HW 2
Please give complete, well written solutions to the following exercises.
1. (4 pts) Find all unit vectors which are perpendicular to h8, 3i.
(by perpendicular, we mean that the vectors meet at a right angle)
Solution: We use the formula h8, 3i u = |h
HW 1
Please give complete, well written solutions to the following exercises.
2 2
.
1. (5 pts) Let f (x, y) = ln x +y
x
(a) (3 pts) What is the domain of f ? Very briefly justify your answer.
2
2
> 0 and
Solution: (x, y) is in the domain of f if and only
Assignment 3, Math for Economics III (28 points)
Due Friday, September 25, in recitation
Write neat solutions for the problems below. Show all your work. If you only write the answer with no work, you
will not be given any credit. Please do not forget to
HW 7
Due Friday, March 27
Please give complete, well written solutions to the following exercises.
1. Assume an economics model where we have three industries with the following consumption matrix
0 0.1 0.4
4
C = 0.5 0.4 0.2 ,
and final demand d = 8 ,
0
HW 4
Due Friday, February 27
Please give complete, well written solutions to the following exercises.
1. (3 pts) The figures below show the optimal point (marked with a dot)
in three optimization problems with the same constraint. Arrange the
correspondin
HW 3
Please give complete, well written solutions to the following exercises.
1. (2 pts) Find the differential of the function P (K, L, M ) = KM eL
Solution: dP = PK dK + PL dL + PM dM , so
dP = M eL
2
M 2
dK 2KLM eL
2
M 2
dL + (KeL
2
M 2
2
M 2
2KM 2 eL
MFE III
NYU Courant Institute of Mathematical Sciences
Review problems for Midterm 1
Selin Kalaycioglu
n Sheet: 1 Page: 1 (September 25, 2013 20 : 38) [16-2h18-main]
1. In Exercises below write
f dA as an iterated integral for the shaded region R.
R
1
y
1
MFE III
NYU Courant Institute of Mathematical Sciences
Review problems for Midterm 1
Selin Kalaycioglu
n Sheet: 1 Page: 1 (September 25, 2013 20 : 38) [16-2h18-main]
882
Chapter Sixteen INTEGRATING FUNCTIONS OF SEVERAL VARIABLES
1. In Exercises below writ
Sample computations and problems
In the first couple of weeks we have covered new types of integrals. Here I will list them,
the main steps in solving them, the physical intuition for them, and some sample problems
with numerical answers. All the integral
Assignment 5, Math for Economics III (28 points)
Due Friday, October 16, in recitation
Write neat solutions for the problems below from our e-book, and the additional problem. To be clear: our e-book is
the version of Stewarts Essential Calculus that we a
Line Integrals and Vector Fields
(13.1, 13.2 in e-book)
January 27, 2016
Line Integrals and Vector Fields(13.1, 13.2 in e-book)
Line integrals with respect to arc length
1
2
3
4
Split the curve C into n pieces using the parametrization.
Take sample value
Assignment 7, Math for Economics III (28 points)
Due Friday, October 30, in recitation
Write neat solutions for the problems below from Stewarts e-book, and the one additional problem. To be clear: our
e-book is the version of Stewarts Essential Calculus
Series
(8.2 in e-book)
March 21, 2016
Series(8.2 in e-book)
Motivation
A number like ,
= 3.141592653.
can be expressed as a sum of terms, each corresponding to a
decimal place.
=3+
4
1
1
5
9
+
+
+
+
+ .
10 102 102 104 105
Series(8.2 in e-book)
Halfway a
MFE III
Name:
Sample questions for the first midterm
Read the problem statements carefully and write your answers clearly. Show your
work. Where the problem requires intermediate steps, explain them. Good answers will
include some computations and, if nec
Assignment 2, Math for Economics III (28 points)
Due Friday, September 18, in recitation
Write neat solutions for the problems below. Show all your work. If you only write the answer with no work, you will
not be given any credit. Please do not forget to:
The Fundamental Theorem for Line Integrals
(13.3 in e-book)
February 8, 2016
The Fundamental Theorem for Line Integrals(13.3 in e-book)
Recall the Evaluation Theorem (part of the FT of Calc):
If F 0 is a continuous function, then
Z b
F 0 (x) dx = F (b) F
Assignment 3, Math for Economics III (28 points)
Write neat solutions for the problems below. Show all your work. If you only write the answer with no work, you
will not be given any credit. Please do not forget to write your name and recitation section n
Assignment 9, Math for Economics III (28 points)
Due Friday, November 13, in recitation
Write neat solutions for the problems below from Stewarts e-book, and the one additional problem. To be clear: our
e-book is the version of Stewarts Essential Calculus
Double Integrals (12.1, 12.2 in e-book)
April 29, 2015
Double Integrals (12.1, 12.2 in e-book)
We painstakingly defined areas for general regions in the xy -plane
using integrals.
What about the volume of a solid in R3 ? We will start by using
double inte
Assignment 4, Math for Economics III (28 points)
Due Friday, October 2, in recitation
Write neat solutions for the problems below. Show all your work. If you only write the answer with no work, you
will not be given any credit. Please do not forget to wri
Greens Theorem
(13.4 in e-book)
February 8, 2016
Greens Theorem(13.4 in e-book)
Convention
For a simple closed curve C which encloses a region D, we consider
the counterclockwise traversal to be the positive orientation of C .
So, D is on the left (from t
The Integral and Comparison Tests
(8.3 in e-book)
March 23, 2016
The Integral and Comparison Tests(8.3 in e-book)
Example 1
Does
X
1
converge? Well, lim an = 0 which tells us nothing.
n
n2
n=1
Since the partial sums
sn = 1 +
1
1
+ + 2
22
n
X
X
1
1
are inc
Power Series
(8.5 in e-book)
March 23, 2016
Power Series(8.5 in e-book)
A power series is a series of form
X
cn (x a)n = c0 + c1 (x a) + c2 (x a)2 + .
n=0
a: the center of the power series
cn : constants
Convention: (x a)0 = 1 (even when x = a).
The inter
Functions as Power Series;
Taylor and Maclaurin Series
(8.6, 8.7 in e-book)
April 20, 2016
Functions as Power Series;Taylor and Maclaurin Series(8.6, 8.7 in
Introduction
The series
X
x n is a power series centered at 0, with R = 1,
n=0
which diverges when
Assignment 1, Math for Economics III (28 points)
Write neat solutions for the problems below. Show all your work. If you only write the answer with no work, you will
not be given any credit. Please do not forget to:
Write your name and recitation section
HW 8
Due Friday, April 3
Please give complete, well written solutions to the following exercises.
1. (5 points) Find the area between the x-axis and the curve y = x3 from
x = 0 to x = 1 by finding the general formula for Rn and then taking
lim Rn .
n
n
X
HW 6
Due Friday, March 13
Please give complete, well written solutions to the following exercises.
1. (3 pts) Solve the following systems of linear equations
2x1 + x2 + 2x3 3x4
=
1
x1 + x2 x3 + 2x4
=
2
Solution: if x3 , x4 are chosen free, then x2 = 3 + 4
HW
5 solution
Due Friday, March 6
Please give complete, well written solutions to the following exercises.
1. Use Gaussian elimination to solve the system of equations or show that
its inconsistent
(a) (3 pts)
4x + 2y z
=
1
5x + 3y 2z
=
2
3x + 2y 3z
=
0
(
Assignment 2, Math for Economics III (28 points)
Write neat solutions for the problems below. Show all your work. If you only write the answer with no work, you
will not be given any credit. Please do not forget to write your name and recitation section n
Introduction to Linear Programming
(17.1-17.3 in the softcover MFE book by
Sydsaeter and Hammond)
April 20, 2016
Introduction to Linear Programming(17.1-17.3 in the softcover M
Motivation, Example 1
A baker has 150 kg of flour, 22 kg of sugar, and 27.5 kg
Linear Programming: The Simplex Method
April 20, 2016
Linear Programming: The Simplex Method
Example 1, Hot tubs
A company has two warehouses and four stores. The New Model hot tub
is sold at all four stores, and each store has placed an order with the
co
Game Theory
May 4, 2016
Game Theory
What game theory is
Three main ingredients, given some assumptions.
1
Participants (the players)
2
Strategies for each participant
3
Outcomes (payoffs) depending upon the strategies
Game Theory
Basic assumptions
In two
Double Integrals (12.1, 12.2 in e-book)
February 8, 2016
Double Integrals (12.1, 12.2 in e-book)
We painstakingly defined areas for general regions in the xy -plane
using integrals.
Double Integrals (12.1, 12.2 in e-book)
We painstakingly defined areas fo
More Convergence Tests
(8.4 in e-book)
March 23, 2016
More Convergence Tests(8.4 in e-book)
Alternating series
An alternating series is one whose terms alternate signs. Two
examples:
1 1 1
1 + +
2 3 4
2
4
2
5 +
+
7 57 3
Alternating Series Test: If the