MAT237 Problem sessions week of Oct28
Please note the solutions to these questions are not posted but they are only presented in the
Problems sessions Tues. 12 UC244, and Thurs 23 AH107. Please attend the Problem sessions to
improve your understanding a
MAT237 Problem sessions week of March 4
1. Change of Variables in a Double integral:
ZZ
x+y
)dA where S is the region of the plane
xy
S
bounded by the curves x y = 1, y = x + 5 and the coordinate axis.
ZZ
b) A linear example (which looks nonlinear): Comp
MAT237 Problem sessions Week of Jan. 13
1. section 2.9 questions 7,8 (make sure to consider the use of theorem 2.83)
2. Use Lagrange multiplier method to find the dimensions of the rectangular box with the fixed
surface area of 10 square meters which has
MAT237 Problem sessions, week of Feb. 24
1. Consider the ultimate step function, f defined on [0, 1] as follows: f (0) = 0, and f (x) = n1 for
1
< x n1 for all n = 1, 2, 3, . . . . Prove this function is integrable on the interval [0, 1] in many
n+1
ways:
MAT237 Problem sessions, week of Feb. 10
1. Consider the ultimate step function, f defined on [0, 1] as follows: f (0) = 1, and f (x) = n1 for
1
< x n1 for all n = 1, 2, 3, . . . . Use lemma 4.5 to prove that f is Riemann integrable on the
n+1
interval [0
MAT237 Problem sessions, week of Jan. 28
1. Section 3.2: question 4,
2. Read question 6 of 3.2, then examine the following example, draw the set S and verify the claims
made in parts (a) and (b): S = cfw_(x, y) : (x y 2 )(y x3 ) = 0.
3. Section 3.3: Do qu
1.2
topology
Closure of a set
Closure of a set S, denoted by S is the union of the set S together with boundary points of S: S = S S.
Interior and closure of a set are in a dynamic relationship, and most of the topological analysis takes place in
this rel
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MAT 237, Hints for PS2
Please note that problem sets are designed to encourage you to study the textbook. If you ignore the
textbook and try to use your own wisdom (of MAT137) then you have defeated the purpose of the problem
sets. So please try to follow
MAT237 Problem sessions week of March 3
1. Fubinis theorem: Do the deatils of question 13 section 4.3
2. Iterated integrals
a) Application of double integral in claculating single integrals: Assume
never learned about
Z we
1
x cos xdx. First note
the tech
MAT237 Problem sessions Week of Jan. 6
1. Consider the function f (x, y, z) = x2 + y 2 + z 2 + 2xyz. Determine the critical points of f and use
theorem 2.81 to determine the nature of the critical points. If a critical point is a saddle point
decide along
MAT237 Problem sessions week of Oct14
Focus of this weeks problem sessions is section 1.5 and early 1.6
1. Theorem 1.20: Without mentioning 3 construct a sequence of numbers which converges to 3
(One should be able to have all the terms of the sequence ch
MAT237 Problem sessions week of Oct7
Please note the solutions to these questions are not posted but they are only presented in the
Problems sessions Tues. 12 at UC244, and Thurs 23 AH107. Please attend the Problem sessions to
improve your understanding
MAT237 Problem sessions week of Oct21
Please note the solutions to these questions are not posted but they are only presented in the
Problems sessions. Please attend the Problem sessions to improve your understanding as well as your
style of writing mathe
MAT237 Problem sessions week of March 31
1. Let S be the part of the surface z = x2 + y 2 inside the cylinder (x 1)2 + y 2 = 1. (choose the
orientation to be downward if you need to.) Present a parametrization G and use the surface
integral to calculate t
MAT237 Problem sessions week of Nov. 18
Please note the solutions to these questions are not posted but they are only presented in the
Problems sessions Tues. 121 SS 2127, and Thurs 910 SS1069. Please attend the Problem sessions to
improve your understa
MAT237 Problem sessions week of Nov. 4
Please note the solutions to these questions are not posted but they are only presented in the
Problems sessions Tues. 12 UC244, and Thurs 23 AH107. Please attend the Problem sessions to
improve your understanding
MAT237 Problem sessions week of Nov25
Please note the solutions to these questions are not posted but they are only presented in the
Problems sessions Tues. 121 SS 2127, and Thurs 910 SS1069. Please attend the Problem sessions to
improve your understand
MAT237 Problem sessions week of Nov. 11
Please note the solutions to these questions are not posted but they are only presented in the
Problems session Thurs. 12. Please attend the Problem sessions to improve your understanding as well
as your style of w
MAT237 Problem sessions week of Mar. 18
1. Consider the following two paths that connect point A(0, 0) to B(1, 1):
C1 : g (t) = (x(t), y(t) = (t, t2 ), 0 t 1 and C2 : h (t) = (x(t), y(t) = (12t, 4t2 4t+1) 0 t
a) RDraw these Rcurves (indicate the orientat
MAT 237, PS3

Due, Friday Nov. 15, 2:00 pm.
in SS Math Aid Center
FAMILY NAME:
FIRST NAME:
STUDENT ID:
Please note:
1. Your problem set must be submitted on this form. Please provide your nal, polished solutions in
the spaces provided. Remember, it is
Partial Big List Solutions
MAT 237 Advanced Calculus Summer 2015
Solutions
# 1.6
Let f : A B be a map of sets, and let cfw_Xi iI be an indexed collection of subsets of A.
(a) Prove that f (iI Xi ) = iI f (Xi )
(b) Prove that f (iI Xi ) iI f (Xi )
(c) When
MAT237  Tutorial 2  26 May 2015
1
Coverage
Heres what theyve covered in class between last Tuesdays tutorial and this one, which roughly
comprises the material covered in this tutorial.
Sequences and Completeness: All of Chapter 4 of their lecture notes
MAT237  Tutorial 1
1
Coverage
Heres what theyve covered in class so far, which roughly comprises the material covered in this
tutorial.
Set Theory: Set containment, indexed families of sets, intersection, union, complement,
functions, image, preimage
Top
University of Toronto
Faculty of Arts and Science
Term Test
MAT2371Y  Advanced Calculus
Duration  1 hour 50 minutes
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/R4 T5/R5 T5/R5
Chris
Anne
Ivan
SS1074 SS1070 BA1240
University of Toronto
Faculty of Arts and Science
Quiz 2
MAT2371Y  Advanced Calculus
Duration  50 minutes
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/T5 R4/R5 T5/R5
Chris
Anne
Ivan
SS1074 SS1070 BA1240
This exam
University of Toronto
Faculty of Arts and Science
Quiz 3
MAT2371Y  Advanced Calculus
Duration  50 minutes
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/T5 R4/R5 T5/R5
Chris
Anne
Ivan
SS1074 SS1070 BA1240
This exam
University of Toronto
Faculty of Arts and Science
Quiz 6
MAT2371Y  Advanced Calculus
Duration  50 minutes
No Aids Permitted
Surname:
First Name:
Student Number:
Tutorial:
T0101
T5101
T5102
T4/R4 T5/R5 T5/R5
SS1074 SS1070 BA1240
This exam contains 4 page