDEN).
1. P157: #18, 24
2. P 164: #10, 16
3. P 170: #8, 24
4.
University of Toronto Scarborough
Computer & Mathematical Sciences
MAT 0158 Winter 2014
Problem Set #4
Due: Thursday, March 13, 2014 at the start of lecture
All problems are from text (VANDEN EYN DEN).
1. P 123—124: #24, 36, 50
2. P130-131: #8, 38
3. P 13
University of Toronto Scarborough
Computer 85 Mathematical Sciences
MAT 0158 Winter 2014
Problem Set #3
Due: Thursday, February 27, 2014 at the start of lecture
All problems are from text (VANDEN EYNDEN)‘
1. P 82: #35, 41
2. P 88—89: #8, 22, 27
3. P 9293:
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MAT 0158 Winter 2015
Egblem Set #3
Due: Monday, February 23, 2015 at the start of lecture
All problems are from text (VANDEN EYNDEN).
1. P 82: # 24, 42
2. P 88—89: # 9, 13, 1
University'of Toronto Scarborough
Department of Computer 86 Mathematical Sciences
MAT 0158 Winter 2015
Problem Set #2
Due: Monday, February 2, 2015 at the start of lecture
All problems are from text (VANDEN EYNDEN).
1. P 50—51: # 2, 24, 39
2. P 5%: # 24,
University of Toronto Scarborough
Computer 85 Mathematical Sciences
MAT 01-53 _ ' Winter 2014
Problem Set #1
Due: Thursday, January 23, 2014 at the start. of lecture
All problems; are [r0111 text (VANDEK EYN
1‘ P 1647: # 4. 36
P2192: #1114, 30
3. P ‘26
University of Toronto Scarborough
Department of Computer 85 Mathematical Sciences
MAT C158 Winter 2015
Problem Set #1
Due: Monday, January 19, 2015 at the gtart of lecture
All problems are from text (VANDEN EYNDEN).
1. P 16-17: #14, 24, 28
P 21.22: #8, 16
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MATA30: Calculus I - Midterm Test
Examiner: Sophie Chrysostomou Date: Monday, October 26, 2009.
Duration: 110 minutes
DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO.
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University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
MAT A30 Final Exam
Examiners: Sophie Chrysostomou
Date: December 9, 2011
Duration: 170 minutes
Brian Pike
Family Name:
Given Name(s):
Student Number:
Your signature:
DO
HLTB03 Summer 2012
General Paper Guidelines Version 3
OPTION 1:
Your task is to choose from one of three recent articles (posted on Blackboard) that make a claim about a
health issue and have been recently published in the popular press. Each of these art
Section 8. Calculus: Limits and Derivatives 65
8. Calculus: Limits and Derivatives
This section contains review material on:
0 Limits
0 Derivatives
Limits. We do not intend to go into theoretical considerations about limits and other concepts of
calculu
Section 6. Trigonometry
6. Trigonometry
This section contains review material on:
o Trigonometric ratios and trigonometric functions
0 Trigonometric identities and trigonometric equations
Angles. Recall that a positive angle is measured counterclockwise
Section 7. Exponential and Logarithmic Functions
7. Exponential and Logarithmic Functions
This section contains review material on:
o Exponential functions and the natural exponential function
0 Logarithmic functions and the natural logarithmic function
30
Mathematics Review Manual
5. Functions
This section contains review material on:
0 Denition of a function, domain, range
0 Graph of a function
0 Important graphs
0 Creating new functions from old
Function. A function f is a rule that assigns, to each
Section 4. Elements of Analytic Geometry
4. Elements of Analytic Geometry
This section contains review material on:
o Cartesian coordinate system and distance between points
0 Equations of a line
0 Graphs of second degree equations: circle, ellipse, par
Section 3. Equations and Inequalities
3. Equations and Inequalities
This section contains review material on:
0 Solving linear equations in one variable
0 Solving inequalities in one variable
0 Solving equations and inequalities involving absolute value
Section 1. Basic Algebra
1. Basic Algebra
This section contains review material on:
0 Real numbers, intervals and absolute value
0 Polynomials
o Radicals and rational expressions
0 Fractional expressions
Real Numbers and Its Famous Subsets. In Calculus,
Section 2. Basic Formulas from Geometry
2. Basic Formulas from Geometry
This section contains review material on:
0 Plain geometry
0 Geometry in threedimensional space.
Plane Geometry. Usually, we use lowercase letters a, b, c, . . to denote the sides (