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
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
C
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 som
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
%Question 2
clear all
close all
clc
pin = 7000*1000*1e-5;
pout = 101.325*1000*1e-5;
Tin = 370;
sv6 = Xsteam('sV_P',pout);
sl6 = Xsteam('sL_P',pout);
s5 = Xsteam('s_pt',pin,Tin);
X = (s5-sl6)/(sv6 -sl6
WP
Wcxrw
//
ta
\mwv & g*%
( paw oquvapn Hammm
mmum) wjm (0
WW mam WWW
no mamad
WWW] M2 Halffam ' W
M 1 gm?
w aunpw}
9M Mod.
(' Moi cafe")
Wedge (4034 add mould Mam20
of w M3"-
H 0' WW ON)0W VU
W3
Gupta W
' For M mo canauatC-vow' maus-
uMOWriWl-OS
- w Mith mm m mm M} paw ~ch MM.
if WWW PW
- eaw 10W? :5 mm equal!
eam (andIaaK IS heard @17an
- 1/140"qu
- 91W A nomad W mm, amda WW mutat
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
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
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 rema
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 justificati
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 justificatio
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
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
clear all
close all
clc
% 1
T_condensor = 54;
T1 = 280; % Temperature at state 1
T2 = 165; % separator temperature
ng = 1;
h1 = Xsteam('hL_T',T1); % Enthalpy at State 1
h2 = h1; % h2 = h1
P2 = Xsteam(
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 justificati
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 in