Math for Econ II, Written Assignment 1 (28 points)
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.
Math for Econ II, Written Assignment 4 (28 points)
Due Friday, October 3
Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work,
you will not
Assignment 1, Calculus 1, Lecture 001
Due Wednesday, February 6th, in class
Covers material on Sections 1.1, 1.2
Do the following problems. Please make sure you write out the solutions neatly. Please
Math for Econ II, Written Assignment 4 (28 points)
Due Friday, March 6
Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work,
you will not b
Math for Econ II, Written Assignment 2 (28 points)
Due Friday, February 13, in recitation
Spring 2015
Please write neat solutions for the problems below. Show all your work. If you only write the answ
Math for Econ II, Written Assignment 10 (28 points)
Due Friday, May 1
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
Directional derivatives, gradients, tangent planes, and differentials (11.4, 11.6
of Stewart)
Munn, Math for Economics II
Average rate of change in a direction
Suppose we want to compute the average r
The Magic Formula
Integration by parts
(9.5 in MFE book; 6.1 in e-book
Math for Economics II Product rule: If u(x) and v(x) are differentiable,
uv0 + vu0 = (uv)0, uv0 = (uv)0 vu0.
Now integrate both s
The Definite Integral
(9.2, 9.3 in MFE book; 5.2 in Stewart )
Math for Economics II As we saw at the end of last sections:
The definite integral
R
a
b
f (x) dx represents the area between the x-axis a
Vectors
(10.2, 10.3 of Stewart)
Munn, Math for Economics II
Foreshadowing directional derivatives: beyond fx and fy
The contour diagram of f (x,y) below represents the temperature at the point (x,y).
Double Integrals (12.1, 12.2 in e-book)
Math for Economics 2 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 b
Leontief Input-Output Model (16.9)
Math for Economics II
Basic Setup
Consider an economy with n sectors. Their outputs x1, x2, ., xn are listed as a column vector.
Intermediate demand: the demand from
Optimization for functions of multiple variables (Ch. 13 in MFE textbook;
Section 11.7 in e-book)
Munn, Math for Economics I
Let f be a function of multiple variables with domain D.
f has a local maxi
Partial Derivatives (11.2, 11.3, 11.8)
Munn, Math for Economics I In the past weve fixed one variable and varied the
other a bit.
Now, we fix a variable and differentiate with respect to another:
Exam
Fundamental Theorem of Calculus (9.2 in MFE book; 5.4 in Stewart)
Math for Economics II
Example 1
Let
12
if 0 t 1
f
(t
)
=
2t
if 1 < t < 2 4 t if 2 t 3
dt. Lets find formulas for g(x) and g0(x).
Defin
Introduction to Differential Equations
(9.8, 9.9 in MFE book)
Math for Economics 2 Differential equations allow us to find the general formula
for a function by using properties of its derivatives.
Ex
Integration by Substitution
(9.6 in MFE book, 5.5 in e-book )
Math for Economics II
Warm-Up
Lets find
Z
x
2
2xe
dx.
What we CANT do:
R
R
R
R
2xex2 dx 6= 2x dx ex2
R
R
dx ( (f g) 6= f g)
R
Also note ex
Matrices and Systems of Equations
Munn, Math for Economics II
Munn, Math for Economics II
Matrices and Systems of Equations
Systems of equations
In the past weve solved equations like:
x + y = 5,
3x y
Example 1
Condensed Review, Functions of Several Variables
(MFE text: 11.1, 11.2, 11.3, 11.7)
Munn, Math for Economics II
There are comprehensive review slides posted on NYU Classes for functions of m
Multivariable Optimization: Lagrange Multipliers (11.8 in e-book; 14.1-14.6
in MFE textbook)
Munn, Math for Economics II
Tools from Math for Econ 1
1
Maxes, mins, and saddle points on the interior of
Present and Future Values (10.3, 10.5 in e-book)
Math for Economics 2
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 i
Functions of Several Variables (Ch. 11)
Munn, Math for Economics I We used to deals with f (x), g(y), (q), etc.
What if more than one independent variable?
Airline revenue is a function of full price
Antiderivatives and Areas: Intro to Integration
(9.1, 9.2 in MFE book; 5.1 in Stewart)
Math for Economics II
A function F is an antiderivative of f on an interval I ifs
F0(x) = f (x)
for all x in I.
K
HW #3
Please give complete, well written solutions to the following exercises.
1. Find all the values for a so that
x+1 xa
x2
x>a
f (x) =
is continuous.
Solution:
Clearly f is always continuous at poi
HW #4
Please give complete, well written solutions to the following exercises.
1. Section 2.1 Problem 6.
Solution:
2. Section 2.1 Problem 20. Solution:
4
2
2
2
2
4
1
4
6
3. Section 2.1 Problem 30.
1
l
HW #5
Please give complete, well written solutions to the following exercises.
1. Section calc1
HW5 - 2.3 Problem 62
Solution:
2. Given p(x) = xn x, nd the intervals over which p is a decreasing
funct
Calculus 1, Lecture 16
New York University, Jankowski
Exam 1, Spring 2013
Name:
Recitation Section:
Read all of the following information before starting the exam:
Show all work, clearly and in order
Worksheet 1
Precalculus review
Andres Muoz
n
Week 1
1. Lets start with some polynomial expansions and factorizations. For each of the following,
factorize if the polynomial is expanded and expand if t
Math for Econ II, Written Assignment 5 (28 points)
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.
Math for Econ II, Written Assignment 7 (28 points)
Due Friday, Nov 6
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