Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 4
Date Due: 8:00 AM, Thursday, the 22nd of October 2015
Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict
reasoning.
John Locke

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 2
Date Due: 10:00 AM, Thursday, the 9th of October 2014
Investigation may be likened to the long months of pregnancy, and solving a problem to the day of birth. To
investigate a problem

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 4
Date Due: 10:00 AM, Thursday, the 23rd of October 2014
Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict
reasoning.
John Locke

Honors Mathematics II
Functions of a Single Variable
Dr. Horst Hohberger
University of Michigan - Shanghai Jiaotong University
Joint Institute
Fall Term 2014
Dr. Hohberger (UM-SJTU JI)
Functions of a Single Variable
Fall 2014
1 / 621
Introduction
Oce Hour

Vv186 Honors Mathematics II
Sample Exercises for the First Midterm Exam
The following exercises are sample exercises of a diculty comparable to those found the rst
midterm exam. Most of these have been used in actual exams in the last few years. Solutions

Vv186 Honors Mathematics 1
Sample Exercises for the First Midterm Exam - Solutions
Exercise 1.
i) Consider the set U R, where U = A B C with
A = cfw_x R : 0 < x 1,
B = cfw_x R : x = 2 1/n, n N \ cfw_0,
C = cfw_x R : x = 1/n, n N \ cfw_0.
State (without pr

Vv186 Honors Mathematics II
Sample Exercises for the Second Midterm Exam - Solutions
Exercise 1. Test the following series for convergence:
k+5
k=1
5k
,
k=1
3k k 2
5k
,
4k + 3k
k=1
k!
.
(3 Marks)
Solution.
i) Applying the ratio test,
k+1+5 1 k + 6
1
13

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 4
Date Due: 10:00 AM, Thursday, the 24th of October 2013
Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict
reasoning.
John Locke

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 9
Date Due: 10:00 AM, Thursday, the 28th of November 2013
Trigonometry, so far as this, is most valuable to every man; there is scarcely a day in which he will not resort
to it for some

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 11
Date Due: 10:00 AM, Thursday, the 12th of December 2013
Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and
forthwith it is somethi

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 8
Date Due: 10:00 AM, Thursday, the 21st of November 2013
Logic moves in one direction, the direction of clarity, coherence and structure. Ambiguity moves in the other
direction, that o

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 12
This assignment contains additional exercises for your own enjoyment. All
marks on this assignment are imaginary; they will not contribute to the course
grade and the assignment will

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 3
Date Due: 10:00 AM, Thursday, the 17th of October 2013
Obvious is the most dangerous word in mathematics
Eric Temple Bell
Exercise 1. Let (xn ) be a bounded real sequence.1 Let
yn = s

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 1
Date Due: 12:00 PM (noon), Sunday, the 29th of September 2013
Mathematics is learned by doing it, not by watching other people do it.
Michael Reed and Barry Simon, Methods of Modern M

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 2
Date Due: 10:00 AM, Thursday, the 10th of October 2013
Investigation may be likened to the long months of pregnancy, and solving a problem to the day of birth. To
investigate a proble

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 1
Date Due: 10:00 PM, Tuesday, the 30th of September 2014
Mathematics is learned by doing it, not by watching other people do it.
Michael Reed and Barry Simon, Methods of Modern Mathema

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 3
Date Due: 10:00 AM, Thursday, the 16th of October 2014
Obvious is the most dangerous word in mathematics.
Eric Temple Bell
Exercise 1. Let (xn ) be a bounded real sequence.1 Let
yn =

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 10
Date Due: 10:00 AM, Thursday, the 4th of December 2014
I often say that when you can measure what you are speaking about, and express it in numbers, you know
something about it; but

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 5
Date Due: 8:00 AM, Thursday, the 29th of October 2015
Mathematics is the part of physics where experiments are cheap.
V.I. Arnold, On teaching mathematics, March 7th , 19971
Exercise

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 3
Date Due: 8:00 AM, Thursday, the 15th of October 2015
Obvious is the most dangerous word in mathematics.
Eric Temple Bell
Exercise 1. Find the limits of the sequences given by the exp

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 8
Date Due: 8:00 AM, Thursday, the 19th of November 2015
Logic moves in one direction, the direction of clarity, coherence and structure. Ambiguity moves in the other
direction, that of

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 7
Date Due: 8:00 AM, Thursday, the 12th of November 2015
Until now the theory of innite series in general has been very badly grounded. One applies all the operations
to innite series a

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 11
Date Due: 8:00 AM, Thursday, the 10th of December 2015
I often say that when you can measure what you are speaking about, and express it in numbers, you know
something about it; but

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 9
Date Due: 8:00 AM, Thursday, the 26th of November 2015
Trigonometry, so far as this, is most valuable to every man; there is scarcely a day in which he will not resort
to it for some

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 12
Date Due: 8:00 AM, Tuesday, the 15th of December 2015
Doing mathematics is a long-term thing. [] Id describe it as like running a marathon. You cant just sprint
right through it. You

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 5
Date Due: 10:00 AM, Thursday, the 30th of October 2014
Mathematics is the part of physics where experiments are cheap.
V.I. Arnold, On teaching mathematics, March 7th , 19971
Exercise

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 6
Date Due: 10:00 AM, Thursday, the 6th of November 2014
There are no such things as applied sciences, only applications of science.
Louis Pasteur, Address, September 11th , 1872
Exerci

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 7
Date Due: 10:00 AM, Thursday, the 13th of November 2014
Until now the theory of innite series in general has been very badly grounded. One applies all the operations
to innite series

Vv186 Honors Mathematics II
Functions of a Single Variable
Assignment 9
Date Due: 10:00 AM, Thursday, the 27th of November 2014
Trigonometry, so far as this, is most valuable to every man; there is scarcely a day in which he will not resort
to it for some