Math for Econ II, Written Assignment 2 (30 points)
Due Friday, September 18th, in recitation
Please 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.
Write your n

Math for Economics II, Written Assignment 1 (30 points)
Due Friday, September 18th, in recitation
Fall 2015
Please 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

785
14.4 GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE PLANE
Math for Econ II Homework Assignment 2
New York University
Due in Recitation, Friday, September 23
rcises and Problems for Section 14.4
cises
Please write neat solutions for the problems below.

Heres an exercise we did in class on September 20th, with solutions:
Let f (x, y) = xexy .
(a) Find f (1, 0).
Solution. Recall that f (a, b) = hfx (a, b), fy (a, b)i. So,
f (x, y) = hexy + xyexy , x2 exy i,
and
f (1, 0) = h1, 1i.
(b) Find the directional

Invertible Matrices, Determinants, and Cramers
Rule (16.1-16.8)
Trush, Math for Economics II
February 26, 2015
Trush, Math for Economics II
Invertible Matrices, Determinants, and Cramers Rule (16.1-16.8)
Consider
3x1 + 4x2 = 3
5x1 + 6x2 = 7
This is the ma

Fundamental Theorem of Calculus
(9.2 in MFE book; 5.4 in Stewart)
Jankowski, Math for Economics II
December 9, 2014
Jankowski, Math for Economics II
Fundamental Theorem of Calculus(9.2 in MFE book; 5.4 in Stew
Example 1
Let
f (t) =
8
<
1
2
:
4
t
t2
if 0 t

Partial Fractions (6.3 in e-book)
November 30, 2015
Partial Fractions (6.3 in e-book)
Sometimes we have to integrate one polynomial divided by
another, e.g.
Z
5x 7
dx.
2
x 3x + 2
Partial Fractions (6.3 in e-book)
Sometimes we have to integrate one polynom

Multivariable Optimization: Lagrange Multipliers
(11.8 in e-book; 14.1-14.6 in MFE textbook)
Jankowski, Math for Economics II
February 21, 2015
Jankowski, Math for Economics II
Multivariable Optimization: Lagrange Multipliers (11.8 in e-book
Tools from Ma

The Definite Integral
(9.2, 9.3 in MFE book; 5.2 in Stewart)
April 7, 2015
The Definite Integral(9.2, 9.3 in MFE book; 5.2 in Stewart)
As we saw at the end of last section:
Rb
The definite integral a f (x) dx represents the area between the
x-axis and the

Matrices and Systems of Equations
Jankowski, Math for Economics II
February 24, 2015
Jankowski, Math for Economics II
Matrices and Systems of Equations
Why care about matrices?
Organize and manipulate data.
Solve systems like
x 2y + z = 5,
2x y z = 4,
x +

Directional derivatives, gradients, tangent planes,
and dierentials (11.4, 11.6 of Stewart)
Trush, Math for Economics II
February 4, 2015
Trush, Math for Economics II
Directional derivatives, gradients, tangent planes, and dierentials
Average rate of chan

Matrices and Systems of Equations
Trush, Math for Economics II
February 17, 2015
Trush, Math for Economics II
Matrices and Systems of Equations
Systems of equations
In the past weve solved equations like:
x + y = 5,
3x
y = 7.
What about more complicated s

Antiderivatives and Areas: Intro to Integration
(9.1, 9.2 in MFE book; 5.1 in Stewart)
March 25, 2015
Antiderivatives and Areas: Intro to Integration(9.1, 9.2 in MFE b
A function F is an antiderivative of f on an interval I if
F 0 (x) = f (x)
for all x in

The Definite Integral
(9.2, 9.3 in MFE book; 5.2 in Stewart)
Jankowski, Math for Economics II
April 12, 2015
Jankowski, Math for Economics II
The Definite Integral(9.2, 9.3 in MFE book; 5.2 in Stewart)
As we saw at the end of last section:
Rb
The definite

Integration by Substitution
(9.6 in MFE book, 5.5 in e-book)
November 23, 2015
Integration by Substitution(9.6 in MFE book, 5.5 in e-book)
Warm-Up
Lets find
Z
2
2xe x dx.
What
Integration by Substitution(9.6 in MFE book, 5.5 in e-book)
Warm-Up
Lets find
Z

Invertible Matrices, Determinants, and Cramers
Rule (16.1-16.8)
Jankowski, Math for Economics II
February 26, 2015
Jankowski, Math for Economics II
Invertible Matrices, Determinants, and Cramers Rule (16.1-16.8)
Consider
3x1 + 4x2 = 3
5x1 + 6x2 = 7
Jankow

PROBLEMS FOR SECTION 13.1
2 _ ~ .
1. The function f dened for all (x, y) by f (x, y) = *2x2 y + 4x + 4y 3 has a maxrmum
Find the corresponding values of x and y.
2. (a) The function f dened for all (x, y) by f(x, y) = x2 + y2 6x + By + 35 has a minimum
po

'/Price Ceilings and Price Floors
PROBLEM SET #2: UNGRADED WORKSHEET
Instructions: Answer all the questions in this worksheet. Youll use some of the answers to complete
Problem Set #2 - Selected Questions online.
Suggestion: Either print this document and

Worksheet 2, solutions
Multivariable optimization using Lagrange multipliers
1. Consider f (x, y) = 2xy under the constraint 5x + 4y = 100. Does f have a minimum value subject to the
constraint? Find all points which satisfy the Lagrange multiplier condit

Quiz 6
MFE II, Section 006, Fall 2016
Recitation Section:
Name:
Netid:
Problem 1.(5 points.)
Solve the following initial value problem. Give your answer in explicit form.
y 0 + 9ey+3x = 0
y(0) = 0
Problem 2.(5 points.)
Find the general solution to
dy
+ y

Integration by parts
(9.5 in MFE book; 6.1 in e-book)
Jankowski, Math for Economics II
April 9, 2015
Jankowski, Math for Economics II
Integration by parts(9.5 in MFE book; 6.1 in e-book)
The Magic Formula
Product rule: If u(x) and v (x) are differentiable

Vectors (10.2, 10.3 of Stewart)
Jankowski, Math for Economics II
February 5, 2015
Jankowski, Math for Economics II
Vectors (10.2, 10.3 of Stewart)
Foreshadowing directional derivatives: beyond fx and fy
The contour diagram of f (x, y ) below represents th

Condensed Review, Functions of Several Variables
(MFE text: 11.1, 11.2, 11.3, 11.7)
Jankowski, Math for Economics I
February 1, 2015
Jankowski, Math for Economics I
Condensed Review, Functions of Several Variables(MFE text: 11.
There are comprehensive rev

1. Compute the following integrals.
Z
2
dx
x+
(a)
x
Solution:
Z
Z
xex
(b)
2 1
x+
2
x2
dx =
+ 2 ln|x| + C.
x
2
dx
Solution: Let u = x2 1. Then du = 2x dx. Making the substitution, we have
Z
Z
1
1
1 2
x2 1
xe
dx =
eu du = eu + C = ex 1 + C.
2
2
2
8
Z
(c)

Multivariable Optimization: Lagrange Multipliers
(11.8 in e-book; 14.1-14.6 in MFE textbook)
Jankowski, Math for Economics II
February 21, 2015
Jankowski, Math for Economics II
Multivariable Optimization: Lagrange Multipliers (11.8 in e-book
Tools from Ma

Introduction to Dierential Equations
(9.8, 9.9 in MFE book)
December 2, 2015
Introduction to Dierential Equations(9.8, 9.9 in MFE book)
Dierential equations allow us to find the general formula for a
function by using properties of its derivatives.
Introd

Integration by parts
(9.5 in MFE book; 6.1 in e-book)
Jankowski, Math for Economics II
April 9, 2015
Jankowski, Math for Economics II
Integration by parts(9.5 in MFE book; 6.1 in e-book)
The Magic Formula
Product rule: If u(x) and v (x) are dierentiable,

Present and Future Values (10.3, 10.5 in e-book)
April 27, 2015
Present and Future Values (10.3, 10.5 in e-book)
Present vs. Future Values
The future value, B, of a payment P, is the amount to which P
would have grown (at some specified time in the future

Math-UA.212, Section 011
Math for Economics II, Spring 2016
Syllabus
Instructor
Email
Office
Office hours
Dr. Michael Munn
munn@nyu.edu
WWH 725
Mon 5-7pm, Tues 5-6pm,
and by appointment
Lecture
Classroom
Course Page
Mon/Wed 11:00-12:15pm
12 Waverly Pl, G0

Math-UA.212, Section 006
Math for Economics II, Spring 2016
Syllabus
Instructor
Email
Office
Office hours
Dr. Michael Munn
munn@nyu.edu
WWH 725
Mon 5-7pm, Tues 5-6pm,
and by appointment
Lecture
Classroom
Course Page
Mon/Wed 3:30-4:45pm
Silver Center, Room

Integration by parts
(9.5 in MFE book; 6.1 in e-book
November 23, 2015
Integration by parts(9.5 in MFE book; 6.1 in e-book
The Magic Formula
Product rule: If u(x) and v (x) are dierentiable,
uv 0 + vu 0 = (uv )0 ,
uv 0 = (uv )0
vu 0 .
Now integrate both s

Evaluating Definite Integrals
(9.2, 9.4 in MFE book; 5.3 in Stewart)
April 7, 2015
Evaluating Definite Integrals(9.2, 9.4 in MFE book; 5.3 in Stewar
We now learn how to do definite integrals in an easier way.
Definite integrals are very closely related to

Worksheet 2, solutions
Multivariable optimization using Lagrange multipliers
1. Consider f (x, y) = 2xy under the constraint 5x + 4y = 100. Does f have a minimum value subject to the
constraint? Find all points which satisfy the Lagrange multiplier condit

Math for Econ II Homework Assignment 4
New York University
Due in Recitation, Friday, October 21st
Please 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.
Do not

Math for Econ II Homework Assignment 5
New York University
Due in Recitation, Friday, October 28
Please 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.
Do not f