Frances Wong
Math 004 Test II
1) a. Calculate the Shapley-Shubik power indices for every voter in the
weighted voting system [52; 25, 35, 21, 19]
Answer:
A
A
A
A
A
A
B
B
B
B
B
B
C
C
C
C
C
C
D
D
D
D
D
D
Permutations
B
C
B
D
C
B
C
D
D
B
D
C
A
C
A
D
C
A
C
D
Frances Wong
MATH-004
Take Home Final
Take Home Final
1. Page 521; #2
Calculus I
Calculus II
Calculus III
Calculus IV
Total
512
111
333
175
1131
Hamiltons Method:
There are total of 20 TAs available
Course
Calc I
Calc II
Calc III
Calc IV
Total
Enrollment
Real Analysis Final Solutions
Math 112 Harvard University Spring 2002
1. Let f : R R be a C 2 function. Prove that
f 00 (x) = lim
t0
f (x + t) 2f (x) + f (x t)
t2
Proof 1. By Taylors formula with remainder we have
f (x + t) = f (x) + tf 0 (x) + (t2 /2)f 0
Solutions to Term Test 2
(1) (20 pts) Let F (x, y) be given by the formula
Z y
F (x, y) =
ex cos(t2 x)dt
0
(a) Show that F is C 1 on R2 .
(b) Let c = F (0, 2). Compute c and prove that near
(0, 2) the level set cfw_F (x, y) = c can be written
as a graph o
Solutions
Real Analysis Midterm
Math 112 Harvard University Spring 2002
1. True or false: for any open set A R, we have int(A) = A. Justify your
answer.
Answer: This is false. For example if A = (0, 1) (1, 2) R, then
A = [0, 2] and int(A) = (0, 2) 6= A.
2
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Math 146: solutions for homework 2
R
Notation: For convenience, the expression f (x1 , . . . , xn ) d(x1 ,R. . . , xn ) (note the
differential d(x1 , . . . , xn ) is also used to denote the integral f .
1. a) Throughout this question, consider the half-li
Math 146: solutions for homework 1
1. Let C X be a compact subspace of a metric space X.
Let x X. Since C is compact, the open cover cfw_Bn (x) : n N has a finite
subcover. Therefore, C Bn (x) for some n, and so C is bounded.
Let x X \ C, and define the o
Math 146: homework 3
Due on November 27, 2012
Notes:
- You may use any result which was stated in class, unless you are asked to prove
it, or explicitly instructed otherwise.
- Give a full justification for everything you are asked to prove.
Notation: Giv
Math 146: homework 1
Due on October 18, 2012
Notes:
- You may use any result which was stated in class, unless you are asked to prove
it, or explicitly instructed otherwise.
- Give a full justification for everything you are asked to prove.
1. Recall that
Math 146: homework 2
Due on November 13, 2012
Notes:
- You may use any result which was stated in class, unless you are asked to prove
it, or explicitly instructed otherwise.
- Give a full justification for everything you are asked to prove.
R
Notation: F
MAT 257Y
Solutions to Term Test 1
(1) (15 pts) Give the definitions of the following notions.
(a) an open set in Rn ;
(b) a boundary point of a set A Rn ;
(c) a function f : Rn Rm differentiable at a point
p;
(d) a directional derivative of a function f :
MAT 257Y
Solutions to Practice Term Test 1
(1) Find the partial derivatives of the following functions
(a) f (x, y, z) = sin(x sin(y sin z)
2
(b) f (x, y, z) = xyz
Solution
f
(a) x (x, y, z) = (cos(x sin(y sin z)(sin(y sin z)
f
y (x, y, z) = (cos(x sin(y
Math 004 Test II
This test is for individual work. No collaboration is allowed.
The due time at the beginning of the class on 12/9.
Must write in complete sentences.
Typed answers are required.
The instructor cannot answer directly on how to solve these t