Syllabus - Math 042 Spring 2016 - Calculus III
Tufts University Department of Mathematics
Sections and Instructors
01 (block B) and 04 (block F)- Jessica Dyer <jessica.dyer@tufts.edu>
02 (block C) - Andrew Sanchez <andrew.sanchez@tufts.edu>
03 (block E
Math 42
Calculus III
Tufts University
Department of Mathematics
Final Exam
Monday, May 9, 2016
8:3010:30am
No calculators, books or notes are allowed on the exam. All electronic devices must be turned
off and put away. You must show all your work in the s
Math 42
Calculus III
Tufts University
Department of Mathematics
Exam 1
Monday, February 22, 2016
12:001:20pm
No calculators, books or notes are allowed on the exam. All electronic devices must be turned
o and put away. You must show all your work in the b
Example
Let
2
A = 0
0
0
1
0
0
1 .
1
What is the general solution to the system of ODEs D~x = A~x?
Answer:
Clearly, the eigenvalues of A are
=2
=1
with multiplicity 1,
with multiplicity 2.
1
If = 2, it is easy to see that an eigenvector satisfying (A I)~
Example
Consider the system of ODEs D~x = A~x, where
3
A = 4
0
2
1
0
2
0
3
Answer:
Computing the characteristic polynomial det(A I) gives
det(A I) = ( 3)(2 2 + 5)
Clearly, one eigenvalue is
= 3.
The other eigenvalues are obtained by finding the roots of
Example 1
Let be a real number, and consider the ODE
dx
= x.
dt
(1)
Observe
1. We know that a solution to (1) is h(t) = et . (Plug it in.)
2. The function h(t) is not the general solution. Why? Because h(t) cant satisfy all initial conditions.
For example
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Chapter 7
Monotonicity
In the preceding chapters, I argued that a good winner selection method sh
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2009/11/17
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Chapter 6
Schulzes Beatpath Method
In this chapter, we describe a particularly attractive a prior
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Chapter 3
Election Spoilers
In Chapter 1, we saw that candidates C and D spoiled the elec
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page 3
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Chapter 1
Winner Selection
Elections can serve various purposes. One may want to select a
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Chapter 9
Irrelevant Comparisons and the
MullerSatterthwaite Theorem
We started out, in C
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page 11
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Chapter 2
Rule of the Majority
Anybody would agree that in a democracy the majority shoul
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Chapter 8
Elections with Many or
Few Voters
When the number N of voters is very large, on
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page 31
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Chapter 5
Smith-Fairness and the
No-Weak-Spoiler Criterion
The notion of the Smith candid
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Chapter 4
The Smith Set
In Chapter 3, we argued that no reasonable winner selection metho
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borgers
2009/11/17
page 71
Chapter 10
Strategic Voting and the
GibbardSatterthwaite
Theorem
In the United States, supporters
Math 32
Calculus I
All sections
TUFTS UNIVERSITY
Department of Mathematics
Exam I
February 29, 2016
12-1:20 pm
No books, notes, or calculators. TURN OFF YOUR CELL PHONE. ANYONE CAUGHT WITH
THEIR CELL PHONE ON WILL BE GIVEN A 10 POINT DEDUCTION. Cross out
Math 32
Calculus I
All sections
TUFTS UNIVERSITY
Department of Mathematics
Exam II
April 11, 2016
12-1:20 pm
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THEIR CELL PHONE ON WILL BE GIVEN A 10 POINT DEDUCTION. Cross out wh
Tufts University
Department of Mathematics
Math 11, Final Exam
Thursday, December 11, 2008
8:3010:30 a.m.
No books, notes or electronic devices of any kind are allowed. Please do all of your work in the blue book. Remember
to sign your exam book. With you
Tufts University
Math 11
Department of Mathematics
Exam II
November 5, 2007
No books, notes, or calculators are allowed on the exam. SHOW ALL WORK. Remember to sign your exam
book. With your signature, you are pledging that you have neither given nor rece
ECON 303 Chapter 3 Study Questions
1)
Money is
(a) anything that is generally accepted in payment for goods and services or in the repayment of debt.
(b) frequentlybut incorrectlyused synonymously with wealth.
(c) a flow of earnings per unit of time.
(d)
{LEVI
Math 34 TUFTS UNIVERSITY November 9, 2015
Calculus II Department of Mathematics All sections
Exam 2
No books, notes, or calculators. TURN OFF YOUR CELL PHONE. ANYONE CAUGHT WITH
THEIR CELL PHONE ON WILL BE GIVEN A 10 POINT DEDUCTION. Cross out what
(E.
an .54 TUFTS NIVERSITY October 5, 2015
Calculus II Department of Mathematics Exam I
12-1:20
want us to grade. You must show work to receive full credit. Please try to write neatly. You need not simplify
your answers unless asked to do so. You should e
Math 226
NUMERICAL ANALYSIS
Midterm
October 23, 2015
Due October 30, 2015
Show all of your work! For problems that include programming,
please include the code and all outputted gures and tables. Please
label these clearly and refer to them appropriately
MATH 226, HOMEWORK 4, DUE DEC. 11, 2015
SHENG XU
For problems that include programming, please include the code and all outputted gures
and tables. Please label these clearly and refer to them appropriately in your answers to the
questions.
(1) Derive an
MATH 226, HOMEWORK 2, DUE OCT. 16, 2015
For problems that include programming, please include the code and all outputted
gures and tables. Please label these clearly and refer to them appropriately in your
answers to the questions.
(1) Derive the Peano ke
MATH 226, HOMEWORK 1, DUE OCT. 2, 2015
For problems that include programming, please include the code and all outputted
gures and tables. Please label these clearly and refer to them appropriately in your
answers to the questions.
(1) Consider the Bernste
Solutions to Homework 7
3.1, 1 Using ordinary addition of integers as the operation, show that the
set of even integers forms a group but that the set of odd integers
does not.
Proof: Addition is an associative binary operation on Z. The
sum 3 + 5 = 8 is