Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 15
Assigned Tuesday January 14; due Friday January 24, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW15/HW15.html
Required reading
All
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 4
Assigned Tuesday October 1; due Friday October 11, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW04/HW04.html
Required reading
All o
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 5
Assigned Tuesday October 8; due Friday October 18, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW05/HW05.html
Required reading
All o
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 1
assigned Sep. 10; due Sep. 24, 2PM at SS 1071
Required reading
All of Spivak Chapter 1.
To be handed in
From Spivak Chapter 1:
Hand in
Dont hand in
11 even parts
7
12 even part
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Math 157 Analysis I Solution of Term Exam 4
web version:
http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/TermExam4/Solution.html
Problem 1. Is there a non-zero polynomial p(x) dened on t
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Math 157 Analysis I Solution of the Final Exam
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/Final/Solution.html
Problem 1. Let f and g denote functions dened on some set
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Math 157 Analysis I Solution of Term Exam 1
web version:
http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/TermExam1/Solution.html
Problem 1.
1. Prove directly from the postulates for the
Chapter 5
1, Find the following limits.
ii) lim
x2
x3 8
x2
solution:
(
)
(x 2) x 2 + 2 x + 4 = lim x 2 + 2 x + 4
x3 8
= lim
x2 x 2
x2
x2
x2
2
= lim x + lim 2 x + lim 4 = 4 + 4 + 4 = 12
lim
x2
iv) lim
x y
x2
(
)
x2
xn yn
x y
solution:
(x y ) x n1i y i
n 1
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Math 157 Analysis I Solution of Term Exam 3
web version:
http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/TermExam3/Solution.html
Problem 1. Suppose that f is nondecreasing on [a, b]. Not
Chapter 4
3. Draw the set of all point (x,y) satisfying the following conditions.
ii) x + a > y + b
iv) y x
2
vi) x + y < 1
1
is an integer.
x+ y
viii)
x) x < y < x
2
4
Notes: the border is not included, neither the portion between (1,1) and (1,1).
7
Chapter 2
1, Prove the following formulas by induction.
(i) 1 + L + n =
2
2
n(n + 1)(2n + 1)
6
Proof:
1 (1 + 1)(2 1 + 1) 6
= = 1 then, checked.
6
6
k (k + 1)(2k + 1)
2
2
If for n = k, we know 1 + L + k =
, then
6
For n=1, we know 1 = 1 also
2
(k + 1)(k (2
Chapter 6
1. For which of the following functions
f is there a continuous function F with domain
such that F ( x) = f ( x) for all x in the domain of f ?
x2 4
, Yes. There is such a function g ( x ) = x + 2
x2
I.
II.
f (x ) =
III.
f ( x ) = 0,
IV.
3.
f (
Chapter 1:
11: Find all numbers x for which
(ii) x 3 < 8
solution:
x 3 < 8 8 < x 3 < 8 8 + 3 < x < 8 + 3 5 < x < 11
(iv) x 1 + x 2 > 1
solution:
x ( ,1)
x 1 + x 2 = 1 x + 2 x = 3 2x
x 1 + x 2 = x 1 + 2 x = 1 x [1,2]
x 1 + x 2 = x 1 + x 2 = 2 x 3 x (2, )
x
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 3
Assigned Tuesday September 24; due Friday October 4, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW03/HW03.html
Required email
The c
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 17
Assigned Tuesday January 28; due Friday February 7, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW17/HW17.html
Required reading
All
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 24
Assigned Tuesday March 25; due Friday April 4, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW24/HW24.html
Required reading. All of
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Partial Solution of Homework Assignment 1
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/Sol01/Sol01.html
Warning: The solutions presented here are partial at best. They ar
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 18
Assigned Tuesday February 4; not to be submitted.
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW18/HW18.html
Required reading
Reread the handout A
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 9
Assigned Tuesday November 5; due Friday November 15, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW09/HW09.html
Required reading
All
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Welcome Back!
armed with the known, we sail to explore the yet unknown
web version:
http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/WelcomeBack/WelcomeBack.html
The Known:
Setting. f bou
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 12a
Assigned Thursday December 5; not to be submitted.
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW12a/HW12a.html
Required reading
Read, reread, re
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Integration
web version:
http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/Integration/Integration.html
The setting: f bounded on [a, b], P : a = t0 <P
t1 < < tn = b a partition of
b], mi
Math 157 Analysis I Sample Final Exam
University of Toronto, April 16, 2003
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/Final/Sample.html
last yerss exam: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/Final/LastY
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 19
Assigned Tuesday February 4; due Monday February 24 at the tutorials
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW19/HW19.html
Required reading
T
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 16
Assigned Tuesday January 21; due Friday January 31, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW16/HW16.html
Required reading
All
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 21
Assigned Tuesday March 4; due Friday March 14, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW21/HW21.html
Required reading. All of
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 20
Assigned Tuesday February 24; due Friday March 7, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW20/HW20.html
Required reading
All o
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 2
Assigned Tuesday September 17; due Friday September 27, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW02/HW02.html
Required reading
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Math 157 Analysis I Solution of Term Exam 2
web version:
http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/TermExam2/Solution.html
Problem 1. Prove that there is a real number x so that
15
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 3
Assigned Tuesday September 24; due Friday October 4, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW03/HW03.html
Required email
The c
Name:
0
University of Toronto
Department of Mathematics
Math 157 Exam 1
Monday, October 22, 2001
One hour and fifty minutes
There are five problems, each worth 20 points although they do not have
equal difficulty. Write your answer in the space below the
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 6
Assigned Tuesday October 15; not to be submitted
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW06/HW06.html
Required reading
All of Spivak Chapter
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 7
Assigned Tuesday October 22; due Friday November 1, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW07/HW07.html
Required reading
All
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 11
Assigned Tuesday November 19; due Friday November 29, 2PM at SS 1071
web version:
http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW11/HW11.html
Required reading
A
Dror Bar-Natan: Classes: 2002-03: Math 157 - Analysis I:
Homework Assignment 25
Assigned Tuesday April 1; due Friday April 11, 2PM at SS 1071
web version: http:/www.math.toronto.edu/drorbn/classes/0203/157AnalysisI/HW25/HW25.html
God created the integers,