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.

Math for Econ II, Written Assignment 3 (30 points)
Due Friday, October 2, in recitation
1. (5 points, review of MFE 1) A rm produces and sells two commodities. When the rm produces x tons of the
rst commodity, it is able to sell the commodity at a price o

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

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

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

'/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

Present and Future Values (10.3, 10.5 in e-book)
Jankowski, Math for Economics 2
April 21, 2015
Jankowski, Math for Economics 2
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 t

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

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

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

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

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

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

Antiderivatives and Areas: Intro to Integration
(9.1, 9.2 in MFE book; 5.1 in Stewart)
Jankowski, Math for Economics II
March 24, 2015
Jankowski, Math for Economics II
Antiderivatives and Areas: Intro to Integration(9.1, 9.2 in MFE b
A function F is an an

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
This i

Leontief Input-Output Model (16.9)
Jankowski, Math for Economics II
March 10, 2015
Jankowski, Math for Economics II
Leontief Input-Output Model (16.9)
Basic Setup
Consider an economy with n sectors. Their outputs x1 , x2 , ., xn
are listed as a column vec

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
0.5
0.4
0.2
8 ,
C=
,
and final demand d =
0.

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
3
3

785
14.4 GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE PLANE
ms for Section 14.4
14-4w27 29. grad f = (2x + 3ey )%i + 3xey%
j
adient of the function. Assume
domain on which the function
ins14-4w28-29
In Exercises 3031, find grad f from the differential.
HW

HW 3
Please give complete, well written solutions to the following exercises.
2
1. (2 pts) Find the differential of the function P (K, L, M ) = KM eL
M 2
2. (5 pts, review of MFE 1) A firm produces and sells two commodities.
When the firm produces x tons

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 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
2. (5 pts) Solve the following systems of linear equa

HW 5
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
x + 3y 3z

HW 10
Due Friday, April 17
Please give complete, well written solutions to the following exercises.
Z
/4
cos2 (2) sin(2)d
1. (2 pts) Find
0
Z
2. (3 pts) Find
Z
1
dx
1+ 3x
2
3. (3 pts) Find
7
y ln y dy
i
1
Z
4. (3 pts) Find
x2 sin(5x)dx
Z
e2z cos(3z) dz
Z

HW 11
Due Friday, May 1
Please give complete, well written solutions to the following exercises.
1. (2 pts) An account has been dormant for many years earning interest at
the constant rate of 4% per year, compounded annually. Now the amount
is $100, 000.