EC 1535, Fall 2015
Answers to Problem Set #3
1850: Ireland, Germany, United Kingdom, Canada, France
1880: Germany, Ireland, United Kingdom, Canada, Sweeden
1910: Germany, Ireland, Italy, Canada, Russia
1950: Italy, Germany, United Kingdom, Canada, Poland
Economics 1535
International Trade and Investment
Fall 2015
Pol Antrs
Problem Set 2
This problem set is due on Monday, October 5th.
Exercise 1
Chinese works earn only $.50 an hour; if we allow China to export as much as it
likes, U.S. workers will be forc
Economics 1535
International Trade and Investment
Spring 2013
Pol Antrs
Final Exam Suggested Solutions
I. Short Questions (30 points)
1. True. In a model with two industries, one homogenous good on dierentiated good,
intra- and inter-industry trade happen
Economics 1535
International Trade and Investment
Spring 2014
Pol Antrs
Final Exam
Instructions: You have 90 minutes to answer all the questions. The total number of points
is 90, so the number of points per question also indicates how much time you shoul
0
MATH132 Calculus
Dr Ji Li
E7A 211, 0481594222
[email protected]
11
Mean Values
Let f : [a, b] ! R be Riemann integrable on [a, b]. We call
F =
1
Z b
b a a
the mean value of f on [a, b].
f (x) dx
Theorem (Mean Value Theorem for Integrals)
Let f : [a, b] !
0
MATH132 Calculus
Based on the notes of Dr. Bon Clarke, and the slides of Dr.
Stuart C. Hawkins
Dr Ji Li
E7A 211, 0481594222
[email protected]
6
Physical models: average velocity, instantaneous velocity
6
Economic models: Marginal cost
In economics, margin
MATH133 Test 2 Solutions
Test A
1. Let f : R2 R be a differentiable function such that
f
f
(2, 1) = 3 ,
(2, 1) = 5
r
s
Let
h(x, y) = f (x2 + y2 , xy).
h
h
(1, 1) and (1, 1).
x
y
Solution: By Chain rule we find that
h f r f
=
+
x
r x s
h f r f
=
+
y
r y s
0
MATH132 Calculus
Based on the notes of Dr. Bon Clarke, and the slides of Dr.
Stuart C. Hawkins
Dr Ji Li
E7A 211, 0481594222
[email protected]
0
MATH132 Calculus
We will cover topics such as
Real Numbers
Functions
Differentiable functions
Integration
T
MACQUARIE
UNIVERSITY
ENDOFYEAR EXAMINATION 2012
Unit Code and Name: MATH133 a Mathematics IB (Advanced)
Date: Friday 23 November at 13:20
Time allowed: THREE (3) hours, plus 10 minutes reading time.
Total number of questions: SIX (6) QUESTIONS
Instruction
MACQUARIE )
UNIVERSITY /
SESSION 2 EXAMINATIONS 2013
Unit code and name: MATH133 Mathematics IB (Advanced)
Time allowed: THREE (3) hours, plus 10 minutes reading time.
Total Number of questions: Six (6) QUESTIONS
Instructions: All questions may be attempt
MACQUARIE j]
UNIVERSlTY _
This question paper may be retained by candidates.
SESSION 2 FORMAL EXAMINATIONS - NOVEMBER/DECEMBER 2014
EXAMINATION DETAILS:
1 . .1
i Unit Code: i MATH133
i
ii . i
I Unit Name: Mathematics lB (Advanced)
3 Duration of exam (incl
0
MATH132 Calculus
Based on the notes of Dr. Bon Clarke, and the slides of Dr.
Stuart C. Hawkins
Dr Ji Li
E7A 211, 0481594222
[email protected]
2
The real numbers
A satisfactory description of the real numbers is relatively
recent (the 19th Century).
The im
0
MATH132 Calculus
Dr Ji Li
E7A 211, 0481594222
[email protected]
0
Local maximum, minimum
0
Fermats theorem (on stationary points)
One way to state Fermats theorem is that whenever you
compute the derivative of a functions local maximum or
minimum, the res
MACQUARIE l,
UNIVERSITY . ,.
SESSION 1 EXAMINATIONS - JUNE 2013
Unit code and name: MATH132 Mathematics IA (Advanced)
T ime allowed: THREE (3) hours, plus 10 minutes reading time.
Total Number of questions: Six (6) QUESTIONS
Instmctz'om: All questions may
0
MATH132 Calculus
Dr Ji Li
E7A 211, 0481594222
[email protected]
9
Antiderivative
In calculus, an antiderivative, primitive integral or indefinite
integral of a function f is a dierentiable function F whose
derivative is equal to the original function f .
Economics 1535
International Trade and Investment
Spring 2014
Pol Antr`as
Final Exam
Instructions: You have 90 minutes to answer all the questions. The total number of points
is 90, so the number of points per question also indicates how much time you sho
MATH132
WEEK 1 TUTORIAL EXERCISES
(1) Factorize 12x2 x 6.
x2 y 2
.
x1 y 1
Suppose that f (t) = a bt , for constants a and b and variable t. Find a and b so that f (2) = 5
and f (2) = 1.
Find the exact solution to 2x = 3x1 .
cos
Show that tan +
= sec .
1
0
MATH132 Calculus
Dr Ji Li
E7A 211, 0481594222
[email protected]
10
Integration
The integral is an important concept in mathematics. Given a
function f of a real variable x and an interval [a, b] in the real
line, the definite integral
Z b
a
f (x) dx
is de
0
MATH132 Calculus
Dr Ji Li
E7A 211, 0481594222
[email protected]
7
The dierential of a function f at the point x
In calculus, the dierential represents the linear principal part of
the change in a function y = f (x) with respect to changes in the
variable
0
MATH132 Calculus
Based on the notes of Dr. Bon Clarke, and the slides of Dr.
Stuart C. Hawkins
Dr Ji Li
E7A 211, 0481594222
[email protected]
5
The squeeze principle
Let g(x) f (x) h(x) for all x near a (but x 6= a).
Suppose limx!a g(x) = L and limx!a h(x
0
MATH132 Calculus
Dr Ji Li
E7A 211, 0481594222
[email protected]
0
Applications of definite integrals
1. Areas
2. Volumes
3. Arc Length
0
1. Areas
Example: Evaluate the area of the region R bounded by the
2
graphs of the functions f (x) = e x cos(x) and g(
MACQUARI E
UNIVERSITY
0)!
ENDOF-YEAR EXAMINATION 2010
Unit: MATI-I133 - Mathematics IB (Advanced)
Date: Monday 22 November, 9:20 am
Time allowed: THREE (3) hours, plus 10 minutes reading time.
Number of questions: SEVEN (7) QUESTIONS
Instructions: All q
0
MATH132 Calculus
Based on the notes of Dr. Bon Clarke, and the slides of Dr.
Stuart C. Hawkins
Dr Ji Li
E7A 211, 0481594222
[email protected]
4
Limit of functions
Example
Consider the function f (x) =
sufficiently large).
x
1
10
100
1000
1
near 1 (or say
0
MATH132 Calculus
Based on the notes of Dr. Bon Clarke, and the slides of Dr.
Stuart C. Hawkins
Dr Ji Li
E7A 211, 0481594222
[email protected]
3
Limit of a sequence
Formal Definition:
We call x the limit of the sequence cfw_xn if the following
condition ho
MACQUARIE )1
UNIVERSITY I
This question paper may be retained by candidates
SESSION 1 FORMAL EXAMINATIONS JUNE/JULY 2014
EXAMINA'I10N DETAILS:
I Unit Code: I MAJH.132,
I Unit Name: I MATHEMATICS 1A (Advanced)
I.$i22iff'_'lffiITEIITIT._.'_ . ITEEEIIIII
2007
H I G H E R S C H O O L C E R T I F I C AT E
E X A M I N AT I O N
Economics
Total marks 100
Section I
General Instructions
Reading time 5 minutes
Working time 3 hours
Write using black or blue pen
Board-approved calculators may
be used
Write you