Math 274 Quiz 4 (15.03.2016)
Solution
1. The goal of this exercise is to find general solutions of y x2 y = 0 as power series in x.
(a) [2] Consider a solution y(x) =
an2
for n 2.
(n+2)(n+1)
P
Let y(x) = n0 an xn . Then
y (x) =
X
P
n0
an xn . Show that a2
Math 274 Quiz 3 (25.02.2016)
Solution
1. [6] Solve the initial value problem y + y 2y = et + et with y(0) = 0, y (0) = 0.
The characteristic equation of y + y 2y = 0 is r 2 + r 2 = (r 1)(r + 2) = 0 which
has the two characteristic roots r = 1 and r = 2. T
Math 274 Quiz 2 (04.02.2016)
Solution
1. [5] Solve the following
initial value problem explicitly and specify the interval of existence
(
y = xy 2 ,
of the solution:
y(0) = 2 .
The equation is separable and we get
1
x2
1
dy
=
x
dx
=
+ C y = x2
.
2
y
y
2
+
Math 274 Quiz 1 (21.01.2016)
Solution
For the differential equation
dy
= y + et answer the following questions.
dt
1. [3] Is this equation ordinary? Linear? What is its order?
The equation is ordinary, linear, of order 1.
2. [5] Find the general solution
Assignment 7 (due Monday, October 19)
Math 310, Fall 2015
For this assignment, you may use any probability tables in the textbook or any
statistical softwares to answer relevant questions.
Question 0. Read the textbook (section 5.6 5.8 and 6.2, 6.3 and 6.
Assignment 8 (due Friday, October 23)
Math 310, Fall 2015
This assignment covers Lecture 20-23 (section 6.5, 6.7 and 6.8.)
Question 1. State if the following statements are true or false.
(a) If X1 , , X100 N (0, 1) and independent, then
X1 + + X100
10S
i
Year 9 Economics and Maths Research Task
In this task you will use the knowledge, understanding and skills that you have
developed in your Math and Economics classes. You will undertake an investigation into
the relationship between price and demand. The
Properties of Real Functions: Boudedness
Definition
Let f : X ! R where X R. We say f is bounded above if
is bounded above.
f (X ) = cfw_r 2 R : f (x) = r for some x 2 X
Definition
Let f : X ! R where X R. Similarly, we say f is bounded below if
is bound
CHAPTER 2 7
OPTIMIZATION
This chapter is meant to serve as a review of various facts about opti-
mization. Typically students will have spent several weeks studying these
methods in a mathematics course prior to studying this text.
27.1 Single variable op
Mathematical Economics
Linear Algebra
September 22, 2015
Contents
1 Linear computations in Rn
1.1 Linear systems and Gauss method . . .
1.2 Matrices and vectors of Rn . . . . . . . .
1.3 Product of a matrix by a vector . . . . .
1.4 Gaussian operations ap
Number Concept
A satisfactory discussion of the main concepts of analysis (such as
convergence, continuity, dierentiation, and integration) must be based on an
accurately defined number concept. (Rudin)
What does this mean? All those concepts use some ide
Vector Space Over the Reals
Definition
A vector space V over R is a 4-tuple (V , R, +, ) where V is a set, + : V V ! V
is vector addition, : R V ! V is scalar multiplication, satisfying:
1
2
3
4
5
6
7
8
9
Closure under addition: 8x, y 2 V , (x + y) 2 V
As
Simple Logic
P means P is true, P means P is false (or not P).
P ^ Q means P is true and Q is true.
P _ Q means P is true or Q is true (or possibly both).
P ^ Q stands for (P) ^ Q; P _ Q stands for (P) _ Q.
P ) Q means whenever P is true, Q also holds. (i
September 11, 2015
Mathematical Economics
Open and closed sets
Exercises
Exercise 1. Let B (a, r) be an open ball of center a and radius r in the normed space
(E, k k) . Prove that B (a, r) = a + r B (0, 1).
Exercise 2. Let (xk )k be a sequence on Rn that
Open and Closed Sets
A set is open if at any point we can find a neighborhood of that point
contained in the set.
Definition
Let (X , d) be a metric space. A set A X is open if
8x 2 A 9" > 0
such that
B" (x) A
Remember that B" (x) = cfw_y 2 X : d(y , x) <
September 10, 2015
Mathematical Economics
Problem Set 1
(due on Tuesday 15 September 2015)
Exercise 1. (a) Give an example of finite subset of N.
(b) Give an example of a proper subset of N which is infinite.
Exercise 2. What are the subsets of cfw_x R :
December 12, 2015
Mathematical Economics
Final Exam
This exam is worth 50 points (6 + 6 + 9 + 6 + 13 + 10). Answer all problems in as thorough
detail as possible. Partial credit will be given even if the answer is not fully correct. If you believe
there i
September 10, 2015
Mathematical Economics
Normed Spaces
Exercises
Exercise 1.
Let k k1 and k k2 be two norms on the vector space E. Prove that
k k1 + k k2 is a norm on E.
Exercise 2. Let (E, k k1 ) be a normed real vector space. Let c R be such that c > 0
ECON512-Midterm Exam
Dinara Alpysbayeva
Exercise 1
f : Rn R
A Rn
a) x A is global minimum of f on A if
x A, f (x) < f (x)
b) x A is local minimum of f on A if
rx > 0
x B(x, r), f (x) f (x)
c)f has maxima on A: x A such that x A, f (x) > f (x)
Negate the a